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12000 | https://www.youtube.com/watch?v=oP_QlqF3Q3M | Single line Drawings Without Lifting the Pencil 🧠 [CHALLENGES]
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👉 Drawing games with a single stroke. These challenges consist of drawing, without lifting the pencil from the paper, figures such as a square with an X inside, a 5-pointed star inside a circle, combinations of interlocking squares and other easy shapes to solve quickly in a few seconds and so on. increase concentration when drawing. Includes an extra game to connect points with straight lines.
These types of mental activities stimulate creativity and teach us tricks to solve and respond to ingenuity challenges.
In this same channel you can find more challenges, riddles, logic games and interesting hobbies, also suitable for children.
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12001 | https://virtualnerd.com/algebra-2/quadratics/complex-numbers/imaginary-complex-numbers/solve-complex-numbers-equal | Real math help.
How Do You Find Values for x and y to Make Two Complex Numbers Equal?
How Do You Find Values for x and y to Make Two Complex Numbers Equal?
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Solving complex numbers isn't as complex as you might think. Just match up the real parts and the imaginary parts and solve! This tutorial shows you how to solve an equation involving complex numbers for specific variables.
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12002 | https://www.quora.com/The-coefficient-of-the-5th-6th-and-7th-terms-in-the-binomial-expansion-of-1-x-%E2%81%BF-in-ascending-powers-of-x-are-in-the-linear-sequence-What-are-the-possible-values-of-n | The coefficient of the 5th, 6th and 7th terms in the binomial expansion of (1+ x) ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n? - Quora
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The coefficient of the 5th, 6th and 7th terms in the binomial expansion of (1+ x) ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
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Richard Elston
B Sc in Mathematics&Computer Science, Imperial College London (Graduated 1970) · Author has 4.5K answers and 3.1M answer views
·3y
The coefficient of the 5th, 6th and 7th terms in the binomial expansion of (1+x)ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
(1+x)ⁿ = 1 + nx + n(n-1)/2 x² + n(n-1)(n-2)/3! x³ + n(n-1)(n-2)(n-3)/4! x⁴ + n(n-1)(n-2)(n-3)(n-4)/5! x⁵ + n(n-1)(n-2)(n-3)(n-4)(n-5)/6! x⁶ + …
So you require n(n-1)(n-2)(n-3)/4!, n(n-1)(n-2)(n-3)(n-4)/5!, n(n-1)(n-2)(n-3)(n-4)(n-5)/6! to be what you call a linear sequence, which I assume is the same as an arithmetic sequence.
Multiplying or dividing an arithmetic sequence by a constant gives another arithmetic sequence.
So, divi
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The coefficient of the 5th, 6th and 7th terms in the binomial expansion of (1+x)ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
(1+x)ⁿ = 1 + nx + n(n-1)/2 x² + n(n-1)(n-2)/3! x³ + n(n-1)(n-2)(n-3)/4! x⁴ + n(n-1)(n-2)(n-3)(n-4)/5! x⁵ + n(n-1)(n-2)(n-3)(n-4)(n-5)/6! x⁶ + …
So you require n(n-1)(n-2)(n-3)/4!, n(n-1)(n-2)(n-3)(n-4)/5!, n(n-1)(n-2)(n-3)(n-4)(n-5)/6! to be what you call a linear sequence, which I assume is the same as an arithmetic sequence.
Multiplying or dividing an arithmetic sequence by a constant gives another arithmetic sequence.
So, dividing through by n(n-1)(n-2)(n-3), we find that 1/4!, (n-4)/5!, (n-4)(n-5)/6! is an arithmetic sequence.
And then, multiplying through by 6!, we find that 30, 6(n-4), (n-4)(n-5) is an arithmetic sequence.
Multiplying out brackets, 30, 6n-24, n²-9n+20 is an arithmetic sequence.
So (n²-9n+20) - (6n-24) = (6n-24) - (30)
So n² - 15n + 44 = 6n - 54
So n² - 21n + 98 = 0
So (n-7)(n-14) = 0
So n=7 or n=14
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The co-efficient of the 5th, 6th, and 7th terms in the binomial expansion of (1+x) ^n in ascending powers of x are in arithmetic progression. What are the possible values of n and hence the first term?
The coefficient of 5th, 6th and the 7th term in the binomial expansion of (1+x) ^n, in ascending powers of x are in linear sequence. What are the possible values of n?
What is the binomial expansion of (1-x) ^-n?
What is the binomial expansion of (1-x) ^-n and (1+x) ^n?
What is the binomial expansion of (1-X) ^-1?
Elaine Dawe
BMath, in Mathematics&Computer Science, University of Waterloo (Graduated 1985) · Author has 5.4K answers and 6.6M answer views
·1y
Originally Answered: The co-efficient of the 5th, 6th, and 7th terms in the binomial expansion of (1+x) ^n in ascending powers of x are in arithmetic progression. What are the possible values of n and hence the first term? ·
Binomial expansion in ascending powers of x x:
(1+x)n=n∑k=0(n k)x k=1+⋯+(n 4)x 4+(n 5)x 5+(n 6)x 6+⋯+x n(1+x)n=∑k=0 n(n k)x k=1+⋯+(n 4)x 4+(n 5)x 5+(n 6)x 6+⋯+x n
Before we proceed, we can see that whatever the value of n n, the first term is always 1.1.
So I’m not sure why that was asked.
Anyway, since coefficients of 5 t h,6 t h,5 t h,6 t h, and 7 t h 7 t h terms are in arithmetic progression, we get:
(n 5)−(n 4)=(n 6)−(n 5)(n 5)−(n 4)=(n 6)−(n 5)
\qquad\dfrac{n(n-1)(n-2)(n-3)(n-4)}{5!}-\dfrac{n(n-1)(n-2)(n-3)}{\qquad\dfrac{n(n-1)(n-2)(n-3)(n-4)}{5!}-\dfrac{n(n-1)(n-2)(n-3)}{
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Binomial expansion in ascending powers of x x:
(1+x)n=n∑k=0(n k)x k=1+⋯+(n 4)x 4+(n 5)x 5+(n 6)x 6+⋯+x n(1+x)n=∑k=0 n(n k)x k=1+⋯+(n 4)x 4+(n 5)x 5+(n 6)x 6+⋯+x n
Before we proceed, we can see that whatever the value of n n, the first term is always 1.1.
So I’m not sure why that was asked.
Anyway, since coefficients of 5 t h,6 t h,5 t h,6 t h, and 7 t h 7 t h terms are in arithmetic progression, we get:
(n 5)−(n 4)=(n 6)−(n 5)(n 5)−(n 4)=(n 6)−(n 5)
n(n−1)(n−2)(n−3)(n−4)5!−n(n−1)(n−2)(n−3)4!=n(n−1)(n−2)(n−3)(n−4)5!−n(n−1)(n−2)(n−3)4!=
n(n−1)(n−2)(n−3)(n−4)(n−5)6!−n(n−1)(n−2)(n−3)(n−4)5!n(n−1)(n−2)(n−3)(n−4)(n−5)6!−n(n−1)(n−2)(n−3)(n−4)5!
Multiply both sides by 6!6!
6 n(n−1)(n−2)(n−3)(n−4)−30 n(n−1)(n−2)(n−3)=6 n(n−1)(n−2)(n−3)(n−4)−30 n(n−1)(n−2)(n−3)=
n(n−1)(n−2)(n−3)(n−4)(n−5)−6 n(n−1)(n−2)(n−3)(n−4)n(n−1)(n−2)(n−3)(n−4)(n−5)−6 n(n−1)(n−2)(n−3)(n−4)
Since there are at least 7 7 terms, then n≥6 n≥6.
So we can divide both sides of equation by n(n−1)(n−2)(n−3)n(n−1)(n−2)(n−3)
6(n−4)−30=(n−4)(n−5)−6(n−4)6(n−4)−30=(n−4)(n−5)−6(n−4)
6 n−24−30=n 2−9 n+20−6 n+24 6 n−24−30=n 2−9 n+20−6 n+24
n 2−21 n+98=0 n 2−21 n+98=0
n 2−7 n−14 n+98=0 n 2−7 n−14 n+98=0
n(n−7)−14(n−7)=0 n(n−7)−14(n−7)=0
(n−7)(n−14)=0(n−7)(n−14)=0
(n=7 or n=14(n=7 or n=14
When n=7 When n=7
(1+x)7=1+7 x+21 x 2+35 x 3+35 x 4+21 x 5+7 x 6+x 7(1+x)7=1+7 x+21 x 2+35 x 3+35 x 4+21 x 5+7 x 6+x 7
Coefficients of 5 t h,6 t h,5 t h,6 t h, and 7 t h 7 t h are 35,21,7,35,21,7, which are clearly in arithmetic progression with common difference =−14=−14
When n=14 When n=14
(1+x)14=1+14 x+91 x 2+364 x 3+1001 x 4+2002 x 5+3003 x 6+3432 x 7(1+x)14=1+14 x+91 x 2+364 x 3+1001 x 4+2002 x 5+3003 x 6+3432 x 7
+3003 x 8+2002 x 9+1001 x 10+364 x 11+91 x 12+14 x 13+x 14+3003 x 8+2002 x 9+1001 x 10+364 x 11+91 x 12+14 x 13+x 14
Coefficients of 5 t h,6 t h,5 t h,6 t h, and 7 t h 7 t h are 1001,2002,3003,1001,2002,3003, which are clearly in arithmetic progression with common difference =1001=1001
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Bernard Montaron
Studied Mathematics&Counting (Combinatorics) at Pierre and Marie Curie University (Graduated 1980) · Upvoted by
Alon Amit
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·Updated 3y
Originally Answered: The co-efficient of the 5th, 6th, and 7th terms in the binomial expansion of (1+x) ^n in ascending powers of x are in arithmetic progression. What are the possible values of n and hence the first term? ·
We are looking for n n such that (n 5)−(n 4)=(n 6)−(n 5)(n 5)−(n 4)=(n 6)−(n 5) which, after simplifications, gives n 2–21 n+98=0 n 2–21 n+98=0 i.e. n=7 n=7 or n=14 n=14. Voila!
Note that we also have the trivial solutions corresponding to n(n−1)(n−2)(n−3)=0 n(n−1)(n−2)(n−3)=0 giving arithmetic progressions with common difference 0.
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Let the binomial expansion (1 + x)^n = Σ C(n, k) x^k with k starting at 0. The 5th, 6th and 7th terms correspond to k = 4, 5, 6, so the coefficients are
C1 = C(n,4), C2 = C(n,5), C3 = C(n,6).
They are in arithmetic progression (linear sequence), so 2·C2 = C1 + C3.
Use Pascal identity C(n,k+1) = C(n,k)·(n−k)/(k+1). Compute ratios:
C2 = C1 · (n−4)/5,
C3 = C2 · (n−5)/6 = C1 · (n−4)(n−5)/(5·6).
Plug into 2·C2 = C1 + C3 and divide by C1 (nonzero for n ≥ 6):
2·(n−4)/5 = 1 + (n−4)(n−5)/30.
Multiply by 30:
12(n−4) = 30 + (n−4)(n−5).
Expand:
12n − 48 = 30 + (n^2 − 9n + 20)
12n − 48 = 30 + n^2 − 9n + 20
12n
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Let the binomial expansion (1 + x)^n = Σ C(n, k) x^k with k starting at 0. The 5th, 6th and 7th terms correspond to k = 4, 5, 6, so the coefficients are
C1 = C(n,4), C2 = C(n,5), C3 = C(n,6).
They are in arithmetic progression (linear sequence), so 2·C2 = C1 + C3.
Use Pascal identity C(n,k+1) = C(n,k)·(n−k)/(k+1). Compute ratios:
C2 = C1 · (n−4)/5,
C3 = C2 · (n−5)/6 = C1 · (n−4)(n−5)/(5·6).
Plug into 2·C2 = C1 + C3 and divide by C1 (nonzero for n ≥ 6):
2·(n−4)/5 = 1 + (n−4)(n−5)/30.
Multiply by 30:
12(n−4) = 30 + (n−4)(n−5).
Expand:
12n − 48 = 30 + (n^2 − 9n + 20)
12n − 48 = 30 + n^2 − 9n + 20
12n − 48 = n^2 − 9n + 50
Bring all terms to one side:
0 = n^2 − 9n + 50 − 12n + 48
0 = n^2 − 21n + 98
Solve quadratic:
Discriminant D = 21^2 − 4·98 = 441 − 392 = 49, sqrt = 7.
n = (21 ± 7)/2 → n = 14 or n = 7.
Validity: coefficients defined for k ≤ n, so require n ≥ 6. Both 7 and 14 satisfy this.
Answer: n = 7 or n = 14.
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Sankaran Murthy
Retired Engineer, Teacher & Small Businessman · Author has 2.2K answers and 1.3M answer views
·3y
Originally Answered: The co-efficient of the 5th, 6th, and 7th terms in the binomial expansion of (1+x) ^n in ascending powers of x are in arithmetic progression. What are the possible values of n and hence the first term? ·
The fifth term in the binomial expansion = n! /(4! (n-4)!)
The 6th term = n! /(5! (n-5)!)
The 7th term = n! /(6! (n-6)!)
Given that n! /(6! (n-6)!) - n! /(5! (n-5)!) = n! /(5! (n-5)!) - n! /(4! (n-4)!)
Divide out by the common factor n! /(4! (n-6)!) → 1/30 - 1/(5(n-5)) = 1/(5(n-5)) - 1/((n-5)(n-4)) →
(n-5) (n-4) - 6(n-4) = 6(n-4) - 30 →
n^2 - 21n + 98 = 0. Solving the quadratic equation, n = 14 or 7.
When n = 7, the 5th, 6th and 7th terms are 35, 21, 7. Common difference = -14.
When n = 14, the 5th, 6th and 7th terms are 1001, 2002, 3003. Common difference = 1001.
It is not clear
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The fifth term in the binomial expansion = n! /(4! (n-4)!)
The 6th term = n! /(5! (n-5)!)
The 7th term = n! /(6! (n-6)!)
Given that n! /(6! (n-6)!) - n! /(5! (n-5)!) = n! /(5! (n-5)!) - n! /(4! (n-4)!)
Divide out by the common factor n! /(4! (n-6)!) → 1/30 - 1/(5(n-5)) = 1/(5(n-5)) - 1/((n-5)(n-4)) →
(n-5) (n-4) - 6(n-4) = 6(n-4) - 30 →
n^2 - 21n + 98 = 0. Solving the quadratic equation, n = 14 or 7.
When n = 7, the 5th, 6th and 7th terms are 35, 21, 7. Common difference = -14.
When n = 14, the 5th, 6th and 7th terms are 1001, 2002, 3003. Common difference = 1001.
It is not clear what “first term” (of what?) means.
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There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you?
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John K WilliamsSon
BASIC programming expert teaching myself PYTHON !! at Home Office, Retired (2018–present) · Author has 9K answers and 23.4M answer views
·Updated 3y
Originally Answered: The co-efficient of the 5th, 6th, and 7th terms in the binomial expansion of (1+x) ^n in ascending powers of x are in arithmetic progression. What are the possible values of n and hence the first term? ·
I decided to attack this problem by generating Pascal’s Triangle in Python, and let that program identify rows that contained arithmetic sequences, no matter where those sequences were in those rows.
If you want to modify this program to generate even more rows, you can find the program here:
But be sure to expand the Pad5 function to even more digits than 5 (as I
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I decided to attack this problem by generating Pascal’s Triangle in Python, and let that program identify rows that contained arithmetic sequences, no matter where those sequences were in those rows.
If you want to modify this program to generate even more rows, you can find the program here:
But be sure to expand the Pad5 function to even more digits than 5 (as I did when I increased the padding from 5 to 6), and you’ll also to increase the padding in pad100, as I did, increasing it to 121, as I increase the number of rows.
This is a very quick and dirty program. Instead of figuring out why a code did not work properly, I simply kludged it to force it to work. Perh...
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Ernest Leung
B.Sc. (Hons.) in Chemistry Honors&Mathematics, The Chinese University of Hong Kong · Author has 11.9K answers and 5.8M answer views
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The coefficient of the 5th, 6th and 7th terms in the binomial expansion of
(1 + x) ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
The (m + 1)th term in the binomial expansion of (1 + x)ⁿ = ₙCₘ (1)ⁿ⁻ᵐ (x)ᵐ
Hence, the coefficient of the (m + 1)th term = ₙCₘ
Let M = n!/[(n - 4)!6!]
The coefficient of the 5th term = ₙC₄ = n!/[(n - 4)!4!]
= (6)(5)n!/[(n - 4)!(6)(5)4!] = 30n!/[(n - 4)!6!] = 30M
The coefficient of the 6th term = ₙC₅ = n!/[(n - 5)!5!]
= (n - 4)(6)n!/[(n - 4)(n - 5)!(6)5!] = 6(n - 4)n!/ [(n - 4)!6!] = (6n - 24)M
The coef
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The coefficient of the 5th, 6th and 7th terms in the binomial expansion of
(1 + x) ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
The (m + 1)th term in the binomial expansion of (1 + x)ⁿ = ₙCₘ (1)ⁿ⁻ᵐ (x)ᵐ
Hence, the coefficient of the (m + 1)th term = ₙCₘ
Let M = n!/[(n - 4)!6!]
The coefficient of the 5th term = ₙC₄ = n!/[(n - 4)!4!]
= (6)(5)n!/[(n - 4)!(6)(5)4!] = 30n!/[(n - 4)!6!] = 30M
The coefficient of the 6th term = ₙC₅ = n!/[(n - 5)!5!]
= (n - 4)(6)n!/[(n - 4)(n - 5)!(6)5!] = 6(n - 4)n!/ [(n - 4)!6!] = (6n - 24)M
The coefficient of the 7th term = ₙC₆ = n!/[(n - 6)!6!]
= (n - 4)(n - 5)n!/[(n - 4)(n - 5)(n - 6)!6!] = (n² - 9n - 20)n!/[(n - 4)! 6!] = (n² - 9n - 20)M
Difference bet'n the 5th and 6th terms = Difference bet'n the 6th and 6th terms
(6n - 24)M - 30M = (n² - 9n - 20)M - (6n - 24)M
(6n - 24) - 30 = (n² - 9n - 20) - (6n - 24)
n² - 21n + 98 = 0
(n - 7)(n - 14) = 0
n = 7 or n = 14
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What are the all the functions of a printer?
Modern printers have evolved far beyond basic document reproduction. Today’s models offer a range of built-in features that streamline both personal and professional tasks. Depending on the device, a printer can serve as a multifunctional hub for printing, scanning, copying and even faxing. Enhanced connectivity allows seamless operation from mobile devices and cloud-based platforms, while efficiency-focused options such as duplex printing and high-yield cartridges help reduce both cost and waste. For users balancing home office duties with general everyday printing, choosing the right combina
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Modern printers have evolved far beyond basic document reproduction. Today’s models offer a range of built-in features that streamline both personal and professional tasks. Depending on the device, a printer can serve as a multifunctional hub for printing, scanning, copying and even faxing. Enhanced connectivity allows seamless operation from mobile devices and cloud-based platforms, while efficiency-focused options such as duplex printing and high-yield cartridges help reduce both cost and waste. For users balancing home office duties with general everyday printing, choosing the right combination of features matters just as much as print quality itself.
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9 2
Mike Hirschhorn
Honorary Associate Professor of Mathematics at UNSW · Author has 8.1K answers and 2.7M answer views
·Updated 2y
Originally Answered: The coefficient of 5th, 6th and the 7th term in the binomial expansion of (1+x) ^n, in ascending powers of x are in linear sequence. What are the possible values of n? ·
(n 4)+(n 6)=2(n 5),(n 4)+(n 6)=2(n 5),
n!4!(n−4)!+n!6!(n−6)!=2 n!5!(n−5)!,n!4!(n−4)!+n!6!(n−6)!=2 n!5!(n−5)!,
multiply by 6!(n−4)!n!,multiply by 6!(n−4)!n!,
30+(n−4)(n−5)=2×6(n−4),30+(n−4)(n−5)=2×6(n−4),
n 2–21 n+98=0,n 2–21 n+98=0,
n=7 or 14.n=7 or 14.
Upvote ·
9 2
Enrico Gregorio
Associate professor in Algebra · Author has 18.4K answers and 16M answer views
·3y
I assume you want the coefficients of x 4,x 5,x 6 x 4,x 5,x 6. You want that
(n 6)+(n 4)=2(n 5)(n 6)+(n 4)=2(n 5)
After expanding and collecting alike terms, we get
n(n−1)(n−2)(n−3)4!((n−4)(n−5)6⋅5+1)=2 n(n−1)(n−2)(n−3)(n−4)5!n(n−1)(n−2)(n−3)4!((n−4)(n−5)6⋅5+1)=2 n(n−1)(n−2)(n−3)(n−4)5!
So either 0≤n≤3 0≤n≤3, or
n 2−9 n+20+30=12(n−4)n 2−9 n+20+30=12(n−4)
hence
n 2−21 n+98=0 n 2−21 n+98=0
and therefore either n=14 n=14 or n=7 n=7.
Upvote ·
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Alan Silver
Studied Electrical Engineering (Graduated 1984) · Author has 1.4K answers and 1.5M answer views
·1y
Originally Answered: The coefficient of a 5;6;7 in the binomial expansion of (1+x) n in theanding pawers of x are in linear sequence. What is the possible value of n furthermore? ·
Explain what you meant better.
Are you saying the x^5, x^6, and x^7 terms are in a linear sequence ( or equally spaced )
or that the 5th (x^4) , 6th (x^5) , and 7th (x^6) terms are in a linear sequence.
The 2nd case is easier.
Answer to an earlier question says the answers are n = 7 and n = 14
They come to their answers by analytical methods. Instead of restating or copying their arguments.
I will link to that question. I didn’t merge the questions because you didn’t
use the exact some wording.
The coefficient of the 5th, 6th and 7th terms in the binomial expansion of (1+ x) ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
By trial and error going from n =1 to n = 7 shows you that
for n = 7 ,
5th
Continue Reading
Explain what you meant better.
Are you saying the x^5, x^6, and x^7 terms are in a linear sequence ( or equally spaced )
or that the 5th (x^4) , 6th (x^5) , and 7th (x^6) terms are in a linear sequence.
The 2nd case is easier.
Answer to an earlier question says the answers are n = 7 and n = 14
They come to their answers by analytical methods. Instead of restating or copying their arguments.
I will link to that question. I didn’t merge the questions because you didn’t
use the exact some wording.
The coefficient of the 5th, 6th and 7th terms in the binomial expansion of (1+ x) ⁿ in ascending powers of x, are in the linear sequence. What are the possible values of n?
By trial and error going from n =1 to n = 7 shows you that
for n = 7 ,
5th term = (7 4) = 35
6th term = (7 5 ) = 21
7th term (7 6) = 7
slope = d = -14
Pascal triangle gives you this :
1
1 2 1 (^2 )
1 3 3 1 (^3 )
1 4 6 4 1 (^4)
1 5 10 10 5 1 (^5 )
1 6 15 20 15 6 1 (^6 )
1 7 21 35 35 21 7 1 (^7 )
I am looking for an analytical to determine this easily
For n = 14
(14 4 ) = 14 ! / (4 ! 10! ) = 14131211/ 24 = 1001
(14 5 ) = = 1413121110/ 120 = 2002
(14 6 ) = 14131211109 / 720 = 3003
so d = 1001
SO the other answers are right .
Upvote ·
Reuven Harmelin
Studied Mathematics at טכניון (Graduated 1978) · Author has 2.3K answers and 1.9M answer views
·May 2
Related
The coefficient of the 5th, 6th, and 7th terms in the binomial expansion of (1 + x) ^n are in arithmetic progression (AP). What are the possible values of n?
Trivial answers are n=1,2,3,4 in which cases all the three binomial coefficients above the 4-th degree are equal to zero and in particular the 5th, 6th, and 7th coefficients are are an arithmetic progression of zeros.
For powers n>4 you are looking for positive integral solutions (if they exist at all) of the equation
Divide that equation by the common none-zero factor n! and multiply by the common denominator 7!(n-5)! to get the following quadratic equation
Unfortunately the two real roots of that quadratic equation are none integers:
and therefore, if you are looking for traditional finite bino
Continue Reading
Trivial answers are n=1,2,3,4 in which cases all the three binomial coefficients above the 4-th degree are equal to zero and in particular the 5th, 6th, and 7th coefficients are are an arithmetic progression of zeros.
For powers n>4 you are looking for positive integral solutions (if they exist at all) of the equation
Divide that equation by the common none-zero factor n! and multiply by the common denominator 7!(n-5)! to get the following quadratic equation
Unfortunately the two real roots of that quadratic equation are none integers:
and therefore, if you are looking for traditional finite binomial expansions, there are none except for the trivial ones. On the other hand, if you accept the infinite Taylor series version of the binomial expansion, then these two roots are the none-trivial answers to your question.
Upvote ·
9 1
9 1
Junaid Khan
Junaid Khan at Junaid Jamshed (Recording Artist) (2025–present)
·May 2
Related
The coefficient of the 5th, 6th, and 7th terms in the binomial expansion of (1 + x) ^n are in arithmetic progression (AP). What are the possible values of n?
We are given that the 5th, 6th, and 7th terms in the binomial expansion of are in arithmetic progression (AP). We need to find the possible values of .
We are given that the 5th, 6th, and 7th terms in the binomial expansion of are in arithmetic progression (AP). We need to find the possible values of .
Step 1: General binomial expansion term
In the expansion of , the general term is:
T_{r+1} = \binom{n}{r} x^r
So:
5th term corresponds to
6th term:
7th term:
We're told that the coefficients of these terms are in AP:
\binom{n}{5} - \binom{n}{4} = \binom{n}{6} - \binom{n}{5}
Step 2: Set up the equation
\bino
Continue Reading
We are given that the 5th, 6th, and 7th terms in the binomial expansion of are in arithmetic progression (AP). We need to find the possible values of .
We are given that the 5th, 6th, and 7th terms in the binomial expansion of are in arithmetic progression (AP). We need to find the possible values of .
Step 1: General binomial expansion term
In the expansion of , the general term is:
T_{r+1} = \binom{n}{r} x^r
So:
5th term corresponds to
6th term:
7th term:
We're told that the coefficients of these terms are in AP:
\binom{n}{5} - \binom{n}{4} = \binom{n}{6} - \binom{n}{5}
Step 2: Set up the equation
\binom{n}{5} - \binom{n}{4} = \binom{n}{6} - \binom{n}{5}
Multiply both sides by 1 (optional), and simplify:
2 \binom{n}{5} = \binom{n}{4} + \binom{n}{6}
Now use the binomial coefficient formula:
\binom{n}{r} = \frac{n!}{r!(n-r)!}
But to avoid factorials, it’s easier to try small integer values of to check when this equation holds.
Step 3: Try small values of
Let’s test increasing values of :
n = 6 , , LHS: , RHS: → not equal
n = 7 , , LHS: , RHS: → match!
So, is a solution.
Try next few to see if more values work:
n = 8 , , LHS: , RHS: → not equal
n = 9 , , LHS: , RHS: → not equal
So only works.
Would you like me to show a proof that this is the only solution?
Step 1: General binomial expansion term
In the expansion of , the general term is:
T_{r+1} = \binom{n}{r} x^r
So:
5th term corresponds to
6th term:
7th term:
We're told that the coefficients of these terms are in AP:
\binom{n}{5} - \binom{n}{4} = \binom{n}{6} - \binom{n}{5}
Step 2: Set up the equation
\binom{n}{5} - \binom{n}{4} = \binom{n}{6} - \binom{n}{5}
Multiply both sides by 1 (optional), and simplify:
2 \binom{n}{5} = \binom{n}{4} + \binom{n}{6}
Now use the binomial coefficient formula:
\binom{n}{r} = \frac{n!}{r!(n-r)!}
But to avoid factorials, it’s easier to try small integer values of to check when this equation holds.
Step 3: Try small values of
Let’s test increasing values of :
n = 6
, ,
LHS: , RHS: → not equal
n = 7
, ,
LHS: , RHS: → match!
So, is a solution.
Try next few to see if more values work:
n = 8
, ,
LHS: , RHS: → not equal
n = 9
, ,
LHS: , RHS: → not equal
So only works.
Would you like me to show a proof that this is the only solution?
Upvote ·
Ravi Sharma
Former Group A Officer From Indian Railways (1973–2009) · Author has 15.1K answers and 3.8M answer views
·May 2
Related
The coefficient of the 5th, 6th, and 7th terms in the binomial expansion of (1 + x) ^n are in arithmetic progression (AP). What are the possible values of n?
GENERAL TERM OF (1+x)^n= nCr× x^r
T5(r=4)= nC/4x⁴, Coefficient= nC4
SIMILARLY COEFFICIENT OF
T6= nC5, T7= nC6
AS T5, T6 AND T7 ARE IN AP
T5+T7= 2×T6
nC4+nC6= 2nC5
n!/{4!(n—4)!}+n!/{6!(n—6)!}=2×n!/{5!(n—5)!}
1/{(n—4)(n—5)}+1/(6×5)= 2/{5(n—5)}
2/{5(n—5)}—1/{(n—4)(n—5)= 1/30
{2(n—4)—5}/{5(n—4)(n—5)}=1/30
6(2n—13)= n²—9n+20
12n—78= n²—9n+20
n²—21n+98=0
n²—14n—7n+98= 0
n(n—14)—7(n—14)=0
(n—7)(n—14)=0
n= 7 OR n=14
Upvote ·
9 1
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12003 | https://journalaim.com/PDF/17_9_1.pdf | Archives of Iranian Medicine, Volume 15, Number 3, March 2012 162 Original Article Abstract Background: Our aim was to compare different thresholds of middle cerebral artery peak systolic velocity (MCA-PSV) and amniotic Àuid delta optical density (De lta-OD) with fetal hemoglobin (Hb) during ¿rst and second intrauterine transfusions (IUT). Methods: We determined serial MCA-PSV and Delta-OD in 27 red blood cell alloimmunized fetuses who needed IUT. Before the second IUT, MCA-PSV was measured. The sensitivity and speci¿city of MCA-PSV and Delta-OD were calculated and compared with fetal hemo-globin levels. Results: From 27 fetuses, ¿rst time IUT MCA-PSV with a normal median value (MOM) cutoff of > 1.29 detected 60% of the moderate and 100% of the severe anemia cases. MCA-PSV of MOM > 1.5 detected none of the moderate and 93% of severe anemia cases. De lta-OD detected 50% of moderate anemic and 80% of severe anemic cases. At the second IUT, 91% of severe anemia cases were con¿rmed by MCA-PSV with MOM > 1.5 whereas MCA-PSV with MOM > 1.29 con¿rmed all cases. One case of moderate anemia was detected by MCA-PSV of MOM > 1.29 and none were detected by MCA-PSV with MOM > 1.5. Conclusion: Different thresholds of MCA-PSV have higher sensitivity and speci¿city for detecting moderate and severe fetal anemia compared with Delta-OD. It also has a high sensitivity at the second IUT.
Introduction M orbidity and mortality due to red blood cell alloimmuni-zation have steadily declined because use of routine ante-natal anti-Rh prophylaxis and the development of inva-sive and noninvasive investigations for monitoring alloimmu-nized pregnancies. However, red blood cell alloimmunization re-mains a major problem in several areas of the world. 1–5 Amniocentesis and fetal blood sampling are used for screening and diagnosis of fetal anemia. Umbilical fetal blood sampling is the gold standard for diagnosis of fetal anemia. These procedures are invasive and associated with risks such as infection, fetal de-mise, worsening of maternal immunization, preterm labor, and abortion.6–10 On the other hand; invasive procedures in all alloim-munized pregnant women expose 75% of the fetuses to unneces-sary intervention. Thus, a noninvasive screening method would avoid the potential complications of such methods. 8,9 Doppler ultrasound measurement of the human fetal middle cerebral artery peak systolic velocity (MCA-PSV) is a noninva-sive diagnostic method. MCA-PSV has a predictable relationship with fetal hemoglobin (Hb) and hematocrit concentrations. Sev-eral studies have documented that Doppler MCA-PSV is useful in identifying severely anemic fetuses before the ¿rst intrauterine blood transfusion (IUT). However, MCA-PSV may be less accu-rate in the assessment of fetuses that have already been transfused, because in such fetuses the association between blood velocity and Hb concentration is weaker.10, 11 The purpose of our study is to compare the ability of the MCA-PSV in the prediction of moderate and severe anemia in red blood cell alloimmunized fetuses with the Delta-OD of amniotic Àuid. We assess the validity of this technique at the time of subsequent transfusions, and whether there is a correlation between MCA-PSV normal median value (MOM) and fetal Hb. Materials and Methods The Maternal-Fetal and Neonatal Research Center, Women Hospital, Tehran University of Medical Science, is a tertiary re-ferral center for complicated pregnancies in Tehran, Iran. The study population consisted of 27 singletons, red blood cell alloim-munized pregnancies referred to our center from November 2006 through June 2010. The inclusion criteria were: obstetric history of red blood cell alloimmunization and increased indirect Coombs titers 1:16, past history of IUT, hydrops fetalis and having an in-fant that needed exchange transfusion after birth. Cases with evi-dence of hydrops (skin edema, pleural effusion, or ascites) were excluded from the study because these ¿ndings are a clear indica-tion of severe anemia.
For anemia screening, we measured serial MCA-PSV levels at 7-day intervals and Delta-OD by amniocentesis every 1 – 3 weeks from 16 weeks of pregnancy. Cordocentesis was performed when either MCA-PSV, Delta-OD, or both showed the necessity for IUT. In all cases, fetal Hb concentrations were measured before the ¿rst IUT. In 17 cases, cordecentesis was done before the sec-Prediction of Fetal Anemia by Different Thresholds of MCA-PSV and Delta-OD in First and Second Intrauterine Transfusions Fatemeh Rahimi-Sharbaf MD•1, Ma mak Shariat MD2 , Fatemeh Mirzaie MD1, Padideh Dehghan MD2, Ebrahim Dastgardy MD3, Khadijeh Adabi MD1 Authors’ af¿liations: 1 Department of Obstetrics and Gynecology, Women Hos-pital, Tehran Universi ty of Medical Sciences, Tehran, Iran, 2 Maternal-Fetal and Neonatal Research Center, Tehran University of Medical Sciences, Tehran, Iran, 3 Department of Pediatrics, Women Hospital, Tehran University of Medical Sci-ences, Tehran, Iraan.
•Corresponding author and reprints: Fatemeh Rahimi-Sharbaf MD, Depart-ment of Maternal-Fetal Medicine, Women Hospital, Nejatollahi St., Karim Khan Ave., Tehran 15978, Iran. Tel: +98-912-113-4105, Fax: +98-218-891-5959, E-mail: rahimish@sina.tums.ac.ir; f_rahimis@yahoo.com Accepted for publication: 15 March 2011 Keywords: Alloimmunization, amniocentesis, doppler ultrasonography, intrauterine transfusion, fetal anemia, middle cerebral artery MCA-PSV in Fetal Anemia Cite this article as: Rahimi-Sharbaf F, Shariat M , Mirzaie F, Dehghan P, Dastgardy E, Adabi K. Prediction of Fetal Anemia by Different Thresholds of MCA-PSV and Delta-OD in First and Second Intrauterine Transfusions. Arch Iran Med. 2012; 15(3): 162 – 165.
Archives of Iranian Medicine, Volume 15, Number 3, March 2012 163 ond IUT. Maternal medical records, as well as present and past obstetrical histories were collected. Gestational age was calculated according to the ¿rst day of the last menstrual period and matched with ¿rst trimester ultrasound.
Measurements of Delta-OD were taken the day before cordo-centesis and fetal Doppler MCA-PSV measured within six hours before cordocentesis. For measurement of MCA-PSV, color Àow mapping was used to identify the MCA at the level of the circle of Willis, the pulsed wave Doppler gate was placed on the proximal one third of the MCA and the angle of insonation was kept at less than 20 degrees. Three consecutive wave forms in the absence of fetal body or breathing movements were then recorded, and the highest point of the Doppler envelope was considered as the PSV (cm/s). The measurement was repeated three times, with the maximum value considered for PSV. Attention was taken to avoid pressure on the fetal head. MCA-PSV expressed as multiples of the normal median (MOM) value were measured by the reference curve provided by Mari11 and adjusted for gestational age. We used cutoff values of MCA-PSV above 1.29 and 1.5. For Delta-OD, the Queenan curve was used with the cutoff of the upper af-fected zone (zone 3) considered and adjusted for gestational age.12 In normal fetuses, Hb levels increase with gestational age, there-fore a ¿xed Hb cutoff point cannot be used to de¿ne fetal anemia. The reference test for the diagnosis of fetal anemia was measure-ment of the Hb level in umbilical cord blood with the use of the reference range published by Nicolaides et al. in 1988.1 3Moderate anemia was de¿ned as Hb between 2 SD-5 SD below the mean for gestational age and severe anemia was de¿ned as an Hb level more than 5 SD below the mean (one SD is approximately 1g/ dl).13,14 This study was approved by the Research and Ethics Committee of the Maternal, Fetal and Neonatal Research Center of Tehran University of Medical Sciences. Patients signed informed con-sents after being informed about the study. Chi-square and Fisher exact tests were used. Sensitivity, speci¿c-ity, positive predictive value (PPV), and negative predictive value (NPV) were calculated by standard formulas for a binominal pro-portion, generated for speci¿c thresholds of MOM and Delta-OD. A P value < 0.05 was considered statistically signi¿cant. Results Past obstetric history and patient data are shown in Table 1. De-mographic and characteristic details of patients at the time of the second IUT are shown in Table 2.
Maternal age (mean ± SD) 33.66±4.49[ years] Gravidity (mean ± SD) 4.6 ± 2.4 Living Child (mean ± SD) 1.5 ± 1.3 History of IUT (N, %) 2, 6.5% History of Hydrops (N, %) 19, 61.3% History of IUFD (N, %) 16, 51.6% History of neonatal Exchange (N, %) 6, 19.4% Gestational age at 1st IUT weeks (mean ± SD) 21.54 ± 5.03 [weeks] Fetal Hb before ¿rst IUT (mean ± SD) 6.36 ± 3.62 g/dl Table 1. Characteristic details of patients at ¿rst IUT.
5.22 ± 2.59 Gravidity (Mean ± SD) 1.81 ± 1.46 Living child (Mean ± SD) 12, 68% History of hydrops (N, %) 10, 59% History of IUFD (N, %) 4, 23% History of neonatal exchange (N, %) 24.50 ± 4.02 [weeks] Gestational age at second IUT (mean ± SD) 6.48 ± 3.39 g/dl Fetal Hb before second IUT (mean ± SD) Table 2. Characteristic details of patients at second IUT.
Gold standard test Sensitivity Speci¿city PPV NPV P value MCA-PSV ( MOM > 1.29) Severe anemia 100 50 71 100 0.015 Moderate anemia 60 60 50 50 0.015 MCA-PSV ( MOM > 1.5) Severe anemia 93 90 93 90 < 0.0001 Moderate anemia 0 25 0 < 0.0001 Delta –OD Severe anemia 80 65 72 30 0.04 Moderate anemia 50 75 70 40 0.04 Table 3. Diagnostic values of MCA-PSV & Delta-OD to detect anemia in the ¿rst IUT.
Gold standard test Sensitivity Speci¿city PPV NPV P value MCA-PSV (MOM > 1.29) Severe anemia 100 80 84 100 0.006 Moderate anemia 33 50 50 100 0.006 MCA-PSV (MOM > 1.5 ) Severe anemia 91 100 100 100 < 0.001 Moderate anemia 0 100 100 < 0.001 Table 4. Diagnostic values of MCA-PSV to detect anemia in the second IUT.
F. Rahimi-Sharbaf, M. Shariat, F. Mirzaie, et al.
Archives of Iranian Medicine, Volume 15, Number 3, March 2012 164 The study group consisted of 27 fetuses. At the ¿rst IUT, 5 of 27 cases (20%) were moderately anemic whereas 15 (60%) had se-vere anemia. The MCA-PSV threshold MOM of > 1.29 detected 60% of moderate and 100% of severe cases (P < 0.015), whereas the MCA-PSV with an MOM of > 1.5 detected none of the mod-erate cases and 93% of severe ones (P < 0.0001). In this group, Delta-OD was measured in 18 cases, of which 10 were severe and 4 were moderately anemic. Of these, Delta-OD detected 80% of the severe and 50% of the moderate cases (P < 0.04). Sensitivity, speci¿city, PPV and NPV are shown in Table 3.
In 17 cases that underwent a second IUT, 11 (65%) were severe and 3 (18%) moderately anemic. Of these, 91% of the severely anemic cases were con¿rmed by the MCA-PSV threshold MOM of > 1.5 and all were con¿rmed by the MCA-PSV cutoff MOM of > 1.29 (P < 0.001). One case of moderate anemia was detected by MCA-PSV with an MOM > 1.29 and none were detected by MCA-PSV with an MOM > 1.5. Sensitivity, speci¿city, PPV and NPV for the second IUT are shown in Table 4.
Discussion Our data demonstrated that different thresholds of MCA-PSV had higher sensitivities and speci¿cities for the detection of mod-erate and severe fetal anemia compared with Delta-OD. At the ¿rst IUT for the correlation between severe anemia and MCA-PSV with a cutoff MOM of > 1.5, there was a 93% sensitivity and 90% speci¿city, whereas Delta-OD showed 80% sensitivity and 65% speci¿city. For the detection of moderate anemia, the Delta-OD had 50% sensitivity and 75% speci¿city, whereas the MCA-PSV with an MOM of > 1.29 had 60% sensitivity and speci¿city. MCA-PSV with MOM > 1.5 showed 0% sensitivity and 25% speci¿city. Therefore, the MCA-PSV with a threshold MOM of > 1.29 was more accurate than the MCA-PSV MOM of > 1.5 and Delta-OD for the detection of moderate anemia. Additionally there was a high sensitivity at the second IUT for the detection of severe anemia.
Oepkes et al.14 in a multicenter prospective study on 47 severe anemic fetuses compared MCA- PSV with amniocentesis and showed that the Doppler measurement was signi¿cantly more accurate than Delta-OD. They compared severe anemia with the MCA-PSV of MOM > 1.5, but moderate anemia was not consid-ered. In a small, retrospective study by Pereira et al.15 the research-ers compared MCA-PSV with conventional management. In their study, there were ¿ve anemic fetuses, of which two were hy-dropic. They reported a Delta-OD with 80% sensitivity and 78% speci¿city for moderate to severe anemia. MCA-PSV of MOM > 1.5 had 100% sensitivity and 91% speci¿city for moderate to severe anemia. they considered Hb under 9 g/dL as both moderate and severe anemia. The researchers have concluded that MCA-PSV may have a better predictive accuracy for moderate to severe fetal anemia in red blood cell alloimmunization. Our study revealed two separate thresholds for the diagnosis of moderate and severe anemia in fetuses at the ¿rst and second transfusions. MCA-PSV with an MOM > 1.5 cutoff could better detect severely anemic patients and for the detection of milder anemia, a lower cutoff (> 1.29) for MOM should be considered. These results were comparable to a study by Mari et al.10 that showed signi¿cant association between the degree of anemia and the mean velocity of blood Àow in the middle cerebral artery. Moderate and severe anemia can be detected noninvasively by Doppler ultrasonography based on an increase in the peak velocity of systolic blood Àow in the middle cerebral artery. Our ¿ndings show that for detection of moderate anemia the lower MCA-PSV of MOM > 1.29 should be used. These results have con¿rmed the results of other studies by Mari16,17 that measured MCA-PSV in 18 fetuses before an initial cordocentesis. He demonstrated that the correlation between Hb and MCA-PSV was more accurate as the severity of anemia increased. A study by Scheier et al.3 has shown that in patients who re-ceived one previous IUT the prediction of fetal anemia provided by MCA-PSV was useful, but an increase in the cutoff was neces-sary. Our study also con¿rmed that at the second IUT for detec-tion of severe anemia, the MCA-PSV of > 1.29 MOM had 100% sensitivity and 80% speci¿city. MCA-PSV with a > 1.5 MOM had 91% sensitivity and 100% speci¿city. Compared with the ¿rst IUT, a similar cut off had lower sensitivity.
Bartha et al.18 have reported that the prediction of severe fetal anemia after one transfusion is less accurate than in non-trans-fused fetuses. The MCA-PSV is not useful in predicting severe anemia in fetuses that have previously undergone two transfu-sions. Our data suggest a good correlation between MCA-PSV and Hb in fetuses that have undergone one previous transfusion. In the second IUT the MCA-PSV cutoff MOM of > 1.5 could weakly detect true moderate cases but it detected severely anemic cases.
Maciuleviciene et al.19 showed that the sensitivity of the MCA-PSV test decreased in less anemic fetuses from 77.3% in the sub-group with moderate anemia to 32% in the subgroup with mild anemia. This was in agreement with our study where the sensitiv-ity of the MCA-PSV cutoff of > 1.29 was 60% for detection of moderate anemia. A study by Alshimmiri et al. in 2003 has shown that although MCA-PSV is highly speci¿c, negative values do not rule out fetal anemia20 and has emphasized the need for further re-search before it can be recommended in clinical practice. Another study performed by Rimon et al. proved that in the management of Kell isoimmunization, invasive procedures may be avoided by implementing MCA-PSV measurements.21 De lineation of appro-priate intervals between reassessments, the reliability of MCA-PSV following repeated IUTs and cutoff values for fetal blood sampling await further study.
In conclusion, MCA-PSV thresholds of MOM > 1.29 and > 1.5 in the diagnosis of severe anemia have a high sensitivity in ¿rst and second IUT. MCA-PSV is a non-invasive method for the de-tection of fetal anemia. We strongly recommend it as a screening test before the ¿rst and second IUT. Our results suggest that good correlation between MCA-PSV and fetal Hb remains following ¿rst IUT.
References 1. S eeds JW, Chescheir NC, Bowes WA Jr, Owl-Smith FA. Fet al death as a complication of intrauterine intravascular transfusion. Obstet Gy-necol. 19 89; 74: 1073 – 1075.
2. Liley AW. Liquor amnii analysis in the pregnancy complicated by rhe-sus isoimmunization. Am J Obstet Gynecol. 1961; 82: 1359 – 1370.
3. Nicolaides KH, Rodeck CH, Millar DS, Mibashan RS. Fetal hematol-ogy in rhesus isoimmunisatin. BMJ. 1985; 1: 661 – 663.
4. Scheir M, Hernandez-Andrade E, Carmo A, Dezerega V, Nicolaides KH. Prediction of fetal anemia in rhesus disease by measurement of fetal middle cerebral artery peak systolic velocity. Ultrasound Obstet Gynecol. 2004; 23: 432 – 436. 5. Scheier M, Hernandez-Andrade E, Eduardo B, Fonseca EB, Nicolai-MCA-PSV in Fetal Anemia Archives of Iranian Medicine, Volume 15, Number 3, March 2012 165 des KH. Prediction of severe anemia in red blood cell alloimmuniza-tion after previous intrauterine transfusions. Am J Obstet Gynecol. 2006; 195: 1550 – 1556. 6. MacGregor SN, Silver RK, Sholl JS. Enhanced sensitization after cor-doce ntesis in a rhesus-isoimmunized pregnancy. Am J Obstet Gyne-col. 1991; 165: 382 – 383. 7. Deren O, Onderoglu L. The value of middle cerebral artery systolic velocity for initial and subsequent management in fetal anemia. Eu J Obstet Gynecol. 2002; 101: 26 – 30.
8. Ghidini A, Sepulveda W, Lockwood CJ, Romero R. Complications of fetal blood sampling. Am J Obstet Gynecol. 1993; 168: 1339 – 1344.
9. Tongsong T, Wanapirak C, Kunavikatikul C, Sirirchotiyakul S, Pi-yamongkol W, Chanprapaph P. Fetal loss rate associated with cordo-centesis at midgestation. Am J Obstet Gynecol. 2001; 184: 719 – 723.
10. Mari G, Deter RL, Carpenter RL, Rahman F, Zimmerman R, Moise KJ Jr, et al. Noninvasive diagnosis by Doppler ultrasonography of fe-tal anemia due to maternal red-cell alloimmunization. N Eng J Med. 2000; 342: 9 – 14.
11. Teixeira J, Duncan K, Letsky E, Fisk NM. Middle cerebral artery peak systolic velocity in the prediction of fetal anemia. Ultrasound Obstet Gynecol. 2000; 15: 205 – 208.
12. Queenan JT, Tomai TP, Ural SH, King JC. Deviation in amniotic Àuid optical density at a wavelength of 450 nm in Rh-immunized pregnan-cies from 14 to 40 weeks’ gestation: a proposal for clinical manage-ment. Am J Obstet Gyncol. 1993; 168: 1370 – 1376.
13. Nicolaides KH, Soothill PW, Clewell WH, Rodeck CH, Mibashan RS, Campbell S. Fetal haemoglobin measurement in the assessment of red cell isoimmunisation. Lancet. 1988; 1: 1073 – 1075.
14. Oepkes D, Seaward G, Vandenbussche F, Windrim R, Kingdom J, Beyene J, et al. Doppler ultrasonography vs. amniocentesis to predict fetal anemia. N Eng J Med. 2006; 355: 156 – 164.
15. Pereira L, Jenkins T, Berghella V. Conventional management of ma-ternal red cell alloimmunization compared with management by Dop-pler assessment of middle cerebral artery peak systolic velocity. Am J Obstet Gyn. 2003; 189: 1002 – 1006.
16. Mari G, Detti L, Utku Oz, Zimmerman R, Duerig P, Stefos T. Ac-curate prediction of fetal hemoglobin by Doppler ultrasonography. Obstet & Gyncol. 2002; 99: 589 – 593.
17. Mari G, Zimmermann R, Morise KJ, Deter RL. Correlation between middle cerebral artery peak systolic velocity and fetal hemoglobin af-ter 2 previous intrauterine transfusions. Am J Obestet Gynecol. 2005; 193: 1117 – 1120. 18. Bartha JL, Abdel-Fattah SA, Hunter A, Denbow M, Kyle P, Soothill PW. Optimal interval between middle cerebral artery velocity mea-surements when monitoring pregnancies complicated by red cell al-loimmunization. Fetal Diag Ther. 2006; 21: 22 – 25.
19. Maciuleviciene R, Gaurilcikas A, Simanaviciute D, Nadisauskiene RJ, Gintautas V, Vaitkiene D, et al. Fetal middle cerebral artery Dop-pler velocimetry in cases of rhesus alloimmunization. J Matern Fetal Neonatal Med. 2008; 21: 361 – 365.
20. Alshimmiri MM, Hamoud MS, Al-Saleh EA, Mujaibel KY, Al-Harmi JA, Thalib L. Prediction of fetal anemia by middle cerebral artery peak systolic velocity in pregnancies complicated by rhesus isoimmuniza-tion. J Perinatol. 2003; 23: 536 – 540.
21. Rimon E, Peltz R, Gamzu R, Yagel S, Feldman B, Chayen B, et al. Management of kell isoimmunization–evaluation of a Doppler-guided approach. Ultrasound Obstet Gynecol. 2006; 28: 814 – 820.
F. Rahimi-Sharbaf, M. Shariat, F. Mirzaie, et al. |
12004 | https://www.britannica.com/topic/Amazon-Web-Services | Amazon Web Services
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12005 | https://www.youtube.com/watch?v=obG7mfsWjss | Solving Linear Diophantine Equations
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if the greatest common divisor of A and B is D then Bezos claims that there are integer Solutions X and Y to the linear equation we can find these Solutions using the extended ukian algorithm but remember a good way to understand mathematics is to solve it again let's find another way using modular arithmetic so suppose ax plus b Y is D to introduce modular arithmetic we need to choose a modulus if we choose mod b we can eliminate the Y terms so a solution to ax plus b yals d would also be a solution to ax congruent to D mod b by the same argument it would also be a solution to B Y congruent to D mod a so for example let's try to solve 7x + 11 y = to 1 this equation corresponds to the congruences solving them gives us since X is congruent to 8 mod 11 then X is 8 plus any multiple of 11 and since Y is congruent to 2 mod 7 then Y is equal to 2 plus any multiple of 7 and so our Solutions are or are they while it's true that any solution to our linear equation will be a solution to the congruences it's not true that any solution to the congruences will solve the original equation so how do we find a solution we'll need to do some algebra so remember you can assume anything you want as long as it's explicit A solution to this equation will have the form for some values of s and t so we can substitute those values into our equation and we find so we can solve for one of the variables in terms of the other let's solve this for T and so this value for T will allow us to rewrite our equation for y so if x = 8 + 11s is a solution then Y Must be5 - 7 s and as with everything trust but verify this gives us Solutions if we choose a value for S so if s equals z we get and we can verify that 7 8 + 11 5 is in fact 1 if s is1 we get and again we can verify this approach worked in this case but will it always work suppose we want to solve our linear equation if the greatest common divisor of A and B is not a divisor of D then this is actually unsolvable but if the greatest common divisor is a divisor of D we could remove it to get an equation of the form and Bezos LMA guarantees a solution exists so suppose we want to solve linear combination equal to the greatest common divisor as before we can reduce these mod both coefficients we know we can solve these congruences and so our Solutions will be something of the form something plus a multiple of B and something plus a multiple of a if we substitute these into our equation we get and now let's try to get rid of this a term in our expression for y so we'll solve this for a and so our Solutions will be of the form notice that X will be an integer as long as s is an integer but y has this fractional component and so y will be an integer as long as this expression is an integer and this would occur if this expression is a multiple of B so note that this expression is going to be congruent to D minus ax knot mod b but remember ax knot is supposed to be congruent to D mod b it was a solution to our congruence and so we know that zero is congruent to the difference so we know that b does in fact divide this expression consequently and so this expression is an integer as well so let's find integer solutions to this linear equation since the greatest common divisor of the coefficients is three and three does in fact divide the constant this equation does have integer Solutions and we might as well just remove that common factor to begin with and now our coefficients are relatively prime so now we can solve 4X congruent to 16 mod 5 and 5 y congruent to 16 mod 4 and these have Solutions and if we substitute these into our equations we find and we can solve this for t to get and we can now generate a few Solutions |
12006 | https://personal.morris.umn.edu/~mcquarrb/teachingarchive/Precalculus/Animations/ConicSectionsAnimationsHyperbola.html | Barry McQuarrie
Animations of the Reflection Property for Hyperbolas
The blue line is a hyperbola with foci at the red dots.
Reflection Property for Hyperbolas: any particle that is directed towards a focus of a hyperbola will be reflected towards the other focus.
This reflection property makes hyperbolas useful in telescopes, where a hyperbola shaped mirror can be used to direct light towards a point. There is a nice diagram of this in our text-note the reflecting telescope uses parabolas, ellipses, and hyperbolas!
Nice Animation (it may take awhile to load) that shows this in a different way. You may have to watch it a few times to understand it! |
12007 | https://www.themathdoctors.org/max-and-min-of-a-cubic-without-calculus/ | Max and Min of a Cubic Without Calculus – The Math Doctors
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Max and Min of a Cubic Without Calculus
December 19, 2018 January 16, 2024 / Algebra, AQOTW, Calculus / Alternatives / By Dave Peterson
Last time we looked at various ways to find tangent lines to a parabola without using calculus. Another standard calculus task is to find the maximum or minimum of a function; this is commonly done in the case of a parabola (quadratic function) using algebra, but can it be done with a cubic function? Yes, if you’re a little adventurous!
Here is a question from 2003:
Max and Min of Functions without Derivative
I was curious to know if there is a general way to find the max and min of cubic functions without using derivatives. Our book does this with the use of graphing calculators, but I was wondering if there is a way to find the critical points without derivatives. Any help is greatly appreciated!
I don’t think I’d ever thought about this before, but ideas such as we saw last time suggested a way to do it. I replied:
Yes, there is a way, and it may be very instructive!
Look at the graph of a cubic, and recall that if a polynomial has a double root, it will be tangent to the x-axis (at Q here):
(A double root is one that corresponds to a squared factor.)
So given a general cubic, if we shift it vertically by the right amount, it will have a double root at one of the turning points.
In this picture, the solid line represents the given cubic, and the broken line is the result of shifting it down some amount D, so that the turning point is on the x-axis. D, clearly, is the y-coordinate of the turning point.
So, given an equation
y = ax^3 + bx^2 + cx + d
any turning point will be a double root of the equation
ax^3 + bx^2 + cx + d - D = 0
for some D, meaning that that equation can be factored as
a(x-p)(x-q)^2 = 0
Our goal now is to find the value(s) of D for which this is true. We don’t yet know what p, q, or D might be.
(You might have been expecting us to use a discriminant. If so, think about why this is related to that idea …)
Buckle your seatbelt and hang on while we do some algebra:
Equating these and expanding, we have
a(x-p)(x-q)^2 = ax^3 + bx^2 + cx + d - D
a(x-p)(x^2 - 2qx + q^2) = ax^3 + bx^2 + cx + d - D
a(x^3 - (p+2q)x^2 + q(2p+q)x - pq^2) = ax^3 + bx^2 + cx + d - D
ax^3 - a(p+2q)x^2 + aq(2p+q)x - apq^2 = ax^3 + bx^2 + cx + d - D
The left-hand and right-hand sides must represent the same polynomial.
Since these must be equal for ALL x, we can equate coefficients:
a = a
-a(p+2q) = b
aq(2p+q) = c
-apq^2 = d-D
We have three equations (ignoring the first, which is simply the reason I multiplied my factored form by a in the first place) in three unknowns, p, q, and D, so we can solve to find the double root q, which is the location of the turning point.
Note that this is a system of non-linear equations, which many students are unfamiliar with; but they can be solved by substitution just like a linear system. Note also that D appears only in the fourth equation, so we will be leaving that for last.
To do this, we'll eliminate p by solving the second equation above for p:
p = -(b/a + 2q)
and putting this into the third equation:
aq(-2(b/a + 2q) + q) = c
This simplifies to
-2bq - 3aq^2 = c
3aq^2 + 2bq + c = 0
(Note that this is the derivative of the cubic we are working with. The rest of the work is just what we would do if we were using calculus, but with different reasoning.)
That was a surprise! We accidentally recreated the derivative (evaluated for x=q) without having slopes in mind at all. But this equation, as I said, is just what we’d have written using calculus, setting the derivative at x=q to zero.
Now we solve this for q using the quadratic formula:
-2b +- sqrt(4b^2 - 12ac) -b +- sqrt(b^2 - 3ac)
q = ------------------------ = ---------------------
6a 3a
This gives both turning points, since there are two ways to make a double root. Our last equation gives the value of D, the y-coordinate of the turning point:
D = apq^2 + d = -a(b/a + 2q)q^2 + d = -2aq^3 - bq^2 + d
= (aq^3 + bq^2 + cq + d) - (3aq^2 + 2bq + c)q
= aq^3 + bq^2 + cq + d
(since 3aq^2 + 2bq + c = 0), as we would expect given that x = q; so we don't really have to carry out this step.
Another surprise — or was it? Once we know q, we find the y-coordinate of the turning point just by evaluating the original equation at x=q. We didn’t really need to use this fourth equation at all.
This is the sort of work that had to be done before calculus was invented. In many cases, calculus is really just a shortcut for algebra.
Thanks for asking the question, because I'd never considered approaching it without calculus before!
Roy wrote back,
Thanks for the quick response, Dr. Peterson. That is just what I was looking for! I think my students will definitely benefit from seeing this approach and appreciate it once they get into calculus.
Repeated, concretely
Thirteen years later, Yousuf read that page, and wrote asking for clarification:
In [the conversation above], you said this:
Any turning point will be a double root of the equation
ax^3 + bx^2 + cx + d - D = 0
for some D, meaning that that equation can be factored as
a(x - p)(x - q)^2 = 0
The '- D' is a little confusing to understand as to where you got it from. How (or where?) did you obtain it to form that first equation? What significance does it play in this analogy?
I think I do understand the basic gist of what you're trying to achieve here. I understand the analogy that (I THINK) we picture first a general cubic graph which has no turning points touching the x-axis. If we can translate such a graph by the right amount vertically, what we get is a cubic graph where one of the turning points is touching the x-axis, making the axis a tangent to that point (we get a double root).
So with this said (daft as it may seem, but I may have answered my own question), is the 'D' the difference of the vertical component of the vector used to translate the general cubic graph to attain the second graph, which has at least one turning point on the x-axis?
People do often answer their own questions when they write them out! The fact that I hadn’t included the second picture above in the original probably accounts for his not having quite seen the point. I responded with such a graph to confirm his understanding.
But he wanted to see how it all works in a specific case, which is often a good idea:
For example, given a graph of x^3 - 7x^2 + 15x - 11. If we translated this graph using the vector (0, 2) -- so, 2 units in the positive direction of the y-axis, i.e., upwards -- we would get the equation
y = x^3 - 7x^2 + 15x - 9
= (x - 1)(x - 3)^2
Here, the double root emerges; and therefore, the x-axis is a tangent to the curve at x = 3, producing a minimum. Is the 'D' somehow tied into this analogy? Is D the difference between the original intercept of the graph (-11) and the translated graph (-9)? So would D (in this particular example) be -11 - (-9) = -2?
But having said all that, Doctor Math ... if the above analogy is correct, it was achieved through pure luck: I deliberately chose the equation (x - 1)(x - 3)^2, and just manipulated it. If we had such an equation as, say, x^3 - 7x^2 + 15x - 11, how would you know that translating it by 2 units upwards would achieve a similar graph containing a double root? and therefore a max/min point?
I'm really keen to understand if it is possible to calculate maximum and minimum points of cubic graphs WITHOUT the use of calculus. However easy it is, I want to explore this possibility.
I'd be most grateful if you would be able to please clarify this predicament.
Here is his function, before and after translating it upward:
Like my graphs above, he worked backward to find a nice example, and wants to see the forward process actually carried out. I responded further:
Let's do it.
The original conversation, above, answers your question didactically, showing how to find D ... eventually; but looking at it concretely would help anyone fully grasp it.
We have the cubic equation
y = x^3 - 7x^2 + 15x - 11
We translate this downward by an unknown distance D (the y-coordinate of the extremum in question), with the goal of having a double zero:
y = x^3 - 7x^2 + 15x - 11 - D
That is, we want, for some p and q,
(x - p)(x - q)^2 == x^3 - 7x^2 + 15x - 11 - D
Here, "==" represents "is identically equal to." That is, they must be equal for all x. Note that our "a" is 1.
So far, this is identical to what I did in the abstract. Our main goal is to find q, the x-coordinate of one of the turning points.
By the way, another way to say this is that we want to take this ...
x^3 - 7x^2 + 15x - 11
... and rewrite it as this:
(x - p)(x - q)^2 + D
Now we dig into the algebra, which will be a little easier to follow with ordinary numerical coefficients:
We expand the left side, as I showed, resulting in
x^3 - (p + 2q)x^2 + q(2p + q)x - pq^2
== x^3 - 7x^2 + 15x - 11 - D
We conclude that we need
-(p + 2q) = -7
q(2p + q) = 15
-pq^2 = -11 - D
We solve the first equation for p:
p + 2q = 7
p = 7 - 2q
We put this into the second equation:
q(2(7 - 2q) + q) = 15
q(14 - 3q) = 15
3q^2 - 14q + 15 = 0
We solve this by factoring (since it has rational solutions, we don't need to use the quadratic formula):
(3q - 5)(q - 3) = 0
q = 5/3 or 3
This part was easier this time.
These will be the x-coordinates of the two turning points. To find the amount by which we have to translate to get x = 3, we put q = 3 and p = 7 - 2(3) = 1 into the last of our equations:
-pq^2 = -11 - D
D = pq^2 - 11
= (1)(3)^2 - 11 = -2
This tells us that our original equation ...
y = x^3 - 7x^2 + 15x - 11
... can be written as
y = (x - 1)(x - 3)^2 - 2
This last bit looks quite different from what I did in the general case, because I don't have crowds of letters everywhere and could use a different approach, which makes some parts feel more natural, but hides the fact that D = aq^3 + bq^2 + cq + d.
So we translated the graph up 2 units to touch the x-axis.
Now, the OTHER extremum is found by taking q = 5/3, so that p = 11/3. Then D turns out to be
D = pq^2 - 11
= (11/3)(5/3)^2 - 11
= 275/27 - 11
= -22/27
So we can also write our equation as
y = (x - 11/3)(x - 5/3)^2 - 22/27
You can check this out to see that it is true.
I hope that helps! It was interesting revisiting this old discussion and looking at it from a different direction.
Yousuf replied,
I want to say a HUUUGE thank you for helping clarify this article in the most subtle way. You explained it in the example and it all clicked in perfectly well. Now I can associate the points in the example seamlessly with the algebraic proof -- all thanks to your help!!
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12008 | https://m.chuangshi.qq.com/read/54060170/7 | 第2章 函数_高考数学真题分级精刷（2024年版）最新章节在线阅读-创世中文网官网
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高考数学真题分级精刷（2024年版） ------------------
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第2章 函数
函数贯穿于高中数学始终，高考数学中，通常先单独出一道5分的小题，然后将函数结合导数、三角函数、数列、推理、逻辑、不等式、统计、解析几何等，出一道小题和大题，大概20多分。共计30分左右。
本章要求：简单函数题必须满分；综合其他知识点的复杂函数题量力而行。高考中如果实在不会不要纠结，与其浪费太多时间在不确定上，不如节省时间保证其他题目的正确率。
2.1 考点及常考结论
（1）求定义域常见三题型
① 分母；② 根号；③ 真数log a M（M>0）。
（2）函数的奇偶性
① 奇函数：设 y=f（x），x∈A，如果对于任意 x∈A，都有 f（−x）=−f（x），则称 y=f（x）为奇函数。y=f（x）是奇函数⇔y=f（x）的图像关于原点对称。
② 偶函数：设 y=f（x），x∈A，如果对于任意 x∈A，都有 f（−x）=f（x），则称 y=f（x）为偶函数。y=f（x）是偶函数⇔y=f（x）的图像关于 y 轴对称。
③ 偶函数在定义域内关于原点对称的两个区间上单调性相反，奇函数在定义域内关于原点对称的两个区间上单调性相同。
④ 若函数 f（x）的定义域关于原点对称，则它可表示为一个奇函数与一个偶函数之和。
⑤ 奇±奇=奇 偶±偶=偶 奇×奇=偶 偶×偶=偶 奇×偶=奇。
⑥ 对于复合函数
若 g（x）是偶函数，则 F（x）是偶函数；
若 g（x）是奇函数且 f（x）是奇函数，则 F（x）是奇函数；
若 g（x）是奇函数且 f（x）是偶函数，则 F（x）是偶函数。
（3）函数的单调性
① 若 f（x）、g（x）均为增函数，则 f（x）+g（x）仍为增函数。
② 若 f（x）为增函数，则−f（x）为减函数。
③ 互为反函数的两个函数有相同的单调性。
④ 设是定义在 N 上的函数
若 f（x）与 g（x）的单调性相反，则在 N 上是减函数；
若 f（x）与 g（x）的单调性相同，则在 N 上是增函数。
（4）指数结论
① 正整数指数幂。
② 零指数幂 a 0=1（a≠0）。
③ 负整数指数幂。
④ 正分数指数幂。
⑤ 负分数指数幂。
⑥ 0的正分数指数幂等于0，0的负分数指数幂没有意义。
⑦ a r a s=a r+s（a>0；r，s∈Q）。
⑧
⑨（ab）r=a r b r（a>0，b>0，r∈Q）。
（5）对数结论
① 对数的概念
如果 a b=N（a>0，a≠1），那么 b 叫作以 a 为底 N 的对数，记为 b=log a N（a>0，a≠1）。
② 对数的性质
零与负数没有对数 log a 1=0 log a a=1
③ 对数的运算性质
a>0，a≠0，M>0，N>0
log a MN=log a M+log a N
log a M n=n log a M
④ 对数换底公式
⑤ 对数的降幂公式
（6）函数的图像变换
① 平移变换（左正右负，上正下负）
② 对称变换（对称谁，谁不变，对称原点都要变）
③ 伸缩变换
④ 综合变换
若将函数 y=f（x）的图像右移 a、上移 b 个单位，得到函数 y=f（x−a）+b 的图像。
若将曲线 f（x，y）=0的图像右移 a、上移 b 个单位，得到曲线 f（x−a，y−b）=0的图像。
（7）函数周期常用结论
对于 f（x）定义域内任一自变量的值 x（a>0，c 为常数）：
若 f（x+a）=−f（x），或 f（x+a）=−f（x）+c，或，或，则 T=2 a。
2.2 历年真题
2.2.1 函数的定义域值域
2-1．（2022北京11★）函数的定义域是______。
答案见此处
2-2．（2022浙江7★★）已知2 a=5，log 8 3=b，则4 a−3 b=（ ）
A．25
B．5
C．
D．
答案见此处
2-3．（2022北京4★★）已知函数，则对任意实数 x，有（ ）
A．f（−x）+f（x）=0
B．f（−x）−f（x）=0
C．f（−x）+f（x）=1
D．
答案见此处
2-4．（2021全国乙卷文8★★★）下列函数中最小值为4的是（ ）
A．y=x 2+2 x+4
B．
C．y=2 x+2 2−x
D．
答案见此处
2-5．（2019江苏4★）函数的定义域是_____。
答案见此处
2-6．（2015山东理14★★）已知函数 f（x）=a x+b（a>0，a≠1）的定义域和值域都是[−1，0]，则 a+b=___。
答案见此处
2.2.2 函数的奇偶性单调性周期性
2-7．（2023新课标Ⅰ卷4★★）设函数 f（x）=2 x （x−a） 在区间（0，1）单调递减，则 a 的取值范围是（ ）
A．（−∞，−2]
B．[−2，0）
C．（0，2]
D．[2，+∞）
答案见此处
2-8．（2023新课标Ⅱ卷4★★）若为偶函数，则 a=（ ）
A．−1
B．0
C．
D．1
答案见此处
2-9．（2022全国乙卷文16★★★）若是奇函数，则 a=_，b_=__。
答案见此处
2-10．（2021全国甲卷文4★）下列函数中是增函数的为（ ）
A．f（x）=−x
B．
C．f（x）=x 2
D．
答案见此处
2-11．（2021全国乙卷理4文9★★）设函数，则下列函数中为奇函数的是（ ）
A．f（x−1）−1
B．f（x−1）+1
C．f（x+1）−1
D．f（x+1）+1
答案见此处
2-12．（2021全国甲卷文12★★）记 f（x）是定义域为R的奇函数，且 f（1+x）=f（−x）。若，则
A．
B．
C．
D．
答案见此处
2-13．（2021全国新高考Ⅰ卷13★★）已知函数 f（x）=x 3（a·2 x−2−x）是偶函数，则 a=____。
答案见此处
2-14．（2021全国新高考Ⅱ卷8★★★）已知函数 f（x）的定义域为R，f（x+2）为偶函数，f（2 x+1）为奇函数，则（ ）
A．
B．f（−1）=0
C．f（2）=0
D．f（4）=0
答案见此处
2-15．（2021全国甲卷理12★★★）设函数 f（x）的定义域为R，f（x+1）为奇函数，f（x+2）为偶函数，当 x∈[1，2]时，f（x）=ax 2+b。若 f（0）+f（3）=6，则
A．
B．
C．
D．
答案见此处
2-16．（2020全国Ⅱ文10★★）设函数，则 f（x）（ ）
A．是奇函数，且在（0，+∞）单调递增
B．是奇函数，且在（0，+∞）单调递减
C．是偶函数，且在（0，+∞）单调递增
D．是偶函数，且在（0，+∞）单调递减
答案见此处
2-17．（2020全国Ⅱ理9★★★）设函数 f（x）=ln|2 x+1|−ln|2 x−1|，则 f（x）（ ）
A．是偶函数，且在单调递增
B．是奇函数，且在单调递减
C．是偶函数，且在单调递增
D．是奇函数，且在单调递减
答案见此处
2-18．（2019全国Ⅱ理14★★）已知 f（x）是奇函数，且当 x<0时，f（x）=−e ax。若 f（ln 2）=8，则 a=____。
答案见此处
2-19．（2019全国Ⅱ文6★★）设 f（x）为奇函数，且当 x≥0时，f（x）=e x−1，则当 x<0时，f（x）=（ ）
A．e−x−1
B．e−x+1
C．−e−x−1
D．−e−x+1
答案见此处
2-20．（2019北京理13★★）设函数 f（x）=e x+a e−x（a 为常数）。若 f（x）为奇函数，则 a=_；若 f_（x）是R上的增函数，则 a 的取值范围是__。
答案见此处
2-21．（2018上海7★★）已知，若幂函数 f（x）=x a 为奇函数，且在（0，+∞）上递减，则 a=____。
答案见此处
2-22．（2018全国Ⅱ理11★★）已知 f（x）是定义域为（−∞，+∞）的奇函数，满足 f（1−x）=f（1+x）。若 f（1）=2，则 f（1）+f（2）+f（3）+…+f（50）=（ ）
A．−50
B．0
C．2
D．50
答案见此处
2-23．（2017北京理5★★）已知函数，则 f（x）（ ）
A．是奇函数，且在R上是增函数
B．是偶函数，且在R上是增函数
C．是奇函数，且在R上是减函数
D．是偶函数，且在R上是减函数
答案见此处
2-24．（2015新课标Ⅰ理13★★）若函数为偶函数，则 a=____。
答案见此处
2-25．（2015福建理2★★）下列函数为奇函数的是（ ）
A．
B．
C．=cos x
D．y=e x−e−x
答案见此处
2-26．（2015广东理3★★）下列函数中，既不是奇函数，也不是偶函数的是（ ）
A．
B．
C．
D．y=x+e x
答案见此处
2-27．（2015湖南理5★★）设函数 f（x）=ln（1+x）−ln（1−x），则 f（x）是（ ）
A．奇函数，且在（0，1）上是增函数
B．奇函数，且在（0，1）上是减函数
C．偶函数，且在（0，1）上是增函数
D．偶函数，且在（0，1）上是减函数
答案见此处
2-28．（2014北京文2★★）下列函数中，定义域是R且为增函数的是（ ）
A．y=e−x
B．y=x 3
C．y=ln x
D．
答案见此处
2-29．（2014湖南理3★★）已知 f（x），g（x）分别是定义在R上的偶函数和奇函数，且 f（x）−g（x）=x 3+x 2+1，f（1）+g（1）＝（ ）
A．−3
B．−1
C．1
D．3
答案见此处
2-30．（2014新课标Ⅱ文15★★）偶函数 f（x）的图像关于直线 x=2对称，f（3）=3，则 f（−1）=____。
答案见此处
2-31．（2014湖南文15★★）若 f（x）=ln（e 3 x+1）+ax 是偶函数，则 a=____。
答案见此处
2-32．（2017新课标Ⅰ理5★★）函数 f（x）在（−∞，+∞）单调递减，且为奇函数。若 f（1）=−1，则满足−1≤f（x−2）≤1的 x 的取值范围是（ ）
A．[−2，2]
B．[−1，1]
C．[0，4]
D．[1，3]
答案见此处
2-33．（2016山东理9★★）已知函数 f（x）的定义域为R。当 x<0时，f（x）=x 3−1；当−1≤x≤1时，f（−x）=−f（x）；当时，。则 f（6）=（ ）
A．−2
B．−1
C．0
D．2
答案见此处
2-34．（2014重庆文4★）下列函数为偶函数的是（ ）
A．f（x）=x−1
B．f（x）=x 2+x
C．f（x）=2 x−2−x
D．f（x）=2 x+2−x
答案见此处
2-35．（2014新课标Ⅰ理3★）设函数 f（x），g（x）的定义域都为R，且 f（x）是奇函数，g（x）是偶函数，则下列结论正确的是（ ）
A．f（x）g（x）是偶函数
B．是奇函数
C．是奇函数
D．是奇函数
答案见此处
2.2.3 函数的图像
2-36．（2022全国甲卷理5文7★★）函数 y=（3 x−3−x）cos x 在区间的图像大致为（ ）
答案见此处
2-37．（2022全国乙卷文8★★）如图是下列四个函数中的某个函数在区间[−3，3]的大致图像，则该函数是（ ）
A．
B．
C．
D．
答案见此处
2-38．（2021浙江7★★）已知函数，g（x）=sin x，则图像为如图2-1所示的函数可能是（ ）
图2-1
A．
B．
C．y=f（x）g（x）
D．
答案见此处
2-39．（2019全国Ⅰ理文5★★）函数在[−π，π]的图像大致为（ ）
答案见此处
2-40．（2019浙江6★★）在同一直角坐标系中，函数（a>0且 a≠1）的图像可能是（ ）
答案见此处
2-41．（2019全国Ⅲ理7★★）函数在[−6，6]的图像大致为（ ）
答案见此处
2-42．（2018全国Ⅲ理7★★）函数 y=−x 4+x 2+2的图像大致为（ ）
答案见此处
2-43．（2018全国Ⅱ理3★★）函数的图像大致为（ ）
答案见此处
2-44．（2018浙江5★★）函数 y=2|x| sin 2 x 的图像可能是（ ）
答案见此处
2-45．（2016全国Ⅰ理7★★）函数 y=2 x 2−e|x|在[–2，2]的图像大致为（ ）
答案见此处
2-46．（2015安徽理9★★）函数的图像如图2-2所示，则下列结论成立的是（ ）
图2-2
A．a>0，b>0，c<0
B．a<0，b>0，c>0
C．a<0，b>0，c<0
D．a<0，b<0，c<0
答案见此处
2-47．（2014浙江理7★★）在同一直角坐标系中，函数 f（x）=x a（x≥0），g（x）=log a x 的图像可能是（ ）
答案见此处
2-48．（2014山东文6★★）已知函数 y=log a（x+c）（a、c 为常数，其中 a>0，a≠1）的图像如图2-3所示，则下列结论成立的是（ ）
图2-3
A．a>1，c>1
B．a>1，0<c<1
C．0<a<1，c>1
D．0<a<1，0<c<1
答案见此处
2.2.4 比较大小
2-49．（2022全国甲卷理12★★★）已知，则（ ）
A．c>b>a
B．b>a>c
C．a>b>c
D．a>c>b
答案见此处
2-50．（2022全国新高考Ⅰ卷7★★★）设 a=0.1e 0.1，，c=−ln0.9，则（ ）
A．a<b<c
B．c<b<a
C．c<a<b
D．a<c<b
答案见此处
2-51．（2022全国甲卷文12★★★）已知9 m=10，a=10 m−11，b=8 m−9，则（ ）
A．a>0>b
B．a>b>0
C．b>a>0
D．b>0>a
答案见此处
2-52．（2021全国新高考Ⅰ卷7★★）若过点（a，b）可以作曲线 y=e x 的两条切线，则（ ）
A．e b<a
B．e a<b
C．0<a<e b
D．0<b<e a
答案见此处
2-53．（2021全国新高考Ⅱ卷7★★）已知 a=log 5 2，b=log 8 3，，则下列判断正确的是（ ）
A．c<b<a
B．b<a<c
C．a<c<b
D．a<b<c
答案见此处
2-54．（2021全国乙卷理11★★★★）设 a=2ln 1.01，b=ln 1.02，，则（ ）
A．a<b<c
B．b<c<a
C．b<a<c
D．c<a<b
答案见此处
2-55．（2020全国Ⅰ理12★★★★）若2 a+log 2 a=4 b+2log 4 b，则（ ）
A．a>2 b
B．a<2 b
C．a>b 2
D．a<b 2
答案见此处
2-56．（2020全国Ⅲ文10★★）设 a=log 3 2，b=log 5 3，，则（ ）
A．a<c<b
B．a<b<c
C．b<c<a
D．c<a<b
答案见此处
2-57．（2020全国Ⅲ理12★★★）已知5 5<8 4，13 4<8 5。设 a=log 5 3，b=log 8 5，c=log 13 8，则（ ）
A．a<b<c
B．b<a<c
C．b<c<a
D．c<a<b
答案见此处
2-58．（2020全国Ⅱ理11文12★★★）若2 x−2 y<3−x−3−y，则（ ）
A．ln（y−x+1）>0
B．ln（y−x+1）<0
C．ln|x−y|>0
D．ln|x−y|<0
答案见此处
2-59．（2019全国Ⅱ理6★★）若 a>b，则（ ）
A．ln（a−b）>0
B．3 a<3 b
C．a 3−b 3>0
D．|a|>|b|
答案见此处
2-60．（2019全国Ⅲ理11★★）若 f（x）是定义域 R 的偶函数，且在（0，+∞）单调递减，则（ ）
A．
B．
C．
D．
答案见此处
2-61．（2019全国Ⅰ理文3★★）已知 a=log 2 0.2，b=2 0.2，c=0.2 0.3，则（ ）
A．a<b<c
B．a<c<b
C．c<a<b
D．b<c<a
答案见此处
2-62．（2019天津理6★★）已知 a=log 5 2，b=log 0.5 0.2，c=0.5 0.2，则 a、b、c 的大小关系为（ ）
A．a<c<b
B．a<b<c
C．b<c<a
D．c<a<b
答案见此处
2-63．（2018全国Ⅲ理12★★）设 a=log 0.2 0.3，b=log 2 0.3，则（ ）
A．a+b<ab<0
B．ab<a+b<0
C．a+b<0<ab
D．ab<0<a+b
答案见此处
2-64．（2018天津理5★★）已知 a=log 2 e，b=ln 2，，则 a、b、c 的大小关系为（ ）
A．a>b>c
B．b>a>c
C．c>b>a
D．c>a>b
答案见此处
2-65．（2017天津理6★★）已知奇函数 f（x）在R上是增函数，g（x）=xf（x）。若 a=g（−log 2 5.1），b=g（2 0.8），c=g（3），则 a、b、c 的大小关系为（ ）
A．a<b<c
B．c<b<a
C．b<a<c
D．b<c<a
答案见此处
2-66．（2017新课标Ⅰ理11★★）设 x、y、z 为正数，且2 x=3 y=5 z，则（ ）
A．2 x<3 y<5 z
B．5 z<2 x<3 y
C．3 y<5 z<2 x
D．3 y<2 x<5 z
答案见此处
2-67．（2016新课标Ⅲ理6★★）已知，，，则（ ）
A．b<a<c
B．a<b<c
C．b<c<a
D．c<a<b
答案见此处
2-68．（2015天津理7★★★）已知定义在R上的函数（m 为实数）为偶函数，记 a=f（log 0.5 3），b=f（log 2 5），c=f（2 m），则 a、b、c 的大小关系为（ ）
A．a<b<c
B．a<c<b
C．c<a<b
D．c<b<a
答案见此处
2-69．（2014安徽文5★★）设 a=log 3 7，b=2 3.3，c=0.8，则（ ）
A．b<a<c
B．c<a<b
C．c<b<a
D．a<c<b
答案见此处
2.2.5 分段函数
2-70．（2022北京14★★）设函数。若 f（x）存在最小值，则 a 的一个取值为_；a_ 的最大值为__。
答案见此处
2-71．（2022浙江14★★★）已知函数，则；若当 x∈[a，b]时，1≤f（x）≤3，则 b−a 的最大值是____。
答案见此处
2-72．（2021浙江12★★）已知 a∈R，函数，若，则 a=____。
答案见此处
2-73．（2018江苏9★★）函数 f（x）满足 f（x+4）=f（x）（x∈R），且在区间（−2，2]上，，则 f（f（15））的值为____。
答案见此处
2-74．（2017新课标Ⅲ理15文16★★★）设函数，则满足的 x 的取值范围是____。
答案见此处
2-75．（2015福建理14★★）若函数（a>0且 a≠1）的值域是[4，+∞），则实数 a 的取值范围是____。
答案见此处
2-76．（2015浙江理10★★）已知函数，则 f（f（−3））_，f_（x）的最小值是__。
答案见此处
2-77．（2016江苏11★★）设 f（x） 是定义在R上且周期为2的函数，在区间[−1，1）上，，其中 a∈R，若，则 f（5 a）的值是____。
答案见此处
2-78．（2016四川理9★★★）设直线 l 1、l 2 分别是函数图像上点 P 1、P 2 处的切线，l 1 与 l 2 垂直相交于点 P，且 l 1、l 2 分别与 y 轴相交于点 A、B，则△PAB 的面积的取值范围是（ ）
A．（0，1）
B．（0，2）
C．（0，+∞）
D．（1，+∞）
答案见此处
2-79．（2014福建理7★★）已知函数，则下列结论正确的是（ ）
A．f（x）是偶函数
B．f（x）是增函数
C．f（x）是周期函数
D．f（x）的值域为[−1，+∞）
答案见此处
2-80．（2014四川理12★★）设 f（x）是定义在R上的周期为2的函数，当 x∈[−1，1）时，，则。
答案见此处
2-81．（2014辽宁文10★★★）已知 f（x）为偶函数，当 x≥0时，f（x）=，则不等式的解集为（ ）
A．
B．
C．
D．
答案见此处
2.2.6 函数与方程
2-82．（2021北京15★★★）已知 f（x）=|lg x|−kx−2，给出下列四个结论：
（1）若 k=0，则 f（x）有两个零点；
（2）∃k<0，使得 f（x）有一个零点；
（3）∃k<0，使得 f（x）有三个零点；
（4）∃k>0，使得 f（x）有三个零点。
以上正确结论的序号是____。
答案见此处
2-83．（2019全国Ⅱ理12★★★★）设函数 f（x）的定义域为R，满足 f（x+1）=2 f（x），且当 x∈（0，1]时，f（x）=x（x−1）。若对任意 x∈（−∞，m]，都有，则 m 的取值范围是（ ）
A．
B．
C．
D．
答案见此处
2-84．（2019浙江9★★★★）已知 a、b∈R，函数 f（x）=，若函数 y=f（x）−ax−b 恰有三个零点，则（ ）
A．a<−1，b<0
B．a<−1，b>0
C．a>−1，b<0
D．a>−1，b>0
答案见此处
2-85．（2019江苏14★★★★）设 f（x），g（x）是定义在R上的两个周期函数，f（x）的周期为4，g（x）的周期为2，且 f（x）是奇函数。当 x∈（0，2]时，，其中 k>0。若在区间（0，9]上，关于 x 的方程 f（x）=g（x）有8个不同的实数根，则 k 的取值范围是____。
答案见此处
2-86．（2018全国Ⅰ理9★★★）已知函数，g（x）=f（x）+x+a。若 g（x）存在2个零点，则 a 的取值范围是（ ）
A．[−1，0）
B．[0，+∞）
C．[−1，+∞）
D．[1，+∞）
答案见此处
2-87．（2018天津理14★★★）已知 a>0，函数 f（x）=。若关于 x 的方程 f（x）=ax 恰有2个互异的实数解，则 a 的取值范围是____。
答案见此处
2-88．（2018浙江15★★★）已知 λ∈R，函数，当 λ=2时，不等式 f（x）<0的解集是_。若函数 f_（x）恰有2个零点，则 λ 的取值范围是__。
答案见此处
2-89．（2018江苏11★★★★）若函数 f（x）=2 x 3−ax 2+1（a∈R）在（0，+∞）内有且只有一个零点，则 f（x）在[−1，1]上的最大值与最小值的和为____。
答案见此处
2-90．（2017山东理10★★★）已知当 x∈[0，1]时，函数 y=（mx−1）2 的图像与的图像有且只有一个交点，则正实数 m 的取值范围是（ ）
A．
B．（0，1]∪[3，+∞）
C．
D．
答案见此处
2-91．（2017江苏14★★★★）设 f（x）是定义在R且周期为1的函数，在区间[0，1）上，，其中集合，则方程 f（x）−lg x=0的解的个数是____。
答案见此处
2-92．（2017新课标Ⅲ理11★★★）已知函数 f（x）=x 2−2 x+a（e x−1+e−x+1）有唯一零点，则 a=（ ）
A．
B．
C．
D．1
答案见此处
2-93．（2016全国Ⅱ理12★★）已知函数 f（x）（x∈R）满足 f（−x）=2−f（x），若函数与 y=f（x）图像的交点为（x 1，y 1），（x 2，y 2），…，（x m，y m），则
A．0
B．m
C．2 m
D．4 m
答案见此处
2-94．（2016天津理8★★★★）已知函数（a>0，且 a≠1）在R上单调递减，且关于 x 的方程|f（x）|=2−x 恰好有两个不相等的实数解，则 a 的取值范围是（ ）
A．
B．
C．
D．
答案见此处
2-95．（2016山东理15★★★★）已知函数，其中 m>0，若存在实数 b，使得关于 x 的方程 f（x）=b 有三个不同的根，则 m 的取值范围是____。
答案见此处
2-96．（2015湖北理12★★★）函数|ln（x+1）|的零点个数为____。
答案见此处
2-97．（2015天津理8★★★★）已知函数函数 g（x）=b−f（2−x），其中 b∈R，若函数 y=f（x）−g（x）恰有4个零点，则 b 的取值范围是（ ）
A．
B．
C．
D．
答案见此处
2-98．（2015北京理14★★★）设函数，x≥1
（1）若 a=1，则 f（x）的最小值为____；
（2）若 f（x）恰有2个零点，则实数 a 的取值范围是____。
答案见此处
2-99．（2014山东理8★★★）已知函数，g（x）=kx。若方程 f（x）=g（x）有两个不相等的实根，则实数 k 的取值范围是（ ）
A．
B．
C．（1，2）
D．（2，+∞）
答案见此处
2-100．（2014江苏13★★★）已知 f（x） 是定义在R上且周期为3的函数，当 x∈[0，3）时，。若函数 y=f（x）−a 在区间[−3，4]上有10个零点（互不相同），则实数 a 的取值范围是____。
答案见此处
2-101．（2014福建文15★★）函数的零点个数是____。
答案见此处
2-102．（2014北京文6★★）已知函数，在下列区间中，包含 f（x）零点的区间是（ ）
A．（0，1）
B．（1，2）
C．（2，4）
D．（4，+∞）
答案见此处
2-103．（2014天津理14★★★）已知函数 f（x）=|x 2+3 x|，x∈R。若方程 f（x）−a|x−1|=0恰有4个互异的实数根，则实数 a 的取值范围为____。
答案见此处
2.2.7 函数综合题及其他
2-104．（2022全国新高考Ⅱ卷8★★★）若函数 f（x）的定义域为R，且 f（x+y）+f（x−y）=f（x）f（y），f（1）=1，则
A．−3
B．−2
C．0
D．1
答案见此处
2-105．（2022全国乙卷理12★★★★）已知函数 f（x）、g（x）的定义域均为R，且 f（x）+g（2−x）=5，g（x）−f（x−4）=7。若 y=g（x）的图像关于直线 x=2对称，g（2）=4，则
A．−21
B．−22
C．−23
D．−24
答案见此处
2-106．（2021全国新高考Ⅱ卷14★★）写出一个同时具有下列性质①②③的函数 f（x）：____。
① f（x 1 x 2）=f（x 1）f（x 2）；②当 x∈（0，+∞）时，f′（x）>0；③ f′（x）是奇函数。
答案见此处
2-107．（2021上海5★）已知，则 f−1（1）=____。
答案见此处
2-108．（2021天津7★★）2 a=5 b=10，则。
答案见此处
2-109．（2019全国Ⅰ理11★★★）关于函数 f（x）=sin|x|+|sin x|有下述四个结论：
① f（x）是偶函数
② f（x）在区间（，π）单调递增
③ f（x）在[−π，π]有4个零点
④ f（x）的最大值为2
其中所有正确结论的编号是（ ）
A．①②④
B．②④
C．①④
D．①③
答案见此处
2-110．（2018北京理13★★）能说明“若 f（x）>f（0）对任意的 x∈（0，2]都成立，则 f（x）在[0，2]上是增函数”为假命题的一个函数是____。
答案见此处
2-111．（2017浙江17★★★★）已知 a∈R，函数在区间[1，4]上的最大值是5，则 a 的取值范围是____。
答案见此处
2-112．（2017山东理15★★★★）若函数e x f（x）（e=2.71828…，是自然对数的底数）在 f（x）的定义域上单调递增，则称函数 f（x）具有 M 性质，下列函数中具有 M 性质的是（ ）。
① f（x）=2−x
② f（x）=3−x
③ f（x）=x 3
④ f（x）=x 2+2
答案见此处
2-113．（2017浙江5★★★）若函数 f（x）=x 2+ax+b 在区间[0，1]上的最大值是 M，最小值是 m，则 M−m（ ）
A．与 a 有关，且与 b 有关
B．与 a 有关，但与 b 无关
C．与 a 无关，且与 b 无关
D．与 a 无关，但与 b 有关
答案见此处
2-114．（2015湖北理6★★）已知符号函数，f（x）是R上的增函数，g（x）=f（x）−f（ax）（a>1），则（ ）
A．sgn[g（x）]=sgn x
B．sgn[g（x）]=−sgn x
C．sgn[g（x）]=sgn[f（x）]
D．sgn[g（x）]=−sgn[f（x）]
答案见此处
2-115．（2014江西理3★★）已知函数，g（x）=ax 2−x（a∈R），若，则 a=（ ）
A．1
B．2
C．3
D．−1
答案见此处
2-116．（2014浙江理6★★）已知函数 f（x）=x 3+ax 2+bx+c，且0<f（−1）=f（−2）=f（−3）≤3，则（ ）
A．c≤3
B．3<c≤6
C．6<c≤9
D．c>9
答案见此处
2-117．（2016上海理22★★★★）已知 a∈R，函数。
（1）当 a=5时，解不等式 f（x）>0；
（2）若关于 x 的方程 f（x）−log 2[（a−4）x+2 a−5]=0的解集中恰好有一个元素，求 a 的取值范围；
（3）设 a>0，若对任意，函数 f（x）在区间[t，t+1]上的最大值与最小值的差不超过1，求 a 的取值范围。
答案见此处
查漏补缺
2.2.1 函数的定义域值域
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
2.2.2 函数的奇偶性单调性周期性
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
2.2.3 函数的图像
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
2.2.4 比较大小
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
2.2.5 分段函数
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
2.2.6 函数与方程
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
2.2.7 函数综合题及其他
完全没问题，无论什么时候做都能快速做对，题号：___；
能做对，但是用时较长，题号：___；
本来会做，但由于某种原因，结果错了，题号：___；
不太会，题号：___；
完全不会，题号：___。
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12009 | https://medium.com/i-math/intro-to-modular-arithmetic-34ad9d4537d1 | Intro to Modular Arithmetic. Equivalence Classes and Circular… | by Brett Berry | Math Hacks | Medium
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Intro to Modular Arithmetic
Equivalence Classes and Circular Counting
Brett Berry
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4 min read
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May 24, 2017
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This article is useless, wanna know why? Because you already know how to do modular arithmeticeven if you’ve never heard of it before. In fact, I bet you use it all the time.
Let me show you.
The Old-Fashioned Way
First I want you to go back…way, way, waaaaaay back to when you very first learned long-hand division. Now I bet your primary school teacher didn’t walk in and say “hey kiddos, today I’m going to show you how long division is also modular arithmetic,” but it’s the truth.
Let’s start with a simple problem.
To solve this problem all you need to do is divide 11 by 4 the old-fashioned way.
old-school long division
Remember how your teacher used to have you write “R3” at the top for “remainder 3." Later on you learned more sophisticated ways of expressing that extra amount like decimals and fractions, but for modular arithmetic that little ole remainder guy is exactly what we want.
Yep, A mod B is the same as saying “how much is left over when you divide A by B.”
That’s all there is to it! Simple right?
Modulus Time
See that clock on the wall?
It’s modulus 12.
Why? Because the standard method for telling time is to split the day into two 12 hour segments. Instead of counting up to 24, we count to 12 twice.
In fact, circular countingis a fundamental representation of modular arithmetic.
Furthermore when you convert between military time and standard time, you’re performing modular arithmetic. For example, we know that 15:00 is the same as 3:00pm because when we divide 15 by 12, we’re left with 3 as a remainder.
So let me posit this: what time would it be right now in a universe that used modulus 8 in their time system?
As I’m writing this it’s 10pm. In a universe that uses modulus 8, the time would be 6 o’clock.
(solution: 10pm is the 22nd hour of the day. So we take 22 mod 8. Eight of course goes into 22 twice with a remainder of 6. That means22 mod 8 = 6.)
Equivalence Classes
Now your probably thinking that modular arithmetic is kinda useless because you keep getting the same answers over and over again.
You’re right! In fact that’s the beauty of modular arithmetic. It gives us a new way to relate numbers to one another.
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Check this out.
Let’s represent modulus 4with the following circle diagram.
Recall that when you divide by 4, you have 4 possible remainders: 0 (aka no remainder), 1, 2, and 3.
Let’s calculate 0, 1, 2 and 3 mod 4:
Place the numbers in their respective sections of the modulus 4 diagram.
And continue calculating:
Press enter or click to view image in full size
Add these numbers to the diagram:
Each of these sections represents an equivalence class.
For example we could say:
Press enter or click to view image in full size
read: 1 is congruent to 5 in modulus 4
Since 1 and 5 are in the same equivalence class (i.e. they both have a remainder of 1 when divided by 4) they’re congruent.
Furthermore, we can write formulas for each equivalence class of modulus 4:
Values in the “0” equivalence class are multiples of 4 → 4x
Values in the “1” equivalence class are multiples of 4 plus 1 → 4x + 1
Values in the “2” equivalence class are multiples of 4 plus 2 → 4x + 2
Values in the “3” equivalence class are multiples of 4 plus 3 → 4x + 3
where x = 0, 1, -1, 2, -2, and so forth
There you go! The basics of modular arithmetic, and you see it’s actually quite elementary. Till next time! Happy math-ing!!
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Piyush Raj
May 25, 2017
The example of splitting clock into 8 was refreshing.
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CEO CEO
Jul 11, 2019
Thanks Brett, you just made me understand Arithmetic modulo better. It’s really beyond the surface as the surface appears too easy, going further exposes one to the main Modulo which you’ve made me have a grasp.
Never a dull moment studying from…more
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Gil Sousa
Jun 4, 2017
Great explanation. I have been using circular counting when I can after discovering it for the first in a in a ceaser cypher algorithm:
(x+n) mod 26
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Derivative of Arcsin
Derivative of Arcsin x – Formula & Proof using Quotient Rule, Chain Rule, and First Principle
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The derivative of arcsin x is 1/√1-x². It is written as d/dx(arcsin x) = 1/√1-x². Arcsin function is the inverse of the sine function and is a pure trigonometric function. We will learn how to differentiate arcsin x by using various differentiation rules like the first principle of derivative, differentiate arcsin x using chain rule and differentiate arcsin x using the quotient rule. Arcsin of x is defined as the inverse sine function of x when -1 ≤ x ≤ 1.
What is Derivative of arcsin x?
Derivative of arcsin function is denoted by d/dx(arcsin x) and its value is 1/√1-x². It returns the angle whose sine is a given number.
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When the sine of y is equal to x:
sin y = x
Then the arcsine of x is equal to the inverse sine function of x, which is equal to y:
a r c s i n x=s i n−1 x=y a r c s i n x=s i n−1 x=y
Example: a r c s i n 1=s i n−1 1=π/2 r a d=90°a r c s i n 1=s i n−1 1=π/2 r a d=90°
Graph of Arcsine: Arcsin x can be represented in graphical form as follows:
Values of Arcsin
x arcsin(x) (rad)arcsin(x) (°)
-1-π/2-90°
−3√2−3 2-π/3-60°
−2√2−2 2-π/4-45°
−1 2−1 2-π/6-30°
0 0 0°
1 2 1 2 π/6 30°
2√2 2 2 π/4 45°
3√2 3 2 π/3 60°
1 π/2 90°
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Proof of Derivative of Arcsin x
We will learn how to differentiate arcsin x by using various differentiation rules:
Proof of Derivative of Arcsin by Quotient Rule
Proof of Derivative of Arcsin by first principle of derivative
Proof of Derivative of Arcsin by differentiating arcsin x using chain rule
Proof of Derivative of Arcsin by Quotient Rule
We can prove the derivative of arcsin by quotient rule using the following steps:
Step 1: Write sin y = x,
Step 2: Differentiate both sides of this equation with respect to x.
d d x s i n y=d d x x c o s y d d x y=1 d d x s i n y=d d x x c o s y d d x y=1
Step 3:Solve for d y d x d y d x
d d x y=1 c o s y d d x y=1 c o s y
Step 4: Define cos y in terms of x using a reference triangle.
From the reference triangle, the adjacent side is (1–x 2)−−−−−−√(1–x 2) and the hypotenuse is 1. Thus, c o s y=(1–x 2)√1 c o s y=(1–x 2)1 which means 1 c o s y=1(1–x 2)√1 c o s y=1(1–x 2)
Step 5: Substitute for cos y.
d d x y=1 c o s y=1(1–x 2)√d d x y=1 c o s y=1(1–x 2)
Step 6: Define arcsine.
Now we can define arcsine as:
y=s i n−1 x y=s i n−1 x
Step 7:Differentiate and write the result.
d d x s i n−1 x=d y d x=1(1–x 2)√d d x s i n−1 x=d y d x=1(1–x 2)
Proof of Derivative of Arcsin by Chain Rule
We can prove the derivative of arcsin by Chain rule using the following steps:
Let y=a r c s i n x Taking sin on both sides,s i n y=s i n(a r c s i n x)By the definition of an inverse function, we have,s i n(a r c s i n x)=x So the equation becomes s i n y=x Differentiating both sides with respect to x,d d x(s i n y)=d d x(x)c o s y d d x y=1 d d x y=1 c o s y Using one of the trigonometric identities s i n y+c o s y=1∴c o s y=1–s i n 2 y−−−−−−−√=1–x 2−−−−√d y d x=1(1–x 2)√Substituting y = arcsin x back d d x(a r c s i n x)=a r c s i n′x=1 1−x 2√Let y=a r c s i n x Taking sin on both sides,s i n y=s i n(a r c s i n x)By the definition of an inverse function, we have,s i n(a r c s i n x)=x So the equation becomes s i n y=x Differentiating both sides with respect to x,d d x(s i n y)=d d x(x)c o s y d d x y=1 d d x y=1 c o s y Using one of the trigonometric identities s i n y+c o s y=1∴c o s y=1–s i n 2 y=1–x 2 d y d x=1(1–x 2)Substituting y = arcsin x back d d x(a r c s i n x)=a r c s i n′x=1 1−x 2
Proof of Derivative of Arcsin by First Principle
We can prove the derivative of arcsin by First Principle using the following steps:
f′(x)=d y d x=lim h→0 f(x+h)–f(x)h f(x)=a r c s i n x f(x+h)=a r c s i n(x+h)f(x+h)–f(x)=t a n(x+h)–t a n(x)=a r c s i n(x+h)–a r c s i n x f(x+h)–f(x)h=a r c s i n(x+h)–a r c s i n x h lim h→0 f(x+h)–f(x)h=lim h→0 a r c s i n(x+h)–a r c s i n x h Assume that arcsin (x + h) = A and arcsin x = B s i n A=x+h s i n B=x s i n A–s i n B=(x+h)–x s i n A–s i n B=h I f h → 0, (sin A – sin B) → 0 sin A → sin B or A → B or A – B → 0 lim A−B→0 f(x+h)–f(x)h=lim A−B→0(A–B)(s i n A–s i n B)sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2]lim A−B→0 f(x+h)–f(x)h=lim A−B→0(A–B)[2 s i n[(A–B)/2]c o s[(A+B)/2]]A – B → 0, we can have (A – B)/2 → 0 lim A−B 2→0 f(x+h)–f(x)h=lim A−B 2→0 1(s i n[(A–B)/2][(A–B)/2])lim A−B→0 c o s[(A+B)/2]f′(x)=c o s[(B+B)/2]=c o s B s i n B=x c o s B=1–s i n 2 B−−−−−−−√=1–x ²−−−−√lim h→0 f(x+h)–f(x)h=1(1–x 2)√f′(x)=d y d x=1(1–x 2)√f′(x)=d y d x=lim h→0 f(x+h)–f(x)h f(x)=a r c s i n x f(x+h)=a r c s i n(x+h)f(x+h)–f(x)=t a n(x+h)–t a n(x)=a r c s i n(x+h)–a r c s i n x f(x+h)–f(x)h=a r c s i n(x+h)–a r c s i n x h lim h→0 f(x+h)–f(x)h=lim h→0 a r c s i n(x+h)–a r c s i n x h Assume that arcsin (x + h) = A and arcsin x = B s i n A=x+h s i n B=x s i n A–s i n B=(x+h)–x s i n A–s i n B=h I f h → 0, (sin A – sin B) → 0 sin A → sin B or A → B or A – B → 0 lim A−B→0 f(x+h)–f(x)h=lim A−B→0(A–B)(s i n A–s i n B)sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2]lim A−B→0 f(x+h)–f(x)h=lim A−B→0(A–B)[2 s i n[(A–B)/2]c o s[(A+B)/2]]A – B → 0, we can have (A – B)/2 → 0 lim A−B 2→0 f(x+h)–f(x)h=lim A−B 2→0 1(s i n[(A–B)/2][(A–B)/2])lim A−B→0 c o s[(A+B)/2]f′(x)=c o s[(B+B)/2]=c o s B s i n B=x c o s B=1–s i n 2 B=1–x ² lim h→0 f(x+h)–f(x)h=1(1–x 2)f′(x)=d y d x=1(1–x 2)
Properties of Arcsine
Rule name Rule
Sine of arcsine sin( arcsin x ) = x
Arcsine of sine arcsin( sin x ) = x+2 k π, when k∈ℤ (k is integer)
Arcsin of negative argument arcsin(-x) = – arcsin x
Complementary angles arcsin x = π/2 – arccos x = 90° – arccos x
Arcsin sum a r c s i n α+a r c s i n β=a r c s i n(α(1−β 2)−−−−−−−√+β(1−α 2)−−−−−−−√)a r c s i n α+a r c s i n β=a r c s i n(α(1−β 2)+β(1−α 2))
Arcsin difference a r c s i n α–a r c s i n β=a r c s i n(α(1−β 2)−−−−−−−√–β(1−α 2)−−−−−−−√)a r c s i n α–a r c s i n β=a r c s i n(α(1−β 2)–β(1−α 2))
Cosine of arcsine c o s(a r c s i n x)=s i n(a r c c o s x)=1−x 2−−−−−√c o s(a r c s i n x)=s i n(a r c c o s x)=1−x 2
Tangent of arcsine t a n(a r c s i n x)=x 1−x 2√t a n(a r c s i n x)=x 1−x 2
Derivative of arcsine d d x(a r c s i n x)=a r c s i n′x=1 1−x 2√d d x(a r c s i n x)=a r c s i n′x=1 1−x 2
Indefinite integral of arcsine∫(a r c s i n x)d x=x(a r c s i n x)+1−x 2−−−−−√+C∫(a r c s i n x)d x=x(a r c s i n x)+1−x 2+C
Corollary d arcsin x a d x=1 a 2–x 2−−−−−√d arcsinx a d x=1 a 2–x 2
Solved Examples of Derivative of Arcsin x
Example 1: What is the derivative of arcsin(x − 1)?
Solution: Derivative of inverse trigonometric functions. The general formula to differentiate the arcsin functions is
∫s i n−1 u=1(1–u 2)√d u d x d d x s i n−1(x−1)=1(1−(x−1)2)√×d(x−1)d x d d x s i n−1(x−1)=1(1−(x−1)2)√∫s i n−1 u=1(1–u 2)d u d x d d x s i n−1(x−1)=1(1−(x−1)2)×d(x−1)d x d d x s i n−1(x−1)=1(1−(x−1)2)
Example 2: What is the derivative of arcsin(x\a)?
Solution:
To start off, let’s set this function equal to y
y=s i n−1(x a)s i n y=(x a)Multiply a to both sides and taking the derivative d d x[a s i n y]=d d x d y d x a c o s y=1 d y d x=1 a c o s y Divide both sides to isolate d y d x d y d x=1 a s e c y s e c y=1 c o s y from the image below s e c y=a a 2−x 2√We will now substitute this value back into the answer for our derivative:d y d x=1 a s e c y d y d x=1 a a a 2−x 2√Canceling out the a in the numerator and denominator, we are left with our final answer:d y d x=1 a 2−x 2√y=s i n−1(x a)s i n y=(x a)Multiply a to both sides and taking the derivative d d x[a s i n y]=d d x d y d x a c o s y=1 d y d x=1 a c o s y Divide both sides to isolate d y d x d y d x=1 a s e c y s e c y=1 c o s y from the image below s e c y=a a 2−x 2 We will now substitute this value back into the answer for our derivative:d y d x=1 a s e c y d y d x=1 a a a 2−x 2 Canceling out the a in the numerator and denominator, we are left with our final answer:d y d x=1 a 2−x 2
If you are checking Derivative of Arcsin x article, also check the related maths articles:
Applications of DerivativesPartial Derivative
Basics of DerivativesDerivative Rules
Derivative of xDerivative of cot x
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FAQs For Derivative of Arcsin
What is Derivative of arcsin x?
Derivative of arcsin function is denoted by d/dx(arcsin x) and its value is 1/√1-x². It returns the angle whose sine is a given number.
What is Arcsin function?
Arcsin function is the inverse of the sine function and is a pure trigonometric function.
How do you find the derivative of a arcsin x function?
Let y=a r c s i n x Taking sin on both sides,s i n y=s i n(a r c s i n x)By the definition of inverse function, we have,s i n(a r c s i n x)=x So the equation becomes s i n y=x Differentiating both sides with respect to x,d d x(s i n y)=d d x(x)c o s y d d x y=1 d d x y=1 c o s y Using one of the trigonometric identities s i n y+c o s y=1∴c o s y=1−s i n 2 y−−−−−−−−√=1−x 2−−−−−√d y d x=1(1−x 2)√Substituting y = arcsin x back d d x(a r c s i n x)=a r c s i n′x=1 1−x 2√Let y=a r c s i n x Taking sin on both sides,s i n y=s i n(a r c s i n x)By the definition of inverse function, we have,s i n(a r c s i n x)=x So the equation becomes s i n y=x Differentiating both sides with respect to x,d d x(s i n y)=d d x(x)c o s y d d x y=1 d d x y=1 c o s y Using one of the trigonometric identities s i n y+c o s y=1∴c o s y=1−s i n 2 y=1−x 2 d y d x=1(1−x 2)Substituting y = arcsin x back d d x(a r c s i n x)=a r c s i n′x=1 1−x 2
What is the derivative of arcsin of square root x?
The derivative of arcsin x is d d x(a r c s i n x)=a r c s i n′x=1 1−x 2√d d x(a r c s i n x)=a r c s i n′x=1 1−x 2. By using this formula and chain rule, we can find the derivative of a r c s i n x−−√a r c s i n x.d d x a r c s i n x−−√=1 1−x√2√×d d x x−−√d d x a r c s i n x−−√=1 1−x√2√×d d x x−−√d d x a r c s i n x−−√=1 1−x√2√×d d x x(1/2)d d x a r c s i n x−−√=1 1−x√2√×(1/2)x(−1/2)d d x a r c s i n x−−√=1 1−x√2√×1 2 x√d d x a r c s i n x=1 1−x 2×d d x x d d x a r c s i n x=1 1−x 2×d d x x d d x a r c s i n x=1 1−x 2×d d x x(1/2)d d x a r c s i n x=1 1−x 2×(1/2)x(−1/2)d d x a r c s i n x=1 1−x 2×1 2 x
What are the properties of arcsin x?
Sine of arcsine: sin( arcsin x ) = xArcsine of sine: arcsin( sin x ) = x+2kπ, when k∈ℤ (k is integer)Arcsin of negative argument: arcsin(-x) = - arcsin xComplementary angles: arcsin x = π/2 - arccos x = 90° - arccos xArcsin sum: a r c s i n α+a r c s i n β=a r c s i n(α(1−β 2)−−−−−−−√+β(1−α 2)−−−−−−−√)a r c s i n α+a r c s i n β=a r c s i n(α(1−β 2)+β(1−α 2))Arcsin difference: a r c s i n α−a r c s i n β=a r c s i n(α(1−β 2)−−−−−−−√−β(1−α 2)−−−−−−−√)a r c s i n α−a r c s i n β=a r c s i n(α(1−β 2)−β(1−α 2))Cosine of arcsine: c o s(a r c s i n x)=s i n(a r c c o s x)=1−x 2−−−−−√c o s(a r c s i n x)=s i n(a r c c o s x)=1−x 2 Tangent of arcsine: t a n(a r c s i n x)=x 1−x 2√t a n(a r c s i n x)=x 1−x 2 Derivative of arcsine: d d x(a r c s i n x)=a r c s i n′x=1 1−x 2√d d x(a r c s i n x)=a r c s i n′x=1 1−x 2 Indefinite integral of arcsine: ∫(a r c s i n x)d x=x(a r c s i n x)+1−x 2−−−−−√+C∫(a r c s i n x)d x=x(a r c s i n x)+1−x 2+C Corollary: d arcsin x a d x=1 a 2−x 2−−−−−−√d arcsinx a d x=1 a 2−x 2
Why do we use the inverse trigonometric function arcsin(x)?
The arcsin(x) function helps us find the angle whose sine is x. It’s used in calculus, geometry, and physics where we need to reverse the sine function.
What does the graph of arcsin(x) tell us about its derivative?
The graph of arcsin(x) increases slowly as x approaches ±1, where the slope becomes steeper. This matches the derivative, which approaches infinity near x = ±1 due to the square root in the denominator.
Is the derivative of arcsin(x) continuous over its domain?
Yes, it is continuous and differentiable in the open interval (−1,1) but it becomes undefined at x = ±1.
In which real-world applications is the derivative of arcsin(x) used?
It appears in problems involving wave motion, pendulum physics, angle measurements in navigation, and inverse kinematics in robotics.
Can the Chain Rule be applied to arcsin(f(x))?
Yes. If y = arcsin(f(x)), then using the Chain Rule: dy/dx = f ′(x) / √(1 - [f(x)]²)
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Sinusoidal Steady-State Power Calculations
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Technical Articles
Sinusoidal Steady-State Power Calculations
Technical Article
Sinusoidal Steady-State Power Calculations
November 25, 2015 by Donald Krambeck
Let's delve into AC power concepts, how to calculate instantaneous power, average power, reactive power, complex power, and the power factor. We'll also talk about the relationship each concept has to one another.
Instantaneous Power
We begin our exploration of sinusoidal power calculations with the genaric circuit in Fig. 1.1. In here, vand i are steady-state sinusoidal signals. By using the passive sign convention (PSC), the power at any instant of time is given by:
$$p=vi$$ (1.1)
Figure 1.1 Representation of a circuit used for calculating power.
Equation 1.1 describes instantaneous power. Recall that if the reference direction of the current is in the direction of the voltage rise, Eq. 1.1 must be written with a minus sign. Instantaneous power is always measured in watts when the voltage is measured in volts and the current is measured in amperes. Two expressions of phase angles of vand iare written as
$$v=V_{m}\cos (\omega t + \theta _{v}),$$ (1.2)
$$i=I_{m}\cos (\omega t + \theta _{i}),$$ (1.3)
In these two expressions, $$\theta _{v}$$ is the voltage phase angle, and $$\theta _{i}$$ is the current phase angle.
While working in the sinusoidal steady state, a convenient reference for zero time may be chosen. Engineers who design systems that transfer large amounts of power have found it convenient to use a zero time that corresponds to the instant the current is passing through a positive maximum. By choosing such a reference time, a shift of both voltage and current by $$\theta _{i}$$ is required. Now, Eqs. 1.2 and 1.3 become
$$v=V_{m}\cos (\omega t + \theta _{v} - \theta _{i})$$ (1.4)
$$i= I_{m}\cos (\omega t)$$ (1.5)
If Eqs. 1.4 and 1.5 are substituted into Eq. 1.1, the expression for the instantaneous power now becomes
$$p= V_{m}I_{m}\cos (\omega t + \theta _{v} - \theta _{i})\cos (\omega t)$$ (1.6)
Eq. 1.6 can be used to solve for average power the way it is; however, by applying a few simple trigonometric identities the instantaneous power equation can be simplified. Using cosine's product identity gives
$$\cos (\alpha )\cos (\beta)=\frac{1}{2}\cos (\alpha -\beta )+\frac{1}{2}\cos (\alpha +\beta )$$
Letting $$\alpha =\omega t+\theta _{v}-\theta _{i}$$ and $$\beta =\omega t$$ provides
$$p=\frac{V_{m}I_{m}}{2}\cos (\theta _{v}-\theta _{i})+\frac{V_{m}I_{m}}{2}\cos (2\omega t+\theta _{v}-\theta _{i})$$ (1.7)
Lastly, using the cosine angle-sum identity
$$\cos (\alpha +\beta )=\cos (\alpha )\cos (\beta )-\sin (\alpha )\sin (\beta )$$
to expand the second term on the right-hand side of Eq 1.7, which gives
$$p=\frac{V_{mI_{m}}}{2}\cos (\theta _{v}-\theta _{i})+\frac{V_{m}I_{m}}{2}\cos (\theta _{v}-\theta _{i})\cos (2\omega t) -\frac{V_{m}I_{m}}{2}\sin (\theta _{v}-\theta _{i})\sin (2\omega t)$$ (1.8)
Relationship Between Current, Power, and Voltage
Figure 1.2 below depicts the relationship between i, v, and p, assuming that $$\theta _{v}=60^{\circ}$$ and $$\theta _{i}=0^{\circ}$$. The frequency of the instantaneous power is twice the frequency of the current or voltage. This depiction also follows from the second two terms on the right side of Eq. 1.8. This means that the instantaneous power goes through two complete cycles for every cycle of either the current or the voltage. If you look at Fig. 1.2, the instantaneous power can be negative for a portion of each cycle, even if the network between the terminals is passive. In a passive network, this negative power implies that the energy being stored in the inductors or capacitors is now being extracted. While the instantaneous power varies with time in the sinusoidal steady-state of a circuit, this causes some vibration in some motor-driven appliances. Due to this vibration in these appliances, resilient motor mountings are required to reduce any excessive vibration.
Figure 1.2 Instantaneous power, current, and voltage vs. angular frequency
Average and Reactive Power
Eq. 1.8 can now be utilized to find the average power at the terminals of the circuit, as well as establish the concept of reactive power. Noting that the equation has three terms, it can be rewritten as
$$p=P+P\cos (2\omega t)-Q\sin (2\omega t),$$ (1.9)
Where
Average (real) power$$P=\frac{V_{m}I_{m}}{2}\cos (\theta _{v}-\theta _{i})$$ (1.10)
Reactive power$$Q=\frac{V_{m}I_{m}}{2}\sin (\theta _{v}-\theta _{i})$$ (1.11)
P is called the average power, and Q is called the reactive power. Average power is also known as real power, because it is the actual power in a circuit that is transformed from electric to nonelectric energy. The average power associated with sinusoidal signals is the average of the instantaneous power over one period, or
$$P=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}pdt,$$ (1.12)
Where T is the period of the sinusoidal varying function. The bounds of the integral indicate that integration can be made at any convenient time $$t_{0}$$ and integration must be bounded exactly one period later. To grasp a better understanding of all the terms in Eq. 1.9 and the relationships they hold, we will need to examine the power in circuits that are purely resistive, purely inductive, and purely capacitive.
Purely Resistive Circuits
If a circuit between terminals is purely resistive, the current and voltage are in phase $$(\theta _{v}=\theta _{i})$$. Thus, Eq. 1.9 can be reduced to
$$p=P+P\cos (2\omega t)$$ (1.13)
This is referred to as the instantaneous real power. Figure 1.3 is a graph of the instantaneous real power for a purely resistive circuit, assuming $$\omega = 377 \mathrm{ rad/s}$$. The average power, P, is the average of p, over one period. This can be seen by looking at the graph where P=1 for the circuit. From the Fig 1.3, instantaneous real power can never be negative; in other words, power cannot be removed from a purely resistive network. While the power cannot be removed, it is however, dissipated in the form of thermal energy.
Figure 1.3 Instantaneous real power and average power of a purely resistive circuit
Purely Inductive Circuits
Now, if the circuit between the terminals is purely inductive, the current and voltage are out of phase by $$90^{\circ}.$$ The current of the circuit lags the voltage by $$90^{\circ}$$ $$(\theta _{i}=\theta _{v}-90^{\circ}).$$ The instantaneous power equation can be reduced to
$$p=-Q\sin (2\omega t)$$ (1.14)
In this purely inductive circuit, the average power is zero. This means that no transformation of energy from electric to nonelectric energy takes place. The power at the terminals is continually exchanged between the circuit and the power source driving the circuit at a frequency of $$2\omega .$$ What this means, is that when p is positive, energy is stored in the magnetic fields associated with the inductive elements, and when p is negative, energy is being removed from the magnetic fields.
Power associated with purely inductive circuits is known as the reactive power Q. Reactive power comes from the characterization of an inductor as a reactive element. To differentiate between average power and reactive power, units watt (W) for average power and var (volt-amp reactive, or VAR) for reactive power are used. Figure 1.4 depicts the instantaneous power for a purely inductive circuit, assuming $$\omega =377 \mathrm{ rads/s}$$ and Q= 1 VAR.
Figure 1.4 Instantaneous real power, average power, and reactive power for a purely inductive circuit
Purely Capacitive Circuits
In this purely capacitive circuit, the current and voltage are $$90^{\circ}$$ out of phase with each other. In this case, the current leads the voltage by exactly $$90^{\circ}$$ $$(\theta _{i}=\theta _{v}+90^{\circ})$$. The expression of this instantaneous power is given by
$$p=-Q\sin (2\omega t)$$ (1.15)
In this circuit, there is no transformation of energy from electric to nonelectric energy because the average power is zero. In a purely capacitive circuit, the power is continually transferred between the source delivering power and to the electric field associated with the capacitive elements. Figure 1.5 depicts the instantaneous power for a purely capacitive circuit, assuming $$\omega =377 \mathrm{ rads/s}$$ and Q = -1 VAR.
Figure 1.5 Instantaneous real power and average power for a purely inductive circuit
Understanding the Power Factor
This angle $$(\theta_{v}-\theta _{i})$$ has a significant role in computing both the average and the reactive power and is known as the power factor angle. Taking the cosine of this angle gives what is known as the power factor, shortened to pf, and taking the sine of this angle is known as the reactive factor, shortened to rf. This can be denoted as:
$$\mathrm{pf}=\cos (\theta _{v}-\theta _{i})$$ (1.16)
$$\mathrm{rf}=\sin (\theta _{v}-\theta _{i})$$ (1.17)
To completely describe the power factor angle, either lagging power factoror leading power factor terms are used. If the power factor lags, the current lags voltage (i.e. an inductive load is present). On the other hand, if the power factor leads, the current leads voltage (i.e. a capacitive load is present).
Calculating AC Power Concepts
A load comprising a 480 $$\Omega $$ resistor in parallel with a $$\frac{5}{9}\mu F$$ capacitor is connected across the terminals of a sinusoidal varying voltage source $$v_{g}$$, where $$v_{g}=240\cos (5000t) \mathrm{ V}$$
A) What is the peak value of the instantaneous power delivered by the power source?
$$P=(\frac{V_{m}}{\sqrt{2}})^{2}(\frac{1}{R})$$
$$=\frac{(240)^{2}}{2\cdot 480}$$
$$=60 \mathrm{ W}$$
Calculating the capacitive reactance:
$$X_{C}=\frac{1}{\omega c}$$
$$=\frac{-9000}{25}$$
$$=-360 \Omega $$
Calculating reactive power gained by the source:
$$Q=(\frac{V_{m}}{\sqrt{2}})^{2}(\frac{1}{X_{C}})$$
$$=\frac{(240)^{2}}{2\cdot (-360)}$$
$$=-80 \mathrm{ VAR}$$
Calculating the peak value of instantaneous power delivered:
$$p_{max}=P+\sqrt{P^{2}+Q^{2}}$$
$$=60 + \sqrt{(60)^{2}+(80)^{2}}$$
$$p_{max}=160 \mathrm{ W}$$
B) What is the peak value of the instantaneous power absorbed by the source?
$$p_{min}=P-\sqrt{P^{2}+Q^{2}}$$
$$=60-\sqrt{(60)^{2}+(80)^{2}}$$
$$p_{min}=-40 \mathrm{ W}$$
C) What is the average power delivered to the load?
Using the power equation from part A, $$= \frac{(240)^{2}}{2\cdot 480}$$
The average power is $$P= 60 \mathrm{ V}$$
D) What is the reactive power delivered to the load?
Using the reactive power equation in part A, $$=\frac{(240)^{2}}{2\cdot (-360)}$$
$$=\frac{(240)^{2}}{2\cdot (-360)}$$
$$Q=-80 \mathrm{ VAR}$$
E) Is the load absorbing or generating magnetizing vars?
Using the reactive power equation from part A, $$=\frac{(240)^{2}}{2\cdot (-360)}$$
$$Q=-80 \mathrm{ VAR}$$ A negative value means that the load is generating magnetizing vars.
F) What is the power factor?
Using the power factor equation, $$=\frac{V}{R}+\frac{V}{X_{C}}$$
$$I=\frac{240}{480}+\frac{240}{-j360}$$
$$=0.836\angle 53.267^{\circ}\mathrm{ A}$$
Therefore, the power factor is $$pf=0.6$$ leading
G) What is the reactive factor?
Using the reactive factor equation, $$\sin (0^{\circ}-53.267^{\circ})$$
Therefore, the reactive factor is $$rf=-0.8$$
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Comments
3 Comments
J
Jusgast December 01, 2015
Good elaboration, refreshes my school time!Found 1 typo: “The average power is P=60V”
Like. Reply
X
Xptpcrewx February 11, 2017
This is an interesting read. I have some questions however; Can you provide references and reasoning for shifting the current angle to time zero for the instantaneous power derivation, and also provide references for uses of the “rf” reactive factor terminology? Thanks.
Like. Reply
M
merlynski February 15, 2017
CAPTION ERROR: Figure 1.5 Instantaneous real power and average power for a purely ‘INDUCTIVE’ circuit. Should be ‘CAPACITIVE’
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12012 | https://brainly.com/question/33247492 | [FREE] The Sun has a radius of 7. \times 10^5 kilometers. Calculate the surface area of the Sun in square - brainly.com
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The Sun has a radius of 7.×1 0 5 kilometers. Calculate the surface area of the Sun in square meters.
Note that you can approximate the Sun to be a sphere with a surface area of:
A=4 π R 2
where R is the radius (the distance from the center to the edge) of the Sun. In this class, approximating π≈3 is perfectly fine, so we can approximate the formula for surface area to be:
A≈12 R 2
Hint:
1 km 2=1(km)2=1×(1 0 3)2 m 2=1×1 0 6 m 2
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The surface area of the Sun is approximately 6.07 x 10¹² square meters.
To calculate the surface area of the Sun, we can use the formula A = 4πR², where R is the radius of the Sun. Given that the radius of the Sun is 7.105 kilometers, we need to convert it to meters before substituting it into the formula.
1 kilometer (km) is equal to 1000 meters (m). Therefore, the radius of the Sun in meters (Ro) is:
R₀ = 7.105 km∗1000 m/km
R₀ = 7,105 meters
Now, we can substitute the value of R₀ into the formula:
A = 4π(7,105)²
A = 4π(50,441,025)
A ≈ 201,764,100π
Since we can approximate π to 3, the surface area can be further simplified:
A ≈ 201,764,100 3
A ≈ 605,292,300 square meters
The surface area of the Sun is approximately 6.07 x 10¹² square meters.
Learn more about surface area
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The surface area of the Sun can be calculated using the formula A=4 π R 2. After converting the radius from kilometers to meters and approximating π as 3, the surface area is found to be approximately 5.88×1 0 18 square meters.
Explanation
To calculate the surface area of the Sun, we can use the formula for the surface area of a sphere given by:
A=4 π R 2
where R is the radius of the sphere. For the Sun, the radius R is approximately 7.0×1 0 5 kilometers.
First, we need to convert this radius from kilometers to meters:
R=7.0×1 0 5 km×1000 m/km=7.0×1 0 8 m
Now we can substitute this value into the surface area formula. As per the approximation mentioned, we can simplify the use of π by using π≈3, which gives us:
A≈12 R 2
Substituting for R:
A≈12×(7.0×1 0 8)2
Calculating R 2:
R 2=(7.0×1 0 8)2=49.0×1 0 16 m 2
Now plug this back into the equation for area:
A≈12×49.0×1 0 16=588.0×1 0 16 m 2
Finally, we can write this value in scientific notation:
A≈5.88×1 0 18 m 2
Therefore, the surface area of the Sun is approximately 5.88×1 0 18 square meters.
Examples & Evidence
For example, by using the formula to calculate the surface area of other spheres, you can see how the radius affects the total area significantly. For instance, a sphere with half the radius will have only one fourth of the area, illustrating the R 2 relationship.
The formula for the surface area of a sphere and the conversion from kilometers to meters are widely accepted mathematical principles. The calculations follow standard arithmetic rules, ensuring the results are accurate.
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The surface area of the Sun is approximately 5.88 × 10¹⁸ square meters.
To find the surface area of the Sun in square meters, we will start with the given radius of the Sun and use the formula for the surface area of a sphere. Here's the step-by-step calculation:
Given Data:
Radius of the Sun (R) : 7 × 10⁵ kilometers
Surface area formula for a sphere: A = 4πR²
Approximation:
In this problem, we are allowed to approximate π (pi) as 3.
Therefore, the formula becomes: A ≈ 12R²
Converting kilometers to meters:
1 kilometer = 1000 meters
Radius in meters: 7 × 10⁵ km × 1000 m/km = 7 × 10^8 meters
Calculating the Surface Area:
Substitute the radius value into the approximated formula: A ≈ 12R²
A ≈ 12 × (7 × 10⁸ m)²
A ≈ 12 × (49 × 10¹⁶ m²)
A ≈ 588 × 10¹⁶ m²
A ≈ 5.88 × 10¹⁸ m²
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Community Answer Calculate the average power radiated by each square meter of the Sun's surface. (Hint: The formula for the surface area of a sphere is A = 4tr.) O 2 x 1045 W/m^2 O 5 x 1016 W/m^2 O 6 x 10? W/m^2O 2 x 108 W/m^2
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Classical Mechanics
Overview
Mechanics is the study of movement. Kinematics, mechanical forces, work, power, energy, and matter are all part of mechanics.
Kinematics is the study of (relative) motion - displacement, velocity, acceleration etc. The two relations at the heart of kinematics are: and where is displacement at time , is velocity, is acceleration, and is time. Uniform rectilinear motion, projectile motion, uniform circular motion, and simple harmonic motion are some of the types of problems studied in kinematics.
The rules of physics are almost fully summarized by the three famous laws of motion formulated by Isaac Newton:
A body continues to be in its state of uniform rectilinear motion until it is disturbed by an external force. This property is known as inertia.
The rate of change of momentum of a body with respect to time is directly proportional to the force acting on it.
Every action has an equal and opposite reaction.
Mass is one of the two most basic intrinsic properties of a body. It is a measure of its inertia. Momentum is defined as the product of the mass and velocity of a body. Force is something that changes or tends to change the momentum of a body, or, informally, "a push or pull".
Mechanical work is defined by the relation where is work done, is force, is displacement, and subscripts and denote the initial and final states respectively. Similarly, mechanical power is defined as where is power delivered and is velocity. Energy is the other basic intrinsic property of a body. Mechanical energy is simply the capacity of a body to do mechanical work.
Among the various properties of matter are elasticity, surface tension, and viscosity. The most important one is gravity. Gravity is indeed considered one of the most mysterious things not only in physics but in science as a whole.
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12014 | https://thecontentauthority.com/blog/detente-vs-rapprochement | Skip to Content
Detente vs Rapprochement: Which One Is The Correct One?
When it comes to international relations, the terms detente and rapprochement are often used interchangeably. However, there are subtle differences between the two concepts that are worth exploring further.
Detente refers to a period of reduced tension between two opposing sides. It is often used in reference to the Cold War era, when the United States and Soviet Union were engaged in a prolonged arms race that threatened to escalate into a full-blown nuclear war. During detente, both sides agreed to limit their military buildup and engage in diplomatic talks to resolve their differences.
Rapprochement, on the other hand, refers to the process of two estranged parties coming together to reconcile their differences. This can involve a range of activities, from informal talks to formal negotiations, with the ultimate goal of establishing a more positive and productive relationship.
In this article, we will explore the differences between detente and rapprochement, and examine the various factors that can influence the success of each approach in different contexts.
Define Detente
Detente is a term used to describe a period of relaxation or easing of tensions between two opposing nations or groups. It is a French word that means “relaxation” or “loosening”. The concept of detente gained popularity during the Cold War when the United States and the Soviet Union sought to reduce the risk of nuclear war by establishing a more cooperative relationship.
Detente is characterized by a reduction in military activity, diplomatic efforts to resolve conflicts, and an increase in cultural and economic exchanges. It is a strategy that seeks to avoid direct confrontation and promote peaceful coexistence.
Define Rapprochement
Rapprochement is a term used to describe a process of reconciliation or the reestablishment of friendly relations between two nations or groups that have been estranged or at odds with each other. It is a French word that means “reconciliation” or “coming together”.
Rapprochement is often used to describe a shift in foreign policy that seeks to improve relations between two nations or groups through diplomatic efforts, negotiations, and other forms of engagement. It is a strategy that seeks to promote mutual understanding and cooperation while reducing tensions and conflicts.
Unlike detente, which focuses on reducing tensions and avoiding direct confrontation, rapprochement seeks to address the underlying causes of conflict and promote long-term stability and cooperation.
How To Properly Use The Words In A Sentence
When it comes to international relations, the terms detente and rapprochement are often used interchangeably. However, they have distinct meanings and should be used appropriately to convey the intended message. Here’s how to use detente and rapprochement in a sentence:
How To Use Detente In A Sentence
Detente refers to a period of reduced tension or hostility between two countries. It is often used in the context of Cold War politics, where the United States and the Soviet Union engaged in a series of detente efforts to ease tensions and avoid nuclear war. Here are some examples of how to use detente in a sentence:
After years of hostility, the two countries finally reached a detente and agreed to reduce military spending.
The summit was seen as a major step toward detente between the two nations.
The policy of detente was designed to prevent a nuclear war.
As you can see, detente is typically used to describe a period of reduced tension or hostility between two countries. It can also refer to a specific policy or strategy aimed at achieving this goal.
How To Use Rapprochement In A Sentence
Rapprochement, on the other hand, refers to the reestablishment of friendly relations between two countries or groups after a period of estrangement or hostility. It is often used to describe a process of reconciliation or normalization of relations between two former enemies. Here are some examples of how to use rapprochement in a sentence:
The two countries announced a rapprochement after years of bitter conflict.
The visit was seen as a gesture of rapprochement between the two nations.
The treaty marked a significant step toward rapprochement between the two former enemies.
As you can see, rapprochement is typically used to describe the process of reestablishing friendly relations between two countries or groups after a period of estrangement or hostility. It can also refer to a specific event or action that marks the beginning of this process.
More Examples Of Detente & Rapprochement Used In Sentences
As previously discussed, detente and rapprochement are two terms that are often used in international relations. In this section, we will provide more examples of how these terms are used in sentences.
Examples Of Using Detente In A Sentence
The two countries reached a detente after years of political tension.
The detente between the United States and Cuba was a historic moment in diplomatic relations.
The detente between the two rival factions was a welcome relief to the citizens who had been caught in the crossfire.
The detente between the two superpowers helped to ease global tensions during the Cold War.
The detente between the two neighboring countries paved the way for increased trade and economic cooperation.
The detente between the two warring factions was fragile, and any misstep could lead to renewed hostilities.
The detente between the two leaders was short-lived, and soon they were back to their old ways of bickering and name-calling.
The detente between the two nations was seen as a positive step towards lasting peace in the region.
The detente between the two political parties was a sign that compromise and cooperation were still possible in the face of ideological differences.
The detente between the two rival companies was a strategic move to consolidate their market share and fend off competition from new entrants.
Examples Of Using Rapprochement In A Sentence
The rapprochement between the two former enemies was a remarkable achievement in the history of diplomacy.
The rapprochement between the two estranged siblings was a heartwarming reunion after years of bitter feuding.
The rapprochement between the two factions was a significant step towards ending the civil war.
The rapprochement between the two former lovers was a nostalgic trip down memory lane.
The rapprochement between the two neighboring countries was a welcome sign of progress towards regional integration and cooperation.
The rapprochement between the two political rivals was a surprising turn of events that caught many by surprise.
The rapprochement between the two superpowers was a major breakthrough in arms control negotiations.
The rapprochement between the two trading partners was a boost to their respective economies and a win-win for both sides.
The rapprochement between the two warring factions was a glimmer of hope in an otherwise bleak and violent conflict.
The rapprochement between the two world leaders was a symbolic gesture of goodwill and a commitment to peaceful coexistence.
Common Mistakes To Avoid
When it comes to international diplomacy, the terms detente and rapprochement are often used interchangeably. However, this is a common mistake that should be avoided in order to accurately convey the intended meaning.
Highlighting Common Mistakes
The first mistake people make is assuming that detente and rapprochement are synonymous. While both terms refer to the easing of tensions between two countries, there are important differences between them. Detente refers to a period of relaxed tensions between two countries, while rapprochement refers to a process of reconciliation and the establishment of normal diplomatic relations.
The second mistake is using the terms in the wrong context. For example, some people may use detente to refer to a process of reconciliation between two countries, when in fact it is only a temporary period of reduced tensions. Similarly, using rapprochement to describe a short-term easing of tensions between two countries is also incorrect.
Tips For Avoiding Common Mistakes
To avoid making these common mistakes, it is important to understand the nuances of the terms detente and rapprochement. Here are some tips to help:
Read up on the history of detente and rapprochement to gain a better understanding of their meanings and contexts.
Use detente to describe a temporary period of reduced tensions between two countries, and rapprochement to describe a process of reconciliation and the establishment of normal diplomatic relations.
Be mindful of the context in which you are using these terms, and ensure that you are using them accurately and appropriately.
By avoiding these common mistakes, you can communicate more effectively and accurately when discussing international diplomacy and foreign relations.
Context Matters
When it comes to international relations, the choice between detente and rapprochement can depend heavily on the context in which they are used. While both terms refer to a lessening of tensions between countries, they differ in their approach and goals. Understanding the context of the situation at hand is crucial in determining which approach is most appropriate.
Examples Of Different Contexts
One example of a context in which detente might be the preferred approach is during a time of heightened tensions between two countries. In this situation, detente would involve a deliberate effort to reduce tensions and avoid conflict through diplomatic means. For instance, during the Cold War, the United States and the Soviet Union engaged in detente to prevent the escalation of nuclear war.
On the other hand, rapprochement might be more appropriate in a context where two countries have been estranged for a long period of time and are seeking to establish a new relationship. In this case, rapprochement would involve a gradual process of building trust and cooperation. For example, the United States and Cuba engaged in rapprochement in 2014 after more than 50 years of hostility.
Another context in which the choice between detente and rapprochement might change is when dealing with a country that has a history of aggression or human rights violations. In this situation, detente might be seen as appeasement and not an acceptable option. Rapprochement, however, could be seen as a way to encourage positive change and promote human rights. This was the case with South Africa during apartheid, where rapprochement was used to promote democracy and end racial segregation.
Ultimately, the choice between detente and rapprochement depends on the specific context of the situation. Both approaches have their advantages and disadvantages, and it is up to policymakers to determine which approach is most appropriate for a given situation. By carefully considering the context, policymakers can make informed decisions that promote peace, stability, and cooperation between nations.
Exceptions To The Rules
While detente and rapprochement are often used in international relations, there are certain exceptions where their rules might not apply. Here, we will identify some of those exceptions and offer explanations and examples for each case.
Exception 1: Ideological Differences
One exception to the rules of using detente and rapprochement is when there are ideological differences between two countries. In such cases, the two countries might not be able to find common ground and may be unwilling to compromise. For example, the United States and North Korea have very different political ideologies, which has made it difficult for them to establish detente or rapprochement.
Exception 2: Strategic Interests
Another exception to the rules of using detente and rapprochement is when a country’s strategic interests are at stake. In such cases, a country may prioritize its own interests over the need for detente or rapprochement. For example, India and Pakistan have had a long-standing conflict over the disputed territory of Kashmir. Despite several attempts at detente and rapprochement, the conflict remains unresolved due to the strategic interests of both countries.
Exception 3: Domestic Politics
Domestic politics can also be an exception to the rules of using detente and rapprochement. In some cases, leaders may be reluctant to establish detente or rapprochement with another country due to domestic political pressure. For example, in the United States, some politicians have been critical of detente with Russia due to concerns about Russian interference in the 2016 presidential election.
Exception 4: Historical Animosity
Finally, historical animosity can be an exception to the rules of using detente and rapprochement. In some cases, countries may have a long history of conflict or animosity, which can make it difficult to establish detente or rapprochement. For example, Japan and South Korea have had a difficult relationship due to Japan’s actions during World War II, which has made it difficult for the two countries to establish detente or rapprochement.
Practice Exercises
Now that you have learned about the differences between detente and rapprochement, it’s time to put your knowledge into practice. Below are some exercises to help you improve your understanding and use of these terms in sentences.
Exercise 1: Fill In The Blank
Choose the correct word (detente or rapprochement) to fill in the blank in the following sentences:
The two countries have been in a state of __________ for years, but recent talks have led to a possible breakthrough in their relationship.
After years of tension, the leaders of the two countries agreed to a policy of __________, which included a reduction in military spending.
The __________ between the two countries was short-lived, as a disagreement over trade policies led to renewed tensions.
Answer Key:
rapprochement
detente
detente
Exercise 2: Sentence Writing
Write a sentence using each of the following words:
Detente
Rapprochement
Cold War
Foreign Policy
Answer Key:
The two countries reached a detente after years of conflict.
The rapprochement between the two leaders was a sign of progress.
The Cold War was a period of intense rivalry between the United States and the Soviet Union.
The president’s foreign policy was focused on promoting peace and stability in the region.
By practicing these exercises, you can improve your understanding and use of detente and rapprochement in your writing and conversations.
Conclusion
After exploring the differences between detente and rapprochement, it is clear that these two terms have distinct meanings in the context of international relations. Detente refers to a temporary easing of tensions between two opposing sides, while rapprochement is a more permanent reconciliation that involves a shift in attitudes and policies.
It is important to use these terms correctly in order to accurately convey the nature of diplomatic relationships. Confusing detente with rapprochement could lead to misunderstandings and misinterpretations of political events.
Key Takeaways
Detente refers to a temporary easing of tensions between two opposing sides.
Rapprochement is a more permanent reconciliation that involves a shift in attitudes and policies.
Using these terms correctly is important for accurately conveying the nature of diplomatic relationships.
Overall, understanding the nuances of grammar and language use is essential for effective communication. By continuing to learn and improve our language skills, we can better express our thoughts and ideas to others.
Shawn Manaher
Shawn Manaher is the founder and CEO of The Content Authority. He’s one part content manager, one part writing ninja organizer, and two parts leader of top content creators. You don’t even want to know what he calls pancakes. |
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Units of Measure
Quantifying Nature
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Tue, 29 Aug 2023 08:47:12 GMT
Metric Prefixes - from yotta to yocto
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Metric Prefixes - from yotta to yocto
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Table of contents
1. Binary Prefixes
2. Contributors and Attributions
In introductory chemistry we use only a few of the most common metric prefixes, such as milli, centi, and kilo. Our various textbooks and lab manuals contain longer lists of prefixes, but few if any contain a complete list. There is no point of memorizing this, but it is nice to have a place to look them up. You will find prefixes from throughout the range as you read the scientific literature. In particular, the smaller prefixes such as nano, pico, femto, etc., are becoming increasingly common as analytical chemistry and biotechnology develop more sensitive methods. To help you visualize the effect of these prefixes, there is a column "a sense of scale", which gives some examples of the magnitudes represented.
| prefix | abbreviation (upper and lower case are important) | meaning | example | a sense of scale (for some) Most are approximate. |
--- ---
| yotta | Y | 10 24 | yottagram, 1 Yg = 10 24 g | mass of water in Pacific Ocean ~ 1 Yg energy given off by the sun in 1 second ~ 400 YJ volume of earth ~ 1 YL mass of earth ~ 6000 Yg |
| zetta | Z | 10 21 | zettameter, 1 Zm = 10 21 m | radius of Milky Way galaxy ~ 1 Zm volume of Pacific Ocean ~ 1 ZL world energy production per year, ~ 0.4 ZJ |
| exa | E | 10 18 | exasecond, 1 Es = 10 18 s | age of universe ~ 0.4 Es (12 billion yr) |
| peta | P | 10 15 | petameter, 1 Pm = 10 15 m | 1 light-year (distance light travels in one year) ~ 9.5 Pm The dinosaurs vanished ~ 2 Ps ago. |
| tera | T | 10 12 | terameter, 1 Tm = 10 12 m | distance from sun to Jupiter ~ 0.8 Tm |
| giga | G | 10 9 | gigasecond, 1 Gs = 10 9 s | human life expectancy ~ 1 century ~ 3 Gs 1 light-second (distance light travels in one second) ~ 0.3 Gm |
| mega | M | 10 6 | megasecond, 1 Ms = 10 6 s | 1 Ms ~ 11.6 days |
| kilo | k | 10 3 | kilogram, 1 kg = 10 3 g |
| hecto | h | 10 2 | hectogram, 1 hg = 10 2 g |
| deka (or deca) | da | 10 = 10 1 | dekaliter, 1 daL = 10 1 L |
| deci | d | 10-1 | deciliter, 10 1 dL = 1 L |
| centi | c | 10-2 | centimeter, 10 2 cm = 1 m |
| milli | m | 10-3 | millimole, 10 3 mmol = 1 mol |
| micro | μ (Greek letter "mu") | 10-6 | microliter, 10 6 μL = 1 L | 1 μL ~ a very tiny drop of water |
| nano | n | 10-9 | nanometer, 10 9 nm = 1 m | radius of a chlorine atom in Cl 2 ~ 0.1 nm or 100 pm |
| pico | p | 10-12 | picogram, 10 12 pg = 1 g | mass of bacterial cell ~ 1 pg |
| femto | f | 10-15 | femtometer, 10 15 fm = 1 m | radius of a proton ~ 1 fm |
| atto | a | 10-18 | attosecond, 10 18 as = 1 s | time for light to cross an atom ~ 1 as bond energy for one C=C double bond ~ 1 aJ |
| zepto | z | 10-21 | zeptomole, 10 21 zmol = 1 mol | 1 zmol ~ 600 atoms or molecules "A picture is worth about 1.7 zmol of words." |
| yocto | y | 10-24 | yoctogram, 10 24 yg = 1 g | 1.7 yg ~ mass of a proton or neutron |
Binary Prefixes
You have probably heard words such as kilobyte, in the context of computers. What does it mean? It might seem to mean 1000 bytes, since kilo means 1000. But in the computer world it often means 1024 bytes. That is 2 10 - a power of two very close to 1000. Now, in common usage it often does not matter whether the intent was 1000 bytes or 1024 bytes. But they are different numbers and sometimes it does matter. So, a new set of "binary prefixes", distinguished by "bi" in the name or "i" in the abbreviation, was introduced in 1998. By this new system, 1024 bytes would be properly called a kibibyte or KiB. (Sounds like something you would feed the dog.)
This new system of binary prefixes has been endorsed by the International Electrotechnical Commission (IEC) for use in electrical technology. See the NIST page at Whether these will catch on remains to be seen, but at least if you see such an unusual prefix you might want to be aware of this.
Contributors and Attributions
Robert Bruner (
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Reference #1
Research Article
Originally Published 12 February 2014
Free Access
Complex Left Atrial Appendage Morphology and Left Atrial Appendage Thrombus Formation in Patients With Atrial Fibrillation
Masayoshi Yamamoto, MD, Yoshihiro Seo, MD, Naoto Kawamatsu, MD, Kimi Sato, MD, Akinori Sugano, MD, Tomoko Machino-Ohtsuka, MD, Ryo Kawamura, MD, Hideki Nakajima, PhD, Miyako Igarashi, MD, Yukio Sekiguchi, MD, Tomoko Ishizu, MD, and Kazutaka Aonuma, MDAuthor Info & Affiliations
Circulation: Cardiovascular Imaging
Volume 7, Number 2
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Abstract
Background—
In patients with atrial fibrillation (AF), most thrombus forms in the left atrial appendage (LAA). However, the relation of LAA morphology with LAA thrombus is unknown.
Methods and Results—
We prospectively enrolled 633 consecutive patients who were candidates for catheter ablation for symptomatic drug-resistant AF. Transesophageal echocardiography (TEE) was performed to assess LAA thrombus. LAA structure was assessed by 3-dimensional TEE. LAA orifice area, depth, volume, and number of lobes were measured on reconstructed 3-dimensional images. Clinical characteristics and echocardiographic measures were compared to determine variables predicting LAA thrombus. Excluded were 69 (10.9%) patients who met the exclusion criteria. Finally, this study comprised 564 patients, of whom LAA thrombus was observed in 36 (6.4%) patients. Multivariate analysis revealed CHADS 2 (Congestive heart failure, Hypertension Age>75, Diabetes mellitus and prior Stroke or transient ischemic attack) score (P=0.002), left ventricular ejection fraction (P=0.01), degree of spontaneous echo contrast (P=0.02), left atrial volume (P=0.02), and number of LAA lobes (P<0.001) to be independently associated with thrombus formation. Most patients with LAA thrombus (32/34, 94.4%) had ≥3 LAA lobes, whereas LAA thrombus was observed in only 2 (0.7%) of 296 patients with 1 or 2 lobes. LAA volume significantly decreased in patients maintaining sinus rhythm after catheter ablation (P=0.0009). Number of LAA lobes did not change in any patient.
Conclusions—
Complex LAA morphology characterized by an increased number of LAA lobes was associated with the presence of LAA thrombus independently of clinical risk and blood stasis. This study suggests that LAA morphology might be a congenital risk factor for LAA thrombus formation in patients with AF.
Introduction
In patients with atrial fibrillation (AF), most thrombus formation occurs in the left atrial appendage (LAA) because the saccate and complex morphology of the LAA induces stasis of blood flow.1 The CHADS 2 (Congestive heart failure, Hypertension Age>75, Diabetes mellitus and prior Stroke or transient ischemic attack) and CHA 2 DS 2-VaSc scores are indices of thromboembolic risk for patients with AF.2,3 Of the echocardiographic indices, left ventricular systolic dysfunction, dense spontaneous echo contrast (SEC), low LAA peak flow velocities, and complex aortic plaque were reported to be related to thromboembolic risk.4,5 However, whether LAA morphology itself is a risk factor for LAA thrombus formation has not been well studied. Recently, Di Biase et al6 assessed the LAA with computed tomography (CT) or magnetic resonance imaging (MRI) and reported the types of morphology related to thromboembolic risk. That report strongly suggests that LAA morphology is a risk factor for LAA thrombus formation; however, the relation between presence of LAA thrombus and types of LAA morphology has not been clarified. Although CT and MRI have been used to assess LAA morphology,6–8 initial studies with three-dimensional (3D) transesophageal echocardiography (TEE) have also shown the potential to assess LAA morphology.9,10 Recently, our coworkers reported that 3D-TEE could evaluate the LAA volume, orifice area, depth, and number of LAA lobes from reconstructed 3D LAA images.11 Therefore, the aim of our study was to clarify the relation between LAA morphology evaluated with 3D-TEE and LAA thrombus in patients with AF in comparison with conventional indices.
Clinical Perspective on p 343
Methods
Study Population and Study Protocol
Between April 2008 and February 2012, a total of 633 consecutive patients who were candidates for catheter ablation for symptomatic drug-resistant AF at the University of Tsukuba Hospital were prospectively enrolled in this study (Table 1). Among them, patients with inadequate TEE imaging quality and severe mitral or aortic valvular diseases and those undergoing hemodialysis or not receiving anticoagulation therapy with warfarin were excluded.
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Table 1. Baseline Characteristics and Echocardiographic Measurements
| Variable Value (N=633) |
| Men/women | 512/121 |
| Age, y | 62±11 |
| Heart rate | 67±16 |
| Comorbidities |
| Paroxysmal atrial fibrillation | 373 (59%) |
| Nonparoxysmal atrial fibrillation | 260 (41%) |
| Rhythm at examination |
| Normal sinus rhythm | 365 (58%) |
| Atrial fibrillation | 268 (42%) |
| Measurements on 2D transthoracic echocardiography |
| Left ventricular end-diastolic volume, ml | 101±36 |
| Left ventricular end-systolic volume, ml | 38±26 |
| Left ventricular ejection fraction, % | 63±11 |
| Left atrial volume, ml | 63±29 |
Expand Table
Data are expressed as mean±SD or as number (percentage).
On the day before catheter ablation, transthoracic echocardiography and TEE examinations were performed to identify the presence of LAA thrombus in all patients. Anticoagulation management of warfarin was in accordance with the guideline of The Japanese Circulation Society.12 In the guideline, the target international normalized ratio (INR) is 2.0 to 3.0 for patients under 70 years of age and 1.6 to 2.6 for patients 70 years of age or older because an INR >2.6 increases the risk for serious bleeding complications in the Japanese elderly population. All patients received therapeutic anticoagulation therapy at least during the 3 weeks before TEE.
Sequential TEE examinations were also performed in patients who had maintained sinus rhythm for ≥1 year after catheter ablation to clarify the contribution of AF to left atrium and LAA morphology and function. Ethical approval of the present study was obtained from the local review committee, and all patients provided their written informed consent.
Echocardiographic Studies
Standard 2D transthoracic echocardiographic examinations were performed with an iE33 ultrasound system and S5-2 probe (Philips Medical Systems, Andover, MA). Left ventricular end-diastolic volume, end-systolic volume, and ejection fraction (LVEF) were measured using the modified Simpson’s method from the apical view.13 The left atrial (LA) volume was measured using the modified Simpson’s method from the apical view.14
TEE was performed with an iE33 ultrasound system and S7-2 probe (Philips Medical Systems). SEC was visually classified into 4 grades by careful attention to the gain settings adjusted to distinguish background white noise.15 The severity of SEC was scored as follows: 0, absence of echogenicity; 1+, mild (minimal echogenicity detectable in only a part of the LA cavity with high gain settings); 2+, moderate (denser swirling during the entire cardiac cycle); and 3+, severe (intense echodensity and slow swirling patterns in the LAA usually with similar density in the main cavity) as defined in a previous report. Reproducibility of the SEC grade was evaluated between 2 observers in 50 patients selected at random, and the concordance rate was 92% (46/50). Based on the recommendations from the American Society of Echocardiography,16 blood stasis was quantified by LAA flow velocities, which were measured at ≈1 cm below the outlet of the LAA cavity using pulsed Doppler. LAA emptying flow velocity was measured in the basal short-axis view from the transverse scan (45° views). The LAA emptying and filling flow velocities were measured as the average of 3 consecutive cardiac cycles in patients with normal sinus rhythm and 5 consecutive cardiac cycles in patients with AF. Full-volume mode examinations were performed from 45° views during apnea at end-expiration. To obtain these data sets, 6 sectors were scanned with gating to the electrocardiographic R wave and were automatically integrated into a wide-angle (76×69 degrees) pyramidal data image covering the entire LAA. The frame rate of each image was set at ≈20 to 30 frames/s. In patients with AF during the examination, zoom mode, which magnified the pyramidal scan by 1 cardiac beat, was used. The frame rate of each image was set at ≈10 frames/s.
Quantification of LAA Morphology
Quantification of LAA morphology was performed with QLAB GI-3DQ software (Philips Medical Systems). First, multireconstruction planes of the LAA were obtained from 3D data sets (pyramidal images) including the LAA at end-systole. The method of determining LAA orifice size is shown in Figure 1A and 1B.17,18 We measured the LAA orifice long and short diameters, orifice area (Figure 1C), and depth of the orifice to a lobe tip (Figure 1B).
Open in Viewer
Figure 1. Determination of the left atrial appendage (LAA) orifice. A, LAA long-axis view at the level including the mitral valve annulus, left coronary artery (white arrow), and lateral ridge of the left superior pulmonary vein (white arrowhead). B, LAA long-axis view at the level including the aortic valve annulus (white arrows). The dashed line with double-headed white arrows in both (A) and (B) corresponds to the line used to determine the orifice of the LAA as shown in (C). In (B), the dashed line shows LAA depth. C, LAA orifice area and long and short diameters.
The inner border of the LAA was manually traced within the distal area of the orifice, and the transverse images of the longest axis of the LAA trace area were automatically sliced at 10 levels from the orifice to the most distal site. On each sliced transverse image, the inner border was manually traced and reconstructed into a 3D image (Figure 2). On a reconstructed image, we measured LAA volume and the number of LAA lobes, which was assessed based on the definitions by Veinot et al19 as follows: (1) LAA lobe was a visible out-pouching from the main tubular body of the LAA, usually demarcated by an external crease; (2) it was internally capable of admitting a 2-mm probe (ie, it was not simply a tag of external adipose tissue); (3) it was occasionally but not necessarily associated with a change in direction of the main tubular body of the LAA; (4) it could lie in a different anatomic plane than the main tubular body; and (5) by definition, the LAA must have ≥1 lobe. Intraobserver and interobserver variability of these parameters was <10% in our previous study.11
Open in Viewer
Figure 2. Representative reconstructed 3D images of the left atrial appendage (LAA). The reconstructed 3D image is of an LAA with a single lobe (A), an LAA with 2 lobes (B), and an LAA with 3 lobes (C). The green line indicates the coronal section line for ultrasonic beam direction, and the red line indicates the section line for a sagittal section.
Clinical Risk Stratification
We calculated CHADS 2 score for clinical risk stratification of stroke in patients with AF.2 Some clinical studies have reported a relation between inflammation and AF,20 and C-reactive protein (CRP) is a representative marker of vascular inflammation. B-type natriuretic peptide (BNP) is a marker that increases in patients with structural heart disease, heart failure, and lone AF. It has been reported that the atrium, and not the ventricle, is the main source of BNP in patients with AF,21 and BNP is higher in patients with a history of thromboembolism than in patients without this complication.22 Therefore, blood samples to measure plasma high-sensitivity CRP, BNP concentration, and prothrombin time/INR were obtained at the time of TEE examinations.
Catheter Ablation
Three long sheaths were advanced into the LA. After pulmonary vein angiography, 2 decapolar ring catheters (Lasso, Biosense Webster, Diamond Bar, CA) were placed in the superior and inferior pulmonary veins on one side at a time. An open-irrigation, 3.5-mm tip deflectable catheter (ThermoCool; Biosense Webster) was used for mapping and ablation. The LA and pulmonary veins were constructed with a 3D electroanatomic mapping system (CARTO; Biosense Webster). The ipsilateral pulmonary vein antrum was circumferentially ablated under fluoroscopic, electrogram, and CARTO guidance. Radiofrequency energy was delivered at a power of 20 to 35 W. The end point of ablation was the elimination of all pulmonary vein potentials.
Statistical Analysis
Results are expressed as number (%) or mean±SD. Comparisons between the 2 groups were performed with the Student t test for continuous variables and the Fisher exact test for categorical variables. One-way analysis of variance with the post hoc Tukey–Kramer test was used to compare variables in ≥3 groups. Multiple logistic analysis was performed to identify independent risk factors for LAA thrombus. A P value of <0.05 was considered to indicate statistical significance. These analyses were performed with SPSS version 17.0 for Windows (SPSS Inc, Chicago, IL).
Results
TEE examinations were successfully performed in all patients. We excluded 69 (10.9%) patients because of inadequate imaging quality for LAA analysis (n=42), not receiving anticoagulation therapy with warfarin (n=12), severe mitral regurgitation (n=8), undergoing hemodialysis (n=5), and severe mitral stenosis (n=2). Finally, this study comprised 564 patients. Among them, LAA thrombus was observed in 36 (6.4%) patients. Comparisons between patients with and without LAA thrombus are summarized in Table 2. Patients with LAA thrombus had a significantly higher prevalence of nonparoxysmal AF (P<0.001). Patients with LAA thrombus were significantly older (P=0.003), and their LVEF was lower (P<0.001) and CHADS 2 score higher (P<0.001) than those in patients without LAA thrombus. There was no significant difference in prothrombin time/INR between the 2 groups. In patients with LAA thrombus, plasma high-sensitivity CRP (P<0.001) and BNP (P<0.001) concentrations were significantly higher than those in patients without LAA thrombus. In patients with LAA thrombus, LAA volume (P<0.001) and LA volume (P<0.001) were significantly larger with larger LAA orifice area (P<0.001) and deeper LAA depth (P<0.001) in comparison with those in patients without LAA thrombus. In addition, the number of LAA lobes was significantly higher (P<0.001) than that of the patients without LAA thrombus. Degree of SEC was significantly higher (P<0.001) with lower LAA emptying velocities (P<0.001) compared with those in patients without LAA thrombus.
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Table 2. Baseline Characteristics According to Presence of LAA Thrombus
| Variables | No Thrombus(n=528) | Thrombus(n=36) | P Value |
--- --- |
| Age, y | 61±11 | 67±8 | 0.003 |
| AF type (nonparoxysmal AF) | 186 (35%) | 26 (72%) | <0.001 |
| Sex (male) | 424 (80%) | 33 (92%) | 0.12 |
| PT-INR | 1.92±0.31 | 2.15±0.42 | 0.08 |
| hs-CRP, mg/dL | 0.18±0.42 | 0.44±0.65 | <0.001 |
| BNP, mg/dL | 97.1±139.1 | 291.2±288.8 | <0.001 |
| CHADS 2 score | 1.1±1.1 | 2.2±1.3 | <0.001 |
| Left ventricular ejection fraction, % | 64±10 | 51±18 | <0.001 |
| Left ventricular end-diastolic volume, ml | 99±32 | 129±68 | <0.001 |
| LA volume, ml | 61±26 | 99±45 | <0.001 |
| Degree of spontaneous echo contrast | 0.6±1.0 | 2.1±1.1 | <0.001 |
| LAA emptying velocity, cm/s | 46±21 | 28±17 | <0.001 |
| LAA orifice area, cm 2 | 4.4±3.1 | 6.3±3.0 | <0.001 |
| LAA depth, mm | 34±11 | 42±12 | <0.001 |
| LAA volume, ml | 7.6±7.1 | 12.1±5.8 | <0.001 |
| Number of LAA lobes | 2.5±0.8 | 3.4±0.8 | <0.001 |
Expand Table
Data are expressed as mean±SD or as number (percentage).
AF indicates atrial fibrillation; BNP, B-type natriuretic peptide; CHADS 2, Congestive heart failure, Hypertension Age>75, Diabetes mellitus and prior Stroke or transient ischemic attack; hs-CRP, high-sensitivity C-reactive protein; LA, left atrium; LAA, left atrial appendage; and PT-INR, prothrombin time/international normalized ratio.
Risk Factors for LAA Thrombus
Various factors had a significant relation with the presence of LAA thrombus as shown in Table 3. In a multivariate logistic analysis, number of LAA lobes (odds ratio [OR], 2.469; 95% confidence interval [CI {1.495–4.078}]; P<0.001) was identified as an independent risk factor for presence of LAA thrombus, as were CHADS 2 score (OR, 1.752; 95% CI [1.237–2.483]; P=0.002), LVEF (OR, 0.962; 95% CI [0.934–0.992]; P=0.01), LA volume (OR, 1.018; 95% CI [1.003–1.032]; P=0.02), and degree of SEC (OR, 1.783; 95% CI [1.102–2.740]; P=0.02). Prevalence of the number of LAA lobes is shown in Figure 3. The majority of patients with LAA thrombus (32/34, 94.4%) had ≥3 LAA lobes. In contrast, LAA thrombus was observed in only 2 (0.7%) of 296 patients with 1 or 2 lobes. In a multiple logistic regression analysis model, as compared with an LAA with 1 or 2 lobes, an LAA with 3 lobes was 8.6 times (OR, 8.6; 95% CI [1.9–39.8]; P=0.006), 4 or 5 lobes was 10 times (OR, 10.0; 95% CI [2.2–42.1]; P=0.004), and ≥3 lobes was 9.2 times (OR, 9.2; 95% CI [2.0–41.1]; P=0.004) more likely to have thrombus.
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Table 3. Univariate and Multivariate Analyses for Presence of LAA Thrombus
| | Univariate | Multivariate |
--- |
| Variables | OR (95% CI) | P Value | OR (95% CI) | P Value |
| AF type (nonparoxysmal AF) | 4.785 (2.26–10.10) | <0.001 | … | 0.41 |
| CHADS 2 score | 1.915 (1.486–2.467) | <0.001 | 1.752 (1.237–2.483) | 0.002 |
| Degree of spontaneous echo contrast | 3.128 (2.262–4.326) | <0.001 | 1.783 (1.102–2.740) | 0.02 |
| Left ventricular ejection fraction, % | 0.935 (0.914–0.956) | <0.001 | 0.962 (0.934–0.992) | 0.01 |
| LA volume, ml | 1.031 (1.021–1.041) | <0.001 | 1.018 (1.003–1.032) | 0.02 |
| LAA emptying velocity, cm/s | 0.947 (0.925–0.970) | <0.001 | … | 0.60 |
| LAA volume, ml | 1.038 (1.007–1.070) | 0.02 | … | 0.86 |
| Number of LAA lobes | 3.318 (2.179–5.052) | <0.001 | 2.469 (1.495–4.078) | <0.001 |
Expand Table
AF indicates atrial fibrillation; CHADS 2, Congestive heart failure, Hypertension Age>75, Diabetes mellitus and prior Stroke or transient ischemic attack; CI, confidence interval; LA, left atrium; LAA, left atrial appendage; and OR, odds ratio.
Open in Viewer
Figure 3. Prevalence of number of left atrial appendage (LAA) lobes between patients with and without thrombus. The graphs show the difference in the prevalence of number of LAA lobes between patients without LAA thrombus (A) and with LAA thrombus (B). Most patients in the group without LAA thrombus (A) had 2 LAA lobes, whereas most patients in the group with LAA thrombus had 3 or more lobes.
The relation of number of LAA lobes with degree of SEC and LAA empting velocity is shown in Figure 4. In patients with ≥3 LAA lobes, a higher degree of SEC (Figure 4A) and lower LAA empting velocity (Figure 4B) were observed as compared with those in patients with 1 or 2 LAA lobes.
Open in Viewer
Figure 4. Degree of blood stasis according to number of left atrial appendage (LAA) lobes. A, Comparisons of the degree of spontaneous echo contrast according to numbers of LAA lobes. B, Comparisons of LAA emptying velocity. P<0.05 vs 1 lobe group; †P<0.05 vs 2 lobes group; ‡P<0.05 vs 3 lobes group.
Risk Factors for LAA Thrombus in Patients with Low CHADS 2 Score
Patients with LAA thrombus included 13 (13/198, 6.6%) with a CHADS 2 score of 1 and 1 (1/185, 0.5%) patient with a CHADS 2 score of 0. In a multiple logistic regression analysis model for LA thrombus in only limited patients with CHADS 2 score 0/1, number of LAA lobes (OR, 2.8; 95% CI [1.3–6.1]; P=0.008) was identified as a significant predictor as were SEC (OR, 3.1; 95% CI [1.5–6.6]; P=0.003) and LVEF (OR, 0.9; 95% CI [0.89–0.98]; P=0.022). Unlike the analysis in the overall population, LA volume was not a significant predictor.
Functional and Morphological Changes After Catheter Ablation
Repeat TEE examinations were performed in 46 patients who maintained sinus rhythm during 1 year after catheter ablation. Average term from catheter ablation to next TEE examination was 19.2±5.3 months. A representative case is shown in Figure 5. The LAA volume decreased after catheter ablation from 17.8 to 10.3 mL, whereas the number of lobes and fundamental morphology were maintained. Comparison of clinical characteristics and echocardiographic measurements between baseline and after catheter ablation are summarized in Table 4. After catheter ablation, plasma high-sensitivity CRP (P=0.04) and BNP (P<0.001) concentrations were decreased, and LVEF (P=0.002) was increased. In addition, significant reverse remodeling of both the LA (P<0.001) and LAA (P<0.001) were observed with improvements of blood stasis represented as SEC (P<0.001) and LAA emptying velocity (P=0.02). However, the number of lobes did not change in any of the patients.
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Table 4. Clinical Characteristics and Echocardiographic Measurements in Patients Who Maintained Sinus Rhythm After Radiofrequency Catheter Ablation
| Variables | Baseline | After RFCA | P Value |
--- --- |
| Men/women | 36/10 | … | … |
| Age, y | 62±9 | … | … |
| BNP, mg/dL | 124±135 | 44±52 | <0.001 |
| hs-CRP, mg/dL | 0.13±0.14 | 0.09±0.10 | 0.04 |
| Left ventricular ejection fraction, % | 65±10 | 69±8 | 0.002 |
| Left ventricular end-diastolic volume, ml | 103±33 | 104±27 | 0.91 |
| Left ventricular end-systolic volume, ml | 37±20 | 33±12 | 0.03 |
| E | 78±20 | 72±20 | 0.1 |
| E′ | 9.5±2.6 | 13±9 | 0.001 |
| E/E′ | 8.6±2.9 | 8.3±3.3 | 0.19 |
| LA volume, ml | 73±25 | 63±24 | <0.001 |
| LAA emptying velocity, cm/s | 38±19 | 44±19 | 0.02 |
| LAA filling velocity, cm/s | 43±22 | 46±18 | 0.45 |
| LAA orifice area, cm 2 | 4.4±1.4 | 4.0±1.2 | 0.003 |
| LAA depth, mm | 37±8 | 36±7 | 0.002 |
| LAA volume, ml | 7.4±2.9 | 6.7±2.6 | <0.001 |
| Number of LAA lobes | 2.6±0.7 | 2.6±0.7 | … |
| Degree of spontaneous echo contrast | 1.3±1.2 | 0.70±0.9 | <0.001 |
Expand Table
Data are expressed as mean±SD or as number (percentage).
BNP indicates B-type natriuretic peptide; E, peak velocity of early transmitral flow; E′, peak velocity of early mitral annular motion; hs-CRP, high-sensitivity C-reactive protein; LA, left atrium; LAA, left atrial appendage; and RFCA, radiofrequency catheter ablation.
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Figure 5. Representative case of left atrial appendage (LAA) remodeling after catheter ablation. A, Baseline LAA images (top) and (B) shows the corresponding images (bottom) after catheter ablation. The image in the red frame shows a short-axis view of the apex of the LAA lobe, that in the blue frame shows an LAA long-axis view (sagittal plane), and that in the green frame shows an LAA long-axis view (horizontal plane). The rightmost image shows a 3-dimensional (3D) LAA image. The green line in the 3D LAA image indicates the horizontal section line for ultrasonic beam direction, the red line indicates a coronal section, and the blue line indicates a sagittal section. The LAA was reduced in size after catheter ablation (LAA volume, 17.8 to 10.3 mL; length of LAA neck, 23 to 14 mm), while the number of lobes () and fundamental morphology were maintained.
Discussion
This is the first study, to our knowledge, to examine the relation between LAA morphology assessed by 3D-TEE and LAA thrombus formation in patients with AF. We found a significant relation between LAA morphology and the prevalence of LAA thrombus formation; in particular, number of LAA lobes was identified as an independent risk factor for LAA thrombus. The LAA with multiple lobes is thought to have a more complex morphology. Therefore, we think that complex LAA morphology should be considered as a novel risk factor for the formation of LAA thrombus in patients with AF, as well as CHADS 2 score, LA volume, LVEF, and degree of SEC, which are other well-known risk factors of LAA thrombus.
In patients who maintained sinus rhythm during the first year after catheter ablation, LA and LAA volumes were decreased, but the number of LAA lobes did not change. This finding indicates that the number of LAA lobes is a congenital characteristic and is not influenced by LAA remodeling. Accordingly, the complexity of the LAA, which is represented by the number of LAA lobes, may be a congenital and specific factor in each individual. Several previous studies have reported on the diversity of LAA lobes and their geometry.6–10,19,23 In a large autopsy series comprising 500 normal human hearts (age range, 0–100 years), the distribution of the number of LAA lobes was 2 (54% of hearts), 3 (23%), 1 (20%), and 4 (3%).19 In our study, the distribution was 2 (42.2%), 3 (34.9%), 1 (10.3%), 4 (11.3%), and 5 (1.2%). The mean number of LAA lobes is 2.1 in the autopsy study and 2.5 in our study, and there was a statistically significant difference by χ 2 test (P<0.01). The difference might be caused by study populations, namely, our study consisted of patients with AF, in contrast to the normal hearts in the autopsy study.
Di Biase et al6 categorized LAA into 4 different morphologies and reported that the Chicken Wing LAA morphology was less likely to produce an embolic event. We did not analyze morphology type in the present study; however, morphology such as the Chicken Wing type is the simplest among the Di Biase et al classification and may correspond to the LAA with 1 or 2 lobes in our study. Di Biase et al did not discuss the reason that simple LAA morphology was related to a lower embolic event rate. Because it is assumed that a simple morphology is less likely to induce blood stasis, the present study clearly revealed that an increase in the number of LAA lobes was related to a high degree of SEC and low LAA empting velocity. The findings suggest that complex LAA morphology characterized by an increased number of lobes is likely to induce blood stasis, which is a fundamental cause of thrombus formation.
CT and MRI are useful methods of characterizing the LAA; however, image quality is severely deteriorated by stitching artifact in patients with AF during the examination. We used zoom mode, which magnified the pyramidal scan by 1 cardiac beat in patients with AF during the examination, so we could avoid stitching artifact. Moreover, CT has a problem of radiation exposure, and both methods require use of a contrast agent, so it is difficult to use in patients with severe renal dysfunction. In most hospitals, TEE was generally performed before catheter ablation for AF to assess LAA thrombus. TEE has the advantage of assessing LAA morphology in routine clinical practice.
Because we managed anticoagulation based on the guideline of the Japanese Circulation Society, the target ranges differ slightly from those in the AHA guidelines or those of other Western countries. The Japanese Circulation Society guideline recommends a basic target INR of 2.0 to 3.0. However, in patients ≥70 years of age, the recommended target INR is 1.6 to 2.6 because an INR of >2.6 increases the risk for serious bleeding complications in the Japanese elderly population. In our patients without thrombus, 482 (482/528, 91.3%) achieved the target INR, and in patients with thrombus, 33 (33/36, 91.7%) achieved the target INR. In the 3 patients with thrombus who did not achieve the target INR, thrombus formation might be because of poor warfarin control. However, the INR in these 3 patients was not much lower than the target INR (every patient exceeded an INR of 1.5) because they were all >70 years of age. Therefore, most patients with thrombus were treated within the target INR.
Plasma CRP concentrations in patients with LAA thrombus were increased over those in patients without LAA thrombus. Previous studies reported that CRP and interleukin-6 were elevated in patients with AF, and elevated inflammatory markers are related to embolic events.24,25 Our findings support these results, but multivariate analysis could not identify the independent strength of the association of these variables with LAA thrombus as compared with that of LAA morphology and blood stasis. Because a significant relation between inflammation and LA remodeling has been observed, the inflammation process may indirectly contribute to LAA thrombus formation via LA and LAA remodeling.
Clinical Implications
In this study, we clarified the relation between LAA morphology and thrombus formation. In the clinical setting, we have used the CHADS 2 or CHA 2 DS 2-VASc score for thromboembolic risk stratification. However, in cases of high bleeding risk or low CHADS 2 score (0 or 1), both bleeding risk because of anticoagulation therapy and thromboembolic risk should be considered. Furthermore, in patients who have maintained sinus rhythm for a long time after catheter ablation, we wonder whether anticoagulation therapy should be continued. Knowledge of the number of LAA lobes would be helpful in making clinical decisions on antithrombotic therapy in such controversial cases. However, thrombus is a cause of stroke, and thrombus formation is no more than a surrogate for stroke risk. In the future, long-term follow-up studies identifying how the number of LAA lobes together with the CHADS 2 score influences the risk of stroke as an end point are needed.
Study Limitations
Forty-two patients (6.6%), most of whom were in an AF rhythm at the time of TEE examination, were excluded because of inadequate 3D LAA image quality. Because the full-volume imaging method derived from 6 cardiac beats was not available in patients with AF, the single-beat image acquisition method (zoom-mode imaging) was used, which may exacerbate image quality in assessing LAA morphology in patients with AF. From the statistical aspect, there is a great difference between the number of patients with thrombus (n=528) and without thrombus (n=36). Therefore, comparisons between the 2 groups are limited statistically. Furthermore, our study was a single-center study, and the study population was limited to patients undergoing catheter ablation and anticoagulation therapy with warfarin for AF. To confirm whether this study is applicable to a wider population not undergoing catheter ablation or taking warfarin, a multicenter study consisting of a larger number of participants with diverse diagnoses is necessary.
Conclusions
Complex LAA morphology that was characterized by an increased number of LAA lobes was associated with the presence of LAA thrombus, independently of clinical risk and blood stasis. Our study suggested that LAA morphology might be a congenital risk factor for LAA thrombus formation in patients with AF. Accordingly, analysis of LAA morphology may provide additional information in the diagnosis of LAA thrombus and in decision making and formulation of medical strategies including anticoagulation management.
CLINICAL PERSPECTIVE
In patients with atrial fibrillation (AF), most thrombus formation occurs in the left atrial appendage (LAA) in part because the complex morphology of the LAA predisposes to blood flow stasis. Three-dimensional transesophageal echocardiography (3D-TEE) allows detailed imaging of the LAA and could help evaluate morphological characteristics that predispose to thrombus formation. In the present study, 564 patients who were candidates for catheter ablation for symptomatic drug-resistant AF were studied, and LAA thrombus was observed in 36 patients (6.4%). Independent of clinical risk factors and blood stasis, the presence of LAA thrombus was strongly associated with complex LAA morphology characterized by more numbers of LAA lobes. In patients who maintained sinus rhythm during 1 year after catheter ablation, LA and LAA volumes decreased, but, not unexpectedly, the number of LAA lobes did not change. This finding is consistent with the fact that the number of LAA lobes is a structural feature and is not influenced by LAA remodeling. This study contributes to our understanding of risk stratification for LAA thrombus formation and may provide additional information to be incorporated into decisions around anticoagulation in patients with AF. Long-term follow-up studies will be required to assess how the LAA morphology influences stroke risk, and whether this information provides incremental benefit over traditional risk scoring systems.
References
1.
Mügge A, Kühn H, Nikutta P, Grote J, Lopez JA, Daniel WG. Assessment of left atrial appendage function by biplane transesophageal echocardiography in patients with nonrheumatic atrial fibrillation: identification of a subgroup of patients at increased embolic risk. J Am Coll Cardiol. 1994;23:599–607.
Go to Citation
Crossref
PubMed
Google Scholar
2.
Fuster V, Ryden LE, Cannom DS, Crijns HJ, Curtis AB, Ellenbogen KA, Halperin JL, Le Heuzey JY, Kay GN, Lowe JE, Olsson SB, Prystowsky EN, Tamargo JL, Wann S, Smith SC, Jacobs AK, Adams CD, Anderson JL, Antman EM, Halperin JL, Hunt SA, Nishimura R, Ornato JP, Page RL, Riegel B, Priori SG, Blanc JJ, Budaj A, Camm AJ, Dean V, Deckers JW, Despres C, Dickstein K, Lekakis J, McGregor K, Metra M, Morais J, Osterspey A, Tamargo JL, Zamorano JL. ACC/GBPA/ESC 2006 Guidelines for the Management of Patients with Atrial Fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the European Society of Cardiology Committee for Practice Guidelines (Writing Committee to Revise the 2001 Guidelines for the Management of Patients with Atrial Fibrillation): developed in collaboration with the European Heart Rhythm Association and the Heart Rhythm Society. Circulation. 2006;114:e257–e354.
Crossref
PubMed
Google Scholar
a [...] thromboembolic risk for patients with AF.
b [...] of stroke in patients with AF.
3.
European Heart Rhythm Association, European Association for Cardio-Thoracic SurgeryCamm AJ, Kirchhof P, Lip GY, Schotten U, Savelieva I, Ernst S, Van Gelder IC, Al-Attar N, Hindricks G, Prendergast B, Heidbuchel H, Alfieri O, Angelini A, Atar D, Colonna P, De Caterina R, De Sutter J, Goette A, Gorenek B, Heldal M, Hohloser SH, Kolh P, Le Heuzey JY, Ponikowski P, Rutten FH. Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC). Eur Heart J. 2010;31:2369–2429.
Go to Citation
Crossref
PubMed
Google Scholar
4.
Zabalgoitia M, Halperin JL, Pearce LA, Blackshear JL, Asinger RW, Hart RG; Stroke Prevention in Atrial Fibrillation III Investigators. Transesophageal echocardiographic correlates of clinical risk of thromboembolism in nonvalvular atrial fibrillation. J Am Coll Cardiol. 1998;31:1622–1626.
Go to Citation
Crossref
PubMed
Google Scholar
5.
Atrial Fibrillation Investigators. Echocardiographic predictors of stroke in patients with atrial fibrillation: a prospective study of 1066 patients from 3 clinical trials. Arch Intern Med. 1998;158:1316–1320.
Go to Citation
Crossref
PubMed
Google Scholar
6.
Di Biase L, Santangeli P, Anselmino M, Mohanty P, Salvetti I, Gili S, Horton R, Sanchez JE, Bai R, Mohanty S, Pump A, Cereceda Brantes M, Gallinghouse GJ, Burkhardt JD, Cesarani F, Scaglione M, Natale A, Gaita F. Does the left atrial appendage morphology correlate with the risk of stroke in patients with atrial fibrillation? Results from a multicenter study. J Am Coll Cardiol. 2012;60:531–538.
Crossref
PubMed
Google Scholar
a [...] been well studied. Recently, Di Biase et al
b [...] have been used to assess LAA morphology,
c [...] diversity of LAA lobes and their geometry.
d [...] Di Biase et al
7.
Lacomis JM, Goitein O, Deible C, Moran PL, Mamone G, Madan S, Schwartzman D. Dynamic multidimensional imaging of the human left atrial appendage. Europace. 2007;9:1134–1140.
Crossref
PubMed
Google Scholar
a [...] have been used to assess LAA morphology,
b [...] diversity of LAA lobes and their geometry.
8.
Heist EK, Refaat M, Danik SB, Holmvang G, Ruskin JN, Mansour M. Analysis of the left atrial appendage by magnetic resonance angiography in patients with atrial fibrillation. Heart Rhythm. 2006;3:1313–1318.
Crossref
PubMed
Google Scholar
a [...] have been used to assess LAA morphology,
b [...] diversity of LAA lobes and their geometry.
9.
Shah SJ, Bardo DM, Sugeng L, Weinert L, Lodato JA, Knight BP, Lopez JJ, Lang RM. Real-time three-dimensional transesophageal echocardiography of the left atrial appendage: initial experience in the clinical setting. J Am Soc Echocardiogr. 2008;21:1362–1368.
Crossref
PubMed
Google Scholar
a [...] the potential to assess LAA morphology.
b [...] diversity of LAA lobes and their geometry.
10.
Agoston I, Xie T, Tiller FL, RGBPman AM, GBPmad M. Assessment of left atrial appendage by live three-dimensional echocardiography: early experience and comparison with transesophageal echocardiography. Echocardiography. 2006;23:127–132.
Crossref
PubMed
Google Scholar
a [...] the potential to assess LAA morphology.
b [...] diversity of LAA lobes and their geometry.
11.
Nakajima H, Seo Y, Ishizu T, Yamamoto M, Machino T, Harimura Y, Kawamura R, Sekiguchi Y, Tada H, Aonuma K. Analysis of the left atrial appendage by three-dimensional transesophageal echocardiography. Am J Cardiol. 2010;106:885–892.
Crossref
PubMed
Google Scholar
a [...] LAA lobes from reconstructed 3D LAA images.
b [...] parameters was <10% in our previous study.
12.
JCS Joint Working Group. Guidelines for pharmacotherapy of atrial fibrillation (JCS 2008): digest version. Circ J. 2010;74:2479–2500.
Go to Citation
Crossref
PubMed
Google Scholar
13.
Nosir YF, Vletter WB, Boersma E, Frowijn R, Ten Cate FJ, Fioretti PM, Roelandt JR. The apical long-axis rather than the two-chamber view should be used in combination with the four-chamber view for accurate assessment of left ventricular volumes and function. Eur Heart J. 1997;18:1175–1185.
Go to Citation
Crossref
PubMed
Google Scholar
14.
Lang RM, Bierig M, Devereux RB, Flachskampf FA, Foster E, Pellikka PA, Picard MH, Roman MJ, Seward J, Shanewise JS, Solomon SD, Spencer KT, Sutton MS, Stewart WJ; Chamber Quantification Writing Group; American Society of Echocardiography’s Guidelines and Standards Committee; European Association of Echocardiography. Recommendations for chamber quantification: a report from the American Society of Echocardiography’s Guidelines and Standards Committee and the Chamber Quantification Writing Group, developed in conjunction with the European Association of Echocardiography, a branch of the European Society of Cardiology. J Am Soc Echocardiogr. 2005;18:1440–1463.
Go to Citation
Crossref
PubMed
Google Scholar
15.
Vincelj J, Sokol I, Jaksić O. Prevalence and clinical significance of left atrial spontaneous echo contrast detected by transesophageal echocardiography. Echocardiography. 2002;19:319–324.
Go to Citation
Crossref
PubMed
Google Scholar
16.
Stroke Prevention in Atrial Fibrillation Investigators Committee on Echocardiography. Transesophageal echocardiography in atrial fibrillation: standards for acquisition and interpretation and assessment of interobserver variability. J Am Soc Echocardiogr. 1996;9:556–566.
Go to Citation
PubMed
Google Scholar
17.
Cabrera JA, Ho SY, Climent V, Sánchez-Quintana D. The architecture of the left lateral atrial wall: a particular anatomic region with implications for ablation of atrial fibrillation. Eur Heart J. 2008;29:356–362.
Go to Citation
Crossref
PubMed
Google Scholar
18.
Budge LP, Shaffer KM, Moorman JR, Lake DE, Ferguson JD, Mangrum JM. Analysis of in vivo left atrial appendage morphology in patients with atrial fibrillation: a direct comparison of transesophageal echocardiography, planar cardiac CT, and segmented three-dimensional cardiac CT. J Interv Card Electrophysiol. 2008;23:87–93.
Go to Citation
Crossref
PubMed
Google Scholar
19.
Veinot JP, Harrity PJ, Gentile F, Khandheria BK, Bailey KR, Eickholt JT, Seward JB, Tajik AJ, Edwards WD. Anatomy of the normal left atrial appendage: a quantitative study of age-related changes in 500 autopsy hearts: implications for echocardiographic examination. Circulation. 1997;96:3112–3115.
Crossref
PubMed
Google Scholar
a [...] based on the definitions by Veinot et al
b [...] diversity of LAA lobes and their geometry.
c [...] of hearts), 3 (23%), 1 (20%), and 4 (3%).
20.
Boos CJ, Anderson RA, Lip GY. Is atrial fibrillation an inflammatory disorder? Eur Heart J. 2006;27:136–149.
Go to Citation
Crossref
PubMed
Google Scholar
21.
Inoue S, Murakami Y, Sano K, Katoh H, Shimada T. Atrium as a source of brain natriuretic polypeptide in patients with atrial fibrillation. J Card Fail. 2000;6:92–96.
Go to Citation
Crossref
PubMed
Google Scholar
22.
Shimizu H. High plasma brain natriuretic polypeptide level as a marker of risk for thromboembolism in patients with nonvalvular atrial fibrillation. Stroke. 2002;33:1005–1010.
Go to Citation
Crossref
PubMed
Google Scholar
23.
Agmon Y, Khandheria BK, Gentile F, Seward JB. Echocardiographic assessment of the left atrial appendage. J Am Coll Cardiol. 1999;34:1867–1877.
Crossref
PubMed
Google Scholar
24.
Psychari SN, Apostolou TS, Sinos L, Hamodraka E, Liakos G, Kremastinos DT. Relation of elevated C-reactive protein and interleukin-6 levels to left atrial size and duration of episodes in patients with atrial fibrillation. Am J Cardiol. 2005;95:764–767.
Go to Citation
Crossref
PubMed
Google Scholar
25.
Conway DS, Buggins P, Hughes E, Lip GY. Prognostic significance of raised plasma levels of interleukin-6 and C-reactive protein in atrial fibrillation. Am Heart J. 2004;148:462–466.
Go to Citation
Crossref
PubMed
Google Scholar
Show all references
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Circulation: Cardiovascular Imaging
Volume 7 • Number 2 • March 2014
Pages: 337 - 343
PubMed: 24523417
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Received: 26 June 2013
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Keywords
anticoagulants
atrial appendage
atrial fibrillation
catheter ablation
echocardiography
stroke
thrombosis
Subjects
Epidemiology
Authors
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Masayoshi Yamamoto, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Yoshihiro Seo, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Naoto Kawamatsu, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Kimi Sato, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Akinori Sugano, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Tomoko Machino-Ohtsuka, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Ryo Kawamura, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Hideki Nakajima, PhD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Miyako Igarashi, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Yukio Sekiguchi, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Tomoko Ishizu, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Kazutaka Aonuma, MD
From the Cardiovascular Division, Institute of Clinical Medicine, Graduate School of Comprehensive Human Sciences, University of Tsukuba, Ibaraki, Japan (M.Y., Y. Seo, N.K., K.S., A.S., T.M.-O., R.K., M.I., Y. Sekiguchi, T.I., K.A.); and Department of Clinical Laboratory, Tsukuba University Hospital, Ibaraki, Japan (H.N.).
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Notes
Correspondence to Yoshihiro Seo, MD, Cardiovascular Division, Faculty of Clinical Medicine, University of Tsukuba 1-1-1 Tennodai, Tsukuba 305–8575, Japan. E-mail yo-seo@md.tsukuba.ac.jp
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Figure 1. Determination of the left atrial appendage (LAA) orifice. A, LAA long-axis view at the level including the mitral valve annulus, left coronary artery (white arrow), and lateral ridge of the left superior pulmonary vein (white arrowhead). B, LAA long-axis view at the level including the aortic valve annulus (white arrows). The dashed line with double-headed white arrows in both (A) and (B) corresponds to the line used to determine the orifice of the LAA as shown in (C). In (B), the dashed line shows LAA depth. C, LAA orifice area and long and short diameters.
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Figure 2. Representative reconstructed 3D images of the left atrial appendage (LAA). The reconstructed 3D image is of an LAA with a single lobe (A), an LAA with 2 lobes (B), and an LAA with 3 lobes (C). The green line indicates the coronal section line for ultrasonic beam direction, and the red line indicates the section line for a sagittal section.
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Figure 3. Prevalence of number of left atrial appendage (LAA) lobes between patients with and without thrombus. The graphs show the difference in the prevalence of number of LAA lobes between patients without LAA thrombus (A) and with LAA thrombus (B). Most patients in the group without LAA thrombus (A) had 2 LAA lobes, whereas most patients in the group with LAA thrombus had 3 or more lobes.
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Figure 4. Degree of blood stasis according to number of left atrial appendage (LAA) lobes. A, Comparisons of the degree of spontaneous echo contrast according to numbers of LAA lobes. B, Comparisons of LAA emptying velocity. P<0.05 vs 1 lobe group; †P<0.05 vs 2 lobes group; ‡P<0.05 vs 3 lobes group.
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Figure 5. Representative case of left atrial appendage (LAA) remodeling after catheter ablation. A, Baseline LAA images (top) and (B) shows the corresponding images (bottom) after catheter ablation. The image in the red frame shows a short-axis view of the apex of the LAA lobe, that in the blue frame shows an LAA long-axis view (sagittal plane), and that in the green frame shows an LAA long-axis view (horizontal plane). The rightmost image shows a 3-dimensional (3D) LAA image. The green line in the 3D LAA image indicates the horizontal section line for ultrasonic beam direction, the red line indicates a coronal section, and the blue line indicates a sagittal section. The LAA was reduced in size after catheter ablation (LAA volume, 17.8 to 10.3 mL; length of LAA neck, 23 to 14 mm), while the number of lobes () and fundamental morphology were maintained.
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Table 1. Baseline Characteristics and Echocardiographic Measurements
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Table 2. Baseline Characteristics According to Presence of LAA Thrombus
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Table 3. Univariate and Multivariate Analyses for Presence of LAA Thrombus
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Table 4. Clinical Characteristics and Echocardiographic Measurements in Patients Who Maintained Sinus Rhythm After Radiofrequency Catheter Ablation
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References
References
1.
Mügge A, Kühn H, Nikutta P, Grote J, Lopez JA, Daniel WG. Assessment of left atrial appendage function by biplane transesophageal echocardiography in patients with nonrheumatic atrial fibrillation: identification of a subgroup of patients at increased embolic risk. J Am Coll Cardiol. 1994;23:599–607.
Go to Citation
Crossref
PubMed
Google Scholar
2.
Fuster V, Ryden LE, Cannom DS, Crijns HJ, Curtis AB, Ellenbogen KA, Halperin JL, Le Heuzey JY, Kay GN, Lowe JE, Olsson SB, Prystowsky EN, Tamargo JL, Wann S, Smith SC, Jacobs AK, Adams CD, Anderson JL, Antman EM, Halperin JL, Hunt SA, Nishimura R, Ornato JP, Page RL, Riegel B, Priori SG, Blanc JJ, Budaj A, Camm AJ, Dean V, Deckers JW, Despres C, Dickstein K, Lekakis J, McGregor K, Metra M, Morais J, Osterspey A, Tamargo JL, Zamorano JL. ACC/GBPA/ESC 2006 Guidelines for the Management of Patients with Atrial Fibrillation: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and the European Society of Cardiology Committee for Practice Guidelines (Writing Committee to Revise the 2001 Guidelines for the Management of Patients with Atrial Fibrillation): developed in collaboration with the European Heart Rhythm Association and the Heart Rhythm Society. Circulation. 2006;114:e257–e354.
Crossref
PubMed
Google Scholar
a [...] thromboembolic risk for patients with AF.
b [...] of stroke in patients with AF.
3.
European Heart Rhythm Association, European Association for Cardio-Thoracic SurgeryCamm AJ, Kirchhof P, Lip GY, Schotten U, Savelieva I, Ernst S, Van Gelder IC, Al-Attar N, Hindricks G, Prendergast B, Heidbuchel H, Alfieri O, Angelini A, Atar D, Colonna P, De Caterina R, De Sutter J, Goette A, Gorenek B, Heldal M, Hohloser SH, Kolh P, Le Heuzey JY, Ponikowski P, Rutten FH. Guidelines for the management of atrial fibrillation: the Task Force for the Management of Atrial Fibrillation of the European Society of Cardiology (ESC). Eur Heart J. 2010;31:2369–2429.
Go to Citation
Crossref
PubMed
Google Scholar
4.
Zabalgoitia M, Halperin JL, Pearce LA, Blackshear JL, Asinger RW, Hart RG; Stroke Prevention in Atrial Fibrillation III Investigators. Transesophageal echocardiographic correlates of clinical risk of thromboembolism in nonvalvular atrial fibrillation. J Am Coll Cardiol. 1998;31:1622–1626.
Go to Citation
Crossref
PubMed
Google Scholar
5.
Atrial Fibrillation Investigators. Echocardiographic predictors of stroke in patients with atrial fibrillation: a prospective study of 1066 patients from 3 clinical trials. Arch Intern Med. 1998;158:1316–1320.
Go to Citation
Crossref
PubMed
Google Scholar
6.
Di Biase L, Santangeli P, Anselmino M, Mohanty P, Salvetti I, Gili S, Horton R, Sanchez JE, Bai R, Mohanty S, Pump A, Cereceda Brantes M, Gallinghouse GJ, Burkhardt JD, Cesarani F, Scaglione M, Natale A, Gaita F. Does the left atrial appendage morphology correlate with the risk of stroke in patients with atrial fibrillation? Results from a multicenter study. J Am Coll Cardiol. 2012;60:531–538.
Crossref
PubMed
Google Scholar
a [...] been well studied. Recently, Di Biase et al
b [...] have been used to assess LAA morphology,
c [...] diversity of LAA lobes and their geometry.
d [...] Di Biase et al
7.
Lacomis JM, Goitein O, Deible C, Moran PL, Mamone G, Madan S, Schwartzman D. Dynamic multidimensional imaging of the human left atrial appendage. Europace. 2007;9:1134–1140.
Crossref
PubMed
Google Scholar
a [...] have been used to assess LAA morphology,
b [...] diversity of LAA lobes and their geometry.
8.
Heist EK, Refaat M, Danik SB, Holmvang G, Ruskin JN, Mansour M. Analysis of the left atrial appendage by magnetic resonance angiography in patients with atrial fibrillation. Heart Rhythm. 2006;3:1313–1318.
Crossref
PubMed
Google Scholar
a [...] have been used to assess LAA morphology,
b [...] diversity of LAA lobes and their geometry.
9.
Shah SJ, Bardo DM, Sugeng L, Weinert L, Lodato JA, Knight BP, Lopez JJ, Lang RM. Real-time three-dimensional transesophageal echocardiography of the left atrial appendage: initial experience in the clinical setting. J Am Soc Echocardiogr. 2008;21:1362–1368.
Crossref
PubMed
Google Scholar
a [...] the potential to assess LAA morphology.
b [...] diversity of LAA lobes and their geometry.
10.
Agoston I, Xie T, Tiller FL, RGBPman AM, GBPmad M. Assessment of left atrial appendage by live three-dimensional echocardiography: early experience and comparison with transesophageal echocardiography. Echocardiography. 2006;23:127–132.
Crossref
PubMed
Google Scholar
a [...] the potential to assess LAA morphology.
b [...] diversity of LAA lobes and their geometry.
11.
Nakajima H, Seo Y, Ishizu T, Yamamoto M, Machino T, Harimura Y, Kawamura R, Sekiguchi Y, Tada H, Aonuma K. Analysis of the left atrial appendage by three-dimensional transesophageal echocardiography. Am J Cardiol. 2010;106:885–892.
Crossref
PubMed
Google Scholar
a [...] LAA lobes from reconstructed 3D LAA images.
b [...] parameters was <10% in our previous study.
12.
JCS Joint Working Group. Guidelines for pharmacotherapy of atrial fibrillation (JCS 2008): digest version. Circ J. 2010;74:2479–2500.
Go to Citation
Crossref
PubMed
Google Scholar
13.
Nosir YF, Vletter WB, Boersma E, Frowijn R, Ten Cate FJ, Fioretti PM, Roelandt JR. The apical long-axis rather than the two-chamber view should be used in combination with the four-chamber view for accurate assessment of left ventricular volumes and function. Eur Heart J. 1997;18:1175–1185.
Go to Citation
Crossref
PubMed
Google Scholar
14.
Lang RM, Bierig M, Devereux RB, Flachskampf FA, Foster E, Pellikka PA, Picard MH, Roman MJ, Seward J, Shanewise JS, Solomon SD, Spencer KT, Sutton MS, Stewart WJ; Chamber Quantification Writing Group; American Society of Echocardiography’s Guidelines and Standards Committee; European Association of Echocardiography. Recommendations for chamber quantification: a report from the American Society of Echocardiography’s Guidelines and Standards Committee and the Chamber Quantification Writing Group, developed in conjunction with the European Association of Echocardiography, a branch of the European Society of Cardiology. J Am Soc Echocardiogr. 2005;18:1440–1463.
Go to Citation
Crossref
PubMed
Google Scholar
15.
Vincelj J, Sokol I, Jaksić O. Prevalence and clinical significance of left atrial spontaneous echo contrast detected by transesophageal echocardiography. Echocardiography. 2002;19:319–324.
Go to Citation
Crossref
PubMed
Google Scholar
16.
Stroke Prevention in Atrial Fibrillation Investigators Committee on Echocardiography. Transesophageal echocardiography in atrial fibrillation: standards for acquisition and interpretation and assessment of interobserver variability. J Am Soc Echocardiogr. 1996;9:556–566.
Go to Citation
PubMed
Google Scholar
17.
Cabrera JA, Ho SY, Climent V, Sánchez-Quintana D. The architecture of the left lateral atrial wall: a particular anatomic region with implications for ablation of atrial fibrillation. Eur Heart J. 2008;29:356–362.
Go to Citation
Crossref
PubMed
Google Scholar
18.
Budge LP, Shaffer KM, Moorman JR, Lake DE, Ferguson JD, Mangrum JM. Analysis of in vivo left atrial appendage morphology in patients with atrial fibrillation: a direct comparison of transesophageal echocardiography, planar cardiac CT, and segmented three-dimensional cardiac CT. J Interv Card Electrophysiol. 2008;23:87–93.
Go to Citation
Crossref
PubMed
Google Scholar
19.
Veinot JP, Harrity PJ, Gentile F, Khandheria BK, Bailey KR, Eickholt JT, Seward JB, Tajik AJ, Edwards WD. Anatomy of the normal left atrial appendage: a quantitative study of age-related changes in 500 autopsy hearts: implications for echocardiographic examination. Circulation. 1997;96:3112–3115.
Crossref
PubMed
Google Scholar
a [...] based on the definitions by Veinot et al
b [...] diversity of LAA lobes and their geometry.
c [...] of hearts), 3 (23%), 1 (20%), and 4 (3%).
20.
Boos CJ, Anderson RA, Lip GY. Is atrial fibrillation an inflammatory disorder? Eur Heart J. 2006;27:136–149.
Go to Citation
Crossref
PubMed
Google Scholar
21.
Inoue S, Murakami Y, Sano K, Katoh H, Shimada T. Atrium as a source of brain natriuretic polypeptide in patients with atrial fibrillation. J Card Fail. 2000;6:92–96.
Go to Citation
Crossref
PubMed
Google Scholar
22.
Shimizu H. High plasma brain natriuretic polypeptide level as a marker of risk for thromboembolism in patients with nonvalvular atrial fibrillation. Stroke. 2002;33:1005–1010.
Go to Citation
Crossref
PubMed
Google Scholar
23.
Agmon Y, Khandheria BK, Gentile F, Seward JB. Echocardiographic assessment of the left atrial appendage. J Am Coll Cardiol. 1999;34:1867–1877.
Crossref
PubMed
Google Scholar
24.
Psychari SN, Apostolou TS, Sinos L, Hamodraka E, Liakos G, Kremastinos DT. Relation of elevated C-reactive protein and interleukin-6 levels to left atrial size and duration of episodes in patients with atrial fibrillation. Am J Cardiol. 2005;95:764–767.
Go to Citation
Crossref
PubMed
Google Scholar
25.
Conway DS, Buggins P, Hughes E, Lip GY. Prognostic significance of raised plasma levels of interleukin-6 and C-reactive protein in atrial fibrillation. Am Heart J. 2004;148:462–466.
Go to Citation
Crossref
PubMed
Google Scholar
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12018 | https://archive.org/details/schaumsoutlineof0000lips_y8t2 | Schaum's outline of theory and problems of discrete mathematics : Lipschutz, Seymour : Free Download, Borrow, and Streaming : Internet Archive
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Schaum's outline of theory and problems of discrete mathematics
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Schaum's outline of theory and problems of discrete mathematics
byLipschutz, Seymour
Publication date2007TopicsAlgebra, Abstract -- Outlines, syllabi, etc, Combinatorial analysis -- Outlines, syllabi, etc, Logic, Symbolic and mathematical -- Outlines, syllabi, etc, Algebra, Abstract, Combinatorial analysis, Logic, Symbolic and mathematical, Diskrete MathematikPublisher New York : McGraw-Hill Collectioninternetarchivebooks; inlibrary; printdisabledContributorInternet ArchiveLanguageEnglishItem Size 1.1G
xi, 474 pages : 28 cm
This edition features hundreds of fully-solved problems, contains new chapters on computer arithmetic and cryptology, and covers all course fundamentals, therefore supplementing any course text
Includes index
Set theory -- Relations -- Functions and algorithms -- Logic and propositional calculus -- Techniques of counting -- Advanced counting techniques, recursion -- Probability -- Graph theory -- Directed graphs -- Binary trees -- Properties of the integers -- Languages, automata, grammars -- Finite state machines and turing machines -- Ordered sets and lattices -- Boolean algebra -- Vectors and matrices -- Algebraic systems
Access-restricted-item true Addeddate 2022-02-10 15:07:56 Associated-names Lipson, Marc Bookplateleaf 0004 Boxid IA40359907 Camera USB PTP Class Camera Collection_set printdisabled External-identifierurn:oclc:record:1311131488
urn:lcp:schaumsoutlineof0000lips_y8t2:lcpdf:85e8c1d0-ae93-4b28-8f78-74f54c650bbf
urn:lcp:schaumsoutlineof0000lips_y8t2:epub:d9d454bd-3e4f-4b6b-8459-854b6f7523aaFoldoutcount 0 Identifier schaumsoutlineof0000lips_y8t2 Identifier-ark ark:/13960/s2gbbbhc69x Invoice 1652 Isbn 9780071470384
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12019 | https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119066767.ch24 | Tissue and blood protozoa
Frank E. Berkowitz,
Frank E. Berkowitz
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Robert C. Jerris,
Robert C. Jerris
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Book Author(s):Frank E. Berkowitz,
Frank E. Berkowitz
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Robert C. Jerris,
Robert C. Jerris
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First published: 18 December 2015
Summary
The tissue and blood protozoa include the apicomplexa (Plasmodium spp., Babesia spp., and Toxoplasma gondii), the flagellates (Leishmania spp., Trypanosoma spp., and Trichomonas vaginalis), and the free-living amebae (Naegleria fowleri, Acanthamoeba spp., Balamuthia madrillaris, and Sappinia spp.). Sarcocystis spp., which can invade tissue in humans, are considered with the gut protozoa. Plasmodium causes malaria. Plasmodia are endemic in many tropical areas and some temperate areas. They are transmitted by the bite of a female mosquito of the genus Anopheles. Hemoflagellates, which infect blood and tissues other than the intestine, belong to the order Kinetoplastida. They are transmitted by arthropod vectors. Trichomonas vaginalis is a flagellate that causes trichomoniasis, a common sexually transmitted infection causing symptoms mainly in females. There are four known genera of free-living amebae that can cause invasive disease in animals and humans: Acanthamoeba, Balamuthia, Naegleria, and Sappinia.
Further reading
Bern C (2011) Antitrypanosomal therapy for chronic Chagas’ disease. N Engl J Med 364: 2527–34.
10.1056/NEJMct1014204
CAS PubMed Web of Science® Google Scholar
Boelaert M, Verdonck K, Meaten J, et al. (2014) Rapid tests for the diagnosis of visceral leishmaniasis in patients with suspected disease. Cochrane Database Syst Rev 6: CD009135.
PubMed Web of Science® Google Scholar
Brun R, Blum J, Chappuis F, et al. (2010) Human African trypanosomiasis. Lancet 375: 148–59.
10.1016/S0140-6736(09)60829-1
PubMed Web of Science® Google Scholar
Chappuis F, Loutan L, Simarro P, et al. (2005) Options for field diagnosis of human African trypanosomiasis. Clin Microbiol Rev 18: 133–46.
10.1128/CMR.18.1.133-146.2005
PubMed Web of Science® Google Scholar
Crawley J, Chu C, Mtove G, et al. (2010) Malaria in children. Lancet 375: 1468–81.
10.1016/S0140-6736(10)60447-3
PubMed Web of Science® Google Scholar
Elmahallawy EK, Martinez AS, Rodriquez-Granger J, et al. (2014) Diagnosis of leishmaniasis. J Infect Dev Ctries 8: 961–72.
10.3855/jidc.4310
PubMed Web of Science® Google Scholar
Fox MC, Lakdawala N, Miller AL, et al. (2013) A patient with syncope. N Engl J Med 369: 966–72.
10.1056/NEJMcps1300093
CAS PubMed Web of Science® Google Scholar
Hunfeld KP, Hildebrand A, Gray JS (2008) Babesiosis: recent insights into an ancient disease. Int J Parasitol 38: 1219–37.
10.1016/j.ijpara.2008.03.001
CAS PubMed Web of Science® Google Scholar
Huppert JS (2009) Trichomoniasis in teens: an update. Curr Opin Obstet Gynecol 21: 371–8.
10.1097/GCO.0b013e32832e0827
PubMed Web of Science® Google Scholar
Kantele A, Jokiranta TS (2011) Review of cases with the emerging fifth human malaria parasite, Plasmodium knowlesi
. Clin Infect Dis 52: 1356–62.
10.1093/cid/cir180
PubMed Web of Science® Google Scholar
Lescure FX, LeLoup G, Freilij H, et al. (2010) Chagas disease: changes in knowledge and management. Lancet Infect Dis 10: 556–70.
10.1016/S1473-3099(10)70098-0
PubMed Web of Science® Google Scholar
McAuley JB (2008) Toxoplasmosis in children. Pediatr Infect Dis J 27: 161–2.
10.1097/INF.0b013e3181658abb
PubMed Web of Science® Google Scholar
Montoya JG, Liesenfeld O (2004) Toxoplasmosis. Lancet 363: 1965–76.
10.1016/S0140-6736(04)16412-X
CAS PubMed Web of Science® Google Scholar
Murray HW, Berman JD, Davies CR, et al. (2005) Advances in leishmaniasis. Lancet 366: 1561–77.
10.1016/S0140-6736(05)67629-5
CAS PubMed Web of Science® Google Scholar
Opsteegh M, Kortbeek TM, Havelaar AH, et al. (2015) Intervention strategies to reduce human Toxoplasma gondii disease burden. Clin Infect Dis 60: 101–7.
10.1093/cid/ciu721
PubMed Web of Science® Google Scholar
Petersen E (2007) Toxoplasmosis. Semin Fetal Neonat Med 12: 214–23.
10.1016/j.siny.2007.01.011
PubMed Web of Science® Google Scholar
Rassi A Jr, Rassi A, Marin-Neto JA (2010) Chagas disease. Lancet 375: 1388–402.
10.1016/S0140-6736(10)60061-X
PubMed Web of Science® Google Scholar
Rosenblatt JE (2009) Laboratory diagnosis of infections due to blood and tissue parasites. Clin Infect Dis 49: 1103–8.
10.1086/605574
PubMed Web of Science® Google Scholar
Travassos MA, Laufer MK (2009) Resistance to antimalarial drugs: molecular, pharmacologic, and clinical considerations. Pediatr Res 65(5 pt 2): 64R–70R.
10.1203/PDR.0b013e3181a0977e
PubMed Web of Science® Google Scholar
Vannier E, Krause PJ (2012) Human babesiosis. N Engl J Med 366: 2397–407.
10.1056/NEJMra1202018
CAS PubMed Web of Science® Google Scholar
Vannier E, Gewurz BE, Krause PJ (2008) Human babesiosis. Infect Dis Clin North Am 22: 469–88.
10.1016/j.idc.2008.03.010
PubMed Web of Science® Google Scholar
Visvesvara GS (2010) Amebic meningoencephalitis and keratitis: challenges in diagnosis and treatment. Curr Opin Infect Dis 23: 590–4.
10.1097/QCO.0b013e32833ed78b
CAS PubMed Web of Science® Google Scholar
Visvesvara GS, Moura H, Schuster FL (2007) Pathogenic and opportunistic free-living amoebae: Acanthamoeba spp., Balamuthia mandrillaris, Naegleria fowleri, and Sappinia diploidea. FEMS Immunol Med Microbiol 50: 1–26.
10.1111/j.1574-695X.2007.00232.x
CAS PubMed Web of Science® Google Scholar
White NJ, Pukrittayakamee S, Hien TT, et al. (2014) Malaria. Lancet 383: 723–35.
10.1016/S0140-6736(13)60024-0
updates
This document has been updated
Click for further information.
PubMed Web of Science® Google Scholar
Wilson ML (2013) Laboratory diagnosis of malaria. Arch Pathol Lab Med 137: 805–11.
10.5858/arpa.2011-0602-RA
PubMed Web of Science® Google Scholar
Yager PH, Luginbuhl LM, Dekker JP (2014) Case 6-2014: a 35-day-old boy with fever, vomiting, mottled skin, and severe anemia. N Engl J Med 370: 753–62.
10.1056/NEJMcpc1208155
CAS PubMed Web of Science® Google Scholar
Practical Medical Microbiology for Clinicians
## References
## Related
## Information
Recommended
Blood and Tissue Protozoa
Cases in Human Parasitology,
Intestinal protozoa
Practical Medical Microbiology for Clinicians,
Section: Parasite Classification and Relevant Body Sites
Practical Guide to Diagnostic Parasitology,
Protozoan Pathogens of Humans
Heinz Mehlhorn,
Encyclopedia of Life Sciences,
Plasmodium
Irwin W Sherman,
Encyclopedia of Life Sciences,
Details
Copyright © 2016 John Wiley & Sons, Inc.
Keywords
amebae
apicomplexa
blood protozoa
flagellates
tissue protozoa
Publication History
18 December 2015
23 February 2016
ISBN Information
Online ISBN: 9781119066767
Print ISBN: 9781119066743
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12020 | https://math.stackexchange.com/questions/1356027/permutations-without-repetitions-exclude-repeated-permutations | Skip to main content
Permutations without repetitions (exclude repeated permutations) [duplicate]
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The formula to calculate all permutations without repetitions of the set {1,2,3} is n!(n−r)! But how to calculate it if the set (or rather array in programming terms) includes repeated numbers {1,2,2,3,3} so that you don't add up same permutations?
combinatorics
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edited Jul 10, 2015 at 5:55
Arty
asked Jul 10, 2015 at 5:31
ArtyArty
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It is not clear to me what exactly you want. You talk of permutations but use the combination formula !
true blue anil
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true blue anil
2015-07-10 05:54:07 +00:00
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Jul 10, 2015 at 5:54
you are right, fixed
Arty
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Arty
2015-07-10 05:55:37 +00:00
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Jul 10, 2015 at 5:55
1
There is r in your formula, but there isn't anything that seems to be r in your question.
Marc van Leeuwen
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Marc van Leeuwen
2015-07-10 06:57:40 +00:00
Commented
Jul 10, 2015 at 6:57
What about filtering only distinct values in the array and then count permutations? This seems to me like the fastest and the simplest solution. Otherwise you'd presumably have to count how many times each element is repeated.
Sergiu
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Sergiu
2017-03-15 14:05:23 +00:00
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Mar 15, 2017 at 14:05
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First, to clear something up: the formula n!r!(n−r)! is for a combination (i.e., the number of ways you can choose r elements from a set of n disregarding order), not a permutation. The formula for the number of permutations of a set with n elements is simply n!.
Now, let's say we have your set {1,21,22,31,32}. It has 5 elements, so there are 5!=120 ways of permuting it − or there would be, if we could tell every element apart. However, we don't actually want to consider two orderings different if they only differ by the swapping around of 21 and 22 and/or 31 and 32. So first let's say that we only care about the sequences where 21 comes before 22. Then we can throw away half of our 120 permutations (because every permutation that has 21 before 22 comes with a partner that is the same except with the two swapped), so we're down to 60 permutations. We can use the same logic with 31 and 32 to cut our answer down to 30 unique permutations, which is our final answer.
In general, if we have a set {a1,a2,…,an}, for each group of k indistinguishable elements, we need to divide our initial result of n! permutations by k! (which you can justify using the same "ordering" argument as presented in the paragraph above).
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answered Jul 10, 2015 at 6:03
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If the set includes repeated terms it is actually not that different.
Take for example you want to find all different combinations of 'aabbcc'. What you would do is 6!2!2!2!, the numerator comes from the number of letters in 'aabbcc' and the denominator comes from all the repeated letters (we have 2 a's, 2 b's, and 2 c's).
You can do the same for numbers. To calculate the number of different permutations of {1,1,2,2,3,3}. It is just 6!2!2!2!, which is 90 different permutations.
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answered Jul 10, 2015 at 5:45
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There is a standard formula to deal with such cases.
If n total objects have k1 of one kind, k2 of another, ......
number of permutations = n!k1!⋅k2!...
For your particular case, for instance, it would be 5!1!2!2!
The 1's in the denominator can conveniently be left out. so 5!2!2!
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answered Jul 10, 2015 at 6:05
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7 Days
Spring, Summer, Fall, Winter
Dali
Dali
AboutItineraryHotelsPhotosDetailsReviews
About this Journey
As chronicled in Travels Through Dali: With a Leg of Ham by WildChina founder Mei Zhang, Yunnan — at China’s southernmost tip — is a cradle of cultural and culinary diversity. This food tour traces a route through its storied heartlands, beginning in Dali and continuing to the historic towns of Weishan and Shaxi.
Along the way, taste the region’s celebrated cuisine — from famed ham and rare mountain herbs to little-known cheese-making traditions and a rich variety of mushrooms. Wander Bai minority villages, visit ancient market towns on the Tea and Horse Road, and meet the people who have kept these traditions alive for centuries.
What we love
Catching the sunrise over Erhai Lake as fishermen set out with their cormorants
Learning the Bai people’s centuries-old tie-dye techniques in Zhoucheng Village
Sipping green tea with a Taoist master on sacred Weibao Mountain
Map
Itinerary
Day 1 Arrive in Dali
First Stop: Dali, Yunnan Province
Yunnan Province — whose name literally means “South of the Clouds” due to its location just south of the Tibetan Plateau — is home to some of the most diverse cultures, ecology, and terrain in China. Verdant valleys, snow-capped mountains, rustic towns, and a lively mix of ethnic minority communities have long inspired poets and travelers alike.
Tucked between Cang Mountain and Erhai Lake, Dali is a Ming-era town that has preserved much of its historical character. Once the chief city of Yunnan and the capital of the Nanzhao Kingdom — an empire that, at its peak, stood up to the imperial Chinese armies — it remains deeply connected to its roots. The Bai ethnic minority, who formed the backbone of the Nanzhao Kingdom, still make up the majority of Dali’s population and those of the surrounding villages.
Your Journey Begins
Today is the day, the start of your WildChina adventure. Step off the plane in Dali and meet your local WildChina guide in the airport arrivals hall. They’ll be waiting to welcome you and will have your private vehicle on standby, ready to whisk you off to your hotel for check in.
Welcome Dinner in Dali
Join your fellow guests for a Welcome Dinner. During dinner, you’ll get to know each other, your WildChina guide, and learn more about the journey ahead. Your guide will give a short lecture on the history of Dali and introduce you to the local cultures and customs that we’ll see throughout the trip.
Meals included:dinner
Day 2 Dali
Sunrise at Erhai Lake
Nestled between mountains at 2,000 meters above sea level, the shimmering expanse of Erhai Lake is so vast it is known in Chinese as a “sea.” A vital source of fish for the Bai people — who still practice cormorant fishing — Erhai is especially enchanting at sunrise, when the first rays of light turn its reflective waters into a palette of rich, shifting hues. Begin the day early and watch as life stirs along the lakeshore.
Flower Cake Lesson
At an organic rose plantation tucked beneath the protective slopes of Mount Cangshan, learn the art of making Dali’s signature flower cakes. From May to November, the fields are in full bloom, and you can pick petals straight from the stems; at other times of year, explore how Dali’s unique climate nurtures its roses. Then, under expert guidance, transform the petals into fragrant cakes — shaping, baking, and tasting your creation in a hands-on, two-hour masterclass.
Bai Tea Ceremony
Step into the world of the Bai Tea Ceremony, a time-honored ritual performed at weddings and festivals, guided by strict etiquette. Water is boiled and tea steeped in a clay pot before you, then served in three distinct brews — bitter, sour, and sweet — a symbolic journey through life’s stages from hardship to happiness.
Zhoucheng Village
At the base of Cangshan Mountain lies Zhoucheng, a vibrant artisan village renowned for its intricate Bai tie-dye. Here, work alongside local craftspeople, using indigo dyes, wooden sticks, and rubber bands to create striking patterns on white cotton — the same techniques passed down for generations.
Meals included:breakfast, lunch, and dinner
Day 3 Dali - Weishan
Drive to Weishan
The ride from Dali to Weishan will take around 2 hours.
On the Way: Donglianhua Muslim Village
While Islamic culture is often associated with China’s arid northwest, the fertile valleys of western Yunnan are home to a beautifully preserved Muslim village. Its residents descend from soldiers who settled here after the Mongol army conquered the region over 700 years ago. Stroll the lanes, meet traditionally dressed locals, and visit Yunnan’s largest mosque, where warm welcomes are extended to the village’s rare international guests.
Weishan Old Town
Amid the rapid development seen in much of Yunnan, Weishan Old Town retains its authentic charm. Once a cozy outpost along the ancient Tea and Horse Road, its architecture and artisan traditions remain much as they were centuries ago.
Meals included:breakfast, lunch, and dinner
Day 4 Weishan
Visit Weishan’s Famous Pickle Empire
For decades, Yin Family Pickles has ruled the roost when it comes to pickled vegetables. From daikon radish and black beans in winter to broad beans and string beans in summer, the family business has a recipe for every season. Go behind the scenes to discover what it takes to keep the pickle venture booming, gazing upon rows of jars working their magic and family members busily snipping away at colorful vegetables (and, of course, tasting). You may even be lucky enough to happen upon the pickle empress herself: the entrepreneurial octogenarian Madame Chen.
One Noodle Soup
Today you’ll get a taste of the town’s history by learning the process of making “one noodle soup”, and then devour your very own bowl.
Meditate on Weibao Mountain
Step back in time on Weibaoshan, a sacred mountain dotted with ancient Taoist temples, just 70 kilometers from Dali and virtually untouched by modern tourism. Enjoy a moderate hike up the picturesque mountain (2 – 2.5 hrs) before finding tranquility at a temple atop the mountain. Here, sip green tea with a Taoist master as he unveils the mysteries of his religion, before finding out how to balance your yin and yang with a round of tai chi in the spiritual grounds of a temple hidden deep within the forest. Hike the 45 minutes back to your private transport or, for the hiking enthusiasts who want to immerse further in the area’s pristine nature, complete the five-hour return journey to the summit and back.
Meals included:breakfast, lunch, and dinner
Day 5 Weishan - Shaxi
Drive to Shaxi
The ride from Weishan to Shaxi will take around 3 hours.
Next Stop: Shaxi
Shaxi is a beautifully preserved market town set in the tranquil countryside between Lijiang and Dali. Once a bustling outpost on the Tea and Horse Road, it was a key stop where horses, tea, and other goods were traded en route to Burma and Tibet. Today, its Buddhist grottoes, lively weekend markets, and simple yet flavorful rural cuisine — sourced straight from surrounding farms — offer a rich blend of cultural heritage and pastoral charm.
Make Authentic Shaxi Cheese
Step into the home of a local cheesemaker and discover how the region developed its own cheese-making tradition, a legacy of earlier Mongolian influence. Watch as milk is steamed over a wood-fired stove before the curds are separated from the whey, then cool and shape your own snack. As you wait, enjoy a conversation with your host and learn about daily life in Shaxi.
Meals included:breakfast, lunch, and dinner
Day 6 Shaxi
haxi Market
Friday is market day in Shaxi. Each week, Yi, Bai, Lisu, Naxi, Hui, and Tibetan communities from the surrounding mountains gather to buy and sell produce, livestock, and handicrafts. Many arrive in traditional, brightly colored clothing, carrying large baskets or leading pack horses and mules through the narrow lanes toward the town’s central square.
Hike Through Shibaoshan
The 1,300-year-old Shibaoshan Grottoes feature remarkable rock carvings that illustrate the spread of Mahayana Buddhism along the Tea and Horse Caravan Trail from Tibet into Yunnan. On a 90-minute hike, explore each of the five grottoes, discovering new statues at every stop. The route ends at the Stone Bell Temple, its curved roofs and red walls dramatically perched against the mountainside.
Stone Dragon Village
In Stone Dragon Village, “Those who can walk, dance; and those who can talk, sing.” Meet the region’s Folk Music King and enjoy a captivating performance — a blend of longing, passion, and exuberance that reflects the village’s spirited traditions.
Bai Feast
Bai cuisine blends Buddhist and Han influences, balancing spicy, sour, and sweet flavors with hearty stews, peppered ham, and sizzling hot pots. Gather for a traditional meal while learning about Bai customs and their cultural connections to the Tea and Horse Road.
Meals included:breakfast, lunch, and dinner
Day 7 Shaxi to Dali for Departure
Drive to Dali Airport
The ride from Shaxi to Dali airport will take around 2 hours.
Your Flight Back Home
Your WildChina guide will escort you to the airport and help you check in for your flight home.
Meals included:breakfast
Hotels
Sky Valley Heritage Boutique Hotel, Dali
Boutique
This boutique hotel in the quiet town of Xizhou, 20 km from Dali, sits among well-preserved Bai architecture. Rooms feature antique furnishings with mountain or courtyard views, while spacious suites suit couples and families.
The Dawn of Yihe Luxury Hotel, Weishan
Boutique
Set in Weishan, once the cultural center of the Nanzhao Kingdom, this boutique hotel blends Ming Dynasty architecture with minimalist interiors. Surrounded by bamboo and historic buildings, it offers a spa, gym, and gardens for a serene retreat on the Tea-Horse Road.
Sunyata Hotel, Shaxi
Boutique
Near Shaxi Old Town, this 20-room boutique blends Bai design with modern Chinese garden style. Its courtyards and warm, natural interiors offer a comfortable retreat, with your guide available to assist as English ability may vary.
Photos
Details
What’s Included
Services of an English-speaking WildChina local guide
Accommodation based on double occupancy with breakfasts included
All overland transfers with private chauffeur and in-transport refreshments
Private chauffeur service to and from airport at start and end of your trip
All admission fees and activity expenses, as noted in the itinerary
Meals as noted in the itinerary with complimentary drinking water
What’s Excluded
International and domestic flights, domestic trains, plus relevant taxes
Chinese tourist visa, which is required for many foreign passport holders
Travel and medical insurance
Meals, apart from those included in the itinerary, and alcohol
Expenses of a personal nature
Excursions and activities not included in the itinerary
Discretionary gratuities for guides and drivers
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12022 | https://www.chempap.org/file_access.php?file=596ba500.pdf | Dimensionless Characteristics of Centrifugal Pump P. TIMÁR Department of Chemical and Biochemical Engineering, Faculty of Chemical and Food Technology, Slovak University of Technology, SK-812 37 Bratislava e-mail: pavel.timar@stuba.sk Received 1 April 2005 Centrifugal pump characteristics (i.e. specific energy gH, power P, and efficiency η curves plotted vs. flow capacity Q) at different pump operating speeds are important to the successful operation of pumps in various areas of chemical technology. In the present paper the pump characteristics are treated in a dimensionless form. The main advantage of this approach is the fact that only one characteristic curve is used for all possible operating speeds of the tested pump in the frame of suitable dimensionless numbers. Also, the obtained relation between dimensionless numbers allows to estimate the main pump characteristics in the case when the properties (ρ and µ) of the liquid were varied.
Dimensional analysis is a useful and powerful tool applied mainly for investigation of complex problems.
In recent years, Zlokarnik , Szirtes , and Baren-blatt used this approach for solving various techni-cal or physical problems.
According to the authors, first it is necessary to determine all variables connected with the problem solved. This process is finished with establishing the dependent variables. In order to complete the dimen-sional analysis successfully, the most important condi-tion is to find all variables influencing the system stud-ied (neither exceeding nor missing variables). Based on the list of influencing variables, it is possible to con-struct the so-called dimensional matrix . This ma-trix is divided into two parts: quadratic core matrix and residual matrix. By rearranging the core matrix to the unity matrix and using the same arithmetic oper-ations for the residual matrix, one gets dimensionless numbers corresponding to the system studied.
During determination of relevant variables one has to keep in mind that only one dependent variable can be chosen for each individual problem. In the case of evaluation of centrifugal pump characteristics, the chosen variables are the specific energy of pump (gH) and the pump efficiency (η). Then, all independent variables influencing the target variable should be de-termined and the so-called relevance list of variables must be formed. For the chosen centrifugal pump the following independent variables are given in the rel-evance list: n – rotational speed of the pump, Q – volumetric flow rate, ρ – density, and µ – dynamic viscosity of the pumped liquid. Besides this, different geometric quantities of the pump should be taken into account: D – diameter of the pump impeller, L1, L2, . . ., Ln – various length scales describing the pump geometry.
It is clear that only one length parameter will be important in a future analysis, because all other lengths will create simplexes of geometrical similarity.
Hence, diameter of the pump impeller D was chosen as a characteristic length in this case. Mathematical for-mulation of the functional dependence between target variable and independent variables for the centrifugal pump is given by the following equations F (gH, Q, µ, ρ, D, n) = 0 (1) G (η, Q, µ, ρ, D, n) = 0 (2) Based on the aforementioned, the dimensional ma-trix consisting of a core and a residual matrix can be expressed as ρ D n m/kg 1 0 0 l/m −3 1 0 t/s 0 0 −1 core matrix gH Q µ 0 0 1 2 3 −1 −2 −1 −1 residual matrix (3) where m is the mass, l the length, and t the time.
It should be noted here that each variable involved in the residual matrix would occur in the resulting di-mensionless numbers separately. If, for example, the rotational speed of a pump impeller n will be in-volved only in one dimensionless variable, it is needful Presented at the 32nd International Conference of the Slovak Society of Chemical Engineering, Tatranské Matliare, 23—27 May 2005.
500 Chem. Pap. 59(6b)500—503 (2005) DIMENSIONLESS CHARACTERISTICS OF CENTRIFUGAL PUMP to interchange column n by column µ in the dimen-sional matrix (3). The dimensional matrix reveals that for 6 dimensional variables only 3 dimensions (mass, length, and time) exist. Then, maximally 3 indepen-dent dimensionless variables may be obtained. To ob-tain these dimensionless numbers it is necessary to recalculate the above-mentioned matrix (3) by linear operations, the result of which is to transform the core matrix to the unity matrix. By doing so one gets ρ D n m/kg 1 0 0 l/m 0 1 0 t/s 0 0 1 core matrix gH Q µ 0 0 1 2 3 2 2 1 1 residual matrix (4) Thus, according to the method of author fol-lowing dimensionless numbers resulting from the di-mensional analysis may (matrix (4)) now be defined π1 = gH n2D2 (5) π2 = Q nD3 (6) π3 = µ ρnD2 (7) The dimensionless numbers defined by eqs (5) and (6) are the so-called head coefficient and the flow co-efficient of the centrifugal pump, respectively. The re-ciprocal value of the remaining dimensionless number π3 is in the literature [1, 2] the well-known Reynolds number.
Following functional relations between dimension-less variables may subsequently be written π1 = F1 π2, π3, L1 D , L2 D , L3 D , · · · , Ln D (8) η = F2 π2, π3, L1 D , L2 D , L3 D , · · · , Ln D (9) If only one centrifugal pump or a series of geometri-cally similar pumps are taken into account (geometri-cal simplexes are the same), the geometrical simplexes L1 D , L2 D , L3 D , · · · , Ln D remain constant and can be ex-cluded from the relevance list, because their equality is fulfilled automatically. Then, simple relationships between dimensionless numbers are obtained π1 = gH n2D2 = F1 Q nD3 , Re (10) η = F2 Q nD3 , Re (11) The final form of functions F1 and F2 can be ob-tained only on the basis of experimental measure-ments.
EXPERIMENTAL Experiments have been carried out with a labora-tory one-stage centrifugal pump having the impeller diameter D = 220 mm. This equipment was used to deliver water under laboratory conditions (tempera-ture 18◦C). The centrifugal pump was equipped with a drive, which was able to change continuously the ro-tational speed of the pump. The volumetric flow rate was controlled by a valve and measured by a rotame-ter. A differential mercury manometer was used to measure the pressure difference between output and input sides of the centrifugal pump. The rotational speed corresponded to the speed of electric motor and during experiments was changed from 1200 min−1 to 2000 min−1. All measured data were transformed into the form of dimensionless numbers according to eqns (5—7).
RESULTS AND DISCUSSION Fig. 1 is the basic characteristic of the centrifu-gal pump: the curve head on dimensionless number π1 connected with specific energy of centrifugal pump vs. dimensionless number π2 including liquid flow rate.
We can see that in the space of dimensionless variables defined by eqns (5) and (6) only one curve exists for all tested rotational speeds n. Variation of the pump im-peller rotational speed caused changes of the Reynolds number (7) ranging from 410 000 to 680 000. These changes have no influence on the pump characteristic depicted in Fig. 1.
A similar picture can be seen for the variation of the pump efficiency with the volumetric flow rate of the liquid. This dependence is shown in Fig. 2 in the frame of derived dimensionless numbers. Only one curve for all measurements of the pump efficiency at any rotational speed (Reynolds number) exists. This Fig. 1. Head coefficient as a function of the flow coefficient for centrifugal pump.
2000 min−1, Re = 681 800; 1800 min−1, Re = 613 600; ▲1600 min−1, Re = 545 500; ◦1400 min−1, Re = 477 300; 1200 min−1, Re = 409 100.
Chem. Pap. 59(6b)500—503 (2005) 501 P. TIMÁR Fig. 2. Efficiency as a function of the flow coefficient for cen-trifugal pump.
2000 min−1, Re = 681 800; 1800 min−1, Re = 613 600; ▲1600 min−1, Re = 545 500; ◦1400 min−1, Re = 477 300; 1200 min−1, Re = 409 100.
Fig. 3. Output power as a function of the flow coefficient for centrifugal pump.
2000 min−1, Re = 681 800; 1800 min−1, Re = 613 600; ▲1600 min−1, Re = 545 500; ◦1400 min−1, Re = 477 300; 1200 min−1, Re = 409 100.
conclusion, however, can be drawn only for the range of measured Reynolds numbers mentioned above. The maximum efficiency of the tested pump was approx-imately 50 %. At this value the flow coefficient was about 0.015.
By multiplying the dimensionless criteria π1 and π2, one obtains the so-called power coefficient of the pump, a further useful characteristic of the centrifugal pump. This new dimensionless variable is known as the Newton number, Ne [1, 2].
π1 · π2 = Ne = gH n2D2 Q nD3 ρ ρ = = P ρn3D5 = F3 Q nD3 , Re (12) The plot of the Newton number vs. the flow coeffi-cient is drawn in Fig. 3. From this figure it is evident that the Newton number is a function of the flow coef-ficient and the drawn curve exhibits a maximum sim-Fig. 4. Head coefficient as a function of the liquid viscosity.
2000 min−1, Re = 681 800; 1800 min−1, Re = 613 600; ▲1600 min−1, Re = 545 500; ◦1400 min−1, Re = 477 300; 1200 min−1, Re = 409 100.
ilar to that of the efficiency curve depicted in Fig. 2.
The maximum power output of the tested pump is lo-cated between the flow coefficient values from 0.015 to 0.020 for the whole range of rotational speeds of pump impeller. Up to this maximum point the dependence between dimensionless numbers is linear.
To estimate the pump characteristics for the trans-port of another liquid it is useful to rearrange the de-rived dimensionless numbers in such a way that the properties of the liquid are present explicitly in the dimensionless number with the flow rate of the liquid.
For this purpose, a product of pump flow coefficient defined by eqn (6) and the Reynolds number is suit-able Q nD3 ρnD2 µ = Qρ µD = Q νD = πLIQUID (13) Variation of the head coefficient (5) and the new dimensionless number πLIQUID defined by eqn (13) is depicted in Fig. 4. Estimation of pump characteristics for a new delivered liquid is possible, if the kinematic viscosity of liquid is known.
SYMBOLS D diameter of the pump impeller m F, G function gH specific energy of the centrifugal pump m2 s−2 L1, L2, . . . , Ln length scales describing the pump geometry m n rotational speed of the pump impeller s−1 P output power of the pump kg m2 s−3 Q volumetric flow rate of the liquid m3 s−1 Re Reynolds number (= (ρnD2)/µ, eqn (7)) Ne Newton number (= P/(ρn3D5), eqn (12)) Greek Letters ρ density of the liquid kg m−3 µ dynamic viscosity of the liquid kg m−1 s−1 502 Chem. Pap. 59(6b)500—503 (2005) DIMENSIONLESS CHARACTERISTICS OF CENTRIFUGAL PUMP ν kinematic viscosity m2 s−1 η pump efficiency % π dimensionless number REFERENCES 1. Zlokarnik, M., Scale-Up in Chemical Engineering.
Wiley—VCH, Weinheim, 2002.
2. Szirtes, T., Applied Dimensional Analysis and Model-ing. McGraw-Hill, New York, 1997.
3. Barenblatt, G. I., Dimensional Analysis and Interme-diate Asymptotics. Cambridge University Press, Cam-bridge, 1996.
Chem. Pap. 59(6b)500—503 (2005) 503 |
12023 | https://tutorial.math.lamar.edu/extras/commonerrors/algebraerrors.aspx | Paul's Online Notes
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2. Algebra Errors
The topics covered here are errors that students often make in doing algebra, and not just errors typically made in an algebra class. I’ve seen every one of these mistakes made by students in all level of classes, from algebra classes up to senior level math classes! In fact, a few of the examples in this section will actually come from calculus.
If you have not had calculus you can ignore these examples. In every case where I’ve given examples I’ve tried to include examples from an algebra class as well as the occasion example from upper level courses like Calculus.
I’m convinced that many of the mistakes given here are caused by people getting lazy or getting in a hurry and not paying attention to what they’re doing. By slowing down, paying attention to what you’re doing and paying attention to proper notation you can avoid the vast majority of these mistakes!
Division by Zero
Everyone knows that 02=0 the problem is that far too many people also say that 20=0 or 20=2! Remember that division by zero is undefined! You simply cannot divide by zero so don’t do it!
Here is a very good example of the kinds of havoc that can arise when you divide by zero. See if you can find the mistake that I made in the work below.
a=bWe’ll start assuming this to be true.
ab=a2Multiply both sides by a.
ab−b2=a2−b2Subtract b2 from both sides.
b(a−b)=(a+b)(a−b)Factor both sides.
b=a+bDivide both sides by a−b.
b=2bRecall we started off assuming a=b.
1=2Divide both sides by b.
So, we’ve managed to prove that 1 = 2! Now, we know that’s not true so clearly we made a mistake somewhere. Can you see where the mistake was made?
The mistake was in step 5. Recall that we started out with the assumption a=b. However, if this is true then we have a−b=0! So, in step 5 we are really dividing by zero!
That simple mistake led us to something that we knew wasn’t true, however, in most cases your answer will not obviously be wrong. It will not always be clear that you are dividing by zero, as was the case in this example. You need to be on the lookout for this kind of thing.
Remember that you CAN’T divide by zero!
Bad/lost/Assumed Parenthesis
This is probably error that I find to be the most frustrating. There are a couple of errors that people commonly make here.
The first error is that people get lazy and decide that parenthesis aren’t needed at certain steps or that they can remember that the parenthesis are supposed to be there. Of course, the problem here is that they often tend to forget about them in the very next step!
The other error is that students sometimes don’t understand just how important parentheses really are. This is often seen in errors made in exponentiation as my first couple of examples show.
Example 1 Square 4x.
CorrectIncorrect
(4x)2=(4)2(x)2=16x2
4x2
Note the very important difference between these two! When dealing with exponents remember that only the quantity immediately to the left of the exponent gets the exponent. So, in the incorrect case, the x is the quantity immediately to the left of the exponent so we are squaring only the x while the 4 isn’t squared. In the correct case the parenthesis is immediately to the left of the exponent so this signifies that everything inside the parenthesis should be squared!
Parenthesis are required in this case to make sure we square the whole thing, not just the x, so don’t forget them!
Example 2 Square -3.
CorrectIncorrect
(−3)2=(−3)(−3)=9
−32=−(3)(3)=−9
This one is similar to the previous one, but has a subtlety that causes problems on occasion. Remember that only the quantity to the left of the exponent gets the exponent. So, in the incorrect case ONLY the 3 is to the left of the exponent and so ONLY the 3 gets squared!
Many people know that technically they are supposed to square -3, but they get lazy and don’t write the parenthesis down on the premise that they will remember them when the time comes to actually evaluate it. However, it’s amazing how many of these folks promptly forget about the parenthesis and write down -9 anyway!
Example 3 Subtract 4x−5 from x2+3x−5.
CorrectIncorrect
x2+3x−5−(4x−5)=x2+3x−5−4x+5=x2−x
x2+3x−5−4x−5=x2−x−10
Be careful and note the difference between the two! In the first case I put parenthesis around then 4x−5 and in the second I didn’t. Since we are subtracting a polynomial we need to make sure we subtract the WHOLE polynomial! The only way to make sure we do that correctly is to put parenthesis around it.
Again, this is one of those errors that people do know that technically the parenthesis should be there, but they don’t put them in and promptly forget that they were there and do the subtraction incorrectly.
Example 4 Convert √7x to fractional exponents.
CorrectIncorrect
√7x=(7x)12
√7x=7x12
This comes back to same mistake in the first two. If only the quantity to the left of the exponent gets the exponent. So, the incorrect case is really 7x12=7√x and this is clearly NOT the original root.
Example 5 Evaluate −3∫6x−2dx.
CorrectIncorrect
−3∫6x−2dx=−3(3x2−2x)+c=−9x2+6x+c
−3∫6x−2dx=−3⋅3x2−2x+c=−9x2−2x+c
Note the use of the parenthesis. The problem states that it is -3 times the WHOLE integral not just the first term of the integral (as is done in the incorrect example).
Improper Distribution
Be careful when using the distribution property! There two main errors that I run across on a regular basis.
Example 1 Multiply 4(2x2−10).
CorrectIncorrect
4(2x2−10)=8x2−40
4(2x2−10)=8x2−10
Make sure that you distribute the 4 all the way through the parenthesis! Too often people just multiply the first term by the 4 and ignore the second term. This is especially true when the second term is just a number. For some reason, if the second term contains variables students will remember to do the distribution correctly more often than not.
Example 2 Multiply 3(2x−5)2.
CorrectIncorrect
3(2x−5)2=3(4x2−20x+25)=12x2−60x+75
3(2x−5)2=(6x−15)2=36x2−180x+225
Remember that exponentiation must be performed BEFORE you distribute any coefficients through the parenthesis!
Additive Assumptions
I didn’t know what else to call this, but it’s an error that many students make. Here’s the assumption. Since 2(x+y)=2x+2y then everything works like this. However, here is a whole list in which this doesn’t work.
(x+y)2≠x2+y2
√x+y≠√x+√y
1x+y≠1x+1y
cos(x+y)≠cosx+cosy
It’s not hard to convince yourself that any of these aren’t true. Just pick a couple of numbers and plug them in! For instance,
(1+3)2≠12+32(4)2≠1+916≠10
You will find the occasional set of numbers for which one of these rules will work, but they don’t work for almost any randomly chosen pair of numbers.
Note that there are far more examples where this additive assumption doesn’t work than what I’ve listed here. I simply wrote down the ones that I see most often. Also, a couple of those that I listed could be made more general. For instance,
(x+y)n≠xn+ynfor any integer n≥2
n√x+y≠n√x+n√yfor any integer n≥2
Canceling Errors
These errors fall into two categories. Simplifying rational expressions and solving equations. Let’s look at simplifying rational expressions first.
Example 1 Simplify 3x3−xx (done correctly).
3x3−xx=x(3x2−1)x=3x2−1
Notice that in order to cancel the x out of the denominator I first factored an x out of the numerator. You can only cancel something if it is multiplied by the WHOLE numerator and denominator, or if IS the whole numerator or denominator (as in the case of the denominator in our example).
Contrast this with the next example which contains a very common error that students make.
Example 2 Simplify 3x3−xx (done incorrectly).
Far too many students try to simplify this as,
3x2−xOR3x3−1
In other words, they cancel the x in the denominator against only one of the x’s in the numerator (i.e. cancel the x only from the first term or only from the second term). THIS CAN’T BE DONE!!!!! In order to do this canceling you MUST have an x in both terms.
To convince yourself that this kind of canceling isn’t true consider the following number example.
Example 3 Simplify 8−32.
This can easily be done just be doing the arithmetic as follows
8−32=52=2.5
However, let’s do an incorrect cancel similar to the previous example. We’ll first cancel the two in the denominator into the eight in the numerator. This is NOT CORRECT, but it mirrors the canceling that was incorrectly done in the previous example. This gives,
8−32=4−3=1
Clearly these two aren’t the same! So you need to be careful with canceling!
Now, let’s take a quick look at canceling errors involved in solving equations.
Example 4 Solve 2x2=x (done incorrectly).
Too many students get used to just canceling (i.e. simplifying) things to make their life easier. So, the biggest mistake in solving this kind of equation is to cancel an x from both sides to get,
2x=1⇒x=12
While, x=12 is a solution, there is another solution that we’ve missed. Can you see what it is? Take a look at the next example to see what it is.
Example 5 Solve 2x2=x (done correctly).
Here’s the correct way to solve this equation. First get everything on one side then factor!
2x2−x=0x(2x−1)=0
From this we can see that either,
x=0OR2x−1=0
In the second case we get the x=12 we got in the first attempt, but from the first case we also get x=0 that we didn’t get in the first attempt. Clearly x=0 will work in the equation and so is a solution!
We missed the x=0 in the first attempt because we tried to make our life easier by “simplifying” the equation before solving. While some simplification is a good and necessary thing, you should NEVER divide out a term as we did in the first attempt when solving. If you do this, you WILL lose solutions.
Proper Use of Square Root
There seems to be a very large misconception about the use of square roots out there. Students seem to be under the misconception that
√16=±4
This is not correct however. Square roots are ALWAYS positive or zero! So the correct value is
√16=4
This is the ONLY value of the square root! If we want the -4 then we do the following
−√16=−(√16)=−(4)=−4
Notice that I used parenthesis only to make the point on just how the minus sign was appearing! In general, the middle two steps are omitted. So, if we want the negative value we have to actually put in the minus sign!
I suppose that this misconception arises because they are also asked to solve things like x2=16. Clearly the answer to this is x=±4 and often they will solve by “taking the square root” of both sides. There is a missing step however. Here is the proper solution technique for this problem.
x2=16x=±√16x=±4
Note that the ± shows up in the second step before we actually find the value of the square root! It doesn’t show up as part of taking the square root.
I feel that I need to point out that many instructors (including myself on occasion) don’t help matters in that they will often omit the second step and by doing so seem to imply that the ± is showing up because of the square root.
So, remember that square roots ALWAYS return a positive answer or zero. If you want a negative you’ll need to put it in a minus sign BEFORE you take the square root.
Ambiguous Fractions
This is more a notational issue than an algebra issue. I decided to put it here because too many students come out of algebra classes without understanding this point. There are really three kinds of “bad” notation that people often use with fractions that can lead to errors in work.
The first is using a “/” to denote a fraction, for instance 2/3. In this case there really isn’t a problem with using a “/”, but what about 2/3x? This can be either of the two following fractions.
23xOR23x
It is not clear from 2/3x which of these two it should be! You, as the student, may know which one of the two that you intended it to be, but a grader won’t. Also, while you may know which of the two you intended it to be when you wrote it down, will you still know which of the two it is when you go back to look at the problem when you study?
You should only use a “/” for fractions when it will be clear and obvious to everyone, not just you, how the fraction should be interpreted.
The next notational problem I see fairly regularly is people writing,
23x
It is not clear from this if the x belongs in the denominator or the fraction or not. Students often write fractions like this and usually they mean that the x shouldn’t be in the denominator. The problem is on a quick glance it often looks like it should be in the denominator and the student just didn’t draw the fraction bar over far enough.
If you intend for the x to be in the denominator then write it as such that way, 23x, i.e. make sure that you draw the fraction bar over the WHOLE denominator. If you don’t intend for it to be in the denominator then don’t leave any doubt! Write it as 23x.
The final notational problem that I see comes back to using a “/” to denote a fraction, but is really a parenthesis problem. This involves fractions like
a+bc+d
Often students who use “/” to denote fractions will write this is fraction as
a+b/c+d
These students know that they are writing down the original fraction. However, almost anyone else will see the following
a+bc+d
This is definitely NOT the original fraction. So, if you MUST use “/” to denote fractions use parenthesis to make it clear what is the numerator and what is the denominator. So, you should write it as
(a+b)/(c+d)
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17.2: Nonhomogeneous Linear Equations
Last updated
: Sep 1, 2025
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17.1E: Exercises for Section 17.1
17.2E: Exercises for Section 17.2
Page ID
: 2627
Gilbert Strang & Edwin “Jed” Herman
OpenStax
( \newcommand{\kernel}{\mathrm{null}\,})
Learning Objectives
Write the general solution to a nonhomogeneous
differential equation
.
Solve a nonhomogeneous
differential equation
by the
method of undetermined coefficients
.
Solve a nonhomogeneous
differential equation
by the
method of variation of parameters
.
In this section, we examine how to solve nonhomogeneous
differential
equations. The terminology and methods are different from those we used for homogeneous equations, so let’s start by defining some new terms.
General Solution to a Nonhomogeneous Linear Equation
Consider the nonhomogeneous
linear
differential equation
a2(x)y″+a1(x)y′+a0(x)y=r(x).
a2(x)y''+a1(x)y'+a0(x)y=r(x).
The associated homogeneous equation
a2(x)y″+a1(x)y′+a0(x)y=0
a2(x)y''+a1(x)y'+a0(x)y=0
is called the complementary equation. We will see that solving the
complementary equation
is an important step in solving a nonhomogeneous
differential equation
.
Definition: Particular Solution
A solution yp(x) of a
differential equation
that contains no arbitrary constants is called a particular solution to the equation.
GENERAL Solution TO A NONHOMOGENEOUS EQUATION
Let yp(x) be any
particular solution
to the nonhomogeneous
linear
differential equation
a2(x)y″+a1(x)y′+a0(x)y=r(x).
Also, let c1y1(x)+c2y2(x) denote the general solution to the
complementary equation
. Then, the general solution to the nonhomogeneous equation is given by
y(x)=c1y1(x)+c2y2(x)+yp(x).
Proof
To prove y(x) is the general solution, we must first show that it solves the
differential equation
and, second, that any solution to the
differential equation
can be written in that form. Substituting y(x) into the
differential equation
, we have
a2(x)y″+a1(x)y′+a0(x)y=a2(x)(c1y1+c2y2+yp)″+a1(x)(c1y1+c2y2+yp)′+a0(x)(c1y1+c2y2+yp)=[a2(x)(c1y1+c2y2)″+a1(x)(c1y1+c2y2)′+a0(x)(c1y1+c2y2)]+a2(x)yp″+a1(x)yp′+a0(x)yp=0+r(x)=r(x).
So y(x) is a solution.
Now, let z(x) be any solution to a2(x)y″+a1(x)y′+a0(x)y=r(x). Then
a2(x)(z−yp)″+a1(x)(z−yp)′+a0(x)(z−yp)=(a2(x)z″+a1(x)z′+a0(x)z)−(a2(x)yp″+a1(x)yp′+a0(x)yp)=r(x)−r(x)=0,
so z(x)−yp(x) is a solution to the
complementary equation
. But, c1y1(x)+c2y2(x) is the general solution to the
complementary equation
, so there are constants c1 and c2 such that
z(x)−yp(x)=c1y1(x)+c2y2(x).
Hence, we see that
z(x)=c1y1(x)+c2y2(x)+yp(x).
Example 17.2.1: Verifying the General Solution
Given that yp(x)=x is a
particular solution
to the
differential equation
y″+y=x, write the general solution and check by verifying that the solution satisfies the equation.
Solution
The
complementary equation
is y″+y=0, which has the general solution c1cosx+c2sinx. So, the general solution to the nonhomogeneous equation is
y(x)=c1cosx+c2sinx+x.
To verify that this is a solution, substitute it into the
differential equation
. We have
y′(x)=−c1sinx+c2cosx+1
and
y″(x)=−c1cosx−c2sinx.
Then
y″(x)+y(x)=−c1cosx−c2sinx+c1cosx+c2sinx+x=x.
So, y(x) is a solution to y″+y=x.
Exercise 17.2.1
Given that yp(x)=−2 is a
particular solution
to y″−3y′−4y=8, write the general solution and verify that the general solution satisfies the equation.
Hint
: Find the general solution to the
complementary equation
.
Answer
: y(x)=c1e−x+c2e4x−2
In the preceding section, we learned how to solve homogeneous equations with constant coefficients. Therefore, for nonhomogeneous equations of the form ay″+by′+cy=r(x), we already know how to solve the
complementary equation
, and the problem boils down to finding a
particular solution
for the nonhomogeneous equation. We now examine two techniques for this: the
method of undetermined coefficients
and the
method of variation of parameters
.
Undetermined Coefficients
The method of undetermined coefficients involves making educated guesses about the form of the
particular solution
based on the form of r(x). When we take derivatives of polynomials, exponential functions, sines, and cosines, we get polynomials, exponential functions, sines, and cosines. So when r(x) has one of these forms, it is possible that the solution to the nonhomogeneous
differential equation
might take that same form. Let’s look at some examples to see how this works.
Example 17.2.2: Undetermined Coefficients When r(x) Is a Polynomial
Find the general solution to y″+4y′+3y=3x.
Solution
The
complementary equation
is y″+4y′+3y=0, with general solution c1e−x+c2e−3x. Since r(x)=3x, the
particular solution
might have the form yp(x)=Ax+B. If this is the case, then we have yp′(x)=A and yp″(x)=0. For yp to be a solution to the
differential equation
, we must find values for A and B such that
y″+4y′+3y=3x0+4(A)+3(Ax+B)=3x3Ax+(4A+3B)=3x.
Setting coefficients of like terms equal, we have
3A=34A+3B=0.
Then, A=1 and B=−43, so yp(x)=x−43 and the general solution is
y(x)=c1e−x+c2e−3x+x−43.
In Example 17.2.2, notice that even though r(x) did not include a constant
term
, it was necessary for us to include the constant
term
in our guess. If we had assumed a solution of the form yp=Ax (with no constant
term
), we would not have been able to find a solution. (Verify this!) If the
function
r(x) is a polynomial, our guess for the
particular solution
should be a polynomial of the same
degree
, and it must include all lower-order terms, regardless of whether they are present in r(x).
Example 17.2.3: Undetermined Coefficients When r(x) Is an Exponential
Find the general solution to y″−y′−2y=2e3x.
Solution
The
complementary equation
is y″−y′−2y=0, with the general solution c1e−x+c2e2x. Since r(x)=2e3x, the
particular solution
might have the form yp(x)=Ae3x. Then, we have yp′(x)=3Ae3x and yp″(x)=9Ae3x. For yp to be a solution to the
differential equation
, we must find a value for A such that
y″−y′−2y=2e3x9Ae3x−3Ae3x−2Ae3x=2e3x4Ae3x=2e3x.
So, 4A=2 and A=1/2. Then, yp(x)=(12)e3x, and the general solution is
y(x)=c1e−x+c2e2x+12e3x.
Exercise 17.2.2
Find the general solution to y″−4y′+4y=7sint−cost.
Hint
: Use yp(t)=Asint+Bcost as a guess for the
particular solution
.
Answer
: y(t)=c1e2t+c2te2t+sint+cost
In the previous checkpoint, r(x) included both sine and cosine terms. However, even if r(x) included a sine
term
only or a cosine
term
only, both terms must be present in the guess. The
method of undetermined coefficients
also works with products of polynomials, exponentials, sines, and cosines. Some of the key forms of r(x) and the associated guesses for yp(x) are summarized in Table 17.2.1.
Table 17.2.1: Key Forms for the
Method of Undetermined Coefficients
| r(x) | Initial guess for yp(x) |
| k (a constant) | A (a constant) |
| ax+b | Ax+B (Note: The guess must include both terms even if b=0.) |
| ax2+bx+c | Ax2+Bx+C (Note: The guess must include all three terms even if b or c are zero.) |
| Higher-order polynomials | Polynomial of the same order as r(x) |
| aeλx | Aeλx |
| acosβx+bsinβx | Acosβx+Bsinβx (Note: The guess must include both terms even if either a=0 or b=0.) |
| aeαxcosβx+beαxsinβx | Aeαxcosβx+Beαxsinβx |
| (ax2+bx+c)eλx | (Ax2+Bx+C)eλx |
| (a2x2+a1x+a0)cosβx+(b2x2+b1x+b0)sinβx | (A2x2+A1x+A0)cosβx+(B2x2+B1x+B0)sinβx |
| (a2x2+a1x+a0)eαxcosβx+(b2x2+b1x+b0)eαxsinβx | (A2x2+A1x+A0)eαxcosβx+(B2x2+B1x+B0)eαxsinβx |
Keep in mind that there is a key pitfall to this method. Consider the
differential equation
y″+5y′+6y=3e−2x. Based on the form of r(x), we guess a
particular solution
of the form yp(x)=Ae−2x. But when we substitute this expression into the
differential equation
to find a value for A,we run into a problem. We have
yp′(x)=−2Ae−2x
and
y″p=4Ae−2x,
so we want
y″+5y′+6y=3e−2x4Ae−2x+5(−2Ae−2x)+6Ae−2x=3e−2x4Ae−2x−10Ae−2x+6Ae−2x=3e−2x0=3e−2x,
which is not possible.
Looking closely, we see that, in this case, the general solution to the
complementary equation
is c1e−2x+c2e−3x. The exponential
function
in r(x) is actually a solution to the
complementary equation
, so, as we just saw, all the terms on the left side of the equation cancel out. We can still use the
method of undetermined coefficients
in this case, but we have to alter our guess by multiplying it by x. Using the new guess, yp(x)=Axe−2x, we have
yp′(x)=A(e−2x−2xe−2x
and
y″p(x)=−4Ae−2x+4Axe−2x.
Substitution gives
y″+5y′+6y=3e−2x(−4Ae−2x+4Axe−2x)+5(Ae−2x−2Axe−2x)+6Axe−2x=3e−2x−4Ae−2x+4Axe−2x+5Ae−2x−10Axe−2x+6Axe−2x=3e−2xAe−2x=3e−2x.
So, A=3 and yp(x)=3xe−2x. This gives us the following general solution
y(x)=c1e−2x+c2e−3x+3xe−2x.
Note that if xe−2x were also a solution to the
complementary equation
, we would have to multiply by x again, and we would try yp(x)=Ax2e−2x.
PROBLEM-SOLVING STRATEGY: METHOD OF UNDETERMINED COEFFICIENTS
Solve the
complementary equation
and write down the general solution.
2. Based on the form of r(x), make an initial guess for yp(x).
3. Check whether any
term
in the guess foryp(x) is a solution to the
complementary equation
. If so, multiply the guess by x. Repeat this step until there are no terms in yp(x) that solve the
complementary equation
.
4. Substitute yp(x) into the
differential equation
and equate like terms to find values for the unknown coefficients in yp(x).
5. Add the general solution to the
complementary equation
and the
particular solution
you just found to obtain the general solution to the nonhomogeneous equation.
Example 17.2.4: Solving Nonhomogeneous Equations
Find the general solutions to the following
differential
equations.
y″−9y=−6cos3x
x″+2x′+x=4e−t
y″−2y′+5y=10x2−3x−3
y″−3y′=−12t
Solution
The
complementary equation
is y″−9y=0, which has the general solution c1e3x+c2e−3x(step 1). Based on the form of r(x)=−6cos3x, our initial guess for the
particular solution
is yp(x)=Acos3x+Bsin3x (step 2). None of the terms in yp(x) solve the
complementary equation
, so this is a valid guess (step 3).
Now we want to find values for A and B, so substitute yp into the
differential equation
. We have
yp′(x)=−3Asin3x+3Bcos3x and yp″(x)=−9Acos3x−9Bsin3x,
so we want to find values of A and B such that
y″−9y=−6cos3x−9Acos3x−9Bsin3x−9(Acos3x+Bsin3x)=−6cos3x−18Acos3x−18Bsin3x=−6cos3x.
Therefore,
−18A=−6−18B=0.
This gives A=13 and B=0, so yp(x)=(13)cos3x (step 4).
Putting everything together, we have the general solution
y(x)=c1e3x+c2e−3x+13cos3x.
2. The
complementary equation
is x″+2x′+x=0, which has the general solution c1e−t+c2te−t (step 1). Based on the form r(t)=4e−t, our initial guess for the
particular solution
is xp(t)=Ae−t (step 2). However, we see that this guess solves the
complementary equation
, so we must multiply by t, which gives a new guess: xp(t)=Ate−t (step 3). Checking this new guess, we see that it, too, solves the
complementary equation
, so we must multiply by t again, which gives xp(t)=At2e−t (step 3 again). Now, checking this guess, we see that xp(t) does not solve the
complementary equation
, so this is a valid guess (step 3 yet again).
We now want to find a value for A, so we substitute xp into the
differential equation
. We have
xp(t)=At2e−t, soxp′(t)=2Ate−t−At2e−t
and xp″(t)=2Ae−t−2Ate−t−(2Ate−t−At2e−t)=2Ae−t−4Ate−t+At2e−t.
Substituting into the
differential equation
, we want to find a value of A so that
x″+2x′+x=4e−t2Ae−t−4Ate−t+At2e−t+2(2Ate−t−At2e−t)+At2e−t=4e−t2Ae−t=4e−t.
This gives A=2, so xp(t)=2t2e−t (step 4). Putting everything together, we have the general solution
x(t)=c1e−t+c2te−t+2t2e−t.
3. The
complementary equation
is y″−2y′+5y=0, which has the general solution c1excos2x+c2exsin2x (step 1). Based on the form r(x)=10x2−3x−3, our initial guess for the
particular solution
is yp(x)=Ax2+Bx+C (step 2). None of the terms in yp(x) solve the
complementary equation
, so this is a valid guess (step 3). We now want to find values for A, B, and C, so we substitute yp into the
differential equation
. We have yp′(x)=2Ax+B and yp″(x)=2A, so we want to find values of A, B, and C such that
y″−2y′+5y=10x2−3x−32A−2(2Ax+B)+5(Ax2+Bx+C)=10x2−3x−35Ax2+(5B−4A)x+(5C−2B+2A)=10x2−3x−3.
Therefore,
5A=105B−4A=−35C−2B+2A=−3.
This gives A=2, B=1, and C=−1, so yp(x)=2x2+x−1 (step 4). Putting everything together, we have the general solution
y(x)=c1excos2x+c2exsin2x+2x2+x−1.
4. The
complementary equation
is y″−3y′=0, which has the general solution c1e3t+c2 (step 1). Based on the form r(t)=−12t,r(t)=−12t, our initial guess for the
particular solution
is yp(t)=At+B (step 2). However, we see that the constant
term
in this guess solves the
complementary equation
, so we must multiply by t, which gives a new guess: yp(t)=At2+Bt (step 3). Checking this new guess, we see that none of the terms in yp(t) solve the
complementary equation
, so this is a valid guess (step 3 again). We now want to find values for A and B, so we substitute yp into the
differential equation
. We have yp′(t)=2At+B and yp″(t)=2A, so we want to find values of AA and BB such that
y″−3y′=−12t2A−3(2At+B)=−12t−6At+(2A−3B)=−12t.
Therefore,
−6A=−122A−3B=0.
This gives A=2 and B=4/3, so yp(t)=2t2+(4/3)t (step 4). Putting everything together, we have the general solution
y(t)=c1e3t+c2+2t2+43t.
Exercise 17.2.3
Find the general solution to the following
differential
equations.
y″−5y′+4y=3ex
y″+y′−6y=52cos2t
Hint
: Use the problem-solving strategy.
Answer a
: y(x)=c1e4x+c2ex−xex
Answer b
: y(t)=c1e−3t+c2e2t−5cos2t+sin2t
Variation of Parameters
Sometimes, r(x) is not a combination of polynomials, exponentials, or sines and cosines. When this is the case, the
method of undetermined coefficients
does not
work
, and we have to use another approach to find a
particular solution
to the
differential equation
. We use an approach called the method of variation of parameters.
To simplify our calculations a little, we are going to divide the
differential equation
through by a, so we have a leading coefficient of 1. Then the
differential equation
has the form
y″+py′+qy=r(x),
where p and q are constants.
If the general solution to the
complementary equation
is given by c1y1(x)+c2y2(x), we are going to look for a
particular solution
of the form
yp(x)=u(x)y1(x)+v(x)y2(x).
In this case, we use the two
linearly independent
solutions to the
complementary equation
to form our
particular solution
. However, we are assuming the coefficients are functions of x, rather than constants. We want to find functions u(x) and v(x) such that yp(x) satisfies the
differential equation
. We have
yp=uy1+vy2yp′=u′y1+uy1′+v′y2+vy2′yp″=(u′y1+v′y2)′+u′y1′+uy1″+v′y2′+vy2″.
Substituting into the
differential equation
, we obtain
yp″+pyp′+qyp=[(u′y1+v′y2)′+u′y1′+uy1″+v′y2′+vy2″]+p[u′y1+uy1′+v′y2+vy2′]+q[uy1+vy2]=u[y1″+py1′+qy1]+v[y2″+py2′+qy2]+(u′y1+v′y2)′+p(u′y1+v′y2)+(u′y1′+v′y2′).
Note that y1 and y2 are solutions to the
complementary equation
, so the first two terms are zero. Thus, we have
(u′y1+v′y2)′+p(u′y1+v′y2)+(u′y1′+v′y2′)=r(x).
If we simplify this equation by imposing the additional condition u′y1+v′y2=0, the first two terms are zero, and this reduces to u′y1′+v′y2′=r(x). So, with this additional condition, we have a system of two equations in two unknowns:
u′y1+v′y2=0u′y1′+v′y2′=r(x).
Solving this system gives us u′ and v′, which we can integrate to find u and v.
Then, yp(x)=u(x)y1(x)+v(x)y2(x) is a
particular solution
to the
differential equation
. Solving this system of equations is sometimes challenging, so let’s take this opportunity to review Cramer’s rule, which allows us to solve the system of equations using determinants.
RULE: CRAMER’S RULE
The system of equations
a1z1+b1z2=r1a2z1+b2z2=r2
has a unique solution if and only if the determinant of the coefficients is not zero. In this case, the solution is given by
z1=r1b1r2b2a1b1a2b2andz2=a1r1a2r2a1b1a2b2.
Example 17.2.5: Using Cramer’s Rule
Use Cramer’s rule to solve the following system of equations.
x2z1+2xz2=0z1−3x2z2=2x
Solution
We have
a1(x)=x2a2(x)=1b1(x)=2xb2(x)=−3x2r1(x)=0r2(x)=2x.
Then,a1b1a2b2=x22x1−3x2=−3x4−2x
and
r1b1r2b2=02x2x−3x2=0−4x2=−4x2.
Thus,
z1=r1b1r2b2a1b1a2b2=−4x2−3x4−2x=4x3x3+2.
In addition,
a1r1a2r2=x2012x=2x3−0=2x3.
Thus,
z2=a1r1a2r2a1b1a2b2=2x3−3x4−2x=−2x23x3+2.
Exercise 17.2.4
Use Cramer’s rule to solve the following system of equations.
2xz1−3z2=0x2z1+4xz2=x+1
Hint
: Use the process from the previous example.
Answer
: z1=3x+311x2,z2=2x+211x
PROBLEM-SOLVING STRATEGY: METHOD OF VARIATION OF PARAMETERS
Solve the
complementary equation
and write down the general solution c1y1(x)+c2y2(x).
2. Use Cramer’s rule or another suitable technique to find functions u′(x) and v′(x) satisfying u′y1+v′y2=0u′y1′+v′y2′=r(x).
3. Integrate u′ and v′ to find u(x) and v(x). Then, yp(x)=u(x)y1(x)+v(x)y2(x) is a
particular solution
to the equation.
4. Add the general solution to the
complementary equation
and the
particular solution
found in step 3 to obtain the general solution to the nonhomogeneous equation.
Example 17.2.6: Using the Method of Variation of Parameters
Find the general solution to the following
differential
equations.
y″−2y′+y=ett2
y″+y=3sin2x
Solution
The
complementary equation
is y″−2y′+y=0 with associated general solution c1et+c2tet. Therefore, y1(t)=et and y2(t)=tet. Calculating the derivatives, we get y1′(t)=et and y2′(t)=et+tet (step 1). Then, we want to find functions u′(t) and v′(t) so that
u′et+v′tet=0u′et+v′(et+tet)=ett2.
Applying Cramer’s rule (Equation 17.2.1), we have
u′=0tetett2et+tetettetetet+tet=0−tet(ett2)et(et+tet)−ettet=−e2tte2t=−1t
and
v′=et0etett2ettetetet+tet=et(ett2)e2t=1t2(step 2).
Integrating, we get
u=−∫1tdt=−ln|t|v=∫1t2dt=−1t
Then we have
yp=−etln|t|−1ttet=−etln|t|−et.
The et
term
is a solution to the
complementary equation
, so we don’t need to carry that
term
into our general solution explicitly. The general solution is
y(t)=c1et+c2tet−etln|t|
2. The
complementary equation
is y″+y=0 with associated general solution c1cosx+c2sinx. So, y1(x)=cosx and y2(x)=sinx (step 1). Then, we want to find functions u′(x) and v′(x) such that
u′cosx+v′sinx=0−u′sinx+v′cosx=3sin2x.
Applying Cramer’s rule, we have
u′=0sinx3sin2xcosxcosxsinx−sinxcosx=0−3sin3xcos2x+sin2x=−3sin3x
and
v′=cosx0−sinx3sin2xcosxsinx−sinxcosx=3sin2xcosx1=3sin2xcosx(step 2).
Integrating first to find u, we get
u=∫−3sin3xdx=−3[−13sin2xcosx+23∫sinxdx]=sin2xcosx+2cosx.
Now, we integrate to find v. Using substitution (with w=sinx), we get
v=∫3sin2xcosxdx=∫3w2dw=w3=sin3x.
yp=(sin2xcosx+2cosx)cosx+(sin3x)sinx=sin2xcos2x+2cos2x+sin4x=2cos2x+sin2x(cos2x+sin2x)(step 4).=2cos2x+sin2x=cos2x+1
The general solution is
y(x)=c1cosx+c2sinx+1+cos2x(step 5).
Exercise 17.2.5
Find the general solution to the following
differential
equations.
y″+y=secx
x″−2x′+x=ett
Hint
: Follow the problem-solving strategy.
Answer a
: y(x)=c1cosx+c2sinx+cosxln|cosx|+xsinx
Answer b
: x(t)=c1et+c2tet+tetln|t|
Key Concepts
To solve a nonhomogeneous
linear
second-order
differential equation
, first find the general solution to the
complementary equation
, then find a
particular solution
to the nonhomogeneous equation.
Let yp(x) be any
particular solution
to the nonhomogeneous
linear
differential equation
a2(x)y″+a1(x)y′+a0(x)y=r(x),and let c1y1(x)+c2y2(x) denote the general solution to the
complementary equation
. Then, the general solution to the nonhomogeneous equation is given by y(x)=c1y1(x)+c2y2(x)+yp(x).
When r(x) is a combination of polynomials, exponential functions, sines, and cosines, use the
method of undetermined coefficients
to find the
particular solution
. To use this method, assume a solution in the same form as r(x), multiplying by x as necessary until the assumed solution is
linearly independent
of the general solution to the
complementary equation
. Then, substitute the assumed solution into the
differential equation
to find values for the coefficients.
When r(x) is not a combination of polynomials, exponential functions, or sines and cosines, use the
method of variation of parameters
to find the
particular solution
. This method involves using Cramer’s rule or another suitable technique to find functions and v′(x) satisfying u′y1+v′y2=0u′y1′+v′y2′=r(x).Then, yp(x)=u(x)y1(x)+v(x)y2(x) is a
particular solution
to the
differential equation
.
Key Equations
Complementary equation
a2(x)y″+a1(x)y′+a0(x)y=0
General solution to a nonhomogeneous
linear
differential equation
y(x)=c1y1(x)+c2y2(x)+yp(x)
Glossary
complementary equation
: for the nonhomogeneous
linear
differential equation
a+2(x)y″+a1(x)y′+a0(x)y=r(x), the associated homogeneous equation, called the complementary equation, is a2(x)y″+a1(x)y′+a0(x)y=0
method of undetermined coefficients
: a method that involves making a guess about the form of the
particular solution
, then solving for the coefficients in the guess
method of variation of parameters
: a method that involves looking for particular solutions in the form yp(x)=u(x)y1(x)+v(x)y2(x), where y1 and y2 are
linearly independent
solutions to the complementary equations, and then solving a system of equations to find u(x) and v(x)
particular solution
: a solution yp(x) of a
differential equation
that contains no arbitrary constants
17.1E: Exercises for Section 17.1
17.2E: Exercises for Section 17.2 |
12025 | https://www.radfordmathematics.com/calculus/Differentiation/stationary-points/stationary-points.html | Online Mathematics Book
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Stationary Points - Part 1
(Definition & How to Find Stationary Points)
A stationary point, or critical point, is a point at which the curve's gradient equals to zero. Consequently if a curve has equation (y=f(x)) then at a stationary point we'll always have: [f'(x)=0] which can also be written: [\frac{dy}{dx} = 0] In other words the derivative function equals to zero at a stationary point.
Different Types of Stationary Points
There are three types of stationary points:
local (or global) maximum points
local (or global) minimum points
horizontal (increasing or decreasing) points of inflexion.
It is worth pointing out that maximum and minimum points are often called turning points.
Turning Points
A turning point is a stationary point, which is either:
a local (or global) minimum
a local (or global) maximum
each of which are illustared in the graphs shown here, where the horizontal tangent is shown in orange:
Horizontal Points of Inflection
A horizontal point of inflection is a stationary point, which is either:
a increasing horizontal point of inflection
a decreasing horizontal point of inflection
each of which are illustared in the graphs shown here, where the horizontal tangent is shown in orange:
Method: finding stationary points
Given a function (f(x)) and its curve (y=f(x)), to find any stationary point(s) we follow three steps:
Step 1: find (f'(x))
Step 2: solve the equation (f'(x)=0), this will give us the (x)-coordinate(s) of any stationary point(s).
Step 3 (if needed/asked): calculate the (y)-coordinate(s) of the stationary point(s) by plugging the (x) values found in step 2 into (f(x)).
Tutorial 1: How to Find Stationary Points
In the following tutorial we illustrate how to use our three-step method to find the coordinates of any stationary points, by finding the stationary point(s) of the curves:
(y = x^2 - 4x + 1), and
(y = 2x^3 + 3x^2 - 12x - 1)
Example 1
Given the function defined by the equation: [y = x^2 - 4x+5] find the coordinates of any stationary point(s).
Solution
Following our three-step method:
Step 1: find (\frac{dy}{dx}). For (y = x^2 - 4x+5), we find: [\frac{dy}{dx} = 2x-4]
Step 2: solve the equation (\frac{dy}{dx}=0). That's: [2x - 4 = 0] Solving this leads to: [2x = 4] Finally: [x = 2] At this stage we can state the curve (y=x^2 - 4x + 5) has one stationary point whose (x)-coordinate is (x=2).
Step 3: calculate the sationary point's (y)-coordinate. To do this we replace (x) by (2) in the curve's equation (y = x^2 - 4x + 5). That's: [\begin{aligned} y & = 2^2 - 4 \times 2 + 5 \ & = 4 - 8 + 5 \ & = -4 + 5 \ y & = 1 \end{aligned}] So the stationary point has (y)-coordinate (y=1).
We can therefore state that the curve (y = x^2 - 4x + 5) has one stationary point with coordinates (\begin{pmatrix}2,1 \end{pmatrix}).
This result is confirmed, using our graphical calculator and looking at the curve (y=x^2 - 4x+5):
We can see quite clearly that the curve has a global minimum point, which is a stationary point, at (\begin{pmatrix}2,1 \end{pmatrix}).
Example 2
Find the coordinates of any stationary point(s) of the function defined by: [y = 2x^3 + 3x^2 - 12x+1]
Solution
Following our three-step method:
Step 1: find (\frac{dy}{dx}). Using the power rule for differentiation we find: [\begin{aligned} \frac{dy}{dx} & = 3\times 2x^{3-1} + 2\times 3x^{2-1} - 12 \ & = 6x^2+6x^1 - 12 \ \frac{dy}{dx} & = 6x^2+6x - 12 \end{aligned}]
Step 2: solve the equation (\frac{dy}{dx}=0). Since (\frac{dy}{dx} =6x^2+6x - 12) we need to solve the quadratic equation: [6x^2+6x-12 = 0] Doing this either using our graphical calculator, the quadratic formula, or by factoring, we find two solutions: [x = -2 \quad \text{and} \quad x = 1] So at this stage we can state that the function (y = 2x^3+3x^2-12x+1) has two stationary points. One with (x)-coordinate (x=-2) and the other with (x)-coordinate (x=1).
Step 3: calculate the sationary point's (y)-coordinate. Since we found two values of (x), in step 2, there are two (y)-coordinates to calculate, one for each value of (x).
when (x=-2):Replacing (x) by (-2) in (y = 2x^3+3x^2-12x+1), we find: [\begin{aligned} y & = 2\times (-2)^3+3\times (-2)^2 - 12 \times (-2) + 1 \ & = 2\times (-8) + 3\times 4 - (-24)+1\ & = -16 + 12 + 24 + 1 \ y & = 21 \end{aligned}] So the function has a stationary point at: [\begin{pmatrix}-2,21 \end{pmatrix}]
when (x=1):Replacing (x) by (1) in (y = 2x^3+3x^2-12x+1), we find: [\begin{aligned} y & = 2\times 1^3+3\times 1^2 - 12 \times 1 + 1 \ & = 2\times 1 + 3\times 1 - 12+1\ & = 2+3-12+1 \ y & = -6 \end{aligned}] So the function has its second stationary point at: [\begin{pmatrix}1,-6 \end{pmatrix}]
We can see both of these stationary points on the graph shown below:
We can see quite clearly that the stationary point at (\begin{pmatrix}-2,21\end{pmatrix}) is a local maximum and the stationary point at (\begin{pmatrix}1,-6\end{pmatrix}) is a local minimum.
Example 3
Given the function defined by: [y = x^3-6x^2+12x-12] Find the coordinates of any stationary point(s) along this function's curve's length.
Solution
Following our three-step method:
Step 1: find (\frac{dy}{dx}). Since (y = x^3-6x^2+12x-12), we use the power rule for differentiation, to find this function's derivative: [\begin{aligned} \frac{dy}{dx} & = 3\times x^{3-1} - 2 \times 6x^{2-1} + 12x^{1-1} + 0 \ & = 3x^2 - 12x^1 + 12x^0 \ \frac{dy}{dx} & = 3x^2 - 12x + 12 \end{aligned}]
Step 2: solve the equation (\frac{dy}{dx}=0). Since (\frac{dy}{dx} = 3x^2 - 12x + 12) we have to solve the quadratic equation: [3x^2 - 12x + 12= 0 ] We can solve this using the quadratic formula or by factoring or even using our graphical calculator. In doing so we find one solution: [x = 2] So, at this stage, we can state that this function has one stationary point whose (x)-coordinate is (x = 2).
Step 3: calculate the sationary point's (y)-coordinate. Since we found that the stationary point had (x)-coordinate (x = 2), to find its (y)-coordinate we replace (x) by (2) in the function's equation (y = x^3-6x^2+12x-12). That's: [\begin{aligned} y & = 2^3 - 6\times 2^2 + 12 \times 2 - 12 \ & = 8 - 6\times 4 + 24 - 12 \ & = 8 - 24 + 24 - 12 \ y& = -4 \end{aligned}] So this function has a stationary point with coordinates: [\begin{pmatrix}2,-4\end{pmatrix}]
This result is confirmed when we look at the graph of (y = x^3 - 6x^2 + 12x - 12):
Looking at this graph, we can see that this curve's stationary point at (\begin{pmatrix}2,-4\end{pmatrix}) is an increasing horizontal point of inflection.
Exercise 1
Find the coordinates of any stationary point(s) along the length of each of the following curves:
(y = x^2 - 2x - 8)
(y = -x^2-6x-8)
(y = 2x^3 - 12x^2 - 30x - 10)
(y = -2x^3 + 3x^2 + 36x - 6)
(y = x^3+3x^2+3x-2)
Note: this exercise can be downloaded as a worksheet to practice with: worksheet
Answers Without Working
We find the derivative to be (\frac{dy}{dx} = 2x-2) and this curve has one stationary point: [\begin{pmatrix} 1,-9\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = -2x-6) and this curve has one stationary point: [\begin{pmatrix} -3,1\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = 2x^3 - 12x^2 - 30x- 10) and this curve has two stationary points: [\begin{pmatrix} -1,6\end{pmatrix}] and [\begin{pmatrix} 5,-210\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = -2x^3+3x^2+36x - 6) and this curve has two stationary points: [\begin{pmatrix} -2,-50\end{pmatrix}] and [\begin{pmatrix} 3,75\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = x^3+3x^2+3x-2) and this curve has one stationary point: [\begin{pmatrix} -1,-3\end{pmatrix}]
Answers with Working
Select the question number you'd like to see the working for:
Tutorial 2: How to Find Stationary Points
In the following tutorial we illustrate how to use our three-step method to find the coordinates of any stationary points, by finding the stationary point(s) along the curve:
[y = x + \frac{9}{x}]
Example 4
Given the function defined by: [y = x+\frac{4}{x}] find the coordinates of any stationary points along this curve's length.
Solution
Following our three-step method:
Step 1: find (\frac{dy}{dx}). We can re-write (y=x+\frac{4}{x}), using negative exponents: [y = x+4.x^{-1}] We can now use the power rule for differentiation to find the derivative: [\begin{aligned} \frac{dy}{dx} & = 1+(-1).4.x^{-1-1} \ & = 1-4x^{-2} \ \frac{dy}{dx} & = 1 - \frac{4}{x^2} \end{aligned}]
Step 2: solve the equation (\frac{dy}{dx}=0). Since (\frac{dy}{dx} =1 - \frac{4}{x^2}) we need to solve the equation: [1 - \frac{4}{x^2} = 0] To solve this equation by hand we start by writing the entire left hand side over (x^2) using the fact that (1 = \frac{x^2}{x^2}) so that: [1 - \frac{4}{x^2} = 0] can be written: [\frac{x^2}{x^2} - \frac{4}{x^2} = 0] and therefore we have to solve: [\frac{x^2-4}{x^2} = 0] This will equal to (0) if and only if the numerator, (x^2-4), equals to (0). So all we need to solve is: [x^2 - 4 = 0] That's: [x^2 = 4] which leads to two solutions: [x = -2 \quad \text{and} \quad x = 2]
Step 3: calculate the sationary point's (y)-coordinate. Since we found two values of (x), in step 2, there are two (y)-coordinates to calculate, one for each value of (x).
when (x = -2):replacing (x) by (-2) in (y = x+\frac{4}{x}), we find: [\begin{aligned} y & = -2 + \frac{4}{-2} \ & = -2 + (-2) \ & = -2 - 2 \ y & = -4 \end{aligned}] So one of this function's stationary points is: [\begin{pmatrix}-2,-4 \end{pmatrix}]
when (x = 2):replacing (x) by (2) in (y = x+\frac{4}{x}), we find: [\begin{aligned} y & = 2 + \frac{4}{2} \ & = 2 + 2 \ y & = 4 \end{aligned}] So the function's second stationary point has coordinates: [\begin{pmatrix}2,4 \end{pmatrix}]
We can see both of these stationary points on the graph shown below:
We can see quite clearly that the stationary point at (\begin{pmatrix}-2,-4\end{pmatrix}) is a local maximum and the stationary point at (\begin{pmatrix}2,4\end{pmatrix}) is a local minimum.
Exercise 2
Find the coordinates of any stationary point(s) along the length of each of the following curves:
( y = 2x + \frac{8}{x})
( y = -x - \frac{1}{x})
( y = 3x + \frac{27}{x})
( y = -2x - \frac{72}{x})
( y = x + \frac{25}{x})
Note: this exercise can be downloaded as a worksheet to practice with: worksheet
Answers Without Working
We find the derivative to be (\frac{dy}{dx} = 2 - \frac{8}{x^2}) and this curve has two stationary points: [\begin{pmatrix} -2,-8\end{pmatrix}] and [\begin{pmatrix} 2,8\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = -1 + \frac{1}{x^2}) and this curve has two stationary points: [\begin{pmatrix} -1,2\end{pmatrix}] and [\begin{pmatrix} 1,-2\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = 3 - \frac{27}{x^2}) and this curve has two stationary points: [\begin{pmatrix} -3,-18\end{pmatrix}] and [\begin{pmatrix} 3,18\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = -22 + \frac{72}{x^2}) and this curve has two stationary points: [\begin{pmatrix} -6,48\end{pmatrix}] and [\begin{pmatrix} 6,-48\end{pmatrix}]
We find the derivative to be (\frac{dy}{dx} = 1 - \frac{25}{x^2}) and this curve has two stationary points: [\begin{pmatrix} -5,-10\end{pmatrix}] and [\begin{pmatrix} 5,10\end{pmatrix}]
Answers with Working
Select the question number you'd like to see the working for: |
12026 | https://artofproblemsolving.com/wiki/index.php/2015_AIME_I_Problems/Problem_8?srsltid=AfmBOooasg9FMs5ekCdVhQze7p_Zdw8R4Mg4_wkPbKoL_T6Lzpszyb0n | Art of Problem Solving
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2015 AIME I Problems/Problem 8
Contents
[hide]
1 Problem
2 Solution 1
3 Solution 2
4 Modification for Solution 2
5 Solution 3
6 Solution 4
7 Solution 5
8 Solution 6 (Fast)
9 Solution 7 (Bad Solution)
10 Solution 8 (not realistic but works)
11 Video Solution
12 See also
Problem
For positive integer , let denote the sum of the digits of . Find the smallest positive integer satisfying .
Solution 1
You know whatever is, it has to have 3 digits, because if it had only two, the maximum of is 18 and because in AIME all answers are up to three digits.
Now let
So first we know, . Okay now we have to split into cases based on which digit gets carried. This meaning, when you add a 3 digit number to 864, we have to know when to carry the digits. Note that if you don't understand any of the steps I take, just try adding any 3-digit number to 864 regularly (using the old-fashioned "put one number over the other" method, not mental calculation), and observe what you do at each step.
(1) None of the digits get carried over to the next space: So this means and . So
So it doesn't work. Now:
(2) is the only one that carries over So this means and . So
(3) is the only one that carries over. So
(4)The first and second digit carry over (but not the third)
Aha! This case works but we still have to make sure it's possible for (We assumed this is true, so we have to find a number that works.) Since only the second and first digit carry over, and . The smallest value we can get with this is 695. Let's see if we can find a smaller one:
(5)The first and third digit carry over (but not the second)
The largest value for the middle digit is 2, so the other digits have to be both 9's. So the smallest possible value is 929
(6) All the digits carry over
So the answer is which after a quick test, does indeed work.
Note: This problem is VERY easy to bash out. I did this when I mocked this test, it gave me the answer in 5 min. Basically you just bash out all of the three digit numbers whose digit sum is 20 and you'll find (quickly) that the answer is
Note to author: Since we have , this implies that must be at least (since ), and therefore the first digit must ALWAYS carry. This is presumably the reason where the case where the second and third digits were carried but not the first was omitted, but for this reason cases (1) and (3) are also unnecessary.
Solution 2
First, it is easy to verify that works and that no other numbers beginning with the digit 6 work (i.e. do not work).
Suppose by contradiction that there is a smaller valid , where the leading digit of the three-digit number is 5 or less. (Two-digit obviously do not work because 9 + 9 < 20.) Clearly because the smallest three-digit number whose digits sum to 20 is . Also, because the second digit is at most 9, the units digit is at least 6, which means that the addition regroups in the ones place. Then the units digit of is clearly less than 4. But as , the sum of the thousands digit and the hundredth digit is at most 5. Because the second digit is at most 9, the sum of the digits of is at most , contradiction. Hence is the answer.
Modification for Solution 2
During the real test, I immediately noticed that must be less than 1000 (AIME problem) and that must be a three-digit number. Therefore, I began casework on the leading digit of . The casework was not intensive (how many ways are there to have digits sum to 20?) and I eventually got 695 as my answer. The rigorous proof that 695 was the smallest came afterwards.
Solution 3
First of all, notice that the smallest with is . Also, if , (because subtracting from the number removes the in the thousands place). After checking for various with , we see that we need to have a carry when subtracting . To have this, we must either have a in the tens place or a in the units place. The minimum for the former is , and for the latter it is . We check and see that , so our answer is .
Solution 4
Observation (Lemma): If r is the number of regroups in the addition of n+k, .
Proof: When you add two numbers, and you do a carry, you are taking away 10 from 1 column, and adding 1 to another column, giving a net loss of 9 to the total.
Thus, we can see that we need to regroup exactly twice when we add 864. And, the lowest possible n is 299, so let's start from there.
299 gets three regroups, so we are going to need to take away from digits, and dump the excess in the hundred's place, since the hundreds are going to regroup anyways.
So, if we take away from the tens digit, we need to take away until we get 2 in the tens digit(since the ones will regroup). So, we get the number 929, which works (929+864 = 1793), but is not the smallest.
If we take away from the ones digit, we only have to take away 4, turning the unit's place to 5. 5+4 is 9, so it won't regroup. Dump the ones into the hundred's place, and we get the number
-AlexLikeMath
Solution 5
Although this solution doesn't directly solve the problem, it greatly hastens the bashing process.
Call the three digits a, b, and c.
When you add each of 8, 6, and 4 to a, b, and c the resultant will either get smaller or larger, depending on the original number.
e.g. If c is 7, then adding 4 will reduce the 7 to a 1, whilst leaving a one for b.
If c is 3, then adding 4 will simply add four to the total, and make the 3 a 7.
Each of 8, 6, and 4 all can reduce the original number by a certain amount and can increase the original number by a certain amount.
8 can reduce by 2 for all numbers greater than 1, 6 can reduce numbers by 4, and 4 can reduce numbers by 6.
Possibilities:
-2, +8, (although this becomes obviously impossible later on) -4, +6, -6, +4
Also, realize that if the number is reduced, then a one will be carried to the following decimal place on the left, consequently reducing that amount they reduce. It's like a puzzle! Within no time you should find that if you add 4 to c, subtract 4 from b, subtract 1 from a, and leave a 1 in the thousands place, the total is equated to zero. This is optimal because most of the addition is kept to the left, where the effect to real value is less. (e.g. 299 is smaller than 992)
Now you have +1, -1, -4, +4 in the decimals, and a VERY fast trial and error gives
-Jackshi2006
Postscript by Jackshi2006
When I revisited the problem I realized that you can actually list out every possible number for n. 695 stands out very clearly as the smallest, because the only other possibility is 595, which doesn’t add to 20 to begin with.
Solution 6 (Fast)
First, note that to compute (for any positive integers and ), one can simply find the sum of and minus 9 times the number of times one regroups when adding to . One can see why this is by noticing that if one were to "forget" to regroup, and leave, say, a 10 in the ones' place, the sum of the digits would be 9 higher than if one did regroup. Anyway, one can see that the smallest 3-digit number (on AIME, all the answers are integers from 0 to 999) whose digits sum to 20 is 299. If we add this number to 864, we have to regroup 3 times, so will be smaller than . We want this difference to be 0, so we need to find a way to only regroup two times.
We now notice that regrouping the hundreds is inevitable, so we must either prevent regrouping the ones or the tens. Preventing regrouping the tens would require moving many of the tens to the hundreds' place (the ones' place is already full), which is bad when we are trying to minimize the number, but preventing regrouping the ones requires moving fewer ones to the hundreds' place.
We find that to preventing regrouping the ones, the ones' place of our number must be at most 5 (a larger number would sum to ten when added to 4). Because we want to minimize the number of ones we move to the hundreds' place, we leave exactly 5 by moving four ones to the hundreds' place: .
~ SymbolicPermutation
Solution 7 (Bad Solution)
Bashing out modulo and getting lucky we get that if the and carry over when adding and , that such that and after maximizing and such that to not make the carry over to minimize we get that is our answer after confirming there are no lesser solutions.
Note: If the sum of the digits are equal, they are also the same modulo . ~SirAppel
Solution 8 (not realistic but works)
Alternatively, you can use python to solve the problem:
def sum_of_digits(number):
total = 0
while number > 0:
digit = number % 10
total += digit
number //= 10
return total
for i in range(1000):
digitsum = sum_of_digits(i)
newdigitsum = sum_of_digits(i+864)
if digitsum == newdigitsum:
if digitsum == 20:
print(i)
~yeetdayeet
Video Solution
~savannahsolver
See also
2015 AIME I (Problems • Answer Key • Resources)
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12027 | https://brainly.com/question/21105026 | [FREE] What is the molar mass of nitrogen gas, N_2(g)? How many moles of nitrogen molecules are in 56 grams? - brainly.com
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What is the molar mass of nitrogen gas, N 2(g)? How many moles of nitrogen molecules are in 56 grams? Explain.
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28g/mol
2 moles of nitrogen are present
Explanation
The molar mass is defined as the mass of a chemical when 1 mole of this substance is present and is used to convert mass of the substance to moles or vice versa. Is widely used in stoichiometry problems.
1 atom of nitrogen has a molar mass of 14g/mol. N₂(g) -The molecule that contains 2 atoms of nitrogen- has a molar mass of 28g. That is 28g/mol.
56g of N₂ are:
56g (1mol / 28g) =
2 moles of nitrogen are present
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Basics of General, Organic, and Biological Chemistry - David W. Ball, John W. Hill, Rhonda J. Scott
The Basics of General, Organic, and Biological Chemistry - David W Ball
University Physics Volume 2 - Samuel J. Ling, William Moebs, Jeff Sanny
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The molar mass of nitrogen gas (N 2) is approximately 28.02 g/mol. In 56 grams of N 2, there are about 2 moles of nitrogen molecules. This can be calculated by dividing the mass by the molar mass.
Explanation
To find the molar mass of nitrogen gas, denoted as N 2(g), we first need to look at the atomic mass of a single nitrogen atom.
The atomic mass of nitrogen (N) is approximately 14.01 grams per mole (g/mol).
Since nitrogen gas is diatomic (meaning it exists as N 2), we calculate the molar mass of N 2 by adding the masses of the two nitrogen atoms:
M N 2=2×14.01 g/m o l=28.02 g/m o l
Thus, the molar mass of nitrogen gas is approximately 28.02 g/mol.
Next, to determine how many moles of nitrogen molecules are present in 56 grams of N 2:
Use the formula for converting mass to moles:
Moles=molar mass(g/mol)mass(g)
Substitute the values into the formula:
Moles=28.02 g/m o l 56 g≈1.996 moles
For practical purposes, we can say there are approximately 2 moles of nitrogen gas in 56 grams.
In summary, the molar mass of N 2 is about 28.02 g/mol, and 56 grams of N 2 corresponds to approximately 2 moles of nitrogen gas.
Examples & Evidence
For example, if you had 56 grams of oxygen gas (O 2), you would perform a similar calculation using the molar mass of O 2 (which is about 32 g/mol) to determine the number of moles present. Similarly, you can do this for any gas by knowing its molar mass, enabling you to convert between grams and moles easily.
The molar mass of elements and compounds is widely accepted in chemistry. Standard references like the periodic table provide the atomic masses used to compute the molar mass of gases, ensuring these calculations are accurate in a scientific context.
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Community Answer 6 How many moles are in 75.0 grams of nitrogen gas, N2?
Community Answer 5.0 2 Based on the equation below, how many grams of nitrogen gas (N2) will be produced from the decomposition of one mole of sodium azide (NaN3)? Use a molar mass of 28.0 grams for nitrogen gas (N2). 2NaN3 → 2Na + 3N2 A. 56.0 grams B. 42.0 grams C. 28.0 grams D. 18.7 grams
Community Answer Air is about 78% nitrogen gas (N2) by mass. The molar mass of nitrogen gas is 28.0 g/mol. A 100.0-g sample of air contains how many moles of nitrogen?
Community Answer Nitrogen gas, N2, reacts with hydrogen gas, H2 to form ammonia, NH3. 3 H2(g)+ N2(g) → 2 NH3(g) If 5.40 L of ammonia gas is collected at 25.60 °C and 761.0 mmHg, how many grams of nitrogen gas are consumed? The molar mass of nitrogen gas is 28.0134 g/mol.
Community Answer Identify the calculations possible using only 28.02 g/mol as a conversion factor. Select one or more: Calculate the grams of N2 in 10.58 liters of nitrogen gas Calculate the grams of N2 in 5.03 x 1020 moles of nitrogen molecules Calculate the moles of N2 molecules in 3.94 grams of nitrogen Calculate the moles of N2 molecules in 4.73 liters of nitrogen gas
Community Answer 5.0 3 NEED HELP PLSS:( Values for the molar mass of nitrogen, oxygen, and nitrogen dioxide molecules are given in the table below. What mass of nitrogen dioxide molecules is formed when 1 mole of nitrogen reacts with 2 moles of oxygen and forms nitrogen dioxide? Molecule Molar mass (g/mol) N2 32 .00 NO2 46.01 O A. 92.02 g O B. 23.00 g O C. 2.00 g O D. 46.01 g
Community Answer 4.2 19 A drink that contains 4 1/2 ounces of a proof liquor… approximately how many drinks does this beverage contain?
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Community Answer 4.3 189 1. Holding 100mL of water (ebkare)__2. Measuring 27 mL of liquid(daudgtear ldnreiyc)____3. Measuring exactly 43mL of an acid (rtube)____4. Massing out120 g of sodium chloride (acbnela)____5. Suspending glassware over the Bunsen burner (rwei zeagu)____6. Used to pour liquids into containers with small openings or to hold filter paper (unfenl)____7. Mixing a small amount of chemicals together (lewl letpa)____8. Heating contents in a test tube (estt ubet smalcp)____9. Holding many test tubes filled with chemicals (estt ubet karc) ____10. Used to clean the inside of test tubes or graduated cylinders (iwer srbuh)____11. Keeping liquid contents in a beaker from splattering (tahcw sgasl)____12. A narrow-mouthed container used to transport, heat or store substances, often used when a stopper is required (ymerereel kslaf)____13. Heating contents in the lab (nuesnb bneurr)____14. Transport a hot beaker (gntos)____15. Protects the eyes from flying objects or chemical splashes(ggloges)____16. Used to grind chemicals to powder (tmraor nda stlepe) __
Community Answer Food waste, like a feather or a bone, fall into food, causing contamination. Physical Chemical Pest Cross-conta
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Consider the chemical equations shown here. C H 4(g)+2 O 2(g)→C O 2(g)+2 H 2O(g)Δ H 1=−802 k J 2 H 2O(g)→2 H 2O(l)Δ H 2=−88 k J Which equation shows how to calculate Δ H r x n for the equation below? C H 4(g)+2 O 2(g)→C O 2(g)+2 H 2O(l) What is Δ H r x n for the overall reaction?
Consider the chemical equations shown here. 2 H 2(g)+O 2(g)→2 H 2O(g)Δ H 1=−483.6 k J+2=−241.8 k J/m o l 3 O 2(g)→2 O 3(g)Δ H 2=284.6 k J+2=142.3 k J/m o l What is the overall enthalpy of reaction for the equation shown below? 3 H 2(g)+O 3(g)→3 H 2O(g)□ kJ
Consider the chemical equations shown here. P 4(s)+3 O 2(g)→P 4O 6(s)Δ H 1=−1,640.1 k J P 4O 10(s)→P 4(s)+5 O 2(g)Δ H 2=2,940.1 k J What is the overall enthalpy of reaction for the equation shown below? Round the answer to the nearest whole number. P 4O 6(s)+2 O 2(g)→P 4O 10(s)
Prepare a "Coal and Petroleum" fact file.
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12028 | https://www.youtube.com/watch?v=6xXatAcdRjk | Dividing Decimals by Whole Numbers | Math with Mr. J
Math with Mr. J
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Welcome to How to Divide Decimals by Whole Numbers with Mr. J! Need help with dividing decimals? You're in the right place!
Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for help with how to divide decimals by whole numbers. Mr. J will go through examples of dividing decimals by whole numbers and explain the steps of how to divide decimals by whole numbers.
✅ More Videos on Dividing Decimals
Dividing Decimals by Whole Numbers:
Dividing Decimals by 1-Digit Whole Numbers:
Dividing Decimals by 2-Digit Whole Numbers:
Dividing Decimals by Decimals:
Dividing Whole Numbers by Decimals:
Dividing by Mixed Decimals:
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Transcript:
welcome to math with Mr J [Music] in this video I'm going to cover how to divide decimals by whole numbers now decimals and division show up all throughout math and really throughout life as well so being able to work with decimals and solve division problems involving decimals will be beneficial no matter what class level of math or goal you are working towards whatever your situation may be this should be helpful in better understanding how to divide decimals let's jump into number one where we have 73 and 8 10 divided by three the first thing that we need to do is set this problem up that way we can go through the division process 73 and 8 10 is the dividend the number we are dividing so it goes under the division bar 73 and 8 10 divided by three three is the divisor the number we are dividing by it goes on the outside of the division bar now whenever we have a division problem that involves decimals we always need to check is the divisor a whole number if yes bring the decimal straight up into the answer so for number one our divisor is three that's a whole number so we bring the decimal straight up into the answer now we go through the division process the same process we use when we divide whole numbers so divide multiply subtract bring down repeat we start with divide so we have 7 divided by three how many whole groups of three in seven well two that gets us to 6. now we multiply two times three is six subtract 7 minus six is one and then we bring down now we have 13 and we repeat so we go back to divide 13 divided by three how many whole groups of three in thirteen 4 that gets us to 12. now we multiply 4 times 3 equals 12. subtract 13 minus 12 is 1 and then we bring down the eight now we have 18 and we repeat so we go back to divide 18 divided by three so how many whole groups of three in eighteen six that equals eighteen exactly multiply 6 times 3 18 subtract and we get zero so we went all the way over to the furthest place to the right which was the tenths place for number one and we have a clean cut zero here that tells us that we are done our final answer 24 and six tenths for number two we will see what happens when we do not get that clean cut zero and we need to extend the problem until we do let's move on to number two where we have 46 and 62 hundredths divided by fifteen let's start by setting this problem up 46 and 62 hundredths divided by 15. is the divisor a whole number yes 15 is a whole number that means we bring the decimal straight up into the answer and now we go through the division process so we start with divide we have 4 divided by 15. how many whole groups of 15 and 4 well we can't do that so we need to use the 6 as well and take a look at 46 we have 46 divided by 15. how many whole groups of 15 in 46 well 3 that gets us to 45. make sure that 3 goes above the 6 in 46 not above the 4. since we used both of those digits and we had 46 divided by 15 that 3 needs to go above the six now we multiply 3 times 15 is 45 subtract 46 minus 45 is 1. and then bring down now we have 16 and we repeat so back to divide 16 divided by 15. how many whole groups of 15 in 16 1. now we multiply 1 times 15 is 15 subtract 16 minus 15 is 1. bring down the 2 and now we have 12. so repeat and we go back to divide 12 divided by 15. how many whole groups of 15 in 12 well we can't do that so we need a zero here and now we multiply 0 times 15 is 0. subtract 12 minus zero is 12. so we went all the way over to the furthest place to the right the hundredths place but we have a 12 at the end we do not have a clean cut zero yet that 12 does not mean remainder 12 like when we work with whole numbers this answer is in decimal form so we need to keep it that way what we need to do is extend the problem until we do get a clean cut zero and that will mean the problem is done and we have our final answer we can do this by using placeholder zeros that we can bring down so let's use a placeholder 0 in the thousandths place that we can bring down and continue this problem in order to work towards that final answer now remember zeros to the right of a decimal do not change the value of that decimal of that number so we are able to use this strategy let's bring this zero down and continue the problem so now we have 120 and we repeat so we go back to divide we have 120 divided by 15. how many whole groups of 15 in 120 well eight and that hits 120 exactly now we Multiply eight times fifteen is 120 subtract 120 minus 120 is zero so now we have that clean cut zero and we are done with this problem we have our final answer 3 and 108 thousandths so there you have it there's how you divide decimals by whole numbers remember always check is the divisor a whole number if so bring that decimal straight up into the answer now in this video the divisors were whole numbers I will cover what we do when the divisor is not a whole number in other videos those links are down in the description I hope that helped thanks so much for watching until next time peace |
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12030 | https://tutorial.math.lamar.edu/classes/alg/integerexponents.aspx | Paul's Online Notes
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Section 1.1 : Integer Exponents
We will start off this chapter by looking at integer exponents. In fact, we will initially assume that the exponents are positive as well. We will look at zero and negative exponents in a bit.
Let’s first recall the definition of exponentiation with positive integer exponents. If a is any number and n is a positive integer then,
an=a⋅a⋅a⋅⋯⋅an times
So, for example,
35=3⋅3⋅3⋅3⋅3=243
We should also use this opportunity to remind ourselves about parenthesis and conventions that we have in regard to exponentiation and parenthesis. This will be particularly important when dealing with negative numbers. Consider the following two cases.
(−2)4and−24
These will have different values once we evaluate them. When performing exponentiation remember that it is only the quantity that is immediately to the left of the exponent that gets the power.
In the first case there is a parenthesis immediately to the left so that means that everything in the parenthesis gets the power. So, in this case we get,
(−2)4=(−2)(−2)(−2)(−2)=16
In the second case however, the 2 is immediately to the left of the exponent and so it is only the 2 that gets the power. The minus sign will stay out in front and will NOT get the power. In this case we have the following,
−24=−(24)=−(2⋅2⋅2⋅2)=−(16)=−16
We put in some extra parenthesis to help illustrate this case. In general, they aren’t included and we would write instead,
−24=−2⋅2⋅2⋅2=−16
The point of this discussion is to make sure that you pay attention to parenthesis. They are important and ignoring parenthesis or putting in a set of parenthesis where they don’t belong can completely change the answer to a problem. Be careful. Also, this warning about parenthesis is not just intended for exponents. We will need to be careful with parenthesis throughout this course.
Now, let’s take care of zero exponents and negative integer exponents. In the case of zero exponents we have,
a0=1provided a≠0
Notice that it is required that a not be zero. This is important since 00 is not defined. Here is a quick example of this property.
(−1268)0=1
We have the following definition for negative exponents. If a is any non-zero number and n is a positive integer (yes, positive) then,
a−n=1an
Can you see why we required that a not be zero? Remember that division by zero is not defined and if we had allowed a to be zero we would have gotten division by zero. Here are a couple of quick examples for this definition,
5−2=152=125(−4)−3=1(−4)3=1−64=−164
Here are some of the main properties of integer exponents. Accompanying each property will be a quick example to illustrate its use. We will be looking at more complicated examples after the properties.
Properties
| | |
--- |
| 1. anam=an+m | Example : a−9a4=a−9+4=a−5 |
| 2. (an)m=anm | Example : (a7)3=a(7)(3)=a21 |
| 3. anam=⎧⎨⎩an−m1am−n,a≠0 | Example : a4a11=a4−11=a−7a4a11=1a11−4=1a7=a−7 |
| 4. (ab)n=anbn | Example : (ab)−4=a−4b−4 |
| 5. (ab)n=anbn,b≠0 | Example : (ab)8=a8b8 |
| 6. (ab)−n=(ba)n=bnan | Example : (ab)−10=(ba)10=b10a10 |
| 7. (ab)−n=1(ab)n | Example : (ab)−20=1(ab)20 |
| 8. 1a−n=an | Example : 1a−2=a2 |
| 9. a−nb−m=bman | Example : a−6b−17=b17a6 |
| 10. (anbm)k=ankbmk | Example : (a4b−9)3=a(4)(3)b(−9)(3)=a12b−27 |
| 11. (anbm)k=ankbmk | Example : (a6b5)2=a(6)(2)b(5)(2)=a12b10 |
Notice that there are two possible forms for the third property. Which form you use is usually dependent upon the form you want the answer to be in.
Note as well that many of these properties were given with only two terms/factors but they can be extended out to as many terms/factors as we need. For example, property 4 can be extended as follows.
(abcd)n=anbncndn
We only used four factors here, but hopefully you get the point. Property 4 (and most of the other properties) can be extended out to meet the number of factors that we have in a given problem.
There are several common mistakes that students make with these properties the first time they see them. Let’s take a look at a couple of them.
Consider the following case.
Correct : ab−2=a1b2=ab2Incorrect : ab−2≠1ab2
In this case only the b gets the exponent since it is immediately off to the left of the exponent and so only this term moves to the denominator. Do NOT carry the a down to the denominator with the b. Contrast this with the following case.
(ab)−2=1(ab)2
In this case the exponent is on the set of parenthesis and so we can just use property 7 on it and so both the a and the b move down to the denominator. Again, note the importance of parenthesis and how they can change an answer!
Here is another common mistake.
Correct : 13a−5=131a−5=13a5Incorrect : 13a−5≠3a5
In this case the exponent is only on the a and so to use property 8 on this we would have to break up the fraction as shown and then use property 8 only on the second term. To bring the 3 up with the a we would have needed the following.
1(3a)−5=(3a)5
Once again, notice this common mistake comes down to being careful with parenthesis. This will be a constant refrain throughout these notes. We must always be careful with parenthesis. Misusing them can lead to incorrect answers.
Let’s take a look at some more complicated examples now.
Example 1 Simplify each of the following and write the answers with only positive exponents.
(4x−4y5)3
(−10z2y−4)2(z3y)−5
n−2m7m−4n−3
5x−1y−4(3y5)−2x9
(z−5z−2x−1)6
(24a3b−86a−5b)−2
Show All Solutions Hide All Solutions
Show Discussion
Note that when we say “simplify” in the problem statement we mean that we will need to use all the properties that we can to get the answer into the required form. Also, a “simplified” answer will have as few terms as possible and each term should have no more than a single exponent on it.
There are many different paths that we can take to get to the final answer for each of these. In the end the answer will be the same regardless of the path that you used to get the answer. All that this means for you is that as long as you used the properties you can take the path that you find the easiest. The path that others find to be the easiest may not be the path that you find to be the easiest. That is okay.
Also, we won’t put quite as much detail in using some of these properties as we did in the examples given with each property. For instance, we won’t show the actual multiplications anymore, we will just give the result of the multiplication.
a (4x−4y5)3 Show Solution
For this one we will use property 10 first.
(4x−4y5)3=43x−12y15
Don’t forget to put the exponent on the constant in this problem. That is one of the more common mistakes that students make with these simplification problems.
At this point we need to evaluate the first term and eliminate the negative exponent on the second term. The evaluation of the first term isn’t too bad and all we need to do to eliminate the negative exponent on the second term is use the definition we gave for negative exponents.
(4x−4y5)3=64(1x12)y15=64y15x12
We further simplified our answer by combining everything up into a single fraction. This should always be done.
The middle step in this part is usually skipped. All the definition of negative exponents tells us to do is move the term to the denominator and drop the minus sign in the exponent. So, from this point on, that is what we will do without writing in the middle step.
b (−10z2y−4)2(z3y)−5 Show Solution
In this case we will first use property 10 on both terms and then we will combine the terms using property 1. Finally, we will eliminate the negative exponents using the definition of negative exponents.
(−10z2y−4)2(z3y)−5=(−10)2z4y−8z−15y−5=100z−11y−13=100z11y13
There are a couple of things to be careful with in this problem. First, when using the property 10 on the first term, make sure that you square the “-10” and not just the 10 (i.e. don’t forget the minus sign…). Second, in the final step, the 100 stays in the numerator since there is no negative exponent on it. The exponent of “-11” is only on the z and so only the z moves to the denominator.
c n−2m7m−4n−3 Show Solution
This one isn’t too bad. We will use the definition of negative exponents to move all terms with negative exponents in them to the denominator. Also, property 8 simply says that if there is a term with a negative exponent in the denominator then we will just move it to the numerator and drop the minus sign.
So, let’s take care of the negative exponents first.
n−2m7m−4n−3=m4n3m7n2
Now simplify. We will use property 1 to combine the m’s in the numerator. We will use property 3 to combine the n’s and since we are looking for positive exponents we will use the first form of this property since that will put a positive exponent up in the numerator.
n−2m7m−4n−3=m5n7
Again, the 7 will stay in the denominator since there isn’t a negative exponent on it. It will NOT move up to the numerator with the m. Do not get excited if all the terms move up to the numerator or if all the terms move down to the denominator. That will happen on occasion.
d 5x−1y−4(3y5)−2x9 Show Solution
This example is similar to the previous one except there is a little more going on with this one. The first step will be to again, get rid of the negative exponents as we did in the previous example. Any terms in the numerator with negative exponents will get moved to the denominator and we’ll drop the minus sign in the exponent. Likewise, any terms in the denominator with negative exponents will move to the numerator and we’ll drop the minus sign in the exponent. Notice this time, unlike the previous part, there is a term with a set of parenthesis in the denominator. Because of the parenthesis that whole term, including the 3, will move to the numerator.
Here is the work for this part.
5x−1y−4(3y5)−2x9=5(3y5)2xy4x9=5(9)y10xy4x9=45y6x10
e (z−5z−2x−1)6 Show Solution
There are several first steps that we can take with this one. The first step that we’re pretty much always going to take with these kinds of problems is to first simplify the fraction inside the parenthesis as much as possible. After we do that we will use property 5 to deal with the exponent that is on the parenthesis.
(z−5z−2x−1)6=(z2x1z5)6=(xz3)6=x6z18
In this case we used the second form of property 3 to simplify the z’s since this put a positive exponent in the denominator. Also note that we almost never write an exponent of “1”. When we have exponents of 1 we will drop them.
f (24a3b−86a−5b)−2 Show Solution
This one is very similar to the previous part. The main difference is negative on the outer exponent. We will deal with that once we’ve simplified the fraction inside the parenthesis.
(24a3b−86a−5b)−2=(4a3a5b8b)−2=(4a8b9)−2
Now at this point we can use property 6 to deal with the exponent on the parenthesis. Doing this gives us,
(24a3b−86a−5b)−2=(b94a8)2=b1816a16
Before leaving this section we need to talk briefly about the requirement of positive only exponents in the above set of examples. This was done only so there would be a consistent final answer. In many cases negative exponents are okay and in some cases they are required. In fact, if you are on a track that will take you into calculus there are a fair number of problems in a calculus class in which negative exponents are the preferred, if not required, form.
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12031 | https://www.quora.com/Why-is-sin-%CE%B8-sin-180-%CE%B8 | Something went wrong. Wait a moment and try again.
Angle Sum Identity
Sinusoidal Function
Sine and Cosine
Mathematical Identities
Sine Theta
Mathematical Sciences
Trigonometry Maths
5
Why is sin θ = sin (180 - θ)?
Rodolpho Martins de Brito Cavalcanti
Manager - Fraud Analytics (2019–present)
·
8y
Hi there!
The most effective way to understand trigonometry functions and identities is to draw the unit circle. The unit circle is one such as its radius is 1.
Then, you draw a cartesian axis on top of the circle: your x axis will be cos() and the y axis will be sin(). With that you’re ready to go.
The first step is to place an arbitrary point on top of the circle, in my example I have placed it on the first quadrant for simplicity purposes. This point has a angle t with the cos() axis.
The second step is to place a point 180°-t degrees distant from our cos() axis.
With the two points you’ll then
Hi there!
The most effective way to understand trigonometry functions and identities is to draw the unit circle. The unit circle is one such as its radius is 1.
Then, you draw a cartesian axis on top of the circle: your x axis will be cos() and the y axis will be sin(). With that you’re ready to go.
The first step is to place an arbitrary point on top of the circle, in my example I have placed it on the first quadrant for simplicity purposes. This point has a angle t with the cos() axis.
The second step is to place a point 180°-t degrees distant from our cos() axis.
With the two points you’ll then project those points on the sin() and cos() axis, this is when you’ll figure out the identities:
sin(t) = sin(180-t)
cos(t) = -cos(180-t)
Related questions
What is the relationship between sin(θ) and sin (180° − θ)?
What is the mathematical explanation for the equality of Sin (180-θ) and Cos θ?
How do you verify that sin (180° - θ) = sin(θ).?
What is the distance between the points (sinθ, cosθ) and (cos θ, -sin θ)?
On a Cartesian plane, P (-√7; 3) and PÔR = θ. What is sin(-θ)?
Philip Lloyd
Former Specialist Calculus Teacher and Mentor..
·
Upvoted by
Mike Drake
, B.S. Physics & Mathematics, Saint Joseph's University, Philadelphia and
Ramzan Yousaf
, MSc physics Physics & Mathematics, University of Peshawar (2020) · Author has 6.8K answers and 52.8M answer views
·
Updated 4y
The BEST way to understand this concept is not using established formulae but to use simple diagrams which clearly show HOW and WHY these two values are equal.
Clearly this idea will work for any acute angle θ
We could simp...
The BEST way to understand this concept is not using established formulae but to use simple diagrams which clearly show HOW and WHY these two values are equal.
Clearly this idea will work for any acute angle θ
We could simp...
Girija Warrier
Studied at Sufficiently Educated
·
Author has 5.9K answers and 13.9M answer views
·
Updated 10mo
As angle θ lies in the first quadrant, where, 90° < θ < 0°
& (180° - θ ) lies in the 2nd quadrant.
(positive θ means ray is rotated anticlockwise) , (negative θ means ray is rotated clockwise) as shown in the image..
And we know that sin θ is the ratio of perpendicular( opposite side) to the hypotenuse. & perpendiculars, in both the quadrants( 1st & 2nd quadrant) are positive as these are going upwards. But in 3rd & 4th quadrant, perpendiculars become negative as these will be downwards.
[●★This way, in right triangles, OAB, if theeta = 30°, Sin 30° = +veAB/ hypotenuse = 1/2,
& in right triangle OC
As angle θ lies in the first quadrant, where, 90° < θ < 0°
& (180° - θ ) lies in the 2nd quadrant.
(positive θ means ray is rotated anticlockwise) , (negative θ means ray is rotated clockwise) as shown in the image..
And we know that sin θ is the ratio of perpendicular( opposite side) to the hypotenuse. & perpendiculars, in both the quadrants( 1st & 2nd quadrant) are positive as these are going upwards. But in 3rd & 4th quadrant, perpendiculars become negative as these will be downwards.
[●★This way, in right triangles, OAB, if theeta = 30°, Sin 30° = +veAB/ hypotenuse = 1/2,
& in right triangle OCD, sin(180°-30°) = sin<DOC= +veCD/ hypotenuse = 1/2 ●★ ]
[●★But it is not so in case of cosine ratio
In triangle OAB, cos30° = +veOB/ hypotenuse = √3/2
& in right triangle OCD, cos <DOC = cos(180°-30°) = -veOD/hypotenuse = -√3/2 ●★]
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
6y
Originally Answered: Why does sin (180-theta) equal sin theta, with respect to the diagram?
·
We don’t refer to SOH CAH TOA for this work.
The definitions of sin (θ), cos (θ) and tan (θ) are clearly shown on this diagram.
Cutting to the chase, I will focus on the basic angle of 30 degrees
So if I just refer to a ge...
We don’t refer to SOH CAH TOA for this work.
The definitions of sin (θ), cos (θ) and tan (θ) are clearly shown on this diagram.
Cutting to the chase, I will focus on the basic angle of 30 degrees
So if I just refer to a ge...
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Related questions
What is the value of sin θ when θ = 90 degrees?
What is the difference between sin θ and sin(θ)?
What is θ, if Sin 90- θ=52.3?
In the expression sin(θ°) /θ = π / 180, how do I find the value of θ?
Prove that
cos
(
θ
)
−
sin
(
θ
)
+
1
cos
(
θ
)
+
sin
(
θ
)
−
1
=
cosec
(
θ
)
+
cot
(
θ
)
?
Assistant
Bot
·
1y
The equation sinθ=sin(180∘−θ) is a consequence of the properties of the sine function and its relationship to the unit circle.
Explanation:
Unit Circle Definition: The sine of an angle in the unit circle is defined as the y-coordinate of the point on the circle corresponding to that angle.
Reference Angle: When you have an angle θ, the angle 180∘−θ is in the second quadrant (if θ is in the first quadrant) or another quadrant depending on the value of θ. The y-coordinate (sine value) remains the same for both angles because they share the same ver
The equation sinθ=sin(180∘−θ) is a consequence of the properties of the sine function and its relationship to the unit circle.
Explanation:
Unit Circle Definition: The sine of an angle in the unit circle is defined as the y-coordinate of the point on the circle corresponding to that angle.
Reference Angle: When you have an angle θ, the angle 180∘−θ is in the second quadrant (if θ is in the first quadrant) or another quadrant depending on the value of θ. The y-coordinate (sine value) remains the same for both angles because they share the same vertical position on the unit circle.
Symmetry: The sine function is symmetric about the line y=x for angles in the first and second quadrants. Thus:
For θ in the first quadrant (0° to 90°), sinθ gives a positive value.
For 180∘−θ, which lies in the second quadrant, the sine value is also positive and equal in magnitude to sinθ.
Conclusion:
Thus, the equality sinθ=sin(180∘−θ) holds true for all angles θ in degrees, illustrating the symmetry of the sine function in the unit circle.
Pochu Vinay
Mechanical Engineer at General Electric (company) (2018–present)
·
7y
Look at sin function graph, the value of sin(x) and sin(180-x) will be same.
Look at sin function graph, the value of sin(x) and sin(180-x) will be same.
William Mccoy
Bachelor of Science in 7-12 Education, Miami University (Ohio) (Graduated 1980)
·
Author has 2.6K answers and 27M answer views
·
8y
The given identity, sin θ = sin (180 - θ), is true based upon a proven identity: sin (θ ‒ Ф), the “Difference of Two Angles” formula:
sin (θ ‒ Ф) = sin Ѳ cos Ф ‒ cos Ѳ sin Ф .
Using the “Difference of Two Angles” formula, we can manipulate the right side of the given equation as follows:
sin (180° ‒ Ѳ) = sin 180° cos Ѳ ‒ cos 180° sin Ѳ
= (0) cos Ѳ ‒ (‒1) sin Ѳ
= 0 ‒ (‒ sin Ѳ)
= (‒1)(‒ sin Ѳ)
= sin Ѳ
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Neil Kelcey
Former Public School Math Teacher (retired) at Toronto District School Board (1971–2002)
·
Author has 86 answers and 243.2K answer views
·
6y
Originally Answered: Why does sin (180-theta) equal sin theta, with respect to the diagram?
·
Q posed:
Why does sin (180-theta) equal sin theta, with respect to the diagram?
I am going to offer another graph y = sin(θ), which, in conjunction with Philip’s unit circle diagrams, may help you understand the answer to your question:
Point E has coordinates (θ, sin(θ)). Point F has co=ordinates (180-θ), sin(180-θ)).
Notice that the sine values are the same – ie 0.5 – even though the angle measure are different, but related.
.: sin(θ) = sin(180-θ), for all values of θ, where 0 ≤ θ ≤ 180
Hopefully this will help? Cheers! :)
Q posed:
Why does sin (180-theta) equal sin theta, with respect to the diagram?
I am going to offer another graph y = sin(θ), which, in conjunction with Philip’s unit circle diagrams, may help you understand the answer to your question:
Point E has coordinates (θ, sin(θ)). Point F has co=ordinates (180-θ), sin(180-θ)).
Notice that the sine values are the same – ie 0.5 – even though the angle measure are different, but related.
.: sin(θ) = sin(180-θ), for all values of θ, where 0 ≤ θ ≤ 180
Hopefully this will help? Cheers! :)
Elf Cow
Occupation: High School Student, Status: Too busy
·
Author has 76 answers and 78.1K answer views
·
2y
Originally Answered: Why does sin (θ-180) =sin (180+θ)?
·
As you can see from the unit circle, no matter what angle you choose, if you add or subtract 180 degrees (or equivalently, pi radians), the resulting angles will be equivalent. The official “term” for these equivalent angles are “co-terminal angles”. For example, if you choose theta to equal 90 degrees, 90+180=270 degrees, and 90–180=-90 degrees. As you can see from the unit circle, -90 degrees is exactly the same as 270 degrees! (Thus “co-terminal angle”). This is kind of a not really “deep description” or “deriving” sort of explanation (sorry) and kind of relies on previous knowledge about t
As you can see from the unit circle, no matter what angle you choose, if you add or subtract 180 degrees (or equivalently, pi radians), the resulting angles will be equivalent. The official “term” for these equivalent angles are “co-terminal angles”. For example, if you choose theta to equal 90 degrees, 90+180=270 degrees, and 90–180=-90 degrees. As you can see from the unit circle, -90 degrees is exactly the same as 270 degrees! (Thus “co-terminal angle”). This is kind of a not really “deep description” or “deriving” sort of explanation (sorry) and kind of relies on previous knowledge about trigonometry and the unit circle. Hope this helped!
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Gopal Menon
B Tech in Chemical Engineering, Indian Institute of Technology, Bombay (IITB) (Graduated 1975)
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Author has 10.2K answers and 15.2M answer views
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8y
Do you know the addition formulas for trigonometric ratios? If you do, then the solution is very simple.
sin(a - b) = sin(a) cos(b) - cos(a) sin(b)
Hence, sin(180 - x) = sin(180) cos(x) - cos(180) sin(x)
sin(180) = 0 and cos((180) = -1
Therefore, sin(180 - x) = sin(x)
Mohammad Afzaal Butt
Former Field Operation Manager at ABB (company) (1998–2017)
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Author has 24.6K answers and 22.9M answer views
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6y
sinθ=sin(180∘−θ)
Right hand side
sin(180∘−θ)=sin(180∘)cosθ−cos(180∘)sinθ
Robert Paxson
BSME in Mechanical Engineering, Lehigh University (Graduated 1983)
·
Author has 3.9K answers and 4M answer views
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2y
Originally Answered: Why does sin (θ-180) =sin (180+θ)?
·
The left-hand side simplifies to [math]-\sin{(\theta)}[/math]:
The right-hand side simplifies to [math]-\sin{(\theta)}[/math]:
Since both sides are equal to the same thing, then they are equal to each other:
The left-hand side simplifies to [math]-\sin{(\theta)}[/math]:
[math]\sin{(\theta-180^\circ)}=\sin{(\theta)}\cos{(180^\circ)}-\cos{(\theta)}\sin{(180^\circ)}[/math]
[math]\sin{(\theta-180^\circ)}=\sin{(\theta)}(-1)-\cos{(\theta)}(0)[/math]
[math]\sin{(\theta-180^\circ)}=-\sin{(\theta)}[/math]
The right-hand side simplifies to [math]-\sin{(\theta)}[/math]:
[math]\sin{(180^\circ+\theta)}=\sin{(180^\circ)}\cos{(\theta)}+\cos{(180^\circ)}\sin{(\theta)}[/math]
[math]\sin{(180^\circ+\theta)}=(0)\cos{(\theta)}+(-1)\sin{(\theta)}[/math]
[math]\sin{(180^\circ+\theta)}=-\sin{(\theta)}[/math]
Since both sides are equal to the same thing, then they are equal to each other:
Mark Mushi
Work in field and this is NOT legal advice.
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Author has 335 answers and 1.1M answer views
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8y
Sin= opposite over adjacent for a triangle. An angle of 0 degrees and 180 degrees is essentially not a triangle but a straight line. For an angle of 0 degrees, the opposite side length would be 0 regardless of the length for the adjacent side. Similarly, for an angle of 180 degrees (opposite direction of angle 0) the opposite side length again would be zero regardless of the length. Thus, sine 0 = sine of 180 = Zero. (The length of the supposed opposite side of the triangle)
Related questions
What is the relationship between sin(θ) and sin (180° − θ)?
What is the mathematical explanation for the equality of Sin (180-θ) and Cos θ?
How do you verify that sin (180° - θ) = sin(θ).?
What is the distance between the points (sinθ, cosθ) and (cos θ, -sin θ)?
On a Cartesian plane, P (-√7; 3) and PÔR = θ. What is sin(-θ)?
What is the value of sin θ when θ = 90 degrees?
What is the difference between sin θ and sin(θ)?
What is θ, if Sin 90- θ=52.3?
In the expression sin(θ°) /θ = π / 180, how do I find the value of θ?
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cos
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θ
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−
sin
(
θ
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+
1
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(
θ
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+
sin
(
θ
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−
1
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cosec
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θ
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cot
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How do you prove that
1
sin
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θ
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cos
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sin
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cos
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θ
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2
sin
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cos
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How do you prove that
e^i^θ=cos (θ)+i sin(θ)
?
What is the value of sin θ when θ is equal to 0?
The angle between vector P and vector Q is θ. What is (𝑃̂×𝑄) ̂? (a) sin θ (b) PQ sin θ (c) PQ (d) PQ cos θ
Is the statement "sin(θ) is approximately equal to θ for small angles" true?
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14: Rates of Chemical Reactions
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14.2: Rates of Chemical Reactions
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60725
Robert E. Belford
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Contents
1. Home
2. Campus Bookshelves
3. University of Arkansas Little Rock
4. Chem 1403: General Chemistry 2
5. Text
6. 14: Rates of Chemical Reactions
7. 14.2: Rates of Chemical Reactions
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Text
Front Matter
1: General Information
10: Review
11: Intermolecular Forces and Liquids
12: Solids
13: Solutions
14: Rates of Chemical Reactions
15: Equilibria
16: Acids and Bases
17: Aqueous Equilibria
18: Entropy and Free Energy
19: Electron Transfer Reactions
20: Coordination Chemistry
21: Nuclear Chemistry
Appendix 1: Google Sheets
Back Matter
14.2: Rates of Chemical Reactions
Last updated Sep 17, 2025
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14.1: Prelude
14.3: Reaction Conditions and Rate
Page ID 60725
Robert Belford
University of Arkansas at Little Rock
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Introduction
2. Reaction Graphs
3. Reaction Rate of Hydrolysis of Aspirin
4. Relative Rates
5. In Class Activity
6. Test Yourself
Introduction
The rate of reaction, often called the "reaction velocity" and is a measure of how fast a reaction occurs. As a reaction proceeds in the forward direction products are produced as reactants are consumed, and the rate is how fast this occurs. Sort of like the speed of a car is how its location changes with respect to time, the rate is how the concentration changes over time. This makes sense, because products are produced as the reaction proceeds and they thus get more concentrated, while reactants are consumed and thus become less concentrated. As the balanced equation describes moles of species it is common to use the unit of Molarity (M=mol/l) for concentration and the convention is to use square brackets [ ] to describe concentration of a species. So the concentration of chemical "A" is denoted as:
(14.2.1)[A]with units of mols l for the chemical species "A"
The reaction rate of Chemical "A" is
(14.2.2)R A=Δ[A]Δt
typically in units of M s e c or mol l⋅sec(they mean the same thing), and of course any unit of tim e can be used, depending on how fast the reaction occurs, so an explosion may be on the nanosecond time scale while a very slow nuclear decay may be on a gigayear scale. It should also be mentioned that in the gas phase we often use partial pressure (P A), but for now will stick to M/time.
By convention we say reactants are on the left side of the chemical equation and products on the right
(14.2.3)Reactants→Products
Reactants are consumed, and so their concentrations go down (is negative), while products are produced, and so their concentrations go up. It is important to keep this notation, and maintain the convention that a Δ means the final state minus the initial state. When relating one species to another in a chemical equation we need to introduce a minus sign in front of the reactants
The Rate of Disappearance of Reactants (14.2.4)−Δ[Reactants]Δt Note this is actually positive because it measures the rate of disappearance of the reactants, which is a negative number and the negative of a negative is positive. Note, technically,Δ[Reactants]Δt is the rate of the disappearance of reactants and we use the minus sign when we relate it's change to that of another species in the chemical equation.
The Rate of Formation of Products (14.2.5)Δ[Products]Δt This is the rate at which the products are formed.
Note: It is important to maintain the above convention of using a negative sign in front of the rate of reactants. Later we will see that reactions can proceed in either direction, with "reactants" being formed by "products" (the "back reaction"). When this happens, the actual value of the rate of change of the reactants Δ[Reactants]Δt will be negative, and so eq. 14.2.4 will be positive, as it is taking the negative of a negative. We do not need to worry about that now, but we need to maintain the conventions.
Exercise 14.2.1
Write the rate of reaction for each species in the following generic equation, where capital letters denote chemical species
(14.2.6)A+B→C+D
Answer
rate of disappearance of A rate=−Δ[A]Δt
rate of disappearance of B rate=−Δ[B]Δt
rate of formation of C rate=Δ[C]Δt
rate of formation of D) rate=Δ[D]Δt
Note
If a chemical species is in the gas phase and at constant
temperature
it's concentration can be expressed in terms of its partial pressure. Consider gas "A",
(14.2.7)P AV=n ART[A]=n A V=P A RT
So for systems at constant
temperature
the concentration can be expressed in terms of partial pressure.
Reaction Graphs
It is common to plot the concentration of reactants and products as a function of time. Figure 14.2.1 shows a simple plot for the reaction
(14.2.8)A→B
Note that this reaction goes to completion, and at t=0 the initial concentration of the reactant (purple [A]) was 0.5M and if we follow the reactant curve (purple) it decreases to a bit over 0.1M at twenty seconds and by 60 seconds the reaction is over and all of the reactant had been consumed. On the other hand we could follow the product concentration on the product curve (green) that started at zero, reached a little less than 0.4M after 20 seconds and by 60 seconds the final concentration of 0.5 M was attained.the there was no [B], but after were originally 50 purple particles in the container, which were completely consumed after 60 seconds. You note from eq. 14.1.7 that for stoichiometric coefficients of A and B are the same (one) and so for every A consumed a B was formed and these curves are effectively symmetric.If the reaction had been A→2B then the green curve would have risen at twice the rate of the purple curve and the final concentration of the green curve would have been 1.0M
Figure 14.2.1:Reaction plot of the number of molecules over time
The rate is technically the instantaneous change in concentration over the change in time when the change in time approaches is technically known as the derivative.
(14.2.9)d[A]dt=lim Δt→0Δ[A]Δt
Calculus is not a prerequisite for this class and we can obtain the rate from the graph by drawing a straight line that only touches the curve at one point, the tangent to the curve, as shown by the dashed curves in figure 14.2.1. From this we can calculate the rate of reaction for A and B at 20 seconds
(14.2.10)R A,t=20=−Δ[A]Δt=−0.0M−0.3M 32s−0s=0.009 Ms−1 or 9 mMs−1 and R B,t=20=Δ[B]Δt=0.5M−0.2 32s−0s=0.009 Ms−1 or 9 mMs−1
If needed, review section 1B.5.3on graphing straight line functions and do the following exercise.
Exercise 14.2.2
What is the rate of reaction for the reactant "A" in figure 14.2.1 at 30 seconds?.
Answer
The black line in the figure below is the tangent to the curve for the decay of "A" at 30 seconds. It would have been better to use graph paper with a higher grid
density
that would have allowed us to exactly pick points where the line intersects with the grid lines. Instead, we will estimate the values when the line intersects the axes.
At 30 seconds the slope of the tangent is:
Δ[A]Δt=A 2−A 1 t 2−t 1=(0−18)molecules(42−0)sec=−0.43(molecules second)(14.2.11)R=−Δ[A]Δt=0.43(molecules consumed second)
NOTE:
The value of the rate of consumption of A is a negative number (A 2< A 1)
Since A→B, the curve for the production of B is symmetric to the consumption of A, except that the value of the rate is positive (A 1< A 2)
The Y-axis (50 to 0 molecules) is not realistic, and a more common system would be the molarity (number of molecules expressed as moles inside of a container with a known volume).
You should also note that from figure 14.2.1 that the initial rate is the highest and as the reaction approaches completion the rate goes to zero because no more reactants are being consumed or products are produced, that is, the line becomes a horizontal flat line. We shall see that the rate is a function of the concentration, but it does not always decrease over time like it did in this example.
Exercise 14.2.3
Calculate the rates of reactions for the product curve (B) at 10 and 40 seconds and show that the rate slows as the reaction proceeds.
AnswerTangents to the product curve at 10 and 40 seconds
The red curve represents the tangent at 10 seconds and the dark green curve represents it at 40 seconds.
R B,t=10=0.5−0.1 24−0=20mMs−1 R B,t=40=0.5−0.4 50−0=2mMs−1
and so the reaction is clearly slowing down over time.
Reaction Rate of Hydrolysis of Aspirin
Lets look at a real reaction,the reaction rate for the hydrolysis of aspirin, probably the most commonly used drug in the world,(more than 25,000,000 kg are produced annually worldwide.) Aspirin (acetylsalicylic acid) reacts with water (such as water in body fluids) to give salicylic acid and acetic acid.Because salicylic acid is the actual substance that relieves pain and reduces fever and inflammation, a great deal of research has focused on understanding this reaction and the factors that affect its rate.
Figure 14.2.2: Hydroloysis of aspirin (acetylsalicylic acid) to acetic acid and salicylic acid(Belford cc 0.0)
Data for the hydrolysis of a sample of aspirin are given below and are shown in the adjacent graph.This data were obtained by removing samples of the reaction mixture at the indicated times and analyzing them for the concentrations of the reactant (aspirin) and one of the products (salicylic acid).
Figure 14.2.3: Graph of concentration vs. time for the hydrolysis of aspirin into salicylic acid showing the rate (tangent to the curve) at t=0 and t=200 hrs.
In addition to calculating the rate from the curve we can also calculate the average rate over time from the actual data, and the shorter the time the closer the average rate is to the actual rate. Let's calculate the average rate for the production of salicylic acid between the initial measurement (t=0) and the second measurement (t=2 hr).
Average rate(t=2.0−0.0 h)=[salicylic acid]2−[salicylic acid]0 2.0 h−0.0 h
=0.040×10−3 M−0.000 M 2.0 h−0.0 h=2×10−5 Mh−1=20μMh−1
Exercise 14.2.4
What is the average rate of salicylic acid production between the last two measurements of 200 and 300 hours, and before doing the calculation, would you expect it to be greater or less than the initial rate? (You may look at the graph)
Answer
It should be clear from the graph that the rate decreases.
rate(t=300−200 h)=[salicylic acid]300−[salicylic acid]200 300 h−200 h
=3.73×10−3 M−2.91×10−3 M 100 h=8.2×10−6 Mh−1=8μMh−1
Relative Rates
The storichiometric coefficients of the balanced reaction relate the rates at which reactants are consumed and products are produced .If we want to relate the rate of reaction of two or more species we need to take into account the stoichiometric coefficients, consider the following reaction for the decomposition of ammonia into nitrogen and hydrogen.
(14.2.12)2 NHA 3NA 2+3HA 2
It is clear from the above equation that for mass to be conserved, every time two ammonia are consumed, one nitrogen and three hydrogen are produced. This means that the rate ammonia consumption is twice that of nitrogen production, while the rate of hydrogen production is three times the rate of nitrogen production.
Figure 14.2.2: This graph shows three curves for the concentration of each species in eq. 14.1.8. Note, the slope of the line is drawn in at the tangent to each curve at 500 seconds and they are related to each other by the stoichiometric coefficient of each reactant and product.(Belford)
If you take the value at 500 seconds in figure 14.1.2 and divide by the stoichiometric coefficient of each species, they all equal the same value. Equation 14-1.9 is a generic equation that can be used to relate the rates of production and consumption of the various species in a chemical reaction where capital letter denote chemical species, and small letters denote their stoichiometric coefficients when the equation is balanced.
(14.2.13)aA+bB→cC+dD
We can normalize the above rates by dividing each species by its coefficient, which comes up with a relative rate of reaction
(14.2.14)R relative=−1 aΔ[A]Δt=−1 bΔ[B]Δt=1 cΔ[C]Δt=1 dΔ[D]Δt⏟Relative Rate of Reaction
There are two important things to note here:
There are actually 5 different Rate expressions for the above equation, The relative rate, and the rate of reaction with respect to each chemical species, A, B, C & D. If you can measure any of the species (A,B,C or D) you can use the above equality to calculate the rate of the other species.
In relating the reaction rates, the reactants were multiplied by a negative sign, while the products were not.
Exercise 14.2.5
What is the rate of ammonia production for the Haber process (Equation 14.2.12) if the rate of hydrogen consumption is -0.458M/min?
Answer
−1 3Δ[H 2]Δt=1 2Δ[NH 3]Δt Δ[NH 3]Δt=−2 3Δ[H 2]Δt=−2 3(−0.458M m i n)=0.305mol L⋅min
In Class Activity
ADAPT 14.2.1
Test Yourself
Homework:Section 14.2
Query 14.2.1
Query 14.2.2
Contributors
Robert E. Belford (University of Arkansas Little Rock; Department of Chemistry). The breadth, depth and veracity of this work is the responsibility of Robert E. Belford, rebelford@ualr.edu. You should contact him if you have any concerns. This material has both original contributions, and content built upon prior contributions of the LibreTexts Community and other resources,including but not limited to:
Liliane Poirot
Modifications or content borrowed from
Paul Flowers et al. Open Stax
Joshua Halpern, Scott Sinex, Scott Johnson
This page titled 14.2: Rates of Chemical Reactions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Robert Belford.
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12034 | https://agjsr.agu.edu.bh/publications/paperpdf/209 | A.rab Gulf J. SC~nI. Res . 1(1), pp. 159-164 (1983)
Coset Diagrams and Relations for PSL (2,Z)
G. Higman and Q. Mushtaq
Mathematical Institute, University of Oxford, Oxford, U.K.
ABSTRACT. A diagramatic argument, called coset diagrams for the modular group PSL(2,Z), is used to prove the results stated in this paper.
Let G denote the subgroup of the modular group PSL(2,Z), generated by the linear-fractional transformations x and y where x and y are respectively defined as z-+ -l i z and z-+ (z - 1) / z.
A diagram with n vertices depicts a (transitive) permutation representation of the modular group: fixed points of x and y are defined by heavy dots, and 3-cycles of y by triangles whose vertices are permuted anti-clockwise by y; and any two vertices which are interchanged by x are joined by an edge.
In this paper we have shown that the coset diagram for the action of G on the rational projective line is connected, and transitive. Using these coset dia grams we have shown that the group PSL(2,Z) is generated by the Iinear
2
fractional transformations x and y and that x = i = I are defming relations for PSL(2,Z) .
A diagramatic argument, called coset diagrams for the modular group PSL(2,Z), is used to prove the results in this paper.
Let G denote the subgroup of the modular group, generated by x and y where x and yare respectively defined as z- -liz and z_ (z - 1)/z .
A diagram with n vertices depicts a (transitive) pennutation representation of the modular group: fixed points of x and y are defined by heavy dots , and 3-cycles of yby triangles whose vertices are pennuted anti-clockwise by y; any two vertices are
159 160 G. Higman and Q. Mushtaq
interchanged by x are joined by an edge.
In this paper, we have shown that the coset diagram for the natural action of G, on the rational projective line, is connected and the action is transitive. A new proof is given to show that PSL(2,Z) is generated by the linear-fractional transformations x and y and that x 2 = y3 = I are defining relations for PSL(2,Z).
If t is the transformation Z~ l/z so that t belongs not to the modular group PSL(2,Z) but to PGL(2,Z) then x,y,t satisfy: x 2 = y3= t2 = (xt)2 = (yt)2 = 1 (i) 2Once it is shown that G has x = y3 = I as defining relations and that G is the whole of PSL (2,Z) , it is clear that relations (i) are defining relations for G =
and that G is the whole of PGL(2,Z).
This theory of coset diagrams has also proved useful in determining generators of PSL(2,p) or PGL(2,p), where p is a prime number. Coset diagrams for the action of the modular group, on projective lines over F p , gives some interesting information.
Before we come to our main results we shall make the following remarks:
(i) If k '" 1,0,00 then of the vertices k,ky,k y 2 of a triangle, in a coset diagram for the action of PGL(2,Z) on any subset of the real projective line, one vertex is negative and two are positive.
(ii) Let k = ± a/b where a,b are positive integers with no common factor. For k '
0,00 we define Ilk lt= max(a,b). Clearly , Il kll = Ilkxll and if k is negative, then Ilkll is less than Ilky ll and I lky2 11.
(iii) We shall use arrow head on an edge to indicate its direction from negative to positive vertex .
Theorem 1 The action of PGL(2,Z) on the rational projective line is connected.
Proof
To prove this we need only to show that for any rational number k there is a path joining k to 00. Since one of k,kx is negative, therefore without any loss of generality, we can assume that k is negative .
Let k = ko be a negative rational number. Then kox is positive and if kox =1= 1 then by remark (i) there is just one negative number (vertex), say ki, in the triangle con taining kox, which cannot be kox. That is , we have a fragment of the coset diagram of
one of the forms:
Fig. 1 161 Coset Diagrams and Relations for PSL (2,Z)
By remark (ii ) we note that l ikol l > Ilk,ll. If we now consider k, then k,xy and k, xy 2will be the vertices of the triangle containing k,x as its third vertex . Since k, x is posi tive, therefore as in the case of ko, of k,xy and k,xy2 we get just one negative vertex, say k2 such that Ilk,1 1 > Ilk21 1. If we continue like this and follow the arrows from
k = ko in Fig. 2, we get a sequence of negative rational numbers ko,k"k2" .. such
that Ilkoll > llk, I I>llk& ..
o
...
--
Fig. 2
The decreasing sequence of positive integers must terminate , and it can terminate only because ultimately we reach a triangle with the vertices 1,0 and 00.
Fig. 3
A sequence of negative rational numbers ko,k J ,k2, ... such that Ilkoll > llk l ll > llk 211 .. . shows that there is a path joining k = ko to 00. This implies that every rational num ber occurs in the diagram and that the diagram for the action of G on the rational projective line is connected .
Corollary 2 The action of G on the rational projective line is transitive. Proof
We shall prove transitivity of the action, by showing that, if there is a path from a rational number p to a rationaJ number q then there exists some g in G such that pg = q.
If pg = q then pxxg = q and since one of p,px is negative, we can assume without any loss of generality that p is negative .
Let p = ko and q = ki for some i. Then from the coset diagram (Fig: 2) we note 162 G. Higman and Q. Mushtaq
that each kj .. t is either kjxy or kjxyl. This implies that q = p xyE] xyE2 ... xli where each t j = 1 or 2 . If xyEt x/2 .. , xyEi = g then q = pg, where q is in G. So the action of G on the rational projective line is transitive.
Theorem 3
The group PSL(2,Z) is generated by the linear-fractional transformations
x and y.
Proof
Let h in PSL (2,Z) be such that k = 00 h for a rational number k. By corollary
2, since the action of G on the rational projective line is transitive, therefore kg = c')
for some g in G. So 00 = kg = (ooh)g. Hence 00 is a fixed point of hg . This means that hg is a linear-fractional transformation z ~ (az + b)/(CZ + d) with c = 0. Since ad - bc = I, therefore a = d = ± 1 which then implies that hg is z ~ z ± b and xy: z ~ z + 1 further implies that hg = (xy):!: b . This shows that hg and hence h is in G, proving that x and y are generators of the group G.
Theorem 4 Relations X2=y3= 1 are defining relations for PSL(2,Z).
Proof
Suppose X2=y3= 1 are not defining relations of PSL(2,Z). Then there is a rel ation of the form xyE xyE2 ... xyEi = 1 where m;::' 1, t i = ± 1 and i = · 1,2, .. . ,m . We know that neither x nor y can be I .
The coset diagram (Fig .2) depicts that it does not contain any closed circuit. apart
from the circuit in the triangle containing 00. For if it contains the closed circuit (Fig. 4). and ko,k],k 2.'" ,km are the vertices of the triangles in the diagram such that ko< 0, then this leads to the contradiction Ilkoll>llkdl> ... llkmll>llkoll. So the coset diagram
(Fig. 2) does not contain any other closed circuit , apart from the circuit in the triangle
containing 00 as its vertex.
I<~ - - . " . -__
\<,
Fig. 4
This shows that there are points in the diagram whose distance from the point 00 is arbitrary large. Choose k with kllk 1 11> l lk 21 1 .. '>llkmll and in particular km =1= ko. Thus xyEt xl 2 ... xyEm t= 1and so x 2 = y3 = 1 are defining relations for the modular group PSL(2,Z) .Coset Diagrams and Relations for PSL (2,Z) 163
Acknowledgements
The second author wishes to thank the Royal Commission for the Exhibition of 1851 and ORS-Committee of the Vice-Chancellors and the Principals of the U.K Uni versities and Colleges for their financial support. 164 G . Higman and Q. Mushtaq
PSL(2,Z) o.r}J ..;.,li~
Jl:.!..o ~J ~~ . [.
•.>.>d\ ~\ ~j".o.-s\ '~j".o.-s\ ....... \::.:- ,~y~)\ -4-v'
p ......... ~I ~ ~I ~ ~I ~Lt.,r.ll ~."...k ..:. \Q,.t'- _,I
o r,D ,PSL (2, Z) C~~I Of jLl c) ...':.l.1 ,.::.;\~~ . ~I Ii". J ~ J~I ~ ~I ~\;;jl ~
o..l.ly PSL (2,Z) C~~I Of) I c:r ~~ Of') G ~
JlyJl ~ 4,j.r" y, X ~I ~~I u~~1 ~ c:r
z~(z-l)/z,z~-l/z
ul&-~ r-")I 4.v~ ~\ ~ .:...P..r. ~I Ii". J
~I? ~~ J'LA..... ~I lS~1 .k;!..1 ~ G ~t:J c) ...':.l.1
cJL..'J.I ul&-~ p ..... ) I 4.v~ J~~ .~~J u).,~1 ~ c:r o..l.l y PSL (2,Z) Of) I 0\ ~ .:...P..r.
~ ul.j~ j-:lx 2 =y3=1..!.lJ..i5 J y, X ,~~I ~I
. PSL(2,Z) Of) I |
12035 | https://stackoverflow.com/questions/19753247/area-of-three-points-in-2d-is-equal-to-a-determinant | math - Area of three points in 2d is equal to a determinant - Stack Overflow
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Area of three points in 2d is equal to a determinant
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Asked 11 years, 11 months ago
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suppose we have a line segment(pq) and a point r close to it. How can I show that the area of the triangle pqr is equal to |D|/2 :
if
|1 px py |
D = det |1 qx qy |
|1 rx ry |
p=(px,py), q=(qx,qy), r=(rx,ry).
math
computational-geometry
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edited Nov 3, 2013 at 17:28
unutbu
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asked Nov 3, 2013 at 13:04
abdolahabdolah
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Consider the vectors PQ and PR:
vector PQ = (qx-px, qy-py, 0)
vector PR = (rx-px, ry-py, 0)
The area of the triangle can be written in terms of PQ and PR using the cross product formula:
Area = 1/2 |PR| · |PQ| · sin(theta) # theta = included angle between PR and PQ
= 1/2 |PR ⨯ PQ|
and this cross product can be written in terms of a determinant:
```
|𝐢 𝐣 𝐤|
2·Area = det |rx-px ry-py 0|
|qx-px qy-py 0|
= abs((rx-px)·(qy-py) - (qx-px)·(ry-py))
= abs(rx·qy - rx·py - px·qy + px·py - qx·ry + qx·py + px·ry - px·py)
^^^^^ ^^^^^
= abs(rx·qy - rx·py - px·qy - qx·ry + qx·py + px·ry)
^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^^
term1 term2 term3 term4 term5 term6
```
While on the other hand, the determinant you posted can also be expanded:
|1 px py |
det |1 qx qy | = abs(qx·ry - rx·qy + rx·py - px·ry + px·qy - qx·py)
|1 rx ry | ^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^^
term4 term1 term2 term6 term3 term5
So
|1 px py |
2·Area = det |1 qx qy |
|1 rx ry |
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edited Nov 3, 2013 at 17:29
answered Nov 3, 2013 at 16:38
unutbuunutbu
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If by space you mean area, remember that the area of a triangle is base by height divided by two. The base can be the distance from p to q, an the height, the distance from the rect pq to point r. Write the equations down and you'll get it.
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answered Nov 3, 2013 at 13:25
ChronoTriggerChronoTrigger
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Tourist Attractiveness: The Travel Industry Term to Learn in 2025
by Jason Brasier | 19 Feb 2025
Travel is an ever-evolving industry, with an ever-expanding glossary of terms. We have already seen a number of new trends emerge in 2025 – from slow travel to JOMO. Another travel term that has flashed up on our radar in recent months is ‘tourist attractiveness’. And before you get the wrong idea, it has nothing to do with the attractiveness of a given destinations visitors. Let us explain further…
What is Tourist Attractiveness?
‘Tourist attractiveness’ describes the way in which the travel industry measures how much tourists want to go to different destinations. Encapsulating the overall appeal of a country, city or attraction, it factors in everything from culture, uniqueness, value for money and accessibility. Learn about this term to make more fulfilling travel decisions, thinking clearly about where you want to go, and why.
Think of tourist attractiveness as a measure of how to choose where to go next, based on simple factors which make up how appealing a destination is to travelers. It goes beyond simple lists of the top attractions in the world and considers deeper factors that shape a destination’s overall desirability. It evaluates elements like authenticity, infrastructure, and environmental responsibility to determine why some places stand out as the top tourist spots in the world, while others have relatively low footfall. When thinking about where you’d like to go next, you can consider the factors that make a destination popular, and use that to help inform whether that’s what you value too.
Key factors that contribute to a location’s tourist attractiveness include:
Cultural richness – How unique and immersive is the destination?
Safety and accessibility – Is the destination suitable for different types of travelers?
Value for money – How far does a tourist’s budget stretch?
Sustainability efforts – Is the tourism industry operating responsibly?
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Key elements of Tourist Attractiveness in 2025
A number of evolving factors shape the most popular tourist places in the world in 2025, and in years past:
1. Cultural Authenticity
More travelers seek deep, immersive experiences rather than just visiting landmarks. A country’s ability to preserve and showcase its traditions, cuisine, and lifestyle makes it stand out. Destinations that maintain their historical sites, traditional practices, and indigenous customs are particularly appealing.
Is the historical legacy of the country well-preserved? Does it have a rich arts scene that you can engage with? Are local markets, food tours, and cultural festivals accessible to visitors? Destinations that foster storytelling and hands-on cultural interactions tend to attract more tourists.
M
One new travel trend of the past few years is ‘Coolcationing’: traveling to milder destinations to escape from the heat.
2. Safety and Accessibility
Safety remains a priority, though this depends on who you are, and how you’re traveling. Some locations may be incredibly safe for families and couples, but less so for solo female travelers. Accessibility also includes ease of transportation, visa policies, and infrastructure that supports tourism.
Governments that invest in better security, emergency services, and crime prevention measures make travelers feel more secure. Additionally, simplified visa processes, improved airport facilities, and well-connected public transport networks increase a destination’s appeal. Digital accessibility, such as seamless mobile payment systems and multilingual navigation apps, also play a crucial role in making some destinations more attractive than others.
3. Tourism Infrastructure
Well-developed transport networks, accommodation options, and digital connectivity significantly enhance a location’s attractiveness. A well-connected transport system, including high-speed rail, efficient public buses, and pedestrian-friendly city planning, enhances the tourist experience. As does the availability of a variety of accommodations—ranging from budget-friendly hostels to luxury eco-lodges—catering to a diverse range travelers. Not to forget the amount of investment and support for the local travel industry, experience providers, and upkeep of national landmarks.
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4. Value for Money
In 2025 travelers are as cost-conscious as ever, so they are looking for destinations where their money goes further. Factors such as exchange rates, affordable experiences, and quality accommodations at reasonable prices play a role. The affordability of dining, transportation, and attractions is of course a key consideration, which can be eased by free cultural experiences, budget-friendly accommodation options, and affordable local cuisine which tend to help a destination score high in value for money. Additionally, flexible pricing models, like city passes that provide discounted entry to multiple attractions make travel destinations more budget-friendly.
5. Sustainability and Eco-Friendly Tourism
Eco-conscious travelers prefer destinations committed to green tourism. Countries that emphasize higher conservation efforts and responsible travel initiatives rank high in tourist attractiveness over others.
More destinations are implementing eco-friendly accommodations, carbon offset programs, and bans on single-use plastics. National parks, marine conservation programs, and eco-tourism initiatives, such as community-led wildlife tours, contribute to a country’s sustainable travel appeal as well providing unique travel experiences that make it more culturally attractive. Destinations that successfully balance tourism growth with environmental preservation and cultural respect will continue to be top choices for mindful travelers.
M
Rising climate consciousness is leading to a rise of train and coach travel, instead of relying on private cars.
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How to Check if Your Tour Itinerary is Eco-Friendly
The impact on global tourism
With changing traveler priorities, destinations worldwide are adapting to enhance their tourist attractiveness in the following ways:
Governments are investing in safety, infrastructure, and sustainable tourism models.
Local economies benefit from tourism but must balance preservation with commercial expansion.
Socially, communities are becoming more involved in shaping tourism experiences.
How travelers can use this concept
To make informed travel decisions, consider these strategies:
Use online resources: Platforms like the Travel & Tourism Development Index and travel advisory sites help assess safety, affordability, and infrastructure.
Compare destinations more effectively: Consider the destinations you want to visit in the next few years. With this concept, you should be able to weigh up places easier, comparing things like cost-effectiveness, cultural conservation, and sustainability practices.
Learn more about why you travel:As you start to take these metrics in account, you may be able to think more deeply about what it is that you value when you travel. Perhaps you care more about sustainability than you do cost, or a unique culture takes precedence over how much tourism infrastructure there is.
Future of Tourist Attractiveness beyond 2025
Tourism trends evolve, and the concept of tourist attractiveness will continue to shift. Some future trends include:
AI-driven travel planning: Smart technology will help travelers make better choices based on real-time data.
Increased focus on sustainability: Destinations that fail to adopt eco-friendly measures may see their attractiveness decline significantly over the years.
Emerging destinations: Countries investing in cultural preservation and tourism infrastructure will rise as the next big travel hotspots. If you want to try identify ‘up-and-coming’ destinations, look for countries investing heavily in tourism infrastructure which haven’t done so in the past.
M
‘Off-peak’ travel is on the rise, with more and more people preferring to travel in spring and autumn to avoid summer’s high prices.
Tourist attractiveness is a powerful concept for travelers and destinations alike. Understanding what makes a place truly appealing allows you to make better choices, while destinations can consider their attractiveness to help shape their tourism strategies. By staying informed and prioritizing factors like authenticity, safety, and sustainability, both travelers and the travel industry can create more meaningful, responsible, and rewarding travel experiences in 2025 and beyond.
Ready to explore the world’s most tourist attractive destinations? Browse through Insight’s premium tours all across the globe.
About Jason Brasier
I'm Jay – born in Italy, raised in South London. Having French sisters and Hungarian ancestors, I've always been fascinated with the world and its cultures, and I carry this curiosity into my writing for Insightful. My favourite destinations I've traveled to so far have been Italy, Peru, France and Brazil.
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12037 | https://arxiv.org/pdf/2504.03259 | I. J. Mathematical Sciences and Computing, 2025, 1, 1-30
Published Online on April 8, 2025 by MECS Press ( DOI: 10.5815/ijmsc.2025.01.01
This work is open access and licensed under the Creative Commons CC BY 4.0 License. Volume 11 (2025), Issue 1
An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Kennedy E. Ehimwenma
Department of Computer Science College of Science, Mathematics and Technology, Wenzhou Kean University, China
Email: kehimwen@kean.edu
ORCID iD: 0000 -0002 -7616 -9342
Corresponding Author
Hongyu Zhou
Department of Computer Science College of Science, Mathematics and Technology, Wenzhou Kean University, China
Email: zhouho@kean.edu
ORCID iD: 0009 -0002 -8762 -0378
Junfeng Wang
Department of Computer Science College of Science, Mathematics and Technology, Wenzhou Kean University, China
Email: wangjunf@kean.edu
ORCID iD: 0009 -0003 -7897 -1480
Ze Zheng
Department of Computer Science College of Science, Mathematics and Technology, Wenzhou Kean University, China
Email: zhenze@kean.edu
ORCID iD: 0009 -0009 -8222 -7826
Received: 19 May, 2024; Revised: 21 August, 2024; Accepted: 12 October, 2024; Published: 08 April, 2025
Abstract: Double-black (DB) nodes have no place in red-black (RB) trees. So when DB nodes are formed, they are immediately removed. The removal of DB nodes that cause rotation and recoloring of other connected nodes poses greater challenges in the teaching and learning of RB trees. To ease this difficulty, this paper extends our previous work on the symbolic arithmetic algebraic (SA) method for removing DB nodes. The SA operations that are given as, Red + Black = Black; Black - Black = Red; Black + Black = DB; and DB – Black = Black removes DB nodes and rebalances black heights in RB trees. By extension, this paper projects three SA mathematical equations, namely, general symbolic arithmetic rule, ; partial symbolic arithmetic rule1, ; and partial symbolic arithmetic rule2, . The removal of a DB node ultimately affects black heights in RB trees. To balance black heights using the SA equations, all the RB tree cases, namely, LR, RL, LL, and RR, were considered in this work; and the position of the nodes connected directly or indirectly to the DB node was also tested. In this study, to balance a RB tree, the issues considered w.r.t. the different cases of the RB tree were i) whether a DB node has an inner, outer, or both inner and outer black nephews; or
ii) ) whether a DB node has an inner, outer or both inner and outer red nephews . The nephews r and x in this work are the children of the sibling s to a DB, and further up the tree, the parent p of a DB is their grandparent g . Thus, r and x
have indirect relationships to a DB at the point of formation of the DB node. The novelty of the SA equations is in their effectiveness in the removal of DB that involves rotation of node s as well as the recoloring of nodes along any simple
path so as to balance black heights in a tree . Our SA methods assert when, where , and how to remove a DB node and
the nodes to recolo r. As shown in this work, the SA algorithms have been demonstrated to be faster in performance
w.r.t. to the number of steps taken to balance a RB tree when compared to the traditional RB algorithm for DB removal.
The simplified and systematic approach of the SA methods has enhanced student learning and understanding of DB
node removal in RB trees.
Index Terms: Data Structures, Symbolic Arithmetic, Algorithm, Red -Black Tree, Double -Black Removal, Node
Rotation, Tree Balance, Computer Science Education An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
2Volume 11 (2025), Issue 1
Introduction
Red black (RB) trees are binary trees and by extension binary search trees (BST) [1, 2, 3] . The color of the node s
in a RB can be either red or black. Any node, whether red or black, can be inserted or deleted as result of data update or
some modification in the RB tree data structure . Modifications in a RB tree can be from the insert ion or d elet ion on
of node. The resultant color of nodes in a balanced tree is a reflection of such modification s. In RB tree s, the removal of
DB nodes follows some procedure and systematic stages that are very complex. The complexity that lies in the removal
of DB node which involves some node rotation in the tree and color change are relatively fluid a nd ambiguous during
classroom teaching . To resolve these ambiguit ies, the questions to ask are i) How is a DB node formed? ii) How can a
DB be removed? ii i) What is the principle behind the formation of DB ? iv ) At what point should a DB be removed if we
recur up the RB tree? v) Wh ich node (s) are involved in color change ? and vi) What are principles governing the
recoloring of nodes ? The foregoing forms part of the many reasons why students encounter difficulty in understanding
the principles that surround s the removal of DB node s. The lack of absolute clarity in th e proce dure for DB removal in
literature is what this paper has addressed using formal logic and symbolic mathematical procedures . This mathematical
procedure is an extension of our initial work in which we state that , Red + Black = Black; Black - Black = Red; Black +
Black = DB; Black + NULLLEAF = DB and DB - Black = Black.
The goal of this work is to project a model that can ease the learning of DB node removal in RB trees. This paper thus proposes three different equations called the symbolic arithmetic rules. These equations, namely, general symbolic arithmetic rule (GSAR) given as ; partial symbolic arithmetic rule1 (PSAR1) as and partial symbolic arithmetic rule2 (PSAR2) as mathematically computes the stages involved in DB removal and recoloring of the affected nodes along any simple path. The stages demystified by the mathematical rules demonstrate to students the step-by-step process of removing a DB node and recoloring the affected nodes. To the best of our knowledge, this SA approach is novel in the use of symbolic-arithmetic equations in DB node removal, recoloring of nodes, and rebalancing of black heights in a RB tree.
1.1 Node Color-Change
Our recent work in projected and demonstrated the SA method for the balancing of RB trees after a delete operation is performed on a node. Ehimwenma et al. states that the deletion of a red (leaf) node and the rebalancing of a RB tree is pretty straightforward. That; with the symbolic expression; Red + Black = Black , and Red + NULL_LEAF = Black ; any deleted red color node is replaced by a black color node. Similarly, Black + NULL_LEAF = DB ; which symbolizes the deletion of a black color node that turns into a NULL_LEAF , then its replacement with another black node, which finally results in the formation of a DB node. The SA method, as demonstrated in were symbolic arithmetic operations performed on DB nodes, their parent p, and sibling s. However, that research work did not cover the rotation of nodes when removing DB . This paper extends that work in through a further refinement of the SA method for DB removal to make it adaptable to node rotations and tree balancing.
1.2 Node Color-Change and Tree Rotation
The deletion of a black node that forms a DB or its replacement node is where lies the complexity of rotation ,
recoloring and rebalancing (3R ) of a RB tree. This paper demonstrates a new and easy approach for handling the complexity of the 3R in RB trees. Our step-by-step procedure is the SA algorithm which involves the use of the symbols:
R (for red), B (for black ), and DB (for double-black ) colors, respectively. Symbolically, R, B, and DB are color
operands . In between the color operands are two operators , namely, “+” ( addition ), and “ -” ( subtration ). Depending on the case, to effect a color-change; the “+” and “-” operators are applied on the color operands in the process of balancing the black heights in a RB tree. Rotation is the first operation that is performed when a DB is formed in order to to balance a RB tree. From the traditional (conventional) algorithm (TA) , rotations in a RB tree are determined by the case at hand, namely, LR, RL, LL, or RR rotation [1, 2, 6]. However, other factors to consider in the process of balancing a RB tree after a delete operation and the subsequent formation of a DB is whether the DB node has: i) an inner, outer or both inner and outer black nephews; or whether it has ii) an inner, outer or both inner and outer red nephews . The nephew of a DB is the child of the sibling of the DB .
1.3 Naming the Derived Symbolic-Algebraic Rule
The work of demonstrates the application of the SA method in the removal of the DB node but did not cover
the rotation of nodes. That work introduced, the subtraction of -B from DB as -B(DB) ; the s ubtraction of -B from
sibling s of DB as -B(s ), and the a ddition of +B to parent p of DB, +B(p ) (Figure 2) . The removal of DB with rotation
is a complex operation. For DB nodes that involves rotation to be effectively deleted and to have a balanced tree; this
paper extends our initial work of by refining the SA methods to obtain three equation, namely
a) General Symbolic Algebraic Rule ( GSAR ),
b) Partial Symbolic Algebraic Rule1 (PSAR1 ), and
c) Partial Symbolic Algebraic Rule 2 ( PSAR2 ). An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 3
From the point of a DB formation to the point where the tree is balanced; the given symbolic rules are deterministic, precise, and unambiguous such that we can list the stages and operations as they occur step-by-step. For instance, i) if a DB is formed, ii) nodes are rotated to form a new sub-tree structure, iii) DB is removed using the general rule, iv) node colors are changed, and v) tree is balanced . The concise operations contained in the SA algorithms can help students and teachers to quickly answer the “what , how, and why ” questions about DB removal in RB trees which were hitherto difficult.
1.4 Contributions
The formation of a DB node from the deletion of a black node or when a black leafnode replaces a deleted internal
Red node causes a reduction in black heights --- the number of black nodes on any simple path in a RB tree. The conventional (traditional) algorithm (TA) for fixing the DB node problem is very complex and fluid and also challenging for a quick grasp in teaching and learning. This in turn affects students’ understanding of DB removal. The contributions of this paper are therefore, 1) to create an unambiguous and concise process for removing DB nodes; 2) to propose a new method called the symbolic-arithmetic algebraic rules for the removal of DB nodes and balancing of RB trees; 3) to present three variants of the symbolic arithmetic rules for removing DB nodes and show their effectiveness on node color-change and tree balancing; 4) to establish a clearer procedure for DB node removal and RB tree balancing using mathematical process; and 5) to improve the teaching and learning of DB node removal in computer science curriculum
1.5 Statement of the Problem
The deletion of a black node or replacement of a black node by another black node unsettles the RB tree. The problems that are occasioned by a delete operation may include: i) a reduction in black heights, ii) tree rotation, iii) tree restructuring, and iv) recoloring of nodes. The traditional algorithm (TA) and research approaches geared towards simplifying the learning of DB removal pose a lot of doubt to students on, i) when to rotate a DB, ii) when to recolor a node, and iii) which node to recolor to rebalance black heights. Thus, our research work revolves around the statement that: There is a mathematically-based system of approach that can be used to precisely and unambiguously analyze, state, and ease the learning of the traditional algorithm of double-black removal, recoloring and rebalancing of nodes in RB trees.
This paper continues with related works in section 2. In section 3 we present our methodology and foundational theory of the concept of SA operation as well as the three derived SA algorithmic methods. Section 4 discusses the SA operation and DB removal using different RB tree structural examples. The section also presented our findings, graphical visualization of the findings, and students’ feedback on their user-experience w.r.t. our SA method compared to the traditional algorithm (TA) for DB removal. Section 5 is conclusions and further works.
Related Works
A RB tree is one of several binary search trees. In data structures, every binary tree has their peculiar properties that define, namely; its structure, height and relationships between nodes, and the tree’s stable state. One of the attributes of the RB tree is the “ color ” field through which every node in a RB tree is assigned a color which is either red or black [1, 2, 7] .
2.1 Properties of Red Black Trees
a) Each node is either red or black.
b) The root node is black.
c) Each leaf node (i.e. NULL_ LEAF) is black.
d) The children of a red node are black.
e) For a node, all simple paths from the node to descendent leave contain the same number of black nodes.
f) Two consecutive nodes cannot be both red.
g) A red black tree is a binary search tree (BST) .
At the point where all these requirements are satisfied, a RB tree is created. The aforementioned properties defined the stable or balanced state of the RB tree. Furthermore, if the insert or delete operation is called on a given node, the structure of the tree and the nodes ’ color also change to reflect the requirements which helps the tree to rebalance itself . Based on node coloration; the children of a black node can; i) both be blacks, ii) red and black, or iii) both reds
(Figure 1) . But for a red node, its tw o children must always be black . This is because a red node must have a black
parent node as well as a black child node. Otherwise, the property that states “no two nodes that are connected side -by -
side with an edge can be red” would be violated. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
4Volume 11 (2025), Issue 1
Fig. 1. A balanced red-black tree showing null leaves. In a Red-Black Tree Visualizer .
After operations like insert, update or delete; tree-based data structures such as the AVL and RB trees must be balanced . A RB tree is balanced if a simple path from the root node to every descendant leafnode has equal number of black nodes (Figure 1). On the other hand; it is unbalanced if there are unequal number of black nodes on every simple path from the root to the leafnodes. The RB tree is a famous data structure for storage and management of data. In the non-teaching fields, several studies e.g. [12, 13, 14, 15] have been conducted on the impact analysis of RB trees on memory management and performance. For instance, conducted a research on the application of RB trees in wireless sensor networks. The use of RB trees for optimising costs in network trees with a prescribed algorithm . As an abstract data type that is aimed at understanding computer storage and software engineering; the operations performed on a RB are insertion, rotation, deletion, and recoloring of nodes.
The RB tree is an important topic in data structures in the study of computer science. Students can find RB trees very challenging and difficult to learn [9, 18, 19]. In the lookout for new approaches to teaching RB trees, Wu et al. proposed a customized platform for learning data structures. After the proposal of a series of steps to learn data structures with-ease there was however no mention of tangible topical areas in the subject of data structures. To improve the efficiency of teaching data structures, conducted some experiments using different student groups e.g. Group1 vs. Group2 to compare the effectiveness of his teaching strategy using visualization tool vs. flipped-classroom
in the subject area of Hashing and Trees (BST, RBT, AVL). The scores obtained in the research were used to determine what group learned the most. King reported how their university data structures & algorithms curriculum was redesigned to reflect experimental analysis as against only theory of concepts in data structures. The revised curriculum and practice, thereafter as reported, enabled students to include experimental analysis in their studies to connect computer science theory with software engineering practice. The methodology in the report lacked algorithmic technique, and the report concluded its findings using the Likert Rating Scale for the collection of metrics.
The report of presented a list of the most conventional topics in data structures & algorithms and emphasized the importance of logic in algorithmic studies --- that algorithms are logics . Furthermore, that data structures curriculum needs to be supported with critical computational thinking and formal proofs and logic . In a top-down insertion method was proposed to address the problem of single and double rotation through a granularity approach to balance a RB after an insert operation. The granularity approach prescribed a step-by-step selection of rules for students to follow and balance a RB tree. But the aforementioned approaches for addressing the teaching and learning of RB trees that involves rotation and recoloring of nodes e.g. [9, 23] has not made the problem any less or easier.
The insert operation of a new node in a RB tree is the simpler of two hard problems. The harder is dealing with the deletion of a black node or the replacement node and the subsequent rebalancing of the tree. In order to address the removal of DB node, used formalized statement s in the description of nodes’ color, recoloring and tree balancing
which (in any way) did not simplify the learning challenges faced by students in the learning of DB removal algorithm. In a parity-seeking delete algorithm was introduced with a goal that is similar to the aim of our work in this paper. They introduced a pedagogically sound and easy way to understand the algorithm for RB tree deletion algorithm. The rationale which was to balance a RB tree by repairing a defective subtree also left a bit of complexities in learning and understanding. Also, the work of presented 2-3 variants RB tree, called the Left-Leaning Red-Black trees (LLRB) and proposed a concise number of deletion algorithms. However, the deletion algorithm of the LLRB was still complex and was not suitable for educational use.
Based on these complexities that are left with the deletion algorithm of the RB tree, the aim of this paper is to simplify the learning, understanding, and teaching of DB node removal using basic symbolic-arithmetic operations after a delete operation. Students have expressed satisfaction in the use of the SA algorithmic rules over the TA algorithm in that the TA is not deterministic nor is it precise in approach for ease of understanding. See the relevant Chapters in [1, 2] and RB tree configuration in visualization tools . Hence, the quest for this new SA approaches. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 5
Fig. 2. The symbolic subtraction of the color black (represented as -B)from two siblings and the addition of the color +Bto their parent v . Thus, -B(leftChild) × -B(rightChild) = +B(parent).
2.2 The Symbolic-Algebraic Arithmetic Rules
In , the SA operations are stated as black + black = double black
(1) - black = black double black
(2) dblack - bla k = re c
(3) + black = black red
(4) kr bed + = lac double black
(5) - black = black red
(6)
null(DB) – black = NULL_LEAF (7)
The given symbolic operations imply the simple addition or subtraction of red or black color to or from an existing node color to obtain the resultant color in the process of DB removal and rebalancing of black heights. Note that the color operands in the formulas above cannot be moved to the other side of the equality sign, otherwise, conflicts may occur.
2.3 The Change Factor
The change factor is the color that is added to or subtracted from an original color. For example, if a node is initially black and an extra black color is added to it, the from Equation (1), we have
Fig. 3. An example of the change factor in Equation (1).
2.3.1 The Transmission Rules of The Change Factor
a) The path, or order, of the recoloring process travels from the deleted node to its adjacent node, both along and against the arrow is allowed ( see connected nodes, Figure 2).
b) The transmission (of the SA rule operation) stops when a tree is balanced; but continues if another DB node is regenerated as a result of the SA operation.
Methodology
This paper presents three refined symbolic arithmetic (SA) algorithm based on the previous works of for DB
node removal. In this presentation, we have adopted the use of formal logic in the form of F : X → Y to depict the
operations on the nodes in a RB tree that involved rotation of nodes when removing a DB node . The use of formal logic
for information representation has enriche d the understanding and learning of the DB removal pro blem, and it has made
the process clearer, precise, concise and explainable. Logic based explanations are still present in modern day system s.
As stated in , formal logic should be used to support the learning and u nderstanding of data structures, because
logical formula support s explainable systems . Our mathematical approach is well simplified and to a very large An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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extent it is also different from other approaches like the use of logical formulas in or functional programming in
. Just as formal logic has found its place in modern explainable artificial intelligence (XAI) systems so it
continues to support software engineering systems as exemplified in this work for teaching the DB node removal .
The three SA rules that we have developed for DB removal revolves around the operation of node recoloration, and tree balancing and they are namely
1) General symbolic arithmetic rule (GSAR) ,
2) Partial symbolic arithmetic rule1 (PSAR1) ,
3) Partial symbolic arithmetic rule2 (PSAR2) .
In RB trees, there are four different types of rotation which are LR, RL, LL, and RR rotation. From our observation in this work, the rotation of nodes in the process of balancing a RB tree is determined by the position and color of the following nodes
the nephew of the DB ,
parent of the DB ,
sibling of the DB , and
the grandparent of the nephew (which is also the parent of the DB )
Consequently, we are using the following parameters to denote the aforementioned nodes such that
u is the deleted node (or value) that formed the DB i.e. ( ) (u)
DB is the node that must be removed
p is the parent of DB, p(DB)
s is the sibling of DB, s(DB)
r is a special nephew of DB, r(DB) : and we called this the VIP (very important personality) node
x is a second nephew of DB, x(DB) : Thus, r and x are cousins; and are children of sibling, s, of the DB
g is the grandparent of r, g(r); and of x, g(x) respectively. Thus g(r) ≡ p(p(r)) and g(x) ≡ p(p(x))
Secondly, by correctly identifying the type of rotation i.e. LL, RR, LR and RL as well as the rotation of nodes in their proper order, say from LR to LL or from RL to RR , we observed that any two of the three SA rules can be combined to effectively resolve the 3R problems in RB trees. The combination order of any two SA rules is the pairs of
a) GSAR & PSAR1 .
b) PSAR1 & GSAR .
c) PSAR1 & PSAR2 .
d) GSAR & PSAR2 .
e) Or just the GSAR originally from the work of .
3.1 Theoretical Foundation
Respectively, Figures 4 and 5 are basic RB tree problems with DB formation. To resolve the 3R problem, we have used a step-by-step transition method to process the changes that occurred in different stages. In the aforementioned Figures, the nodes have been theoretically given the values of v, x, y, and z, respectively. The RB tree is a Binary Search Tree (BST). Thus, these labels are guided by the property of the BST. A RB tree that has a DB node with any of the four types of structure of rotation i.e. LR, RL, LL, or RR problem must, firstly, be rotated before the application of the SA rules. The step-by-step stages, namely, a), b), c), and d) depicts the procedure that we have established in applying the SA rules.
For instance, in Figure 4, the pair of rules to establish the 3R algorithmic procedure is the GSAR & PSAR1 . In Figure 4(a), the DB node has a black parent, B(p(DB(v))), and two black nephews , namely, inner x, B(innerNephew(x)),
and outer z, B(outerNephew(z)) . While x is the black inner nephew to DB that can be stated as
B(innerNephew(x(DB))) , z is then given as the B(outerNephew(z(DB))) . With two nephews to a DB, one of the nephews must be a VIP node. In this color configuration that involves five connected and concerned nodes to a DB i.e. the DB , B(p(DB)) , R(s(DB)) , B(innerNephew(x(DB))), and B(outerNephew(z(DB))) ; our interest will be on the
B(innerNephew(x(DB))) . Thus, where there are two black nephews to DB , we will be interested in changing the color of this B(innerNephew(x(DB))) . The B(innerNephew(x(DB))) is the node that we have called the VIP (very important personality) node. It is the VIP node because it is not the node that determines the case (of rotation), in this case a RR ;but it is the node whose color would change from B(x) to R(x) at stage 4(c) of Figure 4. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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Fig. 4. (a) Depicts a RR case with a DB from a deleted black node. (b) Shows a left rotation on node v --- the parent of DB, and the operation of the
general rule ; (c) The depicts the partial rule1 .(d) A balanced tree.
In Figure 4(a), based on the case which is a RR, we left-rotate the B(p(DB)) . Then, in Figure 4(b), we apply the
general rule GSAR, . The GSAR rule depicts the the subtraction of the color black, -B, from both the DB and its new sibling x which was initially its black inner nephew ; and then the addition of the color black, +B, to their parent v, B(p(v)) . Symbolically, this process is depicted as, -B(DB), -B(innerNephew(x(DB))) , and +B(p(DB)). Now, in Figure 4(c) we have a new DB as a result of the addition of black, +B, to the B(p(DB)) . At the new level of DB which is one level up the RB tree, we then apply the second rule, that is the partial rule1 PSAR1 , , by subtracting one
black, -B, from the new DB(v) , -B(newDB(v)) ; and adding one black, +B , to the Red parent y of the new DB ,
+B(R(p(newDB(y)))), while the new sibling z of the new DB is exempted . In Figure 4(b) and Figure 4(c) the GSAR
and the PSAR1 are symbolically denoted as and respectively . Based on the foregoing analysis, we arrived at Corollary 1 .
Corollary 1 :
In a RB tree, if there are two black nephews to a DB with a black parent, then the case is a LL or RR case decided by the black outer nephew but the VIP node would be the black inner nephew that will have a color change.
Fig. 5. (a) Shows a RR case with a DB from a deleted black node. (b) Depicts a left rotation on node v and the operation of partial rule1
. (c) The general rule . (d) A balanced tree given that node y is a root node. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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In Figure 5(a), the DB also have a black parent p that is given as B(p(DB(v))) but two red nephews , x and z. While
x is the Red inner nephew given as R(innerNephew(x(DB))) ; z is the Red outer nephew that is given as
R(outerNephew(z(DB))) . In this type of color configuration i.e. a black parent of DB , B(p(DB)) , and two red nephews,
R(innerNephew(x(DB))) and R(outerNephew(z(DB))) ; our interest will be on the R(outerNephew(z(DB))) to completely resolve the 3R problem. In Figure 5(b) the first rule to apply will be the partial rule1 PSAR1 , . As mentioned earlier, with two nephews to a DB , one of the nephews must be the VIP node. In this case the VIP node is the outer red nephew z to DB, R(outerNephew(z(DB))) . As such, we will subtract one black, -B, from the DB, -B(DB) ;
and add one black, +B, to its parent v, +B(p(DB) . Note that the red inner nephew x to DB that is given as
R(innerNephew(x(DB))) is exempted from this arithmetic operation. In Figure 5(c) we have a new DB which is one level-up the tree. At the new level of the DB ; we will apply the general rule by subtracting one black, -B, from both the
new DB(v) and its sibling which was the original red outer nephew z to DB , respectively, and then add one black, +B ,
to their new parent y, B(p(y(DB))) . Symbolically, this is stated as -B(newDB) , -B(R(outernephew(z(DB)))) , and
+B(p(y(DB))). The pair of rules applied in this algorithmic procedure are the PSAR1 & GSAR . In Figure 5(b) and Figure 5(c), the PSAR1 and GSAR are shown as and , respectively . Based on the foregoing analysis,
Corollary 2 was drawn.
Corollary 2 :
In a RB tree, if there are two red nephews to a DB with a black parent, then the case is a LL or RR case decided by the red outer nephew but the VIP node would be the red outer nephew that will have a color change.
Note that the VIP nodes such as the B(innerNephew(x(DB))), (in Figure 4); and the R(outerNephew(z(DB))) to
DB (in Figure 5) are VIP-nephew nodes of the two nephews to DB to have a color change. In this paper, the VIP nodes are hereafter labeled as the node r in all the examples and illustrations. In the general rule and partial rule2,
respectively, they are represented by the parameter r. At every stage of the SA operation, the algorithm tests whether the RB tree is balanced or not. If the RB tree is balanced, the process of further rotation or SA application ends. Otherwise, we recur one step, further up, the tree as shown in Figures 4(c) and 5(c). In the following section, we present the detailed procedure of the three SA rules. We have used the RB tree Case 2 and Case 3 configuration to illustrate these procedures. The complete expression of the general rule and the two partial operations as denoted in the partial 1
and partial 2 rules, is given as general rule = partial rule1+ partial rule2
and mathematically as [ , , ] [ , ] [ ] ' '' DB r p DB p r
(8)
3.2 General Rule: GSAR
The general rule, GSAR , denoted that is denoted as is the rule that applies to a DB node and its sibling s,and to their parent p. Thus this rule causes a color change to a parent node and its two children. The GSAR operates on three inter-connected nodes as shown by the delta symbol that has 3 vertices. The vertices indicate all the three nodes that are involved in the arithmetic operation. Since we are dealing with a DB problem, one of the three connected nodes must be a DB , and the DB must be one of the two children of a parent p. In logical form, this rule is given as [ , , ] [ , , ]:DB r p DB r p DB r pB B B
(7)
That is [ , , ] [ , , ]: ( ) ( ( ) ) ( ( ) )DB r p DB r p DB B r DB B p DB B
(8)
which states that, for all DB , r, and p, to remove the DB w.r.t. p(DB) , r(DB) ; subtract -B from the DB , and its nephew r;and add +B to parent p of DB . This translates to the symbolic arithmetic form
DB – B NULL_LEAF
B(r) – B R(r) || R(r) – B B(r)
B(p) + B newDB(p) || R(p) + B B(p) An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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Particularly, where the operation DB – B = NULL_LEAF ; the DB must have been formed from a deleted black node. Given the property(object) formalism; the notation B(r) states that the nephew r has the color black B (or that r is black), and B(p) means that the parent p is black. It is important to state that the resultant color of a node after a subtract, -B , or an add, +B , operation is dependent on the existing color of that node. For instance, the resultant color
B(p) , from the operation R(p) + B = B(P) . The pre-existing color on the node p is Red , R. This is further illustrated with conditional IFs as
IF R(p) THEN
B(p)= R(p) + B
IF B(p) THEN
new DB (p)= B(p) + B
3.2.1 Characteristics of The General Rule
i. The general rule is operated on three connected nodes.
ii. The general rule uses a -B symbolic operator on two children and a +B on their parent.
iii. The general rule applies to a DB whose parent can be a Black node, B(p(DB)) ; or Red node , R(p(DB)) , at the time of formation of the DB ;
iv. The general rule works with either the partial rule1 or the partial rule2 to balance a RB tree.
3.3 Partial Rule1: PSAR1
The partial rule1, denoted as is the rule that partially applies to two nodes at any given instance. From the notation , this rule exempts one node of the three nodes that can be handled by the general rule, GSAR . Hence the name partial from partial differenctial calculus . The partial 1 is the first of the two extended partial rules that we are projecting in this work. The partial rule1 causes color-change on a DB and the parent p of DB as given in the notation
. In this operation, the DB may be the original DB and p the parent of the DB at the point of formation of the
DB ; or a new DB and its new parent p after the general rul e, , operation. The single prime ,
’, symbol in the notation indicates the node that is exempted in the operation of this rule . In logical formalism , this is given as [ , ] [ , ]: 'DB p DB p DB pB B
(9)
That is [ , ] [ , ]: ' ( ) ( ( ) )DB p DB p DB B p DB B
(10)
This states that, for all DB and its parent p; we can remove the DB , by calling the partial rule1 function which subtracts -B from the DB ; and add +B to the parent p of the DB . Now, the color of the p(DB) could be black B or red R.
In symbolic arithmetic (SA) form, we have
B(p) + B DB (p) || R(p) + B B(p).
The operation of the partial rule1, , is dependent on two factors, which are namely, i) the position of the inner and outer nephews, r and x; and ii) the color, whether Red or Black, of the nephews r and x of the DB, respectively (see Figure 4 and 5 ).
3.3.1 Characteristics of The Partial Rule1
i. The partial rule1 operates on a DB that may have a black parent or a red parent: B(p(DB)) or R(p(DB)) .
ii. The partial rule1 can be performed before the general rule or vice versa: This is dependent on the p(DB) and the color and position of the ( VIP node ) nephew r of DB , r(DB) .
iii. Partial rule1 denoted as always comes first before the partial rule2 that is denoted as (next section ).
iv. Partial rule1 keeps one node color constant while it makes changes to the other second nodes’ color, see Figure 4 and 5 .
3.4 Partial Rule2: PSAR2
The partial rule2 which is denoted as is an operation that is performed on only one node in an unbalanced RB tree. From the notation , this rule exempts two nodes of the three connected nodes that are depicted by the general rule . The double prime ,
’’ , symbol in the notation indicates the two exempted nodes . This rule applies to a
DB node that has a Red parent p, i.e. R(p(DB)) . In logic form, we state that An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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(13)
That is
( ( ) )) ( ( ) ) (14)
That, for all r that are the original nephews of DB ; to remove a DB and balance the tree, we then subtract -B from the original nephew r of DB . The node r is the nephew of a DB at the point of formation of the DB. Note that as the tree undergoes transformation as a result of node rotation and the application of the symbolic arithmetic (SA) operation on the respective nodes; the position and color of the original nephew r to the DB also changes, see Figures 13, 14 and 15.
At this point, the partial rule2, , will be acting to change the new and current color of nephew r; not the original
color of r. In symbolically arithmetic
B(r) – B R(r) || R(r) – B B(r)
In conditional statements, we state that
IF B(r) THEN
R(r) = B(r) – B
ELSE
B(r) = R(r) – B
3.4.1 Characteristics of Partial Rule2
i. The Partial rule2 is performed on only one node.
ii. The Partial rule2 u ses only one symbolic operator which is the subtraction operator, -B , from a node.
iii. The partial rule2 is always a secondary rule to the general rule or partial rule1 , respectively.
iv. The Partial rule2 comes into effect in a tree structure where a DB has a Red parent p, R(p(DB)) , at the point of formation of the DB .
3.5 Procedure of the Symbolic Arithmetic Rules
The following section presents the step-by-step procedure for removing a DB node using the SA algorithmic rules.
Procedures 1, 2, and 3 describe the process for a DB that has a black parent p, B(p(DB)); and Procedures 4 and 5 for a
DB node that has a Red parent p, R(p(DB)) .
Procedure 1 [Figure 11]: LL case and RR case: Black Parent of DB and Two Black Nephews of DB
Start
IF a black node is deleted and the deleted node has a black parent and two black nephews , r & x :
WHILE tree is not balanced:
Rotate the parent, p, of DB
Apply the general rule :
END general rule
Apply partial rule1 , , at the level of DB:
END partial rule1
IF the new DB is root:
root / root is always black /
Tree is balanced
END WHILE
END
Procedure 2 [Figure 8 & 9]: LL Case and RR Case: Black Parent of DB and Two Red Nephews of DB
Start
IF a black node is deleted and the deleted node has a black parent and two red nephews , r & x :
WHILE tree is not balanced:
Rotate the parent , p, of DB
Apply partial rule1 , , at the level of DB:
END partial rule1
Apply general rule :An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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END general rule
IF the new DB is root:
root / root is always black /
Tree is balanced
END WHILE
END
Procedure 3 [Figure 6 & 7]: LR Case and RL Case: Black Parent of DB and One Red Inner Nephew of DB
Start
IF a black node is deleted and the deleted node has a black parent and one red inner nephew, r :
WHILE tree is not balanced:
Rotate the sibling , s, of DB
Apply the general rule at the level of DB:
END general rule
Rotate the original parent, p, of DB / new DB /
/ root is always black /
Tree is balanced
END WHILE
END
Procedure 4 [Figure 13 & 14]: LR Case and RL Case: Red Parent of DB and One Red Inner Nephew of DB
Start
IF a black node is deleted and the deleted node has a red parent and one red inner nephew, r :
WHILE tree is not balanced:
Rotate the sibling , s, of DB
Apply the general rule at the level of DB:
END general rule
Rotate the original parent, p, of DB
Apply partial rule2 , , at the level of DB:
END partial rule2
12 . Tree is balanced
END WHILE
END
Procedure 5 [Figure 15]: LL Case and RR Case: Red parent of DB and One Red Outer Nephew of DB
Start
IF a black node is deleted and the deleted node has a red parent and one red outer nephew, r :
WHILE tree is not balanced:
Rotate the parent , p, of DB
Apply the partial rule1 at the level of DB:
END partial rule1
Apply partial rule2 , :
END partial rule2
11 . Tree is balanced
END WHILE
END
Discussions
In this section we present our technical but simple SA methods for the removal of DB nodes with rotation. The illustrative examples demonstrate the general rule , partial rule1 and partial rule2 for the LR, RL, LL, and RR cases of the RB tree. In a BST and by extension RB trees, a parent is a node that has two children: the leftChild and rightChild .Based on the examples provided in this work, the first step to identifying any of the four cases of rotation is to locate the parent p of the DB , p(DB) . Thereafter, start putting labels to the nodes from the p(DB) down through the edges to the An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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sibling s of DB , s(DB) ; and through to the nephews r and x of the DB i.e. r(DB) and x(DB) . In Figure 6 for example, we knew that Figure 6(a) is a LR Case by navigating from node 40 , which is p(DB) ; down to the black node 20 , which is the B(s(DB)) ; and finally to node 30 , the red inner nephew r of DB, R(r(DB)) . The node B(s(DB)) is a leftChild of the p(DB) and R(r(DB)) is the rightChild of the B(s(DB)) . Hence, the LR Case structural navigation.
4.1 Recursive Subtraction and Addition of Colors
Irrespective of whether a DB node is a leftChild or rightChild of its parent, our SA algorithm subtracts 1 black color -B from both children of the same parent p where one of the children is a DB and adds 1 black color +B to their parent. Such as -B(leftChild) , -B(rightChild) and +B(p(DB)) . Technically, the strategy is a simple addition of +B
color to/from an existing Black B node or to Red R node e.g. a parent node; or the subtraction of -B color from an existing Black B or Red R node e.g. from child nodes. Always, the SA algorithm always tests whether a RB tree is balanced or not in the process of addition or subtraction of color B. If the RB tree is not balanced, then we recur up the tree by calling on the next appropriate rule as discussed in the methodology . As we apply the SA rules and recur up the tree, we must apply the GSAR and the PSAR1 rules at the level of the DB node, and the PSAR2 on only the nephew node, r. Also, as we recur up the tree, the traversal must be upward along the path of a given DB node towards the root of the tree. This process recursively continues if there is a re-occurrence of a DB by virtue of +B
addition to another black B node which in turn becomes a new parent p. But if the new DB p is a root node then it turns a single black B because a root node is always black.
4.2 PART I: Examples of DB Removal with a Black Parent
For every RB tree case, the following examples demonstrate the detailed step-by-step process of balancing a RB tree without violating its properties. In this SA operation, we have chosen to use the parameters u, DB , p, s, r, x, and g
as already described in our methodology. Also, in algorithmic representation; we shall use the following notations, such as, B(p) for a black parent node, DB(u) for the double-black node u, B(s) and R(s) for the black or red sibling of a DB
node. For a much more detailed representation, wherever necessary, we would state B(s(DB)) as the black sibling,
R(s(DB)) for the red sibling, and B(p(DB)) for the black parent p of DB, respectively. Also, others are
R(innerNephew(r(DB))) for the red inner nephew r and B(outerNephew(x(DB))) for the black outer nephew x of DB, respectively. Furthermore, as mentioned earlier, the parameter r represents a special node called the VIP nephew of a DB, r(DB) , see section 3 . In addition, g(r) shall be used to represent the grandparent g of the nephew r of a DB, and g(r)
is equivalent to p(p(r)) , the parent of the parent of nephew r, and B(g(r)) as the black grandparent of r. The parameters p,
u, s, and r shall be assigned their values whenever necessary in the process of evaluating the SA algebraic algorithm. For instance, where a black parent node value is 40 , this will be written as B(p(40)) . It is important to note that DB
nodes are NOT only formed from the deletion of black leaf nodes but are also formed in the position of a black leaf node that has moved to replace a deleted internal red node.
4.2.1 Left-Right Case of Double-Black
Example 1 : The Case of a DB u with a black parent and one red inner nephew r.
Our first demo is Figure 6 which presents a DB node formation after the deletion of the node value that is labeled u.To resolve the 3R problem of DB removal, firstly, we have to identify the Case. Starting from the DB(u) , to its black parent, B(p(40)) , then to the black sibling s of DB , B(s(20)), and down to the red inner nephew r,
R(innerNephew(r(30))) ; the case of rotation is a LR Case decided by the R(innerNephew(r(30))) . Note that DB(u) has only one red inner nephew, r(30) . This is the type of nephew we have called the VIP nephew r. It a VIP because it had its color changed from red, r(30) to black r(30) . In addition, the VIP nephew r also decided the case of rotation to be LR.
Fig. 6. A LR case after the deletion of a black node.
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We called the red nephew , r, the VIP node because it is the nephew node that changes its color at the end of the DB removal process or the 3R (removal, recoloring, and rebalancing). More so, if a DB has two nephews, namely, r
and x; amongst the two nephews, the VIP nephew r in this work is the only nephew that would change color. We shall see this in the forthcoming examples.
As a LR Case (Figure 6), we left-rotate the sibling s of DB , B(s(20)) to have a LL Case. Then at the level of DB, we apply the general rule , ; to have a new DB at node B(p(40)) . We then right-rotate the new DB , DB(40) to remove the DB. At this point, the tree is balanced i.e. blackHeight = 2 (the equal number of black nodes along every simple path from the root to any descendants leafnode). Using algebraic notation, we describe this Case and the stages of the 3R resolution from the point of the DB formation to tree balancing as ( ) ( ) . In a list format, we enumerate the operations and the different stages as
()( )
( ( )) ( ) (15)
and in first-order logic representation w.r.t. the red VIP nephew r and the black grandparent of r, B( g(r)) , we state that , : ( ( )) ( ) ( ) g r B g r rR B r
(16)
Therefore, we can generalize that for every red nephew R(r) that has a B(g(r)) , the R(r) must become a black node,
B(r), at the end of the 3R process, see Table 1.
Table 1. LR Case with an Inner Red Nephew to DB Node
Steps Structure
&Case
Rotation Rule Applied Operated Nodes Exempted
Node
DB Removed Tree
Balanced
1. LR case with an inner red nephew r of DB
Left-rotate s(DB)
---s(DB) -No No
2. LL case -Yes --DB, red inner nephew r, and p(DB)
-Yes No
3. LL case Right-rotate new DB, remove DB
---New DB -Yes Yes
Based on the foregoing analysis, we arrived at Corollary 3 .
Corollary 3:
In a RB tree, where there is only one red inner nephew to a DB with a black parent, then the case is a LR or RL
case decided by the only red inner nephew; and as the VIP node, this nephew will have a color change.
4.2.2 Right-Left Case of Double Black
Example 2 : The case of a DB u with a black parent and one red inner nephew r.
This Example 2 in Figure 7 is a mirror of the LR Case of Example 1 in Figure 6. As such, the same generalization that holds for the LR Case also hold for the RL case. Table 2 provides the highlight in this operations. In a list format, we state the different stages involved as
()( )
( ( )) ( ) (17) An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
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Fig. 7. A RL case after the deletion of a black node.
Table 2. RL Case with an Inner Red Nephew to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempted
Node
DB
Removed
Tree
Balanced
1. RL case with an inner red nephew r of DB
Right-rotate s(DB)
---s(DB) -No No
2. RR case -Yes --DB, red inner nephew r, and p(DB)
-Yes No
3. RR case Left-rotate new DB, remove DB
---New DB -No No
Note that the given Corollary 3 above also holds for the RL case of a DB with one red inner nephew r to a DB.
4.2.3 Left-Left Case with Two Red Nephews
Example 3 : The case DB u with a black parent and two red nephews r, and x.
In Figure 8 we have a LL Case in which the DB has a black parent B(p(40)) and two red nephews which are the
red outer nephew r given as R( outerNephew (r(DB))), and the red inner nephew x as R( innerNephew (x(DB))) to DB .Assigning values to parameters r and x, we have, R( outerNephew (r(20))) and R( innerNephew (x(35))) . As mentioned earlier, since there are two red nephews r and x, one of the two nephews must be the VIP nephew . Thus, the
R( outerNephew (r(20))) becomes the VIP node. The node R( outerNephew (r(20))) is the VIP node because i) it was used to determine the case as a LL case, and ii) changed its own color. In Figure 8(a) we have a DB formed. In Figure 8(b) we right-rotate the black parent, p, of DB , B(p(40)) to have BST (binary search tree). Then we apply the partial rule1 . Partial rule1 operates on the DB and its parent , p . Thus, we have DB – B = NULL_LEAF , and B(p(40)) + B = DB(40) . This operations gives the tree a new DB(40) . The new DB(40) is the original parent p of the original
DB(u) . The new DB(40) also has no place in a RB tree. So, to remove it, we recur up the RB tree and apply the general rule at the level of, i) the new DB(40) , ii) R(outerNephew(r (20))) and their new black parent, B(p(30)) which was the original sibling s(30), of the DB(u). From the general rule operation, we had, DB(40) - B = B(40) ,
R(outerNephew(r (20))) – B = B(20) and B(s(30)) + B = DB(30) . Now, since the node DB(30) is a root node, every root is black in a RB tree. Table 3 presents these stages of operation in solving the 3R problem. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 15
Fig. 8. LL case with two red nephews of the double black.
Table 3. LL Case with Two Red Nephews to DB Node
Steps Structure &
Case
Rotation Rule Applied Operated Node sExempted
Node
DB
Removed
Tree
Balanced
1. LL case with two red nephews of DB
Right-rotate p(DB)
---p(DB) -No No
2. LL case with two red nephews of DB
--Yes -DB and p(DB) Inner red nephew x
Yes No
3. Binary search tree
-Yes --New DB, outer nephew r, and new p(DB)
-No No
4. New DB is the root
----Root is always black, then remove DB.
-Yes Yes
In a list format, we show the series of operations and the different stages as ( ), ( , ) [ , ] [ , , ][ , ( ), , ' , , , , ( )] B p TwoR r x LL DB p DB r p DB DB rightRotate p BST newDB newDB newDB root
(18)
Using first-order logic formula w.r.t. the red VIP nephew r and the black grandparent of r, B(g(r)) we state that , : ( ( )) ( ) ( ) g r B g r rR B r
(19)
This LL Case example satisfies Corollary 2 as earlier stated above.
4.2.4 Right-Right Case with Two Red Nephews to DB
In this example, we provide two illustrative diagrams to demonstrate the SA algorithmic methods. While the first is a RR Case with two red nephews, the second is a RR case with one red nephew.
Example 4 : The case of DB u with a black parent and two red nephews x and r.
Figure 9 presents a RR Case which is a mirror of the LL Case in Figure 8. As such, the same generalization holds for both structures. Table 4 depicts the series of operations involved, and the different stages of node s’ transformation before the tree was balanced as ( ), ( , ) [ , ] [ , , ][ , ( ), , ' , , , ( ), ( )] B p TwoR r x RR DB p DB r p DB DB leftRotate p BST newDB newDB root B root
(20)
Using first-order logic formula w.r.t. the red VIP nephew r and black grandparent of r, B(g(r)) , we state that , : ( ( )) ( ) ( ) g r B g r rR B r
(21) An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
16 Volume 11 (2025), Issue 1
Fig. 9. RR case with two red nephews of the double black node.
Table 4. RR Case with Two Red Nephews to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempted
Node
DB
Removed
Tree
Balanced
1. RR ca se with two
red nephews of DB
Left -rotate
p(DB )
---p(DB )-No No
2. Binary search tree --Yes -DB and p(DB )Inner red
nephew x
No No
3.Binary search tree -Yes --New DB, outer
nephew r, and new
p(DB)
-No No
4.New DB is the root ----Root is always
black. Remove
DB.
-Yes Yes
This RR Case example satisfies Corollary 2 as earlier stated above.
Example 5 : The RR Case of DB u with a black parent and one red outer nephew r.
When compared to the RR Case in the preceding Example 4 (Figure 9), Example 5 (Figure 10) has only one red
outer nephew r, given as R(outerNephew(r(DB))), to DB . This means that Example 5 is a similar case to the foregoing case in Example 4. Though in Example 5, the tree has no inner nephew node but their similarity lies in the VIP nephew
r that decided the case as well as taking part in the color change operation of the SA algebraic rules. Suppose there was an red inner nephew in Example 5, it will not play any part in the color change operation. After balancing the tree, Example 4 and Example 5 have equal number of blackHeight = 2 and thus the same generalization. Table 5 summarizes the steps invloved.
Fig. 10. RR case with one red outer nephew to the double-black node.
An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 17
Table 5. RR Case with One Red Outer Nephews to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempte
d Node
DB
Removed
Tree
Balanced
1. RR case with one outer red nephews of DB
Left-rotate p(DB)
---p(DB) -No No
2. Binary search tree --Yes -DB and p(DB) -No No
3. Binary search tree -Yes --New DB, outer nephew r, and new p(DB)
-No No
4. New DB is the root ----Root is always black. Remove DB.
-Yes Yes
Based on the foregoing analysis, we therefore arrived at Corollary 4 .
Corollary 4:
In a RB tree, if there is only one red outer nephew to a DB with a black parent, then the case is a LL or RR case decided by the only red outer nephew; and as the VIP node inclusievely, this nephew will have a color change.
4.2.5 Left-Left Case with Two Black Nephews
Example 6 : The case of DB u with a black parent and two black nephews r, and x.
In this demo, we present a LL Case, with a black parent to DB, B(p(DB)), and two black nephews r and x, namely,
the black outer nephew x, B(outerNephew(x(DB))) , and black inner nephew r, B(innerNephew(r(DB))) in Figure 11. Assigning values to r and x, we have B(outerNephew(x(20))) and B(innerNephew(r(35))) . With two black nephews to the DB, the VIP nephew becomes the B(innerNephew(r(35))) . It is the VIP node because it is the nephew r that had a color change but did not determine the case as an LL to have the tree balanced . In Figure 11(a) a DB was formed. In Figure 11(b) we right-rotate the B(p(DB)) . Then in Figure 11(c) we applied the general rule to i) DB node, ii) the B(innerNephew(r(DB)) ); and iii) the B(p(DB)) . by evaluation, DB – B = NULL_LEAF , B(innerNephew(r(35))) –
B = R(r(35)), and B(p(40)) + B = DB(p(40)) . This operation led to a new DB formation i.e. DB(40) . Note that the VIP
nephew has changed color from black B(r(35)) to red R(r(35)) . With a new DB(40) , we recur up the tree to remove the
new DB(40) . At this point we invoked the partial rule1 on the DB(40) and its black new parent, B(p(30))
(which was the original black sibling s, original B(s(DB)) , of DB). By another evaluation, we had newDB(40) – B = B(40)
and original B(s(30)) + B = newDB(30) that took us to another new DB(30) . However, since the newDB(30) is a root node, according to the properties of the RB tree, every root node is black. So, in the end we had a black root node,
B(root(30)) in Figure 11(e) to have a balanced tree. Table 6 summarizes the operations folowed in resolving the 3R problem. In a list, we also represent these operations and the different stages as
()()
( ) ( ) ( ) (22)
Using first order logic formula w.r.t. to the VIP nephew r and g(r) , we state that , : ( ( )) ( ) ( ) g r B g r B r rR
(23)
Fig. 11. LL case with two black nephews of double-black.
An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
18 Volume 11 (2025), Issue 1
Table 6. LL Case with Two Black Nephews to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempte
d Node
DB
Removed
Tree
Balanced
1. LL case with two black nephews of DB
Right-rotate p(DB)
---p(DB) -No No
2. Binary search tree -Yes --DB(u), black inner nephew r, and p(DB)
-No No
3. Binary search tree --Yes -New DB, and new p(DB) i.e. old s(DB)
Black outer nephew x
No No
4. New DB is the root ----Root is always black, then remove DB.
-Yes Yes
Based on the foregoing analysis, we arrived at Corollary 5 .
Corollary 5:
In a RB tree, if there are two black nephews to a DB with a black parent, the case is a LL or RR case decided by the black outer nephew but the VIP node would be the black inner nephew that will have a color change.
Note that from Examples 1 - 5 (Figures 6 – 10 ), the VIP nephew node r has always performed two functions, which are, i) deciding a rotation case and changing its colors. However, that changed in Example 6 (Figure 11), where we found an exemption . Instead, while the black outer nephew x was the node that determined the case, the black inner nephew r became the node that had a change of color to balance black heights in the tree.
4.2.6 Right-Right Case with Two Black Nephews
Example 7 : The case of DB u with a black parent and two black nephews r, and x.
Figure 12 is a RR Case which is a mirror of the LL Case in Example 6 (Figure 11). As such, the same generalization holds for both examples. Table 7 depicts the series of operations involved. In a list format, we represent these operations and the different stages as ( ), ( , ) [ , , ] [ , ][ , ( ), , , ' , ( )] B p TwoB r x RR DB r p DB p DB DB leftRotate p BST B root
(24)
The first-order logic formula w.r.t. the black VIP nephew r and the black grandparent of r, B(g(r)) states that , : ( ( )) ( ) ( ) g r B g r B r rR
(25)
Fig. 12. RR case with two black nephews of the double black node.
An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 19
Table 7. RR Case with Two Black Nephews to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempted
Node
DB
Removed
Tree
Balanced
1. RR case with two black nephews of DB
Left-rotate p(DB)
---p(DB) -No No
2. Binary search tree -Yes --DB(u), inner black nephew r, and p(DB)
-No No
3. Binary search tree --Yes -New DB, and new p(DB) i.e. old s(DB)
Outer black nephew x
No No
4. New DB is the root ----Root is always black. Remove DB.
-Yes Yes
This RR Case example satisfies Corollary 5 as stated above.
4.3 PART II: Examples of Removing Double-Black with a Red Parent
Having taken several Case examples with black parents of DB, B(p(DB)) ; now we consider the Cases of red parent,
R(p(DB)) to DB. In the following examples, we shall consider the cases in Examples 8 - 10 (Figures 13 -15).
4.3.1 Left-Right Case with one Red Nephew
Example 8 : The case of DB u with a red parent and one red inner nephew r.
Figure 13 presents our first case of a red parent p to DB, R(p(DB)) , in a LR Case. In this case, our VIP node
becomes the red inner nephew r to the DB node, which is symbolized as R( innerNephew (r(DB))) . As the only nephew node, r, to the DB, the R( innerNephew (r(DB))) thus determines the case to be a LR case. To resolve the 3R problem of DB removal, we left-rotate the black sibling s of DB, B(s(DB)) , to take us to a LL case, Figure 13(b) . Whilst the nodes still keep their color, we apply the general rule at the level of DB such that DB - B = NULL_LEAF, R( innerNephew (r(19))) - B = B(19) , and R(p(20)) + B = B(p(20)) , Figure 13(c) . Still, as a LL case, we right-rotate the
B(p(20)) , Figure 13(d) . Then apply the partial rule2, , to the black VIP nephew r, and the tree is balanced, Figure 13(e) . The highlight of this procedure is given in Table 8. In a list format, we represent the different operations and stages as ( ), ( ) [ , , ] [ ] [ , ( ), , , ( ), , '' ]R p InnerR r LR DB r p rDB DB leftRotate s LL rightRotate p BST
(26)
In first-order logic formula w.r.t. the red VIP nephew r at the point of formation of the DB , and the red
grandparent of r, R(g(r)) , we conclude that , ( ( )) ( ( ) ( ) ( ) ( )) BR R Rg r g r r B r r r
(27)
Fig. 13. LR case with one red inner nephew to double-black.
An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
20 Volume 11 (2025), Issue 1
Table 8. LR Case with An Inner Red Nephews to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempted
Node
DB
Removed
Tree
Balanced
LR case with one inner red nephew of DB
Left-rotate s(DB)
s(DB) - No No
LL case - Yes - - DB(u), inner red nephew r, and p(DB)
Yes No
LL case Right-rotate original p(DB)
p(DB) - Yes No
Binary search tree - - - Yes Original nephew r(DB)
r’s two new
children
Yes Yes
Note how the VIP nephew r changed its color as shown in the logic formula, firstly, from red R(r) to black B(r)
and then from black B(r) to red R(r) . Based on the foregoing analysis, we arrived at Corollary 6 .
Corollary 6:
In a RB tree, if there is only one red inner nephew to a DB with a red parent, then the case is a LR or RL case decided by the only red inner nephew; and as the VIP node, this nephew will have a color change.
4.3.2 Right-Left Case with one Red Nephew
Example 9 : The case of DB u with a red parent and one red inner nephew r.
Figure 14 of Example 9 is a RL Case which mirrors the LR case of Example 8 (Figure 13). Therefore, the same generalization holds for both Examples 8 and 9. The highlight of the procedure is also shown in Table 9. In a list format, we state the different stages involved as ( ), ( ) [ , , ] [ ] [ , ( ), , , ( ), , '' ]R p InnerR r RL DB r p rDB DB rightRotate s RR leftRotate p BST
(28)
Fig. 14. RL case with one red inner nephew to double-black.
Table 9. RL Case with An Inner Red Nephews to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempted
Node
DB
Removed
Tree
Balanced
RL case with one inner red nephew of DB
Right-rotate s(DB)
s(DB) - No No
RR case - Yes - - DB, inner red nephew r, and p(DB)
Yes No
RR case Left-rotate original p(DB)
p(DB) - Yes No
Binary search tree - - - Yes Original nephew r(DB)
r’s two new
children
Yes Yes
This RL Case example satisfies Corollary 6 as stated above. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 21
4.3.3 Left-Left Case with one Red Nephew
Example 10 : The case of a DB u with a red parent and one red outer nephew r.
In Figure 15 we have a LL Case. The LL case has a red parent, p, R(p(DB)) , a black sibling, s, B(s(DB)) , and one
red outer nephew r, R(outerNephew(r(DB))), to DB, respectively. The R(out erNephew (r(DB))) is the only nephew node and it doubles as the VIP node, and as the node that must have a color change at the end of the 3R process. To resolve the 3R problem, we right-rotate R(p(30)) to have a RR Case, Figure 15(b) , and then applied the partial rule1 at the level of the DB such that DB - B = NULL_LEAF, and R(p(30)) + B = B(30) , Figure 15(c) . Then, called the partial rule2 on the R(outerNephew(r(15))) where R(outerNephew(r(15))) - B = B(15) to have a balanced RB tree,
Figure 15(d) . The highlight of this procedure is given in Table 10. In a list format, the different stage of this SA operation is stated as ( ), ( ) [ , ] [ ] [ , ( ), , ' , , '' ]R p OuterR r LL DB p rDB DB rightRotate p RR BST
(29)
The first-order logic representation w.r.t. the red VIP nephew r and the red grandparent of r, R(g(r)) is given as , : ( ( )) ( ) ( ) g r g r R rR B r
(30)
Fig. 15. LL case with one red outer nephew to double-black.
Table 10. LL Case with an Outer Red Nephew to DB Node
Steps Structure & Case Rotation Rule Applied Operated Nodes Exempted
Node
DB
Removed
Tree
Balanced
1. LL case with one outer red nephew of DB
Right-rotate p(DB)
---p(DB) -No No
2. RR case --Yes -DB, outer red nephew r, R(r(DB))
-Yes No
3. BST ---Yes Outer red nephew R(r(DB))
Original p(DB) & s(DB)
Yes Yes
Based on the foregoing analysis, we arrived Collary 7:
Collary 7:
In a RB tree, if there is only one red outer nephew to a DB with a red parent, then the case is a LL or RR case decided by the only red outer nephew; and as the VIP node, this nephew will have a color change.
The mirror of this LL case example is the RR Case. Thus, the same generalization for the LL Case also holds for its counterpart RR Case. In a list format, the different stages of the RR case operation is given as ( ), ( ) [ , ] [ ] [ , ( ), , ' , , '' ]R p OuterR r RR DB p rDB DB leftRotate p LL BST
(31)
This LL Case example satisfies Corollary 7 as stated above. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
22 Volume 11 (2025), Issue 1
4.4 Algorithm
This is the algorithm of symbolic-algebraic arithmetic (SA) operation for DB removal with rotation , and
recoloring (3Rs ) of nodes in RB trees.
Algorithm of Double_ Black with Rotation Removal by Algebraic Operation
DoubleBlackWithRotation_Removal(T, p, v, u, r, s, x)
IF root = DB THEN
DB(root) - B => B(root)
ENDIF
IF root = x && deleted(node) == x THEN
color(x) = B
ENDIF
IF color(x) = R
IF deleted(node) == x THEN
R(x) + B => B(x)
ENDIF
ENDIF
IF color(x) = B
IF deleted(node) == x THEN
B(x) + B => DB(u)
WHILE DB(u):
16 / LR case /
IF p(DB(u )) == R && DB(u(hasInnerRedNe phew(r))) && LR_case THEN
method_LR_Rotation (p,u,s,r) //LR Call to line 88
apply_GSAR () //GSAR rule call to line 94
method_LL_Rotatio n(p,u,s,r) //LL Call to line 103
apply_PSAR2( ) //PSAR2 rule call to line 109
ENDIF
/ RL case /
IF p(DB(u) ) == R && DB(u(ha sInnerRedNephew(r))) && RL_case THEN
method_RL_Rotation(p , u, s, r) //RL call to line 91
apply_GSAR() //GSAR rule call to line 94
method_RR_Rotation(p,u,s,r) //RR call to line 106
apply_PSAR2() //PSAR2 rule call to line 109
ENDIF
30 . / LR to LL case /
IF p(DB(u) ) == B && DB(u(ha sInnerRedNephew(r))) && LR_case THEN
method_LR_Rotation(p,u,s,r) //LR Call to line 88
apply_GSAR() //LL Case: GSAR rule call to line 94
method_LL_Rotatio n(p,u,s,r) //LL Call to line 103
newDB(P) - B => B(p) //Tree balanced
ELSE // RL to RR case
method_RL _Rotatio n(p,u,s,r) //RL call to line 91
apply_GSAR(); //RR Case: GSAR rule call to line 94
method_RR_Rotation(p,u,s,r) //RR Case
newDB(P) - B => B(p) //Tree balanced
ENDIF
42 / LL or RR case/
43 IF p(DB(u) ) == B && DB(u(hasInnerRedNephew(x) ) && (hasOuterRedNephew(r))) THEN
//LL
method_LL_Rotation(p,u,s,r) //LL Call to line 103
apply_PSAR1() // PSAR1 rule call to line 112
apply_GSAR() //LL Case: GSAR rule call to line 94
ELSE //RR
method_RR_Rotation(p,u,s,r) //RR call to line 106
apply_PSAR1() // PSAR1 rule call to line 112
apply_GSAR() //RR Case: GSAR rule call to line 94
ENDIF
/ LL or RR case/
IF p(DB(u) ) == B && DB(u(hasOutRedNephew(r))) THEN
method_LL_Rotation(p,u,s,r) //LL Call to line 103
apply_PSAR1() // PSAR1 rule call to line 112
apply_GSAR() //LL Case: GSAR rule call to line 94
ELSE
method_RR_Rotation(p,u,s,r) //RR call to line 106
apply_PSAR1() // PSAR1 rule call to line 109
apply_GSAR() //RR Case: GSAR rule call to li ne 94
ENDIF
/ LL or RR case /
IF p(DB(u) ) == B && DB(u(hasInnerBlackNephew(r)) && (hasOuterBlackNephew(x))) & & LL _Case THEN
method_LL_Rotation(p,u,s,r) //LL Call to line 103
apply_GSAR() //LL Case: GSAR rule call to line 94
apply_PSAR1() //PSAR1 rule call to line 112
ELSE An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 23
method_RR_Rotation(p,u,s,r) //RR call to line 106
apply_GSAR() //RR Case: GSAR rule call to line 94
apply_PSAR1() //PSAR1 rule call to line 112
ENDIF
IF p(DB(u)) == R && DB(u(hasOuterRedNephew(r))) && LL_case THEN
method_LL_Rotation(p,u,s,r)
apply_PSAR1()
apply_PSAR2()
ELSE
method_RR_Rotation(p,u,s,r)
apply_PSAR1()
apply_PSAR2()
ENDIF
ENDWHI LE
ENDIF
ENDIF
Output Tree
/ Methods Definition /
method_LR_Rotation(p,u,s,r){
leftRotate(DB(u)); //LL case 3
}
method_RL_Rotation(p,u,s,r){
rightRotate(s(DB(u))); //RR case 3
}
apply_GSAR(p,u,s,r){
DB(u) - B => B(x);
R(r) - B => B(r);
IF p(DB(x)) = R
R(p(DB(x))) + B => B(p);
ELSE
B(P(DB(x))) + B => newDB(u)
// recur up the tree
}
method_LL_Rotation(p,u,s,r){
rightRotate(B(p)); //original parent of DB
}
method_RR_Rotation(p,u,s,r){
leftRotate(B(p)); //original parent of DB
}
apply_PSAR2(){
B(r) - B => R(r) //Tree is balanced
}
apply_PSAR1(){
DB(u) - B => B(NULL_LEAF) //B(NULL_LEAF) = DB(u) - B
B(p(DB(u))) + B => DB(p) // DB(P) = B(p(DB(u))) + B
115 }
116 . END PROCEDURE
4.5 Findings
Rotations in binary search trees such as AVL and RB trees is part of a tree balancing operation that leads to a balanced state of the trees. In this study, all the structural cases of DB removal that involved rotation; we found that
rotation is the first operation that occurs before the application of node color-change by the SA algorithms. Having looked at the evidence of the SA algorithmic operations on subparts of the RB tree; in Examples 11 – 13, we look at the behavior of SA algorithms on RB trees with bigger depths and nodes using tree diagrams.
Example 11: Pictorial Analysis of a RL Case.
This example in Figure 16 is a RL Case denoted as ( ) ( ) given in eq. (28). This Case is similar to the example in Figure 14 above. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
24 Volume 11 (2025), Issue 1
Fig. 16. RL case with one red inner nephew to double-black.
Example 12: Pictorial Analysis of a LL Case.
This pictorial analysis in Figure 17 is a complex LL Case as denoted by ( ) ( )in eq. (18). We deleted node 70 in Figure 17(a) to form a DB. Then right-rotate the B(p(DB)) i.e. B(p(66)) to have a RL Case, Figure 17(b). Apply the general rule to remove the DB. At this point, a new DB(66) is formed in Figure 17(c) to have a RL Case. As we recur up the tree to balance the tree; we right-rotate the new DB(66) to have a BST; and finally applied the
partial rule2, . Enumerating the different stages and operations, we have ( ), ( , ) [ , , ] [ , , ] [ ] [ , ( ), , , , , '' ]B p TwoB r x LL DB r p DB r p rDB DB rightRotate p RL BST
(32)
This case example is also a special case like that of Example 7 Figure 12 ( ) ( ), see eq. (24), where the Case was not determined by the VIP nephew r but by the outer black nephew x whose node color does not change. Removing the DB in this case and balancing the tree, involved applying the general rule twice and then the
partial rule2 .
Fig. 17. Complex LL case with two black nephews to double-black. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 25
Example 13: Pictorial Analysis of a RL Case.
In Figure 18 is a RL case that fits into the expression ( ) ( ) in eq. (28). As demonstrated in Figure 7 and its mirror case in Figure 6; the inner red nephew r is the VIP node that determined the case as a RL. To balance this tree, only the general rule is applicable.
Fig. 18. RL case with one red inner nephew to double-black.
4.5.1 Sibling Node s versus Nephew node r
On a final note, recall that our initial study in worked on DB removal without rotation of nodes. In the aforementioned work, where the general SA rule was first projected, we applied the rule to nodes [DB, s, p] . Because there was no rotation, the SA operated on the DB node, its sibling s, and its parent p only. In this paper which deals with node rotation, we realized that “no one rule fits all” . This is because of the complexity of handling DB removal with rotation. To handle this complexity, we needed to refine the general SA rule to derive two additional rule, namely, the
partial rule1 and partial rule2.
Rotation results in the change of position in a tree. Now, let’s denote a new sibling of DB as s. Depending on the case, for instance, LR or RL Case; when a DB is rotated, a DB gets a new sibling s, which is the original nephew r to DB. See the examples in Figure 6(b), 7(b), 8(c), 9(c), 10(c), 11(b), 12(b), 13(b), 14(b), 16(c), 17(b), and 18(c). The
general SA rule which is always operated on three connected nodes is operated on the [DB, r, p ] after rotation, where r
becomes the newly adopted sibling s to DB . In a nutshell, since an original nephew such as r has transformed from a
nephew to the sibling s ; then we state that the = as a result of node rotation. Except for Example 10 Figure 15 expression ( ) ( ) , eq. 31; all the other examples have in one way or the other used the rule to balance a RB tree. In this LL Case of Example 10, the DB removal problem combined two partial rules: partial 1,
and partial 2, . Therefore, we conclude that [ , , ] [ , ] [ ] ' '' DB r p DB p r
(33) [ , ] [ , , ] [ ] ' '' DB p DB r p r
(34) [ ] [ , , ] [ , ]'' 'r DB r p DB p
(35)
4.6.1 First Comparative Example of the Traditional RB Algorithm vs. SA Algorithm:
Table 10 compares the performance of the (Conventional) Traditional RB algorithm (TA) against our SA algorithm. As presented by the number of steps in Table 10, both TA and SA methods have shown equal performance based on the same number of steps or runtime of 5 each to balance the given RB tree. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
26 Volume 11 (2025), Issue 1
Table 10. Side-by-Side Performance Comparison of the TA vs. SA with Equal Number of Steps.
Steps Traditional Algorithm Symbolic -Arithmetic Algorithm
1. Delete Delete
2. Rotate Rotate
3. Re-color Re-color
4. Rotate Rotate
5. Re-color (Balanced) Re-color (Balanced)
4.6.2 Second Comparative Example of the Traditional RB Algorithm vs. SA Algorithm
Table 11 is another comparison of the TA with our SA algorithm. The side-by-side pictorial evaluation as itemized by the numbering of the steps taken depicts that the SA balanced the RB tree in 4 steps as against the TA in 5 steps. That is, the SA algorithm finished one step faster than the TA RB algorithm. From the step-by-step analysis, it has been shown in this work that the SA algorithm is clearer and much more precise for teaching and learning the DB removal problem. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 27
Table 11. Side-by-Side Performance Comparison Where SA Performed Faster than TA.
Steps Traditional Algorithm Symbol ic -Arithmetic Algorithm
Delete Delete
Rotate Rotate
Re-color Re-color
Rotate Rotate (Balanced)
Re-color (Balanced)
Figure 19 shows a graphical representation of the TA and SA algorithm from the demostrations presented in Table 10 and 11, respectively.
Runtime Data of the TA versus SA Algorithm for DB
Removal and Tree Balancing
R(p) and
InnerR(r)
R(p) and
InnerR(r)
Traditional Algorithm
(TA) 5 5
Symbolic Algorithm (SA) 5 4
Fig. 19. Runtime visualization of the TA versus SA algorithm. On one RB tree problem, both TA and SA showed equal runtime of 5 steps. On another problem, the SA had a faster runtime of 4 steps as against the 5 steps of the TA.
0
2
4
6
R(p) and innerR(r) R(p) and innerR(r)
Runtime Steps
Runtime of Double -Black Node Removal and
Tree Balance
Traditional Algorithm (TA) Symbolic Algorithm (SA) An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
28 Volume 11 (2025), Issue 1
4.6.3 Feedback from the Symbolic Algebraic Teaching Approach
As first reported in , for over 4 academic years running, this work on the symbolic-algebraic arithmetic (SA) method of teaching DB removal in RB trees has been ongoing. The questions and feedback w.r.t. the SA algorithmic method from students are hereby stated below; where SA represents symbolic-algebra arithmetic, and TA means the traditional (or conventional) algorithm of RB trees.
(a) Question 1: What is your opinion on this new symbolic method in the understanding of the removal of DB and balancing of the tree ?
Table 11. Feedback on the Understanding of the Symbolic-Algebraic Method .
Student Feedback Analysis
For me, the new [symbolic] algebra (SA) algorithm is easier to understand compared to the conventional approach, with less classifications and more symbolic visual aids. But it still requires practicing to get the hang of it.
SA approach is simpler.
SA is visual and animated in process.
SA has less classification.
SA is needed to be learned too.
Symbolic algebraic application to tree balance after rotations and node coloration has a fixed pattern to follow which is lacking in the traditional algorithm.
SA provides a fixed length of steps.
The symbols method provides better understanding for node coloration. It explains what node needs a color change, why it needs to change and how it will change.
SA gives clearer understanding.
SA gives the step and the exact node to color.
(b) Question 2: What are the perceived differen ces to other approaches in literature and the original DB removal algorithm?
Table 12. Feedback on the Conventional RB Tree Algorithm vs. the Symbolic-Algebraic Method.
Student Feedback Analysis
This [symbolic approach] is definitely simpler for me because it simplifies the process inherent in a traditional algorithm (TA). Both the traditional red-black tree algorithm and the new symbolic approach algorithm involves remembering several conditions for deletion and rotation before carrying out corresponding operations. In this case [symbolic approach], less is definitely better, because it's not as complicated as it used to be.
SA is definitely simpler.
Both the TA and new SA have certain conditions to learn (w.r.t. rotation); but the SA is lesser with information.
SA method is not complicated.
Traditional algorithm is somehow complicated. Not knowing where to stop color exchange between nodes sometime. The symbolic operation is a much simplified method of tree balancing after a node removal. The symbolic method uses a step-by-step mathematical approach that is certain about when to end color change.
TA algorithm is complicated.
TA is difficult to predict where color change will end.
SA algorithm is much simpler.
SA is step-by-step.
The symbols approach takes away the complications involved in the heavy information that surrounds the traditional algorithm. This is because there are few information to process as we fix the colors of nodes with the symbols method. The steps are easy to follow.
SA is not complicated.
SA has less information w.r.t. its operation.
The deletion operations of traditional red-black [algorithm] are badly explained in many resources I read including my textbook. These operations are not obvious to understand and confuse many students when they first see it. The new symbolic method provides steps to memorize the operations and makes removal of DB easier to learn."
TA is not clear to understand.
SA is a step-by-step operation.
SA makes DB removal easier to memorize and learn.
Conclusions and Further Work
The removal of a double-black (DB) node mechanically poses a lot of difficulty in teaching, learning, and understanding of red-black (RB) trees. The question of when does rotation occur? Or which node changes its color? Or at what point will a node change its color have all been addressed in this study. This study formulated and introduced three rules that uses symbolic arithmetic (SA) operations for removing DB nodes and balancing RB trees. The simplistic SA formulas which are namely the generic rule , partial rule1 , and partial rule2 were demonstrated with the different RB LR, RL, LL, and RR cases in this paper. With these rules , students are well -equipped to understand when
to remove DB node s, and where and what nodes need recoloring after rotation (s). The use of these equations or rules to
remove DB node s and recolo ring of nodes in RB trees has prove n to be decisive and unambiguous. By combining these
formulas, we have shown that a DB can be easily removed and the process, operations , and the stages involved can also
easily be understood . This paper also showed a side -by -side c omparison of the traditional RB algorithm (TA) with our
SA algorithm. Our findings revealed that the SA algorithm was never slower than the TA algorithm . In stead , the SA has
proven to be faster in some cases and equal in other cases . More so , the paper has engaged the use of first -order logic
statements in de scribing the nodes, their values, and operations. Also, t hese logic expressions have enhanced learning
and made the process of dB removal explicitl y explainable — in the concept of explainable AI. Thus, based on RB tree
image visuals with specified logical formalism, intelligent system AI can learn and apply the SA algorithm and also
explain the reasons behind every step that is taken and the decisions made in the re construction of the RB trees. From
the foregoing, in fu ture work , we shall look at image visuals for RB tre e construction , DB removal , and tree re balancing An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
Volume 11 (2025), Issue 1 29
in the contest of explainable AI in combination with formal logic and the SA rules. Having shown that the SA runtime (number of steps taken) can be faster than the TA, further work will also look at the place of the SA algorithm in terms of time complexity.
Acknowledgement
We would like to acknowledge all past and present students of Data Structures & Algorithms at the Wenzhou Kean University (WKU) for their collaboration, practice, and testing of this simplified symbolic-arithmetic algebraic method for satisfiability of the properties of the red-black tree.
References
Y. D. Liang, Introduction to Java programming and data structures, 11th ed., United Kingdom: Pearson Education, 2019.
M. T. Goodrich, R. Tamassia and M. H. Goldwasser, Data structures and algorithms in Java, John Wiley & Sons, 2014.
C. Sanderson and R. Curtin, "A user-friendly hybrid sparse matrix class in C++," in In International Congress on Mathematical Software, Cham, July 2018.
K. E. Ehimwenma, J. Wang, Z. Zheng and H. Zhou, "A symbolic-arithmetic for teaching double-black node removal in red-black trees," Educational Dimension, vol. 59, pp. 112-129, 2022.
Wikipedia, "Red-black tree," [Online]. Available: [Accessed 7th July 2023].
Z. D. Eddine, "Improving the Red-Black tree delete algorithm," 2021.
S. Kahrs, "Red-black trees with types," Journal of functional programming, vol. 11, no. 4, pp. 425-432, 2001.
E. Fredriksson, "Reducing CPU scheduler latency in Linux," Thesis, UMEA University, 2022.
F. Xhakaj and C. W. Liew, "A new approach to teaching red black tree," in In Proceedings of the 2015 ACM Conference on Innovation and Technology in Computer Science Education, 2015.
D. Galles, "Red Black Tree," 2011. [Online]. Available: [Accessed 04 04 2022].
S. Hanke, "The performance of concurrent red-black tree algorithms," in International Workshop on Algorithm Engineering, 1999.
L. Qiaoyu, L. Jianwei and X. Yubin, "Performance analysis of data organization of the real-time memory database based on red-black tree," In 2010 International Conference on Computing, Control and Industrial Engineering, vol. 1, pp. 428-430, 2010.
H. Zhang and Q. Liang, "Red-black tree used for arranging virtual memory area of Linux," in In 2010 International Conference on Management and Service Science, 2010.
J. Li, Y. Xu and H. Guo, "Memory organization in a real-time database based on red-black tree structure," In Fifth World Congress on Intelligent Control and Automation (IEEE Cat. No. 04EX788) , vol. 5, pp. 3971-3974, June 2004.
M. Jeong and E. Lee, "A Swapping Red-black Tree for Wear-leveling of Non-volatile Memory," The Journal of the Institute of Internet, Broadcasting and Communication, vol. 19, no. 6, pp. 139-144, 2019.
S. L. Deshpande, "Tree-Based Approaches for Improving Energy Efficiency and Life Time of Wireless Sensor Networks (WSN): A Survey and Future Scope for Research," Singapore, 2020.
M. Hasanzadeh, B. Alizadeh and F. Baroughi, "The cardinality constrained inverse center location problems on tree networks with edge length augmentation," Theoretical Computer Science, vol. 865, pp. 12-33, 2021.
W. L. Chun and N. Huy, Using an Intelligent Tutoring System to Teach Red Black Trees., 2019.
D. Wu, P. Guo, C. Zhang, C. Hou, Q. Wang and Z. Yang, "Research and Practice of Data Structure Curriculum Reform Based on Outcome-Based Education and Chaoxing Platform," International Journal of Information and Education Technology, vol. 11, no. 8, 2021.
Z. Seidametova, "Some methods for improving data structure teaching efficiency," Educational Dimension, vol. 58, pp. 164-175, 2022.
J. King, "Combining Theory and Practice in Data Structures & Algorithms Course Projects: An Experience Report," in Proceedings of the 52nd ACM Technical Symposium on Computer Science Education, NY, USA, March 2021.
T. Nipkow, "Teaching algorithms and data structures with a proof assistant (invited talk)," in Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, Denmark, January 2021.
L. Bounif and D. E. Zegour, "A revisited representation of the red-black tree," International Journal of Computer Aided Engineering and Technology, vol. 16, no. 1, pp. 95-118, 2022.
K. Ghiasi-Shirazi, T. Ghandi, A. Taghizadeh and A. Rahimi-Baigi, "A Pedagogically Sound yet Efficient Deletion algorithm for Red-Black Trees: The Parity-Seeking Delete Algorithm.," June 2022.
R. Sedgewick, " Left-leaning red-black trees," In Dagstuhl Workshop on Data Structures, vol. 17, Sept 2008.
D. Galles, "Red Black Tree Visualization," 2011. [Online]. Available: [Accessed 04 04 2022].
S. T. Mueller, E. S. Veinott, R. R. Hoffman, G. Klein, L. Alam, T. Mamun and W. J. Clancey, "Principles of explanation in human-AI systems.," 2021.
J. J. B. Vial, A concurrent red black tree, Pontificia Universidad Catolica de Chile , 2012.
K. Germane and M. Might, "Deletion: The curse of the red-black tree," Journal of Functional Programming, vol. 24, no. 4, pp. 423-433, 2014. An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees
30 Volume 11 (2025), Issue 1
Authors’ Profile s
Kennedy E. Ehimwenma, Ph.D. Depart of Computer Science Wenzhou Kean University, China. Research interest: Multi-agent systems, knowledge representation, ontology, computational logic, IoT, smart systems, student learning, tree based learning.
Hongyu Zhou, Student, Depart of Computer Science Wenzhou Kean University, China. Research interest: Blockchain, generative artificial intelligence, cryptocurrency, tree based learning.
Junfeng Wang, Student, Depart of Computer Science Wenzhou Kean University, China. Research interest: Data structures & algorithms, machine learning, deep learning, computer graphics.
Ze Zheng, Student, Depart of Computer Science Wenzhou Kean University, China. Research interest: Data structures & algorithms.
How to cite this paper: Kennedy E. Ehimwenma, Hongyu Zhou, Junfeng Wang, Ze Zheng, "An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.11, No.1, pp. 1-30, 2025. DOI: 10.5815/ijmsc.2025.01.01 |
12038 | https://www.mataverdedecking.com/blog/can-you-prevent-checks-and-cracks-in-large-timbers | Can you Prevent Checks and Cracks in Large Timbers?
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Can you Prevent Checks and Cracks in Large Timbers?
Written byChris Nolan on Tue, Oct 24, 2017 @ 05:10 AM
Although it is impossible to prevent a certain amount of checking and cracking in heavy or large timbers, there are techniques that can be used to help minimize this naturally occurring phenomena. You may be surprised to learn that some checking can be beneficial to exterior woods.
Wood moves, naturally, throughout its growth cycle, and continues to move throughout its usable life cycle, long after it is harvested, dried, milled and used as lumber. Because thinner boards can be kiln dried, most of the movement has already been minimized before it ever reaches a job site. Thicker materials, on the other hand, are often too big to kiln dry, and may take years to fully dry out.
The larger the board or timber, the more likely it is to move, and check, and crack. Larger timbers always check and crack somewhat. Need an example? Look at any telephone pole you have ever seen – there are a certain amount of checks, cracks and splits that are visible.
Because it takes large timbers and heavier materials a long time to fully acclimate on site, some checks and cracks may occur with 3x and 4x nominal thickness materials over time.
The bigger the timber, the more apparent (deeper, wider, longer) the checking is likely to be. Large timbers act more like a log than a board. When checking naturally occurs in a log when it is in place on a site, this is not necessarily a bad thing. In fact, wood scientists will tell you that when heavier timbers or logs check, it rarely affects the strength of the board.
Natural checking will actually relieve some of the built-up tension inside the wood and make it stronger. In fact, the US Forest Products Laboratory has developed criteria for the allowable amount of checks or cracks to maintain the strength of the beam or structural member.
There is no way to prevent the naturally occurring process of checking from happening, especially when the wood will be used outdoors. Outdoor environments experience the most dramatic range of environmental exposure. From snow, inclement weather, pouring rain, howling winds, direct sunlight, freezing, thawing, high temperatures and low temperatures, wood that’s used outdoors experiences the full range of nature’s punishment.
During the warmer months of June, July and August especially, we occasionally hear comments like, “my 4x4 wood is cracking,” or “my wood seems to be splitting”. No surprise there – warmer weather and lots of UV exposure will often hurry the acclimation process,sometimes too quickly. However, many of these seasonal checks and cracks tend to close up over time.
Best Ways to Help Minimize Checking and Cracking in Heavy Timbers
Proper Acclimation
While you can’t eliminate the checking entirely, you can help minimize it through proper on-sit e wood acclimation practices. Always follow proper acclimation instructions for best results. This includes storing the wood in a shaded area, well away from ground moisture and allow the ends and edges of the boards to breathe.
Download Critical Acclimation Checklist here.
Always Seal Cut Ends of Hardwoods
When should I seal the ends of hardwood boards? Immediately after cutting them. This helps minimize end checking from the most vulnerable area on a piece of wood – the end grains. Water moves quickly along the grains of wood. Slowing down the escape of water from the cut ends is crucial to proper installation techniques and affects the performance of the wood.
Adequate UV Protection
Putting a UV inhibitive finish on the wood, immediately after installation, is an effective way to minimize surface checking. It’s like going to the beach with (or without) suntan lotion. Better to be safe than (painfully) sorry.
Provide Relief Grooves
Another technique to help minimize checking is to cut a kerf or kerfs into the back of a heavy timber. Ancient Japanese carpenters used this technique thousands of years ago and the same holds true today. Modern hardwood flooring producers use the same method. Most wood decking sold in Europe nowadays has a fluted profile on one or both faces.Adding relief grooves on one side of a heavy timber immediately relieves much of the tension in hardwoods and helps minimize potential movement.
Hardwood Decking Installation
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For integers A>0, B>0, N>0, find integers x>0,y>0 such that N-(Ax+By) is smallest non-negative
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Asked 8 years, 11 months ago
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Example :
A=5, B=2, N=12
Then let x=2, y=1, so 12 - (5(2) + 2(1)) = 0.
Another example:
A=5, B=4, N=12
Here x=1, y=1 is the best possible. Note x=2, y=0 would be better except that x=0 is not allowed.
I'm looking for something fast.
Note it's sufficient to find the value of Ax+By. It's not necessary to give x or y explicitly.
algorithm
linear-algebra
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edited Oct 21, 2016 at 12:15
Gene
47.1k 4 4 gold badges 63 63 silver badges 100 100 bronze badges
asked Oct 21, 2016 at 3:39
Lalit VermaLalit Verma
802 11 11 silver badges 25 25 bronze badges
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2 Your question does not make any sense. Please reword it.bikeman868 –bikeman868 2016-10-21 03:45:10 +00:00 Commented Oct 21, 2016 at 3:45
See this now @bikeman868 Lalit Verma –Lalit Verma 2016-10-21 03:56:37 +00:00 Commented Oct 21, 2016 at 3:56
Why 9? 52+40=10. And 54+4(-2)=12.Soonts –Soonts 2016-10-21 04:02:57 +00:00 Commented Oct 21, 2016 at 4:02
Still doesn't make enough sense. This might help though: math.stackexchange.com/questions/20717/…Jared Goguen –Jared Goguen 2016-10-21 04:03:59 +00:00 Commented Oct 21, 2016 at 4:03
i put constraints @Soonts Lalit Verma –Lalit Verma 2016-10-21 04:08:50 +00:00 Commented Oct 21, 2016 at 4:08
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If gcd(A,B)|N, then N is your maximal value. Otherwise, it's the greatest multiple of gcd(A,B) that's smaller than N. Using 4x+2y=13 as an example, that value is gcd(4,2)6=12 realized by 4(2)+2(2)=12 (among many solutions).
As a formula, your maximal value is Floor(N/gcd(A,B))gcd(A,B).
Edit: If both x and y must be positive, this may not work. However, won't even be a solution if A+B>N. Here's an algorithm for you...
```
from math import floor, ceil
def euclid_wallis(m, n):
col1 = [1, 0, m]
col2 = [0, 1, n]
while col2[-1] != 0:
f = -1 (col1[-1] // col2[-1])
col2, col1 = [x2 f + x1 for x1, x2 in zip(col1, col2)], col2
return col1, col2
def positive_solutions(A, B, N):
(x, y, gcf), (cx, cy, _) = euclid_wallis(A, B)
f = N // gcf
while f > 0:
fx, fy, n = fx, fy, fgcf
k_min = (-fx + 0.) / cx
k_max = (-fy + 0.) / cy
if cx < 0:
k_min, k_max = k_max, k_min
if floor(k_min) + 1 <= ceil(k_max) - 1:
example_k = int(floor(k_min) + 1)
return fx + cx example_k, fy + cy example_k, n
if k_max <= 1:
raise Exception('No solution - A: {}, B: {}, N: {}'.format(A, B, N))
f -= 1
print positive_solutions(5, 4, 12) # (1, 1, 9)
print positive_solutions(2, 3, 6) # (1, 1, 5)
print positive_solutions(23, 37, 238) # (7, 2, 235)
```
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edited Oct 21, 2016 at 20:38
answered Oct 21, 2016 at 4:12
Jared GoguenJared Goguen
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Lalit Verma
Lalit VermaOver a year ago
what if A=5,B=3, N=12
2016-10-21T04:50:14.33Z+00:00
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Raymond Chen
Raymond ChenOver a year ago
@LalitVerma 53+3(-1)=12. The original problem statement did not say that x and y had to be positive.
2016-10-21T05:52:51.24Z+00:00
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Lalit Verma
Lalit VermaOver a year ago
Sorry i didn't mention but x,y should be positive integers , the above formula failed for this constraint @RaymondChen
2016-10-21T07:52:31.5Z+00:00
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Jared Goguen
Jared GoguenOver a year ago
@LalitVerma I added an algorithm in Python that accounts for that constraint, should be pretty optimal as there isn't much "force" involved.
2016-10-21T20:40:05.153Z+00:00
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A brute-force O(N^2 / A / B) algorithm, implemented in plain Python3:
```
import math
def axby(A, B, N):
return [A x + B y
for x in range(1, 1 + math.ceil(N / A))
for y in range(1, 1 + math.ceil(N / B))
if (N - A x - B y) >= 0]
def bestAxBy(A, B, N):
return min(axby(A, B, N), key=lambda x: N - x)
```
This matched your examples:
```
In : bestAxBy(5, 2, 12)
Out: 12 # 5 (2) + 2 (1)
In : bestAxBy(5, 4, 12)
Out: 9 # 5 (1) + 4 (1)
```
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answered Oct 21, 2016 at 14:08
Ahmed FasihAhmed Fasih
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Raymond Chen
Raymond ChenOver a year ago
Once N > LCM(A, B), you can stop searching because you will just be seeing the same answers over and over.
2016-10-21T18:05:29.803Z+00:00
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Have no idea what algorithm that might be, but I think you need something like that (C#)
```
static class Program
{
static int solve( int a, int b, int N )
{
if( a <= 0 || b <= 0 || N <= 0 )
throw new ArgumentOutOfRangeException();
if( a + b > N )
return -1; // Even x=1, y=1 still more then N
int x = 1;
int y = ( N - ( x a ) ) / b;
int zInitial = a x + b y;
int zMax = zInitial;
while( true )
{
x++;
y = ( N - ( x a ) ) / b;
if( y <= 0 )
return zMax; // With that x, no positive y possible
int z = a x + b y;
if( z > zMax )
zMax = z; // Nice, found better
if( z == zInitial )
return zMax; // x/y/z are periodical, returned where started, meaning no new values are expected
}
}
static void Main( string[] args )
{
int r = solve( 5, 4, 12 );
Console.WriteLine( "{0}", r );
}
}
```
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answered Oct 21, 2016 at 23:39
SoontsSoonts
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12040 | https://www.youtube.com/watch?v=EJ2ZkAqh25s&pp=0gcJCfwAo7VqN5tD | Combinations with Repetition | Combinatorics
Wrath of Math
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Posted: 27 Nov 2020
How many combinations of k objects can we make from a set of n objects when we allow for reptition? We'll go over an interesting solution to this question in today's combinatorics lesson. We consider a number of spaces in which we can place n-1 bars to completely describe a combination with repetition. Furthermore every possible combination with repetition can be described by a placement of bars, and so we reduce the problem to a normal combination problem, counting how many ways we can place bars in spaces!
The combination formula and why it works:
Permutations with repetition:
Number of 4-Digit Codes with and without Repetition:
Symmetric property of binomial coefficients:
Thanks to Nasser Alhouti, Robert Rennie, Barbara Sharrock, and Lyndon for their generous support on Patreon!
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46 comments
Transcript:
Introduction how many ways can k objects be selected from a collection of n objects with repetition aloud that's what we'll be going over in today's wrath of math lesson this was requested by one of my wonderful patrons if you'd like to consider supporting wrath of math on patreon with a monthly donation there will be a link down in the description and be sure to leave your video requests down in the comments usually when we begin to count combinations we do not allow repetition you may recall that the number of ways k objects can be selected from a collection of n objects is equal to this n factorial divided by k factorial times n minus k factorial again that's the number of combinations of k objects that can be selected from a collection of n objects it's denoted like this this is red n choose k and it's called a binomial coefficient you should already be familiar with this sort of thing before tackling today's problem i'll leave links in the description to lessons all about this combinations without repetition also the problem of counting combinations without repetition and the problem of counting combinations with repetition both of those can be phrased in terms of counting subsets of sets and multisets but in the interest of time today we're pretty much going to not touch on the set formulations of these problems just Solution remember when we're counting combinations we're not worried about the order of the objects we're selecting which is why sets and subsets work well to represent these problems all right no more dilly dallying let's go over a very cool solution to this problem let's say we've got this collection of n objects a1 a2 all the way through a n suppose that i've selected k of these objects and repetition is allowed so i've probably selected some of the objects multiple times how could i communicate the combination of objects that i selected to you one way is like this i could say here are k spaces and i'll fill them in with the objects i selected and since order doesn't matter maybe i'll be nice and list the objects in the most convenient order so if i selected any a1s i'll put the a1s first then i'll put the a2s a3s and so on and then here you go i've written out all the k objects i selected you see i selected two a1s 1 a3 2 a4s and so on and this just describes one of the many ways that i could select k objects from our collection of n objects with repetition aloud and perhaps this seems a bit daunting how we could possibly count the number of ways to fill in these k spaces with selections of these n objects especially because repetition is allowed we might wonder is there another way i could completely communicate the combination of objects that i selected without literally writing out every single one here's an idea maybe i give you these k spaces but i draw bars separating the different sections that i want to be filled with different types of objects and so i give you this i've placed a bar everywhere i want you to stop placing one object and move on to the next let's try filling this in to make it more clear what i mean we begin with the a1s and we see there are two empty spaces where we can place the a1s but then we encounter a bar which means i want you to stop placing the a1s and move on to the a2s but then we have another bar which means i want you to stop placing the a2s and move on to the a3s so i don't want any a2s then we have a space for an a3 and another bar to say stop placing the a3s and move on to the a4s then we have a bar to indicate we should stop placing the a4s then another bar to indicate we should stop placing the a5s and then another bar to indicate we should stop placing the a6s so then we're on to a7 and we'd continue in the same way the bars have completely described the collection of objects that i listed out earlier here i'm pasting it we can see it's the same thing then my point to you is that to count the number of ways we can select k objects from a collection of n objects with repetition allowed to count that we can just count the number of ways that we can place bars in spaces to put it simply because each combination of k objects corresponds to exactly one arrangement of these bars and each arrangement of these bars describes exactly one combination of k objects so between the k objects that we're placing and all of these bars how many things do we have total well we've got the k spaces and then we have n minus 1 bars we have n minus 1 bars because we have n objects total and we place a bar to indicate where we should stop placing each object except for a n we don't need a bar to indicate where we should stop placing the last object because we can just stop when we run out of spaces since it's the final object and so we can answer the question by imagining we have a total of k plus n minus 1 spots to put things again that's k plus n minus 1 spots to put things we're going to have k objects and we're going to have n minus 1 bars so to count the number of combinations of k objects from a collection of n objects with repetition allowed we just need to count the number of ways that we can place n minus 1 bars in a total of k plus n minus 1 spots and that my friends is just k plus n minus 1 choose n minus 1. and that is the answer so once more instead of just having k spaces we can think of it as having a total of k plus n minus 1 spaces so that we have a space for the bars as well and then the number of ways that we can pick n minus 1 spaces for our bars from those k plus n minus 1 spaces is the number of combinations of k objects from a collection of n with repetition aloud now let's just see Examples quick two examples to further convince you of this correspondence between a combination of objects and spaces and bars here we've just got one way of representing a multi-set this set has objects x1 x2 x3 and x4 and it has infinitely many of all of them so we want to pick five and of course we're allowed to pick the same one multiple times remember we should have a total of k plus n minus 1 spots or spaces k is the number of objects we're picking in this case 5 and n is the number of different objects we have to choose from we have four different objects to choose from so 5 plus 4 minus 1 that gives us 8. here we have 8 spaces that have n minus one or three bars placed in them because remember we are picking n minus one spaces for the bars so in this case n minus one is three we've got three bars in the eight spaces how does this describe a combination of five of these objects well we begin with the x ones there's one space for an x one then we encounter a bar so we stop with the x ones and move on to the x twos then we encounter another bar so there won't be any x2s and we move on to the x3s we've got a lot of spaces for these x threes and then we encounter a bar that says stop with the x3s move on to the x4s but there are no more spaces so we're done so again the point is to just make clear how placing n minus 1 bars in k plus n minus 1 spaces describes a combination of our objects with repetition quickly let's just go in the other direction for this example we're told that our collection has two x twos two x threes and one x four how can we use the spaces and the bars to represent this combination of objects well again we should have k plus n minus 1 spaces and we're picking from the same collection of objects so this is still equal to 8 and we have 8 spaces then we just need to place n minus one or three bars within the spaces we see that we haven't selected any x ones so we need the first space to have a bar that says immediately stop with the x ones and move on to the x2s then we have two x2s so those would take up the next two spaces and then we would have a bar to indicate that we should move on to the x3s x3s would take up the next two spaces again and then we would place a bar to indicate that you should move on to the x4s and then there's one space left for the one x4 again we don't need a final bar because that's the last space so we know that's the end so hopefully that helps to convince you that the number of ways we can place n minus 1 bars in k plus n minus 1 spaces is indeed the number of ways that k objects can be selected from a collection of n objects with repetition allowed and that's pretty sweet but it gets even better by the symmetric property of binomial coefficients and i'll leave a link in the description to a proof of that property k plus n minus 1 choose n minus 1 is equal to k plus n minus 1 choose k which i think you'll agree is a bit nicer so the number of ways that k objects can be selected from a collection of n objects with repetition allowed is k plus n minus 1 choose k who'da thought finally let's go back to these examples to actually do the calculation how many ways can we pick five objects from these four distinct objects if repetition wasn't allowed of course there would be no ways to do it you can't pick five objects from a collection of four but since we're allowed repetition we can use the beautiful expression that we just explained the answer is this we have a binomial coefficient the number of objects we're selecting which is five plus the number of distinct objects we have to choose from which is four minus one and then choose five the number of objects we're selecting that of course is equal to eight choose 5 which is 56. take note of the fact that the problem becomes a lot different if we're not allowed to repeat each element as many times as we want for example in our multi-set maybe we only have one copy of x1 three copies of x2 two copies of x3 and one copy of x4 this changes the problem completely we'll leave that for another time but i think that is enough for today so the number of ways that k objects can be selected from a collection of n objects with a repetition allowed is k plus n minus 1 choose n minus 1 which is the same as k plus n minus 1 choose k i hope this video helped you understand how to count combinations with repetition let me know in the comments if you have any questions need anything clarified or have any other video requests thank you very much for watching i'll see you next time and be sure to subscribe for the swankiest math lessons on the internet [Music] you |
12041 | https://chemistrytalk.org/water-polarity-why-polar/ | ChemTalk
Polarity of Water – Why is Water Polar?
Core Concepts
Water is one the most abundant substance on Earth, covering 71% of its surface. In this tutorial, you will learn the polarity of water and how it gives rise to some of water’s most important properties.
Topics Covered in Other Articles
What does it mean for a molecule to be polar?
By definition, a polar molecule has a partially positive end and a partially negative end. The molecule achieves this by having an uneven distribution of electrons between its atom. That is, electrons gather more closely to one atom than the other, thus making one atom slightly more negative than the other. Moreover, these atoms must also be arranged in a way that their partial charges do not cancel each other out.
Is water polar?
As stated above, to be polar, a molecule must have polar covalent bonds and be asymmetrical.
Does water have polar covalent bonds?
The electronegativity of oxygen is 3.5, and the electronegativity of hydrogen is 2.1, making the difference in electronegativity between them 1.4. Atoms in polar covalent bonds generally have a difference in electronegativity between 0.4 and 1.7. Thus, the O–H bonds in water molecules—where the O end is partially negative, and the H end is partially positive—are polar covalent bonds.
Is water asymmetrical?
The oxygen atom in a water molecule bonds to 2 hydrogen atoms and 2 lone pairs, giving it the steric number of 4. According to VSEPR theory, a steric number of 4 means the electron geometry of water is tetrahedral. Moreover, because that steric number includes 2 lone pairs, the molecular geometry of water is bent. A bent structure gives the water molecule its asymmetry.
In conclusion, water is polar because it has polar covalent bonds and is asymmetrical.
What comes out of water’s polarity?
Water as the universal solvent
Recall the golden rule of solubility: like dissolves like. Therefore, water, as a polar solvent, readily dissolves polar solutes. How it dissolves is as follows: the partially positive hydrogen atoms of water latch onto the negatively charged atoms of the solute, while the partially negative oxygen of water attracts the positively charged atoms of the solute. In doing so, water dissociates the covalent or ionic bonds of the solute, separating the solute’s atoms from each other.
However, water can’t dissolve nonpolar solutes like oil or wax (remember the golden rule), which makes its title of the “universal solvent” a little bit misleading.
Hydrogen bonding
The polarity of water allows it to form hydrogen bonds, a stronger version of the usual dipole-dipole intermolecular force (and NOT a type of covalent or ionic bond as its name might imply). Hydrogen bonding occurs between a partially positive H atom (usually bonded to an N, O, or F atom) of one molecule and a partially negative atom (usually N, O, or F) of another molecule. In the case of water, hydrogen bonding occurs between an H atom of one water molecule and an O atom of another water molecule.
Hydrogen bonding gives water cohesion, adhesion, and a high boiling point.
Cohesion
Cohesion refers to the attraction between molecules of the same type. In other words, molecules with strong cohesive forces like to stick to each other. Thanks to its ability to form strong hydrogen bonds, water is one such molecule.
Water molecules like to stick to each other, and such stickiness (i.e. cohesive forces) gives rise to the surface tension of water. Surface tension is the ability of the surface of a liquid to resist an external force by behaving like an elastic film. In the case of water, its surface tension allows small insects to move across it without the need to swim or float, like the water strider pictured below.
Adhesion
Adhesion refers to the attraction between molecules of different types. The polarity of water allows it to stick to other polar molecules. Thus, the more polar a molecule, the better water will stick to it.
The adhesive and cohesive forces of water allow water to perform capillary action. During capillary action, water flows through a narrow space without the help of, or even against, gravity. This is important because many processes in everyday life rely on capillary action. Capillary action helps water to reach the roots of some plants. Capillary action allows paper towels and sponges to soak up water. Moreover, some scientific techniques like thin-layer chromatography utilize capillary action.
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28
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Question Tags: Summary
Question tags are made of an auxiliary verb plus a pronoun, which go at the end of a sentence, either to ask for more information about something, or to confirm something which we believe is true. The auxiliary verb matches the tense of the main verb and the pronoun comes directly from the noun. Question tags can either have a rising or falling intonation depending if you want to genuinely ask for the answer or expect someone to agree with you. Finally, Question tags usually work in opposition. If the main sentence is affirmative, then the question tag is usually negative, so: You do live here, don’t you? and vice versa.
1. Negative or Limiting Adverbs
Sentences which use a negative or limiting adverbs (never, no, hardly, scarcely, little, etc.) are treated as negative by the question tag, even though their construction is positive. Therefore the question tag is positive.
WRONG: They never go on holiday, don’t they?
RIGHT: They never go on holiday, do they?
2. Indefinite Nouns
Sentences with indefinite nouns such as someone, anyone, no one and everyone, use they in the tag.
No one cares, do they?
Everyone left, didn’t they?
Someone's at the door, are they?
With other indefinite nouns such as something and everything, use it:
Everything is OK, isn’t it?
Nothing matters, does it?
Something smells bad, doesn't it?
3. Imperatives
Imperatives are commands, suggestions, offers, advice etc. They are constructed of the bare infinitive in the affirmative and use don't in the negative: Sit down! Don't sit there!
Imperatives don’t have a tense or use an auxiliary verb in the same way as other sentences do. To make a question tag with imperatives, we use won't you? although other modal verbs can be used, such as will, would, can, and could.
Sit down, won’t you?
Open the window, would you?
Don’t go outside, will you?
Keep quiet, won’t you?
Politeness and formality depend upon the choice of question tag and the tone of your voice, although can’t you can come across as quite impatient and annoyed:
Turn the TV down, can’t you?
4. Suggestions with Let's:
When making a suggestion, it is common to use let’s. Let’s stands for let us, for example: Let’s go to the cinema. When we use let’s in a question tag we always use shall we, regardless of whether let’s is affirmative or negative. So for example:
Let’s go to the cinema, shall we?
Let’s not go to the cinema, shall we?
5. Double positives
Double positives are possible. This is where the both the sentence and the question tag are positive, for example:
You're going to become a pilot, are you?
Double positives are common when people are reacting to news, repeating something they have just heard or reacting in an emotional way to something. For example:
You’re getting married, are you?
You've lost your wallet, have you?
6. Opinions
If you start a sentence with I think, don’t use the question tag do/don't I. Make the question tag agree with the main information in the sentence. This does not apply in the second or third person.
I think it's a nice day, isn't it?
I don't think that's a good idea, is it?
You thought you'd be OK, did you? (second person)
He thinks he's going to university, does he? (third person)
However, in some situations, for example, when we're being sarcastic, we do make the tag agree with I think.
A: Yes, yes. Well done. I know you think you are so clever.
B: Oh! So, I think I'm clever, do I? Not at all.
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12043 | https://www.reddit.com/r/explainlikeimfive/comments/1dmhj2m/eli5_why_does_simplifying_radical_equations/ | ELI5: Why does simplifying radical equations sometimes give wrong solutions? : r/explainlikeimfive
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ElegantPoet3386
ELI5: Why does simplifying radical equations sometimes give wrong solutions?
Mathematics
(sorry for bad formatting on mobile). Let’s say we have sqr(x + 14) - 2 = x. If we move 2, we get sqr(x + 14) = x + 2. Square both sides, x + 14 = x^2 + 4x + 4. Then by subtracting, we get x^2 + 3x - 10 = 0. Factored this equals (x + 5) (x-2) = 0. So we get x = -5 , 2 right? But the weird thing is if we put -5 back into the original equation we get sqr(9) - 2 = -5. Which simplifies to 1 = -5 which is completely not true. What’s even weirder is 2 still works perfectly fine getting sqr(16) - 2 = 2 —> 4 - 2 = 2 —-> 2 = 2. So the way I see it either I did something very wrong or this is normal? Im very sorry if this post is extremely stupid but I just am very confused and would like some answers that don’t involve me having to learning calculus or something like that. Edit: yall, if the square root could give a positive and negative version on its own what is the point of the +- symbol?
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12044 | https://fiveable.me/key-terms/intermediate-macroeconomic-theory/discount-rate | Discount rate - (Intermediate Macroeconomic Theory) - Vocab, Definition, Explanations | Fiveable | Fiveable
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All Key Terms
Intermediate Macroeconomic Theory
Discount rate
🥨intermediate macroeconomic theory review
key term - Discount rate
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Definition
The discount rate is the interest rate used by central banks to lend money to commercial banks, often influencing overall economic activity. It plays a crucial role in monetary policy, as changes in the discount rate can affect borrowing costs, consumer spending, and investment decisions. A lower discount rate generally encourages borrowing and spending, while a higher rate tends to cool economic activity.
5 Must Know Facts For Your Next Test
Central banks adjust the discount rate to control inflation and stabilize the economy, responding to changes in economic conditions.
A lower discount rate can stimulate economic growth by making loans cheaper for banks, which can then pass those savings onto consumers and businesses.
The discount rate is one of the primary tools used by central banks, along with open market operations and reserve requirements, to implement monetary policy.
When banks borrow from the central bank at the discount rate, it can signal the central bank's stance on future monetary policy directions.
The impact of changes in the discount rate may take time to affect the broader economy due to lags in consumer behavior and business investment.
Review Questions
How does a change in the discount rate influence the overall economy?
A change in the discount rate directly impacts borrowing costs for commercial banks. When the discount rate is lowered, banks can borrow money at a lower cost, which often leads them to offer cheaper loans to consumers and businesses. This encourages increased spending and investment, stimulating economic growth. Conversely, raising the discount rate makes borrowing more expensive, which can lead to reduced spending and slower economic activity.
Compare the roles of the discount rate in monetary policy versus fiscal policy.
The discount rate is a key component of monetary policy, where it is used by central banks to manage liquidity and control inflation through interest rates. In contrast, fiscal policy involves government spending and taxation decisions that affect overall economic conditions. While monetary policy focuses on influencing borrowing costs via the discount rate, fiscal policy aims to directly influence demand through budgetary decisions. Both approaches work together but operate through different mechanisms.
Evaluate the implications of a persistently low discount rate on long-term economic growth.
A persistently low discount rate can encourage high levels of borrowing and spending in the short term, potentially leading to rapid economic growth. However, if maintained for too long, it could contribute to asset bubbles and excessive debt accumulation. Additionally, businesses might become reliant on cheap credit rather than investing in efficiency or innovation. This reliance can create vulnerabilities in the economy that may lead to instability when interest rates eventually rise or if economic conditions shift unexpectedly.
Related terms
interest rate:The percentage charged on borrowed money or earned on invested funds, which can influence consumer behavior and economic growth.
monetary policy:The process by which a central bank manages the money supply and interest rates to achieve macroeconomic goals like controlling inflation, consumption, and liquidity.
fiscal policy:Government policies regarding taxation and spending that influence economic conditions and are used to manage economic fluctuations.
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12045 | https://askfilo.com/user-question-answers-smart-solutions/find-length-of-chord-with-midpoint-is-3232303636363232 | \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1, find length of..
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{ x ^ { 2 } } { 2 } + { y ^ { 2 } } { 4 } = 1, find length o
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2 x 2+4 y 2=1, find length of chord with midpoint (1,2 1) is
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Updated on: Jan 25, 2025
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Text solutionVerified
Concepts:
Ellipse, Chord of an ellipse, Midpoint of a chord
Explanation:
To find the length of the chord of an ellipse with given midpoint, we use the formula for the chord length. For an ellipse given by a 2 x 2+b 2 y 2=1, and a chord with midpoint (h,k), the length of the chord is given by 2 a 2(1−a 2 h 2)+b 2(1−b 2 k 2).
Step by Step Solution:
Step 1
Identify the given ellipse equation: 2 x 2+4 y 2=1. Here, a 2=2 and b 2=4, so a=2 and b=2.
Step 2
Identify the midpoint of the chord: (h,k)=(1,2 1).
Step 3
Substitute a, b, h, and k into the chord length formula: 2 a 2(1−a 2 h 2)+b 2(1−b 2 k 2).
Step 4
Calculate the terms inside the square root: a 2(1−a 2 h 2)=2(1−2 1 2)=2(1−2 1)=2×2 1=1 and b 2(1−b 2 k 2)=4(1−4(2 1)2)=4(1−16 1)=4×16 15=3.75.
Step 5
Add the results and multiply by 2: 2 1+3.75=2 4.75=2×2.179=4.358.
Final Answer:
The length of the chord is approximately 4.358 units.
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Question Text 2 x 2+4 y 2=1, find length of chord with midpoint (1,2 1) is
Updated On Jan 25, 2025
Topic All topics
Subject Smart Solutions
Class Class 12
Answer Type Text solution:1
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12046 | https://www.quora.com/How-do-I-find-maximum-and-minimum-value-of-a-function | Something went wrong. Wait a moment and try again.
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5
How do I find maximum and minimum value of a function?
·
To find the maximum and minimum values of a function, you can follow these steps:
Identify the Function and Domain
Start with the function f(x) you want to analyze and determine its domain (the set of all possible input values).
Find Critical Points
Take the Derivative: Compute the first derivative f′(x).
Set the Derivative to Zero: Solve f′(x)=0 to find critical points, which are potential locations for maxima and minima.
Check for Undefined Points: Identify where f′(x) is undefined, as these points may also be critical points.
Evaluate the Function at Critical Points
Calculate the value o
To find the maximum and minimum values of a function, you can follow these steps:
Identify the Function and Domain
Start with the function f(x) you want to analyze and determine its domain (the set of all possible input values).
Find Critical Points
Take the Derivative: Compute the first derivative f′(x).
Set the Derivative to Zero: Solve f′(x)=0 to find critical points, which are potential locations for maxima and minima.
Check for Undefined Points: Identify where f′(x) is undefined, as these points may also be critical points.
Evaluate the Function at Critical Points
Calculate the value of the function f(x) at each critical point found in step 2.
Evaluate the Function at the Endpoints
If the domain is closed (e.g., [a, b]), evaluate f(x) at the endpoints a and b.
Use the Second Derivative Test (Optional)
To determine whether each critical point is a maximum, minimum, or neither:
Compute the second derivative f′′(x).
If f′′(x)>0 at a critical point, it is a local minimum.
If f′′(x)<0, it is a local maximum.
If f′′(x)=0, the test is inconclusive.
Compare Values
Compare the function values at all critical points and endpoints to determine the absolute maximum and minimum.
Consider the function f(x)=−x2+4x on the interval [0, 4].
Take the Derivative:
f′(x)=−2x+4
Set the Derivative to Zero:
−2x+4=0⟹x=2
Evaluate at Critical Points:
f(2)=−(2)2+4(2)=8
Evaluate at Endpoints:
f(0)=0
f(4)=−(4)2+4(4)=0
Compare Values:
f(0)=0
f(2)=8
f(4)=0
The maximum value is 8 at x=2, and the minimum value is 0 at x=0 and x=4.
By following these steps, you can systematically find the maximum and minimum values of a function.
Anupreet Choudhary
Engineer,IITian, Trying to become good in mathematics
·
Author has 71 answers and 571.6K answer views
·
9y
There are several methods of doing this. The easiest way is to use differential calculus and try to plot the curve. Tarun Desu has already stated the steps involved but it is only true for functions in single variable like y= f(x).
For functions in multi-variables, like z=f(x,y) & z=f(x,y,z......) things become more complex and it is harder to visualize these functions. Anyway, the concept remains the same.
So, here is the method for two-variable functions:
Find partial differentials of function f(x,y) w.r.t x and y and equate them to zero. There will be two equations - fx=0 and fy=0
Solve t
There are several methods of doing this. The easiest way is to use differential calculus and try to plot the curve. Tarun Desu has already stated the steps involved but it is only true for functions in single variable like y= f(x).
For functions in multi-variables, like z=f(x,y) & z=f(x,y,z......) things become more complex and it is harder to visualize these functions. Anyway, the concept remains the same.
So, here is the method for two-variable functions:
Find partial differentials of function f(x,y) w.r.t x and y and equate them to zero. There will be two equations - fx=0 and fy=0
Solve these two equations simultaneously in x and y. Let (a,b) (c,d)... be the pair of values
Calculate: r=∂2f∂x2 , s=∂2f∂y∂x and t=∂2f∂y2 for (a,b) (c,d)...
Now if rt−s2>0 and r<0 , then f(a,b) is maximum. If rt−s2>0 and r>0 then f(a,b) is minimum. If rt−s2<0 then f(a,b) is a saddle point
You can apply the above steps for the following example -
f(x,y)=x4+y4−2x2+4xy−2y2
The graph of the above function is
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Author has 454 answers and 763.7K answer views
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4y
Originally Answered: How do you determine if a function has a minimum or maximum value?
·
It’s pretty easy to check if a given function has a maximum or minimum value.
Write down the function, in the form y = .
Differentiate the function twice. If the value on the RHS is a positive integer, then the function has a minimum value, or a maximum value if the value is negative.
In some cases, you don’t get an integer on the RHS after second derivative. Don’t
It’s pretty easy to check if a given function has a maximum or minimum value.
Write down the function, in the form y = .
Differentiate the function twice. If the value on the RHS is a positive integer, then the function has a minimum value, or a maximum value if the value is negative.
In some cases, you don’t get an integer on the RHS after second derivative. Don’t worry, we can handle this. Just follow step 4.
Now, to obtain the value, assume that the first derivative is equal to zero. Solve the equation, in most equations, you get the value of x. This is the point at which the function has a minimum/maximum. Functions which are of form, y = , don’t return values but get reduced to expressions (or equations).
If you didn’t get an integer in the second derivative, you can use this value (your result from step 4) to substitute in the second derivative. Then see if the value is positive or negative.
That might seem complex. Here’s an example:
Consider an equation, y = 2x^2 + 4x - 1.
First, we differentiate the equation twice.
The result will be y’’ = 4, which is > 0 (i.e, positive). What does this mean? It means that the function has a minimum value.
Now to find the value, we simply find the first derivative of the equation and equate it to zero.
First derivative will be, y’ = 4x + 4.
Equating it to zero, 4x + 4 = 0.
Hence, x = -1.
No, this isn’t the minimum value, this is the x coordinate at which the equation has a minimum value. To see what the minimum value is, substitute and solve the equation with the result.
y = 2x^2 + 4x - 1
y = 2(-1)^2 + 4(-1) - 1
y = 2 - 4 - 1
y = -3. This is the minimum value of the function.
Here’s one ...
Related questions
How do I find the maximum value of a function?
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Vishakh Rajendran
M.S. in Aerospace and Aeronautical Engineering, Nanyang Technological University
·
Author has 684 answers and 2.9M answer views
·
4y
You can use differential calculus for this purpose. In fact, some of the best known applications of differential calculus is for optimization problems wherein we are supposed to determine the best possible outcome (maximum value) or the worst possible outcome (minimum value) for functions.
The usage of first derivative in combination with the second derivative can be used to establish the maxima and the minima.
The slope of the curve at the exact maxima would be 0. The slope of the portions just before the maxima would be positive and the slope after the maxima would be negative. So maxima is th
You can use differential calculus for this purpose. In fact, some of the best known applications of differential calculus is for optimization problems wherein we are supposed to determine the best possible outcome (maximum value) or the worst possible outcome (minimum value) for functions.
The usage of first derivative in combination with the second derivative can be used to establish the maxima and the minima.
The slope of the curve at the exact maxima would be 0. The slope of the portions just before the maxima would be positive and the slope after the maxima would be negative. So maxima is the point on the curve where the slope behavior changes from being positive to negative. Minima on the other hand is also a case where the slope at the point of minima is 0. Minima is the point where the slope behavior changes from being negative to positive.
This approach of determining the maxima and minima using calculus is what is called as the Fermat’s theorem.
Vishakh Rajendran's answer to What is Fermat's theorem for determining the maxima and the minima?
So the procedure to determine the maxima and minima is -
a. Determine the critical points (points where the derivative = 0 or the derivative does not exist).
b. Determine the value of the function at these critical points.
c. Determine the value of the function at the end points.
d. The largest of the determined values is the maxima and the smallest of the determined values is the minima.
Once you have the critical points, you can determine the maxima and minima without going in for steps b and c and d. For this, you need to use the second derivative test.
Once you have the knowledge about the critical points (c), determine the value of the second derivative of the function at this c.
If f’’(x) > 0 at c where c is a critical point, then the function has a local minima at c.
If f’’(x) < 0 at c where c is a critical point, then the function has a local maxima at c.
Tarun Desu
Undergrad Electronics student, Aiming infinity and beyond
·
9y
There are Various methods in order to find maximum or minimum value of a function.
One of the conventional methods is:
Find the derivative of the function and equate it to zero.
Find the roots of the differentiated equation.
Do double differentiation of original function and substitute the values of roots in the 2nd differentiated expression.
If the value comes out to be negative, At the particular value of the root Maximum occurs. Then substitute the value in original expression to get Maximum of the function.
If the value of double derivative after substituting the root is positive, Minimum occur
There are Various methods in order to find maximum or minimum value of a function.
One of the conventional methods is:
Find the derivative of the function and equate it to zero.
Find the roots of the differentiated equation.
Do double differentiation of original function and substitute the values of roots in the 2nd differentiated expression.
If the value comes out to be negative, At the particular value of the root Maximum occurs. Then substitute the value in original expression to get Maximum of the function.
If the value of double derivative after substituting the root is positive, Minimum occurs. Then substitute the value in original equation to get Minimum value of the function.
If the Second derivative is Zero: Then go for higher derivatives of the function & substitute the value of the root in the nth order derivative expression. If it's positive it would give the Maximum of the function at the particular root.
Hope the answer Helps.
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Satyannarayana Yedla
Subject Matter Expert, Calculus,Fitness Enthusiast.
·
Author has 99 answers and 460K answer views
·
9y
Let us suppose that the function is a single variable function.
You can find the Local maximum and local minimum of the function by the following steps.
Finding critical points: You have derivative the function and set the derivative is equal to zero and solve the equation. You will get the roots of the equation. The roots are called critical points.
At the critical points, the function has either Local maximum or Local minimum, you need to determine the points by using a second derivative test.
Find the second derivative of the function and evaluate it at the critical points by simply substituti
Let us suppose that the function is a single variable function.
You can find the Local maximum and local minimum of the function by the following steps.
Finding critical points: You have derivative the function and set the derivative is equal to zero and solve the equation. You will get the roots of the equation. The roots are called critical points.
At the critical points, the function has either Local maximum or Local minimum, you need to determine the points by using a second derivative test.
Find the second derivative of the function and evaluate it at the critical points by simply substituting the critical point in the second derivative of the function.
The value of the second derivative at the critical point is positive, that means the graph of the function is concave up. So, the function has minimum value at this point. That is Local minimum.
The value of the second derivative at the critical point is negative, that means the graph of the function is concave down. So, the function has minimum value at this point. That is Local maximum.
The value of the second derivative at the critical point is zero, we can not determine what the value that function has. So, its called a Saddle point.
If you have a domain, check the function value at the extremes of the domain and compare the values with the function that we found at critical points . The largest value the maximum and smallest value is minimum.
I hope you can understand now :)
Related questions
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Glen Melnik
Private Tutor (1981–present)
·
Author has 131 answers and 166.9K answer views
·
6y
To find a local max and min value of a function, take the first derivative and set it to zero. To determine where it is a max or min, use the second derivative test to find it concavity.
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Eshan Kulkarni
Studied at D. E. S School
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Author has 203 answers and 168K answer views
·
5y
Originally Answered: How do you find the max and min of a function?
·
This is a concept called Maxima and Minima. So, first of all, we must obtain a function ( I am assuming that you require it for high school studies) in two variables x,y. A function is a Mathematical expression which shows you how a variable or variables are related to a dependent variable. For e.g F(x)= y= x² tells us how x is related to y.
So, the technique is related to calculus. The idea is, if you plot a graph of the above function, y=x², you will find that there exists a point, where the tangent to the point is parallel to the x- axis.
So for some real function, y= F(x), there may exist 'n
This is a concept called Maxima and Minima. So, first of all, we must obtain a function ( I am assuming that you require it for high school studies) in two variables x,y. A function is a Mathematical expression which shows you how a variable or variables are related to a dependent variable. For e.g F(x)= y= x² tells us how x is related to y.
So, the technique is related to calculus. The idea is, if you plot a graph of the above function, y=x², you will find that there exists a point, where the tangent to the point is parallel to the x- axis.
So for some real function, y= F(x), there may exist 'n' number of points, where the tangent is parallel to x-axis.
So, dy/dx, represents the slope of the tangent at that point. Slope of x-axis is 0, therefore,by putting dy/dx is 0, you will get values of x, where slope of tangent is 0. But how to identify minimum and maximum of them?
At this stage, you take the double derivative, i.e after you differentiate F(x) 2 times. Now, for the point for which dy/dx is > 0, is the minima or minimum value of y with respect to x and for dy/dx < 0, is the Maxima or maximum value of y with respect to x.
A small tip, derivative of xⁿ= n.x^(n-1). It will help you in most of the functions at elementary school level.
Sarath
Solving, means a lot.
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Author has 556 answers and 2.3M answer views
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8y
Originally Answered: How do I find the maximum and minimum values of a function by the maxima and minima method?
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By fnding derivatives.
First calculate the derivate of the function and find the values where it becomes zero.
And then keep those values aside
Then find the second derivate of the function.
Then by substituting the previously found values in the second derivative of the function we can find Maxima and minima .
If the second derivative of function is less than zero( i.e negative) the function is maximum at that value (for previously found value)and if it is positive then the function is minimal(at that previously found value)
In this way we can find the maximum and minimal value of function.(by subs
By fnding derivatives.
First calculate the derivate of the function and find the values where it becomes zero.
And then keep those values aside
Then find the second derivate of the function.
Then by substituting the previously found values in the second derivative of the function we can find Maxima and minima .
If the second derivative of function is less than zero( i.e negative) the function is maximum at that value (for previously found value)and if it is positive then the function is minimal(at that previously found value)
In this way we can find the maximum and minimal value of function.(by substituting the values in the main function respectively)
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Murali Krishna
Former Retired Senior Lecturer in DIET at Government (General) (1990–2004)
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Author has 6.5K answers and 6.7M answer views
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5y
Originally Answered: What is the function's minimum or maximum value?
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The maximum value of a function is the place where a function reaches its highest point or vertex on a graph
There are many methods to CALCULATE maximum value of a QUADRATIC eqation
1st way is Graphing Method
2 to determine the maximum value is using the quadratic eqation y=ax^2+bx+c
MAX =C- (b^2/4a)
If x^2 term is positive the vertex point will be a minimum
If the x^2 term is negative the vertex will be a
maximum
Example
-x^2+4x--2
Since the x^2 term is negative
Max = c--(b^2/4a)
= --2 —(16/-4 ×1 ) = --2 +4 = 2
Max value of 2
2nd way
y= a(x-h)^2+k the maximum value will be given by k
-x^2+4x--2
a= -1 b=4 c= -
The maximum value of a function is the place where a function reaches its highest point or vertex on a graph
There are many methods to CALCULATE maximum value of a QUADRATIC eqation
1st way is Graphing Method
2 to determine the maximum value is using the quadratic eqation y=ax^2+bx+c
MAX =C- (b^2/4a)
If x^2 term is positive the vertex point will be a minimum
If the x^2 term is negative the vertex will be a
maximum
Example
-x^2+4x--2
Since the x^2 term is negative
Max = c--(b^2/4a)
= --2 —(16/-4 ×1 ) = --2 +4 = 2
Max value of 2
2nd way
y= a(x-h)^2+k the maximum value will be given by k
-x^2+4x--2
a= -1 b=4 c= --2
y=--1(x--h)^2 +k
K is constant = --2
-2 is the maximum value of quadratic eqation
Minimum value :The minimum value of a function is the lowest point of a vertex If the your QUADRATIC EQUATION has a positive term of x^2 it will have minimum value
You can find this minimum value by graphing the function or using the eqations
min =c --(b^2/4 a)
3x^2+12x-12
x^2 term 3 is + positive so thre will be a minimum point
Min =c —(b^2/4a) = 2–(12^2/4 ×3) = 2-- 12 =--10
Minimum point is -10
Refer Calculus FUNCTIONS
Maxima and Minima chapter
Chaula Dixit
MSc in Mathematics, Christ (deemed university) (Graduated 2020)
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7y
Originally Answered: How can I find maxima and minima of any given function?
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For any function of one variable: f(x)
Step 1- Find f'(x)
Step 2- Find ‘a' for which f'(a)=0 (a is called critical point)
Step 3- Find f”(x)
Step 4- Substitute a in f”(x)
If f”(x) > 0, then f has minimum value f(a)
If f”(x) < 0, then f has maximum value f(a)
For any function of two variables: f(x,y)
Step 1- Find fx (partial derivative w.r.t. x) & fy (partial derivative w.r.t. y)
Step 2- Solve for fx = 0 and fy = 0, call it (a,b)
Step 3- Find fxx, fyy & fxy at (a,b)
Step 4- Find value of equation: t = fxxfyy - (fxy)^2
If t < 0, (a,b) is called a saddle point
If t > 0, f has min or max at (a,b)
(fxx & fyy >
For any function of one variable: f(x)
Step 1- Find f'(x)
Step 2- Find ‘a' for which f'(a)=0 (a is called critical point)
Step 3- Find f”(x)
Step 4- Substitute a in f”(x)
If f”(x) > 0, then f has minimum value f(a)
If f”(x) < 0, then f has maximum value f(a)
For any function of two variables: f(x,y)
Step 1- Find fx (partial derivative w.r.t. x) & fy (partial derivative w.r.t. y)
Step 2- Solve for fx = 0 and fy = 0, call it (a,b)
Step 3- Find fxx, fyy & fxy at (a,b)
Step 4- Find value of equation: t = fxxfyy - (fxy)^2
If t < 0, (a,b) is called a saddle point
If t > 0, f has min or max at (a,b)
(fxx & fyy > 0, f is minimum
fxx & fyy < 0, f is maximum)
Ahilan Palarajah
Diagnosed at 28
·
4y
I see answers that are technically more correct than anything I'm about to say. If you're asking this it's my impression that you want an intuitive answer.
At the minimum or maximum it's probably clear to you why the slope is 0: movement along the graph means less extreme values (the definition implies this). But it's unclear which one of the two it is because both of them have this 0 slope property.
There's a more telling sign of what you're dealing with which is related to the curvature at that point. It turns out at these extremes, which way the graph curves depends entirely on the second der
I see answers that are technically more correct than anything I'm about to say. If you're asking this it's my impression that you want an intuitive answer.
At the minimum or maximum it's probably clear to you why the slope is 0: movement along the graph means less extreme values (the definition implies this). But it's unclear which one of the two it is because both of them have this 0 slope property.
There's a more telling sign of what you're dealing with which is related to the curvature at that point. It turns out at these extremes, which way the graph curves depends entirely on the second derivative (acceleration). When acceleration is negative it curves down, and when it's positive it curves upward. So the easiest way is to use the 0 slope property to find the points of interest and find the second derivative at a point to see if it's minimum or maximum.
Bob Collier
Former EE Designed Specialized Computers for 33 Years.
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Author has 3.1K answers and 1.6M answer views
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2y
3 examples:
Left to right:
Looks like a parabola y(x) = x^2 - 5. Highest power of x is x^2. [Plot that and see that it looks like my curve]
See y(x1) = ‘minimum’. It’s called that because for x just a tad greater than x1 the value of y is greater than y(x1). And, similarly, for just a little bit less than x1, the value of y is, again, greater than y(x1). When all your neighbors are taller than you, you are the minimum.
Next one looks line y(x) = (x-2)(x+3)(x-1) + 5 [cubic: highest power of x is x^3]
Local maximum at y(x2), similarly, because for all x such that x-q<x2<x+q where q is ‘small’ y(x) is
3 examples:
Left to right:
Looks like a parabola y(x) = x^2 - 5. Highest power of x is x^2. [Plot that and see that it looks like my curve]
See y(x1) = ‘minimum’. It’s called that because for x just a tad greater than x1 the value of y is greater than y(x1). And, similarly, for just a little bit less than x1, the value of y is, again, greater than y(x1). When all your neighbors are taller than you, you are the minimum.
Next one looks line y(x) = (x-2)(x+3)(x-1) + 5 [cubic: highest power of x is x^3]
Local maximum at y(x2), similarly, because for all x such that x-q<x2<x+q where q is ‘small’ y(x) is les than y(x2). When you are the tallest one around, you are the max. There is a local min after that at x3. These are called ‘local’ because there are values of x for which Y(x) is much greater than Y(x2) and other values of x for which y is much less than y(x3). These min & max approach ‘infinity’ = ‘are unbounded’.
The last curve is a simple maximum at x4. At this point, nothing else needs to be said about min & max.
But notice that in these smooth, continuous functions, the slope of y(xn) at every bounded min & max is zero.
Follow my arrows on the curve before and after x2. As x approaches x2 form the left [xx2, the slope is negative, the curve is going down. If it’s positive before a point and negative after that point, than for a smooth continuous curve, the slope at that point [x2] MUST be zero.
The slope of any such continuous curve is the first derivative of the curve.
slope(x) = y’(x).
Notice that for maximum, as x goes left to right through x=x2, slope is + then 0 then -.
And w/o further ado, as x goes left to right through x=x3, slope is - then 0 then +. If you will look at the curves you should see that it has to be that way.
So if you don’t have the curves to look at, but you do have/can calculate the derivative, then you can tell which zero-slopes are max and which are min just by looking at the value of the derivative in the neighborhood.
You get the same information form the 2nd derivative. If the slope is increasing at the point of min or max, then the point must be at a local minimum; and conversely for 2nd derivative negative. Look at the curves. You can see all of this in these curves.
When you don’t have the cures, use the derivatives. And if you forget which is which [increasing slope means what?] draw a picture of zero slope, increasing and is that point of zero slope above or below your next point? The picture you draw at this point is just to remind yourself of the rules above. If you know/ ‘remember’ the rules, you don’t need the pictures.
Notice I did not say ‘memorize’. Those who ‘memorize’, never learn.
There is a tricky kind of question about min/max over some fixed interval, say 0 <= x <= 10.
It could well be that Y(0) and or y(10) are local min or max JUST BECAUSE YOU DON’T LOOK FARTHER than those limits. They don’t have to be. It could be that Y(10) is less than Y(x3) and greater than y(x2) so it’s not really a max/min even locally. The key to this aspect is that at these boundary points, the slope may very well NOT BE 0. So if you just look at where the derivative is 0, you will miss these points.
I remember this happening some 60-odd years ago. The question was to find the maximum values over some interval. It just happened that the slope at the end of the interval was zero, so I included [inspection of] that point, but if the derivative had not been zero there, I would have never considered investigating that point. Nobody told me about such things.
Now you are ahead of me at this point in your mathematical career.
Notice: I still remember that. I did not memorize it. The boundary may well not even have been 10!!!
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12047 | https://nntdm.net/volume-12-2006/number-3/10-19/ | Using integer structure to solve Diophantine equations
J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 12, 2006, Number 3, Pages 10–19
Full paper (PDF, 85 Kb)
Details
Authors and affiliations
J. V. Leyendekkers
The University of Sydney, 2006 Australia
A. G. Shannon
Warrane College, The University of New South Wales, Kensington, 1465,
& KB Institute of Technology, North Sydney, NSW 2060, Australia
Abstract
Diophantine equations {ax + by = c; a, b, c ∈ ℤ} are classified according to parity constraints. Various types, so classified, are solved with the theory of integer structure, via the modular ring Z4. The simplest forms are those where one of the variables is confined to a single class. However, the more complex equations have solutions that follow regular (x, y) class patterns. The famous Diophantine equation in Fermat’s Last Theorem is discussed in terms of the factor structure of the sum of two powers.
AMS Classification
11A41
11A07
References
J.H. Clarke, Linear Diophantine Equations Applied to Modular Co-ordination, Australian Journal of Applied Science, 15(4) (1964): 201-4.
J.H. Clarke, Conditions for the Solution of a Linear Diophantine Equation, New Zea- land Mathematics Magazine, 14(1) (1977): 45-47.
J. Hunter, Number Theory. Edinburgh: Oliver and Boyd, 1964.
J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, Integer Class Properties Associated with an Integer Matrix. Notes on Number Theory & Discrete Mathematics. 1 (2) (1995): 53-59.
J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, Analysis of Diophantine Properties Using Modular Rings with Four and Six Classes. Notes on Number Theory & Discrete Mathematics. 3 (2) (1997): 61-74.
J.V. Leyendekkers & A.G. Shannon, Analyses of Row Expansions within the Octic ‘Chess’ Modular Ring, 8 Z. Notes on Number Theory & Discrete Mathematics. 5(3) (1999): 102-114.
Hans Riesel. Prime Numbers and Computer Methods for Factorization. 2nd edition. Progress in Mathematics, Volume 126. Boston: Birkhäuser, 1994.
A.J. van der Poorten, Notes on Fermat’s Last Theorem. New York: Wiley.
S. Yates, The Mystique of Repunits, Mathematics Magazine, 51(1): (1978): 22-28.
Related papers
Cite this paper
Leyendekkers, J. V., and Shannon, A. G. (2006). Using integer structure to solve Diophantine equations. Notes on Number Theory and Discrete Mathematics, 12(3), 10-19.
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12048 | https://www.math.umd.edu/~dolgop/BBNotes2.pdf | LECTURES ON BOUNCING BALLS.
Contents 1.
Introduction.
2 1.1.
Goals of the lectures.
2 1.2.
Main examples.
3 2.
Normal forms.
10 2.1.
Smooth maps close to identity.
10 2.2.
Adiabatic invariants.
12 2.3.
Systems with singularities.
14 2.4.
Accelerating orbits for piecewise smooth maps.
19 2.5.
Birkhoffnormal form.
21 3.
Applications of KAM theory.
23 3.1.
Introduction.
23 3.2.
Theory.
24 3.3.
Applications.
25 4.
Recurrence.
28 4.1.
Poincare Recurrence Theorem.
28 4.2.
Background from ergodic theory.
30 4.3.
Ergodicity and recurrence.
32 4.4.
Proof of the Maximal Ergodic Theorem.
34 4.5.
Ergodic Theorems for L2-functions.
35 4.6.
Ergodic decomposition.
37 4.7.
Proof of Rokhlin Lemma.
37 5.
Statistical properties of hyperbolic sawtooth maps.
38 5.1.
The statement.
38 5.2.
The Hopf argument.
39 5.3.
Long invariant manifolds and ergodicity.
40 5.4.
Growth Lemma.
42 5.5.
Weak Mixing.
45 5.6.
Mixing and equidistribution.
45 5.7.
Another growth lemma.
46 5.8.
Trying to succeed.
47 5.9.
Equidistribution and coupling.
48 5.10.
Coupling, separating, recovering.
50 6.
The Central Limit Theorem for Dynamical Systems.
53 6.1.
Estimating error in Ergodic Theorem.
53 1 2 LECTURES ON BOUNCING BALLS.
6.2.
Bernstein method.
54 6.3.
Moment estimates.
57 6.4.
The case of zero variance.
58 6.5.
Convergence to Brownian Motion.
62 7.
Invariant comes and hyperbolicity.
63 7.1.
Dimension 2.
63 7.2.
Higher dimensions.
65 7.3.
Lyapunov exponents.
68 7.4.
Examples.
69 References 74 1. Introduction.
1.1. Goals of the lectures. The purpose of these lectures is to illus-trate some ideas and techniques of smooth ergodic theory in the setting of simple mechanical systems.
Namely we consider either one or several particles moving on a line either freely or in a field of a force and interacting with each other and with the walls according to the law of elastic collisions.
The main questions we are going to address are the following.
(1) Acceleration. Is it possible to accelerate the particle so that its velocity becomes arbitrary large? If the answer is yes we would like to know how large is the set of such orbits. We would also like to know how quickly a particle can gain energy both in the best (or worst) case scenario and for typical initial conditions. We are also interested to see if the particle will accelerate indefinitely so that its energy tend to infinity or if its energy will drop to its initial value from time to time.
(2) Transitivity. Does the system posses a dense orbit? That is, does there exist an initial condition (Q0, V0) such that for any ε and any ¯ Q, ¯ V there exists t such that |Q(t) −¯ Q| < ε, |V (t) −¯ V | < ε.
A transient system has no open invariant sets. A stronger notion is ergodicity which says that any measurable invariant set either has mea-sure 0 or its complement has measure 0. If the system preserves a finite measure µ and the system is ergodic with respect to this measure then by pointwise ergodic theorem for µ-almost all initial conditions we have 1 T mes(t ∈[0, T] : (Q(t), V (t)) ∈A) →µ(A) as T →∞.
If the measure of the whole system is infinite then we can not make such a simple statement but we have the Ratio Ergodic Theorem which LECTURES ON BOUNCING BALLS.
3 says that for any sets A, B and for almost all initial conditions mes(t ∈[0, T] : (Q(t), V (t)) ∈A) mes(t ∈[0, T] : (Q(t), V (t)) ∈B) →µ(A) µ(B) as T →∞.
The purpose of the introductory lectures is to introduce several exam-ples which will be used later to illustrate various techniques. Most of the material of the early lectures can be found in several textbooks on dynamical systems but it is worth repeating here since it will help us to familiarize ourselves with the main examples. The material of the second part will be less standard and it will be of interest to a wider audience.
1.2. Main examples. Here we describe several simple looking systems which exhibit complicated behavior. At the end of the lectures we will gain some knowledge about the properties of these systems but there are still many open questions which will be mentioned in due course.
(I) Colliding particles. The simplest model of the type mentioned above is the following. Consider two particles on the segment [0, 1] colliding elastically with each other and the walls.
Let m1 and m2 denote the masses of the particles. Recall that a collision is elastic if both energy and momentum are preserved. That is, both P = m1v1 + m2v2 and 2K = m1v2 1 + m2v2 2 are conserved. In particular if P = 0 then 2K = m2v2 2 m2+m1 m1 and so in this case (v+ 2 )2 = (v− 2 )2. Similarly, (v+ 1 )2 = (v− 1 )2, that is, the particles simply change the signs of their velocities. In the general case we can pass to the frame moving with the center of mass. The center of mass’ velocity is u = m1v1+m2v2 m1+m2 so in the new frame we have ˜ v1 = v1 −u = m2(v1 −v2) m1 + m2 and ˜ v2 = v1 −u = m1(v2 −v1) m1 + m2 .
In our original frame of reference we have v+ 1 = u −˜ v1 = m1 −m2 m1 + m2 v− 1 + 2m2 m1 + m2 v− 2 and similarly v+ 2 = u −˜ v2 = m2 −m1 m1 + m2 v− 2 + 2m1 m1 + m2 v− 1 .
The collisions with the walls are described by the same formulas but we consider the walls to be infinitely heavy. Thus if the particle collides with the wall its velocity becomes v+ = 2vwall−v−. In particular, in the present setting the wall is fixed so the particle’s velocity just changes the sign.
4 LECTURES ON BOUNCING BALLS.
Returning to our system introduce (1.1) qj = √mjxj. Thus uj = ˙ qj = √mjvj.
q1 q2 √m1 √m2 Figure 1. Configuration space for two points on the segment The configuration space of the system becomes q1 ≥0, q2 ≤√m2, q1 √m1 ≤ q2 √m2 .
This is a right triangle with hypotenuse lying on the line q1 √m2 −q2 √m1 = 0.
The law of elastic collisions preserves 2K = u2 1 + u2 2 and P = √m1u1 + √m2u2.
In other words if we consider (q1(t), q2(t)) as a trajectory of the parti-cle in our configuration spaces then as the particle reaches hypotenuse its speed is preserved and the angle which its velocity makes with (√m1, √m2) remains the same. Since (√m1, √m2) iscolinear to the boundary this see that the tangential component of the particle veloc-ity is preserved. Since the length of the velocity vector is also conserved we see that the normal componet of the velocity is reversed. Therefore the change of velocity satisfies the law of the elastic reflection. Simi-larly if the particle hits q1 = 0 then u2 remains the same and u1 changes to the opposite which is again in accordance with the elastic collision law. Hence our system is isomorphic to a billiard in a right triangle.
A similar analysis can be performed for three particles on the circle R/Z. In this case there are no walls so the velocity of the mass center LECTURES ON BOUNCING BALLS.
5 is preserved. It is therefore convenient to pass to a frame of reference where this center is fixed at the origin. So we have m1x1 + m2x2 + m3x3 = 0 and m1v1 + m2v2 + m3v3 = 0.
Figure 2. Configuration space of three points on the circle using the distance from the first point as coordi-nates In coordinates from (1.1) the above relation reads √m1q1 + √m2q2 + √m3q3 = 0 and √m1u1 + √m2u2 + √m3u3 = 0.
Thus points are confined to a plane Π and the particle velocity lies in this plane. The collisions of the particles have equations qi √mi − qj √mj = l.
These lines divide Π into triangles. We claim that dynamics restricted to each triangle is a billiard. Consider, for example, the collision of the first two particles. Since (√m1, √m2, √m3) is collinear to the plane P l 12 = { q1 √m1 − q2 √m2 = l} it follows that P l 12 is orthogonal to Π. Next, √m1u1 + √m2u2 is preserved Note that n12 is also colllinear to the plane P l 12. Denoting by ⃗ n∗ 12 the orthogonal projection of ⃗ n12 to Π we 6 LECTURES ON BOUNCING BALLS.
Figure 3. Impact oscillator (left) and Ulam pingpong (right) are two systems fitting into our setting see that ⟨⃗ n∗ 12, ⃗ u+⟩= ⟨⃗ n∗ 12, ⃗ u−⟩ where ⃗ u = (u1, u2, u3). In other words, the tangential componet of the velocity is preserved and since the lenth of the velocity vector is also preserved we have an elastic collision.
We can also consider more particles on a line or a circle and show that that system is isomorphic to a polyhedral billiard.
(II) Particle in a potential. Our second example is a particle moving on the line under the force created by the potential U(x) = gxα and colliding elastically with an infinitely heavy plate. We assume that α > 0 since otherwise the particle can go to infinity after finitely many bounces. Let f(t) denote the height of the plane at time t. We assume that f(t) > 0 for all t so that U(x) is defined for all x > f(t) and that f(t) is periodic. In fact, the case of f(t) = B + A sin t (where A < B) is already quite interesting. Two cases attracted a particular attention in the past.
(a) Gravity (α = 1). In this setting the acceleration question can be posed as follows: how much can one accelerate a tennis ball by periodic motion of a tennis rocket (of course one needs to be in a good fitness condition for the infinitely heavy wall approximation to be reasonable).
(b) Impact oscillator (α = 2). In this case one has a particle attached to a string and colliding with the wall. Apart from an easy mechanical implementation this system is also related to an interesting geometric object-outer billiard.
Outer billiards are defined in an exterior of a closed convex curve Γ on the plane. Given a point A0 ∈R2 −IntΓ, there are two support lines from A0 to Γ. Choose the one for which if one walks from A0 to the point of contact then Γ is to the right of the line. Then we reflect A0 about the point of contact to get its image A1. Applying this procedure repeatedly we obtain the orbit of A0 under the outer billiard LECTURES ON BOUNCING BALLS.
7 map. Outer billiards were popularized by Moser as they provide simple illustration to KAM theory.
A0 A1 A2 B0 B1 Figure 4. Outer billiard We now describe a construction of Boyland which associates to each outer billiard an impact oscillator. To this end we consider a third system (see figure 5). Its phase space consists of a pair (Γ0, A0) where Γ0 is a closed and convex curve and A0 is the point in R2 −IntΓ0 such that the supporting line from A0 to Γ0 is vertical. To describe one iteration of our system one first reflects A0 about the point of contact to get the pair (Γ0, ˜ A1) and then rotates the picture counterclockwise until the second support line becomes vertical. If (Γn, An) is the n-th iteration of our system then clearly there exists a rotation Rn such that Γ0 = RnΓn. Then RnAn = f n Γ0A0 where fΓ0 denotes the outer billiard map about Γ0. On the other hand between the reflections the point evolves according to the ODE ˙ x = v, ˙ v = −x while during the reflection x is unchanged and v+ + v−= 2vtip where vtip denotes the velocity of the rightmost point of Γ(t). One can check that the motion of the tip is given by ¨ x + x = r(x(t)) where r(x) is the radius of curvature of point x. Thus given a curve Γ one can associate to it an impact oscillator with the wall motion given by ¨ f + f = r(f(t)).
Note that in that construct the frequencies of the wall and the spring are the same. Conversely, given an impact oscillator one can consider a curve whose radius of curvature is r(f(t)) = ¨ f + f but the resulting curve need not be either close or convex.
Thus the class of impact oscillators is much larger than the class of outer billiards but the later is an important subclass supplying clear geometric intuition.
While α = 1 and α = 2 are the two most studied cases we will see that the dynamics for α ̸= 1, 2 is quite different. As it was mentioned above one of the main question is large velocity behavior of the model.
Note that different collisions occur at different heights. However if the particle’s velocity is high it takes a very short time to pass between 8 LECTURES ON BOUNCING BALLS.
Figure 5. Outer billiards and Impact Oscillators max f(t) and min f(t). Since the explicit computations of the height of the next collision is usually impossible one often considers a simplified model which is called static wall approximation (SWA). In this model one fixes a height ¯ h and assumes that the next collision occurs at the time tn+1 = tn + T(vn) where T(vn) is the time it takes the particle to return to the height ¯ h. However velocity is still updated as vn+1 = 2 ˙ f(tn+1) −2˜ vn where ˜ vn is velocity of the particle when it returns to ¯ h.
By energy conservation ˜ vn = −vn so SWA takes form tn+1 = tn + T(vn), vn+1 = vn + 2 ˙ f(tn+1).
While SWA provides a good approximation for the actual system in high velocity regime for one or a few collisions, in general, it is not easy to transfer the results between the original model and SWA. However the SWA is an interesting system in its own right. In addition, the SWA and the original system often have similar geometric features and since formulas are often simpler for the SWA we will often present the arguments for the SWA. For example, for α = 1 the SWA takes from (1.2) tn+1 = tn + 2vn g , vn+1 = vn + 2 ˙ f(tn+1).
This system is the celebrated standard map. Phase portraits of the map (1.2) for several values of parameters can be found in Section 2.4 of . (1.2) is defined on R × T but it is a lift of T2 diffeomorphism since the change of v by 2 g commutes with the dynamics.
(III) Fermi-Ulam pingpong. In model (II) the particle has infin-itely many collisions with a moving wall because the force make it to fall down. Another way to enforce infinitely many bounces is to put the second stationary wall with which the particle collides elastically.
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9 This model can be thought as a special case of the previous model where (1.3) U(x) = ( 0, if x ≤¯ h ∞ if x > ¯ h where ¯ h is the height of the stationary wall. Pingpong model was intro-duced by Ulam to study Fermi acceleration. To explain the presence of highly energetic particles in cosmic rays Fermi considered particles passing through several galaxies. If the particle moves towards a galaxy it accelerates while if it goes in the same direction it decelerates. Fermi argued that head-on collisions are more frequent than the overtaking collisions (for the same reason that a driver on a highway sees more cars coming towards her than going in the same direction even though the effect becomes less pronounced if the car’s speed is 3000 m/h) leading to overall acceleration. Pingpong was a simple model designed to test this mechanism. This model was one of the first systems studied by a computer (first experiments were performed by Ulam and Wells around 1960). Since the computers were very slow at that time they chose wall motions which made computations simpler, namely, either wall velocity or interwall distance was piecewise linear. It was quickly realized that the acceleration was impossible for smooth wall motions. The motions studied by Ulam and Wells turned out to be more complicated and there are still many open questions.
All of the above systems can be considered Hamiltonian with poten-tial containing hard core part (1.3). Accordingly these systems preserve measures with smooth densities.
Consider for example models (II) and (III). It is convenient to study the Poincare map corresponding to collision of the particle with the moving wall. One can approximate the hard core systems by a Hamiltonian system with the Hamilton-ian Hε = v2 2 + U(x) + Wε(x −f(t)) where W(d) is zero for d < ε and W(−ε) = 1 ε. One can consider the collision map as the limit of Poincare map corresponding to the cross section x−f(t) = ε. The map preserve the form ω = dH ∧dt−dv ∧dx. On our cross section we have dx = ˙ fdt so the invariant form becomes (1.4) ω = (v −˙ f)dv ∧dt.
One can also directly show that the form (1.4) is invariant without using approximation argument.
Consider for example the pingpong system tn+1 = tn + T(tn, vn), vn+1 = vn + 2 ˙ f(tn+1).
10 LECTURES ON BOUNCING BALLS.
This map is a composition of two maps ¯ tn+1 = tn + T(tn, vn), ¯ vn+1 = vn and tn+1 = ¯ tn+1, vn+1 = ¯ vn+1 + 2 ˙ f(¯ tn+1).
Accordingly the Jacobian of this map equals to ∂tn+1 ∂tn . We have (see figure 6) tn tn + δtn δhn Figure 6. Derivative of pingpong map.
δhn = (vn −˙ fn)δtn, (1.5) δtn+1 = δhn vn + ˙ fn+1 = vn −˙ fn vn+1 −˙ fn+1 δhn.
Thus the Jacobian equals to vn−˙ fn vn+1−˙ fn+1 proving the invariance of ω.
A similar calculation can be done for the model (II) using the fact that autonomous Hamiltonian systems preserve the form dv ∧dx.
2. Normal forms.
2.1. Smooth maps close to identity. Here we discuss the behaviour of highly energetic particles using the methods of averaging theory. The following lemma will be useful.
Lemma 2.1. Consider an area preserving map of the cylinder R × T of the form Rn+1 = Rn + A(Rn, θn), θn+1 = θn + B(Rn, θn) Rn .
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11 Assume that the functions A and B admit the following asymptotic expansion for large R (2.1) A = k X j=0 aj(θ) Rj + O R−(k+1) , B = k X j=0 bj(θ) Rj + O R−(k+1) where b0(θ) > 0 (twist condition).
Then for each k there exists coordinates I(k), φ(k) such that I R is uni-formly bounded from above and below and our map takes form In+1 = O I−(k+1) n , θn+1 = θn + 1 In k X j=0 cj Ij n + O I−(k+1) n !
.
Remark 2.2. I(0) is called adiabatic invariant of the system. I(k) for k > 0 are called improved adiabatic invariants.
Proof. We proceed by induction. First, let I = RΓ(θ), φ = Φ(θ) then In+1 −In = RnΓ′(θn)b0(θn) Rn + a0(θn)Γ(θn) + O 1 Rn .
So if we let Γ′ Γ = −a0 b0 that is Γ(θ0) = exp Z θ 0 −a0(s) b0(s) ds then In+1 −In = O(R−1 n ).
Next φn+1 −φn = Φ′(θn)b0(θn) Rn = Φ′(θn)b0(θn)Γ(θn) In .
We let Φ′(θ) = c b0(θ)Γ(θ) so that Φ(θ) = c Z θ 0 ds b0(s)Γ(s) and c = Z 1 0 ds b0(s)Γ(s) −1 .
Note that Γ(1) = Γ(0) so that Γ is actually a function on the circle.
Indeed if Γ(1) < Γ(0) then there would exist a constant ε such that after one rotation around the cylinder R decreases at least by the factor (1 −ε). So after many windings the orbit would come closer and closer to the origin contradicting the area preservation. If Γ(1) > Γ(0) we would get a similar contradiction moving backward in time.
This completes the base of induction. The inductive step is even easier. Namely if In+1 = In + ˆ a(φn) Ik+1 n + . . . then the changes of variables 12 LECTURES ON BOUNCING BALLS.
J = I + γ(φ) Ik leads to Jn+1 −Jn = ˆ a(φn) + γ′ n(φn)c0 Jk+1 n so we can improve the order of conservation by letting γ′ = −ˆ a c0.
Next, if φn+1 −φn = 1 In Pk−1 j=0 cj Ij n + ˆ b(φn) Ik+1 n then letting ψ = φ + Ψ(φ) Ik we obtain ψn+1 −ψn = 1 In k−1 X j=0 cj Ij n + ˆ b(ψn) + Ψ′(φn)c0 Ik+1 allowing us to eliminate the next term if Ψ′ = ck−ˆ b c0 where ck = R 1 0 ˆ b(s)ds.
□ 2.2. Adiabatic invariants. It is instructive and useful to compute the leading terms in several examples.
(I) Fermi-Ulam pingpong. We have vn+1 −vn ≈2 ˙ f(tn), tn+1 −tn ≈2l(tn) vn where l(t) is the distance between the walls at time t. We have l = ¯ h−f so ˙ f = −˙ l and the above equation is the Euler scheme for the ODE dv dt = −v ˙ l l.
Thus ldv + vdl = 0 so I = lv is an adiabatic invariant. In fact one can check by direct computation that letting Jn = (vn + ˙ l(tn))l(tn) one gets Jn+1 −Jn = O 1 J2 n , tn+1 −tn = 2l2(tn) Jn + O 1 J2 n so Jn is the second order adiabatic invariant.
(II) Outer billiard. If A0 is far from the origin then A1 is close to −A0, however |A0A2| = 2|B0B1| there Bj denotes the point of tangency of AjAj+1 with Γ (see Figure 4) and so |A0A2| ≤2diam(Γ). It fact it is not difficult to see that we get the following approximation when A0 is far from the origin: ⃗ A0A2 ≈2⃗ v(θ) where ⃗ v(θ) is the vector joining two points on Γ whose tangent line have slope θ. Let B0(θ) and B1(θ) denote the tangency points and let Q be the point such that B1Q has slope θ while B0Q is perpendicular to B1Q. Note that |B0Q| = w(θ)-the width of Γ in the direction θ.
Fix a direction θ0 and choose coordinates on the plane so that θ0 is equal to 0. Let Bj = (xj, yj). Then for θ near 0 we have xj(θ) = xj(0) + θξj + . . . , yj(θ) = yj(0) + θ2ηj + . . .
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13 B0 B1 Q Figure 7. Derivative of the support function.
and so (x1 −x0, y1 −y0)(sin θ, cos θ) = −|QB1|θ + . . . .
Therefore the equation of motion takes the following form in polar coordinates (up to lower order terms).
˙ R = −w′(θ), ˙ θ = w(θ) R .
Hence dR dθ = −Rw′(θ) w(θ) or wdR + Rdw = 0.
Accordingly I = Rw is the adiabatic invariant and ˙ θ = w(θ) R = w(θ)R R2 = I R2.
In other words I = R2 ˙ θ, that is the angular momentum is preserved and so the point moves with constant sectoral velocity.
Consider, in particular, the case where Γ is centrally symmetric.
Then w(θ) = 2 supx∈Γ(e⊥(θ), x) and since R = I w(θ) level curves of the limiting equation are rescalings of the right angle rotation of Γ∗where Γ∗= {D(e)e}e∈S1 and D(e) = 1 supx∈Γ(e, x).
Thus if ˆ Γ = Int(Γ) then c Γ∗= {e ∈R2 : |(e, x)| ≤1 for all x ∈ˆ Γ}.
Thus for each x ∈Γ and for all e ∈Γ∗we have |(x, e)| ≤1 and there is unique e ∈Γ∗with (x, e) = 1. Therefore (Γ∗)∗= Γ and so each smooth 14 LECTURES ON BOUNCING BALLS.
convex centrally symmetric curve appears as an invariant curve for motion at infinity for some outer billiard.
2.3. Systems with singularities. Lemma 2.1 describes the normal form for smooth maps, so it is not applicable to systems with discon-tinuities such as Fermi-Ulam pingpongs where ˙ l or ¨ l has jumps or to outer billiards about nonsmooth curves such as circular caps or lenses.
It is turns out that for such maps it is convenient to consider the first return map to a neighbourhood of singularities.
In this section we present the normal form of such first return maps.
Figure 8. Large velocity phase portrait of piecewise smooth pingpong looks similar for different values of time so it makes sense to consider the first return map to a neighbourhood of the singularity.
We assume that the cylinder is divided into a finite union of sectors Sj so that our map is C∞in Int(Sj), has C∞extension to a neighbourhood of Sj, and satisfies the asymptotics (2.1) in each sector. We suppose LECTURES ON BOUNCING BALLS.
15 that the boundaries of Sj are γj and γj+1 where γj = θ = θj0 + θj1 R + θj2 R2 + . . .
.
By Lemma 2.1 we can introduce in each sector action-angle coordinates (I, φ) so that the boundaries of the sector become {φ = 0} and n φ = α0 + α1 R + α2 R2 + . . .
o and the map takes form In+1 = In + O I−k n , φn+1 = φn + 1 In " k X m=0 cm Im n + O I−k n # (we suppress the dependence of αs and cs on j since we will work with a fixed sector for a while).
Let Πj be the fundamental domain bounded by γj and fγj and let Fj be the Poincare map Fj : Πj →Πj+1.
It is convenient to introduce coordinates (I, ψ) in Πj where φ = c0 I + c1 I2 + . . . ck Ik+1 ψ so that ψ changes between 0 and 1+O(I−(k+1)). We first describe Fj in the action-angle variables of Sj and then pass to the new action-angle variables of Sj+1. We have φn −φ0 = c0n I + c1n I2 + . . .
The leading term here is the first one so that for the first n such that φn ∈Sj+1 we have c0n I ≈α0 and hence c1n I2 ≈c1α0 c0I . Therefore φn+1 = c0ψ0 I + c0n I + c1α0 c0I + . . .
Now the condition φn−1 ≤α0 + α1 I ≤φn reduces to α0 + ˜ α1 I −c0ψ0 I + · · · ≤c0n I ≤α0 + ˜ α1 I −c0ψ0 I + c0 I + . . .
where ˜ α1 = α1 −c1α0 c0 . For typical ψ0 this means that n = α0I + ˜ α1 c1 −ψ0 + 1 = α0I + ˜ α1 c1 −ψ0 + 1 − α0I + ˜ α1 c1 −ψ0 .
16 LECTURES ON BOUNCING BALLS.
Then φn = α0+α1 I +c0 1 − α0I + ˜ α1 c0 −ψ0 = α0+α1 I +c0 ψ0 −α0I + ˜ α1 c0 .
Rescaling the angle variable so that it measures the distance from the singularity ¯ ψ = I c0(ψn −α0 −α1 I + . . . ) we get that Fj has form ¯ I = I + . . . , ¯ ψ = ψ0 −α0I + ˜ α1 c1 + . . .
To pass to action coordinate of Sj+1 we note that I(j) = Γ(j)(θ)R + . . . , I(j+1) = Γ(j+1)R + . . .
which implies that that the new adiabatic invariant satisfies J = (1 + ˜ λφ + . . . ).
Thus in terms of the new action-angle coordinates Fj takes the form ˆ J = I + λ ¯ ψ + . . . , ˆ ψ = ¯ ψ (to justify the last equation we note that if we just use the Taylor expansion we would get ˆ ψ = σ ¯ ψ and then we get σ = 1 from the condition that Fj is one-to-one).
In terms of the original values of (I, ψ) in Πj we get ˆ ψ = {ψ −β(j) 0 I −β(j) 1 }, ˆ J = I + λ(j) ˆ ψ.
Note that to find the leading term we used the first order Taylor expan-sion, To compute 1 I -term we need to use the second order expansion, for 1 I2 we need the third order expansion and so on. hence we actually have Lemma 2.3. If the orbit does not pass in O(1/I2) neighbourhood of the singularities then Fj has the following form ψj+1 Ij+1 = {ψj −(β(j) 0 Ij + β(j) 1 )} Ij + λ(j)ψj+1 + 1 [Ij]R2 + 1 [Ij]2R3 + . . .
where Rj are piecewise continuous and on each continuity domain they are polynomials in ({Ij}, ψj) of degree j.
We shall say that a map F is of class A if for each k F ψ I = ψ I +L1 {ψ} {I} + k X j=1 1 nj Pj+1 ({ψ}, {I})+O(n−(k+1)) where L1 is linear, A = dL1 is constant and Pj are piecewise polyno-mials of degree j.
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17 Lemma 2.4. A composition of A maps is a A map.
Proof. We need to show that if Fs(z) = L1,s(z) + k X j=2 1 nj−1Pj,s(z) for s = 1, 2 where Pj,s are polynomials of degree j then F2 ◦F1 is also of the same form.
It is sufficient to consider the case where Pj,s have positive coefficients since in the sign changing case there might be additional cancellations. Observe that Fs(z) = Pk j=1 1 nj−1Pj where Pj are some polynomials then the degree restriction amounts to saying that Gn(u) = 1 nFs(un) is bounded for each u as n →∞. But if Gn,1 and Gn,2 satisfy this condition then the same holds also for their composition.
□ Corollary 2.5. The first return map F : Π1 →Π1 is an A map and the same holds for any power F m.
Remark 2.6. Corollary 2.5 applies in particular in the case where the original map is smooth. In that case the coefficients λ(j) vanish so the linear part is the integrable twist map (2.2) ˆ I = I, ˆ ψ = ψ −β0I −β1.
More generally, λ(j) depend only on the behaviour of the function Γ near the singularities so the normal form (2.2) holds also in the case where a0 and b0 from Lemma 2.1 are continuous (even though the higher order terms may be nontrivial in that case).
We say that the original map f is hyperbolic at infinity if the linear part L1 of the normal form of the first return map F is hyperbolic and say that f is elliptic at infinity if L1 is elliptic. Recall that the ellipticity condition is |Tr(L1)| < 2 and the hyperbolicity condition is |Tr(L1)| > 2.
One can work out several leading terms in our main examples. Namely for outer billiard about the semicircle it is shown in that L1 = L2 where (2.3) L(I, ψ) = (I −4 3 + 8 3{ψ −I}, {ψ −I}).
For Fermi-Ulam pingpongs where the wall velocity has one discontinu-ity at 0 one has (2.4) L1(I, ψ) = (I + ∆ {ψ −I} −1 2 , {ψ −I}) where ∆= l(0)∆˙ l(0) Z 1 0 ds l2(s) 18 LECTURES ON BOUNCING BALLS.
Figure 9. Dynamics of the first return map. Top: hy-perbolic case. Bottom: elliptic case.
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19 and l(s) is the distance between the walls at time s.
For example, for motions studied by Ulam and Wells one has l(s) = b+a({s}−1/2)2. We can choose the units of length so that b = 1, then l(s) > 0 for all s provided that a > −4. Then ∆(a) = −2a(1+a/4)J(a) where J(a) = Z 1 0 ds (1 + a(s −1/2)2)2 = 2 a + 4 + 1 2√ |a| ln 2+√ |a| 2−√ |a| if a < 0 1 √a arctan √a 2 if a > 0.
One can check that f is hyperbolic at infinity if a ∈(−4, ac) or a > 0 and f is elliptic at infinity for a ∈(ac, 0) where ac ≈−2.77927 . . .
Figure 10. ∆(a) for piecewise linear wall velocity 2.4. Accelerating orbits for piecewise smooth maps. Given an A map f we say that p = (¯ I, ¯ ψ) is an accelerating orbit if there exist m, l > 0 such that Lm 1 (p) = p + (l, 0).
Lemma 2.7. Assume that f is elliptic at infinity and has an (m, l) accelerating orbit such that the spectrum of Lm 1 does not contain k-th roots of unity for k ∈{1, 2, 3}. Suppose also that F preserves a smooth measure with density of the form ρ(I, ψ) = Iρ0(ψ)+ρ1(ψ)+o(1). Then f has positive (and hence infinite) measure of orbits such that In →∞.
Proof. Consider a point {IN, ψN} in a small neighborhood of {¯ I+Nl, ¯ φ} and study its dynamics. For n ≥N, we will denote {In, ψn} the point F (n−N)l(IN, ψN). Set Un = In−(¯ I+nL), υn = ψn−¯ ψ. We can introduce a suitable complex coordinate zn = Un + i(aUn + bυn) such that DF l becomes a rotation by angle 2πs near the origin where s ̸∈ 1 kZ for k ∈{1, 2, 3}. In these coordinates F l takes the following form in a 20 LECTURES ON BOUNCING BALLS.
small neighborhood of (0, 0) (2.5) zn+1 = ei2πszn + A(zn) N + O(N−2) where A(z) = w1 + w2z + w3¯ z + w4z2 + w5z¯ z + w6¯ z2.
Lemma 2.8. (a) We have that Re(e−i2πsw2) = 0.
(b) There exists ǫ > 0 and a constant C such that if |zN| ≤ǫ, then for every n ∈[N, N + √ N] |zn| ≤|zN| + CN−1.
Part (b) is the main result of the lemma. Part (a) is an auxiliary statement needed in the proof of (b). Namely, part (a) says that a certain resonant coefficient vanishes (this vanishing is due to the fact that f preserves a measure with smooth density).
Before we prove this lemma, let us observe that it implies that for sufficiently large N, all the points |zN| ≤ǫ/2 are escaping orbits. In-deed by [ √ N] applications of lemma 2.8 there is a constant C such that |zl| ≤ǫ 2 + CN−1 2 for every l ∈[N, 2N]. It now follows by induction on k that if l ∈ [2kN, 2k+1N] then |zl| ≤ǫk where ǫk = ǫ 2 + C √ N k X j=0 1 √ 2 j (N has to be chosen large so that ǫk ≤ǫ for all k).
This proves lemma 2.7.
□ Proof of lemma 2.8. Let ¯ n = n −N. For ¯ n ≤ √ N equation (2.5) gives (2.6) zn = ei2π¯ nszN + 1 N ¯ n−1 X m=0 ei2πmsA(ei2π(¯ n−m−1)szN+¯ n−m) + O(N−3 2) In particular for these values of n we have zn = ei2πs(n−N)zN + O 1 √ N .
Substituting this into (2.6) gives zn = ei2π¯ nszN + 1 N ¯ n−1 X m=0 ei2πmsA(ei2π(¯ n−m−1)szN) + O 1 N .
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21 To compute the sum above expand A as a sum of monomials and observe that ¯ n−1 X m=0 ei2πms ei2π(¯ n−m−1)szN α e−i2π(¯ n−m−1)s¯ zN β is bounded for α + β ≤2 unless α = β + 1 (that is α = 1, β = 0).
Therefore (2.7) zn = ei2π¯ nszN 1 + ˜ w2 ¯ n N + O N−1 where ˜ w2 = e−i2πsw2.
Consider now the disc DN around 0 of radius N−0.4. Let W(z) denote the density of invariant measure in our complex coordinates. Then by (2.7) Area(F ¯ nDN) Area(DN) = 1 + 2Re( ˜ w2) ¯ n N + O N−0.6 .
On the other hand there exists z ∈DN such that denoting z′ = F ¯ nz we have Area(F ¯ nDN) Area(DN) = 1 + W(z)/N 1 + W(z′)/(¯ n + N) + O N−2 = 1 + O N−1.4 since W(z)−W(z′) = O (N−0.4) . Comparing those two expressions for the ratio of areas we obtain that Re( ˜ w2) = 0.
This proves part (a) of Lemma 2.8. Part (b) now follows from (2.7).
□ Corollary 2.9. mes(E) = ∞for the following systems: (a) outer billiards about circular caps with angle close to π; (b) Ulam pingpongs with ∆∈(2, 4).
Proof. For part (a) observe that map (2.3) has accelerating orbit (0, 7 8) and for part (b) observe that map (2.4) has accelerating orbit (0, 1 2+ 1 ∆).
□ Problem 2.10. Does map (2.4) have stable accelerated orbits for all ∆∈(0, 4)?
2.5. Birkhoffnormal form. Here we discuss the normal form of an area preserving difffeomorphism near a periodic point.
Consider an area preserving map f of R2 which has an elliptic fixed point p such that in suitable complex coordinates z near p our map has the following form f(z) = e2πiαz + O(z2).
22 LECTURES ON BOUNCING BALLS.
Lemma 2.11. Suppose that e2πikα ̸= 1 for k = 1, 2, . . . 2s. Then there exists a local diffeomorphism h such that h ◦f ◦h−1 has form rn+1 = rn + O(r2s n ), φn+1 = φn + α + s−1 X j=1 cjr2j n + O(r2s n ).
Proof. If suffices to prove that one can reduce f to the following form rn+1 = rn + s−1 X j=1 djr2j+1 n + O(r2s n ), φn+1 = φn + α + s−1 X j=1 cjr2j n + O(r2s n ) since than area preservation would imply that dj ≡0 since otherwise the orbits will go either further away from 0 or closer to 0 with each iteration contradicting area preservation.
So we would like to conjugate f to g(z) = e2πiαz + 2s−1 X k=2 Gk(z) + O(z2s) by the map h(z) = z + 2s−1 X k=2 Hk(z) + O(z2s) where Gk and Hk are polynomials of degree k in z, ¯ z. Expanding the equation h ◦f = g ◦h into Taylor series we get Hk(e2πiαz, e−2πiα¯ z) + Ak = e2πiαHk(z, ¯ z) + Bk + G(z, ¯ z) where Ak and Bk denote the terms which are determined by the lower order coeffcients of H and G respectively. If Hk = X l1+l2=k hl1,l2zl1¯ zl2, Gk = X l1+l2=k gl1,l2zl1 ¯ zl2 when we get hl1,l2 e2πiα(l1−l2) −e2πiα = gl1,l2 + cl1,l2 where cl1,l2 are determined by Ak and Bk. Hence if l1 −l2 ̸= 1 then we can choose gl1,l2 = 0 and take hl1,l2 = e2πiα(l1−l2) −e2πiα−1 cl1,l2.
On the other hand if l1 −l2 = 1 then we are forced to take gl1,l2 = −cl1,l2. Hence f is conjugated to g(z) = e2πiαzγ(r2) + O(r2s). Writing γ(u) = a(u)e2πib(u) we obtain the result.
□ LECTURES ON BOUNCING BALLS.
23 3. Applications of KAM theory.
3.1. Introduction. In this section we review some applications of Kolmogorov-Arnold-Moser theory to bouncing balls. Our overview will be brief since this material is pretty standard and can be found in sev-eral textbooks. However I would like to emphasize that the brvity of this section does not reflect the importance of this material. In fact, KAM theory is the prime tool for showing lack of acceleration and/or ergodicity. The rest of the course will be devoted to discussing a rele-tively small class of systems where KAM is not applicable with the goal of developing the tools to handle such systems.
A0 B0 A1 Figure 11. The outer caustic has the property that any tangent line to the table cuts offthe segment which is divided into two equal pieces by the tangentcy point.
For one and a half degrees of freedom systems invariant curves pro-vide an easy obstruction to transitivity since the orbit can not pass from one component of R2 −γ to another. One example where it is easy to visualize the invariant curves is given by outer billialrds. In this case the invariant curves are called outer caustics. A curve S is an outer caustic for the outer billiard about a curve Γ if for any tangent line to Γ the points of intersection of that line with S are equidistant from the tangency point. Parametrize Γ by the arclength s and let A(s) denote the area cut from S by the tangent line emanating from Γ(s). We claim that A(s) does not in fact depend on s. Indeed let |A0 ˆ A0| = δs then up to higher order terms we have 24 LECTURES ON BOUNCING BALLS.
A0 ˆ A0 C ˆ B0 B0 A1 ˆ A1 − + Figure 12. Area property of the outer caustics |A0B0| ≈| ˆ A0 ˆ B0| ≈|A0C| ≈| ˆ A0C| ≈|A1C| ≈| ˆ A1C| and ∠A0C ˆ A0 ≈∠A1C ˆ A1 ≈κδs there κ denotes the curvature at the tangency point. Accordingly ∂A ∂s = |A0B0|2κ −|A1B0|2κ = 0.
Therefore given a curve S we can easily given a curve S we can easily construct a curve Γ such that S is an outer caustic for Γ by fixing a paprameter a, considering all segments which cut area a from S and taking the midpoints of those segments. It is more difficult to find outer caustics for a given billiard table Γ. For this we need a full strength of the KAM theory. In particular we need to assume that Γ is sufficiently smooth. We saw in Section 2 that some smoothness is needed but the exact trashhold is currently unknown.
Problem 3.1. Suppose that Γ is piecewise smooth and the first k derivatives at the break points coincede.
For which k must Γ have invariant curves near infinity?
3.2. Theory. Two classical results about invariant curves are Twist Theorem and Small Twist Theorem of Moser.
Proposition 3.2 (Moser Small Twist Theorem). Let Q : R+ →R+ be a C5-function. Then for any numbers a, b such that Q′(r) ̸= 0 for r ∈[a, b] for any K there is ε0 such that if Fε are exact mappings of the annulus R+ × S1 of the form Fε(r, φ) = (r + ε1+δP(r, φ), φ + α + εQ(r) + ε1+δR(r, φ)) where ||P||C5([a,b]×S1) ≤K, ||R||C5([a,b]×S1) ≤K then for ε ≤ε0 Fε has (many) invariant curve(s) separating [a, b] × S1 into two parts. In fact, the set of invariant curves has positive measure.
LECTURES ON BOUNCING BALLS.
25 Proposition 3.3 (Moser Invariant Curve Theorem). Let Q : R+ →R+ be a C5-function. Then for any numbers a, b such that Q′(r) ̸= 0 for r ∈[a, b] there is ε0 such that if F is an exact mapping of the annulus R+ × S1 of the form F(r, φ) = (r + P(r, φ), φ + Q(r) + R(r, φ)) where ||P||C5([a,b]×S1) ≤ε0, ||R||C5([a,b]×S1) ≤ε0 then F has (many) invariant curve(s) separating [a, b] × S1 into two parts. In fact, the set of invariant curves has positive measure.
A classical application of KAM theory is stability of nonresonant elliptic periodic points.
Lemma 3.4. Suppose that p is an elliptic periodic point of an area preserving diffeomorphism f with multiplier e2πα such that e2πkα ̸= 1 for |k| ≤4 and such that the Birkhoffnormal form is non-degenerate.
Then f has a positive measure set of invariant curves near p.
Proof. This follows from Lemma 2.11 and Proposition 3.3.
□ 3.3. Applications. Here we describe some applications of the KAM theory to bouncing balls.
(I) Pingpongs.
Corollary 3.5. Consider Fermi-Ulam pingpong with wall motion of class C6. Then there are KAM curves for arbitrary high velocities. Ac-cordingly all orbits are bounded.
Proof. This follows from Proposition 3.2 and the normal form obtained in Section 2.
□ Corollary 3.6. Consider pingpongs where the wall motion has one discontinuity and the system is elliptic at infinity. Then there is a con-stant C such that for all sufficiently large v there is a positive measure set of orbits such that ¯ v C ≤v(t) ≤C¯ v.
Proof. The map (2.4) has periodic orbit ( 1 2, 0). The non-degeneracy of the Birghoffnormal form is checked in . For the orbits constructed with the help of Lemma 3.4 the adiabatic invariant l(t)v(t) will change little so the oscilations of ln v(t) are of constant order.
□ Problem 3.7. Is Corollary 3.6 valid for systems with several velocity jumps?
26 LECTURES ON BOUNCING BALLS.
We shall see later that the result of Corollary 3.6 is false for pingpongs which are hyperbolic at infinity. In that case the system may even be ergodic so that almost every orbit is dense.
(II) Balls in a potential. Consider a moving in a potential U(x) = gxα and colliding elastically with infinitely heave wall. Suppose that f is C6 and periodic.
Corollary 3.8. If α > 1 and α ̸= 2 then there are KAM curves for arbitrary large velocities. In particular, all orbits are bounded.
Proof. To simplify the formulas we consider the SWA tn+1 = tn + T(vn), vn+1 = vn + 2 ˙ f(tn+1).
An easy calculation using energy conservation shows that T(v) ∼cvσ, T ′(v) ∼cσvσ−1, T ′′(v) ∼cσ(σ −1)vσ−2 where σ = 2 α −1.
Consider first the case α > 2. Take ¯ v ≫1 and suppose that v0 ∼¯ v.
Rescaling un = vn ¯ v we get tn+1 ≈tn + c¯ vσuσ n, un+1 −un = 2 ˙ f(tn) ¯ v .
Since σ > 1 the change of u is much smaller than the change of t and so we can use Proposition 3.2.
Next, consider the case 1 < α < 2. Set zn = vn−v0 vσ−1 . The map takes form tn+1 −tn ≈α0 + Kzn + . . . , zn+1 −zn = 2 ˙ f(tn+1) vσ−1 .
Therefore the statement follows from Proposition 3.3.
□ One can ask what happens for other values of α. Surprisingly for α = 1 one can have a positive measure set of escaping orbits. The proof of that given by Pustylnikov uses KAM theory. It relies on the following non stationary extension of stability of elliptic periodic orbits.
Theorem 3.9. Let fn(z) be a family of real analytic area preserving maps defined near the origin and converging to a limiting value f so that P n ||fn −f|| ≤∞. Suppose that 0 is an elliptic fixed point for f with multiplier e2πα satisfying e2πikα ̸= 0 for k ∈{1, 2, 3, 4}. and that the corresponding Birkhoffnormal form is nondegenerate. Consider a recurrence zn = fnzn−1. Then there is a positive measure set of initial conditions z0 such that zn is bounded.
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27 The proof of Theorem 3.9 proceeds along the line of the proof of Lemma 3.4. We refer the reader to for details.
We now show how this theorem can be used to construct escaping orbits. Consider a two parameter family of SWA tn+1 = tn + 2vn g , vn+1 = vn + 2A ˙ f(tn+1).
Here Af(t) is the height of the ball at time t and g is the gravity strengh.
Consider the orbits where the rocket always hit the ball at the same height. Thus tn+1 = tn mod 1, 2vn g = l. Next d f = 1 0 2A ¨ f 1 1 2 g 0 1 so that Tr(d f) = 2 + 4A ¨ f(t) g . Projecting our orbit to the torus we obtain a fixed point which is elliptic provided that (3.1) −1 < A ¨ f(t) g < 0.
The original orbit on the cylinder is accelerating if (3.2) ˙ f(t) > 0 Next, if we have an accelerating orbit for the SWA, Theorem 3.9 allows to infer stability of the original system. Let us show that we can find the periodic point in our two parameter family of the toral maps satisfying (3.1) and (3.2). Indeed, the periodicity condition amounts to (3.3) ˙ f(tn) = lg 2A If A, g ≫1 then we can arrange tn ≈¯ t for any ¯ t such that ˙ f(¯ t) > 0.
Next, in view of (3.3) condition (3.1) amounts to −2 l < ¨ f(¯ t) ˙ f(¯ t) < 0.
Take an interval (t1, t2) such that ˙ f(t1) = ˙ f(t2) = 0 and ˙ f(¯ t) > 0 for ¯ t ∈(t1, t2). Since R t2 t1 ¨ f(¯ t)d¯ t = 0 the second derivative changes sign on (t1, t2) and so we can find ¯ t satisfying −1 < ¨ f(¯ t) ˙ f(¯ t) < 0 as needed.
The case α = 2 was investigated by Ortega (). He showed that if the periods of the wall and the string are incommensurable then the averaging prevails and there are KAM curves. In the commensuarable case both KAM curves and positve measure of escaping sets are possi-ble. For example, in the case of outer billiards all orbits are bounded.
Corollary 3.10. If Γ is C6 and strictly convex then all orbits are bounded.
28 LECTURES ON BOUNCING BALLS.
Proof. The result follows from the normal form obtained in Section 2 and Proposition 3.2.
□ Problem 3.11. Show that the result is not correct if Γ has points with zero curvature.
Finally in case α > 1 one can always construct a Cantor set of escaping orbits. In fact, it is shown in that HD(E)=2.
Conjecture 3.12. If α < 1 then mes(E) = 0.
We will see in Section 6 that this conjecture is true for very week potentials, that is, for α ≪1.
4. Recurrence.
4.1. Poincare Recurrence Theorem. In this section we describe applications of ergodic theory to the dynamics of bouncing balls.
As it was mentioned before dynamical systems theory strives to de-scribe a long time behavior of a given system. In particular, one can ask if (q(t), x(t)) come close to its initial values for arbitrary large t.
A general result in this direction is the Poincare Recurrence Theorem given below.
Theorem 4.1. Let T be a transformation of a space X preserving a finite measure µ. Then for each set A almost all points from A returns to A in the future.
Proof. Let B = {x ∈A : T nx ̸∈A ∀n > 0}. Then T nB ∩B = ∅ and so T kB ∩T k+nB = T k(B ∩T nB) = ∅. Thus for each N the sets B, TB . . . T N−1B are disjoint and therefore µ(∪N−1 n=0 B) = Nµ(B) ≤µ(X).
Since N is arbitrary we have µ(B) = 0.
□ Poincare Recurrence Theorem need not hold for infinite measure preserving transformations such as m →m + 1 on Z. One can however show that the above map is an only obstacle to Poincare recurrence.
Namely, let T be a transformation of a metric space X preserving an infinite measure µ such that measure of any ball is finite.
Theorem 4.2. X can be represented as a disjoint union X = C ⊔D where (i) D = ∪n∈ZT nB and B is wondering in the sense that T nB∩B = ∅ for n ̸= 0; (ii) C satisfies the Poincare Recurrence Theorem in the sense that for any set A ⊂C almost all points from A visit A.
LECTURES ON BOUNCING BALLS.
29 In abstract ergodic theory C is called conservative part of X and D is called dissipative part of X. However in the setting of smooth dynamical systems this terminology is misleading since D need not be dissipative in the sense that Jac(f) < 1 as the above example of the shift on R shows. Therefore we adopt the terminology of probability theory. That is, we call C recurrent part of X and D transient part of X. If C = X we say that the system is recurrent, if D = X we say that the system is transient.
Introduce coordinates (b, m) on D where b ∈B, m ∈Z and the point (b, m) corresponds to T mb. Then the map takes form (b, m) →(b, m + 1), that is the dynamics on D is isomorphic to the shift on the integers.
Proof. Pick a reference point a. Let B1 = {x ∈B(a, 1) : T nx ̸∈B(a, 1) for n > 0}.
For k > 1 let Bk = {x ∈B(a, k) − ∪k−1 j=1 ∪∞ m=−∞T mBj : T nx ̸∈B(a, k) for n > 0}.
Let B = ∪∞ k=1Bk. Note that the orbits of Bk for different k are disjoint by construction. Next we claim that T nBk ∩Bk = ∅for n ̸= 0. Indeed if x ∈Bk and T nx ∈Bk then T nx ∈B(a, k) by the definition of Bk.
Thus n can not be positive by the definition of Bk. Also n can not be negative since in that case T −n(T nx) = x ∈B(a, k) contradicting the definition of Bk. Thus n = 0 as claimed.
Next let D = ∞ [ n=−∞ T nB and C = X−D. Let A ⊂C. Then A = ∞ [ k=1 Ak where Ak = A ∩B(a, k). Note that by definition of Bk the first return map Rk is well defined on B(a, k) −Bk. Applying Poincare Recurrence Theorem to Rk we see that almost all points from Ak visit Ak, so A satisfies Poincare Recurrence Theorem as claimed.
□ In the setting of bouncing balls the system has nontrivial transient component if the set E = {(t0, v0) : vn →∞} has positive measure. More generally we have the following.
Lemma 4.3. Let T : X →X preserve an infinite measure µ. Suppose that there is a set A such that µ(A) < ∞and an invariant set B such that all points from B visit A. Then B ⊂C. In particular if almost all points from X visit A then T is recurrent.
30 LECTURES ON BOUNCING BALLS.
Proof. Let S ⊂B. For x ∈B let r(x) = min(k ≥0 : T −kx ∈A) so that T r(x)x ∈A. Let ˆ Sk = ∪x∈S:r(x)≤kT r(x)x. It is sufficient to show that almost all points from ˆ Sk visit ˆ Sk infinitely often since if T nx ∈ˆ Sk then T n−jx ∈S for some j ≤k. Note that ˆ Sk ⊂A∩B. By assumption almost all points in T(A ∩B) visit A and so the first return map R : ˆ Sk →ˆ Sk is well defined. Applying Poincare Recurrence Theorem to ( ˆ Sk, R) we obtain our claim.
□ Lemma 4.3 implies that E is indeed the transient part of the phase space since the compliment of E is ∪NZN where ZN = {(t0, v0) : lim inf vn ≤N} and all points from ZN visit {v ≤N + 1}.
While the proof of Lemma 4.3 is very easy there is no general recipe for finding the set A and sometimes it can be tricky. In this section though we present a few examples there the construction of A is rela-tively simple.
Corollary 4.4. mes(E) = 0 for the following systems (a) Fermi-Ulam pingpongs there l and ˙ l are continuous and ¨ l has finitely many jumps; (b) outer billiards around lenses.
Proof. In both cases the return map F : Π1 →Π1 has the following form (I, ψ) →(I, {ψ −a0I −a1}) + O(1/I) (see remark 2.6). That is, after one rotation the adiabatic invariant changes by O(1/I). Therefore each unbounded orbit visits the set A = ∪k |I −3k| < 1 2k .
Since µ(A) < ∞the statement follows from Lemma 4.3.
□ Problem 4.5. Do above systems have escaping orbits? In fact even the existence of unbounded orbits is unknown.
4.2. Background from ergodic theory. To proceed further we need to recall some facts from ergodic theory. Let T : X →X be a map preserving a measure µ. A set A is called invariant if T −1A = A and it is called essentially invariant if µ(T −1A∆A) = 0. T is called ergodic if for any T invariant set we have µ(A) = 0 or µ(Ac) = 0. Next suppose that µ is a probability measure.
LECTURES ON BOUNCING BALLS.
31 Lemma 4.6. The following are equivalent: (a) T is ergodic; (b) If B is an essentially invariant set then µ(B) = 0 or µ(Bc) = 0; (c) If A is a set of positive measure then µ(∪∞ n=1T nA) = 1; (d) If A and B are sets of positive measure then there exists n > 0 such that µ(T nA ∪B) > 0, (e) If φ : X →R is a measurable function such that φ(Tx) = φ(x) almost everywhere when there exists a constant c such that φ = c almost everywhere.
Proof. (a) ⇒(b). Let CN = ∪∞ n=NT −nB. Then since µ(T −n1B∆T −n2B) it follows that µ(CN) = µ(B) for all N. On the other hand CN are nested. Let C = ∪∞ N=0CN. Then µ(C) = µ(B). Since T −1CN = CN+1 we have T −1C = C so µ(C) = 0 or µ(C) = 1 and hence µ(B) = 0 or µ(B) = 1.
(b) ⇒(c). Let B = [ n T nA. Then T −1B ⊃B, so by measure preser-vation, B is essentially invariant and since µ(B) ≥µ(A) it follows that µ(B) = 1.
Exercise 4.7. Prove that (c) ⇒(d) and (d) ⇒(a).
(b) ⇒(e). Suppose that (b) holds and let φ be a T-invariant function.
Then for each t we have µ(x : φ(x) > t) = 0 or 1. Let c = sup(t : µ(x : φ(x) > t) = 0).
Then for each ε µ(x : c −ε < φ(x) < c + ε) = 1 and so φ(x) = c almost everywhere.
(e) ⇒(b). If B is an essentially invariant set let φ = 1B. Then φ = c almost everywhere. Clearly c is either 0 or 1.
□ In this and the following sections we will use the following results.
Theorem 4.8. (Ergodic Theorem) (a) If φ ∈L1(µ) then for almost every x the following limits exist φ±(x) = lim N→∞ 1 N N−1 X n=0 φ(T ±nx).
Moreover for almost every x, φ+(x) = φ−(x) := ¯ φ(x) and Z ¯ φ(x)dµ(x) = Z φ(x)dµ.
(b) If T is ergodic then φ(x) = R φ(x)dµ(x) almost everywhere.
32 LECTURES ON BOUNCING BALLS.
Theorem 4.9. (Maximal Ergodic Theorem) Let Eα = {x : sup N≥1 1 N N−1 X n=0 φ(x) > α}.
Then αµ(Eα) ≤||φ||L1.
Lemma 4.10. (Rokhlin’s Lemma) If T : X →X is an aperiodic transformation then for each n, ε there is a set B such that B, TB, . . . , T n−1B are disjoint and µ X − n−1 [ j=0 T jB !
≤ε.
4.3. Ergodicity and recurrence. Next we consider skew product maps TΦ : (X × R) →(X × R) given by TΦ(x, y) = (Tx, y + Φ(x)) preserving measure dν = dµdx. Denote τm(x, y) = (x, y + m).
Lemma 4.11. (Atkinson, ) Suppose that T is ergodic. If Φ ∈L1(µ) then TΦ is recurrent if µ(Φ) = 0 and transient if µ(Φ) ̸= 0.
Proof. Suppose that µ(Φ) ̸= 0. If C was nontrivial there would exist R such that ν(CR) > 0 where CR = C ∩{|y| ≤R}. Then almost all points from CR would return to CR infinitely often. However by Pointwise Ergodic Theorem yn →∞giving a contradiction.
Our next remark is that TΦ commutes with translations. Hence if (x, y) ∈C then for each ˜ y (x, ˜ y) = τ˜ y−y(x, y) ∈C. Therefore C and D are of the form C = ˜ C × R and D = ˜ D × R where ˜ C and ˜ D are T-invariant. Thus either ˜ C or ˜ D has measure 0.
We now consider the case µ(Φ) = 0. Assume that ˜ C = ∅so that D = X ×R. We shall show that this assumption will lead to a contradiction.
We have that almost all (x, y) with |y| ≤1 visit {|y| ≤2} only finitely many times. Indeed, the set B = {(x0, y0) : |yn| ≤2 infinitely often} is TΦ invariant and all points from B visit A = {|y| ≤2|} so if µ(B) > 0 TΦ would have a nontrivial recurrent part by Lemma 4.3.
Hence for almost all x the set Mx = {n : |Φn| ≤1} is finite where Φn(x) = Pn−1 j=0 Φ(T jx). Let AN = {x : Card(Mx) ≤N}. Pick N such that µ(AN) > 1/2. Take n ≫N. Consider Yn(x) = {y : ∃j ∈[0, n −1] : T jx ∈AN and Φj(x) = y}.
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33 By ergodic theorem applied to the indicator of AN for large n we have Card(Yn(x)) ≥n 2 and for each ¯ y ∈Yn(x) we have Card y ∈Yn : |y −¯ y| < 1 2 ≤(N + 1) since otherwise taking a point from this set with minimal j will lead to a contradiction with the definition of AN. It follows that max j≤n |Φj(x)| ≥ max j≤n,T jx∈AN |Φj(x)| ≥ n 8(N + 1).
On the other hand by ergodic theorem Φj(x) j →0 as j →∞and hence maxj≤n |Φj(x)| n →0 as n →∞contradicting the last displayed inequality.
□ As an application of Lemma 4.11 consider SWA to an impact oscil-lator with ˙ f(t) = ( 1 if {t} ≤1 2 −1 if {t} > 1 2 .
Choose ¯ h = 0. Then f(v, t) = (¯ t, v + ˙ f(¯ t)) where ¯ t = t+ T 2 and T is the period of the spring. Therefore f is recurrent if T is irrational.
On the other hand if ¯ h ̸= 0 then Lemma 4.11 is not directly applicable since ¯ t = t+ T 2 + 2¯ h v + o(1/v) weakly depends on v. To include this case we need another lemma. Let S(x, y) = (T (x, y), y+φ(x, y)) be the map which is well approximated by a skew product at infinity. We assume that S is defined on a subset Ω⊂X × R given by y ≥h(x). We also assume that there exist a map T : X →X and a function Φ : X →R such that T preserves measure µ and that for each k and each function bounded measurable function h supported on X × [−M, M] we have ||h ◦Sk m −h ◦T k Φ||L1(ν) →0 as m →∞ where Sm = τ−m ◦S ◦τm and dν = dµdx.
Lemma 4.12. Assume that (i) T is ergodic; (ii) µ(Φ) = 0; (iii) S preserves a measure ˜ ν having bounded density with respect to ν; (iv) there exists a number K such that φ||L∞(µ) ≤K.
Then S is recurrent.
Proof. Let ¯ Y = X × [0, K] where K is the constant from condition (iv). By Lemma 4.11 TΦ is conservative and hence the first return map 34 LECTURES ON BOUNCING BALLS.
R : ¯ Y →¯ Y is defined almost everywhere. By Rokhlin Lemma applied to R there exists a set Ωε and a number Lε such that ν(Ωε) < ε and ν({(x, y) ∈¯ Y : T j Φ(x, y) ̸∈Ωε for j = 0, 1 . . . Lε −1}) < ε.
It follows that there exists mε > 1/ε such that ν(Aε) < ε where Aε = {(x, y) ∈τmε ¯ Y : Sj(x, y) ̸∈τmεΩε for j = 0, 1 . . . Lε −1}.
In addition we have ˜ ν(Aε) < Cε and ˜ ν(τmεΩε) < Cε. Let A = [ n τm1/n2Ω1/n2 ∪A1/n2 .
Then ν(A) < ∞. Note that every unbounded orbit crosses τm1/n2Ω1/n2 for a sufficiently large n and so it visits A. Therefore S is recurrent by Lemma 4.3.
□ Lemma 4.12 shows recurrence of impact oscillator SWA for all ¯ h. It also implies recurrence of Fermi-Ulam pingpongs in the case where ˙ l has one discontinuity and the corresponding map is hyperbolic at infinity.
This follows from the normal form at infinity derived in Section 2 and the ergodicity of hyperbolic sawtooth map proved in Section 5.
4.4. Proof of the Maximal Ergodic Theorem. We need the fol-lowing result called maximal inequality. Give a function ψ ∈L1(µ) define ˜ S0 = 0, and for k > 0, ˜ Sk = k X n=1 ψ◦T n, ψ∗ N = max 0≤k≤N ˜ Sk, PN = {x : ψ∗ N > 0}.
Lemma 4.13. For all positve N Z PN ψ(x)dµ(x) > 0.
Proof. We have that ψ∗ N(Tx) = max 1≤k≤N+1 k X n=2 ψ(T nx).
Thus ψ∗ N(Tx) + ψ(x) = max 1≤k≤N+1 k X n=1 ψ(T nx).
Since on PN ψ∗ N = max1≤k≤N ˜ Sk it follows that on PN ψ∗ N(Tx) + ψ(x) ≥ψ∗ N(x).
LECTURES ON BOUNCING BALLS.
35 Thus Z PN ψ(x)dµ(x) ≥ Z PN ψ∗ N(x)dµ(x) − Z P ∗ N ψ∗ N(x)dµ(x) = Z PN ψ∗ N(x)dµ(x) − Z X ψ∗ N(x)dµ(x) ≥ Z X ψ∗ N(x)dµ(x) − Z X ψ∗ N(x)dµ(x) = 0 where the second line follows since ψ∗ N = 0 on P c N and the third line follows since φ∗ N ◦T is non negative.
□ Proof of Theorem 4.9. Let ψ = φ −α. Then PN = {x : max k≤N 1 k k X n=1 φ(T nx) > α}.
Taking N →∞in Lemma 4.13 we obtain the statement required.
□ 4.5. Ergodic Theorems for L2-functions. Let I = {φ ∈L2(µ) : φ(Tx) = φ(x)}.
By coboundary we mean a function of the form ψ(x) −ψ(Tx) for an L2 function ψ. Let B denote the closure of the space of coboundaries.
Lemma 4.14. B⊥= I.
Proof. If ψ ∈B⊥then ⟨ψ(x), ψ(x) −ψ(Tx)⟩= 0. Accordingly |ψ(x)ψ(Tx)dµ(x)| = ||ψ||L2 = p ||ψ||L2||ψ ◦T||L2.
By Cauchy Schwartz inequality this is only possible if ψ(Tx) = cψ(x).
Since ||ψ ◦T||L2 = ||ψ||L2 c = ±1. Now it is evident that ⟨ψ(x), ψ(x) −ψ(Tx)⟩= 0 iffc = 1, that is ψ ∈I.
□ Proposition 4.15. If φ ∈L2(µ) then 1 N N−1 X n=0 φ(T nx) →¯ φ := πIφ as N →±∞ almost everywhere and in L2.
36 LECTURES ON BOUNCING BALLS.
Proof. The statement is obvious if φ ∈I, so we amy assume that φ ∈B.
If φ(x) = ψ(x) −ψ(Tx) then 1 N N−1 X n=0 φ(T nx) = ψ(x) −ψ(T nx) N →0 in L2.
Also by Chebyshev inequlity µ(ψ(T nx) > εN) ≤||ψ||L2 ε2N2 so ψ(x)−ψ(T nx) N →0 almost everywhere as well. For general φ ∈B, given ε, we can find ˜ φ, ψ such that φ(x) = ˜ φ(x) + ψ(x) −ψ(T nx) and ||˜ φ||L2 ≤ε. Then ˆ φ±(x) = lim sup N→±∞ 1 N N−1 X n=0 φ(T nx) = lim sup N→±∞ 1 N N−1 X n=0 ˜ φ(T nx) .
Thus by Maximal Ergodic Theorem µ(φ±(x) ≥δ) ≤||˜ φ||L1 δ ≤||˜ φ||L2 δ = ε δ.
Since ε is arbitrary ˆ φ± = 0 so 1 N N−1 X n=0 φ(T nx) →0 almost everywhere. The argument for L2-convergences is similar.
□ Exercise 4.16. Prove Theorem 4.8 for L1 functions.
Hint. Take φ(n) ∈L2 such that ||φ(n) −φ||L1 ≤1 n and show that ¯ φ(n) form a Cauchy sequence. Thus there is a function ¯ φ such that ¯ φ(n) →¯ φ in L1. Show that 1 N N−1 X n=0 φ(T nx) →¯ φ(x) almost everywhere.
Remark 4.17. Ergodic Theorem for L2 functions will be sufficient for all applications given in these lectures.
Corollary 4.18. If ¯ φ is constant almost everywhere for a L2-dense set of functions then T is ergodic.
LECTURES ON BOUNCING BALLS.
37 Proof. If πIφ = Z φ(x)dµ(x) 1 for a dense set of functions then by continuity it holds for all functions.
Thus I consists only of constants and hence T is ergodic.
□ 4.6. Ergodic decomposition. Let T be a map of a separable metric space X preserving a measure µ. Let I be the algebra of T invariant sets. Define a family of measures µx by condition µx(φ) = E(φ|I)(x).
Proposition 4.19.
(a) Z µx(φ)dµ(x) = Z φ(x)dµ(x).
(b) µx is T-invariant for almost every x.
(c) µx is ergodic for almost every x.
Proof. (a) follows from the Law of Total Expectation.
(b) and (c) follow from the formula µx(φ) = πI(φ) and Proposition 4.15.
□ 4.7. Proof of Rokhlin Lemma.
Proof of Lemma 4.10. We first prove the result in case T is ergodic.
Choose a set A with µ(A) < ε/n. Let R(x) be the first return time to A. Let B = ∪j∈A,j<R(x)/nT jx.
Clearly B, TB . . . T n−1B are disjoint.
On the other hand denoting Am = {x ∈A : R(x) = m} we get X −∪n−1 j=0T jB ⊂∪∞ m=1cupn k=1T m−kAm so µ X −∪n−1 j=0T jB ≤ ∞ X m=1 n X k=1 µ(T m−kAm) ≤ ∞ X m=1 nµ(Am) < nε completing the proof for ergodic transformations.
In the non-ergodic case the same argument works provided that we can find A with (4.1) µ x : µx(A) > ε 2n < ε 2.
Take A with µ(A) < ε2 4n. Then (4.1) follows from Proposition 4.19(a) and Markov inequality.
□ 38 LECTURES ON BOUNCING BALLS.
5. Statistical properties of hyperbolic sawtooth maps.
5.1. The statement. We saw in Section 4.3 that ergodicity of hyper-bolic sawtooth maps implies the recurrence of a large class of Fermi-Ulam pingpongs in case velocity has one discontinuity. The required ergodicity is established in this section. In fact, following Chernov we consider a wider class of maps. Let T be a piecewise linear auto-morphism of T2. Let S+ and S−denote the discontinuity lines of T and T −1 respectively. Denote Sn = T n−1S+, S−n = T −(n−1)S−. We assume that (i) A = dT is constant hyperbolic SL2(R)-matrix.
(ii) S± are not parallel to eigendirections of A.
Theorem 5.1. T is ergodic.
In fact, we derive stronger statistical properites of our map T. These results are not needed to establish the recurrence of pingpongs but the techniques introduced here could be used to study several classes of bouncing ball systems.
Recall that a map T preserving a probability measure µ is called mixing if for any L2 functions φ1, φ2, (5.1) ρφ1,φ2(n) →0 as n →∞ where ρφ1,φ2(n) = Z φ1(x)φ2(T nx)dµ(x) − Z φ1(x)dµ(x) Z φ2(x)dµ(x).
Exercise 5.2. (a) T is mixing if (5.1) holds for a dense set of functions φ1, φ2.
(b) T is mixing if for each pair of measurable sets A1, A2 µ(A1 ∩T −nA2) →µ(A1)µ(A2).
(That is, it suffices to check (5.1) for φi = 1Ai.) Thus mixing gives, in particular, that for any measurable set A we have µ(A ∩T −nA) →µ2(A). Thus if A is invariant then µ(A) = µ2(A) and, hence, mixing implies ergodicity.
In case T is a (piecewise smooth map) of a manifold we say that T is exponentially mixing if for some α > 0 there are constants K > 0 and θ < 1 such that (5.2) |ρφ1,φ2(n)| ≤Kθn||φ1||Cα||φ2||Cα.
Exercise 5.3. Show that if for some r |ρφ1,φ2(n)| ≤˜ K ˜ θn||φ1||Cr||φ2||Cr LECTURES ON BOUNCING BALLS.
39 then (5.2) holds. That is, if (5.2) holds for some α then it holds for all α (with K and θ depending on α).
Hint. Approximate Cα functions by Cr functions.
Exercise 5.4. Suppose that T is a smooth linear hyperbolic map of T2. That, is Tx = Ax mod 1 where A ∈SL2(R) and |Tr(A)| > 2.
(a) Show that T is exponentially mixing.
(b) Show that given an arbitrary positive sequence ζn →0 where exists a continuous (but not Holder continuous) function φ such that |ρφ,φ(n)| > ζn for infinitely many n.
Hint. Consider the Fourier series of φ.
Theorem 5.5. Under the assumptions (i) and (ii) above T is expo-nentially mixing.
By the foregoing discussion exponential mixing implies mixing which, in turn, implies ergodicity.
Thus Theorem 5.1 follows from Theo-rem 5.5.
However our proof of Theorem 5.5 relies on Theorem 5.1 so we begin with the proof of Theorem 5.1.
5.2. The Hopf argument. The proof relies on the Hopf argument.
To explain this argument we consider first the case where T is smooth, that is fx = Ax mod 1 and A ∈SL2(Z). Denote W s(x) = {y : d(T nx, T ny) →0 as n →+∞}, W u(x) = {y : d(T −nx, T −ny) →0 as n →+∞}.
It is easy to see that W ∗(x) = {x + ξe∗}ξ∈R where es and eu are con-tracting and expanding eigenvectors of A.
Let R0 be the set of regular points, that is, the points such that for any continuous function Φ we have Φ+(x) = Φ−(x). By Pointwise Ergodic Theorem R0 has full measure in T2. For j > 1 we can define inductively Rj = {x ∈Rj−1 : mes(y ∈W u(x) : y ̸∈Rj−1) = 0 and mes(y ∈W s(x) : y ̸∈Rj−1) = 0}.
Then we can show by induction using Fubini Theorem that Rj has full measure in T2 for all j.
For x ∈R0 and Φ ∈C(T2) let ¯ Φ(x) denote the common value of Φ+(x) and Φ−(x). By Corollary 4.18 it suffices to show that ¯ Φ(x) is constant almost everywhere for every continuous function Φ.
We say that x ∼y if for all continuous Φ we have ¯ Φ(x) = ¯ Φ(y).
Note that if x, y ∈R0 and y ∈W s(x) then for all Φ ∈C(T2) we have 40 LECTURES ON BOUNCING BALLS.
Φ−(x) = Φ−(y) and so x ∼y. Similarly if x, y ∈R0 and y ∈W u(x) then x ∼y, . Given x ∈R2 and ρ ∈R+ let Γρ = [ y∈W u ρ (x) W s(y), ˜ Γρ = [ y∈R1∩W u ρ (x) (W s(y) \ R0).
Then if ρ is large then Γρ = T2 and by Fubini theorem mes(Γρ−˜ Γρ) = 0 so ¯ Φ(z) = ¯ Φ(x) for almost all z. Therefore ¯ Φ is constant almost surely and hence T is ergodic.
5.3. Long invariant manifolds and ergodicity. The Hopf argu-ment has been expanded in several directions. Already Hopf realized that the same argument works for nonlinear systems provided that the stable and unstable foliations are C1. This condition however is too restrictive.
Versions of the Hopf argument under weaker conditions have been presented by Anosov, Pesin, Pugh-Shub, Burns-Wilkinson.
We need a version of the Hopf argument for systems with singularities.
The approach to handle such systems is due to Sinai and it has been extended by Chernov-Sinai and Liverani-Wojtkowski. The proof given here follows the presentation of . A slightly different argument can be found in .
The difficulty in the nonsmooth case is that it is no longer true that W ∗(x) coincides with ˜ W ∗(x) = {x + ξe∗}. Indeed if y ∈˜ W s(x) and x and y belong to the same continuity domain then d(Tx, Ty) = 1 λd(x, y) where λ is the expanding eigenvalue of A. However if Tx and Ty are separated by a singularity then Tx and Ty can be far apart. In fact, there might be points which come so close to the singularities that W s(x) is empty. This is however, an exception rather than a rule. Let ru(x) = max{δ : ˜ W u δ (x) ⊂W u(x)}, rs(x) = max{δ : ˜ W s δ (x) ⊂W s(x)}.
Lemma 5.6.
mes{x ∈T2 : ru(x) ≤ε} ≤Cε, mes{x ∈T2 : rs(x) ≤ε} ≤Cε.
Proof. We prove the second statement, the first one is similar. Note that {rs(x) ≤ε} = S n Sn(ε) where Sn(ε) = n x : d(T nx, S−) ≤ε λn o .
Since our system is measure preserving mes(Sn) = mes n x : d(x, S−) ≤ε λn o ≤¯ C ε λn .
□ The proof of Theorem 5.1 relies on a local version of this result.
Namely, the following statement holds.
LECTURES ON BOUNCING BALLS.
41 Lemma 5.7. Pick y, δ and k such that d(T j ˜ W u(y), S−) ≥ε for j = 0 . . . k. Then mes{x ∈˜ W u δ (y) : rs(x) ≤ε} ≤Cθkε.
A similar statement holds with s and u interchanged.
We first show how Lemma 5.7 can be used to derive Theorem 5.1 and then present the proof of the lemma.
Pick k such that (5.3) Cθk < 0.001.
We first establish local ergodicity. Namely let M be a connected com-ponent of continuity for T k and T −k. We shall show that almost all points in M belong to one equivalence class. This will imply that ev-ery invariant function is constant on M, that is, any invariant set is a unions of continuity domains. Then we conclude the global ergod-icity by noticing that there are no nontrivial invariant sets which are union of continuity components because the boundary would be a col-lection of line segments and this boundary can not be invariant since the segments in Sn have different slopes for different n.
Let us prove local ergodicity. To simplify the exposition we will refer to ˜ W u leaves as horizontal lines and to ˜ W s leaves as vertical lines.
Take a rectangle U ⊂Int(M). It is enough to show that all points are equivalent. Given N consider all squares with sides 1 N and centers in 0.1Z N 2 ∩U.
We say that a points z in a square S is typical if z ∈R2 and both W u(z) and W s(z) cross S completely.
Note that all typical points in S are equivalent. Indeed denote Σ(z) = ∪x∈W u(z)W u(x).
Note that if z1, z2 ∈S then by Lemma 5.7 and (5.3), Σ(zj) ∩S has measure at least 0.999mes(S) and by the Hopf argument almost all points in Σ(zj) are equivalent to zj. Also by Lemma 5.7 the set of typical points in S has measure at least 0.998mes(S). Since for two neighbouring squares we have mes(S1∩S2) = 0.9mes(S1) it follows that all typical points in neighbouring squares are equivalent. Therefore all typical points in all squares in Int(M) are equivalent. On the other hand by Lemma 5.6 for almost all x ∈R2 we have ru(x) > 0 and rs(x) > 0 so such x is typical for sufficiently large N. Local ergodicity follows and Theorem 5.1 is proved.
42 LECTURES ON BOUNCING BALLS.
Figure 13. Each square intersect its neighbours by 0.9 of their area A B C Figure 14. A is typical in S, B is not typical in S but it is typical in a nearby square, C is not typical in any square z Figure 15. Hopf brush Σ(z) 5.4. Growth Lemma. It remains to prove Lemma 5.7. To this end fix a curve γ ⊂˜ W u(x). Due to singularities T n(x) consists of many components. Let rn(x) be the distance from x to the boundary of the component containing x. We claim that there are constants θ < 1 and LECTURES ON BOUNCING BALLS.
43 ˆ C > 0 such that (5.4) P(rn ≤ε) ≤2 θn |γ| + ˆ C ε.
(5.4) implies Lemma 5.7 since it implies that P(Sn) ≤2 θn |γ| + ˆ C ε λn.
Summing this for n ≥k we obtain the statement of Lemma 5.7.
Figure 16. The complexity is determined by the largest number of lines passing through one point since one can always take δ so small that any curve of length less than δ can not come close to two intersection points The proof of (5.4) relies on complexity bound.
Let κn(δ) be the maximal number of continuity components of T n an unstable curve of length less than δ can be cut into.
Set κn = limδ→0 κn(δ). For the case at hand there is a constant K such that κn ≤Kn since the singularities of T n are lines and there at most Kn possibilities for their slopes.
Accordingly there exist numbers n0, δ0 such that κn0(δ0) ≤ 44 LECTURES ON BOUNCING BALLS.
2λn0. Replacing T by T n0 we can assume that this inequality holds for n0 = 1 (clearly it is sufficient to prove (5.4) for ¯ T = T n0 in place of T).
Given a curve γ we define ¯ rn(x) as follows.
Tγ is cut into sev-eral components. Some of them can be longer than δ0. Cut each long component into segments of length between δ0/2 and δ0. For each of the resulting curves γj consider Tγj and repeat this procedure. Let ¯ rn(x) be the distance to the boundary of the new components. Thus ¯ rn(x) ≤rn(x). In fact, ¯ rn equals to rn if each continuity component has width less than δ0 so we can think of ¯ rn as the length of continuity components then we partition T2 into the strips of width δ0 and regard the boundaries of the strips as ”artificial singularities”.
Figure 17. Dynamics of components. The vertical seg-ments are “artificial singularities”.
It suffices to prove (5.4) with rn replaced by ¯ rn. To this end let Zn = sup ε>0 mes(x ∈γ : ¯ rn(x) ≤ε) ε .
Then Z0 = 2 |γ|. We claim that there are constants θ < 1, C > 0 such that Zn+1 ≤θZn + C.
Indeed ¯ rn+1(x) is less than ε if either ¯ rn(x) < ε λ or T n+1x passes near either genuine or artificial singularity. In T n+1x passes near a genuine singularity then T nx is ε λ close to the preimage of singularity. Since each curve is cut into at most κ1(δ0) components, we conclude that each component of T n contributes by less than κ1(δ0)mes x : rn(x) ≤ε λ ≤κ1(δ0) λ Zn.
On the other hand for long curves the relative measure of points with small ¯ rn+1 is less than C(δ0)ε so their contribution is less than C(δ0)ε|γ|.
The result follows.
LECTURES ON BOUNCING BALLS.
45 5.5. Weak Mixing. In fact, the argument used to prove Theorem 5.1 can be used to obtain a stronger result.
Theorem 5.8. T × T is ergodic.
Proof. The proof of Theorem 5.1 was based on the fact, that T is a piecewise linear map satisfying the Growth Lemma and the transver-sality of singularity set to the stable and unstable directions. The map T × T has the same properties. Indeed let Γ1 × Γ2 be a product of un-stable curves in T2 × T2. Defining as before rn(x, y) the distance from (T nx, T n) to the boundary of the component of T nΓ1×T nΓ2 containing that point we have rn(x, y) = min(rn(x), rn(y)) and so P(rn(x, y) ≤ε) ≤P(rn(x) < ε)+P(rn(y) < ε) ≤2θn 1 |Γ1| + 1 |Γ2| + +2 ˆ C ε proving the Growth Lemma for T × T. Now the proof of ergodicity of T × T proceeds along the same line as the proof of ergodicty of T. □ 5.6. Mixing and equidistribution. Here we derive Theorem 5.5 from the following statement. Fix a small constant ¯ δ.
Proposition 5.9. Let Γ be a horizontal segment of length ¯ δ. Then for any φ ∈Cα(T2) we have 1 ¯ δ Z Γ φ(T nx)dx − Z T2 φ(z)dz ≤Cθn||φ||Cα.
This proposition claims that images of horizontal curves of large size become equidistributed. To obtain Theorem 5.5 we need to bootstrap this result to small horizontal curves.
Proposition 5.10. There is a constant K > 0 such that if Γ is an Γ be a horizontal segment of length less than ¯ δ, then for any φ ∈Cα(T2) for any n > K| ln |Γ|| we have 1 |Γ| Z Γ φ(T nx)dx − Z T2 φ(z)dz ≤Cθn−K| ln |Γ||||φ||Cα.
Proof of Theorem 5.5. Let φ1, φ2 ∈Cα(T2). Partition T2 into squares of small size δ. Let ¯ φ be an approximation to φ1 which is constant on the elements of our partition. Denote by ¯ φj the value of ¯ φ on the square Sj. Note that ||φ1 −¯ φ||∞≤||φ1||Cαδα. Let Γj(h) denote the horizontal section of Sj at height h. Then Z T2 φ1(z)φ2(T nz)dz = Z T2 ¯ φ(z)φ2(T nz)dz + O (δα||φ1||Cα||φ2||Cα) 46 LECTURES ON BOUNCING BALLS.
= X j ¯ φj Z Sj φ2(T nz)dz + O (δα||φ1||Cα||φ2||Cα) = X j ¯ φj Z δ 0 Z Γj(h) φ2(T nx)dx !
dh + O (δα||φ1||Cα||φ2||Cα) = X j ¯ φjδ2 Z T2 φ2(z)dz +O (δα||φ1||Cα||φ2||Cα)+O ||φ1||Cα||φ2||Cαθn−K| ln δ| .
= Z T2 φ1(z)dz Z T2 φ2(z)dz+O (δα||φ1||Cα||φ2||Cα)+O ||φ1||Cα||φ2||Cαθn−K| ln δ| .
Choosing δ so that | ln δ| = n 2K completes the proof of Theorem 5.5. □ 5.7. Another growth lemma. To derive Proposition 5.10 from Propo-sition 5.9 we need another Growth Lemma. Recall that the first Growth Lemma (formula (5.4)) tells us that if we start from a horizontal seg-ment of short length δ then the probability that rn(x) is small becomes small for n ≥K| ln δ|). Thus on average most points belong to long components most of the time. Here we discuss the exceptional points which stay in short components for a long time. Let Γ be a segment of length δ. Fix ˜ n ≥K| ln δ|. We define a piecewise constant function ¯ n : Γ →N such that ¯ n ≥˜ n, T ¯ nΓ = [ j Γj and |Γj| = ¯ δ.
¯ n will be defined inductively. Namely, let {Γj0} be the components of T ˜ nΓ. Call Γj0 long if its length is at least 2¯ δ. Each long component will be further decomposed as Γj0 = Lj ⊔(⊔kΓjk) ⊔Rj where |Γjk| = ¯ δ and Lj and Rj are neighbourhood of the left and right endpoints of Γj0 with ¯ δ 2 ≤|Lj| ≤¯ δ, ¯ δ 2 ≤|Rj| ≤¯ δ.
On ∪jkΓjk we let ˜ n = ¯ n. Let ˜ Γ be a component where ¯ n has not yet been defined (thus ˜ Γ = Γj0 where |Γj0| < 2¯ δ or ˜ Γ = Lj or Rj for some long component Γj0) consider T K| ln |˜ Γ||˜ Γ and trim their long components as before. On the long components of the resulting set we set ¯ n = n1 + K| ln |˜ Γ|| while points which stay in the short components on both attempts will be iterated once more. This procedure is continued inductively. The second growth Lemma says that for most ponts we stop after a relatively short time.
Lemma 5.11. P(¯ n > ˜ n + k) ≤Cθk.
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47 Before giving the proof of Lemma 5.11 we show how it helps to derive Proposition 5.10.
Proof of Proposition 5.10. We apply Lemma 5.11 with ˜ n = K| ln δ|.
Let {Γj}j∈J be all components where ¯ n ≤K ln |δ|+n 2 . Let nj denote the value of ¯ n on T −¯ nΓj. Then 1 |Γ| Z Γ φ(T nx)dx = X j∈J |Γj| |Γ| " 1 |Γj| Z Γj φ(T n−njy)dy # +O ||φ||P ¯ n > K ln |δ| + n 2 = X j∈J |Γj| |Γ| Z T2 φ(z)dz + O ||φ||θ(n−K|ln δ|)/2 +O ||φ||P ¯ n > K ln |δ| + n 2 = Z T2 φ(z)dzP ¯ n ≤n + K| ln δ| 2 +O ||φ||θ(n−K|ln δ|)/2 +O ||φ||P ¯ n > K ln |δ| + n 2 = Z T2 φ(z)dzO ||φ||θ(n−K| ln δ|)/2 where the first equality is obtained by changing variables, the second uses Proposition 5.9, the third follows from the definition of J and the fourth follows from Lemma 5.11.
□ 5.8. Trying to succeed. We derive Lemma 5.11 from the following more general result. Let J be a N valued random variable and T = PJ j=1 kj where kj are N valued random variables. Let Fj be a filtration such that k1, . . . kj are Fj-measurable and as well as sets {J = j}.
Lemma 5.12. Suppose that there are constats K > 0, p < 1, θ < 1 such that (5.5) P(J = j + 1|Fj) ≥p, (5.6) P(kj+1 = k|Fj) ≤Kθj.
Then P(T = l) ≤˜ K ˜ θl.
Proof. We use moment generating functions. Let Φj(z) = E zk1+...kj1J≥j .
We claim that there exist numbers r > 1, ζ < 1 such that (5.7) Φ1(z) and Φ2(z)converge for 0 < z ≤r (5.8) For j > 2, Φj(z) < ζΦj−2(z) for z ≤r.
(5.8) implies that Φ2l+1 ≤ζlΦ1(z), Φ2l+2 ≤ζlΦ2(z) 48 LECTURES ON BOUNCING BALLS.
and so P l Φl(z) converges for z ≤r. But then E(zT ) ≤P l Φl(z) is also finite proving Lemma 5.12. It remains to establish (5.7) and (5.8).
The fact that Φ1(z) is bounded for z < r < θ−1 follows from (5.6).
Next Φj(z) = E zk1+···+kj−11J>j−1E zkj|Fj−1 .
We claim that given δ we can take r so close to 1 that (5.9) E zkj|Fj−1 ≤1 + δ.
Hence Φj(z) ≤(1 + δ)E zk1+···+kj−11J>j−1 .
Next E zk1+···+kj−11J>j−1 = E zk1+···+kj−21J≥j−2E zkj−11J̸=j−1|Fj−2 .
We claim that there is a constant ¯ ζ < 1 such that if r is sufficiently close to 1, then (5.10) E zkj−11J̸=j−1|Fj−2 ≤¯ ζ.
Taking δ so small that ζ := ¯ ζ(1 + δ) < 1 we obtain (5.8). Also (5.9) implies that Φ2(z) ≤(1 + δ)Φ1(z) proving (5.7).
It remains to prove (5.9) and (5.10). Since the LHSs of (5.9) and (5.10) are increasing in z it sffices to consider z = r. Then (5.11) E rkj|Fj−1 = 1 + ∂ ∂z E zkj|Fj−1 (z = ˜ z)(r −1) for some ˜ z ∈[1, r].
By (5.6) ∂ ∂z E zkj|Fj−1 (z) ≤K X l lθl is bounded for z ≤r < θ−1. Therefore the second term in (5.11) can be made as close to 1 as we wish by taking r close to 1. This proves (5.9). The proof of (5.10) is similar except that we use that E zkj−11j̸=J−1|Fj−2 ≤1 −p due to (5.5).
□ 5.9. Equidistribution and coupling. The main step in proving Propo-sition 5.9 is the following.
Lemma 5.13. Let Γ1 and Γ2 be two segments of length ¯ δ then 1 ¯ δ Z Γ1 φ(T nx)dx −1 ¯ δ Z Γ2 φ(T nx)dx ≤Cθn||φ||Cα.
LECTURES ON BOUNCING BALLS.
49 Proof of Proposition 5.9. We claim that for any ε > 0 we can de-compose T2 = (⊔jSj) ⊔Z where Sj are rectangles of width ¯ δ and mes(Z) < ε. Indeed let ¯ ∆be a vertical rectange of small length η and let τ be a first return time to ∆by the horizontal flow. Then τ →∞ as η →0. Cutting each piece of time τ orbit into segments of length ¯ δ we obtain the required partition. Let Hj be the height of Sj and Γj(h) be the vertical segment in Sj at height h. Then Z T2 φ(z)dz = Z T2 φ(T nz)dz = X j Z Hj 0 Z Γj φ(T nx)dx !
dh+O (ε||φ||∞) X j Z Hj 0 Z Γ φ(T nx)dx dh + O (θn||φ||Cα) + O (ε||φ||∞) where the last step relies on Lemma 5.13. Since ε is arbitrary the result follows.
□ Note that the above proof shows in particular that for each ε we can find rectangles {Sj} so that the following decomposition holds (5.12) Z T2 φ(z)dz = X j Z Hj 0 Z Γj φ(x)dx !
dh + O (ε||φ||∞) .
This decomposition proves very convenient in the study of statistical properties of T.
Next we describe the idea of the proof of Lemma 5.13 in the smooth case. We have 1 |Γj| Z Γj φ(T nx)dx = 1 |T n/2Γj| Z T n/2Γj φ(T n/2y)dy.
Note that T n/2Γj are segment of length λn/2¯ δ. Since both of the seg-ments are in T2 the distance between their starting points in O(1).
Hence we can represent T n/2Γj = ˜ Γj + ˆ Γj where |˜ Γj| = O(1) and ˆ Γ2 is a projection of Γ1 along es. Denoting this projection by π we have d(y, πy) = O(1) and hence d T n/2y, T n/2πy = O λ−n/2 .
Thus Z ˜ Γ2 φ(T n/2y2)dy2 − Z ˜ Γ1 φ(T n/2y1)dy1 = Z ˜ Γ1 [φ(T n/2y1) −φ(T n/2πy1)dy1] = O λ−αn/2||φ||Cα .
50 LECTURES ON BOUNCING BALLS.
On the other hand Z ˜ Γj φ(T n/2yj)dyj = O(||φ||∞).
Dividing by ¯ δλn/2|T n/2Γj| we obtain the required estimate.
In the nonsmooth case the structure of T n/2Γj is more compicated but we still can split T n/2Γj into pieces which are close to each other.
This is content of the following result.
Lemma 5.14. (Coupling Lemma) There exists a measure preserving map π : Γ1 →Γ2 (coupling map) and a function R : Γ1 →N (coupling time) such that (a) There is a constant η such T R(x)x and T R(x)π(x) belong to the same stable manifold of length less that η and so for n > R(x) d(T nx, T nπx) ≤ηθn−R(x).
(b) P(R ≥k) ≤Cθk.
5.10. Coupling, separating, recovering. Here we describe an algo-rithm for constructing the coupling map π. This will be done recur-sively. Namely, given Γ1, Γ2 as in the Coupling Lemma we define a time of the first attempt k1 : Γ1 ∪Γ2 →N. The coupling map will be defined on a subset L1 ⊂Γ1 so that on L1 we have R(x) = k1. We will arrange that (5.13) |L1| ≥p.
In addition if L2 = πL1 then T k1(Γ1 −Li) = ∪j˜ Γij so that |˜ Γij| = ¯ δ and (5.14) PΓ1(k1 = k) = PΓ2(k1 = k) ≤Cθk.
Then we will try recursively to couple ˜ Γ1j to ˜ Γ2j and so on.
(5.13) shows that repeating the above procedure repeatedly we can define π alomost everywhere. Also (5.13) and (5.14) allow to apply Lemma 5.12 to get an exponential tail bound on the coupling time.
it remains to describe one step of our proceudre verifying (5.13) and (5.14). The coupling algorithm relies on the following estimate.
Lemma 5.15. If ˜ δ, ˜ η are sufficiently small than there is a constant N = N(˜ δ, ˜ η) such that for each pair Γ1, Γ2 with |Γ1| = |Γ2| = ¯ δ there are segments ˆ L1 ⊂Γ1, ˆ L2 ⊂Γ2 and ¯ N < N such that T ¯ N ˆ Li are horizontal segments of length ˜ δ, T ¯ N ˆ L2 is a vertical projection of T ¯ N ˆ L1 and the distance between T ¯ N ˆ L1 and T ¯ N ˆ L2 is less that ˜ η.
LECTURES ON BOUNCING BALLS.
51 During one run of our algorithm π will be defined on a subset L1 ⊂ ˆ L1. But first we explain how to define k1 on Γ1 −ˆ L1. We will use the following fact.
Lemma 5.16. Let G1, G2 be unions of horizontal segments of the same total length such that PGi(r(x) < ε) ≤Zε where PGi is the uniform distribution on Gi and r(x) is the distance from x to the boundary of the segment containing x. Then there is a function k : G1 ∪G2 →N such that T kGi = S j Gij, |Gij| = ¯ δ and PG1(k(x) = k) = PG2(k(x) = k) ≤Cθk−K ln Z.
The proof of Lemma 5.16 is similar to the proof of Lemma 5.11 so it is left to the reader.
Next we consider ˆ L1. We want to set π = T −¯ N ˜ π where ˜ π : T ¯ Nˆ l1 → T ¯ N ˆ L2 is a vertical projection. However, y and ˜ πy need not belong to the same stable manifold. The obstacle is existence of a number n such that T ny and T n˜ πy are separeted by a singularity. In that case d(T ny, ˜ S) ≤ηλ−N, d(T n˜ πy, ˜ S) ≤ηλ−n.
So at time n we remove the points falling into ηλ−n neighbourhood of ˜ S as well as its vertical projection. On the removed set we define separation time s(x) as follows. Consider a component of the set removed at time n. If this component is longer than λ−2n then we let s(x) to be equal to n on that component. Otherwise an endpoint b of this component has been removed at a erlier time and we let s = s(b) on this component. Note that by construction and (5.4)we have (5.15) P(s = k) ≤¯ C ¯ θk.
Also if ˜ η ≪˜ δ then (5.4) show sthat the set where π = T −¯ N ˜ π (that is the set of points which are not removed due to a close approach to the singular set) has a relatively large measure in L1 proving (5.13).
It remains to define k on L1 −ˆ L1. This will be done using Lemma 5.16 pairing the points having the same separation time. Note that by our construction all componets where s = ¯ s have length at least λ−2¯ s, so letting ρ(x) = k(x) −s(x) be the recovery time we get (5.16) P(ρ(x) > ks(x) + l) ≤¯ Cθl.
(5.15) and (5.16) give (5.14) for the separated points since P(k(x) > k) ≤P s(x) > k 2K + P ρ(x) ≥Ks(x) + k 2 52 LECTURES ON BOUNCING BALLS.
and both terms have exponentially small probability.
It remains to prove Lemma 5.15.
Proof of Lemma 5.15. We begin with a simplifying remark. The state-ment requires that N be uniform in Γ1, Γ2 but we note that it suffices to prove it for fixed Γ1, Γ2 (but for all sufficiently small ˜ δ, ˜ η). Indeed take ˜ ˜ δ > ˜ δ, ˜ ˜ η < ˜ η. Then if the statement holds for (Γ1, Γ2, ˜ ˜ δ, ˜ ˜ η) then it also holds for (Γ′ 1, Γ′ 2, ˜ δ, ˜ η) provided that (Γ′ 1, Γ′ 2) is sufficiently close to (Γ1, Γ2). Since the set of pairs is compact we can choose a finite subcover achived the required uniformity.
Let Σj = [ y∈Γj W s ˜ η/3(y).
Note that Σj has a positive measure in T2, hence Σ1×Σ2 has a positive measure in T2 × T2. By ergodicity of T × T given a set Ω∈T2 × T2 almost every point almost every point in Σ1×Σ2 visits Ωwith frequency mes(Ω). Let Ωk = {(x1, x2) : T2×T2 : d(x1, x2) ≤˜ η 3 and T −k is continuous in B(xi, 2˜ δ)}.
Note that if ˜ η, ˜ δ are small than mes(Ωk) ≥η2/10. By the foregoing discussion given ε there is N such that mes ((y1, y2) ∈Σ1 × Σ2 : Card(n ≤N : (T ny1, T ny2) ∈Ωk) ≤N mes(Ωk) 2 ) ≤ε Next by the growth lemma given ˆ ε there eixsts ˆ δ so small that mesΣ1×Σ2((y1, y2) : rn(x1) < ˆ δ or rn(x2) ≤ˆ δ) < ˆ εmes(Σ1 × Σ2) where xi = ˜ πyi. So if ˆ n(y1, y2) = Card(n ≤N : rn(x1) ≥ˆ δ and rn(x2) > ˆ δ) then E(ˆ n) ≥(1 −ˆ ε)N. Since ˆ n ≤N P ˆ n < N 1 −η2 100 ≤100ˆ ε η2 which can be made as small as we wish by choosing ˆ ε small. Note that if d(T ny1, T ny2) < ˜ η/3 then d(T nx1, T nx2) < ˜ η. Therefore given if N is sufficiently large and ε, ˆ ε are sufficiently small then there exist (x1, x2) ∈Σ1 × Σ2 and ¯ N < N such that d(T ¯ Nx1, T ¯ Nx2) < ˜ η, r ¯ N−k(xi) > ˆ δ and T −k is continuous on B(xi, δ) for i ∈{1, 2}. The continuity condition implies that r ¯ N(xi) ≥r ¯ N−kλk ≥˜ δ LECTURES ON BOUNCING BALLS.
53 if k is sufficiently large. Therefore Γ1 and Γ2 contain two segments such that their T ¯ N image has length at least 2˜ δ and the distance between their centers is at most ˜ η. If ˜ η < ˜ δ (which can be assumed without loss of the generality) we can trim those segments so that one is a vertical projection of the other proving the lemma.
□ 6. The Central Limit Theorem for Dynamical Systems.
6.1. Estimating error in Ergodic Theorem. If T is an ergodic map of a space M equipped with a probability measure µ then the Ergodic Theorem says that for φ ∈L1(µ) we have SN(x) N → Z M φ(x)dµ(x) where SN(x) = N−1 X n=0 φ(T nx).
The next natural question is the rate of convergence.
To formulate the question more precisely we need to recall some facts from probability theory. Let S be a Polish metric space, Sn be a sequnce of S valued random variables and S be an S valued random variable. We say that SN converges to S in distribution (written as SN ⇒S) if for any bounded continuous function Φ we have E(Φ(SN)) →E(Φ(S)).
In case S = Rd the following statements are equivalent • SN ⇒S • For each ξ ∈Rd E(eiξSN) →E(eiξS) • Define FS(s) = P(S(1) ≤s1, S(2) ≤s2, . . . S(d) ≤sd) and and let FSN(s) be a similar expression for SN then for all continuity points of FS we have limN→∞FSN(s) = FS(s).
Given a number σ > 0 let N (σ2) denote the normal random variable with zero mean and standard deviation σ. Thus P(N ≤s) = Z s −∞ 1 √ 2πσe−u2/2σ2du and E eiξN = e−σ2ξ2/2.
In case of independent random variables the fluctuations of ergodic sums are of order √ N and the limiting distribution is normal. One can ask if the same is true in the dynamical systems setting. In many cases where one has exponential divergence of nearby trajectories the answer is YES. However one needs to impose some regularity requirement on φ. Without smoothness assumptions, Exercise 5.4(b) shows that one a function φ such that E(S2 N) ≫N.
54 LECTURES ON BOUNCING BALLS.
Theorem 6.1. Let T be as in Theorem 5.1 and φ ∈Cα(T2) be a function with zero mean. Suppose that x is uniformly distributed on T2. Then SN √ N ⇒N (σ2) where (6.1) σ2 = ∞ X p=−∞ Z T2 φ(z)φ(T pz)dz.
In other words, P SN √ N ≤s → Z s −∞ 1 √ 2πσ e−u2/(2σ2)du.
The assumption that φ has zero mean does not cause any loss of generality since we can always replace φ by ¯ φ = φ − R T2 φ(z)dz.
6.2. Bernstein method. By the foregoing discussion we need to show that (6.2) E eiξSN/ √ N →e−σ2ξ2/2.
Before proving (6.2) for toral maps let us recall how (6.2) is es-tablished for independent random variables.
Namely, suppose for a moment that SN = PN−1 n=0 Xn where Xn are indendent identically dis-tributed random variables with zero mean and standard deviation σ.
We have E eiξSN/ √ N = h E eiξX/ √ NiN .
Next E eiξX/ √ N = E 1 + iξX √ N −ξ2X2 2N + o 1 N = 1 −ξ2σ2 2N + o 1 N .
Raising this expression to the N-th power we obtain (6.2) in the case of indendent identically distributed random variables.
In the dynamical system case this method can not be applied di-rectly since φ(T nx) are not independent for different n. However, mix-ing shows that φ(T Nx) and φ(T n+jx) are weakly dependent for large j. One useful technique for handling weakly dependent random vari-able is Bernstein big block small block method which we now describe.
Choose α1 < α2 < 1 4 and divide [0, N] into big blocks of size Nα2 sep-arated by small blocks of size Nα1 starting from a small block. Thus LECTURES ON BOUNCING BALLS.
55 we let ˜ ˜ Sj = Nα1j+Nα2(j−1) X n=(Nα1+Nα2)(j−1)+1 φ(T nx), ˜ Sj = (Nα1+Nα2)j X n=Nα1j+Nα2(j−1)+1 φ(T nx), ˜ S = X j ˜ Sj, ˜ ˜ S = X j ˜ ˜ Sj.
Thus ˜ ˜ S is the contribution of small blocks while ˜ S is the contribution of big blocks. The idea is that the contribution of the small blocks is negligible since the number of terms in ˜ ˜ S is O (N1−α2+α1) while the contribution of different big blocks is almost independent since the blocks are far apart.
Let us give the detailed argument.
First we dispose of ˜ ˜ S. We have E ˜ ˜ S 2 = ∗∗ X n1,n2 Z T2 φ(T n1x)φ(T n2x)dx = ∗∗ X n1,n2 Z T2 φ(x)φ(T n2−n1x)dx = ∗∗ X n1,n2 O θ|n2−n1| = O N1+α1−α2 where P∗∗denotes the sum over the small blocks. Thus E ˜ ˜ S √ N !2 = O Nα1−α2 and so ˜ ˜ S √ N ⇒0.
Exercise 6.2. If S′ N ⇒S, S′′ N ⇒0 then S′ N + S′′ N ⇒S.
Accordingly it is enough to prove the CLT for ˜ S. Due to the decom-position (5.12) we may assume that x is chosen uniformly on a segment Γ of length ¯ δ. Let mj be the center of j small block. Let Γjk denote the components of T mj+1Γjk. We have EΓ (ψ ◦T mj+1) = X k ckEΓjk(ψ) where ck = PΓ(T mj+1x ∈Γjk. Denote Qj = P(j) Φ(T nx) where P(j) means the sum over the j-th big block.
Condier the characterisitc fuctions γj(ξ) = EΓ exp j X l=1 iξQl √ N !!
for j > 1 and γ0(ξ) = 1.
56 LECTURES ON BOUNCING BALLS.
We have γj+1(ξ) = X k ckEΓjk exp " j X l=1 iξQl √ N # + iξQj+1 √ N !!
= ′ X k ckEΓjk exp " j X l=1 iξQl √ N # + iξQj+1 √ N !!
+ O N−100 where P′ k denotes the sum over the components with |Γjk| ≥N−100.
Choose xk ∈T −mj+1Γjk and let qk = Pj l=1 Ql(xk). Note that if ˜ xk ∈ T −mj+1Γjk then d(T mj+1xk, T mj+1˜ xk) ≤1 and so d(T nxk, T n˜ xk) ≤ 1 λmj+1−n ≤ 1 λNα1/2 .
Since φ ∈Cα(T2) we have (6.3) X l [Ql(˜ xk) −Ql(xk)] = O λ−αNα2 and so γj+1(ξ) = ′ X k ckeiξqk/ √ NEΓjk eiξQj+1/ √ N + O N−100 .
Now as in the independent case we can use a decomposition eiξQj+1/ √ N = 1 + iξQj+1 √ N −ξ2Q2 j+1 2N + O |Qj|3 N3/2 .
Lemma 6.3. (a) EΓjk(Qj+1) = O (N−100) .
(b) EΓjk(Q2 j+1) = Nα2σ2 + o(1).
Lemma 6.3 implies that (6.4) EΓjk eiξQj+1 = 1 −Nα2−1σ2ξ2 2 + O 1 N + O EΓjk Q3 j+1 N3/2 !!
.
Next EΓjk |Qj+1|3 ≤Nα2EΓjk |Qj+1|2 = O N2α2 so the last term in (6.4) is O N2α−3/2 and so it is negligible. Using again (5.4) and (6.3) we see that ′ X k ckeiξqk/ √ N = γj(ξ) + O N−100 LECTURES ON BOUNCING BALLS.
57 so that (6.5) γj+1(ξ) = γj(ξ) 1 −Nα2−1σ2ξ2 2 + O 1 N .
Iterating this relation N Nα1+Nα2 times we obtain (6.6) E eiξ ˜ S/ √ N ≈ 1 −Nα2−1σ2ξ2 2 N1−α2 ≈e−σ2ξ2/2 as needed.
Exercise 6.4. Show that (6.5) implies (6.6).
6.3. Moment estimates.
Proof of Lemma 6.3.
(a) EΓjk (Qj+1) = bj+1 X aj+1 EΓjk(φ ◦T n) = O ||φ||CαNα2θNα2 where [aj, bj] is the j-th big block.
(b) EΓjk Q2 j+1 = bj+1 X n1,n2=aj+1 EΓjk((φ ◦T n1)(φ ◦T n2)).
We begin with an a priori bound (6.7) EΓjk((φ ◦T n1)(φ ◦T n2)) = O θ|n2−n1| .
To check (6.7) we assume without the loss of the generality that n2 > n1. Let n3 = n1+n2 2 . We have EΓjk(ψ ◦T n3) = X s csEΓjks(ψ) where Γjks are the components of T n3Γjk. By the Growth Lemma we have EΓjk((φ ◦T n1)(φ ◦T n2)) = ′ X s csEΓjks((φ ◦T −r)(φ ◦T r)) + O(e−εr) where p = n2−n1 2 and P′ s means the sum over components which are longer than e−εr. Choosing xs ∈T −n3Γjks we get that on Γjks φ(T −px)φ(T px) = φs + O λ−αp φ(T px) where φs = φ(T −rxs). Thus EΓjk ((φ ◦T n1)(φ ◦T n2)) = ′ X s csφsEΓjks(φ ◦T p) + O e−εp 58 LECTURES ON BOUNCING BALLS.
and if εK < 1 then Proposition 5.10 gives EΓjks(φ ◦T p) = O(θp) com-pliting the proof (6.7). Now to finish the proof of part (b) it remains to show that for a fixed p (6.8) EΓjk((φ ◦T n)(φ ◦T n+p)) → Z T2 φ(z)φ(T pz)dz as n →∞. Let φp(x) = φ(x)φ(T px). The LHS of (6.8) is EΓjk(φp ◦ T n). Thus if φp were Holder then the result would follow directly from Proposition 5.10. However φp is not Holder since T p is not smooth.
Fortunately, φp can be well approximated by Holder functions. Given ε let φp,ε = 1 πε2 Z B(x,ε) φp(y)dy.
Then φp,ε is uniformly Lipshitz |φp,ε(x′) −φp,ε(x′′)| ≤C ε2d(x′, x′′).
On the other hand if T p is continuous on B(x, ε) then (6.9) |φp,ε(x) −φ(x)| ≤Krεα.
Thus EΓjk(φp ◦T n) = EΓjk(φp,ε ◦T n) + EΓjk([φp −φp,ε] ◦T n) = I + (II).
I = Z T2 φp,ε(z)dz + O θn ε2 since φp,ε is Lipshitz. Next, due to (6.9) we have (6.10) (II) ≤ p−1 X k=0 PΓjk(rn+k ≤ε) + Kpεα ≤Cpε + Kpεα.
Choose ε so that θn = ε3 then both error terms are small.
Combining (6.10) with decomposition (5.12) we see that Z T2 φp,ε(z)dz = Z T2 φp(z)dz + O(pε + εα) so (6.8) follows.
□ 6.4. The case of zero variance. Theorem 6.1 is only intersting if the variance given by formula (6.1) is non-zero. Indeed if σ2 is zero then the Theorem just says that √n is a wrong scaling for ergodic sums.
Here we show that σ2 = 0 only in exceptional cases. Namely we prove the following result.
Let Pk denote the partition into domains of continuity of T k and ˜ Pk = Pk ∨P−k.
LECTURES ON BOUNCING BALLS.
59 Proposition 6.5. Suppose that for some ¯ k A is Lipshitz on each el-ement of ˜ P¯ k and that σ2 = 0. Then there exists k0 and a function B which is Lipshitz on each element of ˜ Pk0 such that (6.11) φ(x) = B(x) −B(Tx).
Corollary 6.6. If there is a periodic point p of period N such that N−1 X n=0 φ(T np) ̸= 0 then σ2 > 0.
Proof. If σ2 = 0 then PN−1 n=0 φ(T np) = B(p) −B(T Np) = 0.
□ The proof of Proposition 6.5 consists of several steps.
Lemma 6.7. If σ2 = 0 then (6.11) admits an L2 solution.
Proof.
||Sn||2 L2 = n−1 X k1,k2=0 µ(φ(T k1x)A(T k2x)) = n−1 X k=0 σn,k where σn,k denotes the sum of the terms with the smallest index k.
Thus σn,k = n−1−k X j=−(n−1−k) µ(φ(x)φ(T jx)) = σ2 + O θn−k = O θn−k since σ2 = 0. Hence ||Sn||2 L2 = n−1 X k=0 σn,k = n−1 X k=0 O θn−k = O(1).
Therefore {Sn} is weakly precompact. Let B = w−lim Snj for some subsequence nj. Then B(x) = φ(x) + w−lim nj→∞Snj(Tx) −w−lim nj→∞φ(T njx).
By mixing for each H ∈L2 we have lim n→∞µ(H(φ ◦T n)) = µ(H)µ(µ) = 0.
That is w−limn→∞φ(T nx) = 0 and so B(x) = φ(x)+B(Tx) as claimed.
□ 60 LECTURES ON BOUNCING BALLS.
Lemma 6.8. (a) For almost all x1 ∈T2, for almost all x2 ∈W s(x1) we have (6.12) B(x2) −B(x1) = ∞ X n=0 [φ(T nx1) −φ(T nx2)].
(b) For almost all x1 ∈T2, for almost all x2 ∈W u(x1) we have (6.13) B(x2) −B(x1) = ∞ X n=1 [φ(T −nx1) −φ(T −nx2)].
Proof. We prove (a). The proof of (b) is similar.
We have B(x2) −B(x1) = N−1 X n=0 [φ(T nx1) −φ(T nx2)] −B(T Nx2) + B(T Nx1).
Given m we can choose ¯ Bm, ¯ ¯ Bm such that B = ¯ Bm + ¯ ¯ Bm, ¯ Bm ∈ C(T2), || ¯ ¯ Bm||L2 ≤2−m. Next choose εm such that if d(y1, y2) < εm then |B(y2) −B(y1)| < 2−m. Now take ¯ Nm such that λ−¯ Nm < εm. Set x2(t, x1) = x1 + tes. Then mes (t, x1) : |B(T ¯ Nmx1) −B(T ¯ Nmx2(t, x1))| < 9 10 m + 1 2 m ≤mes (t, x1) : | ¯ ¯ Bm(T ¯ Nmx1) −¯ ¯ Bm(T ¯ Nmx2(t, x1))| < 9 10 m ≤2 (1/2)m (9/10)2m = 2 50 81 m .
Therefore for almost all x1 ∈T2 for almost all x2 ∈W s(x1) there exists ¯ m such that for m > ¯ m |[B(x2) −B(x1)] − ¯ Nm−1 X n=0 [φ(T nx2) −φ(T nx1)]| ≤ 9 10 m + 1 2 m .
Taking m →∞we obtain the claimed result.
□ Let D = {x ∈T2 : for a. e. ¯ x ∈W s(x) (6.12) holds and for a. e. ¯ x ∈W u(x) (6.13) holds}.
˜ D = {x ∈D : for a.e. ¯ x ∈W s(x) ∪W u(x) ¯ x ∈D}.
Choose k0 so large that if Γ is a horizontal curve of length δ with d(Γ, ∂˜ Pk0) ≥3δ then mes(x ∈Γ : rs(x) ≥3δ) ≥0.99δ LECTURES ON BOUNCING BALLS.
61 and if Γ is a vertical curve of length δ with d(Γ, ∂˜ Pk0) ≥3δ then mes(x ∈Γ : ru(x) ≥3δ) ≥0.99δ.
Let Π be a rectangle containing in one element of ˜ Pk0. Divide Π into squares of size 1/N. Let DN be the set of points x in ˜ D ∩Π such that both W s(x) and W u(x) fully cross the square containing x. Consider two points x1, x2 ∈DN. Assume for a moment that x1, x2 belong to the same row and that the 1/N ≤d(x1, x2) ≤2/N. Let ΓN(xj) = [ y∈ˆ W s(xj)∩˜ D:ru(x)≥3/N (W u 3/N(y) ∩D), where ˆ W s(xj) is the part of W s(xj) containing inside our row. By our choice of k0, ΓN(x1) ∪ΓN(x2) has large measure. In particular, there exist yj ∈ˆ W s(xj)∩˜ D such that y2 ∈W u(y1), and (xj, yj) satisfy (6.12).
We claim that (y1, y2) satisfy (6.13). Indeed since yj ∈D there exists y3 such that (y1, y3) and (y3, y2) satisfy (6.13) but then (y1, y2) also satisfy (6.13). Since B(x2) −B(x1) = [B(x2) −B(y2)] + [B(y2) −B(y1)] + [B(y1) −B(x1)] we conclude from (6.12) and (6.13) that |B(x2)−B(x1)| ≤C/N. Taking arbitrary x1, x2 ∈DN which are in the same row we can find x1 = z0, z1, z2 . . . zl = x2, such that 1/N ≤|zj −zj−1| ≤2/N and l ≤ N|x2 −x1| + 1. Accordingly |B(x2) −B(x1)| ≤ l X j=1 |B(zj) −B(zj−1)| ≤Cl N ≤C(|x2 −x1| + 1 N ).
Hence if x1, x2 ∈D are in the same row and |x2 −x1| ≥1/N then (6.14) |B(x2) −B(x1)| ≤¯ C|x2 −x1|.
A similar conclusion holds if x1 and x2 at the same column. For general x1, x2 we can find z such that x1 and z are at the same row and x2 and z are at the same column and write B(x2) −B(x1) = [B(x2) −B(z)] + [B(z) −B(x1)] to conclude that (6.14) holds for arbitrary x1, x2 ∈Π ∩D with |x2 − x1| ≥1/N. Next if x1, x2 ∈˜ D then for large N we will have xj ∈DN and |x2 −x1| ≥1/N so (6.14) holds for all x1, x2 ∈˜ D ∩Π. In other words B can be modified on the set of measure 0, so that it becomes Lipshitz on Π and (6.11) holds almost everywhere.
Covering every element of ˜ Pk0 by rectangles we see that there exists a version of B which is Lipshitz on each element of ˜ Pk0 such that (6.11) holds almost 62 LECTURES ON BOUNCING BALLS.
everywhere. But then by continuity it should hold everywhere. The proof of Proposition 6.5 is complete.
6.5. Convergence to Brownian Motion. Theorem 6.1 concerns the distribution of Sn for fixed n. Sometimes we would like to know a joint distribution of several Snj at the same time, for example, we may wish to compute P(maxn≤N Sn ≤L). Such questions are addressed by Functional Central Limit Theorem. Recall that a random process B(t) is called a Brownian Motion with variance parameter σ2 if B has continuous paths, B(0) = 0 and given 0 = t0 < t1 < t2 · · · < tk,the increments B(tj+1) −B(tj) have normal distribution with zero mean and varaince σ2(tj+1 −tj) and are independent of each other.
In the setting of Theorem 6.1 let BN(t) = SNt √ N if Nt is integer with linear interpolation in between.
Theorem 6.9. As N →∞, BN(t) converges to the Brownian Motion with variance parameter σ2 given by (6.1).
According to to show that BN(t) ⇒B(t) we need to check two things.
First, we have to establish the convergence of finite dimensional dis-tributions. That is, for each 0 = t0 < t1 < t2 < . . . tk (B(t1) −B(t0), B(t2) −B(t1), . . . B(tk) −B(tk−1)) ⇒(N1, N2 . . . Nk) where N1, N2 . . . Nk are indepepndent normal random variables with zero means and variances σ2(t1 −t0), σ2(t2 −t1), . . . , σ2(tk −tk−1).
Second, we need to prove tightness, that is, to show that for each ε > 0 there exists a compact set Kε ⊂C[0, 1] such that P(BN(t) ∈ Kε) > 1 −ε for large N.
Let Nj = Ntj. To check the convergence of finite dimensional distri-butions we need to show that (6.15) E exp "X m ξm(SNm −SNm−1) #!
→exp " −σ2 2 X m ξ2 m(tm −tm−1) # .
But the proof of (6.15) is similar to (6.6) so it can be left to the reader.
To prove tightness we need an auxillary result.
Lemma 6.10.
E((SN2 −SN1)4) ≤L(n2 −n1)2.
Proof. We have (SN2 −SN1)4) = N2 X n1,n2,n3,n4=N1 φ(T n1x)φ(T n2x)φ(T n3x)φ(T n4x).
LECTURES ON BOUNCING BALLS.
63 To estimate the expectation of the above expression we may assume without the loss of the generality that n1 ≤n2 ≤n3 ≤n4. We claim that (6.16) E(φ(T n1x)φ(T n2x)φ(T n3x)φ(T n4x)) = O(θp) where p = max(n4 −n3, n2 −n1). Indeed if p = n3 −n3 then the proof of (6.16) is similar to the proof of (6.7). If p = n2 −n1 then we use the same argument for T −1.
□ Define KL = ψ ∈C[0, 1] : ψ(0) = 0 and for all l ≥L, k ≤2l : ψ k + 1 2l −ψ k 2l < 2−l/8 Exercise 6.11. Show that KL is compact in C[0, 1].
Thus it reamins to show that if L is large than P(BN ∈KL) is close to 1. Let nk,l(N) = kN 2l . Then P BN k + 1 2l −BN k 2l ≥2−l/8 = P Snk+1,l(N) −Snk,l(N) ≥ √ N2−l/8 ≤ E Snk+1,l(N) −Snk,l(N) 4 N2/2l/2 ≤C N2/22l N2/2l/2 = C2−3l/2.
Accordingly P ∃k ≤2l : BN k + 1 2l −BN k 2l ≥2−l/8 ≤C2−l/2 and P ∃l ≥L, ∃k ≤2l : BN k + 1 2l −BN k 2l ≥2−l/8 ≤˜ C2−l/2 proving tightness. This completes the proof of Theorem 6.9.
7. Invariant comes and hyperbolicity.
7.1. Dimension 2. In Sections 5 and 6 we saw that in order to ensure strong stochasticity we need to construct a cone family K(x) such that this family is invariant: d f(K(x)) ⊂K(x) and d f expands the cones, that is, there is a constant λ > 1 such that for all v ∈K(x) we have ||d f(v)|| ≥λ||v||. Here we shall show that in the area preserving case the mere existence of invariant comes implies expansion.
We begin with the following elementary fact.
Lemma 7.1. If L ∈SL2(R) has positive elements then it is hyperbolic.
64 LECTURES ON BOUNCING BALLS.
This result is quite intuitive. If L has positive elements then the angle between Le1 and Le2 is less than π 2 and since due to area preservation ||Le1||||Le2|| sin ∠(Le1, Le2) = 1 there should be some expansion. The analytic prove is also easy. If L = a b c d then ad = 1 + bc > 1 and so a + d ≥2 √ ad > 2.
The above proof does not show where the expanding direction is located. There is another argument which is equally simple but has an advantage that it works for products of different matrices. This argument is based on the classical notion of Lyapunov function. Let φ0 be the angle which vector (x0, y0) makes with x axis and φ1 be the angle which vector (x1, y1) = L(x0, y0) makes with x axis. Then φ1 = g(φ0) for a continuous function g satisfying 0 < g(0) < g( π 2) < π 2.
By the intermediate value theorem there exists φ such that g(φ) = φ and hence (x1, y1) = λ(x0, y0). To estimate λ let Q(x, y) = xy. Then Q(x1, y1) = λ2x0y0 = x1y1 = acx2 0 + bdy2 0 + (ad + bc)x0y0 > (ad + bc)x0y0 = (1 + 2b0c0)x0y0.
It follows that λ > √ 1 + 2bc > 1.
The previous argument shows that Q increases after the applica-tion of L. The same argument works for compositions.
Namely, if L1, L2 . . . Ln are positive SL2(R) matrices and vn = Ln . . . L2L1v0 then ||vn|| ≥2 p Q(vn) ≥2Q(v0) n Y j=1 Λj where Λj = (1 + 2bjcj).
To get a coordinate free interpretation of this result suppose that f : M2 →M2 preserves a smooth measure given by µ(A) = RR A ω and that there is a family of cones K(x) such that along an orbit xn = f nx0 we have d f(K(xn)) ⊂Kn+1. Choose a basis in TxM so that K(x) = {e = α1e1 + α2e2 : α1 > 0 and α2 > 0} and ω(e1, e2) = 1. Then d f can be represented by an SL2(R) matrix and by the above inequality we have ||d f n(v0)|| ≥2 p Q(v0) Qn−1 j=0 Λj where Λj = 1 + 2bjcj.
LECTURES ON BOUNCING BALLS.
65 7.2. Higher dimensions. Here we present a multidimensional version of this estimate which is due to Wojtkowski. Consider a symplectic space (R2d, ω). Let V1 and V2 be two transversal Lagrangian subspaces (ω|Vj = 0). Than each vector v ∈R2d has a unique decomposition v = v1 + v2, vj ∈Vj. Let Q(v) = ω(v1, v2). We can choose frames in V1 and V2 so that if u1 = (ξ1, η1), u2 = (ξ2, η2) where ξj ∈V1, ηj ∈V2 then ω(v1, v2) = ⟨ξ1, η2⟩−⟨ξ2, η1⟩. Then Q((ξ, η)) = ⟨ξ, η⟩. Define K = {v : Q(v) ≥0}.
Let L be a symplectic matrix. We can write L in the block form with respect to the decomposition R2d = V1 ⊕V2 as L = A B C D .
The symplecticity condition amounts to the equations A∗D −C∗B = I, A∗C = C∗A, D∗B = B∗D.
One important case is ˜ L = I R P C . Then we have P ∗= P, R∗S = S∗R and S −PR = I.
The last two equations give R∗S −R∗PR = R∗ that is (S∗−R∗P)R = R∗.
But S∗−R∗P = (S −PR)∗= I. Therefore the symplecticity of ˜ L amounts to (7.1) R∗= R, P ∗= P, S −PR = I.
We say that L is monotone if LK ⊂K and strictly monotone if LK ⊂ Int(K) ∪{0}.
Lemma 7.2. If L is monotone then LV1 is transversal to V2 and LV2 is transversal to V1.
Proof. Suppose to the contrary that there is 0 ̸= v1 such that Lv1 ∈V2.
Take v2 ∈V2 such that ω(v1, v2) > 0. We have ω(v1, v2) = ω(Lv1, Lv2) = ω(Cv1, Bv2).
Take vε −v1 + εv2. Then vε ∈K for ε > 0. On the hand Q(Lvε) = ⟨εBv2, Cv1 + εDV2⟩= −εω(v1, v2) + ε2ω(Bv2, Dv2) is negative for small positive ε giving a contradiction.
□ Lemma 7.2 implies that A is invertible, so we can consider ˆ L = A 0 0 (A∗)−1 . Note that ˆ L preserves Q since Q(ˆ Lv) = ⟨Aξ, (A∗)−1η⟩= Q(v).
66 LECTURES ON BOUNCING BALLS.
We shall use a decomposition L = ˆ L˜ L where ˜ L = I R P A∗D for some matrices P and R.
Theorem 7.3. L is monotone iffQ(Lv) ≥Q(v) for all v ∈R2d.
L is strictly monotone iffQ(Lv) > Q(v) for all 0 ̸= v ∈R2d.
Proof. We prove the first statement, the second is similar.
Clearly, if L increases Q and v ∈K then Q(Lv) ≥Q(v), so Lv ∈K.
Conversely, suppose K is monotone. Since ˆ L preserves Q, we need to show that Q(˜ Lv) ≥Q(v). Due to (7.1) we have ˜ L(ξ, η) = (ξ + Rη, Pξ + η + PRη) so (7.2) Q(˜ L(ξ, η)) −Q(ξ, η) = ⟨Rη, η⟩+ ⟨Pζ, ζ⟩ where ζ = ξ + Rη. Since Q(˜ L(ξ, 0)) = ⟨Pξ, ξ⟩so P ≥0. Our next goal is to show that R ≥0. To this end consider an eigenvector Rη = λη.
Take ξ = aη. Then (ξ, η) ∈K if a > 0. On the other hand Q(˜ L(ξ, η)) = (a + λ)⟨η, η⟩+ (a + λ)2⟨Pη, η⟩.
Therefore Q(˜ L(ξ, η)) < 0 for a = −λ −ε. Hence −λ < 0, that is λ > 0.
This proves that R ≥0. Now (7.2) gives Q(˜ L(ξ, η)) ≥Q((ξ, η)) as claimed.
□ This proves shows in particular that if L is monotone then it is strictly monotone iffP > 0 and R > 0, that is, iffL(Vj) ⊂Int(K)∪{0}.
Next let L1, L2 . . . Ln be a sequence of monotone maps. Pick c so that ||v|| ≥c p Q(v). Let vn = Ln . . . L2L1v0. Then for v0 ∈Int(K) we have ||vn|| ≥c p Q(vn) ≥c p Q(v0) n Y j=1 Λj where Λj = Λ(Lj) and Λ(L) = minv∈Int(K) q Q(Lv) Q(v) .
To compute Λ(L) we shall use a decomposition R−1/2 0 0 R1/2 I R P I + PR R1/2 0 0 R−1/2 = I I K I + K where K = R1/2PR1/2 = R1/2(PR)R−1/2. Note that PR = A∗D −I = C∗B. Choose an orthogonal matrix F such that F −1KF is diagonal.
Then (7.3) F −1 0 0 F −1 I I K I + K F 0 0 F = I I T I + T LECTURES ON BOUNCING BALLS.
67 where T = F −1KF is diagonal and Sp(T) =Sp(C∗B). We can also assume by choosing F appropriately that the diagonal elements of T are increasing. Denoting by M the RHS of (7.3) we have Λ(M) = Λ(L).
On the other hand Q(Mv) = ⟨ξ, η⟩+ ⟨η, η⟩+ ⟨T(ξ + η), (ξ + η)⟩ = d X j=1 tjξ2 2 + (1 + 2tj)ξjηj + (1 + tj)η2 j X ηj≥0 h p tjξj − p 1 + tjηj 2 p 1 + tj + p tj 2 ξjηj i + X ηj<0 h p tjξj + p 1 + tjηj 2 p 1 + tj − p tj 2 ξjηj i ≥m(L) X j ξjηj = m(L)Q(v) where m(L) = min j ( p 1 + tj − p tj)2 = ( √ 1 + t1 −√t1)2 and t1 ≤t2 ≤· · · ≤td are the eigenvalues of T. The equality is achieved if ξj = ηj = 0 for j ≥2 and √t1ξ1 = √1 + t1η1.
Next, suppose that f : M →M is a symplectic map and there is a transverse family of Lagrangian subspaces V1(x), V2(x) and an orbit xn = f nx such that d f(K(xn)) ⊂K(xn+1) where K(x) are the conses associated with the pair (V1(x), V2(x)). Let Q be the associated qua-dratic form and take small c so that ||v|| ≥c p Q(v). Choose frames so that ω((ξ1, η1)(ξ2, η2)) = ⟨ξ1, η2⟩−⟨ξ2, η1⟩.
Let d f : TxM →TfxM have block form d f = A(x) B(x) C(x) D(x) . Let (7.4) Λ(x) = min t∈Sp(C∗B)( √ t + √ 1 + t).
Then for x ∈K(x0) we have (7.5) ||d f n(v0)|| ≥c n−1 Y j=0 Λ(xj) !
p Q(v0).
68 LECTURES ON BOUNCING BALLS.
7.3. Lyapunov exponents. Now we pass from the individual orbits to typical ones. Recall that given a diffeomorphism f : M →M, a point x and a vector v in Tx, one can define the forward and backward Lyapunov exponents λ±(x, v) = lim n→±∞ 1 n ln ||d f n(x)(v)||.
If f preserves a probability measure µ then, by Multiplicative Ergodic Theorem, for µ−almost all x λ±(x, v) exist for all v and they can take at most dim(M) different values.
In fact, there exists a splitting TxM = ⊕s j=1Ej and numbers λ1 > λ2 > λs such that if v = vi1 + vi2 + . . . vik where i1 < i2 < · · · < ik and 0 ̸= vik ∈Eik then λ+(x, v) = λi1 and λ−(x, v) = λik. If µ is ergodic then λj are constant almost surely.
In case µ is a smooth measure and λj ̸= 0 almost surely (in which case we say that the system has non-zero Lyapunov exponents or that it is (nonuniformly) hyperbolic) there are strong methods to control the statistical properties of f. In particular Pesin theory guarantees the ex-istence of stable and unstable manifolds tangent to E−= ⊕λj<0Ej and E+ = ⊕λj>0Ej respectively. (Pesin theory was extended to systems with singularities by Katok-Strelcyn . The main idea is to show that most orbits do not come to close to the singularities in the spirit of Lemma 5.6 of Section 5.) Also taking Σ(x) = ∪y∈W u(x)W s(x) we obtain a set of positive measure and if x ∈R2 then almost all points in Σ(x) have the same averages for all continuous functions. There-fore the systems with non-zero exponents has almost countable many ergodic components, that is M is a disjoint union M = ∪Bj where Bj are invariant and f restricted to Bj is ergodic. In case they hyperbol-icity comes from invariant cones as we describe below Chernov-Sinai-Wojtkowski-Liverani theory provides sufficient conditions for ergodic-ity. Namely one needs to ensure appropriate transversality conditions between the singularity manifolds and stable/unstable manifolds of f.
Unfortunately those transversality conditions are not easy to verify in practise so the ergodicity is not yet proved in all the examples where we can ensure nonzero exponents.
Returning to the computations of the Lyapunov exponents let us consider the setting of 2d dimensional symplectic manifold.
In this case one can show that (Ej)⊥= P i̸=s−j Ei and so dim(Ej) = dim Es−j.
Therefore in order to prove that the system has nonzero exponents it suffices to check that (7.6) dim(E+) ≥d.
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69 Suppose now that at each point there are transversal Lagrangian subspaces V1(x), V2(x) such that d f is monotone with respect to the cone KV1,V2. Let Λ(x) be defined by (7.4). In order to establish (7.6) we consider the smallest j such that dim(E− j ) > d where E− j = Ej ⊕Ej+1 ⊕Es.
Lemma 7.4. If µ is ergodic then λj ≥ R ln Λ(x)dµ(x).
Proof. Let D = {(ξ, ξ)}ξ∈Rd where we use the coordinates of Theo-rem 7.3. Then E− j ∩D contains a nonzero vector v. For this vector λ+(x, v) ≤λj. On the other hand in view of (7.5) and the Pointwise Ergodic Theorem we have λ+(x, v) ≥lim n→∞ 1 n X j ln Λ(f jx) = Z ln Λ(x)dµ(x).
□ In general it is possible to have Λ(x) ≡1 (consider for example the map (I, φ) →(I, φ + I)). Let G = {x : Λ(x) > 1} = {x : d f(x) is strictly monotone}.
Consider now the smooth invariant measure µ(A) = Z A ω ∧· · · ∧ω.
Note that µ need not be ergodic.
Corollary 7.5. If almost all points visit G then the system has nonzero Lyapunov exponents.
Proof. We apply Lemma 7.4 to each ergodic component of G. The assumption that ν(G) > 0 for each ergodic component implies that R ln Λ(x)dν > 0.
□ 7.4. Examples. Here we present several examples of systems possess-ing invariant cones.
We discuss two dimensional examples in more detail since the computations are simpler in that case.
(I) Dispersing billiards. Consider a particle moving in a domain with piecewise concave boundaries. Let s be the arclenth parameter and φ be the angle with the tangent direction.
Lemma 7.6. d f has the following form in (s, φ) variables κ0τ + sin φ0 sin φ1 τ sin φ1 κ0κ1τ + κ1 sin φ0 + κ0 sin φ1 sin φ1 κ1τ + sin φ1 sin φ1 70 LECTURES ON BOUNCING BALLS.
Figure 18. Two tables with nonzero Lyapunov expo-nents: dispersing billiard on the left and Bunimovich stadium on the right where κ0 (κ1) is the curvature of the boundary at the initial (final) point and τ is the flight length.
Note that f preserves the form ω = sin φds ∧dφ. The above matrix has all elements positive therefore d f increases the quadratic form Q = sin φdsdφ. Moreover the product of the offdiagonal terms with sin φ1 is uniformly bounded from below so Λ(s, φ) is uniformly bounded away from 1.
Proof. We compute ∂s1 ∂s0, the other terms are similar. Consider figure 19. Let |AB| = δs0. We have |CB| ≈sin φ0δs0, |DE| = |BC|, |EF| ≈τ sin ∠FBE, ∠BFE ≈κ0δs0, |DG| ≈δs1 ≈|DF| sin φ1 .
□ For dispersing billiards we have κ0 > 0, κ1 > 0. Another way to make all elements of d f positive is to have κ0, κ1 negative but require that τ ≥sin φ0 |κ0| + sin φ0 |κ0| .
The billiards satisfying the above condition are called defocusing. Per-haps the most famous example of the defocusing billiard is Bunimovich stadium.
Ergodicity of dispersing billiards is shown in . Ergodicity of Buni-movich stadium is shown in . Further properties of dispersing and defocusing billiards are discussed in .
(II) Dispersing pingpongs. Consider pingpong whose wall motion satisfies ¨ f(t) < 0 at all points of continuity.
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71 G A B C D E F v(A) v(B) Figure 19. Computing ∂s1 ∂s0 Lemma 7.7. In (t, v) variables the derivative takes form vn −˙ fn vn + ˙ fn+1 − Ln v2 n(vn + ˙ fn+1) vn −˙ fn vn + ˙ fn+1 ¨ fn+1 1 − Ln ¨ fn+1 v2 n(vn + ˙ fn+1) where Ln is the distance traversed by the particle between n-th and (n + 1)-st collisions.
Note that the offdiagonal entries of the above matrix are negative so the form Q = −dtdv is increasing.
72 LECTURES ON BOUNCING BALLS.
Proof. Let us compute ∂vn+1 ∂tn . Referring to figure 6 we have δhn = (vn −˙ fn)δtn, δtn+1 = δhn vn + ˙ fn , δ ˙ fn+1 = ¨ fn+1δtn+1.
This proves the formula for ∂vn+1 ∂tn . Together with (1.5) this completes the estimate of t derivatives. v derivatives are computed similarly.
□ (III) Balls in gravity field. Consider two balls on the line moving in a gravity field and colliding elastically with each other and the fixed floor. Let m1 be the mass of the bottom ball and m2 be the mass of the top ball. It is convenient to use h and z as variables where h = h1 is the energy of the bottom ball and z = v2 −v1 is the relative velocity of the second ball. We consider the balls at the moments when the bottom particle collides with the floor. During the collisions of the bottom ball with the floor our variables change as follows (¯ h, ¯ z) = F1(h, z) where F1(h, z) = (h, z + c √ h) and c = r 8 m1 .
Next we consider the collision between the walls. Using the formulas of Section 1 we find that the changes of energy and velocity are the following ¯ z = −z, ¯ v = u + 2m2 m1 + m2 z where u is velocity of the first ball at the moment of collision. Accord-ingly ¯ h = h + 2m1m2uz m1 + m2 + 2m1m2 2z2 (m1 + m2)2.
To find u note that u = v1 −τg where τ is the time between collisions of the first ball with the floor and with the second ball. Next, τ = −x z where x is the height of the second ball when the first one hits the floor. Therefore uz = v1z + gx. The energy of the system is E = h+ m2(v1 + z)2 2 +m2gx.
Thus v1z +gx = E m2 −h m1 −h m2 −z2 2 .
Accordingly ¯ h = b −h −az2 where b = 2m1E m1+m2 and a = m1m2 m1 + m2 − 2m1m2 2 (m1 + m2)2.
Therefore if the ball returns to the floor after the collision we have (¯ h, ¯ z) = F1 ◦F2 where F2(h, z) = (b −h −az2, −z).
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73 We assume that m1 > m2 so that a > 0. Note that dF1 = 1 0 c 2 √ h 1 , dF2 = −1 −2az 0 −1 = −I × 1 2az 0 1 .
Both 1 0 c 2 √ h 1 , and 1 2az 0 1 have positive elements so they are monotone with respect to Q = dhdz while −I is Q-isometry. Also note that d(F2 ◦F k 1 ) is strictly monotone for each k and since starting from any initial condition we will eventu-ally have a collision between the balls, Corollary 7.5 implies that this system has nonzero Lyapunov exponents.
Ergodicity of two balls in gravity under the condition m1 > m2 is proved in .
On the other hand if m1 = m2 then the particles just exchange their energy during the collisions so the function I = min(h1, h2) is the first integral of this system. One can also show that for m1 < m2 elliptic islands are present so the system is not ergodic.
One can also construct multidimensional examples satisfying the above criteria.
In particular n particles of the line in gravity field have nonzero exponents provided that m1 > m2 > · · · > mn when the particles are numbered from the bottom up. The monotonicity of this system was proved in while showed that the conditions of Corollary 7.5 are satisfied for this system. One can also consider non-linear potentials. shows that the following conditions are sufficient for nonzero Lyapunov exponents (i) m1 > m2 > · · · > mn; (ii) U′(q) > 0; (iii) U′′(q) < 0.
Figure 20. Wojtkowski wedge Another example is the particle in gravity field moving in a two dimensional domain whose boundary consists of two concave broken lines meeting at a right angle. It is shown in that this system has nonzero Lyapunov exponents.
Problem 7.8. Show ergodicity of the last two examples.
74 LECTURES ON BOUNCING BALLS.
References G. Atkinson Recurrence of co-cycles and random walks, J. London Math. Soc.
13 (1976) 486–488.
P. Billingsley Convergence of probability measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968 xii+253 pp.
P. Boyland Dual billiards, twist maps and impact oscillators, Nonlinearity 9 (1996) 1411–1438.
L. A. Bunimovich On the ergodic properties of nowhere dispersing billiards, Comm. Math. Phys. 65 (1979) 295–312.
J. Cheng, M. Wojtkowski Linear stability of a periodic orbit in the system of falling balls, T MSRI Publ. 22 (1991) 53–71.
N. Chernov Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2-torus, J. Statist. Phys. 69 (1992) 111–134.
N. Chernov, R. Markarian Chaotic billiards, Math. Surv. & Monographs 127 (2006) AMS, Providence, RI. xii+316 pp.
J. de Simoi Stability and instability results in a model of Fermi acceleration, Discrete Contin. Dyn. Syst. 25 (2009) 719–750.
J. de Simoi Fermi Acceleration in anti-integrable limits of the standard map, arXiv:1204.5667.
J. de Simoi, D. Dolgopyat Dynamics of some piecewise smooth Fermi-Ulam models, Chaos 22 (2012) paper 026124.
D. Dolgopyat Bouncing balls in non-linear potentials, Discrete & Continuous Dyn. Sys.– A 22 (2008) 165–182 D. Dolgopyat Limit Theorems for Hyperbolic Systems, Lecture Notes.
D. Dolgopyat, B. Fayad Unbounded orbits for semicircular outer billiard, Ann.
Henri Poincar´ e, 10 (2009) 357–375.
J. Guckenheimer, P. Holmes Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sciences 42 (1990) Springer-Verlag, New York, xvi+459 pp.
P. Holmes The dynamics of repeated impacts with a sinusoidally vibrating table, J. Sound Vibration 84 (1982) 173–189.
A. Katok, J.-M. Strelcyn Invariant manifolds, entropy and billiards; smooth maps with singularities, Lect. Notes in Math. 1222 (1986) Springer-Verlag, Berlin.
C. Liverani, M. Wojtkowski Ergodicity in Hamiltonian Systems, Dynamics Reported 4 (1995) 130–202.
J. Moser Stable and random motions in dynamical systems, Ann. Math. Stud-ies 77 (1973) Princeton University Press, Princeton, N. J. viii+198 pp.
Ortega R. Asymmetric oscillators and twist mappings, J. London Math. Soc.
53 (1996), no. 2, 325–342.
L. D. Pustylnikov Stable and oscillating motions in nonautonomous dynamical systems–II, Proc. Moscow Math. Soc. 34 (1977), 3–103.
L. D. Pustylnikov Poincare models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism, Russian Math. Surveys 50 (1995) 145–189.
M. Shub Global stability of dynamical systems, Springer-Verlag, New York, 1987. xii+150 pp.
LECTURES ON BOUNCING BALLS.
75 N. Simanyi The characteristic exponents of the falling ball model, Comm.
Math. Phys. 182 (1996) 457–468.
Ya. G. Sinai Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Russ. Math. Surv. 25 (1970) 141–192.
S. Tabachnikov Geometry and billiards, Student Math Library 30 (2005) AMS Providence RI.
M. Wojtkowski A system of one dimensional balls with gravity, Comm. Math.
Phys. 126 (1990) 507–533.
M. Wojtkowski A system of one dimensional balls in external field, Comm.
Math. Phys. 127 (1990) 425–432.
M. Wojtkowski Hamiltonian systems with linear potential and elastic con-straints, Fund. Math. 157 (1998) 305–341. |
12049 | https://www.geogebra.org/m/FS9QqXaB | SSA ambiguous case – GeoGebra
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SSA ambiguous case
Author:jriggins
Rotate the endpoints of the 3 options for side a of this triangle to see the triangles that are possible with the given lengths. To change the lengths of side a or b slide the a_1 or b_1 slider. To change the angle move point C The length of a_0 is a critical length to determine whether one or two triangles can be formed. How can the length of a_0 be found? How is this diagram calculating it?
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12050 | https://cad.onshape.com/help/Content/cplane.htm | Plane
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Part Studios>Feature Tools>Plane
Plane
Create a new construction plane. Note that you can use the context menu for a plane to turn Section view on, more information below.
Plane: DesktopCopy link
Video example:OffsetCopy link
Video transcriptCopy link
The Plane feature allows you to create a new construction plane. Click the Plane feature tool on the toolbar. Select an entity on which to base the new plane. By default, Offset the plane from a planar face, Mate connector, or another plane by a specified distance numerically in the dialog, or using the orange arrow in the graphics area. Click the Opposite direction arrow to flip the plane's direction. Check Flip normal to reverse the plane direction relative to the original geometry. Click the green checkmark to accept the new plane.
Video example: Non-offset optionsCopy link
Video transcriptCopy link
This video explores creating planes using non-offset plane options. Click the Plane feature tool on the toolbar. Select Plane point. Select a plane and a point. The new plane passes through the point, parallel to the plane. Select Line angle. This creates a plane that passes through a line at an angle. Use additional reference geometry as the basis for the angle, enter the Angle value in the dialog, or alter the angle using the orange arrow. Click the Opposite direction arrow to flip the angle direction. Select Point Normal. Select a line and a point. The plane passes through the point and is normal to the line. Select Three point. Then select three points in the graphics area. The plane passes through all three points. Select Mid plane. Select two planes in the graphics area. The plane is created at the intersection of both planes. Check Flip alignment to flip the new plane's alignment. Finally, select Curve point. Select the curve defining the normal of the plane, and a point, defining the origin of the plane. The plane normal is always tangent to the curve.
StepsCopy link
Click .
Select the entity or entities on which to base the new plane.
Click the Plane type dropdown and select an option. See the various sections below for examples of each plane type.
Make further specifications where necessary.
Click .
You can create a plane based on the relative position to another entity, including:
Plane - Select another plane, planar face, or Mate connector (implicit or explicit).
Point - Select a vertex, sketch point, Mate connector (implicit or explicit), or the origin
Line - Select a linear edge, sketch line, or cylindrical face to get its axis
Onshape remembers the selection for the plane type and opens the dialog with the last selected option in subsequent operations.
Note that pre-selecting a planar face (solid or plane) and creating a plane defaults to Offset plane.
Once you select the geometry with which to create the plane, you can adjust the normal of the plane to be one direction or the opposite:either check the Flip normal box, or leave it unchecked. You can see which direction is normal by looking at the name of the plane in the graphics area.
The image above shows the plane normal to the normal of the selected face.
The image above shows the plane with the normal flipped (the box is checked), and you can see by the name on the plane that the normal is reversed.
OffsetCopy link
Create a plane a specified distance from another plane (or the XY plane of a Mate connector, implicit or explicit) using a plane and a distance value.
Offset from a planar face:
Offset from a Mate connector:
Use the manipulator to drag the new plane to the desired distance; the numeric distance field in the dialog automatically updates. Click the manipulator arrow to flip the direction.
Offset from another plane:
Plane PointCopy link
Create a plane that passes through a point, parallel to a plane, using a plane and a point:
Line AngleCopy link
Create a plane that passes through a line at an angle, using a line, reference geometry (such as a plane, point, or axis) and an angle value:
Use the manipulator to drag the angle specification; see that the Angle field in the dialog automatically updates. Click the manipulator arrow to flip the direction.
Point NormalCopy link
Create a plane that is normal to the line and passes through the point, using: a straight axis (a straight line segment or anything that defines an axis (circle, arc, cylindrical face, revolved face, etc) and a point (vertex). The point is always the origin of the plane and the axis or line is always the normal of the plane:
Note that pre-selecting a line and a point and creating a plane defaults to Point normal plane.
Three PointCopy link
Create a plane that passes through three points, using three points.
The starting sketches, on two planes:
The resulting third plane:
Mid PlaneCopy link
Create a plane at the intersection of two other planes:
Flip alignment:
You can also create a mid plane that is equally centered between two parallel parent planes:
Flip normal:
Curve PointCopy link
Create a curve point plane that passes through the point, perpendicular to the curve. Use one curve (or edge) defining the normal of the plane and one point (Mate connector or vertex) defining the origin of the plane. The plane normal is always tangent to the curve:
TangentCopy link
Create a plane tangent to the face of a cylinder. Select the face of the cylinder to create the plane tangent to, and then select a point (to be coplanar with the tangent plane), or a plane (to be parallel to the tangent plane).
TipsCopy link
Use the keyboard shortcut, p, to hide/unhide all planes.
Once the plane is created, select it then use the context menu (RMB) and select Section view:
Click the X in the dialog to exit the Section view.
Plane: iOSCopy link
StepsCopy link
Tap Plane tool.
Select entities on which to base the new plane.
Select a plane type (see the subsections below for more information on each type).
Onshape remembers the selection for the plane type and opens the dialog with the last selected option in subsequent operations.
Specify the distance or angle (depending on which plane type you have selected).
Optionally, toggle to switch to the opposite direction.
Optionally, toggle the Flip normal switch to adjust the normal of the plane to be in the opposite direction. You can see which direction is normal by looking at the name of the plane in the graphics area.
Tap the checkmark.
OffsetCopy link
Create a plane a specified distance from another plane using a plane and a distance value. You are able to offset from a planar face or from another plane or Mate connector (implicit or explicit).
Select entity on which to base the new plane and specify the distance.
Plane PointCopy link
Create a plane that passes through a point, parallel to a plane, using a plane and a point (or Mate connector, implicit or explicit).
Select a plane for the new plane to remain parallel with and a point for the new plane to pass through
Line AngleCopy link
Create a plane that passes through a line at an angle, using a line and an angle value.
Select a line for the plane to pass through and specify the angle of the plane. Select an additional point, plane or axis to specify where the angle is to be measured from.
Point NormalCopy link
Create a plane that is normal to the line and passed through the point, using:a straight axis (a straight line segment or anything that defines an axis (circle, arc, cylindrical face, revolved face, etc.)and a point. The point is always the origin of the plane and the axis or line is always the normal of the plane.
Select a line for the plane to pass through and a point for the plane to be normal to.
As shown in the image above, the plane is normal to the point on the corner of the sketch on the Right plane and it is passing through the selected line.
Three PointCopy link
Create a plane that passes through three points, using three points.
Select the three points for the plane to pass through.
As shown in the image below, the plane passes through all three points:
Mid PlaneCopy link
Create a plane at the intersection of two other planes.
Select two planes for the new plane to intersect or a planar face and vertex or Mate connector (implicit or explicit).
Curve PointCopy link
Create a curve-tangent point plane that passes through the point, perpendicular to the curve. Use one curve (or edge)defining the normal of the plane and one point defining the origin of the plane. The plane normal is always tangent to the curve.
Select a curve for the plane to be perpendicular to and select a point to define the origin of the plane.
TangentCopy link
Create a plane tangent to the face of a cylinder. Select the face of the cylinder to create the plane tangent to, and then select a point (to be coplanar with the tangent plane), or a plane (to be parallel to the tangent plane).
Plane: AndroidCopy link
StepsCopy link
Tap Plane tool.
Select entities on which to base the new plane.
Select a plane type (see the subsections below for more information on each type).
Onshape remembers the selection for the plane type and opens the dialog with the last selected option in subsequent operations.
Specify the distance or angle (depending on which plane type you have selected).
Optionally, toggle to switch to the opposite direction.
Optionally, toggle the Flip normal switch to adjust the normal of the plane to be in the opposite direction. You can see which direction is normal by looking at the name of the plane in the graphics area.
Tap the checkmark.
OffsetCopy link
Create a plane a specified distance from another plane using a plane (or Mate connector, implicit or explicit) and a distance value. You are able to offset from a planar face or from another plane.
Select entity on which to base the new plane and specify the distance.
Plane PointCopy link
Create a plane that passes through a point, parallel to a plane, using a plane and a point (or Mate connector, implicit or explicit).
Select a plane for the new plane to remain parallel with and a point for the new plane to pass through.
Line AngleCopy link
Create a plane that passes through a line at an angle, using a line and an angle value.
Select a line for the plane to pass through and specify the angle of the plane.
Point NormalCopy link
Create a plane that is normal to the line and passed through the point, using:a straight axis (a straight line segment or anything that defines an axis (circle, arc, cylindrical face, revolved face, etc.)and a point (or vertex). The point is always the origin of the plane and the axis or line is always the normal of the plane.
Select a line for the plane to pass through and a point for the plane to be normal to.
As shown in the image above, the plane is normal to the point on the corner of the sketch on the Right plane and it is passing through the selected line.
Three PointCopy link
Create a plane that passes through three points, using three points.
Select the three points for the plane to pass through.
As shown in the image below, the plane passes through all three points:
Mid PlaneCopy link
Create a plane at the intersection of two other planes.
Select two planes for the new plane to intersect.
Curve PointCopy link
Create a curve-tangent point plane that passes through the point, perpendicular to the curve. Use one curve (or edge)defining the normal of the plane and one point (or vertex)defining the origin of the plane. The plane normal is always tangent to the curve.
Select a curve for the plane to be perpendicular to and select a point (or vertex) to define the origin of the plane.
TangentCopy link
Create a plane tangent to the face of a cylinder. Select the face of the cylinder to create the plane tangent to, and then select a point (to be coplanar with the tangent plane), or a plane (to be parallel to the tangent plane).
Last Updated: September 25, 2025 |
12051 | https://www.news-medical.net/health/Chancroid-Epidemiology.aspx | Published Time: 2017-10-09T23:20:00-04:00
Chancroid Epidemiology
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2 Currently rated 3.0 by 2 people
1
Chancroid Epidemiology
Download PDF Copy
By Dr. Tomislav Meštrović, MD, Ph.D.Reviewed by Afsaneh Khetrapal, BSc
Chancroid is a sexually-transmitted disease that is caused by the bacterial pathogen Haemophilus ducreyi. The condition is characterized by the presence of non-indurated, ulcerated lesions on the genitalia, usually associated with painful inguinal lymphadenopathy.
Chancroid is a substantial public health problem in many countries of the developing world. The estimates are that approximately seven million cases of this disease are seen worldwide; however, under-reporting and misdiagnoses hamper accurate predictions of its prevalence and spread.
International Epidemiology
Chancroid was once a common isolate in clinics across Africa, Latin America and Asia, where its incidence exceeded all other causes of genital ulcers combined. Nevertheless, after 2000 there was an extensive use of syndromic approaches for the management of sexually transmitted infections caused by bacterial pathogens, which resulted in a rapid decline of chancroid and its near-elimination in certain parts of southern and eastern Africa.
In sub-Saharan Africa, the overall prevalence of chancroid in patients with genital ulcers declined from more than 60 percent in the 1970s to less than 15 percent between 2001 and 2005. Moreover, the disease was undetectable in Zambia and Kenya between 2005 and 2010.
Although the disease is rare in the United States (US), outbreaks have been reported in urban areas. The case numbers are in decline since 1941, with only 24 documented in 2010, and 10 cases in 2013. But although the disease is considered infrequent in the US, it should be taken into account when assessing high-risk individuals presenting with painful genital ulcers.
The low prevalence of chancroid observed in Europe in the last two decades is linked to shifts in both public health and social conditions. Surveillance data has demonstrated Haemophilus ducreyi prevalence between 0.9 and 3 percent in ulcer patients from France and the Netherlands.
Since 2005, Haemophilus ducreyi has been implicated in chronic skin ulcers found in children residing in endemic areas for yaws (a chronic infection by the spirochete bacterium that affects bone, cartilage and skin) of the South Pacific region. Although similar to genital chancroid in appearance, this disease is not transmitted sexually, nor does it give rise to enlarged lymph nodes or bubo formation.
Age, Sex and Racial Differences
Chancroid is most commonly seen in non-white, uncircumcised individuals (which is not surprising considering its endemic prevalence in areas of Asia, Africa and Latin America). Circumcision has been shown protective against chancroid infections, but also for some other sexually transmitted diseases (such as syphilis and human immunodeficiency virus).
The transmission probability from an infected individual to an uninfected one is estimated as 0.35 for a single sexual exposure. The main issue is that in women ulcers tend to be subclinical, resulting in continuous sexual activities and frequent contacts that are needed for the spread of Haemophilus ducreyi within a population.
This disease is usually observed in commercial sex workers, travelers that visited endemic areas, and generally in people of lower socioeconomic standard. Furthermore, chancroid may affect individuals of any age, although it is usually found in younger, sexually-active individuals (aged between 21 and 30 years).
In any case, chancroid is not lethal, and the genital lesions tend to resolve spontaneously after 1-3 months – even if they are left untreated. However, if antimicrobial therapy is not instituted, suppuration of inguinal lymph nodes may ensue, which can even progress to rupture and formation of non-healing inguinal ulcer.
Sources
www.ncbi.nlm.nih.gov/pmc/articles/PMC1744597/pdf/v079p00068.pdf
www.clinicalmicrobiologyandinfection.com/.../pdf
ecdc.europa.eu/.../...-Sexually-Transmitted-Infections-Europe-2010.pdf
Sakuma TH, Dal’Asta Coimbra D, Lupi O. Chancroid. In: Gross G, Tyring SK. Sexually Transmitted Infections and Sexually Transmitted Diseases. Springer Science & Business Media, 2011; pp. 183-190.
Further Reading
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Treatment and Prevention of Chancroid
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Last Updated: Dec 28, 2022
Currently rated 3.0 by 2 people
Written by
Dr. Tomislav Meštrović
Dr. Tomislav Meštrović is a medical doctor (MD) with a Ph.D. in biomedical and health sciences, specialist in the field of clinical microbiology, and an Assistant Professor at Croatia's youngest university - University North. In addition to his interest in clinical, research and lecturing activities, his immense passion for medical writing and scientific communication goes back to his student days. He enjoys contributing back to the community. In his spare time, Tomislav is a movie buff and an avid traveler.
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Meštrović, Tomislav. (2022, December 28). Chancroid Epidemiology. News-Medical. Retrieved on July 29, 2025 from
MLA
Meštrović, Tomislav. "Chancroid Epidemiology". News-Medical. 29 July 2025.
Chicago
Meštrović, Tomislav. "Chancroid Epidemiology". News-Medical. (accessed July 29, 2025).
Harvard
Meštrović, Tomislav. 2022. Chancroid Epidemiology. News-Medical, viewed 29 July 2025,
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12052 | https://www.youtube.com/watch?v=_k-ZyJoqC90 | How to find f(2) for a Function | VCE METHODS
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Learn about function notation and the maths behind it.
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Functions can be used in many ways. The notation of a name is very handy and not limited to the letter "f". f(2) involves finding a y-value for the x-value of 2. This can be done by using the rule of your equation (or function) for f(x). Substitute x=2 into the rule, and you will find the corresponding y-value. That will be your answer for a given equation.
The idea of a function is quite abstract. Imagine a function like a book. "f(x)" is the name of the function, which like a book's title. The x inside the brackets represents the chapter that you may want to continue reading from. To do so, you need the page number of that chapter. You can find this in the table of contents. For our function, that would be the rule on the right hand side of the equation. Using the rule in combination with the x-value is like using the table contents to determine the page number of a particular chapter. Once you have found your chapter, you have found your result. This is known as the y-value.
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Intro okay so here is a function f of x equals negative x squared plus 3x minus 5. and here are some examples of stuff that you can be asked to find f of 2 f of negative 1 f of 0. if you're struggling with this stuff then keep watching because today i'm going to explain how to calculate these and how to do some other operations with functions like multiplication and division and if you stick around to the end of the video i will also show you a special operation that you can do with functions all right let's go How functions work g'day guys this is the math base now before i explain how we're going to do those calculations let's just talk a little bit about how functions even work in the first place so on the right hand side we have negative x squared plus 3x minus 5. this is known as the rule in other words it tells you how to do something and that something is for the stuff on the left hand side f x that is known as the function name or as i like to call it a title here is an example of a title so here's a book that's been written by the same author as lord of the rings the title informs us of which book you would like to read or borrow from the library in other words it's a really easy way to reference what you're looking for in general instead of just trying to explain to you every single word in the book itself for every single chapter i can just refer to it as the title now if you go to the table of contents this tells you where to find each specific chapter in a sense you could think of this as a rule so for example if i wanted to go to chapter 5 i will need to turn to page 80. all right so page 80. here we go and we can basically start reading the entire chapter from here right this can be considered like a result right that this is what we wanted to get to some kind of outcome in the context of a function the outcome we'd be looking for is for something specific like f of two you can think of two as a particular chapter that you're trying to get to and the outcome would be the page number that you need to figure out so if i want to go to chapter 5 i need to go to page 80. if i want to go to chapter 2 i need to go to page 27 or something so now that we Finding f2 understand what we're looking at in terms of the equation let's dive into how to calculate this stuff so first up we have f of 2 the 2 is saying that x is currently equal to 2. so what you want to do is replace all the x's in your rule with 2. when you do all the operations and calculations for this we will find that the result is negative three this is known as a y value which means when x is two y is negative three which means we have a coordinate you could write the coordinate as two comma f of two but nobody really writes it that way it's it's not really going to give you enough information although you can expect stuff like that in a question and so you would have to work that out to get 2 comma negative 3. so we can conclude and say f of 2 is negative 3. as for f of negative 1 you'll do the same thing now we're going to just replace x with negative one working that out brings us to a result of negative nine so if you wanted to write a coordinate you can say negative one comma negative nine as for f of zero we simply need to replace both the x's for zero now giving us a value of negative five and this quadrant would be written as zero comma negative five right which means that if you were to plot this out on the cartesian plane this will be a y-intercept so in general all you really need to do is take the value from the bracket and sub it into x Operations it's all about subs speaking of subs don't forget to subscribe to math base if you haven't done so yet alright so if you guys don't have any trouble with that stuff then let's look at some operations we can do with functions let's say you're asked to find f of two plus f of negative one you've already found the value of f of two we know it's negative three and we've done something similar for f of negative one and so we can just replace those notations those titles for a specific outcome so we can say it's negative 3 plus negative 9 because remember we're not just referring to the title of f of x anymore we're referring to a very specific chapter f 2 and f of negative 1 and their respective page numbers you could say so negative 3 plus negative 9 gives us negative 12. so how about 2 times f of 0 or 2 f of 0 means the same thing f of 0 is negative 5. so all we need to do is replace that bit for negative 5 and multiply by 2 giving us negative 10. and this last one f of 2 over f of 0 looks a little bit difficult but it's really not it's just the same process as before all you need to do is replace f of 2 for the value negative 3 and replace f of 0 for negative 5 giving us 3 over 5 when you simplify it so if you guys have made it this far in the video well done to you i'm going to now talk about a little special operation that you can do with functions and here's what it looks like f of f of 2. it's a bit like that movie inception so you got something inside something you could say this is a composite operation it kind of looks impossible to do but it's really quite straightforward f of 2 is negative 3 right so just replace that for that outcome now we have f of negative 3. and from here you just rinse and repeat what you've been doing this whole time just putting in negative 3 where all the x's are work that out and we'll get negative 23. so we can see that throughout this entire video f of x was a pretty handy notation to do some quick calculations this becomes especially useful when you have another function like g of x equal to three x plus two and you want to do operations like f of two plus g of one this way you know exactly which rule is where and you can begin to apply the calculations and substitutions accordingly with respect to each function it's also very handy when you want to draw multiple graphs on the same set of axes Outro if you guys have any specific function questions you would like me to tackle and show here like a like a work solution let me know in the comments section below so i hope you guys found this helpful be sure to like the video and subscribe to the channel oh and don't forget to ring the bell to get notified of when my next video comes out alright guys have a good one and i'll see you next time |
12053 | https://www.facebook.com/groups/121644125187853/posts/185707758781489/ | Philippine Civil Service Review 2025 | ENGLISH VOCABULARY (Cont.#4) | Facebook
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ジイー ベラスコ
· June 21, 2018 ·
ENGLISH VOCABULARY (Cont.#4)
PULCHRITUDE (puhl-kri-tood)
And here’s the definition: Physical beauty.
Synonyms:
•Allure
•Beauty
•Elegance
JINGOISM (jin-go-iz-uhm)
And here’s the definition: Extreme patriotism for one’s country that often shows itself through aggression towards other nations.
Synonyms:
•Fanaticism
•Nationalism
•Zealotry
HINTERLAND (hin-tur-land)
And here’s the definition: The remote or less developed parts of a country.
Synonyms:
•Outback
•Backcountry
•Boondocks
VALETUDINARIAN (val-ih-tood-n-air-ee-uhn)
And here’s the definition: A person who is unduly anxious about their health.
Synonyms:
•Hypochondriac
•Neurotic
•Paranoid
PORTCULLIS (pohrt-kuhl-is)
And here’s the definition: A sturdy sliding gateway door that moves vertically to open and close.
Synonyms:
•Doorway
•Gate
•Entrance
CLANGOR (klang-er)
And here’s the definition: A continuous noisy banging.
Synonyms:
•Noise
•Blare
•Roar
TURPITUDE (tur-pi-tood)
And here’s the definition: A vile or depraved act.
Synonyms:
•Wrongdoing
•Crime
•Transgression
TOCSIN (tok-sin)
And here’s the definition: An alarm.
Synonyms:
•Alarm
•Siren
•Bell
ACHROMATIC (ak-roh-mat-ik)
And here’s the definition: For something to be colorless usually with black, white and grays.
Synonyms:
•Neutral
•Colorless
•Hueless
AFFABLE (af-uh-bull)
And here’s the definition: pleasantly easy to approach.
Synonyms:
•Amiable
•Friendly
•Genial
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beauty, 32. nationalism, 33. boondocks, 34. hypochondraic, 35. gate, 36. noise, 37. crime, 38. Siren, 39. hueless, 40. friendly
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12054 | http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/alex5.html | | | | | |
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| | | | --- | | Math Central | Quandaries & Queries | |
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| | | | Question from alex, a teacher: I know that (-2)^2 is 4 but what is -2^2? | |
| Hi Alex, When I look at -22 I see two operations, a negation and a power (square). The convention in mathematical expressions is that powers are performed before negations (also before additions, multiplications and divisions) so to evaluate -22 you first square the 2 to get 4 and then apply the negation to arrive at -4. With (-2)2 the parentheses tell you to consider -2 as a single quantity and square it so (-2)2 = (-2) × (-2) = 4. I hope this helps,Penny It's -4. Please note that this is not really a question (or answer) about arithmetic; it is about the convention that the second expression shall be interpreted as -(2^2). There is no deep mathematical reason for this; it has just been found to be more convenient. That said, now that the convention is established, we all need to follow it; it's like driving on the right (or left) side of the road. Good hunting!RD |
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12055 | https://dictionary.cambridge.org/us/dictionary/english/boorish | Cambridge Dictionary +Plus
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Meaning of boorish in English
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rude and not considering other people's feelings:
I found him rather boorish and aggressive.
The boorish behavior of some tourists is embarrassing.
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He is more mature than the boorish lout we remember.
One of the more boorish footballers let out a loud belch.
Why should we tolerate rude and boorish passengers without complaint?
SMART Vocabulary: related words and phrases
Showing arrogance and conceit
(as) proud as Lucifer idiom
airs and graces idiom
arrogant
arrogantly
be too clever by half idiom
egomaniacal
egomaniacally
egotistic
egotistically
fancy-pants
patronizingly
pomposity
pompous
pompously
popinjay
swollen head
swollen-headed
talk down to someone phrasal verb
think the (whole) world revolves around you idiom
throw your weight around idiom
See more results »
(Definition of boorish from the Cambridge Advanced Learner's Dictionary & Thesaurus © Cambridge University Press)
What is the pronunciation of boorish?
Translations of boorish
in Chinese (Traditional)
粗野的，粗魯的，不體諒別人感受的…
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in Chinese (Simplified)
粗野的，粗鲁的，不体谅别人感受的…
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in Spanish
tosco, grosero…
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in Portuguese
grosseiro…
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in more languages
in French
in Turkish
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in Czech
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in Indonesian
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in Polish
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grossier, rustre…
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kaba/görgüsüz, hödük…
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lomp…
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křupanský, neotesaný…
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fræk…
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kampungan, tak tahu adat…
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เหมือนคนชั้นต่ำ…
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thô lỗ…
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gburowaty…
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tölpaktig, bufflig…
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kasar dan tidak tahu adat…
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flegelhaft…
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ubehøvlet…
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невихований, грубий…
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boondocking
boondoggle
boop
boor
boorish
boorishly
boost
boosted
booster
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take something back
to admit that something you said was wrong
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12056 | https://www.purplemath.com/modules/ageprobs.htm | Learn how to set up and solve 'age' word problems. | Purplemath
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"Age" Word Problems
Age ProbsDiophantus
Purplemath
What are "age" word problems?
"Age" type word problems are those which compare two persons' ages, or one person's ages at different times in their lives, or some combination thereof.
Here's an example from my own life:
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Age Word Problems
In January of the year 2000, I was one more than eleven times as old as my son Will. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000?
Obviously, in "real life" you'd have walked up to my kid and asked him how old he was, and he'd have proudly held up three grubby fingers, but that won't help you on your homework.
Here's how you'd figure out his age, if you'd been asked the above question in your math class:
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First, I'll need to name things and translate the English into math.
Since my age was defined in terms of Will's, I'll start with a variable for Will's age. To make it easy for me to remember the meaning of the variable, I will pick W to stand for "Will's age at the start, in the year 2000". Then Will's age in 2009, being nine years later, will be W+9. So I have the following information:
Will's age in 2000: W
Will's age in 2009: W + 9
My age was defined in terms of the above expressions. In the year 2000, I was "eleven times Will's age in the year 2000, plus one more", giving me:
my age in 2000: 11(W) + 1
My age in 2009 was also defined in terms of Will's age in 2009. Specifically, I was "three times Will's age in 2009, plus seven more", giving me:
my age in 2009: 3(W + 9) + 7
But I was also nine years older than I had been in the year 2000, which gives me another expression for my age in 2009:
my age in 2009: [11(W) + 1] + 9
My age in 2009 was my age in 2009. This fact means that the two expressions for "my age in 2009" must represent the same value. And this fact, in turn, allows me to create an equation — by setting the two equal-value expressions equal to each other:
3(W + 9) + 7 = [11(W) + 1] + 9
Solving, I get:
3(W + 9) + 7 = [11(W) + 1] + 9
3W + 27 + 7 = 11W + 1 + 9
3W + 34 = 11W + 10
34 = 8W + 10
24 = 8W
3 = W
Since I set up this equation using expressions for my age, it's tempting to think that 3=W stands for my age. But this is why I picked W to stand for "Will's age"; the variable reminds me that, no, 3=W stands for Will's age, not mine.
And this is exactly what the question had asked in the first place. How old was Will in the year 2000?
Will was three years old.
Note that this word problem did not ask for the value of a variable; it asked for the age of a person. So a properly-written answer reflects this. "W=3" would not be an ideal response.
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What are the steps for solving an age-based word problem?
The important steps for solving an age-based word problem are as follows:
Figure out what is defined in terms of something else
Set up a variable for that "something else" (labelling it clearly with its definition)
Create an expression for the first time frame, and then
Increment the expressions by the required amount (in the example above, this increment was nine years) to reflect the passage of time.
Don't try to use the same variable or expression to stand for two different things! Since, in the above, W stands for Will's age in 2000, then W can not also stand for his age in 2009. Make sure that you are very explicit about this when you set up your variables, expressions, and equations; write down the two sets of information as two distinct situations.
Currently, Andrei is three times Nicolas' age. In ten years, Andrei will be twelve years older than Nicolas. What are their ages now?
Andrei's age in defined in terms of Nicolas' age, so I'll pick a variable for Nicolas' age now; say, "N". This allows me to create an expression for Andrei's age now, which is three times that of Nicolas.
Nicolas' age now: N
Andrei's age now: 3N
In ten years, they each will be ten years older, so I'll add 10 to each of the above for their later ages.
Nicolas' age later: N + 10
Andrei's age later: 3N + 10
But I am also given that, in ten years, Andrei will be twelve years older than Nicolas. So I can create another expression for Andrei's age in ten years; namely, I'll take the expression for Nicolas' age in ten years, and add twelve to that.
Andrei's age later: [N + 10] + 12
Since Andrei's future age will equal his future age, I can take these two expressions for his future age, set them equal (thus creating an equation), and solve for the value of the variable.
3N + 10 = [N + 10] + 12
3N + 10 = N + 22
2N + 10 = 22
2N = 12
N = 6
Okay; I've found the value of the variable. But, looking back at the original question, I see that they're wanting to know the current ages of two people. The variable stands for the age of the younger of the two. Since the older is three times this age, then the older is 18 years old. So my clearly-stated answer is:
Nicolas is 6 years old.
Andrei is 18 years old.
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math
Mathematics
One-half of Heather's age two years from now plus one-third of her age three years ago is twenty years. How old is she now?
This problem refers to Heather's age two years into the future and three years back in the past. Unlike most "age" word problems, this exercise is not comparing two different people's ages at the same point in time, but rather the same person's ages at different points in time.
They ask for Heather's age now, so I'll pick a variable to stand for this unknown; say, H. Then I'll increment this variable in order to create expressions for "two years ago" and "two years from now".
age now: H
age two years from now: H + 2
age three years ago: H − 3
Now I need to create expressions, using the above, which will stand for certain fractions of these ages:
½ of age 2 years from now: 1 2(H+2)\small{ \dfrac{1}{2} \left(H + 2\right) }2 1(H+2)
1 3\small{ \frac{1}{3} }3 1 of age 3 years ago: 1 3(H−3)\small{ \dfrac{1}{3} \left(H - 3\right) }3 1(H−3)
The sum of these two expressions is given as being "20", so I'll add the two expressions, set their sum equal to 20, and solve for the variable:
1 2(H+2)+1 3(H−3)=20\small{ \frac{1}{2} (H + 2) + \frac{1}{3} (H - 3) = 20 }2 1(H+2)+3 1(H−3)=20
1 2 H+1+1 3 H−1=20\small{ \frac{1}{2} H + 1 + \frac{1}{3} H - 1 = 20 }2 1H+1+3 1H−1=20
1 2 H+1 3 H=20\small{ \frac{1}{2} H + \frac{1}{3} H = 20 }2 1H+3 1H=20
3 H+2 H=120\small{ 3H + 2H = 120 }3 H+2 H=120
5 H=120\small{ 5H = 120 }5 H=120
H=24\small{ H = 24 }H=24
Okay; I've found the value of the variable. Now I'll go back and check my definition of that variable (so I see that it stands for Heather's current age), and I'll check for what the exercise actually asked me to find (which was Heather's current age). So my answer is:
Heather is 24 years old.
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Note: Remember that you can always check your answer to any "solving" exercise by plugging that answer back into the original problem. In the case of the above exercise, if Heather is 24 now, then she will be 26 in two years, half of which is 13; three years ago, she would have been 21, a third of which is 7. Adding, I get 13+7 =20, so my solution checks.
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In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one now?
The grandfather's age is defined in terms of Miguel's age, so I'll pick a variable to stand for Miguel's age. Since they're asking me for current ages, my variable will stand for Miguel's current age.
Miguel's age now: m
Now I'll use this variable to create expressions for the various items listed in the exercise.
Miguel's age last year: m − 1
six times Miguel's age last year: 6(m − 1)
Miguel's grandfather's age will, in another three years, be six times what Miguel's age was last year. This means that his grandfather is currently three years less than six times Miguel's age from last year, so:
grandfather's age now: 6(m − 1) − 3
Summing the expressions for the two current ages, and solving, I get:
(m) + [6(m − 1) − 3] = 68
m + [6 m − 6 − 3] = 68
m + [6 m − 9] = 68
7 m − 9 = 68
7 m = 77
m = 11
Looking back, I see that this variable stands for Miguel's current age, which is eleven. But the exercise asks me for the current ages of bother of them, so:
Last year, Miguel would have been ten. In three more years, his grandfather will be six times ten, or sixty. So his grandfather must currently be 60−3=57.
Miguel is currently 11.
His grandfather is currently 57.
The puzzler on the next page is an old one (as in "Ancient Greece" old), but it keeps cropping up in various forms. It's rather intricate.
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12057 | https://webbook.nist.gov/cgi/inchi?ID=C7440575&Mask=20 | gold
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National Institute of Standards and Technology
NIST Chemistry WebBook, SRD 69
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gold
Formula: Au
Molecular weight: 196.966569
IUPAC Standard InChI:InChI=1S/Au Copy
IUPAC Standard InChIKey:PCHJSUWPFVWCPO-UHFFFAOYSA-N Copy
CAS Registry Number: 7440-57-5
Chemical structure:
This structure is also available as a 2d Mol file
Permanent link for this species. Use this link for bookmarking this species for future reference.
Information on this page:
Gas phase ion energetics data
References
Notes
Other data available:
Phase change data
Reaction thermochemistry data
Data at other public NIST sites:
NIST Atomic Spectra Database - Lines Holdings (on physics web site)
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Gas phase ion energetics data
Go To:Top, References, Notes
Data compilation copyright by the U.S. Secretary of Commerce on behalf of the U.S.A. All rights reserved.
Data evaluated as indicated in comments:
L - Sharon G. Lias
Data compiled as indicated in comments:
LL - Sharon G. Lias and Joel F. Liebman
LLK - Sharon G. Lias, Rhoda D. Levin, and Sherif A. Kafafi
RDSH - Henry M. Rosenstock, Keith Draxl, Bruce W. Steiner, and John T. Herron
B - John E. Bartmess
View reactions leading to Au+ (ion structure unspecified)
| Quantity | Value | Units | Method | Reference | Comment |
--- --- --- |
| IE (evaluated) | 9.22567 | eV | N/A | N/A | L |
Electron affinity determinations
| EA (eV) | Method | Reference | Comment |
--- --- |
| 2.3090 ± 0.0010 | N/A | Wu, Qin, et al., 2010 | B |
| 2.30 ± 0.10 | LPES | Gantefor, Kraus, et al., 1998 | B |
| 2.30860 ± 0.00070 | LPD | Hotop and Lineberger, 1973 | B |
| 2.927 ± 0.050 | LPES | Taylor, Pettiettehall, et al., 1992 | EA set as 0.2 eV above onset to correct for unresolved hot bands.; B |
Ionization energy determinations
| IE (eV) | Method | Reference | Comment |
--- --- |
| 9.22567 | EVAL | Lide, 1992 | LL |
| 9.22 | PE | Dyke, Fayad, et al., 1979 | LLK |
| 9.22553 ± 0.00002 | S | Brown and Ginter, 1978 | LLK |
| 8.5 ± 0.8 | EI | Cocke, Gingerich, et al., 1973 | LLK |
| 9.21 ± 0.05 | EI | Cabaud, Hoareau, et al., 1972 | LLK |
| 9.0 ± 0.5 | EI | Gingerich and Pupp, 1971 | LLK |
| 9.22567 | S | Moore, 1970 | RDSH |
References
Go To:Top, Gas phase ion energetics data, Notes
Data compilation copyright by the U.S. Secretary of Commerce on behalf of the U.S.A. All rights reserved.
Wu, Qin, et al., 2010
Wu, X.; Qin, Z.B.; Xie, H.; Cong, R.; Wu, X.H.; Tang, Z.C.; Fan, H.J., Photoelectron Imaging and Theoretical Studies of Group 11 Cyanides MCN (M = Cu, Ag, Au), J. Phys. Chem. A, 2010, 114, 49, 12839-12844, . [all data]
Gantefor, Kraus, et al., 1998
Gantefor, G.; Kraus, S.; Eberhardt, W., Femtosecond photoelectron spectroscopy of the photodissociation of Au-3(-), J. Electron Spectros. Rel. Phenom., 1998, 88, 35-40, . [all data]
Hotop and Lineberger, 1973
Hotop, H.; Lineberger, W.C., Dye-laser Photodetachment Studies of Au-, Pt-, PtN-, and Ag-, J. Chem. Phys., 1973, 58, 6, 2379, . [all data]
Taylor, Pettiettehall, et al., 1992
Taylor, K.J.; Pettiettehall, C.L.; Cheshnovsky, O.; Smalley, R.E., Ultraviolet Photoelectron Spectra of Coinage Metal Clusters, J. Chem. Phys., 1992, 96, 4, 3319, . [all data]
Lide, 1992
Lide, D.R. (Editor), Ionization potentials of atoms and atomic ions in Handbook of Chem. and Phys., 1992, 10-211. [all data]
Dyke, Fayad, et al., 1979
Dyke, J.M.; Fayad, N.K.; Morris, A.; Trickle, I.R., Gas-phase He I photoelectron spectra of some transition metals: Cu, Ag, Au, Cr and Mn, J. Phys. B:, 1979, 12, 2985. [all data]
Brown and Ginter, 1978
Brown, C.M.; Ginter, M.L., Absorption spectrum of Au I between 1300 and 1900 Å, J. Opt. Soc. Am., 1978, 68, 243. [all data]
Cocke, Gingerich, et al., 1973
Cocke, D.L.; Gingerich, K.A.; Kordis, J., Determination of the high bond dissociation energy of the molecule LaRh, High Temp. Sci., 1973, 5, 474. [all data]
Cabaud, Hoareau, et al., 1972
Cabaud, B.; Hoareau, A.; Nounou, P.; Uzan, R., Etude des processus d'ionisation a haute temperature des vapeurs metalliques par couplage d'une cellule de knudsen et d'une source fox. II. Interpretation des processus d'autoionisation des vapeurs metalliques monoatomiques et influence delatemperature sur les courbes d'efficacite d'ionisation., Int. J. Mass Spectrom. Ion Phys., 1972, 8, 181. [all data]
Gingerich and Pupp, 1971
Gingerich, K.A.; Pupp, C., Mass spectrometric determination of the heats of formation atomization of gaseous AuBO, J. Chem. Phys., 1971, 54, 3713. [all data]
Moore, 1970
Moore, C.E., Ionization potentials and ionization limits derived from the analyses of optical spectra, Natl. Stand. Ref. Data Ser., (U.S. Natl. Bur. Stand.), 1970, 34, 1. [all data]
Notes
Go To:Top, Gas phase ion energetics data, References
Symbols used in this document:
EA Electron affinity
IE (evaluated)Recommended ionization energy
Data from NIST Standard Reference Database 69: NIST Chemistry WebBook
The National Institute of Standards and Technology (NIST) uses its best efforts to deliver a high quality copy of the Database and to verify that the data contained therein have been selected on the basis of sound scientific judgment. However, NIST makes no warranties to that effect, and NIST shall not be liable for any damage that may result from errors or omissions in the Database.
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12058 | https://fiveable.me/key-terms/hs-honors-geometry/indirect-measurement | Indirect measurement - (Honors Geometry) - Vocab, Definition, Explanations | Fiveable | Fiveable
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Honors Geometry
Indirect measurement
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key term - Indirect measurement
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Definition
Indirect measurement is a technique used to determine the dimensions or distances of objects that are difficult to measure directly, often by using the properties of similar triangles. This method leverages the concept of ratios and proportions between corresponding sides of similar figures, allowing for calculations based on known measurements. It is particularly useful in scenarios where direct access to an object is restricted or impractical.
5 Must Know Facts For Your Next Test
Indirect measurement relies on the properties of similar triangles to calculate unknown distances or heights without measuring them directly.
Using a method called 'shadow measurement,' you can determine the height of an object by comparing the lengths of its shadow to the length of a shadow cast by a known height.
The concept of ratios is fundamental in indirect measurement, allowing you to set up proportions based on corresponding sides of similar triangles.
Indirect measurement can be applied in real-life scenarios such as surveying land, measuring tall buildings, or estimating distances across bodies of water.
In indirect measurement, accuracy depends on ensuring that the triangles used for comparison remain truly similar in shape and proportion.
Review Questions
How can indirect measurement be applied to determine the height of a tree using similar triangles?
To find the height of a tree using indirect measurement, you can measure the length of the tree's shadow and your own shadow at the same time. By ensuring that both shadows are cast by the same light source (like the sun), you can create a pair of similar triangles. If you know your height and the length of your shadow, you can set up a proportion comparing your height to your shadow and the height of the tree to its shadow, allowing you to solve for the unknown height.
Discuss how proportional relationships are essential in using indirect measurement for real-world applications like surveying.
Proportional relationships are crucial when applying indirect measurement in surveying because they allow surveyors to estimate distances and heights without direct access to every point. For example, if surveyors measure a baseline and use known angles from that baseline, they can create triangles where they know one side and two angles. This information forms proportional relationships that help them calculate other distances indirectly, making it possible to survey large areas efficiently.
Evaluate the effectiveness of indirect measurement compared to direct measurement methods in complex scenarios involving large structures.
Indirect measurement methods often prove more effective than direct measurements when dealing with large structures due to safety concerns, accessibility issues, and practicality. For example, measuring the height of a skyscraper directly might require expensive equipment or safety measures that could be avoided by using indirect methods like triangulation. Indirect measurements can yield accurate results quickly and safely while also minimizing disturbance around large sites. However, it's important to ensure that the assumptions made (like assuming similarity in triangles) hold true for reliable outcomes.
Related terms
Similar Triangles:Triangles that have the same shape but may differ in size, where corresponding angles are equal and the lengths of corresponding sides are proportional.
Proportional Relationships:A relationship between two quantities where their ratio remains constant, often expressed as a fraction or a percentage.
Height of an Object:The vertical measurement from the base of an object to its top, which can be determined using indirect measurement techniques when direct measurement is not feasible.
"Indirect measurement" also found in:
Subjects (1)
Elementary Algebra
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12059 | https://www.gutenberg.org/files/45988/45988-h/45988-h.htm | The Project Gutenberg eBook of Novum Organum, by Lord Bacon
START OF THE PROJECT GUTENBERG EBOOK 45988
Transcriber’s Note
Greek text with transliteration has a dotted underline. To see the transliteration, hover your mouse over the Greek text: βιβλος.
NOVUM ORGANUM
BY
LORD BACON
EDITED BY JOSEPH DEVEY, M.A.
NEW YORK
P. F. COLLIER & SON
MCMII
22
SCIENCE
NOVUM ORGANUM
OR
TRUE SUGGESTIONS FOR THE INTERPRETATION OF NATURE
PREFACE
They who have presumed to dogmatize on nature, as on some well investigated subject, either from self-conceit or arrogance, and in the professorial style, have inflicted the greatest injury on philosophy and learning. For they have tended to stifle and interrupt inquiry exactly in proportion as they have prevailed in bringing others to their opinion: and their own activity has not counterbalanced the mischief they have occasioned by corrupting and destroying that of others. They again who have entered upon a contrary course, and asserted that nothing whatever can be known, whether they have fallen into this opinion from their hatred of the ancient sophists, or from the hesitation of their minds, or from an exuberance of learning, have certainly adduced reasons for it which are by no means contemptible. They have not, however, derived their opinion from true sources, and, hurried on by their zeal and some affectation, have certainly exceeded due moderation. But the more ancient Greeks (whose writings have perished), held a more prudent mean, between the arrogance of dogmatism, and the despair of scepticism; and though too frequently intermingling complaints and indignation at the difficulty of inquiry, and the obscurity of things, and champing, as it were, the bit, have still persisted in pressing their point, and pursuing their intercourse with nature; thinking, as it seems, that the better method was not to dispute upon the very point of the possibility of anything being known, but to put it to the test of experience. Yet they themselves, by only employing the power of the understanding, have not adopted a fixed rule, but have laid their whole stress upon intense meditation, and a continual exercise and perpetual agitation of the mind.
Our method, though difficult in its operation, is easily explained. It consists in determining the degrees of certainty, while we, as it were, restore the senses to their former rank, but generally reject that operation of the mind which follows close upon the senses, and open and establish a new and certain course for the mind from the first actual perceptions of the senses themselves. This, no doubt, was the view taken by those who have assigned so much to logic; showing clearly thereby that they sought some support for the mind, and suspected its natural and spontaneous mode of action. But this is now employed too late as a remedy, when all is clearly lost, and after the mind, by the daily habit and intercourse of life, has come prepossessed with corrupted doctrines, and filled with the vainest idols. The art of logic therefore being (as we have mentioned), too late a precaution, and in no way remedying the matter, has tended more to confirm errors, than to disclose truth. Our only remaining hope and salvation is to begin the whole labor of the mind again; not leaving it to itself, but directing it perpetually from the very first, and attaining our end as it were by mechanical aid. If men, for instance, had attempted mechanical labors with their hands alone, and without the power and aid of instruments, as they have not hesitated to carry on the labors of their understanding with the unaided efforts of their mind, they would have been able to move and overcome but little, though they had exerted their utmost and united powers. And just to pause awhile on this comparison, and look into it as a mirror; let us ask, if any obelisk of a remarkable size were perchance required to be moved, for the purpose of gracing a triumph or any similar pageant, and men were to attempt it with their bare hands, would not any sober spectator avow it to be an act of the greatest madness? And if they should increase the number of workmen, and imagine that they could thus succeed, would he not think so still more? But if they chose to make a selection, and to remove the weak, and only employ the strong and vigorous, thinking by this means, at any rate, to achieve their object, would he not say that they were more fondly deranged? Nay, if not content with this, they were to determine on consulting the athletic art, and were to give orders for all to appear with their hands, arms, and muscles regularly oiled and prepared, would he not exclaim that they were taking pains to rave by method and design? Yet men are hurried on with the same senseless energy and useless combination in intellectual matters, as long as they expect great results either from the number and agreement, or the excellence and acuteness of their wits; or even strengthen their minds with logic, which may be considered as an athletic preparation, but yet do not desist (if we rightly consider the matter) from applying their own understandings merely with all this zeal and effort. While nothing is more clear, than that in every great work executed by the hand of man without machines or implements, it is impossible for the strength of individuals to be increased, or for that of the multitude to combine.
Having premised so much, we lay down two points on which we would admonish mankind, lest they should fail to see or to observe them. The first of these is, that it is our good fortune (as we consider it), for the sake of extinguishing and removing contradiction and irritation of mind, to leave the honor and reverence due to the ancients untouched and undiminished, so that we can perform our intended work, and yet enjoy the benefit of our respectful moderation. For if we should profess to offer something better than the ancients, and yet should pursue the same course as they have done, we could never, by any artifice, contrive to avoid the imputation of having engaged in a contest or rivalry as to our respective wits, excellences, or talents; which, though neither inadmissible nor new (for why should we not blame and point out anything that is imperfectly discovered or laid down by them, of our own right, a right common to all?), yet however just and allowable, would perhaps be scarcely an equal match, on account of the disproportion of our strength. But since our present plan leads up to open an entirely different course to the understanding, and one unattempted and unknown to them, the case is altered. There is an end to party zeal, and we only take upon ourselves the character of a guide, which requires a moderate share of authority and good fortune, rather than talents and excellence. The first admonition relates to persons, the next to things.
We make no attempt to disturb the system of philosophy that now prevails, or any other which may or will exist, either more correct or more complete. For we deny not that the received system of philosophy, and others of a similar nature, encourage discussion, embellish harangues, are employed, and are of service in the duties of the professor, and the affairs of civil life. Nay, we openly express and declare that the philosophy we offer will not be very useful in such respects. It is not obvious, nor to be understood in a cursory view, nor does it flatter the mind in its preconceived notions, nor will it descend to the level of the generality of mankind unless by its advantages and effects.
Let there exist then (and may it be of advantage to both), two sources, and two distributions of learning, and in like manner two tribes, and as it were kindred families of contemplators or philosophers, without any hostility or alienation between them; but rather allied and united by mutual assistance. Let there be in short one method of cultivating the sciences, and another of discovering them. And as for those who prefer and more readily receive the former, on account of their haste or from motives arising from their ordinary life, or because they are unable from weakness of mind to comprehend and embrace the other (which must necessarily be the case with by far the greater number), let us wish that they may prosper as they desire in their undertaking, and attain what they pursue. But if any individual desire, and is anxious not merely to adhere to, and make use of present discoveries, but to penetrate still further, and not to overcome his adversaries in disputes, but nature by labor, not in short to give elegant and specious opinions, but to know to a certainty and demonstration, let him, as a true son of science (if such be his wish), join with us; that when he has left the antechambers of nature trodden by the multitude, an entrance may at last be discovered to her inner apartments. And in order to be better understood, and to render our meaning more familiar by assigning determinate names, we have accustomed ourselves to call the one method the anticipation of the mind, and the other the interpretation of nature.
We have still one request left. We have at least reflected and taken pains in order to render our propositions not only true, but of easy and familiar access to men’s minds, however wonderfully prepossessed and limited. Yet it is but just that we should obtain this favor from mankind (especially in so great a restoration of learning and the sciences), that whosoever may be desirous of forming any determination upon an opinion of this our work either from his own perceptions, or the crowd of authorities, or the forms of demonstrations, he will not expect to be able to do so in a cursory manner, and while attending to other matters; but in order to have a thorough knowledge of the subject, will himself by degrees attempt the course which we describe and maintain; will be accustomed to the subtilty of things which is manifested by experience; and will correct the depraved and deeply rooted habits of his mind by a seasonable, and, as it were, just hesitation: and then, finally (if he will), use his judgment when he has begun to be master of himself.
FOOTNOTE
Because it was idle to draw a logical conclusion from false principles, error being propagated as much by false premises, which logic does not pretend to examine, as by illegitimate inference. Hence, as Bacon says further on, men being easily led to confound legitimate inference with truth, were confirmed in their errors by the very subtilty of their genius.—Ed.
CONTENTS
Preface
Aphorisms—Book I
On the Interpretation of Nature and the Empire of Man
Aphorisms—Book II
On the Interpretation of Nature, or the Reign of Man
APHORISMS—BOOK I
ON THE INTERPRETATION OF NATURE AND THE EMPIRE OF MAN
I. Man, as the minister and interpreter of nature, does and understands as much as his observations on the order of nature, either with regard to things or the mind, permit him, and neither knows nor is capable of more.
II. The unassisted hand and the understanding left to itself possess but little power. Effects are produced by the means of instruments and helps, which the understanding requires no less than the hand; and as instruments either promote or regulate the motion of the hand, so those that are applied to the mind prompt or protect the understanding.
III. Knowledge and human power are synonymous, since the ignorance of the cause frustrates the effect; for nature is only subdued by submission, and that which in contemplative philosophy corresponds with the cause in practical science becomes the rule.
IV. Man while operating can only apply or withdraw natural bodies; nature internally performs the rest.
V. Those who become practically versed in nature are, the mechanic, the mathematician, the physician, the alchemist, and the magician, but all (as matters now stand) with faint efforts and meagre success.
VI. It would be madness and inconsistency to suppose that things which have never yet been performed can be performed without employing some hitherto untried means.
VII. The creations of the mind and hand appear very numerous, if we judge by books and manufactures; but all that variety consists of an excessive refinement, and of deductions from a few well known matters—not of a number of axioms.
VIII. Even the effects already discovered are due to chance and experiment rather than to the sciences; for our present sciences are nothing more than peculiar arrangements of matters already discovered, and not methods for discovery or plans for new operations.
IX. The sole cause and root of almost every defect in the sciences is this, that while we falsely admire and extol the powers of the human mind, we do not search for its real helps.
X. The subtilty of nature is far beyond that of sense or of the understanding: so that the specious meditations, speculations, and theories of mankind are but a kind of insanity, only there is no one to stand by and observe it.
XI. As the present sciences are useless for the discovery of effects, so the present system of logic is useless for the discovery of the sciences.
XII. The present system of logic rather assists in confirming and rendering inveterate the errors founded on vulgar notions than in searching after truth, and is therefore more hurtful than useful.
XIII. The syllogism is not applied to the principles of the sciences, and is of no avail in intermediate axioms, as being very unequal to the subtilty of nature. It forces assent, therefore, and not things.
XIV. The syllogism consists of propositions; propositions of words; words are the signs of notions. If, therefore, the notions (which form the basis of the whole) be confused and carelessly abstracted from things, there is no solidity in the superstructure. Our only hope, then, is in genuine induction.
XV. We have no sound notions either in logic or physics; substance, quality, action, passion, and existence are not clear notions; much less weight, levity, density, tenuity, moisture, dryness, generation, corruption, attraction, repulsion, element, matter, form, and the like. They are all fantastical and ill-defined.
XVI. The notions of less abstract natures, as man, dog, dove, and the immediate perceptions of sense, as heat, cold, white, black, do not deceive us materially, yet even these are sometimes confused by the mutability of matter and the intermixture of things. All the rest which men have hitherto employed are errors, and improperly abstracted and deduced from things.
XVII. There is the same degree of licentiousness and error in forming axioms as in abstracting notions, and that in the first principles, which depend on common induction; still more is this the case in axioms and inferior propositions derived from syllogisms.
XVIII. The present discoveries in science are such as lie immediately beneath the surface of common notions. It is necessary, however, to penetrate the more secret and remote parts of nature, in order to abstract both notions and axioms from things by a more certain and guarded method.
XIX. There are and can exist but two ways of investigating and discovering truth. The one hurries on rapidly from the senses and particulars to the most general axioms, and from them, as principles and their supposed indisputable truth, derives and discovers the intermediate axioms. This is the way now in use. The other constructs its axioms from the senses and particulars, by ascending continually and gradually, till it finally arrives at the most general axioms, which is the true but unattempted way.
XX. The understanding when left to itself proceeds by the same way as that which it would have adopted under the guidance of logic, namely, the first; for the mind is fond of starting off to generalities, that it may avoid labor, and after dwelling a little on a subject is fatigued by experiment. But those evils are augmented by logic, for the sake of the ostentation of dispute.
XXI. The understanding, when left to itself in a man of a steady, patient, and reflecting disposition (especially when unimpeded by received doctrines), makes some attempt in the right way, but with little effect, since the understanding, undirected and unassisted, is unequal to and unfit for the task of vanquishing the obscurity of things.
XXII. Each of these two ways begins from the senses and particulars, and ends in the greatest generalities. But they are immeasurably different; for the one merely touches cursorily the limits of experiment and particulars, while the other runs duly and regularly through them—the one from the very outset lays down some abstract and useless generalities, the other gradually rises to those principles which are really the most common in nature.
XXIII. There is no small difference between the idols of the human mind and the ideas of the Divine mind—that is to say, between certain idle dogmas and the real stamp and impression of created objects, as they are found in nature.
XXIV. Axioms determined upon in argument can never assist in the discovery of new effects; for the subtilty of nature is vastly superior to that of argument. But axioms properly and regularly abstracted from particulars easily point out and define new particulars, and therefore impart activity to the sciences.
XXV. The axioms now in use are derived from a scanty handful, as it were, of experience, and a few particulars of frequent occurrence, whence they are of much the same dimensions or extent as their origin. And if any neglected or unknown instance occurs, the axiom is saved by some frivolous distinction, when it would be more consistent with truth to amend it.
XXVI. We are wont, for the sake of distinction, to call that human reasoning which we apply to nature the anticipation of nature (as being rash and premature), and that which is properly deduced from things the interpretation of nature.
XXVII. Anticipations are sufficiently powerful in producing unanimity, for if men were all to become even uniformly mad, they might agree tolerably well with each other.
XXVIII. Anticipations again, will be assented to much more readily than interpretations, because being deduced from a few instances, and these principally of familiar occurrence, they immediately hit the understanding and satisfy the imagination; while, on the contrary, interpretations, being deduced from various subjects, and these widely dispersed, cannot suddenly strike the understanding, so that in common estimation they must appear difficult and discordant, and almost like the mysteries of faith.
XXIX. In sciences founded on opinions and dogmas, it is right to make use of anticipations and logic if you wish to force assent rather than things.
XXX. If all the capacities of all ages should unite and combine and transmit their labors, no great progress will be made in learning by anticipations, because the radical errors, and those which occur in the first process of the mind, are not cured by the excellence of subsequent means and remedies.
XXXI. It is in vain to expect any great progress in the sciences by the superinducing or ingrafting new matters upon old. An instauration must be made from the very foundations, if we do not wish to revolve forever in a circle, making only some slight and contemptible progress.
XXXII. The ancient authors and all others are left in undisputed possession of their honors; for we enter into no comparison of capacity or talent, but of method, and assume the part of a guide rather than of a critic.
XXXIII. To speak plainly, no correct judgment can be formed either of our method or its discoveries by those anticipations which are now in common use; for it is not to be required of us to submit ourselves to the judgment of the very method we ourselves arraign.
XXXIV. Nor is it an easy matter to deliver and explain our sentiments; for those things which are in themselves new can yet be only understood from some analogy to what is old.
XXXV. Alexander Borgia said of the expedition of the French into Italy that they came with chalk in their hands to mark up their lodgings, and not with weapons to force their passage. Even so do we wish our philosophy to make its way quietly into those minds that are fit for it, and of good capacity; for we have no need of contention where we differ in first principles, and in our very notions, and even in our forms of demonstration.
XXXVI. We have but one simple method of delivering our sentiments, namely, we must bring men to particulars and their regular series and order, and they must for a while renounce their notions, and begin to form an acquaintance with things.
XXXVII. Our method and that of the sceptics agree in some respects at first setting out, but differ most widely, and are completely opposed to each other in their conclusion; for they roundly assert that nothing can be known; we, that but a small part of nature can be known, by the present method; their next step, however, is to destroy the authority of the senses and understanding, while we invent and supply them with assistance.
XXXVIII. The idols and false notions which have already preoccupied the human understanding, and are deeply rooted in it, not only so beset men’s minds that they become difficult of access, but even when access is obtained will again meet and trouble us in the instauration of the sciences, unless mankind when forewarned guard themselves with all possible care against them.
XXXIX. Four species of idols beset the human mind, to which (for distinction’s sake) we have assigned names, calling the first Idols of the Tribe, the second Idols of the Den, the third Idols of the Market, the fourth Idols of the Theatre.
XL. The formation of notions and axioms on the foundation of true induction is the only fitting remedy by which we can ward off and expel these idols. It is, however, of great service to point them out; for the doctrine of idols bears the same relation to the interpretation of nature as that of the confutation of sophisms does to common logic.
XLI. The idols of the tribe are inherent in human nature and the very tribe or race of man; for man’s sense is falsely asserted to be the standard of things; on the contrary, all the perceptions both of the senses and the mind bear reference to man and not to the universe, and the human mind resembles those uneven mirrors which impart their own properties to different objects, from which rays are emitted and distort and disfigure them.
XLII. The idols of the den are those of each individual; for everybody (in addition to the errors common to the race of man) has his own individual den or cavern, which intercepts and corrupts the light of nature, either from his own peculiar and singular disposition, or from his education and intercourse with others, or from his reading, and the authority acquired by those whom he reverences and admires, or from the different impressions produced on the mind, as it happens to be preoccupied and predisposed, or equable and tranquil, and the like; so that the spirit of man (according to its several dispositions), is variable, confused, and as it were actuated by chance; and Heraclitus said well that men search for knowledge in lesser worlds, and not in the greater or common world.
XLIII. There are also idols formed by the reciprocal intercourse and society of man with man, which we call idols of the market, from the commerce and association of men with each other; for men converse by means of language, but words are formed at the will of the generality, and there arises from a bad and unapt formation of words a wonderful obstruction to the mind. Nor can the definitions and explanations with which learned men are wont to guard and protect themselves in some instances afford a complete remedy—words still manifestly force the understanding, throw everything into confusion, and lead mankind into vain and innumerable controversies and fallacies.
XLIV. Lastly, there are idols which have crept into men’s minds from the various dogmas of peculiar systems of philosophy, and also from the perverted rules of demonstration, and these we denominate idols of the theatre: for we regard all the systems of philosophy hitherto received or imagined, as so many plays brought out and performed, creating fictitious and theatrical worlds. Nor do we speak only of the present systems, or of the philosophy and sects of the ancients, since numerous other plays of a similar nature can be still composed and made to agree with each other, the causes of the most opposite errors being generally the same. Nor, again, do we allude merely to general systems, but also to many elements and axioms of sciences which have become inveterate by tradition, implicit credence, and neglect. We must, however, discuss each species of idols more fully and distinctly in order to guard the human understanding against them.
XLV. The human understanding, from its peculiar nature, easily supposes a greater degree of order and equality in things than it really finds; and although many things in nature be sui generis and most irregular, will yet invent parallels and conjugates and relatives, where no such thing is. Hence the fiction, that all celestial bodies move in perfect circles, thus rejecting entirely spiral and serpentine lines (except as explanatory terms). Hence also the element of fire is introduced with its peculiar orbit, to keep square with those other three which are objects of our senses. The relative rarity of the elements (as they are called) is arbitrarily made to vary in tenfold progression, with many other dreams of the like nature. Nor is this folly confined to theories, but it is to be met with even in simple notions.
XLVI. The human understanding, when any proposition has been once laid down (either from general admission and belief, or from the pleasure it affords), forces everything else to add fresh support and confirmation; and although most cogent and abundant instances may exist to the contrary, yet either does not observe or despises them, or gets rid of and rejects them by some distinction, with violent and injurious prejudice, rather than sacrifice the authority of its first conclusions. It was well answered by him who was shown in a temple the votive tablets suspended by such as had escaped the peril of shipwreck, and was pressed as to whether he would then recognize the power of the gods, by an inquiry, But where are the portraits of those who have perished in spite of their vows? All superstition is much the same, whether it be that of astrology, dreams, omens, retributive judgment, or the like, in all of which the deluded believers observe events which are fulfilled, but neglect and pass over their failure, though it be much more common. But this evil insinuates itself still more craftily in philosophy and the sciences, in which a settled maxim vitiates and governs every other circumstance, though the latter be much more worthy of confidence. Besides, even in the absence of that eagerness and want of thought (which we have mentioned), it is the peculiar and perpetual error of the human understanding to be more moved and excited by affirmatives than negatives, whereas it ought duly and regularly to be impartial; nay, in establishing any true axiom the negative instance is the most powerful.
XLVII. The human understanding is most excited by that which strikes and enters the mind at once and suddenly, and by which the imagination is immediately filled and inflated. It then begins almost imperceptibly to conceive and suppose that everything is similar to the few objects which have taken possession of the mind, while it is very slow and unfit for the transition to the remote and heterogeneous instances by which axioms are tried as by fire, unless the office be imposed upon it by severe regulations and a powerful authority.
XLVIII. The human understanding is active and cannot halt or rest, but even, though without effect, still presses forward. Thus we cannot conceive of any end or external boundary of the world, and it seems necessarily to occur to us that there must be something beyond. Nor can we imagine how eternity has flowed on down to the present day, since the usually received distinction of an infinity, a parte ante and a parte post, cannot hold good; for it would thence follow that one infinity is greater than another, and also that infinity is wasting away and tending to an end. There is the same difficulty in considering the infinite divisibility of lines, arising from the weakness of our minds, which weakness interferes to still greater disadvantage with the discovery of causes; for although the greatest generalities in nature must be positive, just as they are found, and in fact not causable, yet the human understanding, incapable of resting, seeks for something more intelligible. Thus, however, while aiming at further progress, it falls back to what is actually less advanced, namely, final causes; for they are clearly more allied to man’s own nature, than the system of the universe, and from this source they have wonderfully corrupted philosophy. But he would be an unskilful and shallow philosopher who should seek for causes in the greatest generalities, and not be anxious to discover them in subordinate objects.
XLIX. The human understanding resembles not a dry light, but admits a tincture of the will and passions, which generate their own system accordingly; for man always believes more readily that which he prefers. He, therefore, rejects difficulties for want of patience in investigation; sobriety, because it limits his hope; the depths of nature, from superstition; the light of experiment, from arrogance and pride, lest his mind should appear to be occupied with common and varying objects; paradoxes, from a fear of the opinion of the vulgar; in short, his feelings imbue and corrupt his understanding in innumerable and sometimes imperceptible ways.
L. But by far the greatest impediment and aberration of the human understanding proceeds from the dulness, incompetence, and errors of the senses; since whatever strikes the senses preponderates over everything, however superior, which does not immediately strike them. Hence contemplation mostly ceases with sight, and a very scanty, or perhaps no regard is paid to invisible objects. The entire operation, therefore, of spirits inclosed in tangible bodies is concealed, and escapes us. All that more delicate change of formation in the parts of coarser substances (vulgarly called alteration, but in fact a change of position in the smallest particles) is equally unknown; and yet, unless the two matters we have mentioned be explored and brought to light, no great effect can be produced in nature. Again, the very nature of common air, and all bodies of less density (of which there are many) is almost unknown; for the senses are weak and erring, nor can instruments be of great use in extending their sphere or acuteness—all the better interpretations of nature are worked out by instances, and fit and apt experiments, where the senses only judge of the experiment, the experiment of nature and the thing itself.
LI. The human understanding is, by its own nature, prone to abstraction, and supposes that which is fluctuating to be fixed. But it is better to dissect than abstract nature: such was the method employed by the school of Democritus, which made greater progress in penetrating nature than the rest. It is best to consider matter, its conformation, and the changes of that conformation, its own action, and the law of this action or motion; for forms are a mere fiction of the human mind, unless you will call the laws of action by that name.
LII. Such are the idols of the tribe, which arise either from the uniformity of the constitution of man’s spirit, or its prejudices, or its limited faculties or restless agitation, or from the interference of the passions, or the incompetence of the senses, or the mode of their impressions.
LIII. The idols of the den derive their origin from the peculiar nature of each individual’s mind and body, and also from education, habit, and accident; and although they be various and manifold, yet we will treat of some that require the greatest caution, and exert the greatest power in polluting the understanding.
LIV. Some men become attached to particular sciences and contemplations, either from supposing themselves the authors and inventors of them, or from having bestowed the greatest pains upon such subjects, and thus become most habituated to them. If men of this description apply themselves to philosophy and contemplations of a universal nature, they wrest and corrupt them by their preconceived fancies, of which Aristotle affords us a single instance, who made his natural philosophy completely subservient to his logic, and thus rendered it little more than useless and disputatious. The chemists, again, have formed a fanciful philosophy with the most confined views, from a few experiments of the furnace. Gilbert, too, having employed himself most assiduously in the consideration of the magnet, immediately established a system of philosophy to coincide with his favorite pursuit.
LV. The greatest and, perhaps, radical distinction between different men’s dispositions for philosophy and the sciences is this, that some are more vigorous and active in observing the differences of things, others in observing their resemblances; for a steady and acute disposition can fix its thoughts, and dwell upon and adhere to a point, through all the refinements of differences, but those that are sublime and discursive recognize and compare even the most delicate and general resemblances; each of them readily falls into excess, by catching either at nice distinctions or shadows of resemblance.
LVI. Some dispositions evince an unbounded admiration of antiquity, others eagerly embrace novelty, and but few can preserve the just medium, so as neither to tear up what the ancients have correctly laid down, nor to despise the just innovations of the moderns. But this is very prejudicial to the sciences and philosophy, and instead of a correct judgment we have but the factions of the ancients and moderns. Truth is not to be sought in the good fortune of any particular conjuncture of time, which is uncertain, but in the light of nature and experience, which is eternal. Such factions, therefore, are to be abjured, and the understanding must not allow them to hurry it on to assent.
LVII. The contemplation of nature and of bodies in their individual form distracts and weakens the understanding; but the contemplation of nature and of bodies in their general composition and formation stupefies and relaxes it. We have a good instance of this in the school of Leucippus and Democritus compared with others, for they applied themselves so much to particulars as almost to neglect the general structure of things, while the others were so astounded while gazing on the structure that they did not penetrate the simplicity of nature. These two species of contemplation must, therefore, be interchanged, and each employed in its turn, in order to render the understanding at once penetrating and capacious, and to avoid the inconveniences we have mentioned, and the idols that result from them.
LVIII. Let such, therefore, be our precautions in contemplation, that we may ward off and expel the idols of the den, which mostly owe their birth either to some predominant pursuit, or, secondly, to an excess in synthesis and analysis, or, thirdly, to a party zeal in favor of certain ages, or, fourthly, to the extent or narrowness of the subject. In general, he who contemplates nature should suspect whatever particularly takes and fixes his understanding, and should use so much the more caution to preserve it equable and unprejudiced.
LIX. The idols of the market are the most troublesome of all, those namely which have entwined themselves round the understanding from the associations of words and names. For men imagine that their reason governs words, while, in fact, words react upon the understanding; and this has rendered philosophy and the sciences sophistical and inactive. Words are generally formed in a popular sense, and define things by those broad lines which are most obvious to the vulgar mind; but when a more acute understanding or more diligent observation is anxious to vary those lines, and to adapt them more accurately to nature, words oppose it. Hence the great and solemn disputes of learned men often terminate in controversies about words and names, in regard to which it would be better (imitating the caution of mathematicians) to proceed more advisedly in the first instance, and to bring such disputes to a regular issue by definitions. Such definitions, however, cannot remedy the evil in natural and material objects, because they consist themselves of words, and these words produce others; so that we must necessarily have recourse to particular instances, and their regular series and arrangement, as we shall mention when we come to the mode and scheme of determining notions and axioms.
LX. The idols imposed upon the understanding by words are of two kinds. They are either the names of things which have no existence (for as some objects are from inattention left without a name, so names are formed by fanciful imaginations which are without an object), or they are the names of actual objects, but confused, badly defined, and hastily and irregularly abstracted from things. Fortune, the primum mobile, the planetary orbits, the element of fire, and the like fictions, which owe their birth to futile and false theories, are instances of the first kind. And this species of idols is removed with greater facility, because it can be exterminated by the constant refutation or the desuetude of the theories themselves. The others, which are created by vicious and unskilful abstraction, are intricate and deeply rooted. Take some word, for instance, as moist, and let us examine how far the different significations of this word are consistent. It will be found that the word moist is nothing but a confused sign of different actions admitted of no settled and defined uniformity. For it means that which easily diffuses itself over another body; that which is indeterminable and cannot be brought to a consistency; that which yields easily in every direction; that which is easily divided and dispersed; that which is easily united and collected; that which easily flows and is put in motion; that which easily adheres to, and wets another body; that which is easily reduced to a liquid state though previously solid. When, therefore, you come to predicate or impose this name, in one sense flame is moist, in another air is not moist, in another fine powder is moist, in another glass is moist; so that it is quite clear that this notion is hastily abstracted from water only, and common ordinary liquors, without any due verification of it.
There are, however, different degrees of distortion and mistake in words. One of the least faulty classes is that of the names of substances, particularly of the less abstract and more defined species (those then of chalk and mud are good, of earth bad); words signifying actions are more faulty, as to generate, to corrupt, to change; but the most faulty are those denoting qualities (except the immediate objects of sense), as heavy, light, rare, dense. Yet in all of these there must be some notions a little better than others, in proportion as a greater or less number of things come before the senses.
LXI. The idols of the theatre are not innate, nor do they introduce themselves secretly into the understanding, but they are manifestly instilled and cherished by the fictions of theories and depraved rules of demonstration. To attempt, however, or undertake their confutation would not be consistent with our declarations. For since we neither agree in our principles nor our demonstrations, all argument is out of the question. And it is fortunate that the ancients are left in possession of their honors. We detract nothing from them, seeing our whole doctrine relates only to the path to be pursued. The lame (as they say) in the path outstrip the swift who wander from it, and it is clear that the very skill and swiftness of him who runs not in the right direction must increase his aberration.
Our method of discovering the sciences is such as to leave little to the acuteness and strength of wit, and indeed rather to level wit and intellect. For as in the drawing of a straight line, or accurate circle by the hand, much depends on its steadiness and practice, but if a ruler or compass be employed there is little occasion for either; so it is with our method. Although, however, we enter into no individual confutations, yet a little must be said, first, of the sects and general divisions of these species of theories; secondly, something further to show that there are external signs of their weakness; and, lastly, we must consider the causes of so great a misfortune, and so long and general a unanimity in error, that we may thus render the access to truth less difficult, and that the human understanding may the more readily be purified, and brought to dismiss its idols.
LXII. The idols of the theatre, or of theories, are numerous, and may, and perhaps will, be still more so. For unless men’s minds had been now occupied for many ages in religious and theological considerations, and civil governments (especially monarchies), had been averse to novelties of that nature even in theory (so that men must apply to them with some risk and injury to their own fortunes, and not only without reward, but subject to contumely and envy), there is no doubt that many other sects of philosophers and theorists would have been introduced, like those which formerly flourished in such diversified abundance among the Greeks. For as many imaginary theories of the heavens can be deduced from the phenomena of the sky, so it is even more easy to found many dogmas upon the phenomena of philosophy—and the plot of this our theatre resembles those of the poetical, where the plots which are invented for the stage are more consistent, elegant, and pleasurable than those taken from real history.
In general, men take for the groundwork of their philosophy either too much from a few topics, or too little from many; in either case their philosophy is founded on too narrow a basis of experiment and natural history, and decides on too scanty grounds. For the theoretic philosopher seizes various common circumstances by experiment, without reducing them to certainty or examining and frequently considering them, and relies for the rest upon meditation and the activity of his wit.
There are other philosophers who have diligently and accurately attended to a few experiments, and have thence presumed to deduce and invent systems of philosophy, forming everything to conformity with them.
A third set, from their faith and religious veneration, introduce theology and traditions; the absurdity of some among them having proceeded so far as to seek and derive the sciences from spirits and genii. There are, therefore, three sources of error and three species of false philosophy; the sophistic, empiric, and superstitious.
LXIII. Aristotle affords the most eminent instance of the first; for he corrupted natural philosophy by logic—thus he formed the world of categories, assigned to the human soul, the noblest of substances, a genus determined by words of secondary operation, treated of density and rarity (by which bodies occupy a greater or lesser space), by the frigid distinctions of action and power, asserted that there was a peculiar and proper motion in all bodies, and that if they shared in any other motion, it was owing to an external moving cause, and imposed innumerable arbitrary distinctions upon the nature of things; being everywhere more anxious as to definitions in teaching and the accuracy of the wording of his propositions, than the internal truth of things. And this is best shown by a comparison of his philosophy with the others of greatest repute among the Greeks. For the similar parts of Anaxagoras, the atoms of Leucippus and Democritus, the heaven and earth of Parmenides, the discord and concord of Empedocles, the resolution of bodies into the common nature of fire, and their condensation according to Heraclitus, exhibit some sprinkling of natural philosophy, the nature of things, and experiment; while Aristotle’s physics are mere logical terms, and he remodelled the same subject in his metaphysics under a more imposing title, and more as a realist than a nominalist. Nor is much stress to be laid on his frequent recourse to experiment in his books on animals, his problems, and other treatises; for he had already decided, without having properly consulted experience as the basis of his decisions and axioms, and after having so decided, he drags experiment along as a captive constrained to accommodate herself to his decisions: so that he is even more to be blamed than his modern followers (of the scholastic school) who have deserted her altogether.
LXIV. The empiric school produces dogmas of a more deformed and monstrous nature than the sophistic or theoretic school; not being founded in the light of common notions (which, however poor and superstitious, is yet in a manner universal, and of a general tendency), but in the confined obscurity of a few experiments. Hence this species of philosophy appears probable, and almost certain to those who are daily practiced in such experiments, and have thus corrupted their imagination, but incredible and futile to others. We have a strong instance of this in the alchemists and their dogmas; it would be difficult to find another in this age, unless perhaps in the philosophy of Gilbert. We could not, however, neglect to caution others against this school, because we already foresee and augur, that if men be hereafter induced by our exhortations to apply seriously to experiments (bidding farewell to the sophistic doctrines), there will then be imminent danger from empirics, owing to the premature and forward haste of the understanding, and its jumping or flying to generalities and the principles of things. We ought, therefore, already to meet the evil.
LXV. The corruption of philosophy by the mixing of it up with superstition and theology, is of a much wider extent, and is most injurious to it both as a whole and in parts. For the human understanding is no less exposed to the impressions of fancy, than to those of vulgar notions. The disputatious and sophistic school entraps the understanding, while the fanciful, bombastic, and, as it were, poetical school, rather flatters it.
There is a clear example of this among the Greeks, especially in Pythagoras, where, however, the superstition is coarse and overcharged, but it is more dangerous and refined in Plato and his school. This evil is found also in some branches of other systems of philosophy, where it introduces abstracted forms, final and first causes, omitting frequently the intermediate and the like. Against it we must use the greatest caution; for the apotheosis of error is the greatest evil of all, and when folly is worshipped, it is, as it were, a plague spot upon the understanding. Yet some of the moderns have indulged this folly with such consummate inconsiderateness, that they have endeavored to build a system of natural philosophy on the first chapter of Genesis, the book of Job, and other parts of Scripture; seeking thus the dead among the living. And this folly is the more to be prevented and restrained, because not only fantastical philosophy, but heretical religion spring from the absurd mixture of matters divine and human. It is therefore most wise soberly to render unto faith the things that are faith’s.
LXVI. Having spoken of the vicious authority of the systems founded either on vulgar notions, or on a few experiments, or on superstition, we must now consider the faulty subjects for contemplation, especially in natural philosophy. The human understanding is perverted by observing the power of mechanical arts, in which bodies are very materially changed by composition or separation, and is induced to suppose that something similar takes place in the universal nature of things. Hence the fiction of elements, and their co-operation in forming natural bodies. Again, when man reflects upon the entire liberty of nature, he meets with particular species of things, as animals, plants, minerals, and is thence easily led to imagine that there exist in nature certain primary forms which she strives to produce, and that all variation from them arises from some impediment or error which she is exposed to in completing her work, or from the collision or metamorphosis of different species. The first hypothesis has produced the doctrine of elementary properties, the second that of occult properties and specific powers; and both lead to trifling courses of reflection, in which the mind acquiesces, and is thus diverted from more important subjects. But physicians exercise a much more useful labor in the consideration of the secondary qualities of things, and the operations of attraction, repulsion, attenuation, inspissation, dilatation, astringency, separation, maturation, and the like; and would do still more if they would not corrupt these proper observations by the two systems I have alluded to, of elementary qualities and specific powers, by which they either reduce the secondary to first qualities, and their subtile and immeasurable composition, or at any rate neglect to advance by greater and more diligent observation to the third and fourth qualities, thus terminating their contemplation prematurely. Nor are these powers (or the like) to be investigated only among the medicines for the human body, but also in all changes of other natural bodies.
A greater evil arises from the contemplation and investigation rather of the stationary principles of things from which, than of the active by which things themselves are created. For the former only serve for discussion, the latter for practice. Nor is any value to be set on those common differences of motion which are observed in the received system of natural philosophy, as generation, corruption, augmentation, diminution, alteration, and translation. For this is their meaning: if a body, unchanged in other respects, is moved from its place, this is translation; if the place and species be given, but the quantity changed, it is alteration; but if, from such a change, the mass and quantity of the body do not continue the same, this is the motion of augmentation and diminution; if the change be continued so as to vary the species and substance, and transfuse them to others, this is generation and corruption. All this is merely popular, and by no means penetrates into nature; and these are but the measures and bounds of motion, and not different species of it; they merely suggest how far, and not how or whence. For they exhibit neither the affections of bodies nor the process of their parts, but merely establish a division of that motion, which coarsely exhibits to the senses matter in its varied form. Even when they wish to point out something relative to the causes of motion, and to establish a division of them, they most absurdly introduce natural and violent motion, which is also a popular notion, since every violent motion is also in fact natural, that is to say, the external efficient puts nature in action in a different manner to that which she had previously employed.
But if, neglecting these, any one were, for instance, to observe that there is in bodies a tendency of adhesion, so as not to suffer the unity of nature to be completely separated or broken, and a vacuum to be formed, or that they have a tendency to return to their natural dimensions or tension, so that, if compressed or extended within or beyond it, they immediately strive to recover themselves, and resume their former volume and extent; or that they have a tendency to congregate into masses with similar bodies—the dense, for instance, toward the circumference of the earth, the thin and rare toward that of the heavens. These and the like are true physical genera of motions, but the others are clearly logical and scholastic, as appears plainly from a comparison of the two.
Another considerable evil is, that men in their systems and contemplations bestow their labor upon the investigation and discussion of the principles of things and the extreme limits of nature, although all utility and means of action consist in the intermediate objects. Hence men cease not to abstract nature till they arrive at potential and shapeless matter, and still persist in their dissection, till they arrive at atoms; and yet were all this true, it would be of little use to advance man’s estate.
LXVII. The understanding must also be cautioned against the intemperance of systems, so far as regards its giving or withholding its assent; for such intemperance appears to fix and perpetuate idols, so as to leave no means of removing them.
These excesses are of two kinds. The first is seen in those who decide hastily, and render the sciences positive and dictatorial. The other in those who have introduced scepticism, and vague unbounded inquiry. The former subdues, the latter enervates the understanding. The Aristotelian philosophy, after destroying other systems (as the Ottomans do their brethren) by its disputatious confutations, decided upon everything, and Aristotle himself then raises up questions at will, in order to settle them; so that everything should be certain and decided, a method now in use among his successors.
The school of Plato introduced scepticism, first, as it were in joke and irony, from their dislike to Protagoras, Hippias, and others, who were ashamed of appearing not to doubt upon any subject. But the new academy dogmatized in their scepticism, and held it as their tenet. Although this method be more honest than arbitrary decision (for its followers allege that they by no means confound all inquiry, like Pyrrho and his disciples, but hold doctrines which they can follow as probable, though they cannot maintain them to be true), yet when the human mind has once despaired of discovering truth, everything begins to languish. Hence men turn aside into pleasant controversies and discussions, and into a sort of wandering over subjects rather than sustain any rigorous investigation. But as we observed at first, we are not to deny the authority of the human senses and understanding, although weak, but rather to furnish them with assistance.
LXVIII. We have now treated of each kind of idols, and their qualities, all of which must be abjured and renounced with firm and solemn resolution, and the understanding must be completely freed and cleared of them, so that the access to the kingdom of man, which is founded on the sciences, may resemble that to the kingdom of heaven, where no admission is conceded except to children.
LXIX. Vicious demonstrations are the muniments and support of idols, and those which we possess in logic, merely subject and enslave the world to human thoughts, and thoughts to words. But demonstrations are in some manner themselves systems of philosophy and science; for such as they are, and accordingly as they are regularly or improperly established, such will be the resulting systems of philosophy and contemplation. But those which we employ in the whole process leading from the senses and things to axioms and conclusions, are fallacious and incompetent. This process is fourfold, and the errors are in equal number. In the first place the impressions of the senses are erroneous, for they fail and deceive us. We must supply defects by substitutions, and fallacies by their correction. Secondly, notions are improperly abstracted from the senses, and indeterminate and confused when they ought to be the reverse. Thirdly, the induction that is employed is improper, for it determines the principles of sciences by simple enumeration, without adopting exclusions and resolutions, or just separations of nature. Lastly, the usual method of discovery and proof, by first establishing the most general propositions, then applying and proving the intermediate axioms according to them, is the parent of error and the calamity of every science. But we will treat more fully of that which we now slightly touch upon, when we come to lay down the true way of interpreting nature, after having gone through the above expiatory process and purification of the mind.
LXX. But experience is by far the best demonstration, provided it adhere to the experiment actually made, for if that experiment be transferred to other subjects apparently similar, unless with proper and methodical caution it becomes fallacious. The present method of experiment is blind and stupid; hence men wandering and roaming without any determined course, and consulting mere chance, are hurried about to various points, and advance but little—at one time they are happy, at another their attention is distracted, and they always find that they want something further. Men generally make their experiments carelessly, and as it were in sport, making some little variation in a known experiment, and then if they fail they become disgusted and give up the attempt; nay, if they set to work more seriously, steadily, and assiduously, yet they waste all their time on probing some solitary matter, as Gilbert on the magnet, and the alchemists on gold. But such conduct shows their method to be no less unskilful than mean; for nobody can successfully investigate the nature of any object by considering that object alone; the inquiry must be more generally extended.
Even when men build any science and theory upon experiment, yet they almost always turn with premature and hasty zeal to practice, not merely on account of the advantage and benefit to be derived from it, but in order to seize upon some security in a new undertaking of their not employing the remainder of their labor unprofitably, and by making themselves conspicuous, to acquire a greater name for their pursuit. Hence, like Atalanta, they leave the course to pick up the golden apple, interrupting their speed, and giving up the victory. But in the true course of experiment, and in extending it to new effects, we should imitate the Divine foresight and order; for God on the first day only created light, and assigned a whole day to that work without creating any material substance thereon. In like manner we must first, by every kind of experiment, elicit the discovery of causes and true axioms, and seek for experiments which may afford light rather than profit. Axioms, when rightly investigated and established, prepare us not for a limited but abundant practice, and bring in their train whole troops of effects. But we will treat hereafter of the ways of experience, which are not less beset and interrupted than those of judgment; having spoken at present of common experience only as a bad species of demonstration, the order of our subject now requires some mention of those external signs of the weakness in practice of the received systems of philosophy and contemplation which we referred to above, and of the causes of a circumstance at first sight so wonderful and incredible. For the knowledge of these external signs prepares the way for assent, and the explanation of the causes removes the wonder; and these two circumstances are of material use in extirpating more easily and gently the idols from the understanding.
LXXI. The sciences we possess have been principally derived from the Greeks; for the addition of the Roman, Arabic, or more modern writers, are but few and of small importance, and such as they are, are founded on the basis of Greek invention. But the wisdom of the Greeks was professional and disputatious, and thus most adverse to the investigation of truth. The name, therefore, of sophists, which the contemptuous spirit of those who deemed themselves philosophers, rejected and transferred to the rhetoricians—Gorgias, Protagoras, Hippias, Polus—might well suit the whole tribe, such as Plato, Aristotle, Zeno, Epicurus, Theophrastus, and their successors—Chrysippus, Carneades, and the rest. There was only this difference between them—the former were mercenary vagabonds, travelling about to different states, making a show of their wisdom, and requiring pay; the latter more dignified and noble, in possession of fixed habitations, opening schools, and teaching philosophy gratuitously. Both, however (though differing in other respects), were professorial, and reduced every subject to controversy, establishing and defending certain sects and dogmas of philosophy, so that their doctrines were nearly (what Dionysius not unaptly objected to Plato) the talk of idle old men to ignorant youths. But the more ancient Greeks, as Empedocles, Anaxagoras, Leucippus, Democritus, Parmenides, Heraclitus, Xenophanes, Philolaus, and the rest (for I omit Pythagoras as being superstitious), did not (that we are aware) open schools, but betook themselves to the investigation of truth with greater silence and with more severity and simplicity, that is, with less affectation and ostentation. Hence in our opinion they acted more advisedly, however their works may have been eclipsed in course of time by those lighter productions which better correspond with and please the apprehensions and passions of the vulgar; for time, like a river, bears down to us that which is light and inflated, and sinks that which is heavy and solid. Nor were even these more ancient philosophers free from the national defect, but inclined too much to the ambition and vanity of forming a sect, and captivating public opinion, and we must despair of any inquiry after truth when it condescends to such trifles. Nor must we omit the opinion, or rather prophecy, of an Egyptian priest with regard to the Greeks, that they would forever remain children, without any antiquity of knowledge or knowledge of antiquity; for they certainly have this in common with children, that they are prone to talking, and incapable of generation, their wisdom being loquacious and unproductive of effects. Hence the external signs derived from the origin and birthplace of our present philosophy are not favorable.
LXXII. Nor are those much better which can be deduced from the character of the time and age, than the former from that of the country and nation; for in that age the knowledge both of time and of the world was confined and meagre, which is one of the worst evils for those who rely entirely on experience—they had not a thousand years of history worthy of that name, but mere fables and ancient traditions; they were acquainted with but a small portion of the regions and countries of the world, for they indiscriminately called all nations situated far toward the north Scythians, all those to the west Celts; they knew nothing of Africa but the nearest part of Ethiopia, or of Asia beyond the Ganges, and had not even heard any sure and clear tradition of the regions of the New World. Besides, a vast number of climates and zones, in which innumerable nations live and breathe, were pronounced by them to be uninhabitable; nay, the travels of Democritus, Plato, and Pythagoras, which were not extensive, but rather mere excursions from home, were considered as something vast. But in our times many parts of the New World, and every extremity of the Old, are well known, and the mass of experiments has been infinitely increased; wherefore, if external signs were to be taken from the time of the nativity or procreation (as in astrology), nothing extraordinary could be predicted of these early systems of philosophy.
LXXIII. Of all signs there is none more certain or worthy than that of the fruits produced, for the fruits and effects are the sureties and vouchers, as it were, for the truth of philosophy. Now, from the systems of the Greeks, and their subordinate divisions in particular branches of the sciences during so long a period, scarcely one single experiment can be culled that has a tendency to elevate or assist mankind, and can be fairly set down to the speculations and doctrines of their philosophy. Celsus candidly and wisely confesses as much, when he observes that experiments were first discovered in medicine, and that men afterward built their philosophical systems upon them, and searched for and assigned causes, instead of the inverse method of discovering and deriving experiments from philosophy and the knowledge of causes; it is not, therefore, wonderful that the Egyptians (who bestowed divinity and sacred honors on the authors of new inventions) should have consecrated more images of brutes than of men, for the brutes by their natural instinct made many discoveries, while men derived but few from discussion and the conclusions of reason.
The industry of the alchemists has produced some effect, by chance, however, and casualty, or from varying their experiments (as mechanics also do), and not from any regular art or theory, the theory they have imagined rather tending to disturb than to assist experiment. Those, too, who have occupied themselves with natural magic (as they term it) have made but few discoveries, and those of small import, and bordering on imposture; for which reason, in the same manner as we are cautioned by religion to show our faith by our works, we may very properly apply the principle to philosophy, and judge of it by its works, accounting that to be futile which is unproductive, and still more so if, instead of grapes and olives, it yield but the thistle and thorns of dispute and contention.
LXXIV. Other signs may be selected from the increase and progress of particular systems of philosophy and the sciences; for those which are founded on nature grow and increase, while those which are founded on opinion change and increase not. If, therefore, the theories we have mentioned were not like plants, torn up by the roots, but grew in the womb of nature, and were nourished by her, that which for the last two thousand years has taken place would never have happened, namely, that the sciences still continue in their beaten track, and nearly stationary, without having received any important increase, nay, having, on the contrary, rather bloomed under the hands of their first author, and then faded away. But we see that the case is reversed in the mechanical arts, which are founded on nature and the light of experience, for they (as long as they are popular) seem full of life, and uninterruptedly thrive and grow, being at first rude, then convenient, lastly polished, and perpetually improved.
LXXV. There is yet another sign (if such it may be termed, being rather an evidence, and one of the strongest nature), namely, the actual confession of those very authorities whom men now follow; for even they who decide on things so daringly, yet at times, when they reflect, betake themselves to complaints about the subtilty of nature, the obscurity of things, and the weakness of man’s wit. If they would merely do this, they might perhaps deter those who are of a timid disposition from further inquiry, but would excite and stimulate those of a more active and confident turn to further advances. They are not, however, satisfied with confessing so much of themselves, but consider everything which has been either unknown or unattempted by themselves or their teachers, as beyond the limits of possibility, and thus, with most consummate pride and envy, convert the defects of their own discoveries into a calumny on nature and a source of despair to every one else. Hence arose the New Academy, which openly professed scepticism, and consigned mankind to eternal darkness; hence the notion that forms, or the true differences of things (which are in fact the laws of simple action), are beyond man’s reach, and cannot possibly be discovered; hence those notions in the active and operative branches, that the heat of the sun and of fire are totally different, so as to prevent men from supposing that they can elicit or form, by means of fire, anything similar to the operations of nature; and again, that composition only is the work of man and mixture of nature, so as to prevent men from expecting the generation or transformation of natural bodies by art. Men will, therefore, easily allow themselves to be persuaded by this sign not to engage their fortunes and labor in speculations, which are not only desperate, but actually devoted to desperation.
LXXVI. Nor should we omit the sign afforded by the great dissension formerly prevalent among philosophers, and the variety of schools, which sufficiently show that the way was not well prepared that leads from the senses to the understanding, since the same groundwork of philosophy (namely, the nature of things), was torn and divided into such widely differing and multifarious errors. And although in these days the dissensions and differences of opinions with regard to first principles and entire systems are nearly extinct, yet there remain innumerable questions and controversies with regard to particular branches of philosophy. So that it is manifest that there is nothing sure or sound either in the systems themselves or in the methods of demonstration.
LXXVII. With regard to the supposition that there is a general unanimity as to the philosophy of Aristotle, because the other systems of the ancients ceased and became obsolete on its promulgation, and nothing better has been since discovered; whence it appears that it is so well determined and founded, as to have united the suffrages of both ages; we will observe—1st.That the notion of other ancient systems having ceased after the publication of the works of Aristotle is false, for the works of the ancient philosophers subsisted long after that event, even to the time of Cicero, and the subsequent ages. But at a later period, when human learning had, as it were, been wrecked in the inundation of barbarians into the Roman empire, then the systems of Aristotle and Plato were preserved in the waves of ages, like planks of a lighter and less solid nature. 2d.The notion of unanimity, on a clear inspection, is found to be fallacious. For true unanimity is that which proceeds from a free judgment, arriving at the same conclusion, after an investigation of the fact. Now, by far the greater number of those who have assented to the philosophy of Aristotle, have bound themselves down to it from prejudice and the authority of others, so that it is rather obsequiousness and concurrence than unanimity. But even if it were real and extensive unanimity, so far from being esteemed a true and solid confirmation, it should even lead to a violent presumption to the contrary. For there is no worse augury in intellectual matters than that derived from unanimity, with the exception of divinity and politics, where suffrages are allowed to decide. For nothing pleases the multitude, unless it strike the imagination or bind down the understanding, as we have observed above, with the shackles of vulgar notions. Hence we may well transfer Phocion’s remark from morals to the intellect: “That men should immediately examine what error or fault they have committed, when the multitude concurs with, and applauds them.” This then is one of the most unfavorable signs. All the signs, therefore, of the truth and soundness of the received systems of philosophy and the sciences are unpropitious, whether taken from their origin, their fruits, their progress, the confessions of their authors, or from unanimity.
LXXVIII. We now come to the causes of errors, and of such perseverance in them for ages. These are sufficiently numerous and powerful to remove all wonder, that what we now offer should have so long been concealed from, and have escaped the notice of mankind, and to render it more worthy of astonishment, that it should even now have entered any one’s mind, or become the subject of his thoughts; and that it should have done so, we consider rather the gift of fortune than of any extraordinary talent, and as the offspring of time rather than wit. But, in the first place, the number of ages is reduced to very narrow limits, on a proper consideration of the matter. For out of twenty-five centuries, with which the memory and learning of man are conversant, scarcely six can be set apart and selected as fertile in science and favorable to its progress. For there are deserts and wastes in times as in countries, and we can only reckon up three revolutions and epochs of philosophy. 1.The Greek. 2.The Roman. 3.Our own, that is the philosophy of the western nations of Europe: and scarcely two centuries can with justice be assigned to each. The intermediate ages of the world were unfortunate both in the quantity and richness of the sciences produced. Nor need we mention the Arabs, or the scholastic philosophy, which, in those ages, ground down the sciences by their numerous treatises, more than they increased their weight. The first cause, then, of such insignificant progress in the sciences, is rightly referred to the small proportion of time which has been favorable thereto.
LXXIX. A second cause offers itself, which is certainly of the greatest importance; namely, that in those very ages in which men’s wit and literature flourished considerably, or even moderately, but a small part of their industry was bestowed on natural philosophy, the great mother of the sciences. For every art and science torn from this root may, perhaps, be polished, and put into a serviceable shape, but can admit of little growth. It is well known, that after the Christian religion had been acknowledged, and arrived at maturity, by far the best wits were busied upon theology, where the highest rewards offered themselves, and every species of assistance was abundantly supplied, and the study of which was the principal occupation of the western European nations during the third epoch; the rather because literature flourished about the very time when controversies concerning religion first began to bud forth. 2.In the preceding ages, during the second epoch (that of the Romans), philosophical meditation and labor was chiefly occupied and wasted in moral philosophy (the theology of the heathens): besides, the greatest minds in these times applied themselves to civil affairs, on account of the magnitude of the Roman empire, which required the labor of many. 3.The age during which natural philosophy appeared principally to flourish among the Greeks, was but a short period, since in the more ancient times the seven sages (with the exception of Thales), applied themselves to moral philosophy and politics, and at a later period, after Socrates had brought down philosophy from heaven to earth, moral philosophy became more prevalent, and diverted men’s attention from natural. Nay, the very period during which physical inquiries flourished, was corrupted and rendered useless by contradictions, and the ambition of new opinions. Since, therefore, during these three epochs, natural philosophy has been materially neglected or impeded, it is not at all surprising that men should have made but little progress in it, seeing they were attending to an entirely different matter.
LXXX. Add to this that natural philosophy, especially of late, has seldom gained exclusive possession of an individual free from all other pursuits, even among those who have applied themselves to it, unless there may be an example or two of some monk studying in his cell, or some nobleman in his villa. She has rather been made a passage and bridge to other pursuits.
Thus has this great mother of the sciences been degraded most unworthily to the situation of a handmaid, and made to wait upon medicine or mathematical operations, and to wash the immature minds of youth, and imbue them with a first dye, that they may afterward be more ready to receive and retain another. In the meantime, let no one expect any great progress in the sciences (especially their operative part), unless natural philosophy be applied to particular sciences, and particular sciences again referred back to natural philosophy. For want of this, astronomy, optics, music, many mechanical arts, medicine itself, and (what perhaps is more wonderful), moral and political philosophy, and the logical sciences have no depth, but only glide over the surface and variety of things; because these sciences, when they have been once partitioned out and established, are no longer nourished by natural philosophy, which would have imparted fresh vigor and growth to them from the sources and genuine contemplation of motion, rays, sounds, texture, and conformation of bodies, and the affections and capacity of the understanding. But we can little wonder that the sciences grow not when separated from their roots.
LXXXI. There is another powerful and great cause of the little advancement of the sciences, which is this; it is impossible to advance properly in the course when the goal is not properly fixed. But the real and legitimate goal of the sciences is the endowment of human life with new inventions and riches. The great crowd of teachers know nothing of this, but consist of dictatorial hirelings; unless it so happen that some artisan of an acute genius, and ambitious of fame, gives up his time to a new discovery, which is generally attended with a loss of property. The majority, so far from proposing to themselves the augmentation of the mass of arts and sciences, make no other use of an inquiry into the mass already before them, than is afforded by the conversion of it to some use in their lectures, or to gain, or to the acquirement of a name, and the like. But if one out of the multitude be found, who courts science from real zeal, and on his own account, even he will be seen rather to follow contemplation, and the variety of theories, than a severe and strict investigation of truth. Again, if there even be an unusually strict investigator of truth, yet will he propose to himself, as the test of truth, the satisfaction of his mind and understanding, as to the causes of things long since known, and not such a test as to lead to some new earnest of effects, and a new light in axioms. If, therefore, no one have laid down the real end of science, we cannot wonder that there should be error in points subordinate to that end.
LXXXII. But, in like manner, as the end and goal of science is ill defined, so, even were the case otherwise, men have chosen an erroneous and impassable direction. For it is sufficient to astonish any reflecting mind, that nobody should have cared or wished to open and complete a way for the understanding, setting off from the senses, and regular, well-conducted experiment; but that everything has been abandoned either to the mists of tradition, the whirl and confusion of argument, or the waves and mazes of chance, and desultory, ill-combined experiment. Now, let any one but consider soberly and diligently the nature of the path men have been accustomed to pursue in the investigation and discovery of any matter, and he will doubtless first observe the rude and inartificial manner of discovery most familiar to mankind: which is no other than this. When any one prepares himself for discovery, he first inquires and obtains a full account of all that has been said on the subject by others, then adds his own reflections, and stirs up and, as it were, invokes his own spirit, after much mental labor, to disclose its oracles. All which is a method without foundation, and merely turns on opinion.
Another, perhaps, calls in logic to assist him in discovery, which bears only a nominal relation to his purpose. For the discoveries of logic are not discoveries of principles and leading axioms, but only of what appears to accord with them. And when men become curious and importunate, and give trouble, interrupting her about her proofs, and the discovery of principles or first axioms, she puts them off with her usual answer, referring them to faith, and ordering them to swear allegiance to each art in its own department.
There remains but mere experience, which, when it offers itself, is called chance; when it is sought after, experiment. But this kind of experience is nothing but a loose fagot; and mere groping in the dark, as men at night try all means of discovering the right road, while it would be better and more prudent either to wait for day, or procure a light, and then proceed. On the contrary, the real order of experience begins by setting up a light, and then shows the road by it, commencing with a regulated and digested, not a misplaced and vague course of experiment, and thence deducing axioms, and from those axioms new experiments: for not even the Divine Word proceeded to operate on the general mass of things without due order.
Let men, therefore, cease to wonder if the whole course of science be not run, when all have wandered from the path; quitting it entirely, and deserting experience, or involving themselves in its mazes, and wandering about, while a regularly combined system would lead them in a sure track through its wilds to the open day of axioms.
LXXXIII. The evil, however, has been wonderfully increased by an opinion, or inveterate conceit, which is both vainglorious and prejudicial, namely, that the dignity of the human mind is lowered by long and frequent intercourse with experiments and particulars, which are the objects of sense, and confined to matter; especially since such matters generally require labor in investigation, are mean subjects for meditation, harsh in discourse, unproductive in practice, infinite in number, and delicate in their subtilty. Hence we have seen the true path not only deserted, but intercepted and blocked up, experience being rejected with disgust, and not merely neglected or improperly applied.
LXXXIV. Again, the reverence for antiquity, and the authority of men who have been esteemed great in philosophy, and general unanimity, have retarded men from advancing in science, and almost enchanted them. As to unanimity, we have spoken of it above.
The opinion which men cherish of antiquity is altogether idle, and scarcely accords with the term. For the old age and increasing years of the world should in reality be considered as antiquity, and this is rather the character of our own times than of the less advanced age of the world in those of the ancients; for the latter, with respect to ourselves, are ancient and elder, with respect to the world modern and younger. And as we expect a greater knowledge of human affairs, and more mature judgment from an old man than from a youth, on account of his experience, and the variety and number of things he has seen, heard, and meditated upon, so we have reason to expect much greater things of our own age (if it knew but its strength and would essay and exert it) than from antiquity, since the world has grown older, and its stock has been increased and accumulated with an infinite number of experiments and observations.
We must also take into our consideration that many objects in nature fit to throw light upon philosophy have been exposed to our view, and discovered by means of long voyages and travels, in which our times have abounded. It would, indeed, be dishonorable to mankind, if the regions of the material globe, the earth, the sea, and stars, should be so prodigiously developed and illustrated in our age, and yet the boundaries of the intellectual globe should be confined to the narrow discoveries of the ancients.
With regard to authority, it is the greatest weakness to attribute infinite credit to particular authors, and to refuse his own prerogative to time, the author of all authors, and, therefore, of all authority. For truth is rightly named the daughter of time, not of authority. It is not wonderful, therefore, if the bonds of antiquity, authority, and unanimity, have so enchained the power of man, that he is unable (as if bewitched) to become familiar with things themselves.
LXXXV. Nor is it only the admiration of antiquity, authority, and unanimity, that has forced man’s industry to rest satisfied with present discoveries, but, also, the admiration of the effects already placed within his power. For whoever passes in review the variety of subjects, and the beautiful apparatus collected and introduced by the mechanical arts for the service of mankind, will certainly be rather inclined to admire our wealth than to perceive our poverty: not considering that the observations of man and operations of nature (which are the souls and first movers of that variety) are few, and not of deep research; the rest must be attributed merely to man’s patience, and the delicate and well-regulated motion of the hand or of instruments. To take an instance, the manufacture of clocks is delicate and accurate, and appears to imitate the heavenly bodies in its wheels, and the pulse of animals in its regular oscillation, yet it only depends upon one or two axioms of nature.
Again, if one consider the refinement of the liberal arts, or even that exhibited in the preparation of natural bodies in mechanical arts and the like, as the discovery of the heavenly motions in astronomy, of harmony in music, of the letters of the alphabet (still unadopted by the Chinese) in grammar; or, again, in mechanical operations, the productions of Bacchus and Ceres, that is, the preparation of wine and beer, the making of bread, or even the luxuries of the table, distillation, and the like; if one reflect also, and consider for how long a period of ages (for all the above, except distillation, are ancient) these things have been brought to their present state of perfection, and (as we instanced in clocks) to how few observations and axioms of nature they may be referred, and how easily, and as it were, by obvious chance or contemplation, they might be discovered, one would soon cease to admire and rather pity the human lot on account of its vast want and dearth of things and discoveries for so many ages. Yet even the discoveries we have mentioned were more ancient than philosophy and the intellectual arts; so that (to say the truth) when contemplation and doctrinal science began, the discovery of useful works ceased.
But if any one turn from the manufactories to libraries, and be inclined to admire the immense variety of books offered to our view, let him but examine and diligently inspect the matter and contents of these books, and his astonishment will certainly change its object: for when he finds no end of repetitions, and how much men do and speak the same thing over again, he will pass from admiration of this variety to astonishment at the poverty and scarcity of matter, which has hitherto possessed and filled men’s minds.
But if any one should condescend to consider such sciences as are deemed rather curious than sound, and take a full view of the operations of the alchemists or magii, he will perhaps hesitate whether he ought rather to laugh or to weep. For the alchemist cherishes eternal hope, and when his labors succeed not, accuses his own mistakes, deeming, in his self-accusation, that he has not properly understood the words of art or of his authors; upon which he listens to tradition and vague whispers, or imagines there is some slight unsteadiness in the minute details of his practice, and then has recourse to an endless repetition of experiments: and in the meantime, when, in his casual experiments, he falls upon something in appearance new, or of some degree of utility, he consoles himself with such an earnest, and ostentatiously publishes them, keeping up his hope of the final result. Nor can it be denied that the alchemists have made several discoveries, and presented mankind with useful inventions. But we may well apply to them the fable of the old man, who bequeathed to his sons some gold buried in his garden, pretending not to know the exact spot, whereupon they worked diligently in digging the vineyard, and though they found no gold, the vintage was rendered more abundant by their labor.
The followers of natural magic, who explain everything by sympathy and antipathy, have assigned false powers and marvellous operations to things by gratuitous and idle conjectures: and if they have ever produced any effects, they are rather wonderful and novel than of any real benefit or utility.
In superstitious magic (if we say anything at all about it) we must chiefly observe, that there are only some peculiar and definite objects with which the curious and superstitious arts have, in every nation and age, and even under every religion, been able to exercise and amuse themselves. Let us, therefore, pass them over. In the meantime we cannot wonder that the false notion of plenty should have occasioned want.
LXXXVI. The admiration of mankind with regard to the arts and sciences, which is of itself sufficiently simple and almost puerile, has been increased by the craft and artifices of those who have treated the sciences, and delivered them down to posterity. For they propose and produce them to our view so fashioned, and as it were masked, as to make them pass for perfect and complete. For if you consider their method and divisions, they appear to embrace and comprise everything which can relate to the subject. And although this frame be badly filled up and resemble an empty bladder, yet it presents to the vulgar understanding the form and appearance of a perfect science.
The first and most ancient investigators of truth were wont, on the contrary, with more honesty and success, to throw all the knowledge they wished to gather from contemplation, and to lay up for use, into aphorisms, or short scattered sentences unconnected by any method, and without pretending or professing to comprehend any entire art. But according to the present system, we cannot wonder that men seek nothing beyond that which is handed down to them as perfect, and already extended to its full complement.
LXXXVII. The ancient theories have received additional support and credit from the absurdity and levity of those who have promoted the new, especially in the active and practical part of natural philosophy. For there have been many silly and fantastical fellows who, from credulity or imposture, have loaded mankind with promises, announcing and boasting of the prolongation of life, the retarding of old age, the alleviation of pains, the remedying of natural defects, the deception of the senses, the restraint and excitement of the passions, the illumination and exaltation of the intellectual faculties, the transmutation of substances, the unlimited intensity and multiplication of motion, the impressions and changes of the air, the bringing into our power the management of celestial influences, the divination of future events, the representation of distant objects, the revelation of hidden objects, and the like. One would not be very wrong in observing with regard to such pretenders, that there is as much difference in philosophy, between their absurdity and real science, as there is in history between the exploits of Cæsar or Alexander, and those of Amadis de Gaul and Arthur of Britain. For those illustrious generals are found to have actually performed greater exploits than such fictitious heroes are even pretended to have accomplished, by the means, however, of real action, and not by any fabulous and portentous power. Yet it is not right to suffer our belief in true history to be diminished, because it is sometimes injured and violated by fables. In the meantime we cannot wonder that great prejudice has been excited against any new propositions (especially when coupled with any mention of effects to be produced), by the conduct of impostors who have made a similar attempt; for their extreme absurdity, and the disgust occasioned by it, has even to this day overpowered every spirited attempt of the kind.
LXXXVIII. Want of energy, and the littleness and futility of the tasks that human industry has undertaken, have produced much greater injury to the sciences: and yet (to make it still worse) that very want of energy manifests itself in conjunction with arrogance and disdain.
For, in the first place, one excuse, now from its repetition become familiar, is to be observed in every art, namely, that its promoters convert the weakness of the art itself into a calumny upon nature: and whatever it in their hands fails to effect, they pronounce to be physically impossible. But how can the art ever be condemned while it acts as judge in its own cause? Even the present system of philosophy cherishes in its bosom certain positions or dogmas, which (it will be found on diligent inquiry) are calculated to produce a full conviction that no difficult, commanding, and powerful operation upon nature ought to be anticipated through the means of art; we instanced above the alleged different quality of heat in the sun and fire, and composition and mixture. Upon an accurate observation the whole tendency of such positions is wilfully to circumscribe man’s power, and to produce a despair of the means of invention and contrivance, which would not only confound the promises of hope, but cut the very springs and sinews of industry, and throw aside even the chances of experience. The only object of such philosophers is to acquire the reputation of perfection for their own art, and they are anxious to obtain the most silly and abandoned renown, by causing a belief that whatever has not yet been invented and understood can never be so hereafter. But if any one attempt to give himself up to things, and to discover something new; yet he will only propose and destine for his object the investigation and discovery of some one invention, and nothing more; as the nature of the magnet, the tides, the heavenly system, and the like, which appear enveloped in some degree of mystery, and have hitherto been treated with but little success. Now it is the greatest proof of want of skill, to investigate the nature of any object in itself alone; for that same nature, which seems concealed and hidden in some instances, is manifest and almost palpable in others, and excites wonder in the former, while it hardly attracts attention in the latter. Thus the nature of consistency is scarcely observed in wood or stone, but passed over by the term solid without any further inquiry about the repulsion of separation or the solution of continuity. But in water-bubbles the same circumstance appears matter of delicate and ingenious research, for they form themselves into thin pellicles, curiously shaped into hemispheres, so as for an instant to avoid the solution of continuity.
In general those very things which are considered as secret are manifest and common in other objects, but will never be clearly seen if the experiments and contemplation of man be directed to themselves only. Yet it commonly happens, that if, in the mechanical arts, any one bring old discoveries to a finer polish, or more elegant height of ornament, or unite and compound them, or apply them more readily to practice, or exhibit them on a less heavy and voluminous scale, and the like, they will pass off as new.
We cannot, therefore, wonder that no magnificent discoveries, worthy of mankind, have been brought to light, while men are satisfied and delighted with such scanty and puerile tasks, nay, even think that they have pursued or attained some great object in their accomplishment.
LXXXIX. Nor should we neglect to observe that natural philosophy has, in every age, met with a troublesome and difficult opponent: I mean superstition, and a blind and immoderate zeal for religion. For we see that, among the Greeks, those who first disclosed the natural causes of thunder and storms to the yet untrained ears of man were condemned as guilty of impiety toward the gods. Nor did some of the old fathers of Christianity treat those much better who showed by the most positive proofs (such as no one now disputes) that the earth is spherical, and thence asserted that there were antipodes.
Even in the present state of things the condition of discussions on natural philosophy is rendered more difficult and dangerous by the summaries and methods of divines, who, after reducing divinity into such order as they could, and brought it into a scientific form, have proceeded to mingle an undue proportion of the contentious and thorny philosophy of Aristotle with the substance of religion.
The fictions of those who have not feared to deduce and confirm the truth of the Christian religion by the principles and authority of philosophers, tend to the same end, though in a different manner. They celebrate the union of faith and the senses as though it were legitimate, with great pomp and solemnity, and gratify men’s pleasing minds with a variety, but in the meantime confound most improperly things divine and human. Moreover, in these mixtures of divinity and philosophy the received doctrines of the latter are alone included, and any novelty, even though it be an improvement, scarcely escapes banishment and extermination.
In short, you may find all access to any species of philosophy, however pure, intercepted by the ignorance of divines. Some in their simplicity are apprehensive that a too deep inquiry into nature may penetrate beyond the proper bounds of decorum, transferring and absurdly applying what is said of sacred mysteries in Holy Writ against those who pry into divine secrets, to the mysteries of nature, which are not forbidden by any prohibition. Others with more cunning imagine and consider, that if secondary causes be unknown, everything may more easily be referred to the Divine hand and wand, a matter, as they think, of the greatest consequence to religion, but which can only really mean that God wishes to be gratified by means of falsehood. Others fear, from past example, lest motion and change in philosophy should terminate in an attack upon religion. Lastly, there are others who appear anxious lest there should be something discovered in the investigation of nature to overthrow, or at least shake, religion, particularly among the unlearned. The last two apprehensions appear to resemble animal instinct, as if men were diffident, in the bottom of their minds and secret meditations, of the strength of religion and the empire of faith over the senses, and therefore feared that some danger awaited them from an inquiry into nature. But any one who properly considers the subject will find natural philosophy to be, after the Word of God, the surest remedy against superstition, and the most approved support of faith. She is, therefore, rightly bestowed upon religion as a most faithful attendant, for the one exhibits the will and the other the power of God. Nor was he wrong who observed, “Ye err, not knowing the Scriptures and the power of God,” thus uniting in one bond the revelation of his will and the contemplation of his power. In the meanwhile, it is not wonderful that the progress of natural philosophy has been restrained, since religion, which has so much influence on men’s minds, has been led and hurried to oppose her through the ignorance of some and the imprudent zeal of others.
XC. Again, in the habits and regulations of schools, universities, and the like assemblies, destined for the abode of learned men and the improvement of learning, everything is found to be opposed to the progress of the sciences; for the lectures and exercises are so ordered, that anything out of the common track can scarcely enter the thoughts and contemplations of the mind. If, however, one or two have perhaps dared to use their liberty, they can only impose the labor on themselves, without deriving any advantage from the association of others; and if they put up with this, they will find their industry and spirit of no slight disadvantage to them in making their fortune; for the pursuits of men in such situations are, as it were, chained down to the writings of particular authors, and if any one dare to dissent from them he is immediately attacked as a turbulent and revolutionary spirit. Yet how great is the difference between civil matters and the arts, for there is not the same danger from new activity and new light. In civil matters even a change for the better is suspected on account of the commotion it occasions, for civil government is supported by authority, unanimity, fame, and public opinion, and not by demonstration. In the arts and sciences, on the contrary, every department should resound, as in mines, with new works and advances. And this is the rational, though not the actual view of the case, for that administration and government of science we have spoken of is wont too rigorously to repress its growth.
XCI. And even should the odium I have alluded to be avoided, yet it is sufficient to repress the increase of science that such attempts and industry pass unrewarded; for the cultivation of science and its reward belong not to the same individual. The advancement of science is the work of a powerful genius, the prize and reward belong to the vulgar or to princes, who (with a few exceptions) are scarcely moderately well informed. Nay, such progress is not only deprived of the rewards and beneficence of individuals, but even of popular praise; for it is above the reach of the generality, and easily overwhelmed and extinguished by the winds of common opinions. It is not wonderful, therefore, that little success has attended that which has been little honored.
XCII. But by far the greatest obstacle to the advancement of the sciences, and the undertaking of any new attempt or department, is to be found in men’s despair and the idea of impossibility; for men of a prudent and exact turn of thought are altogether diffident in matters of this nature, considering the obscurity of nature, the shortness of life, the deception of the senses, and weakness of the judgment. They think, therefore, that in the revolutions of ages and of the world there are certain floods and ebbs of the sciences, and that they grow and flourish at one time, and wither and fall off at another, that when they have attained a certain degree and condition they can proceed no further.
If, therefore, any one believe or promise greater things, they impute it to an uncurbed and immature mind, and imagine that such efforts begin pleasantly, then become laborious, and end in confusion. And since such thoughts easily enter the minds of men of dignity and excellent judgment, we must really take heed lest we should be captivated by our affection for an excellent and most beautiful object, and relax or diminish the severity of our judgment; and we must diligently examine what gleam of hope shines upon us, and in what direction it manifests itself, so that, banishing her lighter dreams, we may discuss and weigh whatever appears of more sound importance. We must consult the prudence of ordinary life, too, which is diffident upon principle, and in all human matters augurs the worst. Let us, then, speak of hope, especially as we are not vain promisers, nor are willing to enforce or insnare men’s judgment, but would rather lead them willingly forward. And although we shall employ the most cogent means of enforcing hope when we bring them to particulars, and especially those which are digested and arranged in our Tables of Invention (the subject partly of the second, but principally of the fourth part of the Instauration), which are, indeed, rather the very object of our hopes than hope itself; yet to proceed more leniently we must treat of the preparation of men’s minds, of which the manifestation of hope forms no slight part; for without it all that we have said tends rather to produce a gloom than to encourage activity or quicken the industry of experiment, by causing them to have a worse and more contemptuous opinion of things as they are than they now entertain, and to perceive and feel more thoroughly their unfortunate condition. We must, therefore, disclose and prefix our reasons for not thinking the hope of success improbable, as Columbus, before his wonderful voyage over the Atlantic, gave the reasons of his conviction that new lands and continents might be discovered besides those already known; and these reasons, though at first rejected, were yet proved by subsequent experience, and were the causes and beginnings of the greatest events.
XCIII. Let us begin from God, and show that our pursuit from its exceeding goodness clearly proceeds from him, the author of good and father of light. Now, in all divine works the smallest beginnings lead assuredly to some result, and the remark in spiritual matters that “the kingdom of God cometh without observation,” is also found to be true in every great work of Divine Providence, so that everything glides quietly on without confusion or noise, and the matter is achieved before men either think or perceive that it is commenced. Nor should we neglect to mention the prophecy of Daniel, of the last days of the world, “Many shall run to and fro, and knowledge shall be increased,” thus plainly hinting and suggesting that fate (which is Providence) would cause the complete circuit of the globe (now accomplished, or at least going forward by means of so many distant voyages), and the increase of learning to happen at the same epoch.
XCIV. We will next give a most potent reason for hope deduced from the errors of the past, and the ways still unattempted; for well was an ill-governed state thus reproved, “That which is worst with regard to the past should appear most consolatory for the future; for if you had done all that your duty commanded, and your affairs proceeded no better, you could not even hope for their improvement; but since their present unhappy situation is not owing to the force of circumstances, but to your own errors, you have reason to hope that by banishing or correcting the latter you can produce a great change for the better in the former.” So if men had, during the many years that have elapsed, adhered to the right way of discovering and cultivating the sciences without being able to advance, it would be assuredly bold and presumptuous to imagine it possible to improve; but if they have mistaken the way and wasted their labor on improper objects, it follows that the difficulty does not arise from things themselves, which are not in our power, but from the human understanding, its practice and application, which is susceptible of remedy and correction. Our best plan, therefore, is to expose these errors; for in proportion as they impeded the past, so do they afford reason to hope for the future. And although we have touched upon them above, yet we think it right to give a brief, bare, and simple enumeration of them in this place.
XCV. Those who have treated of the sciences have been either empirics or dogmatical. The former like ants only heap up and use their store, the latter like spiders spin out their own webs. The bee, a mean between both, extracts matter from the flowers of the garden and the field, but works and fashions it by its own efforts. The true labor of philosophy resembles hers, for it neither relies entirely or principally on the powers of the mind, nor yet lays up in the memory the matter afforded by the experiments of natural history and mechanics in its raw state, but changes and works it in the understanding. We have good reason, therefore, to derive hope from a closer and purer alliance of these faculties (the experimental and rational) than has yet been attempted.
XCVI. Natural philosophy is not yet to be found unadulterated, but is impure and corrupted—by logic in the school of Aristotle, by natural theology in that of Plato, by mathematics in the second school of Plato (that of Proclus and others) which ought rather to terminate natural philosophy than to generate or create it. We may, therefore, hope for better results from pure and unmixed natural philosophy.
XCVII. No one has yet been found possessed of sufficient firmness and severity to resolve upon and undertake the task of entirely abolishing common theories and notions, and applying the mind afresh, when thus cleared and levelled, to particular researches; hence our human reasoning is a mere farrago and crude mass made up of a great deal of credulity and accident, and the puerile notions it originally contracted.
But if a man of mature age, unprejudiced senses, and clear mind, would betake himself anew to experience and particulars, we might hope much more from such a one; in which respect we promise ourselves the fortune of Alexander the Great, and let none accuse us of vanity till they have heard the tale, which is intended to check vanity.
For Æschines spoke thus of Alexander and his exploits: “We live not the life of mortals, but are born at such a period that posterity will relate and declare our prodigies”; as if he considered the exploits of Alexander to be miraculous.
But in succeeding ages Livy took a better view of the fact, and has made some such observation as this upon Alexander: “That he did no more than dare to despise insignificance.” So in our opinion posterity will judge of us, that we have achieved no great matters, but only set less account upon what is considered important; for the meantime (as we have before observed) our only hope is in the regeneration of the sciences, by regularly raising them on the foundation of experience and building them anew, which I think none can venture to affirm to have been already done or even thought of.
XCVIII. The foundations of experience (our sole resource) have hitherto failed completely or have been very weak; nor has a store and collection of particular facts, capable of informing the mind or in any way satisfactory, been either sought after or amassed. On the contrary, learned, but idle and indolent, men have received some mere reports of experience, traditions as it were of dreams, as establishing or confirming their philosophy, and have not hesitated to allow them the weight of legitimate evidence. So that a system has been pursued in philosophy with regard to experience resembling that of a kingdom or state which would direct its councils and affairs according to the gossip of city and street politicians, instead of the letters and reports of ambassadors and messengers worthy of credit. Nothing is rightly inquired into, or verified, noted, weighed, or measured, in natural history; indefinite and vague observation produces fallacious and uncertain information. If this appear strange, or our complaint somewhat too unjust (because Aristotle himself, so distinguished a man and supported by the wealth of so great a king, has completed an accurate history of animals, to which others with greater diligence but less noise have made considerable additions, and others again have composed copious histories and notices of plants, metals, and fossils), it will arise from a want of sufficiently attending to and comprehending our present observations; for a natural history compiled on its own account, and one collected for the mind’s information as a foundation for philosophy, are two different things. They differ in several respects, but principally in this—the former contains only the varieties of natural species without the experiments of mechanical arts; for as in ordinary life every person’s disposition, and the concealed feelings of the mind and passions are most drawn out when they are disturbed—so the secrets of nature betray themselves more readily when tormented by art than when left to their own course. We must begin, therefore, to entertain hopes of natural philosophy then only, when we have a better compilation of natural history, its real basis and support.
XCIX. Again, even in the abundance of mechanical experiments, there is a very great scarcity of those which best inform and assist the understanding. For the mechanic, little solicitous about the investigation of truth, neither directs his attention, nor applies his hand to anything that is not of service to his business. But our hope of further progress in the sciences will then only be well founded, when numerous experiments shall be received and collected into natural history, which, though of no use in themselves, assist materially in the discovery of causes and axioms; which experiments we have termed enlightening, to distinguish them from those which are profitable. They possess this wonderful property and nature, that they never deceive or fail you; for being used only to discover the natural cause of some object, whatever be the result, they equally satisfy your aim by deciding the question.
C. We must not only search for, and procure a greater number of experiments, but also introduce a completely different method, order, and progress of continuing and promoting experience. For vague and arbitrary experience is (as we have observed), mere groping in the dark, and rather astonishes than instructs. But when experience shall proceed regularly and uninterruptedly by a determined rule, we may entertain better hopes of the sciences.
CI. But after having collected and prepared an abundance and store of natural history, and of the experience required for the operations of the understanding or philosophy, still the understanding is as incapable of acting on such materials of itself, with the aid of memory alone, as any person would be of retaining and achieving, by memory, the computation of an almanac. Yet meditation has hitherto done more for discovery than writing, and no experiments have been committed to paper. We cannot, however, approve of any mode of discovery without writing, and when that comes into more general use, we may have further hopes.
CII. Besides this, there is such a multitude and host, as it were, of particular objects, and lying so widely dispersed, as to distract and confuse the understanding; and we can, therefore, hope for no advantage from its skirmishing, and quick movements and incursions, unless we put its forces in due order and array, by means of proper and well arranged, and, as it were, living tables of discovery of these matters, which are the subject of investigation, and the mind then apply itself to the ready prepared and digested aid which such tables afford.
CIII. When we have thus properly and regularly placed before the eyes a collection of particulars, we must not immediately proceed to the investigation and discovery of new particulars or effects, or, at least, if we do so, must not rest satisfied therewith. For, though we do not deny that by transferring the experiments from one art to another (when all the experiments of each have been collected and arranged, and have been acquired by the knowledge, and subjected to the judgment of a single individual), many new experiments may be discovered tending to benefit society and mankind, by what we term literate experience; yet comparatively insignificant results are to be expected thence, while the more important are to be derived from the new light of axioms, deduced by certain method and rule from the above particulars, and pointing out and defining new particulars in their turn. Our road is not a long plain, but rises and falls, ascending to axioms, and descending to effects.
CIV. Nor can we suffer the understanding to jump and fly from particulars to remote and most general axioms (such as are termed the principles of arts and things), and thus prove and make out their intermediate axioms according to the supposed unshaken truth of the former. This, however, has always been done to the present time from the natural bent of the understanding, educated too, and accustomed to this very method, by the syllogistic mode of demonstration. But we can then only augur well for the sciences, when the assent shall proceed by a true scale and successive steps, without interruption or breach, from particulars to the lesser axioms, thence to the intermediate (rising one above the other), and lastly, to the most general. For the lowest axioms differ but little from bare experiment; the highest and most general (as they are esteemed at present), are notional, abstract, and of no real weight. The intermediate are true, solid, full of life, and upon them depend the business and fortune of mankind; beyond these are the really general, but not abstract, axioms, which are truly limited by the intermediate.
We must not then add wings, but rather lead and ballast to the understanding, to prevent its jumping or flying, which has not yet been done; but whenever this takes place, we may entertain greater hopes of the sciences.
CV. In forming axioms, we must invent a different form of induction from that hitherto in use; not only for the proof and discovery of principles (as they are called), but also of minor, intermediate, and, in short, every kind of axioms. The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too small a number of facts, and those only the most obvious. But a really useful induction for the discovery and demonstration of the arts and sciences, should separate nature by proper rejections and exclusions, and then conclude for the affirmative, after collecting a sufficient number of negatives. Now this has not been done, nor even attempted, except perhaps by Plato, who certainly uses this form of induction in some measure, to sift definitions and ideas. But much of what has never yet entered the thoughts of man must necessarily be employed, in order to exhibit a good and legitimate mode of induction or demonstration, so as even to render it essential for us to bestow more pains upon it than have hitherto been bestowed on syllogisms. The assistance of induction is to serve us not only in the discovery of axioms, but also in defining our notions. Much indeed is to be hoped from such an induction as has been described.
CVI. In forming our axioms from induction, we must examine and try whether the axiom we derive be only fitted and calculated for the particular instances from which it is deduced, or whether it be more extensive and general. If it be the latter, we must observe, whether it confirm its own extent and generality by giving surety, as it were, in pointing out new particulars, so that we may neither stop at actual discoveries, nor with a careless grasp catch at shadows and abstract forms, instead of substances of a determinate nature: and as soon as we act thus, well authorized hope may with reason be said to beam upon us.
CVII. Here, too, we may again repeat what we have said above, concerning the extending of natural philosophy and reducing particular sciences to that one, so as to prevent any schism or dismembering of the sciences; without which we cannot hope to advance.
CVIII. Such are the observations we would make in order to remove despair and excite hope, by bidding farewell to the errors of past ages, or by their correction. Let us examine whether there be other grounds for hope. And, first, if many useful discoveries have occurred to mankind by chance or opportunity, without investigation or attention on their part, it must necessarily be acknowledged that much more may be brought to light by investigation and attention, if it be regular and orderly, not hasty and interrupted. For although it may now and then happen that one falls by chance upon something that had before escaped considerable efforts and laborious inquiries, yet undoubtedly the reverse is generally the case. We may, therefore, hope for further, better, and more frequent results from man’s reason, industry, method, and application, than from chance and mere animal instinct, and the like, which have hitherto been the sources of invention.
CIX. We may also derive some reason for hope from the circumstance of several actual inventions being of such a nature, that scarcely any one could have formed a conjecture about them previously to their discovery, but would rather have ridiculed them as impossible. For men are wont to guess about new subjects from those they are already acquainted with, and the hasty and vitiated fancies they have thence formed: than which there cannot be a more fallacious mode of reasoning, because much of that which is derived from the sources of things does not flow in their usual channel.
If, for instance, before the discovery of cannon, one had described its effects in the following manner: There is a new invention by which walls and the greatest bulwarks can be shaken and overthrown from a considerable distance; men would have begun to contrive various means of multiplying the force of projectiles and machines by means of weights and wheels, and other modes of battering and projecting. But it is improbable that any imagination or fancy would have hit upon a fiery blast, expanding and developing itself so suddenly and violently, because none would have seen an instance at all resembling it, except perhaps in earthquakes or thunder, which they would have immediately rejected as the great operations of nature, not to be imitated by man.
So, if before the discovery of silk thread, any one had observed, that a species of thread had been discovered, fit for dresses and furniture, far surpassing the thread of worsted or flax in fineness, and at the same time in tenacity, beauty, and softness; men would have begun to imagine something about Chinese plants, or the fine hair of some animals, or the feathers or down of birds, but certainly would never have had an idea of its being spun by a small worm, in so copious a manner, and renewed annually. But if any one had ventured to suggest the silkworm, he would have been laughed at as if dreaming of some new manufacture from spiders.
So again, if before the discovery of the compass, any one had said, that an instrument had been invented, by which the quarters and points of the heavens could be exactly taken and distinguished, men would have entered into disquisitions on the refinement of astronomical instruments, and the like, from the excitement of their imaginations; but the thought of anything being discovered, which, not being a celestial body, but a mere mineral or metallic substance, should yet in its motion agree with that of such bodies, would have appeared absolutely incredible. Yet were these facts, and the like (unknown for so many ages) not discovered at last either by philosophy or reasoning, but by chance and opportunity; and (as we have observed), they are of a nature most heterogeneous, and remote from what was hitherto known, so that no previous knowledge could lead to them.
We may, therefore, well hope that many excellent and useful matters are yet treasured up in the bosom of nature, bearing no relation or analogy to our actual discoveries, but out of the common track of our imagination, and still undiscovered, and which will doubtless be brought to light in the course and lapse of years, as the others have been before them; but in the way we now point out, they may rapidly and at once be both represented and anticipated.
CX. There are, moreover, some inventions which render it probable that men may pass and hurry over the most noble discoveries which lie immediately before them. For however the discovery of gunpowder, silk, the compass, sugar, paper, or the like, may appear to depend on peculiar properties of things and nature, printing at least involves no contrivance which is not clear and almost obvious. But from want of observing that although the arrangement of the types of letters required more trouble than writing with the hand, yet these types once arranged serve for innumerable impressions, while manuscript only affords one copy; and again, from want of observing that ink might be thickened so as to stain without running (which was necessary, seeing the letters face upward, and the impression is made from above), this most beautiful invention (which assists so materially the propagation of learning) remained unknown for so many ages.
The human mind is often so awkward and ill-regulated in the career of invention that it is at first diffident, and then despises itself. For it appears at first incredible that any such discovery should be made, and when it has been made, it appears incredible that it should so long have escaped men’s research. All which affords good reason for the hope that a vast mass of inventions yet remains, which may be deduced not only from the investigation of new modes of operation, but also from transferring, comparing, and applying these already known, by the method of what we have termed literate experience.
CXI. Nor should we omit another ground of hope. Let men only consider (if they will) their infinite expenditure of talent, time, and fortune, in matters and studies of far inferior importance and value; a small portion of which applied to sound and solid learning would be sufficient to overcome every difficulty. And we have thought right to add this observation, because we candidly own that such a collection of natural and experimental history as we have traced in our own mind, and as is really necessary, is a great and as it were royal work, requiring much labor and expense.
CXII. In the meantime let no one be alarmed at the multitude of particulars, but rather inclined to hope on that very account. For the particular phenomena of the arts and nature are in reality but as a handful, when compared with the fictions of the imagination removed and separated from the evidence of facts. The termination of our method is clear, and I had almost said near at hand; the other admits of no termination, but only of infinite confusion. For men have hitherto dwelt but little, or rather only slightly touched upon experience, while they have wasted much time on theories and the fictions of the imagination. If we had but any one who could actually answer our interrogations of nature, the invention of all causes and sciences would be the labor of but a few years.
CXIII. We think some ground of hope is afforded by our own example, which is not mentioned for the sake of boasting, but as a useful remark. Let those who distrust their own powers observe myself, one who have among my contemporaries been the most engaged in public business, who am not very strong in health (which causes a great loss of time), and am the first explorer of this course, following the guidance of none, nor even communicating my thoughts to a single individual; yet having once firmly entered in the right way, and submitting the powers of my mind to things, I have somewhat advanced (as I make bold to think) the matter I now treat of. Then let others consider what may be hoped from men who enjoy abundant leisure, from united labors, and the succession of ages, after these suggestions on our part, especially in a course which is not confined, like theories, to individuals, but admits of the best distribution and union of labor and effect, particularly in collecting experiments. For men will then only begin to know their own power, when each performs a separate part, instead of undertaking in crowds the same work.
CXIV. Lastly, though a much more faint and uncertain breeze of hope were to spring up from our new continent, yet we consider it necessary to make the experiment, if we would not show a dastard spirit. For the risk attending want of success is not to be compared with that of neglecting the attempt; the former is attended with the loss of a little human labor, the latter with that of an immense benefit. For these and other reasons it appears to us that there is abundant ground to hope, and to induce not only those who are sanguine to make experiment, but even those who are cautious and sober to give their assent.
CXV. Such are the grounds for banishing despair, hitherto one of the most powerful causes of the delay and restraint to which the sciences have been subjected; in treating of which we have at the same time discussed the signs and causes of the errors, idleness, and ignorance that have prevailed; seeing especially that the more refined causes, which are not open to popular judgment and observation, may be referred to our remarks on the idols of the human mind.
Here, too, we should close the demolishing branch of our Instauration, which is comprised in three confutations: 1, the confutation of natural human reason left to itself; 2, the confutation of demonstration; 3, the confutation of theories, or received systems of philosophy and doctrines. Our confutation has followed such a course as was open to it, namely, the exposing of the signs of error, and the producing evidence of the causes of it: for we could adopt no other, differing as we do both in first principles and demonstrations from others.
It is time for us therefore to come to the art itself, and the rule for the interpretation of nature: there is, however, still something which must not be passed over. For the intent of this first book of aphorisms being to prepare the mind for understanding, as well as admitting, what follows, we must now, after having cleansed, polished, and levelled its surface, place it in a good position, and as it were a benevolent aspect toward our propositions; seeing that prejudice in new matters may be produced not only by the strength of preconceived notions, but also by a false anticipation or expectation of the matter proposed. We shall therefore endeavor to induce good and correct opinions of what we offer, although this be only necessary for the moment, and as it were laid out at interest, until the matter itself be well understood.
CXVI. First, then, we must desire men not to suppose that we are ambitious of founding any philosophical sect, like the ancient Greeks, or some moderns, as Telesius, Patricius, and Severinus. For neither is this our intention, nor do we think that peculiar abstract opinions on nature and the principles of things are of much importance to men’s fortunes, since it were easy to revive many ancient theories, and to introduce many new ones; as, for instance, many hypotheses with regard to the heavens can be formed, differing in themselves, and yet sufficiently according with the phenomena.
We bestow not our labor on such theoretical and, at the same time, useless topics. On the contrary, our determination is that of trying, whether we can lay a firmer foundation, and extend to a greater distance the boundaries of human power and dignity. And although here and there, upon some particular points, we hold (in our own opinion) more true and certain, and I might even say, more advantageous tenets than those in general repute (which we have collected in the fifth part of our Instauration), yet we offer no universal or complete theory. The time does not yet appear to us to be arrived, and we entertain no hope of our life being prolonged to the completion of the sixth part of the Instauration (which is destined for philosophy discovered by the interpretation of nature), but are content if we proceed quietly and usefully in our intermediate pursuit, scattering, in the meantime, the seeds of less adulterated truth for posterity, and, at least, commence the great work.
CXVII. And, as we pretend not to found a sect, so do we neither offer nor promise particular effects; which may occasion some to object to us, that since we so often speak of effects, and consider everything in its relation to that end, we ought also to give some earnest of producing them. Our course and method, however (as we have often said, and again repeat), is such as not to deduce effects from effects, nor experiments from experiments (as the empirics do), but in our capacity of legitimate interpreters of nature, to deduce causes and axioms from effects and experiments; and new effects and experiments from those causes and axioms.
And although any one of moderate intelligence and ability will observe the indications and sketches of many noble effects in our tables of inventions (which form the fourth part of the Instauration), and also in the examples of particular instances cited in the second part, as well as in our observations on history (which is the subject of the third part); yet we candidly confess that our present natural history, whether compiled from books or our own inquiries, is not sufficiently copious and well ascertained to satisfy, or even assist, a proper interpretation.
If, therefore, there be any one who is more disposed and prepared for mechanical art, and ingenious in discovering effects, than in the mere management of experiment, we allow him to employ his industry in gathering many of the fruits of our history and tables in this way, and applying them to effects, receiving them as interest till he can obtain the principal. For our own part, having a greater object in view, we condemn all hasty and premature rest in such pursuits as we would Atalanta’s apple (to use a common allusion of ours); for we are not childishly ambitious of golden fruit, but use all our efforts to make the course of art outstrip nature, and we hasten not to reap moss or the green blade, but wait for a ripe harvest.
CXVIII. There will be some, without doubt, who, on a perusal of our history and tables of invention, will meet with some uncertainty, or perhaps fallacy, in the experiments themselves, and will thence perhaps imagine that our discoveries are built on false foundations and principles. There is, however, really nothing in this, since it must needs happen in beginnings. For it is the same as if in writing or printing one or two letters were wrongly turned or misplaced, which is no great inconvenience to the reader, who can easily by his own eye correct the error; let men in the same way conclude, that many experiments in natural history may be erroneously believed and admitted, which are easily expunged and rejected afterward, by the discovery of causes and axioms. It is, however, true, that if these errors in natural history and experiments become great, frequent, and continued, they cannot be corrected and amended by any dexterity of wit or art. If then, even in our natural history, well examined and compiled with such diligence, strictness, and (I might say) reverential scruples, there be now and then something false and erroneous in the details, what must we say of the common natural history, which is so negligent and careless when compared with ours? or of systems of philosophy and the sciences, based on such loose soil (or rather quicksand)? Let none then be alarmed by such observations.
CXIX. Again, our history and experiments will contain much that is light and common, mean and illiberal, too refined and merely speculative, and, as it were, of no use, and this perhaps may divert and alienate the attention of mankind.
With regard to what is common; let men reflect, that they have hitherto been used to do nothing but refer and adapt the causes of things of rare occurrence to those of things which more frequently happen, without any investigation of the causes of the latter, taking them for granted and admitted.
Hence, they do not inquire into the causes of gravity, the rotation of the heavenly bodies, heat, cold, light, hardness, softness, rarity, density, liquidity, solidity, animation, inanimation, similitude, difference, organic formation, but taking them to be self-evident, manifest, and admitted, they dispute and decide upon other matters of less frequent and familiar occurrence.
But we (who know that no judgment can be formed of that which is rare or remarkable, and much less anything new brought to light, without a previous regular examination and discovery of the causes of that which is common, and the causes again of those causes) are necessarily compelled to admit the most common objects into our history. Besides, we have observed that nothing has been so injurious to philosophy as this circumstance, namely, that familiar and frequent objects do not arrest and detain men’s contemplation, but are carelessly admitted, and their causes never inquired after; so that information on unknown subjects is not more often wanted than attention to those which are known.
CXX. With regard to the meanness, or even the filthiness of particulars, for which (as Pliny observes), an apology is requisite, such subjects are no less worthy of admission into natural history than the most magnificent and costly; nor do they at all pollute natural history, for the sun enters alike the palace and the privy, and is not thereby polluted. We neither dedicate nor raise a capitol or pyramid to the pride of man, but rear a holy temple in his mind, on the model of the universe, which model therefore we imitate. For that which is deserving of existence is deserving of knowledge, the image of existence. Now the mean and splendid alike exist. Nay, as the finest odors are sometimes produced from putrid matter (such as musk and civet), so does valuable light and information emanate from mean and sordid instances. But we have already said too much, for such fastidious feelings are childish and effeminate.
CXXI. The next point requires a more accurate consideration, namely, that many parts of our history will appear to the vulgar, or even any mind accustomed to the present state of things, fantastically and uselessly refined. Hence, we have in regard to this matter said from the first, and must again repeat, that we look for experiments that shall afford light rather than profit, imitating the divine creation, which, as we have often observed, only produced light on the first day, and assigned that whole day to its creation, without adding any material work.
If any one, then, imagine such matters to be of no use, he might equally suppose light to be of no use, because it is neither solid nor material. For, in fact, the knowledge of simple natures, when sufficiently investigated and defined, resembles light, which, though of no great use in itself, affords access to the general mysteries of effects, and with a peculiar power comprehends and draws with it whole bands and troops of effects, and the sources of the most valuable axioms. So also the elements of letters have of themselves separately no meaning, and are of no use, yet are they, as it were, the original matter in the composition and preparation of speech. The seeds of substances, whose effect is powerful, are of no use except in their growth, and the scattered rays of light itself avail not unless collected.
But if speculative subtilties give offence, what must we say of the scholastic philosophers who indulged in them to such excess? And those subtilties were wasted on words, or, at least, common notions (which is the same thing), not on things or nature, and alike unproductive of benefit in their origin and their consequences: in no way resembling ours, which are at present useless, but in their consequences of infinite benefit. Let men be assured that all subtile disputes and discursive efforts of the mind are late and preposterous, when they are introduced subsequently to the discovery of axioms, and that their true, or, at any rate, chief opportunity is, when experiment is to be weighed and axioms to be derived from it. They otherwise catch and grasp at nature, but never seize or detain her: and we may well apply to nature that which has been said of opportunity or fortune, that she wears a lock in front, but is bald behind.
In short, we may reply decisively to those who despise any part of natural history as being vulgar, mean, or subtile, and useless in its origin, in the words of a poor woman to a haughty prince, who had rejected her petition as unworthy, and beneath the dignity of his majesty: “Then cease to reign”; for it is quite certain that the empire of nature can neither be obtained nor administered by one who refuses to pay attention to such matters as being poor and too minute.
CXXII. Again, it may be objected to us as being singular and harsh, that we should with one stroke and assault, as it were, banish all authorities and sciences, and that too by our own efforts, without requiring the assistance and support of any of the ancients.
Now we are aware, that had we been ready to act otherwise than sincerely, it was not difficult to refer our present method to remote ages, prior to those of the Greeks (since the sciences in all probability flourished more in their natural state, though silently, than when they were paraded with the fifes and trumpets of the Greeks); or even (in parts, at least) to some of the Greeks themselves, and to derive authority and honor from thence; as men of no family labor to raise and form nobility for themselves in some ancient line, by the help of genealogies. Trusting, however, to the evidence of facts, we reject every kind of fiction and imposture; and think it of no more consequence to our subject, whether future discoveries were known to the ancients, and set or rose according to the vicissitudes of events and lapse of ages, than it would be of importance to mankind to know whether the new world be the island of Atlantis, and known to the ancients, or be now discovered for the first time.
With regard to the universal censure we have bestowed, it is quite clear, to any one who properly considers the matter, that it is both more probable and more modest than any partial one could have been. For if the errors had not been rooted in the primary notions, some well conducted discoveries must have corrected others that were deficient. But since the errors were fundamental, and of such a nature, that men may be said rather to have neglected or passed over things, than to have formed a wrong or false judgment of them, it is little to be wondered at, that they did not obtain what they never aimed at, nor arrive at a goal which they had not determined, nor perform a course which they had neither entered upon nor adhered to.
With regard to our presumption, we allow that if we were to assume a power of drawing a more perfect straight line or circle than any one else, by superior steadiness of hand or acuteness of eye, it would lead to a comparison of talent; but if one merely assert that he can draw a more perfect line or circle with a ruler or compasses, than another can by his unassisted hand or eye, he surely cannot be said to boast of much. Now this applies not only to our first original attempt, but also to those who shall hereafter apply themselves to the pursuit. For our method of discovering the sciences merely levels men’s wits, and leaves but little to their superiority, since it achieves everything by the most certain rules and demonstrations. Whence (as we have often observed), our attempt is to be attributed to fortune rather than talent, and is the offspring of time rather than of wit. For a certain sort of chance has no less effect upon our thoughts than on our acts and deeds.
CXXIII. We may, therefore, apply to ourselves the joke of him who said, that water and wine drinkers could not think alike, especially as it hits the matter so well. For others, both ancients and moderns, have in the sciences drank a crude liquor like water, either flowing of itself from the understanding, or drawn up by logic as the wheel draws up the bucket. But we drink and pledge others with a liquor made of many well-ripened grapes, collected and plucked from particular branches, squeezed in the press, and at last clarified and fermented in a vessel. It is not, therefore, wonderful that we should not agree with others.
CXXIV. Another objection will without doubt be made, namely, that we have not ourselves established a correct, or the best goal or aim of the sciences (the very defect we blame in others). For they will say that the contemplation of truth is more dignified and exalted than any utility or extent of effects; but that our dwelling so long and anxiously on experience and matter, and the fluctuating state of particulars, fastens the mind to earth, or rather casts it down into an abyss of confusion and disturbance, and separates and removes it from a much more divine state, the quiet and tranquillity of abstract wisdom. We willingly assent to their reasoning, and are most anxious to effect the very point they hint at and require. For we are founding a real model of the world in the understanding, such as it is found to be, not such as man’s reason has distorted. Now this cannot be done without dissecting and anatomizing the world most diligently; but we declare it necessary to destroy completely the vain, little and, as it were, apish imitations of the world, which have been formed in various systems of philosophy by men’s fancies. Let men learn (as we have said above) the difference that exists between the idols of the human mind and the ideas of the divine mind. The former are mere arbitrary abstractions; the latter the true marks of the Creator on his creatures, as they are imprinted on, and defined in matter, by true and exquisite touches. Truth, therefore, and utility, are here perfectly identical, and the effects are of more value as pledges of truth than from the benefit they confer on men.
CXXV. Others may object that we are only doing that which has already been done, and that the ancients followed the same course as ourselves. They may imagine, therefore, that, after all this stir and exertion, we shall at last arrive at some of those systems that prevailed among the ancients: for that they, too, when commencing their meditations, laid up a great store of instances and particulars, and digested them under topics and titles in their commonplace books, and so worked out their systems and arts, and then decided upon what they discovered, and related now and then some examples to confirm and throw light upon their doctrine; but thought it superfluous and troublesome to publish their notes, minutes, and commonplaces, and therefore followed the example of builders who remove the scaffolding and ladders when the building is finished. Nor can we indeed believe the case to have been otherwise. But to any one, not entirely forgetful of our previous observations, it will be easy to answer this objection or rather scruple; for we allow that the ancients had a particular form of investigation and discovery, and their writings show it. But it was of such a nature, that they immediately flew from a few instances and particulars (after adding some common notions, and a few generally received opinions most in vogue) to the most general conclusions or the principles of the sciences, and then by their intermediate propositions deduced their inferior conclusions, and tried them by the test of the immovable and settled truth of the first, and so constructed their art. Lastly, if some new particulars and instances were brought forward, which contradicted their dogmas, they either with great subtilty reduced them to one system, by distinctions or explanations of their own rules, or got rid of them clumsily as exceptions, laboring most pertinaciously in the meantime to accommodate the causes of such as were not contradictory to their own principles. Their natural history and their experience were both far from being what they ought to have been, and their flying off to generalities ruined everything.
CXXVI. Another objection will be made against us, that we prohibit decisions and the laying down of certain principles, till we arrive regularly at generalities by the intermediate steps, and thus keep the judgment in suspense and lead to uncertainty. But our object is not uncertainty but fitting certainty, for we derogate not from the senses but assist them, and despise not the understanding but direct it. It is better to know what is necessary, and not to imagine we are fully in possession of it, than to imagine that we are fully in possession of it, and yet in reality to know nothing which we ought.
CXXVII. Again, some may raise this question rather than objection, whether we talk of perfecting natural philosophy alone according to our method, or the other sciences also, such as logic, ethics, politics. We certainly intend to comprehend them all. And as common logic, which regulates matters by syllogisms, is applied not only to natural, but also to every other science, so our inductive method likewise comprehends them all. For we form a history and tables of invention for anger, fear, shame, and the like, and also for examples in civil life, and the mental operations of memory, composition, division, judgment, and the rest, as well as for heat and cold, light, vegetation, and the like. But since our method of interpretation, after preparing and arranging a history, does not content itself with examining the operations and disquisitions of the mind like common logic, but also inspects the nature of things, we so regulate the mind that it may be enabled to apply itself in every respect correctly to that nature. On that account we deliver numerous and various precepts in our doctrine of interpretation, so that they may apply in some measure to the method of discovering the quality and condition of the subject matter of investigation.
CXXVIII. Let none even doubt whether we are anxious to destroy and demolish the philosophy, arts, and sciences, which are now in use. On the contrary, we readily cherish their practice, cultivation, and honor; for we by no means interfere to prevent the prevalent system from encouraging discussion, adorning discourses, or being employed serviceably in the chair of the professor or the practice of common life, and being taken, in short, by general consent as current coin. Nay, we plainly declare, that the system we offer will not be very suitable for such purposes, not being easily adapted to vulgar apprehensions, except by effects and works. To show our sincerity in professing our regard and friendly disposition toward the received sciences, we can refer to the evidence of our published writings (especially our books on the Advancement of Learning). We will not, therefore, endeavor to evince it any further by words; but content ourselves with steadily and professedly premising, that no great progress can be made by the present methods in the theory or contemplation of science, and that they cannot be made to produce any very abundant effects.
CXXIX. It remains for us to say a few words on the excellence of our proposed end. If we had done so before, we might have appeared merely to express our wishes, but now that we have excited hope and removed prejudices, it will perhaps have greater weight. Had we performed and completely accomplished the whole, without frequently calling in others to assist in our labors, we should then have refrained from saying any more, lest we should be thought to extol our own deserts. Since, however, the industry of others must be quickened, and their courage roused and inflamed, it is right to recall some points to their memory.
First, then, the introduction of great inventions appears one of the most distinguished of human actions, and the ancients so considered it; for they assigned divine honors to the authors of inventions, but only heroic honors to those who displayed civil merit (such as the founders of cities and empire legislators, the deliverers of their country from lasting misfortunes, the quellers of tyrants, and the like). And if any one rightly compare them, he will find the judgment of antiquity to be correct; for the benefits derived from inventions may extend to mankind in general, but civil benefits to particular spots alone; the latter, moreover, last but for a time, the former forever. Civil reformation seldom is carried on without violence and confusion, while inventions are a blessing and a benefit without injuring or afflicting any.
Inventions are also, as it were, new creations and imitations of divine works, as was expressed by the poet:
“Primum frugiferos fœtus mortalibus ægris
Dididerant quondam præstanti nomine Athenæ
Et recreaverunt vitam legesque rogarunt.”
And it is worthy of remark in Solomon, that while he flourished in the possession of his empire, in wealth, in the magnificence of his works, in his court, his household, his fleet, the splendor of his name, and the most unbounded admiration of mankind, he still placed his glory in none of these, but declared that it is the glory of God to conceal a thing, but the glory of a king to search it out.
Again, let any one but consider the immense difference between men’s lives in the most polished countries of Europe, and in any wild and barbarous region of the new Indies, he will think it so great, that man may be said to be a god unto man, not only on account of mutual aid and benefits, but from their comparative states—the result of the arts, and not of the soil or climate.
Again, we should notice the force, effect, and consequences of inventions, which are nowhere more conspicuous than in those three which were unknown to the ancients; namely, printing, gunpowder, and the compass. For these three have changed the appearance and state of the whole world: first in literature, then in warfare, and lastly in navigation; and innumerable changes have been thence derived, so that no empire, sect, or star, appears to have exercised a greater power and influence on human affairs than these mechanical discoveries.
It will, perhaps, be as well to distinguish three species and degrees of ambition. First, that of men who are anxious to enlarge their own power in their country, which is a vulgar and degenerate kind; next, that of men who strive to enlarge the power and empire of their country over mankind, which is more dignified but not less covetous; but if one were to endeavor to renew and enlarge the power and empire of mankind in general over the universe, such ambition (if it may be so termed) is both more sound and more noble than the other two. Now the empire of man over things is founded on the arts and sciences alone, for nature is only to be commanded by obeying her.
Besides this, if the benefit of any particular invention has had such an effect as to induce men to consider him greater than a man, who has thus obliged the whole race, how much more exalted will that discovery be, which leads to the easy discovery of everything else! Yet (to speak the truth) in the same manner as we are very thankful for light which enables us to enter on our way, to practice arts, to read, to distinguish each other, and yet sight is more excellent and beautiful than the various uses of light; so is the contemplation of things as they are, free from superstition or imposture, error or confusion, much more dignified in itself than all the advantage to be derived from discoveries.
Lastly, let none be alarmed at the objection of the arts and sciences becoming depraved to malevolent or luxurious purposes and the like, for the same can be said of every worldly good; talent, courage, strength, beauty, riches, light itself, and the rest. Only let mankind regain their rights over nature, assigned to them by the gift of God, and obtain that power, whose exercise will be governed by right reason and true religion.
CXXX. But it is time for us to lay down the art of interpreting nature, to which we attribute no absolute necessity (as if nothing could be done without it) nor perfection, although we think that our precepts are most useful and correct. For we are of opinion, that if men had at their command a proper history of nature and experience, and would apply themselves steadily to it, and could bind themselves to two things: 1, to lay aside received opinions and notions; 2, to restrain themselves, till the proper season, from generalization, they might, by the proper and genuine exertion of their minds, fall into our way of interpretation without the aid of any art. For interpretation is the true and natural act of the mind, when all obstacles are removed: certainly, however, everything will be more ready and better fixed by our precepts.
Yet do we not affirm that no addition can be made to them; on the contrary, considering the mind in its connection with things, and not merely relatively to its own powers, we ought to be persuaded that the art of invention can be made to grow with the inventions themselves.
FOOTNOTES
Bacon uses the term in its ancient sense, and means one who, knowing the occult properties of bodies, is able to startle the ignorant by drawing out of them wonderful and unforeseen changes. See the 85th aphorism of this book, and the 5th cap. book iii. of the De Augmentis Scientiarum, where he speaks more clearly.—Ed.
By this term axiomata, Bacon here speaks of general principles, or universal laws. In the 19th aphorism he employs the term to express any proposition collected from facts by induction, and thus fitted to become the starting-point of deductive reasoning. In the last and more rigorous sense of the term, Bacon held they arose from experience. See Whewell’s “Philosophy of the Inductive Sciences,” vol.i. p.74; and Mill’s “Logic,” vol.i. p.311; and the June “Quarterly,” 1841, for the modern phase of the discussion.—Ed.
Bacon here attributes to the Aristotelian logic the erroneous consequences which sprung out of its abuse. The demonstrative forms it exhibits, whether verbally or mathematically expressed, are necessary to the support, verification, and extension of induction, and when the propositions they embrace are founded on an accurate and close observation of facts, the conclusions to which they lead, even in moral science, may be regarded as certain as the facts wrested out of nature by direct experiment. In physics such forms are absolutely required to generalize the results of experience, and to connect intermediate axioms with laws still more general, as is sufficiently attested by the fact, that no science since Bacon’s day has ceased to be experimental by the mere method of induction, and that all become exact only so far as they rise above experience, and connect their isolated phenomena with general laws by the principles of deductive reasoning. So far, then, are these forms from being useless, that they are absolutely essential to the advancement of the sciences, and in no case can be looked on as detrimental, except when obtruded in the place of direct experiment, or employed as a means of deducing conclusions about nature from imaginary hypotheses and abstract conceptions. This had been unfortunately the practice of the Greeks. From the rapid development geometry received in their hands, they imagined the same method would lead to results equally brilliant in natural science, and snatching up some abstract principle, which they carefully removed from the test of experiment, imagined they could reason out from it all the laws and external appearances of the universe. The scholastics were impelled along the same path, not only by precedent, but by profession. Theology was the only science which received from them a consistent development, and the à priori grounds on which it rested prevented them from employing any other method in the pursuit of natural phenomena. Thus, forms of demonstration, in themselves accurate, and of momentous value in their proper sphere, became confounded with fable, and led men into the idea they were exploring truth when they were only accurately deducing error from error. One principle ever so slightly deflected, like a false quantity in an equation, could be sufficient to infect the whole series of conclusions of which it was the base; and though the philosopher might subsequently deduce a thousand consecutive inferences with the utmost accuracy or precision, he would only succeed in drawing out very methodically nine hundred and ninety-nine errors.—Ed.
It would appear from this and the two preceding aphorisms, that Bacon fell into the error of denying the utility of the syllogism in the very part of inductive science where it is essentially required. Logic, like mathematics, is purely a formal process, and must, as the scaffolding to the building, be employed to arrange facts in the structure of a science, and not to form any portion of its groundwork, or to supply the materials of which the system is to be composed. The word syllogism, like most other psychological terms, has no fixed or original signification, but is sometimes employed, as it was by the Greeks, to denote general reasoning, and at others to point out the formal method of deducing a particular inference from two or more general propositions. Bacon does not confine the term within the boundaries of express definition, but leaves us to infer that he took it in the latter sense, from his custom of associating the term with the wranglings of the schools. The scholastics, it is true, abused the deductive syllogism, by employing it in its naked, skeleton-like form, and confounding it with the whole breadth of logical theory; but their errors are not to be visited on Aristotle, who never dreamed of playing with formal syllogisms, and, least of all, mistook the descending for the ascending series of inference. In our mind we are of accord with the Stagyrite, who propounds, as far as we can interpret him, two modes of investigating truth—the one by which we ascend from particular and singular facts to general laws and axioms, and the other by which we descend from universal propositions to the individual cases which they virtually include. Logic, therefore, must equally vindicate the formal purity of the synthetic illation by which it ascends to the whole, as the analytic process by which it descends to the parts. The deductive and inductive syllogism are of equal significance in building up any body of truth, and whoever restricts logic to either process, mistakes one-half of its province for the whole; and if he acts upon his error, will paralyze his methods, and strike the noblest part of science with sterility.—Ed.
The Latin is, ad ea quæ revera sunt naturæ notiora. This expression, naturæ notiora, naturæ notior, is so frequently employed by Bacon, that we may conclude it to point to some distinguishing feature in the Baconian physics. It properly refers to the most evident principles and laws of nature, and springs from that system which regards the material universe as endowed with intelligence, and acting according to rules either fashioned or clearly understood by itself.—Ed.
This Borgia was Alexander VI., and the expedition alluded to that in which Charles VIII. overran the Italian peninsula in five months. Bacon uses the same illustration in concluding his survey of natural philosophy, in the second book of the “De Augmentis.”—Ed.
Ratio eorum qui acatalepsiam tenuerunt. Bacon alludes to the members of the later academy, who held the ἀκατάληψια, or the impossibility of comprehending anything. His translator, however, makes him refer to the sceptics, who neither dogmatized about the known or the unknown, but simply held, that as all knowledge was relative, πρòς πάντα τι, man could never arrive at absolute truth, and therefore could not with certainty affirm or deny anything.—Ed.
It is argued by Hallam, with some appearance of truth, that idols is not the correct translation of εἴδωλα, from which the original idola is manifestly derived; but that Bacon used it in the literal sense attached to it by the Greeks, as a species of illusion, or false appearance, and not as a species of divinity before which the mind bows down. If Hallam be right, Bacon is saved from the odium of an analogy which his foreign commentators are not far wrong in denouncing as barbarous; but this service is rendered at the expense of the men who have attached an opposite meaning to the word, among whom are Brown, Playfair and Dugald Stewart.—Ed.
We cannot see how these idols have less to do with sophistical paralogisms than with natural philosophy. The process of scientific induction involves only the first elements of reasoning, and presents such a clear and tangible surface, as to allow no lurking-place for prejudice; while questions of politics and morals, to which the deductive method, or common logic, as Bacon calls it, is peculiarly applicable, are ever liable to be swayed or perverted by the prejudices he enumerates. After mathematics, physical science is the least amenable to the illusions of feeling; each portion having been already tested by experiment and observation, is fitted into its place in the system, with all the rigor of the geometrical method; affection or prejudice cannot, as in matters of taste, history or religion, select fragmentary pieces, and form a system of their own. The whole must be admitted, or the structure of authoritative reason razed to the ground. It is needless to say that the idols enumerated present only another interpretation of the substance of logical fallacies.—Ed.
The propensity to this illusion may be viewed in the spirit of system, or hasty generalization, which is still one of the chief obstacles in the path of modern science.—Ed.
Though Kepler had, when Bacon wrote this, already demonstrated his three great laws concerning the elliptical path of the planets, neither Bacon nor Descartes seems to have known or assented to his discoveries. Our author deemed the startling astronomical announcements of his time to be mere theoretic solutions of the phenomena of the heavens, not so perfect as those advanced by antiquity, but still deserving a praise for the ingenuity displayed in their contrivance. Bacon believed a hundred such systems might exist, and though true in their explanation of phenomena, yet might all more or less differ, according to the preconceived notions which their framers brought to the survey of the heavens. He even thought he might put in his claim to the notice of posterity for his astronomical ingenuity, and, as Ptolemy had labored by means of epicycles and eccentrics, and Kepler with ellipses, to explain the laws of planetary motion, Bacon thought the mystery would unfold itself quite as philosophically through spiral labyrinths and serpentine lines. What the details of his system were, we are left to conjecture, and that from a very meagre but naïve account of one of his inventions which he has left in his Miscellany MSS.—Ed.
Hinc elementum ignis cum orbe suo introductum est. Bacon saw in fire the mere result of a certain combination of action, and was consequently led to deny its elementary character. The ancient physicists attributed an orbit to each of the four elements, into which they resolved the universe, and supposed their spheres to involve each other. The orbit of the earth was in the centre, that of fire at the circumference. For Bacon’s inquisition into the nature of heat, and its complete failure, see the commencement of the second book of the Novum Organum.—Ed.
Robert Fludd is the theorist alluded to, who had supposed the gravity of the earth to be ten times heavier than water, that of water ten times heavier than air, and that of air ten times heavier than fire.—Ed.
Diagoras. The same allusion occurs in the second part of the Advancement of Learning, where Bacon treats of the idols of the mind.
A scholastic term, to signify the two eternities of past and future duration, that stretch out on both sides of the narrow isthmus (time) occupied by man. It must be remembered that Bacon lived before the doctrine of limits gave rise to the higher calculus, and therefore could have no conception of different denominations of infinities: on the other hand he would have thought the man insane who should have talked to him about lines infinitely great, inclosing angles infinitely little; that a right line, which is a right line so long as it is finite, by changing infinitely little its direction, becomes an infinite curve, and that a curve may become infinitely less than another curve; that there are infinite squares and infinite cubes, and infinites of infinites, all greater than one another, and the last but one of which is nothing in comparison with the last. Yet half a century sufficed from Bacon’s time, to make this nomenclature, which would have appeared to him the excess of frenzy, not only reasonable but necessary, to grasp the higher demonstrations of physical science.—Ed.
Spinoza, in his letter to Oldenberg (Op. Posth. p.398), considers this aphorism based on a wrong conception of the origin of error, and, believing it to be fundamental, was led to reject Bacon’s method altogether. Spinoza refused to acknowledge in man any such thing as a will, and resolved all his volitions into particular acts, which he considered to be as fatally determined by a chain of physical causes as any effects in nature.—Ed.
Operatio spirituum in corporibus tangibilibus. Bacon distinguished with the schools the gross and tangible parts of bodies, from such as were volatile and intangible. These, in conformity with the scholastic language, he terms spirits, and frequently returns to their operations in the 2d book.—Ed.
Democritus, of Abdera, a disciple of Leucippus, born B.C.470, died 360; all his works are destroyed. He is said to be the author of the doctrine of atoms: he denied the immortality of the soul, and first taught that the milky way was occasioned by a confused light from a multitude of stars. He may be considered as the parent of experimental philosophy, in the prosecution of which he was so ardent as to declare that he would prefer the discovery of one of the causes of natural phenomena, to the possession of the diadem of Persia. Democritus imposed on the blind credulity of his contemporaries, and, like Roger Bacon, astonished them by his inventions.—Ed.
The Latin is actus purus, another scholastic expression to denote the action of the substance, which composes the essence of the body apart from its accidental qualities. For an exposition of the various kinds of motions he contemplates, the reader may refer to the 48th aphorism of the 2d book.—Ed.
The scholastics after Aristotle distinguished in a subject three modes of beings: viz., the power or faculty, the act, and the habitude, or in other words that which is able to exist, what exists actually, and what continues to exist. Bacon means that is necessary to fix our attention not on that which can or ought to be, but on that which actually is; not on the right, but on the fact.—Ed.
The inference to be drawn from this is to suspect that kind of evidence which is most consonant to our inclinations, and not to admit any notion as real except we can base it firmly upon that kind of demonstration which is peculiar to the subject, not to our impression. Sometimes the mode of proof may be consonant to our inclinations, and to the subject at the same time, as in the case of Pythagoras, when he applied his beloved numbers to the solution of astronomical phenomena; or in that of Descartes, when he reasoned geometrically concerning the nature of the soul. Such examples cannot be censured with justice, inasmuch as the methods pursued were adapted to the end of the inquiry. The remark in the text can only apply to those philosophers who attempt to build up a moral or theological system by the instruments of induction alone, or who rush, with the geometrical axiom, and the à priori syllogism, to the investigation of nature. The means in such cases are totally inadequate to the object in view.—Ed.
Gilbert lived toward the close of the sixteenth century, and was court physician to both Elizabeth and James. In his work alluded to in the text he continually asserts the advantages of the experimental over the à priori method in physical inquiry, and succeeded when his censor failed in giving a practical example of the utility of his precepts. His “De Magnete” contains all the fundamental parts of the science, and these so perfectly treated, that we have nothing to add to them at the present day.
Gilbert adopted the Copernican system, and even spoke of the contrary theory as utterly absurd, grounding his argument on the vast velocities which such a supposition requires us to ascribe to the heavenly bodies.—Ed.
The Latin text adds “without end”; but Bacon is scarcely right in supposing that the descent from complex ideas and propositions to those of simple nature, involve the analyst in a series of continuous and interminable definitions. For in the gradual and analytical scale, there is a bar beyond which we cannot go, as there is a summit bounded by the limited variations of our conceptions. Logical definitions, to fulfil their conditions, or indeed to be of any avail, must be given in simpler terms than the object which is sought to be defined; now this, in the case of primordial notions and objects of sense, is impossible; therefore we are obliged to rest satisfied with the mere names of our perceptions.—Ed.
The ancients supposed the planets to describe an exact circle round the south. As observations increased and facts were disclosed, which were irreconcilable with this supposition, the earth was removed from the centre to some other point in the circle, and the planets were supposed to revolve in a smaller circle (epicycle) round an imaginary point, which in its turn described a circle of which the earth was the centre. In proportion as observation elicited fresh facts, contradictory to these representations, other epicycles and eccentrics were added, involving additional confusion. Though Kepler had swept away all these complicated theories in the preceding century, by the demonstration of his three laws, which established the elliptical course of the planets, Bacon regarded him and Copernicus in the same light as Ptolemy and Xenophanes.—Ed.
Empedocles, of Agrigentum, flourished 444 B.C. He was the disciple of Telanges the Pythagorean, and warmly adopted the doctrine of transmigration. He resolved the universe into the four ordinary elements, the principles of whose composition were life and happiness, or concord and amity, but whose decomposition brought forth death and evil, or discord and hatred. Heraclitus held matter to be indifferent to any peculiar form, but as it became rarer or more dense, it took the appearance of fire, air, earth and water. Fire, however, he believed to be the elementary principle out of which the others were evolved. This was also the belief of Lucretius. See book i. 783, etc.
It is thus the Vulcanists and Neptunians have framed their opposite theories in geology. Phrenology is a modern instance of hasty generalization.—Ed.
In Scripture everything which concerns the passing interests of the body is called dead; the only living knowledge having regard to the eternal interest of the soul.—Ed.
In mechanics and the general sciences, causes compound their effects, or in other words, it is generally possible to deduce à priori the consequence of introducing complex agencies into any experiment, by allowing for the effect of each of the simple causes which enter into their composition. In chemistry and physiology a contrary law holds; the causes which they embody generally uniting to form distinct substances, and to introduce unforeseen laws and combinations. The deductive method here is consequently inapplicable, and we are forced back upon experiment.
Bacon in the text is hardly consistent with himself, as he admits in the second book the doctrine, to which modern discovery points, of the reciprocal transmutation of the elements. What seemed poetic fiction in the theories of Pythagoras and Seneca, assumes the appearance of scientific fact in the hands of Baron Caynard.—Ed.
Galileo had recently adopted the notion that nature abhorred a vacuum for an axiomatic principle, and it was not till Torricelli, his disciple, had given practical proof of the utility of Bacon’s method, by the discovery of the barometer(1643) that this error, as also that expressed below, and believed by Bacon, concerning the homœopathic tendencies of bodies, was destroyed.—Ed.
Donec ad materiam potentialem et informem ventum fuerit. Nearly all the ancient philosophers admitted the existence of a certain primitive and shapeless matter as the substratum of things which the creative power had reduced to fixed proportions, and resolved into specific substances. The expression potential matter refers to that substance forming the basis of the Peripatetic system, which virtually contained all the forms that it was in the power of the efficient cause to draw out of it.—Ed.
An allusion to the humanity of the Sultans, who, in their earlier histories are represented as signalizing their accession to the throne by the destruction of their family, to remove the danger of rivalry and the terrors of civil war.—Ed.
The text is “in odium veterum sophistarum, Protagoræ, Hippiæ, et reliquorum.” Those were called sophists, who, ostentationis aut questus causa philosophabantur. (Acad. Prior. ii.72.) They had corrupted and degraded philosophy before Socrates. Protagoras of Abdera(Ἄβδηρα), the most celebrated, taught that man is the measure of all things, by which he meant not only that all which can be known is known only as it related to our faculties, but also that apart from our faculties nothing can be known. The sceptics equally held that knowledge was probable only as it related to our faculties, but they stopped there, and did not, like the sophist, dogmatize about the unknown. The works of Protagoras were condemned for their impiety, and publicly burned by the ædiles of Athens, who appear to have discharged the office of common hangmen to the literary blasphemers of their day.—Ed.
Bacon is hardly correct in implying that the enumerationem per simplicem was the only light in which the ancients looked upon induction, as they appear to have regarded it as only one, and that the least important, of its species. Aristotle expressly considers induction in a perfect or dialectic sense, and in an imperfect or rhetorical sense. Thus if a genus(G), contains four species (A, B, C, D), the syllogism would lead us to infer, that what is true of G, is true of any one of the four. But perfect induction would reason, that what we can prove of A, B, C, D, separately, we may properly state as true of G, the whole genus. This is evidently a formal argument as demonstrative as the syllogism. In necessary matters, however, legitimate induction may claim a wider province, and infer of the whole genus what is only apparent in a part of the species. Such are those inductive inferences which concern the laws of nature, the immutability of forms, by which Bacon strove to erect his new system of philosophy. The Stagyrite, however, looked upon enumerationem per simplicem, without any regard to the nature of the matter, or to the completeness of the species, with as much reprehensive caution as Bacon, and guarded his readers against it as the source of innumerable errors.—Ed.
See Ax.lxi. toward the end. This subject extends to Ax.lxxviii.
Gorgias of Leontium went to Athens in 424 B.C. He and Polus were disciples of Empedocles, whom we have already noticed (Aphorism 63), where he sustained the three famous propositions, that nothing exists, that nothing can be known, and that it is out of the power of man to transmit or communicate intelligence. He is reckoned one of the earliest writers on the art of rhetoric, and for that reason, Plato called his elegant dialogue on that subject after his name.
Chrysippus, a stoic philosopher of Soli in Cilicia, Campestris, born in 280, died in the 143d Olympiad, 208 B.C. He was equally distinguished for natural abilities and industry, seldom suffering a day to elapse without writing 500 lines. He wrote several hundred volumes, of which three hundred were on logical subjects; but in all, borrowed largely from others. He was very fond of the sorites in argument, which is hence called by Persius the heap of Chrysippus. He was called the Column of the Portico, a name given to the Stoical School from Zeno, its founder, who had given his lessons under the portico.
Carneades, born about 215, died in 130. He attached himself to Chrysippus, and sustained with éclat the scepticism of the academy. The Athenians sent him with Critolaus and Diogenes as ambassador to Rome, where he attracted the attention of his new auditory by the subtilty of his reasoning, and the fluency and vehemence of his language. Before Galba and Cato the Censor, he harangued with great variety of thought and copiousness of diction in praise of justice. The next day, to establish his doctrine of the uncertainty of human knowledge, he undertook to refute all his arguments. He maintained with the New Academy, that the senses, the imagination, and the understanding frequently deceive us, and therefore cannot be infallible judges of truth, but that from the impressions produced on the mind by means of the senses, we infer appearances of truth or probabilities. Nevertheless, with respect to the conduct of life, Carneades held that probable opinions are a sufficient guide.
Xenophanes, a Greek philosopher, of Colophon, born in 556, the founder of the Eleatic school, which owes its fame principally to Parmenides. Wild in his opinions about astronomy, he supposed that the stars were extinguished every morning, and rekindled at night; that eclipses were occasioned by the temporary extinction of the sun, and that there were several suns for the convenience of the different climates of the earth. Yet this man held the chair of philosophy at Athens for seventy years.
Philolaus, a Pythagorean philosopher of Crotona, B.C.374. He first supported the diurnal motion of the earth round its axis, and its annual motion round the sun. Cicero (Acad. iv.39) has ascribed this opinion to the Syracusan philosopher Nicetas, and likewise to Plato. From this passage, it is most probable that Copernicus got the idea of the system he afterward established. Bacon, in the Advancement of Human Learning, charges Gilbert with restoring the doctrines of Philolaus, because he ventured to support the Copernican theory.—Ed.
Bacon is equally conspicuous for the use and abuse of analogical illustrations. The levity, as Stuart Mill very properly observes, by which substances float on a stream, and the levity which is synonymous with worthlessness, have nothing beside the name in common; and to show how little value there is in the figure, we need only change the word into buoyancy, to turn the semblance of Bacon’s argument against himself.—Ed.
We have before observed, that the New Academy did not profess skepticism, but the ἀκατάληψια, or incomprehensibility of the absolute essences of things. Even modern physicists are not wanting, to assert with this school that the utmost knowledge we can obtain is relative, and necessarily short of absolute certainty. It is not without an appearance of truth that these philosophers maintain that our ideas and perceptions do not express the nature of the things which they represent, but only the effects of the peculiar organs by which they are conveyed to the understanding, so that were these organs changed, we should have different conceptions of their nature. That constitution of air which is dark to man is luminous to bats and owls.
Owing to the universal prevalence of Aristotelism.
It must be remembered, that when Bacon wrote, algebra was in its infancy, and the doctrine of units and infinitesimals undiscovered.
Because the vulgar make up the overwhelming majority in such decisions, and generally allow their judgments to be swayed by passion or prejudice.
See end of Axiom lxi. The subject extends to Axiom xc.
If we adopt the statement of Herodotus, who places the Homeric era 400 years back from his time, Homer lived about 900 years before Christ. On adding this number to the sixteen centuries of the Christian era which had elapsed up to Bacon’s time, we get the twenty-five centuries he mentions. The Homeric epoch is the furthest point in antiquity from which Bacon could reckon with any degree of certainty. Hesiod, if he were not contemporary, immediately preceded him.
The epoch of Greek philosophy may be included between Thales and Plato, that is, from the 35th to the 88th Olympiad; that of the Roman, between Terence and Pliny. The modern revolution, in which Bacon is one of the central figures, took its rise from the time of Dante and Petrarch, who lived at the commencement of the fourteenth century; and to which, on account of the invention of printing, and the universal spread of literature, which has rendered a second destruction of learning impossible, it is difficult to foresee any other end than the extinction of the race of man.—Ed.
The allusion is evidently to Roger Bacon and Réné Descartes.—Ed.
From the abuse of the scholastics, who mistook the à priori method, the deductive syllogism, for the entire province of logic.—Ed.
See Aphorism xcv.
The incongruity to which Bacon alludes appears to spring from confounding two things, which are not only distinct, but affect human knowledge in inverse proportion, viz., the experience which terminates with life, with that experience which one century transmits to another.—Ed.
The Chinese characters resemble, in many respects, the hieroglyphics of the Egyptians, being adapted to represent ideas, not sounds.
See Axiom 75.
The methods by which Newton carried the rule and compass to the boundaries of creation is a sufficient comment on the sagacity of the text. The same cause which globulizes a bubble, has rounded the earth, and the same law which draws a stone to its surface, keeps the moon in her orbit. It was by calculating and ascertaining these principles upon substances entirely at his disposal that this great philosopher was enabled to give us a key to unlock the mysteries of the universe.—Ed.
See the “Clouds” of Aristophanes, where Socrates is represented as chasing Jupiter out of the sky, by resolving thunderstorms into aërial concussions and whirlwinds.—Ed.
Robespierre was the latest victim of this bigotry. In his younger days he attempted to introduce Franklin’s lightning conductor into France, but was persecuted by those whose lives he sought to protect, as one audaciously striving to avert the designs of Providence.—Ed.
We can hardly agree with the text. The scholastics, in building up a system of divinity, certainly had recourse to the deductive syllogism, because the inductive was totally inapplicable, except as a verificatory process. With regard to the technical form in which they marshalled their arguments, which is what our author aims at in his censure, they owed nothing at all to Aristotle, the conducting a dispute in naked syllogistic fashion having originated entirely with themselves.—Ed.
Bacon cannot be supposed to allude to those divines who have attempted to show that the progress of physical science is confirmatory of revelation, but only to such as have built up a system of faith out of their own refinements on nature and revelation, as Patricius and Emanuel Swedenborg.—Ed.
Daniel xii.4.
Bacon, in this Aphorism, appears to have entertained a fair idea of the use of the inductive and deductive methods in scientific inquiry, though his want of geometrical knowledge must have hindered him from accurately determining the precise functions of each, as it certainly led him in other parts of the Organon (V.Aph.82), to undervalue the deductive, and, as he calls it, the dogmatic method, and to rely too much upon empiricism.—Ed.
The reader may consult the note of the 23d Aphorism for the fault which Bacon censures, and, if he wish to pursue the subject further, may read Plato’s Timæus, where that philosopher explains his system in detail. Bacon, however, is hardly consistent in one part of his censure, for he also talks about the spirit and appetites of inanimate substances, and that so frequently, as to preclude the supposition that he is employing metaphor.—Ed.
Proclus flourished about the beginning of the fifth century, and was the successor of Plotinus, Porphyry and Iamblicus, who, in the two preceding centuries, had revived the doctrines of Plato, and assailed the Christian religion. The allusion in the text must be assigned to Iamblicus, who, in the fourth century, had republished the Pythagorean theology of numbers, and endeavored to construct the world out of arithmetic, thinking everything could be solved by the aid of proportions and geometry. Bacon must not be understood in the text to censure the use but the abuse of mathematics and physical investigations, as in the “De Augmentis” (lib.iv. c.6), he enumerates the multiplicity of demonstration scientific facts admit of, from this source.—Ed.
See Livy, lib.ix. c.17, where, in a digression on the probable effect of a contest between Rome and Alexander the Great, he says: “Non cum Dario rem esse dixisset: quem mulierum ac spadonum agmen trahentem inter purpuram atque aurum, oneratum fortunæ apparatibus, prædam veriùs quam hostem, nihil aliud quam ausus vana contemnere, incruentus devicit.”
The lowest axioms are such as spring from simple experience—such as in chemistry, that animal substances yield no fixed salt by calcination; in music, that concords intermixed with discords make harmony, etc. Intermediate axioms advance a step further, being the result of reflection, which, applied to our experimental knowledge, deduces laws from them, such as in optics of the first degree of generality, that the angle of incidence is equal to the angle of reflection; and in mechanics, Kepler’s three laws of motion, while his general law, that all bodies attract each other with forces proportional to their masses, and inversely as the squares of their distances, may be taken as one of the highest axioms. Yet so far is this principle from being only notional or abstract, it has presented us with a key which fits into the intricate wards of the heavens, and has laid bare to our gaze the principal mechanism of the universe. But natural philosophy in Bacon’s day had not advanced beyond intermediate axioms, and the term notional or abstract is applied to those general axioms then current, not founded on the solid principles of inductive inquiry, but based upon à priori reasoning and airy metaphysics.—Ed.
This hope has been abundantly realized in the discovery of gravity and the decomposition of light, mainly by the inductive method. To a better philosophy we may also attribute the discovery of electricity, galvanism and their mutual connection with each other, and magnetism, the inventions of the air-pump, steam-engine and the chronometer.
As Bacon very frequently cites these authors, a slight notice of their labors may not be unacceptable to the reader. Bernardinus Telesius, born at Cosenza, in 1508, combated the Aristotelian system in a work entitled “De Rerum Natura juxta propria principia,” i.e., according to principles of his own. The proem of the work announces his design was to show that “the construction of the world, the magnitude and nature of the bodies contained in it, are not to be investigated by reasoning, which was done by the ancients, but are to be apprehended by the senses, and collected from the things themselves.” He had, however, no sooner laid down this principle than he departed from it in practice, and pursued the deductive method he so much condemned in his predecessors. His first step was an assumption of principles as arbitrary as any of the empirical notions of antiquity; at the outset of his book he very quietly takes it for granted that heat is the principle of motion, cold of immobility, matter being assumed as the corporeal substratum, in which these incorporeal and active agents carry on their operations. Out of these abstract and ill-defined conceptions Telesius builds up a system quite as complete, symmetrical, and imaginative as any of the structures of antiquity.
Francis Patricius, born at Cherso, in Dalmatia, about 1529, was another physicist who rose up against Aristotle, and announced the dawn of a new philosophy. In 1593 appeared his “Nova de Universis Philosophia.” He lays down a string of axioms, in which scholastic notions, physical discoveries, and theological dogmas, are strangely commingled, and erects upon them a system which represents all the grotesque features of theological empiricism.
Severinus, born in Jutland, in 1529, published an attack on Aristotle’s natural history, but adopted fantasies which the Stagyrite ridiculed in his own day. He was a follower of Paracelsus, a Swiss enthusiast of the fifteenth century, who ignored the ancient doctrine of the four elements for salt, sulphur and mercury, and allied chemistry and medicine with mysticism.—Ed.
Bacon’s apology is sound, and completely answers those German and French critics, who have refused him a niche in the philosophical pantheon. One German commentator, too modest to reveal his name, accuses Bacon of ignorance of the calculus, though, in his day, Wallis had not yet stumbled upon the laws of continuous fractions; while Count de Maistre, in a coarse attack upon his genius, expresses his astonishment at finding Bacon unacquainted with discoveries which were not heard of till a century after his death.—Ed.
Philip of Macedon.
See Plato’s Timæus.
The saying of Philocrates when he differed from Demosthenes.—Ed.
The old error of placing the deductive syllogism in antagonism to the inductive, as if they were not both parts of one system or refused to cohere together. So far from there being any radical opposition between them, it would not be difficult to show that Bacon’s method was syllogistic in his sense of the term. For the suppressed premise of every Baconian enthymeme, viz., the acknowledged uniformity of the laws of nature as stated in the axiom, whatever has once occurred will occur again, must be assumed as the basis of every conclusion which he draws before we can admit its legitimacy. The opposition, therefore, of Bacon’s method could not be directed against the old logic, for it assumed and exemplified its principles, but rather to the abusive application which the ancients made of this science, on turning its powers to the development of abstract principles which they imagined to be pregnant with the solution of the latent mysteries of the universe. Bacon justly overthrew these ideal notions, and accepted of no principle as a basis which was not guaranteed by actual experiment and observation; and so far he laid the foundations of a sound philosophy by turning the inductive logic to its proper account in the interpretation of nature.
This is the opening of the Sixth Book of Lucretius. Bacon probably quoted from memory; the lines are—
“Primæ frugiferos fœtus mortalibus ægris
Dididerunt quondam præclaro nomine Athenæ
Et recreaverunt,” etc.
The teeming corn, that feeble mortals crave,
First, and long since, renowned Athens gave,
And cheered their life—then taught to frame their laws.
Prov. xxv.2.
APHORISMS—BOOK II
ON THE INTERPRETATION OF NATURE, OR THE REIGN OF MAN
I.To generate and superinduce a new nature or new natures, upon a given body, is the labor and aim of human power: while to discover the form or true difference of a given nature, or the nature to which such nature is owing, or source from which it emanates (for these terms approach nearest to an explanation of our meaning), is the labor and discovery of human knowledge; and subordinate to these primary labors are two others of a secondary nature and inferior stamp. Under the first must be ranked the transformation of concrete bodies from one to another, which is possible within certain limits; under the second, the discovery, in every species of generation and motion, of the latent and uninterrupted process from the manifest efficient and manifest subject matter up to the given form: and a like discovery of the latent conformation of bodies which are at rest instead of being in motion.
II. The unhappy state of man’s actual knowledge is manifested even by the common assertions of the vulgar. It is rightly laid down that true knowledge is that which is deduced from causes. The division of four causes also is not amiss: matter, form, the efficient, and end or final cause. Of these, however, the latter is so far from being beneficial, that it even corrupts the sciences, except in the intercourse of man with man. The discovery of form is considered desperate. As for the efficient cause and matter (according to the present system of inquiry and the received opinions concerning them, by which they are placed remote from, and without any latent process toward form), they are but desultory and superficial, and of scarcely any avail to real and active knowledge. Nor are we unmindful of our having pointed out and corrected above the error of the human mind, in assigning the first qualities of essence to forms. For although nothing exists in nature except individual bodies, exhibiting clear individual effects according to particular laws, yet in each branch of learning, that very law, its investigation, discovery, and development, are the foundation both of theory and practice. This law, therefore, and its parallel in each science, is what we understand by the term form, adopting that word because it has grown into common use, and is of familiar occurrence.
III. He who has learned the cause of a particular nature (such as whiteness or heat), in particular subjects only, has acquired but an imperfect knowledge: as he who can induce a certain effect upon particular substances only, among those which are susceptible of it, has acquired but an imperfect power. But he who has only learned the efficient and material cause (which causes are variable and mere vehicles conveying form to particular substances) may perhaps arrive at some new discoveries in matters of a similar nature, and prepared for the purpose, but does not stir the limits of things which are much more deeply rooted; while he who is acquainted with forms, comprehends the unity of nature in substances apparently most distinct from each other. He can disclose and bring forward, therefore (though it has never yet been done), things which neither the vicissitudes of nature, nor the industry of experiment, nor chance itself, would ever have brought about, and which would forever have escaped man’s thoughts; from the discovery of forms, therefore, results genuine theory and free practice.
IV. Although there is a most intimate connection, and almost an identity between the ways of human power and human knowledge, yet, on account of the pernicious and inveterate habit of dwelling upon abstractions, it is by far the safest method to commence and build up the sciences from those foundations which bear a relation to the practical division, and to let them mark out and limit the theoretical. We must consider, therefore, what precepts, or what direction or guide, a person would most desire, in order to generate and superinduce any nature upon a given body: and this not in abstruse, but in the plainest language.
For instance, if a person should wish to superinduce the yellow color of gold upon silver, or an additional weight (observing always the laws of matter) or transparency on an opaque stone, or tenacity in glass, or vegetation on a substance which is not vegetable, we must (I say) consider what species of precept or guide this person would prefer. And, first, he will doubtless be anxious to be shown some method that will neither fail in effect, nor deceive him in the trial of it; secondly, he will be anxious that the prescribed method should not restrict him and tie him down to peculiar means, and certain particular methods of acting; for he will, perhaps, be at loss, and without the power or opportunity of collecting and procuring such means. Now if there be other means and methods (besides those prescribed) of creating such a nature, they will perhaps be of such a kind as are in his power, yet by the confined limits of the precept he will be deprived of reaping any advantage from them; thirdly, he will be anxious to be shown something not so difficult as the required effect itself, but approaching more nearly to practice.
We will lay this down, therefore, as the genuine and perfect rule of practice, that it should be certain, free and preparatory, or having relation to practice. And this is the same thing as the discovery of a true form; for the form of any nature is such, that when it is assigned the particular nature infallibly follows. It is, therefore, always present when that nature is present, and universally attests such presence, and is inherent in the whole of it. The same form is of such a character, that if it be removed the particular nature infallibly vanishes. It is, therefore, absent, whenever that nature is absent, and perpetually testifies such absence, and exists in no other nature. Lastly, the true form is such, that it deduces the particular nature from some source of essence existing in many subjects, and more known (as they term it) to nature, than the form itself. Such, then, is our determination and rule with regard to a genuine and perfect theoretical axiom, that a nature be found convertible with a given nature, and yet such as to limit the more known nature, in the manner of a real genus. But these two rules, the practical and theoretical, are in fact the same, and that which is most useful in practice is most correct in theory.
V. But the rule or axiom for the transformation of bodies is of two kinds. The first regards the body as an aggregate or combination of simple natures. Thus, in gold are united the following circumstances: it is yellow, heavy, of a certain weight, malleable and ductile to a certain extent; it is not volatile, loses part of its substance by fire, melts in a particular manner, is separated and dissolved by particular methods, and so of the other natures observable in gold. An axiom, therefore, of this kind deduces the subject from the forms of simple natures; for he who has acquired the forms and methods of superinducing yellowness, weight, ductility, stability, deliquescence, solution, and the like, and their degrees and modes, will consider and contrive how to unite them in any body, so as to transform it into gold. And this method of operating belongs to primary action; for it is the same thing to produce one or many simple natures, except that man is more confined and restricted in his operations, if many be required, on account of the difficulty of uniting many natures together. It must, however, be observed, that this method of operating (which considers natures as simple though in a concrete body) sets out from what is constant, eternal, and universal in nature, and opens such broad paths to human power, as the thoughts of man can in the present state of things scarcely comprehend or figure to itself.
The second kind of axiom (which depends on the discovery of the latent process) does not proceed by simple natures, but by concrete bodies, as they are found in nature and in its usual course. For instance, suppose the inquiry to be, from what beginnings, in what manner, and by what process gold or any metal or stone is generated from the original menstruum, or its elements, up to the perfect mineral: or, in like manner, by what process plants are generated, from the first concretion of juices in the earth, or from seeds, up to the perfect plant, with the whole successive motion, and varied and uninterrupted efforts of nature; and the same inquiry be made as to a regularly deduced system of the generation of animals from coition to birth, and so on of other bodies.
Nor is this species of inquiry confined to the mere generation of bodies, but it is applicable to other changes and labors of nature. For instance, where an inquiry is made into the whole series and continued operation of the nutritive process, from the first reception of the food to its complete assimilation to the recipient; or into the voluntary motion of animals, from the first impression of the imagination, and the continuous effects of the spirits, up to the bending and motion of the joints; or into the free motion of the tongue and lips, and other accessories which give utterance to articulate sounds. For all these investigations relate to concrete or associated natures artificially brought together, and take into consideration certain particular and special habits of nature, and not those fundamental and general laws which constitute forms. It must, however, be plainly owned, that this method appears more prompt and easy, and of greater promise than the primary one.
In like manner the operative branch, which answers to this contemplative branch, extends and advances its operation from that which is usually observed in nature, to other subjects immediately connected with it, or not very remote from such immediate connection. But the higher and radical operations upon nature depend entirely on the primary axioms. Besides, even where man has not the means of acting, but only of acquiring knowledge, as in astronomy (for man cannot act upon, change, or transform the heavenly bodies), the investigation of facts or truth, as well as the knowledge of causes and coincidences, must be referred to those primary and universal axioms that regard simple natures; such as the nature of spontaneous rotation, attraction, or the magnetic force, and many others which are more common than the heavenly bodies themselves. For let no one hope to determine the question whether the earth or heaven revolve in the diurnal motion, unless he have first comprehended the nature of spontaneous rotation.
VI. But the latent process of which we speak, is far from being obvious to men’s minds, beset as they now are. For we mean not the measures, symptoms, or degrees of any process which can be exhibited in the bodies themselves, but simply a continued process, which, for the most part, escapes the observation of the senses.
For instance, in all generations and transformations of bodies, we must inquire, what is in the act of being lost and escaping, what remains, what is being added, what is being diluted, what is being contracted, what is being united, what is being separated, what is continuous, what is broken off, what is urging forward, what impedes, what predominates, what is subservient, and many other circumstances.
Nor are these inquiries again to be made in the mere generation and transformation of bodies only, but in all other alterations and fluctuations we must in like manner inquire; what precedes, what succeeds, what is quick, what is slow, what produces and what governs motion, and the like. All which matters are unknown and unattempted by the sciences, in their present heavy and inactive state. For, since every natural act is brought about by the smallest efforts, or at least such as are too small to strike our senses, let no one hope that he will be able to direct or change nature unless he have properly comprehended and observed these efforts.
VII. In like manner, the investigation and discovery of the latent conformation in bodies is no less new, than the discovery of the latent process and form. For we as yet are doubtless only admitted to the antechamber of nature, and do not prepare an entrance into her presence-room. But nobody can endue a given body with a new nature, or transform it successfully and appropriately into a new body, without possessing a complete knowledge of the body so to be changed or transformed. For he will run into vain, or, at least, into difficult and perverse methods, ill adapted to the nature of the body upon which he operates. A clear path, therefore, toward this object also must be thrown open, and well supported.
Labor is well and usefully bestowed upon the anatomy of organized bodies, such as those of men and animals, which appears to be a subtile matter, and a useful examination of nature. The species of anatomy, however, is that of first sight, open to the senses, and takes place only in organized bodies. It is obvious, and of ready access, when compared with the real anatomy of latent conformation in bodies which are considered similar, particularly in specific objects and their parts; as those of iron, stone, and the similar parts of plants and animals, as the root, the leaf, the flower, the flesh, the blood, and bones, etc. Yet human industry has not completely neglected this species of anatomy; for we have an instance of it in the separation of similar bodies by distillation, and other solutions, which shows the dissimilarity of the compound by the union of the homogeneous parts. These methods are useful, and of importance to our inquiry, although attended generally with fallacy: for many natures are assigned and attributed to the separate bodies, as if they had previously existed in the compound, which, in reality, are recently bestowed and superinduced by fire and heat, and the other modes of separation. Besides, it is, after all, but a small part of the labor of discovering the real conformation in the compound, which is so subtile and nice, that it is rather confused and lost by the operation of the fire, than discovered and brought to light.
A separation and solution of bodies, therefore, is to be effected, not by fire indeed, but rather by reasoning and true induction, with the assistance of experiment, and by a comparison with other bodies, and a reduction to those simple natures and their forms which meet, and are combined in the compound; and we must assuredly pass from Vulcan to Minerva, if we wish to bring to light the real texture and conformation of bodies, upon which every occult and (as it is sometimes called) specific property and virtue of things depends, and whence also every rule of powerful change and transformation is deduced.
For instance, we must examine what spirit is in every body, what tangible essence; whether that spirit is copious and exuberant, or meagre and scarce, fine or coarse, aëriform or igniform, active or sluggish, weak or robust, progressive or retrograde, abrupt or continuous, agreeing with external and surrounding objects, or differing from them, etc. In like manner must we treat tangible essence (which admits of as many distinctions as the spirit), and its hairs, fibres, and varied texture. Again, the situation of the spirit in the corporeal mass, its pores, passages, veins, and cells, and the rudiments or first essays of the organic body, are subject to the same examination. In these, however, as in our former inquiries, and therefore in the whole investigation of latent conformation, the only genuine and clear light which completely dispels all darkness and subtile difficulties, is admitted by means of the primary axioms.
VIII. This method will not bring us to atoms, which takes for granted the vacuum, and immutability of matter (neither of which hypotheses is correct), but to the real particles such as we discover them to be. Nor is there any ground for alarm at this refinement as if it were inexplicable, for, on the contrary, the more inquiry is directed to simple natures, the more will everything be placed in a plain and perspicuous light, since we transfer our attention from the complicated to the simple, from the incommensurable to the commensurable, from surds to rational quantities, from the indefinite and vague to the definite and certain; as when we arrive at the elements of letters, and the simple tones of concords. The investigation of nature is best conducted when mathematics are applied to physics. Again, let none be alarmed at vast numbers and fractions, for in calculation it is as easy to set down or to reflect upon a thousand as a unit, or the thousandth part of an integer as an integer itself.
IX. From the two kinds of axioms above specified, arise the two divisions of philosophy and the sciences, and we will use the commonly adopted terms which approach the nearest to our meaning, in our own sense. Let the investigation of forms, which (in reasoning at least, and after their own laws), are eternal and immutable, constitute metaphysics, and let the investigation of the efficient cause of matter, latent process, and latent conformation (which all relate merely to the ordinary course of nature, and not to her fundamental and eternal laws), constitute physics. Parallel to these, let there be two practical divisions; to physics that of mechanics, and to metaphysics that of magic, in the purest sense of the term, as applied to its ample means, and its command over nature.
X. The object of our philosophy being thus laid down, we proceed to precepts, in the most clear and regular order. The signs for the interpretation of nature comprehend two divisions; the first regards the eliciting or creating of axioms from experiment, the second the deducing or deriving of new experiments from axioms. The first admits of three subdivisions into ministrations. 1.To the senses. 2.To the memory. 3.To the mind or reason.
For we must first prepare as a foundation for the whole, a complete and accurate natural and experimental history. We must not imagine or invent, but discover the acts and properties of nature.
But natural and experimental history is so varied and diffuse, that it confounds and distracts the understanding unless it be fixed and exhibited in due order. We must, therefore, form tables and co-ordinations of instances, upon such a plan, and in such order that the understanding may be enabled to act upon them.
Even when this is done, the understanding, left to itself and to its own operation, is incompetent and unfit to construct its axioms without direction and support. Our third ministration, therefore, must be true and legitimate induction, the very key of interpretation. We must begin, however, at the end, and go back again to the others.
XI. The investigation of forms proceeds thus: a nature being given, we must first present to the understanding all the known instances which agree in the same nature, although the subject matter be considerably diversified. And this collection must be made as a mere history, and without any premature reflection, or too great degree of refinement. For instance; take the investigation of the form of heat.
Instances agreeing in the Form of Heat
The rays of the sun, particularly in summer, and at noon.
The same reflected and condensed, as between mountains, or along walls, and particularly in burning mirrors.
Ignited meteors.
Burning lightning.
Eruptions of flames from the cavities of mountains, etc.
Flame of every kind.
Ignited solids.
Natural warm baths.
Warm or heated liquids.
Warm vapors and smoke; and the air itself, which admits a most powerful and violent heat if confined, as in reverberating furnaces.
Damp hot weather, arising from the constitution of the air, without any reference to the time of the year.
Confined and subterraneous air in some caverns, particularly in winter.
All shaggy substances, as wool, the skins of animals, and the plumage of birds, contain some heat.
All bodies, both solid and liquid, dense and rare (as the air itself), placed near fire for any time.
Sparks arising from the violent percussion of flint and steel.
All bodies rubbed violently, as stone, wood, cloth, etc., so that rudders, and axles of wheels, sometimes catch fire, and the West Indians obtain fire by attrition.
Green and moist vegetable matter confined and rubbed together, as roses, peas in baskets; so hay, if it be damp when stacked, often catches fire.
Quicklime sprinkled with water.
Iron, when first dissolved by acids in a glass, and without any application to fire; the same of tin, but not so intensely.
Animals, particularly internally; although the heat is not perceivable by the touch in insects, on account of their small size.
Horse dung, and the like excrement from other animals, when fresh.
Strong oil of sulphur and of vitriol exhibit the operation of heat in burning linen.
As does the oil of marjoram, and like substances, in burning the bony substance of the teeth.
24. Strong and well rectified spirits of wine exhibit the same effects; so that white of eggs when thrown into it grows hard and white, almost in the same manner as when boiled, and bread becomes burned and brown as if toasted.
Aromatic substances and warm plants, as the dracunculus [arum], old nasturtium, etc., which, though they be not warm to the touch (whether whole or pulverized), yet are discovered by the tongue and palate to be warm and almost burning when slightly masticated.
Strong vinegar and all acids, on any part of the body not clothed with the epidermis, as the eye, tongue, or any wounded part, or where the skin is removed, excite a pain differing but little from that produced by heat.
Even a severe and intense cold produces a sensation of burning.
“Nec Boreæ penetrabile frigus adurit.”
28. Other instances.
We are wont to call this a table of existence and presence.
XII. We must next present to the understanding instances which do not admit of the given nature, for form (as we have observed) ought no less to be absent where the given nature is absent, than to be present where it is present. If, however, we were to examine every instance, our labor would be infinite.
Negatives, therefore, must be classed under the affirmatives, and the want of the given nature must be inquired into more particularly in objects which have a very close connection with those others in which it is present and manifest. And this we are wont to term a table of deviation or of absence in proximity.
Proximate Instances wanting the Nature of Heat
The rays of the moon, stars, and comets, are not found to be warm to the touch, nay, the severest cold has been observed to take place at the full of the moon. Yet the larger fixed stars are supposed to increase and render more intense the heat of the sun, as he approaches them, when the sun is in the sign of the Lion, for instance, and in the dog-days.
The rays of the sun in what is called the middle region of the air give no heat, to account for which the commonly assigned reason is satisfactory; namely, that that region is neither sufficiently near to the body of the sun whence the rays emanate, nor to the earth whence they are reflected. And the fact is manifested by snow being perpetual on the tops of mountains, unless extremely lofty. But it is observed, on the other hand, by some, that at the Peak of Teneriffe, and also among the Andes of Peru, the tops of the mountains are free from snow, which only lies in the lower part as you ascend. Besides, the air on the summit of these mountains is found to be by no means cold, but only thin and sharp; so much so, that in the Andes it pricks and hurts the eyes from its extreme sharpness, and even excites the orifice of the stomach and produces vomiting. The ancients also observed, that the rarity of the air on the summit of Olympus was such, that those who ascended it were obliged to carry sponges moistened with vinegar and water, and to apply them now and then to their nostrils, as the air was not dense enough for their respiration; on the summit of which mountain it is also related, there reigned so great a serenity and calm, free from rain, snow, or wind, that the letters traced upon the ashes of the sacrifices on the altar of Jupiter, by the fingers of those who had offered them, would remain undisturbed till the next year. Those even, who at this day go to the top of the Peak of Teneriffe, walk by night and not in the daytime, and are advised and pressed by their guides, as soon as the sun rises, to make haste in their descent, on account of the danger (apparently arising from the rarity of the atmosphere), lest their breathing should be relaxed and suffocated.
The reflection of the solar rays in the polar regions is found to be weak and inefficient in producing heat, so that the Dutch, who wintered in Nova Zembla, and expected that their vessel would be freed about the beginning of July from the obstruction of the mass of ice which had blocked it up, were disappointed and obliged to embark in their boat. Hence the direct rays of the sun appear to have but little power even on the plain, and when reflected, unless they are multiplied and condensed, which takes place when the sun tends more to the perpendicular; for, then, the incidence of the rays occurs at more acute angles, so that the reflected rays are nearer to each other, while, on the contrary, when the sun is in a very oblique position, the angles of incidence are very obtuse, and the reflected rays at a greater distance. In the meantime it must be observed, that there may be many operations of the solar rays, relating, too, to the nature of heat, which are not proportioned to our touch, so that, with regard to us, they do not tend to produce warmth, but, with regard to some other bodies, have their due effect in producing it.
Let the following experiment be made. Take a lens the reverse of a burning-glass, and place it between the hand and the solar rays, and observe whether it diminish the heat of the sun as a burning-glass increases it. For it is clear, with regard to the visual rays, that in proportion as the lens is made of unequal thickness in the middle and at its sides, the images appear either more diffused or contracted. It should be seen, therefore, if the same be true with regard to heat.
Let the experiment be well tried, whether the lunar rays can be received and collected by the strongest and best burning-glasses, so as to produce even the least degree of heat. But if that degree be, perhaps, so subtile and weak, as not to be perceived or ascertained by the touch, we must have recourse to those glasses which indicate the warm or cold state of the atmosphere, and let the lunar rays fall through the burning-glass on the top of this thermometer, and then notice if the water be depressed by the heat.
Let the burning-glass be tried on warm objects which emit no luminous rays, as heated but not ignited iron or stone, or hot water, or the like; and observe whether the heat become increased and condensed, as happens with the solar rays.
Let it be tried on common flame.
The effect of comets (if we can reckon them among meteors) in augmenting the heat of the season is not found to be constant or clear, although droughts have generally been observed to follow them. However, luminous lines, and pillars, and openings, and the like, appear more often in winter than in summer, and especially with the most intense cold but joined with drought. Lightning, and coruscations, and thunder, however, rarely happen in winter; and generally at the time of the greatest heats. The appearances we term falling stars are generally supposed to consist of some shining and inflamed viscous substance, rather than of violently hot matter; but let this be further investigated.
Some coruscations emit light without burning, but are never accompanied by thunder.
Eructations and eruptions of flame are to be found in cold climates as well as in hot, as in Iceland and Greenland; just as the trees of cold countries are sometimes inflammable and more pitchy and resinous than in warm, as the fir, pine, and the like. But the position and nature of the soil, where such eruptions are wont to happen, is not yet sufficiently investigated to enable us to subjoin a negative instance to the affirmative.
All flame is constantly more or less warm, and this instance is not altogether negative; yet it is said that the ignis fatuus (as it is called), and which sometimes is driven against walls, has but little heat; perhaps it resembles that of spirits of wine, which is mild and gentle. That flame, however, appears yet milder, which in some well authenticated and serious histories is said to have appeared round the head and hair of boys and virgins, and instead of burning their hair, merely to have played about it. And it is most certain that a sort of flash, without any evident heat, has sometimes been seen about a horse when sweating at night, or in damp weather. It is also a well known fact, and it was almost considered as a miracle, that a few years since a girl’s apron sparkled when a little shaken or rubbed, which was, perhaps, occasioned by the alum or salts with which the apron was imbued, and which, after having been stuck together and incrusted rather strongly, were broken by the friction. It is well known that all sugar, whether candied or plain, if it be hard, will sparkle when broken or scraped in the dark. In like manner sea and salt water is sometimes found to shine at night when struck violently by the oar. The foam of the sea when agitated by tempests also sparkles at night, and the Spaniards call this appearance the sea’s lungs. It has not been sufficiently ascertained what degree of heat attends the flame which the ancient sailors called Castor and Pollux, and the moderns call St.Ermus’ fire.
Every ignited body that is red-hot is always warm, although without flame, nor is any negative instance subjoined to this affirmative. Rotten wood, however, approaches nearly to it, for it shines at night, and yet is not found to be warm; and the putrefying scales of fish which shine in the same manner are not warm to the touch, nor the body of the glowworm, or of the fly called Lucciola.
The situation and nature of the soil of natural warm baths has not been sufficiently investigated, and therefore a negative instance is not subjoined.
To the instances of warm liquids we may subjoin the negative one of the peculiar nature of liquids in general; for no tangible liquid is known that is at once warm in its nature and constantly continues warm; but their heat is only superinduced as an adventitious nature for a limited time, so that those which are extremely warm in their power and effect, as spirits of wine, chemical aromatic oils, the oils of vitriol and sulphur, and the like, and which speedily burn, are yet cold at first to the touch, and the water of natural baths, poured into any vessel and separated from its source, cools down like water heated by the fire. It is, however, true that oily substances are rather less cold to the touch than those that are aqueous, oil for instance than water, silk than linen; but this belongs to the table of degrees of cold.
In like manner we may subjoin a negative instance to that of warm vapor, derived from the nature of vapor itself, as far as we are acquainted with it. For exhalations from oily substances, though easily inflammable, are yet never warm unless recently exhaled from some warm substance.
The same may be said of the instance of air; for we never perceive that air is warm unless confined or pressed, or manifestly heated by the sun, by fire, or some other warm body.
A negative instance is exhibited in weather by its coldness with an east or north wind, beyond what the season would lead us to expect, just as the contrary takes place with the south or west winds. An inclination to rain (especially in winter) attends warm weather, and to frost cold weather.
A negative instance as to air confined in caverns may be observed in summer. Indeed, we should make a more diligent inquiry into the nature of confined air. For in the first place the qualities of air in its own nature with regard to heat and cold may reasonably be the subject of doubt; for air evidently derives its heat from the effects of celestial bodies, and possibly its cold from the exhalation of the earth, and in the mid region of air (as it is termed) from cold vapors and snow, so that no judgment can be formed of the nature of air by that which is out of doors and exposed, but a more correct one might be derived from confined air. It is necessary, however, that the air should be inclosed in a vessel of such materials as would not imbue it with heat or cold of themselves, nor easily admit the influence of the external atmosphere. The experiment should be made, therefore, with an earthen jar, covered with folds of leather to protect it from the external air, and the air should be kept three or four days in this vessel well closed. On opening the jar, the degree of heat may be ascertained either by the hand or a graduated glass tube.
There is a similar doubt as to whether the warmth of wool, skins, feathers, and the like, is derived from a slight inherent heat, since they are animal excretions, or from their being of a certain fat and oily nature that accords with heat, or merely from the confinement and separation of air which we spoke of in the preceding paragraph; for all air appears to possess a certain degree of warmth when separated from the external atmosphere. Let an experiment be made, therefore, with fibrous substances of linen, and not of wool, feathers, or silk, which are animal excretions. For it is to be observed that all powders (where air is manifestly inclosed) are less cold than the substances when whole, just as we imagine froth (which contains air) to be less cold than the liquid itself.
We have here no exactly negative instance, for we are not acquainted with any body tangible or spirituous which does not admit of heat when exposed to the fire. There is, however, this difference, that some admit it more rapidly, as air, oil, and water, others more slowly, as stone and metals. This, however, belongs to the table of degrees.
No negative is here subjoined, except the remark that sparks are not kindled by flint and steel, or any other hard substance, unless some small particles of the stone or metal are struck off, and that the air never forms them by friction, as is commonly supposed; besides, the sparks from the weight of the ignited substance have a tendency to descend rather than to rise, and when extinguished become a sort of dark ash.
We are of opinion that here again there is no negative; for we are not acquainted with any tangible body which does not become decidedly warm by friction, so that the ancients feigned that the gods had no other means or power of creating heat than the friction of air, by rapid and violent rotation. On this point, however, further inquiry must be made, whether bodies projected by machines (as balls from cannon) do not derive some degree of heat from meeting the air, which renders them somewhat warm when they fall. The air in motion rather cools than heats, as in the winds, the bellows, or breath when the mouth is contracted. The motion, however, in such instances is not sufficiently rapid to excite heat, and is applied to a body of air, and not to its component parts, so that it is not surprising that heat should not be generated.
We must make a more diligent inquiry into this instance; for herbs and green and moist vegetables appear to possess a latent heat, so small, however, as not to be perceived by the touch in single specimens, but when they are united and confined, so that their spirit cannot exhale into the air, and they rather warm each other, their heat is at once manifested, and even flame occasionally in suitable substances.
Here, too, we must make a more diligent inquiry; for quicklime, when sprinkled with water, appears to conceive heat, either from its being collected into one point (as we observed of herbs when confined), or from the irritation and exasperation of the fiery spirit by water, which occasions a conflict and struggle. The true reason will more readily be shown if oil be used instead of water, for oil will equally tend to collect the confined spirit, but not to irritate. The experiment may be made more general, both by using the ashes and calcined products of different bodies and by pouring different liquids upon them.
A negative instance may be subjoined of other metals which are more soft and soluble; for leaf gold dissolved by aqua regia, or lead by aqua fortis, are not warm to the touch while dissolving, no more is quicksilver (as far as I remember), but silver excites a slight heat, and so does copper, and tin yet more plainly, and most of all iron and steel, which excite not only a powerful heat, but a violent bubbling. The heat, therefore, appears to be occasioned by the struggle which takes place when these strong dissolvents penetrate, dig into, and tear asunder the parts of those substances, while the substances themselves resist. When, however, the substances yield more easily, scarcely any heat is excited.
There is no negative instance with regard to the heat of animals, except in insects (as has been observed), owing to their small size; for in fishes, as compared with land animals, a lower degree rather than a deprivation of heat is observable. In plants and vegetables, both as to their exudations and pith when freshly exposed, there is no sensible degree of heat. But in animals there is a great difference in the degree, both in particular parts (for the heat varies near the heart, the brain, and the extremities) and in the circumstances in which they are placed, such as violent exercise and fevers.
Here, again, there is scarcely a negative instance. I might add that the excrements of animals, even when they are no longer fresh, possess evidently some effective heat, as is shown by their enriching the soil.
Such liquids (whether oily or watery) as are intensely acrid exhibit the effects of heat, by the separation and burning of bodies after some little action upon them, yet they are not at first warm to the touch, but they act according to their affinity and the pores of the substances to which they are applied; for aqua regia dissolves gold but not silver—on the contrary, aqua fortis dissolves silver but not gold; neither of them dissolves glass, and so of the rest.
Let spirits of wine be tried on wood, or butter, wax, or pitch, to see if this will melt them at all by their heat; for the twenty-fourth instance shows that they possess properties resembling those of heat in causing incrustation. Let an experiment also be made with a graduated glass or calendar, concave at the top, by pouring well-rectified spirits of wine into the cavity, and covering it up in order that they may the better retain their heat, then observe whether their heat make the water descend.
Spices and acrid herbs are sensibly warm to the palate, and still more so when taken internally; one should see, therefore, on what other substances they exhibit the effects of heat. Now, sailors tell us that when large quantities of spices are suddenly opened, after having been shut up for some time, there is some danger of fever and inflammation to those who stir them or take them out. An experiment might, therefore, be made whether such spices and herbs, when produced, will, like smoke, dry fish and meat hung up over them.
There is an acrid effect and a degree of penetration in cold liquids, such as vinegar and oil of vitriol, as well as in warm, such as oil of marjoram and the like; they have, therefore, an equal effect in causing animated substances to smart, and separating and consuming inanimate parts. There is not any negative instance as to this, nor does there exist any animal pain unaccompanied by the sensation of heat.
There are many effects common to cold and heat, however different in their process; for snowballs appear to burn boys’ hands after a little time, and cold no less than fire preserves bodies from putrefaction—besides both heat and cold contract bodies. But it is better to refer these instances and the like to the investigation of cold.
XIII. In the third place we must exhibit to the understanding the instances in which that nature, which is the object of our inquiries, is present in a greater or less degree, either by comparing its increase and decrease in the same object, or its degree in different objects; for since the form of a thing is its very essence, and the thing only differs from its form as the apparent from the actual object, or the exterior from the interior, or that which is considered with relation to man from that which is considered with relation to the universe; it necessarily follows that no nature can be considered a real form which does not uniformly diminish and increase with the given nature. We are wont to call this our Table of Degrees, or Comparative Instances.
Table of the Degrees or Comparative Instances of Heat
We will first speak of those bodies which exhibit no degree of heat sensible to the touch, but appear rather to possess a potential heat, or disposition and preparation for it. We will then go on to others, which are actually warm to the touch, and observe the strength and degree of it.
There is no known solid or tangible body which is by its own nature originally warm; for neither stone, metal, sulphur, fossils, wood, water, nor dead animal carcasses are found warm. The warm springs in baths appear to be heated accidentally, by flame, subterraneous fire (such as is thrown up by Etna and many other mountains), or by the contact of certain bodies, as heat is exhibited in the dissolution of iron and tin. The degree of heat, therefore, in inanimate objects is not sensible to our touch; but they differ in their degrees of cold, for wood and metal are not equally cold. This, however, belongs to the Table of Degrees of Cold.
But with regard to potential heat and predisposition to flame, we find many inanimate substances wonderfully adapted to it, as sulphur, naphtha, and saltpetre.
Bodies which have previously acquired heat, as horse dung from the animal, or lime, and perhaps ashes and soot from fire, retain some latent portion of it. Hence distillations and separations of substances are effected by burying them in horse dung, and heat is excited in lime by sprinkling it with water (as has been before observed).
In the vegetable world we know of no plant, nor part of any plant (as the exudations or pith) that is warm to man’s touch. Yet (as we have before observed) green weeds grow warm when confined, and some vegetables are warm and others cold to our internal touch, i.e., the palate and stomach, or even after a while to our external skin (as is shown in plasters and ointments).
We know of nothing in the various parts of animals, when dead or detached from the rest, that is warm to the touch; for horse dung itself does not retain its heat, unless it be confined and buried. All dung, however, appears to possess a potential heat, as in manuring fields; so also dead bodies are endued with this latent and potential heat to such a degree, that in cemeteries where people are interred daily the earth acquires a secret heat, which consumes any recently deposited body much sooner than pure earth; and they tell you that the people of the East are acquainted with a fine soft cloth, made of the down of birds, which can melt butter wrapped gently up in it by its own warmth.
Manures, such as every kind of dung, chalk, sea-sand, salt and the like, have some disposition toward heat.
All putrefaction exhibits some slight degree of heat, though not enough to be perceptible by the touch; for neither the substances which by putrefaction are converted into animalculæ, as flesh and cheese, nor rotten wood which shines in the dark, are warm to the touch. The heat, however, of putrid substances displays itself occasionally in a disgusting and strong scent.
The first degree of heat, therefore, in substances which are warm to the human touch appears to be that of animals, and this admits of a great variety of degrees, for the lowest (as in insects) is scarcely perceptible, the highest scarcely equals that of the sun’s rays in warm climates and weather, and is not so acute as to be insufferable to the hand. It is said, however, of Constantius, and some others of a very dry constitution and habit of body, that when attacked with violent fevers, they became so warm as to appear almost to burn the hand applied to them.
Animals become more warm by motion and exercise, wine and feasting, venery, burning fevers, and grief.
In the paroxysm of intermittent fevers the patients are at first seized with cold and shivering, but soon afterward become more heated than at first—in burning and pestilential fevers they are hot from the beginning.
Let further inquiry be made into the comparative heat of different animals, as fishes, quadrupeds, serpents, birds, and also of the different species, as the lion, the kite, or man; for, according to the vulgar opinion, fishes are the least warm internally, and birds the most, particularly doves, hawks, and ostriches.
Let further inquiry be made as to the comparative heat in different parts and limbs of the same animal; for milk, blood, seed, and eggs are moderately warm, and less hot than the outward flesh of the animal when in motion or agitated. The degree of heat of the brain, stomach, heart, and the rest, has not yet been equally well investigated.
All animals are externally cold in winter and cold weather, but are thought to be internally warmer.
The heat of the heavenly bodies, even in the warmest climates and seasons, never reaches such a pitch as to light or burn the driest wood or straw, or even tinder without the aid of burning-glasses. It can, however, raise vapor from moist substances.
Astronomers tell us that some stars are hotter than others. Mars is considered the warmest after the Sun, then Jupiter, then Venus. The Moon and, above all, Saturn, are considered to be cold. Among the fixed stars Sirius is thought the warmest, then Cor Leonis or Regulus, then the lesser Dog-star.
The sun gives out more heat as it approaches toward the perpendicular or zenith, which may be supposed to be the case with the other planets, according to their degree of heat; for instance, that Jupiter gives out more heat when situated beneath Cancer or Leo than when he is beneath Capricorn and Aquarius.
It is to be supposed that the sun and other planets give more heat in perigee, from their approximation to the earth, than when in apogee. But if in any country the sun should be both in its perigee and nearer to the perpendicular at the same time, it must necessarily give out more heat than in a country where it is also in perigee, but situated more obliquely; so that the comparative altitude of the planets should be observed, and their approach to or declination from the perpendicular in different countries.
The sun and other planets are thought also to give out more heat in proportion as they are nearer to the larger fixed stars, as when the sun is in Leo he is nearer Cor Leonis, Cauda Leonis, Spica Virginis, Sirius, and the lesser Dog-star, than when he is in Cancer, where, however, he approaches nearer to the perpendicular. It is probable, also, that the quarters of the heavens produce a greater heat (though not perceptibly), in proportion as they are adorned with a greater number of stars, particularly those of the first magnitude.
On the whole, the heat of the heavenly bodies is augmented in three ways: 1.The approach to the perpendicular; 2.Proximity or their perigee; 3.The conjunction or union of stars.
There is a very considerable difference between the degree of heat in animals, and even in the rays of the heavenly bodies (as they reach us), and the heat of the most gentle flame, and even of all ignited substances, nay, liquids, or the air itself when unusually heated by fire. For the flame of spirit of wine, though diffused and uncollected, is yet able to set straw, linen, or paper on fire, which animal heat, or that of the sun, will never accomplish without a burning-glass.
There are, however, many degrees of strength and weakness in flame and ignited bodies: but no diligent inquiry has been made in this respect, and we must, therefore, pass it hastily over. Of all flames, that of spirits of wine appears to be the most gentle, except perhaps the ignis fatuus, or the flashes from the perspiration of animals. After this we should be inclined to place the flame of light and porous vegetables, such as straw, reeds, and dried leaves; from which the flame of hair or feathers differs but little. Then, perhaps, comes the flame of wood, particularly that which contains but little rosin or pitch; that of small wood, however (such as is usually tied up in fagots), is milder than that of the trunks or roots of trees. This can be easily tried in iron furnaces, where a fire of fagots or branches of trees is of little service. Next follows the flame of oil, tallow, wax, and the like oily and fat substances, which are not very violent. But a most powerful heat is found in pitch and rosin, and a still greater in sulphur, camphor, naphtha, saltpetre, and salts (after they have discharged their crude matter), and in their compounds; as in gunpowder, Greek fire (vulgarly called wild fire), and its varieties, which possess such a stubborn heat as scarcely to be extinguished by water.
We consider that the flame which results from some imperfect metals is very strong and active; but on all these points further inquiry should be made.
23. The flame of vivid lightning appears to exceed all the above, so as sometimes to have melted even wrought iron into drops, which the other flames cannot accomplish.
In ignited bodies there are different degrees of heat, concerning which, also, a diligent inquiry has not been made. We consider the faintest heat to be that of tinder, touchwood, and dry rope match, such as is used for discharging cannon. Next follows that of ignited charcoal or cinders, and even bricks, and the like; but the most violent is that of ignited metals, as iron, copper, and the like. Further inquiry, however, must be made into this also.
Some ignited bodies are found to be much warmer than some flames; for instance, red hot iron is much warmer, and burns more than the flame of spirits of wine.
Some bodies even not ignited, but only heated by the fire, as boiling water, and the air confined in reverberatories, surpass in heat many flames and ignited substances.
Motion increases heat, as is shown in the bellows and the blowpipe; for the harder metals are not dissolved or melted by steady quiet fire, without the aid of the blowpipe.
28. Let an experiment be made with burning-glasses; in which respect I have observed, that if a glass be placed at the distance of ten inches, for instance, from the combustible object, it does not kindle or burn it so readily, as if the glass be placed at the distance of five inches (for instance), and be then gradually and slowly withdrawn to the distance of ten inches. The cone and focus of the rays, however, are the same, but the mere motion increases the effect of the heat.
29. Conflagrations, which take place with a high wind, are thought to make greater way against than with the wind, because when the wind slackens, the flame recoils more rapidly than it advances when the wind is favorable.
30. Flame does not burst out or arise unless it have some hollow space to move and exert itself in, except in the exploding flame of gunpowder, and the like, where the compression and confinement of the flame increase its fury.
31. The anvil becomes so hot by the hammer, that if it were a thin plate it might probably grow red, like ignited iron by repeated strokes. Let the experiment be tried.
32. But in ignited bodies that are porous, so as to leave room for the fire to move itself, if its motion be prevented by strong compression, the fire is immediately extinguished; thus it is with tinder, or the burning snuff of a candle or lamp, or even hot charcoal or cinders; for when they are squeezed by snuffers, or the foot, and the like, the effect of the fire instantly ceases.
The approach toward a hot body increases heat in proportion to the approximation; a similar effect to that of light, for the nearer any object is placed toward the light, the more visible it becomes.
The union of different heats increases heat, unless the substances be mixed; for a large and small fire in the same spot tend mutually to increase each other’s heat, but lukewarm water poured into boiling water cools it.
The continued neighborhood of a warm body increases heat. For the heat, which perpetually passes and emanates from it, being mixed with that which preceded it, multiplies the whole. A fire, for instance, does not warm a room in half an hour as much as the same fire would in an hour. This does not apply to light, for a lamp or candle placed in a spot gives no more light by remaining there, than it did at first.
The irritation of surrounding cold increases heat, as may be seen in fires during a sharp frost. We think that this is owing not merely to the confinement and compression of the heat (which forms a sort of union), but also by the exasperation of it, as when the air or a stick are violently compressed or bent, they recoil, not only to the point they first occupied, but still further back. Let an accurate experiment, therefore, be made with a stick, or something of the kind, put into the flame, in order to see whether it be not sooner burned at the sides than in the middle of it.
There are many degrees in the susceptibility of heat. And, first, it must be observed how much a low gentle heat changes and partially warms even the bodies least susceptible of it. For even the heat of the hand imparts a little warmth to a ball of lead or other metal held a short time in it; so easily is heat transmitted and excited, without any apparent change in the body.
38. Of all bodies that we are acquainted with, air admits and loses heat the most readily, which is admirably seen in weather-glasses, whose construction is as follows: Take a glass with a hollow belly, and a thin and long neck; turn it upside down, and place it with its mouth downward into another glass vessel containing water; the end of the tube touching the bottom of the vessel, and the tube itself leaning a little on the edge, so as to be fixed upright. In order to do this more readily, let a little wax be applied to the edge, not, however, so as to block up the orifice, lest, by preventing the air from escaping, the motion, which we shall presently speak of, and which is very gentle and delicate, should be impeded.
Before the first glass be inserted in the other, its upper part (the belly) should be warmed at the fire. Then upon placing it as we have described, the air (which was dilated by the heat), after a sufficient time has been allowed for it to lose the additional temperature, will restore and contract itself to the same dimensions as that of the external or common atmosphere at the moment of immersion, and the water will be attracted upward in the tube to a proportionate extent. A long narrow slip of paper should be attached to the tube, divided into as many degrees as you please. You will then perceive, as the weather grows warmer or colder, that the air contracts itself into a narrower space in cold weather and dilates in the warm, which will be exhibited by the rising of the water as the air contracts itself, and its depression as the air dilates. The sensibility of the air with regard to heat or cold is so delicate and exquisite, that it far exceeds the human touch, so that a ray of sunshine, the heat of the breath, and much more, that of the hand placed on the top of the tube, immediately causes an evident depression of the water. We think, however, that the spirit of animals possesses a much more delicate susceptibility of heat and cold, only that it is impeded and blunted by the grossness of their bodies.
After air, we consider those bodies to be most sensible of heat, which have been recently changed and contracted by cold, as snow and ice; for they begin to be dissolved and melt with the first mild weather. Next, perhaps, follows quicksilver; then greasy substances, as oil, butter, and the like; then wood; then water; lastly, stones and metals, which do not easily grow hot, particularly toward their centre. When heated, however, they retain their temperature for a very long time; so that a brick or stone, or hot iron, plunged in a basin of cold water, and kept there for a quarter of an hour or thereabout, retains such a heat as not to admit of being touched.
The less massive the body is, the more readily it grows warm at the approach of a heated body, which shows that heat with us is somewhat averse to a tangible mass.
41. Heat with regard to the human senses and touch is various and relative, so that lukewarm water appears hot if the hand be cold, and cold if the hand be hot.
XIV. Any one may readily see how poor we are in history, since in the above tables, besides occasionally inserting traditions and report instead of approved history and authentic instances (always, however, adding some note if their credit or authority be doubtful), we are often forced to subjoin, “Let the experiment be tried—Let further inquiry be made.”
XV. We are wont to term the office and use of these three tables the presenting a review of instances to the understanding; and when this has been done, induction itself is to be brought into action. For on an individual review of all the instances a nature is to be found, such as always to be present and absent with the given nature, to increase and decrease with it, and, as we have said, to form a more common limit of the nature. If the mind attempt this affirmatively from the first (which it always will when left to itself), there will spring up phantoms, mere theories and ill-defined notions, with axioms requiring daily correction. These will, doubtless, be better or worse, according to the power and strength of the understanding which creates them. But it is only for God (the bestower and creator of forms), and perhaps for angels and intelligences, at once to recognize forms affirmatively at the first glance of contemplation: man, at lest, is unable to do so, and is only allowed to proceed first by negatives, and then to conclude with affirmatives, after every species of exclusion.
XVI. We must, therefore, effect a complete solution and separation of nature; not by fire, but by the mind, that divine fire. The first work of legitimate induction, in the discovery of forms, is rejection, or the exclusive instances of individual natures, which are not found in some one instance where the given nature is present, or are found in any one instance where it is absent, or are found to increase in any one instance where the given nature decreases, or the reverse. After an exclusion correctly effected, an affirmative form will remain as the residuum, solid, true, and well defined, while all volatile opinions go off in smoke. This is readily said; but we must arrive at it by a circuitous route. We shall perhaps, however, omit nothing that can facilitate our progress.
XVII. The first and almost perpetual precaution and warning which we consider necessary is this; that none should suppose from the great part assigned by us to forms, that we mean such forms as the meditations and thoughts of men have hitherto been accustomed to. In the first place, we do not at present mean the concrete forms, which (as we have observed) are in the common course of things compounded of simple natures, as those of a lion, an eagle, a rose, gold, or the like. The moment for discussing these will arrive when we come to treat of the latent process and latent conformation, and the discovery of them as they exist in what are called substances, or concrete natures.
Nor again, would we be thought to mean (even when treating of simple natures) any abstract forms or ideas, either undefined or badly defined in matter. For when we speak of forms, we mean nothing else than those laws and regulations of simple action which arrange and constitute any simple nature, such as heat, light, weight, in every species of matter, and in a susceptible subject. The form of heat or form of light, therefore, means no more than the law of heat or the law of light. Nor do we ever abstract or withdraw ourselves from things, and the operative branch of philosophy. When, therefore, we say (for instance) in our investigation of the form of heat, Reject rarity, or, Rarity is not of the form of heat, it is the same as if we were to say, Man can superinduce heat on a dense body, or the reverse, Man can abstract or ward off heat from a rare body.
But if our forms appear to any one to be somewhat abstracted, from their mingling and uniting heterogeneous objects (the heat, for instance, of the heavenly bodies appears to be very different from that of fire; the fixed red of the rose and the like, from that which is apparent in the rainbow, or the radiation of opal or the diamond; death by drowning, from that by burning, the sword, apoplexy, or consumption; and yet they all agree in the common natures of heat, redness, and death), let him be assured that his understanding is inthralled by habit, by general appearances and hypotheses. For it is most certain that, however heterogeneous and distinct, they agree in the form or law which regulates heat, redness, or death; and that human power cannot be emancipated and freed from the common course of nature, and expanded and exalted to new efficients and new modes of operation, except by the revelation and invention of forms of this nature. But after this union of nature, which is the principal point, we will afterward, in its proper place, treat of the divisions and ramifications of nature, whether ordinary or internal and more real.
XVIII. We must now offer an example of the exclusion or rejection of natures found by the tables of review, not to be of the form of heat; first premising that not only each table is sufficient for the rejection of any nature, but even each single instance contained in them. For it is clear from what has been said that every contradictory instance destroys a hypothesis as to the form. Still, however, for the sake of clearness, and in order to show more plainly the use of the tables, we redouble or repeat the exclusive.
An Example of the Exclusive Table, or of the Rejection of Natures from the Form of Heat
On account of the sun’s rays, reject elementary (or terrestrial) nature.
On account of common fire, and particularly subterranean fires (which are the most remote and secluded from the rays of the heavenly bodies), reject celestial nature.
On account of the heat acquired by every description of substances (as minerals, vegetables, the external parts of animals, water, oil, air, etc.) by mere approximation to the fire or any warm body, reject all variety and delicate texture of bodies.
On account of iron and ignited metals, which warm other bodies, and yet neither lose their weight nor substance, reject the imparting or mixing of the substance of the heating body.
On account of boiling water and air, and also those metals and other solid bodies which are heated, but not to ignition, or red heat, reject flame or light.
On account of the rays of the moon and other heavenly bodies (except the sun), again reject flame or light.
On account of the comparison between red-hot iron and the flame of spirits of wine (for the iron is more hot and less bright, while the flame of spirits of wine is more bright and less hot), again reject flame and light.
On account of gold and other ignited metals, which are of the greatest specific density, reject rarity.
On account of air, which is generally found to be cold and yet continues rare, reject rarity.
On account of ignited iron, which does not swell in bulk, but retains the same apparent dimension, reject the absolute expansive motion of the whole.
On account of the expansion of the air in thermometers and the like, which is absolutely moved and expanded to the eye, and yet acquires no manifest increase of heat, again reject absolute or expansive motion of the whole.
On account of the ready application of heat to all substances without any destruction or remarkable alteration of them, reject destructive nature or the violent communication of any new nature.
On account of the agreement and conformity of the effects produced by cold and heat, reject both expansive and contracting motion as regards the whole.
On account of the heat excited by friction, reject principal nature, by which we mean that which exists positively, and is not caused by a preceding nature.
There are other natures to be rejected; but we are merely offering examples, and not perfect tables.
None of the above natures are of the form of heat; and man is freed from them all in his operation upon heat.
XIX. In the exclusive table are laid the foundations of true induction, which is not, however, completed until the affirmative be attained. Nor is the exclusive table perfect, nor can it be so at first. For it is clearly a rejection of simple natures; but if we have not as yet good and just notions of simple natures, how can the exclusive table be made correct? Some of the above, as the notion of elementary and celestial nature, and rarity, are vague and ill defined. We, therefore, who are neither ignorant nor forgetful of the great work which we attempt, in rendering the human understanding adequate to things and nature, by no means rest satisfied with what we have hitherto enforced, but push the matter further, and contrive and prepare more powerful aid for the use of the understanding, which we will next subjoin. And, indeed, in the interpretation of nature the mind is to be so prepared and formed, as to rest itself on proper degrees of certainty, and yet to remember (especially at first) that what is present depends much upon what remains behind.
XX. Since, however, truth emerges more readily from error than confusion, we consider it useful to leave the understanding at liberty to exert itself and attempt the interpretation of nature in the affirmative, after having constructed and weighed the three tables of preparation, such as we have laid them down, both from the instances there collected, and others occurring elsewhere. Which attempt we are wont to call the liberty of the understanding, or the commencement of interpretation, or the first vintage.
The First Vintage of the Form of Heat
It must be observed that the form of anything is inherent (as appears clearly from our premises) in each individual instance in which the thing itself is inherent, or it would not be a form. No contradictory instance, therefore, can be alleged. The form, however, is found to be much more conspicuous and evident in some instances than in others; in those (for example) where its nature is less restrained and embarrassed, and reduced to rule by other natures. Such instances we are wont to term coruscations, or conspicuous instances. We must proceed, then, to the first vintage of the form of heat.
From the instances taken collectively, as well as singly, the nature whose limit is heat appears to be motion. This is chiefly exhibited in flame, which is in constant motion, and in warm or boiling liquids, which are likewise in constant motion. It is also shown in the excitement or increase of heat by motion, as by bellows and draughts: for which see Inst.29, Tab.3, and by other species of motion, as in Inst.28 and 31, Tab.3. It is also shown by the extinction of fire and heat upon any strong pressure, which restrains and puts a stop to motion; for which see Inst.30 and 32, Tab.3. It is further shown by this circumstance, namely, that every substance is destroyed, or at least materially changed, by strong and powerful fire and heat: whence it is clear that tumult and confusion are occasioned by heat, together with a violent motion in the internal parts of bodies; and this gradually tends to their dissolution.
What we have said with regard to motion must be thus understood, when taken as the genus of heat: it must not be thought that heat generates motion, or motion heat (though in some respects this be true), but that the very essence of heat, or the substantial self of heat, is motion and nothing else, limited, however, by certain differences which we will presently add, after giving some cautions for avoiding ambiguity.
Sensible heat is relative, and regards man, not universe; and is rightly held to be merely the effect of heat on animal spirit. It is even variable in itself, since the same body (in different states of sensation) excites the feeling of heat and of cold; this is shown by Inst.41, Tab.3.
Nor should we confound the communication of heat or its transitive nature, by which a body grows warm at the approach of a heated body, with the form of heat; for heat is one thing and heating another. Heat can be excited by friction without any previous heating body, and, therefore, heating is excluded from the form of heat. Even when heat is excited by the approach of a hot body, this depends not on the form of heat, but on another more profound and common nature; namely, that of assimilation and multiplication, about which a separate inquiry must be made.
The notion of fire is vulgar, and of no assistance; it is merely compounded of the conjunction of heat and light in any body, as in ordinary flame and red-hot substances.
Laying aside all ambiguity, therefore, we must lastly consider the true differences which limit motion and render it the form of heat.
I. The first difference is, that heat is an expansive motion, by which the body strives to dilate itself, and to occupy a greater space than before. This difference is principally seen in flame, where the smoke or thick vapor is clearly dilated and bursts into flame.
It is also shown in all boiling liquids, which swell, rise, and boil up to the sight, and the process of expansion is urged forward till they are converted into a much more extended and dilated body than the liquid itself, such as steam, smoke, or air.
It is also shown in wood and combustibles where exudation sometimes takes place, and evaporation always.
It is also shown in the melting of metals, which, being very compact, do not easily swell and dilate, but yet their spirit, when dilated and desirous of further expansion, forces and urges its thicker parts into dissolution, and if the heat be pushed still further, reduces a considerable part of them into a volatile state.
It is also shown in iron or stones, which though not melted or dissolved, are however softened. The same circumstance takes place in sticks of wood, which become flexible when a little heated in warm ashes.
It is most readily observed in air, which instantly and manifestly expands with a small degree of heat, as in Inst.38, Tab.3.
It is also shown in the contrary nature of cold; for cold contracts and narrows every substance; so that in intense frosts nails fall out of the wall and brass cracks, and heated glass exposed suddenly to the cold cracks and breaks. So the air, by a slight degree of cold, contracts itself, as in Inst.38, Tab.3. More will be said of this in the inquiry into cold.
Nor is it to be wondered at if cold and heat exhibit many common effects (for which see Inst.32, Tab.2), since two differences, of which we shall presently speak, belong to each nature: although in the present difference the effects be diametrically opposed to each other. For heat occasions an expansive and dilating motion, but cold a contracting and condensing motion.
II. The second difference is a modification of the preceding, namely, that heat is an expansive motion, tending toward the exterior, but at the same time bearing the body upward. For there is no doubt that there be many compound motions, as an arrow or dart, for instance, has both a rotatory and progressive motion. In the same way the motion of heat is both expansive and tending upward.
This difference is shown by putting the tongs or poker into the fire. If placed perpendicularly with the hand above, they soon burn it, but much less speedily if the hand hold them sloping or from below.
It is also conspicuous in distillations per descensum, which men are wont to employ with delicate flowers, whose scent easily evaporates. Their industry has devised placing the fire above instead of below, that it may scorch less; for not only flame but all heat has an upward tendency.
Let an experiment be made on the contrary nature of cold, whether its contraction be downward, as the expansion of heat is upward. Take, therefore, two iron rods or two glass tubes, alike in other respects, and warm them a little, and place a sponge, dipped in cold water, or some snow, below the one and above the other. We are of opinion that the extremities will grow cold in that rod first where it is placed beneath, as the contrary takes place with regard to heat.
III. The third difference is this; that heat is not a uniform expansive motion of the whole, but of the small particles of the body; and this motion being at the same time restrained, repulsed, and reflected, becomes alternating, perpetually hurrying, striving, struggling, and irritated by the repercussion, which is the source of the violence of flame and heat.
But this difference is chiefly shown in flame and boiling liquids, which always hurry, swell, and subside again in detached parts.
It is also shown in bodies of such hard texture as not to swell or dilate in bulk, such as red-hot iron, in which the heat is most violent.
It is also shown by the fires burning most briskly in the coldest weather.
It is also shown by this, that when the air is dilated in the thermometer uniformly and equably, without any impediment or repulsion, the heat is not perceptible. In confined draughts also, although they break out very violently, no remarkable heat is perceived, because the motion affects the whole, without any alternating motion in the particles; for which reason try whether flame do not burn more at the sides than in its centre.
It is also shown in this, that all burning proceeds by the minute pores of bodies—undermining, penetrating, piercing, and pricking them as if with an infinite number of needle-points. Hence all strong acids (if adapted to the body on which they act) exhibit the effects of fire, from their corroding and pungent nature.
The difference of which we now speak is common also to the nature of cold, in which the contracting motion is restrained by the resistance of expansion, as in heat the expansive motion is restrained by the resistance of contraction.
Whether, therefore, the particles of matter penetrate inward or outward, the reasoning is the same, though the power be very different, because we have nothing on earth which is intensely cold.
IV. The fourth difference is a modification of the preceding, namely, that this stimulating or penetrating motion should be rapid and never sluggish, and should take place not in the very minutest particles, but rather in those of some tolerable dimensions.
It is shown by comparing the effects of fire with those of time. Time dries, consumes, undermines, and reduces to ashes as well as fire, and perhaps to a much finer degree; but as its motion is very slow, and attacks very minute particles, no heat is perceived.
It is also shown in a comparison of the dissolution of iron and gold; for gold is dissolved without the excitement of any heat, but iron with a vehement excitement of it, although most in the same time, because in the former the penetration of the separating acid is mild, and gently insinuates itself, and the particles of gold yield easily, but the penetration of iron is violent, and attended with some struggle, and its particles are more obstinate.
It is partially shown, also, in some gangrenes and mortifications of flesh, which do not excite great heat or pain, from the gentle nature of the putrefaction.
Let this suffice for a first vintage, or the commencement of the interpretation of the form of heat by the liberty of the understanding.
From this first vintage the form or true definition of heat (considered relatively to the universe and not to the sense) is briefly thus—Heat is an expansive motion restrained, and striving to exert itself in the smaller particles. The expansion is modified by its tendency to rise, though expanding toward the exterior; and the effort is modified by its not being sluggish, but active and somewhat violent.
With regard to the operative definition, the matter is the same. If you are able to excite a dilating or expansive motion in any natural body, and so to repress that motion and force it on itself as not to allow the expansion to proceed equally, but only to be partially exerted and partially repressed, you will beyond all doubt produce heat, without any consideration as to whether the body be of earth (or elementary, as they term it), or imbued with celestial influence, luminous or opaque, rare or dense, locally expanded or contained within the bounds of its first dimensions, verging to dissolution or remaining fixed, animal, vegetable, or mineral, water, or oil, or air, or any other substance whatever susceptible of such motion. Sensible heat is the same, but considered relatively to the senses. Let us now proceed to further helps.
XXI. After our tables of first review, our rejection or exclusive table, and the first vintage derived from them, we must advance to the remaining helps of the understanding with regard to the interpretation of nature, and a true and perfect induction, in offering which we will take the examples of cold and heat where tables are necessary, but where fewer instances are required we will go through a variety of others, so as neither to confound investigation nor to narrow our doctrine.
In the first place, therefore, we will treat of prerogative instances; 2.Of the supports of induction; 3.Of the correction of induction; 4.Of varying the investigation according to the nature of the subject; 5.Of the prerogative natures with respect to investigation, or of what should be the first or last objects of our research; 6.Of the limits of investigation, or a synopsis of all natures that exist in the universe; 7.Of the application to practical purposes, or of what relates to man; 8.Of the preparations for investigation; 9.And lastly, of the ascending and descending scale of axioms.
XXII. Among the prerogative instances we will first mention solitary instances. Solitary instances are those which exhibit the required nature in subjects that have nothing in common with any other subject than the nature in question, or which do not exhibit the required nature in subjects resembling others in every respect except that of the nature in question; for these instances manifestly remove prolixity, and accelerate and confirm exclusion, so that a few of them are of as much avail as many.
For instance, let the inquiry be the nature of color. Prisms, crystalline gems, which yield colors not only internally but on the wall, dews, etc., are solitary instances; for they have nothing in common with the fixed colors in flowers and colored gems, metals, woods, etc., except the color itself. Hence we easily deduce that color is nothing but a modification of the image of the incident and absorbed light, occasioned in the former case by the different degrees of incidence, in the latter by the various textures and forms of bodies. These are solitary instances as regards similitude.
Again, in the same inquiry the distinct veins of white and black in marble, and the variegated colors of flowers of the same species, are solitary instances; for the black and white of marble, and the spots of white and purple in the flowers of the stock, agree in every respect but that of color. Thence we easily deduce that color has not much to do with the intrinsic natures of any body, but depends only on the coarser and as it were mechanical arrangement of the parts. These are solitary instances as regards difference. We call them both solitary or wild, to borrow a word from the astronomers.
XXIII. In the second rank of prerogative instances we will consider migrating instances. In these the required nature passes toward generation, having no previous existence, or toward corruption, having first existed. In each of these divisions, therefore, the instances are always twofold, or rather it is one instance, first in motion or on its passage, and then brought to the opposite conclusion. These instances not only hasten and confirm exclusion, but also reduce affirmation, or the form itself, to a narrow compass; for the form must be something conferred by this migration, or, on the contrary, removed and destroyed by it; and although all exclusion advances affirmation, yet this takes place more directly in the same than in different subjects; but if the form (as it is quite clear from what has been advanced) exhibit itself in one subject, it leads to all. The more simple the migration is, the more valuable is the instance. These migrating instances are, moreover, very useful in practice, for since they manifest the form, coupled with that which causes or destroys it, they point out the right practice in some subjects, and thence there is an easy transition to those with which they are most allied. There is, however, a degree of danger which demands caution, namely, lest they should refer the form too much to its efficient cause, and imbue, or at least tinge, the understanding with a false notion of the form from the appearance of such cause, which is never more than a vehicle or conveyance of the form. This may easily be remedied by a proper application of exclusion.
Let us then give an example of a migrating instance. Let whiteness be the required nature. An instance which passes toward generation is glass in its entire and in its powdered state, or water in its natural state, and when agitated to froth; for glass when entire, and water in its natural state, are transparent and not white, but powdered glass and the froth of water are white and not transparent. We must inquire, therefore, what has happened to the glass or water in the course of this migration; for it is manifest that the form of whiteness is conveyed and introduced by the bruising of the glass and the agitation of the water; but nothing is found to have been introduced but a diminishing of the parts of the glass and water and the insertion of air. Yet this is no slight progress toward discovering the form of whiteness, namely, that two bodies, in themselves more or less transparent (as air and water, or air and glass), when brought into contact in minute portions, exhibit whiteness from the unequal refraction of the rays of light.
But here we must also give an example of the danger and caution of which we spoke; for instance, it will readily occur to an understanding perverted by efficients, that air is always necessary for producing the form of whiteness, or that whiteness is only generated by transparent bodies, which suppositions are both false, and proved to be so by many exclusions; nay, it will rather appear (without any particular regard to air or the like), that all bodies which are even in such of their parts as affect the sight exhibit transparency, those which are uneven and of simple texture whiteness, those which are uneven and of compound but regular texture all the other colors except black, but those which are uneven and of a compound irregular and confused texture exhibit blackness. An example has been given, therefore, of an instance migrating toward generation in the required nature of whiteness. An instance migrating toward corruption in the same nature is that of dissolving froth or snow, for they lose their whiteness and assume the transparency of water in its pure state without air.
Nor should we by any means omit to state, that under migrating instances we must comprehend not only those which pass toward generation and destruction, but also those which pass toward increase or decrease, for they, too, assist in the discovery of the form, as is clear from our definition of a form and the Table of Degrees. Hence paper, which is white when dry, is less white when moistened (from the exclusion of air and admission of water), and tends more to transparency. The reason is the same as in the above instances.
XXIV. In the third rank of prerogative instances we will class conspicuous instances, of which we spoke in our first vintage of the form of heat, and which we are also wont to call coruscations, or free and predominant instances. They are such as show the required nature in its bare substantial shape, and at its height or greatest degree of power, emancipated and free from all impediments, or at least overcoming, suppressing, and restraining them by the strength of its qualities; for since every body is susceptible of many united forms of natures in the concrete, the consequence is that they mutually deaden, depress, break, and confine each other, and the individual forms are obscured. But there are some subjects in which the required nature exists in its full vigor rather than in others, either from the absence of any impediment, or the predominance of its quality. Such instances are eminently conspicuous. But even in these care must be taken, and the hastiness of the understanding checked, for whatever makes a show of the form, and forces it forward, is to be suspected, and recourse must be had to severe and diligent exclusion.
For example, let heat be the required nature. The thermometer is a conspicuous instance of the expansive motion, which (as has been observed) constitutes the chief part of the form of heat; for although flame clearly exhibits expansion, yet from its being extinguished every moment, it does not exhibit the progress of expansion. Boiling water again, from its rapid conversion into vapor, does not so well exhibit the expansion of water in its own shape, while red-hot iron and the like are so far from showing this progress, that, on the contrary, the expansion itself is scarcely evident to the senses, on account of its spirit being repressed and weakened by the compact and coarse particles which subdue and restrain it. But the thermometer strikingly exhibits the expansion of the air as being evident and progressive, durable and not transitory.
Take another example. Let the required nature be weight. Quicksilver is a conspicuous instance of weight; for it is far heavier than any other substance except gold, which is not much heavier, and it is a better instance than gold for the purpose of indicating the form of weight; for gold is solid and consistent, which qualities must be referred to density, but quicksilver is liquid and teeming with spirit, yet much heavier than the diamond and other substances considered to be most solid; whence it is shown that the form of gravity or weight predominates only in the quantity of matter, and not in the close fitting of it.
XXV. In the fourth rank of prerogative instances we will class clandestine instances, which we are also wont to call twilight instances; they are as it were opposed to the conspicuous instances, for they show the required nature in its lowest state of efficacy, and as it were its cradle and first rudiments, making an effort and a sort of first attempt, but concealed and subdued by a contrary nature. Such instances are, however, of great importance in discovering forms, for as the conspicuous tend easily to differences, so do the clandestine best lead to genera, that is, to those common natures of which the required natures are only the limits.
As an example, let consistency, or that which confines itself, be the required nature, the opposite of which is a liquid or flowing state. The clandestine instances are such as exhibit some weak and low degree of consistency in fluids, as a water bubble, which is a sort of consistent and bounded pellicle formed out of the substance of the water. So eaves’ droppings, if there be enough water to follow them, draw themselves out into a thin thread, not to break the continuity of the water, but if there be not enough to follow, the water forms itself into a round drop, which is the best form to prevent a breach of continuity; and at the moment the thread ceases, and the water begins to fall in drops, the thread of water recoils upward to avoid such a breach. Nay, in metals, which when melted are liquid but more tenacious, the melted drops often recoil and are suspended. There is something similar in the instance of the child’s looking-glass, which little boys will sometimes form of spittle between rushes, and where the same pellicle of water is observable; and still more in that other amusement of children, when they take some water rendered a little more tenacious by soap, and inflate it with a pipe, forming the water into a sort of castle of bubbles, which assumes such consistency, by the interposition of the air, as to admit of being thrown some little distance without bursting. The best example is that of froth and snow, which assume such consistency as almost to admit of being cut, although composed of air and water, both liquids. All these circumstances clearly show that the terms liquid and consistent are merely vulgar notions adapted to the sense, and that in reality all bodies have a tendency to avoid a breach of continuity, faint and weak in bodies composed of homogeneous parts (as is the case with liquids), but more vivid and powerful in those composed of heterogeneous parts, because the approach of heterogeneous matter binds bodies together, while the insinuation of homogeneous matter loosens and relaxes them.
Again, to take another example, let the required nature be attraction or the cohesion of bodies. The most remarkable conspicuous instance with regard to its form is the magnet. The contrary nature to attraction is non-attraction, though in a similar substance. Thus iron does not attract iron, lead lead, wood wood, nor water water. But the clandestine instance is that of the magnet armed with iron, or rather that of iron in the magnet so armed. For its nature is such that the magnet when armed does not attract iron more powerfully at any given distance than when unarmed; but if the iron be brought in contact with the armed magnet, the latter will sustain a much greater weight than the simple magnet, from the resemblance of substance in the two portions of iron, a quality altogether clandestine and hidden in the iron until the magnet was introduced. It is manifest, therefore, that the form of cohesion is something which is vivid and robust in the magnet, and hidden and weak in the iron. It is to be observed, also, that small wooden arrows without an iron point, when discharged from large mortars, penetrate further into wooden substances (such as the ribs of ships or the like), than the same arrows pointed with iron, owing to the similarity of substance, though this quality was previously latent in the wood. Again, although in the mass air does not appear to attract air, nor water water, yet when one bubble is brought near another, they are both more readily dissolved, from the tendency to contact of the water with the water, and the air with the air. These clandestine instances (which are, as has been observed, of the most important service) are principally to be observed in small portions of bodies, for the larger masses observe more universal and general forms, as will be mentioned in its proper place.
XXVI. In the fifth rank of prerogative instances we will class constitutive instances, which we are wont also to call collective instances. They constitute a species or lesser form, as it were, of the required nature. For since the real forms (which are always convertible with the given nature) lie at some depth, and are not easily discovered, the necessity of the case and the infirmity of the human understanding require that the particular forms, which collect certain groups of instances (but by no means all) into some common notion, should not be neglected, but most diligently observed. For whatever unites nature, even imperfectly, opens the way to the discovery of the form. The instances, therefore, which are serviceable in this respect are of no mean power, but endowed with some degree of prerogative.
Here, nevertheless, great care must be taken that, after the discovery of several of these particular forms, and the establishing of certain partitions or divisions of the required nature derived from them, the human understanding do not at once rest satisfied, without preparing for the investigation of the great or leading form, and taking it for granted that nature is compound and divided from its very root, despise and reject any further union as a point of superfluous refinement, and tending to mere abstraction.
For instance, let the required nature be memory, or that which excites and assists memory. The constitutive instances are order or distribution, which manifestly assists memory: topics or commonplaces in artificial memory, which may be either places in their literal sense, as a gate, a corner, a window, and the like, or familiar persons and marks, or anything else (provided it be arranged in a determinate order), as animals, plants, and words, letters, characters, historical persons, and the like, of which, however, some are more convenient than others. All these commonplaces materially assist memory, and raise it far above its natural strength. Verse, too, is recollected and learned more easily than prose. From this group of three instances—order, the commonplaces of artificial memory, and verses—is constituted one species of aid for the memory, which may be well termed a separation from infinity. For when a man strives to recollect or recall anything to memory, without a preconceived notion or perception of the object of his search, he inquires about, and labors, and turns from point to point, as if involved in infinity. But if he have any preconceived notion, this infinity is separated off, and the range of his memory is brought within closer limits. In the three instances given above, the preconceived notion is clear and determined. In the first, it must be something that agrees with order; in the second, an image which has some relation or agreement with the fixed commonplaces; in the third, words which fall into a verse: and thus infinity is divided off. Other instances will offer another species, namely, that whatever brings the intellect into contact with something that strikes the sense (the principal point of artificial memory), assists the memory. Others again offer another species, namely, whatever excites an impression by any powerful passion, as fear, shame, wonder, delight, assists the memory. Other instances will afford another species: thus those impressions remain most fixed in the memory which are taken from the mind when clear and least occupied by preceding or succeeding notions, such as the things we learn in childhood, or imagine before sleep, and the first time of any circumstance happening. Other instances afford the following species: namely, that a multitude of circumstances or handles assist the memory, such as writing in paragraphs, reading aloud, or recitation. Lastly, other instances afford still another species: thus the things we anticipate, and which rouse our attention, are more easily remembered than transient events; as if you read any work twenty times over, you will not learn it by heart so readily as if you were to read it but ten times, trying each time to repeat it, and when your memory fails you looking into the book. There are, therefore, six lesser forms, as it were, of things which assist the memory: namely—1, the separation of infinity; 2, the connection of the mind with the senses; 3, the impression in strong passion; 4, the impression on the mind when pure; 5, the multitude of handles; 6, anticipation.
Again, for example’s sake, let the required nature be taste or the power of tasting. The following instances are constitutive: 1.Those who do not smell, but are deprived by nature of that sense, do not perceive or distinguish rancid or putrid food by their taste, nor garlic from roses, and the like. 2.Again, those whose nostrils are obstructed by accident (such as a cold) do not distinguish any putrid or rancid matter from anything sprinkled with rose-water. 3.If those who suffer from a cold blow their noses violently at the very moment in which they have anything fetid or perfumed in their mouth, or on their palate, they instantly have a clear perception of the fetor or perfume. These instances afford and constitute this species or division of taste, namely, that it is in part nothing else than an internal smelling, passing and descending through the upper passages of the nostrils to the mouth and palate. But, on the other hand, those whose power of smelling is deficient or obstructed, perceive what is salt, sweet, pungent, acid, rough, and bitter, and the like, as well as any one else: so that the taste is clearly something compounded of the internal smelling, and an exquisite species of touch which we will not here discuss.
Again, as another example, let the required nature be the communication of quality, without intermixture of substance. The instance of light will afford or constitute one species of communication, heat and the magnet another. For the communication of light is momentary and immediately arrested upon the removal of the original light. But heat, and the magnetic force, when once transmitted to or excited in another body, remain fixed for a considerable time after the removal of the source.
In fine, the prerogative of constitutive instances is considerable, for they materially assist the definitions (especially in detail) and the divisions or partitions of natures, concerning which Plato has well said, “He who can properly define and divide is to be considered a god.”
XXVII. In the sixth rank of prerogative instances we will place similar or proportionate instances, which we are also wont to call physical parallels, or resemblances. They are such as exhibit the resemblances and connection of things, not in minor forms (as the constitutive do), but at once in the concrete. They are, therefore, as it were, the first and lowest steps toward the union of nature; nor do they immediately establish any axiom, but merely indicate and observe a certain relation of bodies to each other. But although they be not of much assistance in discovering forms, yet they are of great advantage in disclosing the frame of parts of the universe, upon whose members they practice a species of anatomy, and thence occasionally lead us gently on to sublime and noble axioms, especially such as relate to the construction of the world, rather than to simple natures and forms.
As an example, take the following similar instances: a mirror and the eye; the formation of the ear, and places which return an echo. From such similarity, besides observing the resemblance (which is useful for many purposes), it is easy to collect and form this axiom. That the organs of the senses, and bodies which produce reflections to the senses, are of a similar nature. Again, the understanding once informed of this, rises easily to a higher and nobler axiom; namely, that the only distinction between sensitive and inanimate bodies, in those points in which they agree and sympathize, is this: in the former, animal spirit is added to the arrangement of the body, in the latter it is wanting. So that there might be as many senses in animals as there are points of agreement with inanimate bodies, if the animated body were perforated, so as to allow the spirit to have access to the limb properly disposed for action, as a fit organ. And, on the other hand, there are, without doubt, as many motions in an inanimate as there are senses in the animated body, though the animal spirit be absent. There must, however, be many more motions in inanimate bodies than senses in the animated, from the small number of organs of sense. A very plain example of this is afforded by pains. For, as animals are liable to many kinds and various descriptions of pains (such as those of burning, of intense cold, of pricking, squeezing, stretching, and the like), so is it most certain, that the same circumstances, as far as motion is concerned, happen to inanimate bodies, such as wood or stone when burned, frozen, pricked, cut, bent, bruised, and the like; although there be no sensation, owing to the absence of animal spirit.
Again, wonderful as it may appear, the roots and branches of trees are similar instances. For every vegetable swells and throws out its constituent parts toward the circumference, both upward and downward. And there is no difference between the roots and branches, except that the root is buried in the earth, and the branches are exposed to the air and sun. For if one take a young and vigorous shoot, and bend it down to a small portion of loose earth, although it be not fixed to the ground, yet will it immediately produce a root, and not a branch. And, vice versâ, if earth be placed above, and so forced down with a stone or any hard substance, as to confine the plant and prevent its branching upward, it will throw out branches into the air downward.
The gums of trees, and most rock gems, are similar instances; for both of them are exudations and filtered juices, derived in the former instance from trees, in the latter from stones; the brightness and clearness of both arising from a delicate and accurate filtering. For nearly the same reason, the hair of animals is less beautiful and vivid in its color than the plumage of most birds, because the juices are less delicately filtered through the skin than through the quills.
The scrotum of males and matrix of females are also similar instances; so that the noble formation which constitutes the difference of the sexes appears to differ only as to the one being internal and the other external; a greater degree of heat causing the genitals to protrude in the male, while the heat of the female being too weak to effect this, they are retained internally.
The fins of fishes and the feet of quadrupeds, or the feet and wings of birds, are similar instances; to which Aristotle adds the four folds in the motion of serpents; so that in the formation of the universe, the motion of animals appears to be chiefly effected by four joints or bendings.
The teeth of land animals, and the beaks of birds, are similar instances, whence it is clear, that in all perfect animals there is a determination of some hard substance toward the mouth.
Again, the resemblance and conformity of man to an inverted plant is not absurd. For the head is the root of the nerves and animal faculties, and the seminal parts are the lowest, not including the extremities of the legs and arms. But in the plant, the root (which resembles the head) is regularly placed in the lowest, and the seeds in the highest part.
Lastly, we must particularly recommend and suggest, that man’s present industry in the investigation and compilation of natural history be entirely changed, and directed to the reverse of the present system. For it has hitherto been active and curious in noting the variety of things, and explaining the accurate differences of animals, vegetables, and minerals, most of which are the mere sport of nature, rather than of any real utility as concerns the sciences. Pursuits of this nature are certainly agreeable, and sometimes of practical advantage, but contribute little or nothing to the thorough investigation of nature. Our labor must therefore be directed toward inquiring into and observing resemblances and analogies, both in the whole and its parts, for they unite nature, and lay the foundation of the sciences.
Here, however, a severe and rigorous caution must be observed, that we only consider as similar and proportionate instances, those which (as we first observed) point out physical resemblances; that is, real and substantial resemblances, deeply founded in nature, and not casual and superficial, much less superstitious or curious; such as those which are constantly put forward by the writers on natural magic (the most idle of men, and who are scarcely fit to be named in connection with such serious matters as we now treat of), who, with much vanity and folly, describe, and sometimes too, invent, unmeaning resemblances and sympathies.
But leaving such to themselves, similar instances are not to be neglected, in the greater portions of the world’s conformation; such as Africa and the Peruvian continent, which reaches to the Straits of Magellan; both of which possess a similar isthmus and similar capes, a circumstance not to be attributed to mere accident.
Again, the New and Old World are both of them broad and expanded toward the north, and narrow and pointed toward the south.
Again, we have very remarkable similar instances in the intense cold, toward the middle regions (as it is termed) of the air, and the violent fires which are often found to burst from subterraneous spots, the similarity consisting in both being ends and extremes; the extreme of the nature of cold, for instance, is toward the boundary of heaven, and that of the nature of heat toward the centre of the earth, by a similar species of opposition or rejection of the contrary nature.
Lastly, in the axioms of the sciences, there is a similarity of instances worthy of observation. Thus the rhetorical trope which is called surprise, is similar to that of music termed the declining of a cadence. Again—the mathematical postulate, that things which are equal to the same are equal to one another, is similar to the form of the syllogism in logic, which unites things agreeing in the middle term. Lastly, a certain degree of sagacity in collecting and searching for physical points of similarity, is very useful in many respects.
XXVIII. In the seventh rank of prerogative instances, we will place singular instances, which we are also wont to call irregular or heteroclite (to borrow a term from the grammarians). They are such as exhibit bodies in the concrete, of an apparently extravagant and separate nature, agreeing but little with other things of the same species. For, while the similar instances resemble each other, those we now speak of are only like themselves. Their use is much the same with that of clandestine instances: they bring out and unite nature, and discover genera or common natures, which must afterward be limited by real differences. Nor should we desist from inquiry, until the properties and qualities of those things, which may be deemed miracles, as it were, of nature, be reduced to, and comprehended in, some form or certain law; so that all irregularity or singularity may be found to depend on some common form; and the miracle only consists in accurate differences, degree, and rare coincidence, not in the species itself. Man’s meditation proceeds no further at present, than just to consider things of this kind as the secrets and vast efforts of nature, without an assignable cause, and, as it were, exceptions to general rules.
As examples of singular instances, we have the sun and moon among the heavenly bodies; the magnet among minerals; quicksilver among metals; the elephant among quadrupeds; the venereal sensation among the different kinds of touch; the scent of sporting dogs among those of smell. The letter S, too, is considered by the grammarians as sui generis, from its easily uniting with double or triple consonants, which no other letter will. These instances are of great value, because they excite and keep alive inquiry, and correct an understanding depraved by habit and the common course of things.
XXIX. In the eighth rank of prerogative instances, we will place deviating instances, such as the errors of nature, or strange and monstrous objects, in which nature deviates and turns from her ordinary course. For the errors of nature differ from singular instances, inasmuch as the latter are the miracles of species, the former of individuals. Their use is much the same, for they rectify the understanding in opposition to habit, and reveal common forms. For with regard to these, also, we must not desist from inquiry, till we discern the cause of the deviation. The cause does not, however, in such cases rise to a regular form, but only to the latent process toward such a form. For he who is acquainted with the paths of nature, will more readily observe her deviations; and, vice versâ, he who has learned her deviations will be able more accurately to describe her paths.
They differ again from singular instances, by being much more apt for practice and the operative branch. For it would be very difficult to generate new species, but less so to vary known species, and thus produce many rare and unusual results. The passage from the miracles of nature to those of art is easy; for if nature be once seized in her variations, and the cause be manifest, it will be easy to lead her by art to such deviation as she was at first led to by chance; and not only to that but others, since deviations on the one side lead and open the way to others in every direction. Of this we do not require any examples, since they are so abundant. For a compilation, or particular natural history, must be made of all monsters and prodigious births of nature; of everything, in short, which is new, rare and unusual in nature. This should be done with a rigorous selection, so as to be worthy of credit. Those are most to be suspected which depend upon superstition, as the prodigies of Livy, and those perhaps, but little less, which are found in the works of writers on natural magic, or even alchemy, and the like; for such men, as it were, are the very suitors and lovers of fables; but our instances should be derived from some grave and credible history, and faithful narration.
XXX. In the ninth rank of prerogative instances, we will place bordering instances, which we are also wont to term participants. They are such as exhibit those species of bodies which appear to be composed of two species, or to be the rudiments between the one and the other. They may well be classed with the singular or heteroclite instances; for in the whole system of things, they are rare and extraordinary. Yet from their dignity, they must be treated of and classed separately, for they point out admirably the order and constitution of things, and suggest the causes of the number and quality of the more common species in the universe, leading the understanding from that which is, to that which is possible.
We have examples of them in moss, which is something between putrescence and a plant; in some comets, which hold a place between stars and ignited meteors; in flying fishes, between fishes and birds; and in bats, between birds and quadrupeds. Again,
Simia quam similis turpissima bestia nobis.
We have also biformed fœtus, mingled species and the like.
XXXI. In the tenth rank of prerogative instances, we will place the instances of power, or the fasces (to borrow a term from the insignia of empire), which we are also wont to call the wit or hands of man. These are such works as are most noble and perfect, and, as it were, the masterpieces in every art. For since our principal object is to make nature subservient to the state and wants of man, it becomes us well to note and enumerate the works, which have long since been in the power of man, especially those which are most polished and perfect: because the passage from these to new and hitherto undiscovered works, is more easy and feasible. For if any one, after an attentive contemplation of such works as are extant, be willing to push forward in his design with alacrity and vigor, he will undoubtedly either advance them, or turn them to something within their immediate reach, or even apply and transfer them to some more noble purpose.
Nor is this all: for as the understanding is elevated and raised by rare and unusual works of nature, to investigate and discover the forms which include them also, so is the same effect frequently produced by the excellent and wonderful works of art; and even to a greater degree, because the mode of effecting and constructing the miracles of art is generally plain, while that of effecting the miracles of nature is more obscure. Great care, however, must be taken, that they do not depress the understanding, and fix it, as it were, to earth.
For there is some danger, lest the understanding should be astonished and chained down, and as it were bewitched, by such works of art, as appear to be the very summit and pinnacle of human industry, so as not to become familiar with them, but rather to suppose that nothing of the kind can be accomplished, unless the same means be employed, with perhaps a little more diligence, and more accurate preparation.
Now, on the contrary, it may be stated as a fact, that the ways and means hitherto discovered and observed, of effecting any matter or work, are for the most part of little value, and that all really efficient power depends, and is really to be deduced from the sources of forms, none of which have yet been discovered.
Thus (as we have before observed), had any one meditated on ballistic machines, and battering rams, as they were used by the ancients, whatever application he might have exerted, and though he might have consumed a whole life in the pursuit, yet would he never have hit upon the invention of flaming engines, acting by means of gunpowder; nor would any person, who had made woollen manufactories and cotton the subject of his observation and reflection, have ever discovered thereby the nature of the silkworm or of silk.
Hence all the most noble discoveries have (if you observe) come to light, not by any gradual improvement and extension of the arts, but merely by chance; while nothing imitates or anticipates chance (which is wont to act at intervals of ages) but the invention of forms.
There is no necessity for adducing any particular examples of these instances, since they are abundant. The plan to be pursued is this: all the mechanical, and even the liberal arts (as far as they are practical), should be visited and thoroughly examined, and thence there should be formed a compilation or particular history of the great masterpieces, or most finished works in each, as well as of the mode of carrying them into effect.
Nor do we confine the diligence to be used in such a compilation to the leading works and secrets only of every art, and such as excite wonder; for wonder is engendered by rarity, since that which is rare, although it be compounded of ordinary natures, always begets wonder.
On the contrary, that which is really wonderful, from some specific difference distinguishing it from other species, is carelessly observed, if it be but familiar. Yet the singular instances of art should be observed no less than those of nature, which we have before spoken of: and as in the latter we have classed the sun, the moon, the magnet, and the like, all of them most familiar to us, but yet in their nature singular, so should we proceed with the singular instances of art.
For example: paper, a very common substance, is a singular instance of art; for if you consider the subject attentively, you will find that artificial substances are either woven by straight and transverse lines, as silk, woollen, or linen cloth, and the like; or coagulated from concrete juices, such as brick, earthenware, glass, enamel, porcelain and the like, which admit of a polish if they be compact, but if not, become hard without being polished; all which latter substances are brittle, and not adherent or tenacious. On the contrary, paper is a tenacious substance, which can be cut and torn, so as to resemble and almost rival the skin of any animal, or the leaf of vegetables, and the like works of nature; being neither brittle like glass, nor woven like cloth, but having fibres and not distinct threads, just as natural substances, so that scarcely anything similar can be found among artificial substances, and it is absolutely singular. And in artificial works we should certainly prefer those which approach the nearest to an imitation of nature, or, on the other hand, powerfully govern and change her course.
Again, in these instances which we term the wit and hands of man, charms and conjuring should not be altogether despised, for although mere amusements, and of little use, yet they may afford considerable information.
Lastly, superstition and magic (in its common acceptation) are not to be entirely omitted; for although they be overwhelmed by a mass of lies and fables, yet some investigation should be made, to see if there be really any latent natural operation in them; as in fascination, and the fortifying of the imagination, the sympathy of distant objects, the transmission of impressions from spirit to spirit no less than from body to body, and the like.
XXXII. From the foregoing remarks, it is clear that the last five species of instances (the similar, singular, deviating and bordering instances, and those of power) should not be reserved for the investigation of any given nature, as the preceding and many of the succeeding instances must, but a collection of them should be made at once, in the style of a particular history, so that they may arrange the matter which enters the understanding, and correct its depraved habit, for it is necessarily imbued, corrupted, perverted and distorted by daily and habitual impressions.
They are to be used, therefore, as a preparative, for the purpose of rectifying and purifying the understanding; for whatever withdraws it from habit, levels and planes down its surface for the reception of the dry and pure light of true notions.
These instances, moreover, level and prepare the way for the operative branch, as we will mention in its proper place when speaking of the practical deductions.
XXXIII. In the eleventh rank of prerogative instances we will place accompanying and hostile instances. These are such as exhibit any body or concrete, where the required nature is constantly found, as an inseparable companion, or, on the contrary, where the required nature is constantly avoided, and excluded from attendance, as an enemy. From these instances may be formed certain and universal propositions, either affirmative or negative; the subject of which will be the concrete body, and the predicate the required nature. For particular propositions are by no means fixed, when the required nature is found to fluctuate and change in the concrete, either approaching and acquired, or receding and laid aside. Hence particular propositions have no great prerogative, except in the case of migration, of which we have spoken above. Yet such particular propositions are of great use, when compared with the universal, as will be mentioned in its proper place. Nor do we require absolute affirmation or negation, even in universal propositions, for if the exceptions be singular or rare, it is sufficient for our purpose.
The use of accompanying instances is to narrow the affirmative of form; for as it is narrowed by the migrating instances, where the form must necessarily be something communicated or destroyed by the act of migration, so it is narrowed by accompanying instances, where the form must necessarily be something which enters into the concretion of the body, or, on the contrary, is repugnant to it; and one who is well acquainted with the constitution or formation of the body, will not be far from bringing to light the form of the required nature.
For example: let the required nature be heat. Flame is an accompanying instance; for in water, air, stone, metal, and many other substances, heat is variable, and can approach or retire; but all flame is hot, so that heat always accompanies the concretion of flame. We have no hostile instance of heat; for the senses are unacquainted with the interior of the earth, and there is no concretion of any known body which is not susceptible of heat.
Again, let solidity be the required nature. Air is a hostile instance; for metals may be liquid or solid, so may glass; even water may become solid by congelation, but air cannot become solid or lose its fluidity.
With regard to these instances of fixed propositions, there are two points to be observed, which are of importance. First, that if there be no universal affirmative or negative, it be carefully noted as not existing. Thus, in heat, we have observed that there exists no universal negative, in such substances, at least, as have come to our knowledge. Again, if the required nature be eternity or incorruptibility, we have no universal affirmative within our sphere, for these qualities cannot be predicated of any bodies below the heavens, or above the interior of the earth. Secondly, to our general propositions as to any concrete, whether affirmative or negative, we should subjoin the concretes which appear to approach nearest to the non-existing substances; such as the most gentle or least-burning flames in heat, or gold in incorruptibility, since it approaches nearest to it. For they all serve to show the limit of existence and non-existence, and circumscribe forms, so that they cannot wander beyond the conditions of matter.
XXXIV. In the twelfth rank of prerogative instances, we will class those subjunctive instances, of which we spoke in the last aphorism, and which we are also wont to call instances of extremity or limits; for they are not only serviceable when subjoined to fixed propositions, but also of themselves and from their own nature. They indicate with sufficient precision the real divisions of nature, and measures of things, and the “how far” nature effects or allows of anything, and her passage thence to something else. Such are gold in weight, iron in hardness, the whale in the size of animals, the dog in smell, the flame of gunpowder in rapid expansion, and others of a like nature. Nor are we to pass over the extremes in defect, as well as in abundance, as spirits of wine in weight, the touchstone in softness, the worms upon the skin in the size of animals, and the like.
XXXV. In the thirteenth rank of prerogative instances we will place those of alliance or union. They are such as mingle and unite natures held to be heterogeneous, and observed and marked as such in received classifications.
These instances show that the operation and effect, which is considered peculiar to some one of such heterogeneous natures, may also be attributed to another nature styled heterogeneous, so as to prove that the difference of the natures is not real nor essential, but a mere modification of a common nature. They are very serviceable, therefore, in elevating and carrying on the mind, from differences to genera, and in removing those phantoms and images of things, which meet it in disguise in concrete substances.
For example: let the required nature be heat. The classification of heat into three kinds, that of the celestial bodies, that of animals, and that of fire, appears to be settled and admitted; and these kinds of heat, especially one of them compared with the other two, are supposed to be different, and clearly heterogeneous in their essence and species, or specific nature, since the heat of the heavenly bodies and of animals generates and cherishes, while that of fire corrupts and destroys. We have an instance of alliance, then, in a very common experiment, that of a vine branch admitted into a building where there is a constant fire, by which the grapes ripen a whole month sooner than in the air; so that fruit upon the tree can be ripened by fire, although this appear the peculiar effect of the sun. From this beginning, therefore, the understanding rejects all essential difference, and easily ascends to the investigation of the real differences between the heat of the sun and that of fire, by which their operation is rendered dissimilar, although they partake of a common nature.
These differences will be found to be four in number. 1.The heat of the sun is much milder and gentler in degree than that of fire. 2.It is much more moist in quality, especially as it is transmitted to us through the air. 3.Which is the chief point, it is very unequal, advancing and increased at one time, retiring and diminished at another, which mainly contributes to the generation of bodies. For Aristotle rightly asserted, that the principal cause of generation and corruption on the surface of the earth was the oblique path of the sun in the zodiac, whence its heat becomes very unequal, partly from the alternation of night and day, partly from the succession of summer and winter. Yet must he immediately corrupt and pervert his discovery, by dictating to nature according to his habit, and dogmatically assigning the cause of generation to the approach of the sun, and that of corruption to its retreat; while, in fact, each circumstance indifferently and not respectively contributes both to generation and corruption; for unequal heat tends to generate and corrupt, as equable heat does to preserve. 4.The fourth difference between the heat of the sun and fire is of great consequence; namely, that the sun, gradually, and for a length of time, insinuates its effects, while those of fire (urged by the impatience of man) are brought to a termination in a shorter space of time. But if any one were to pay attention to the tempering of fire, and reducing it to a more moderate and gentle degree (which may be done in various ways), and then were to sprinkle and mix a degree of humidity with it; and, above all, were to imitate the sun in its inequality; and, lastly, were patiently to suffer some delay (not such, however, as is proportioned to the effects of the sun, but more than men usually admit of in those of fire), he would soon banish the notion of any difference, and would attempt, or equal, or perhaps sometimes surpass the effect of the sun, by the heat of fire. A like instance of alliance is that of reviving butterflies, benumbed and nearly dead from cold, by the gentle warmth of fire; so that fire is no less able to revive animals than to ripen vegetables. We may also mention the celebrated invention of Fracastorius, of applying a pan considerably heated to the head in desperate cases of apoplexy, which clearly expands the animal spirits, when compressed and almost extinguished by the humors and obstructions of the brain, and excites them to action, as the fire would operate on water or air, and in the result produces life. Eggs are sometimes hatched by the heat of fire, an exact imitation of animal heat; and there are many instances of the like nature, so that no one can doubt that the heat of fire, in many cases, can be modified till it resemble that of the heavenly bodies and of animals.
Again, let the required natures be motion and rest. There appears to be a settled classification, grounded on the deepest philosophy, that natural bodies either revolve, move in a straight line, or stand still and rest. For there is either motion without limit, or continuance within a certain limit, or a translation toward a certain limit. The eternal motion of revolution appears peculiar to the heavenly bodies, rest to this our globe, and the other bodies (heavy and light, as they are termed, that is to say, placed out of their natural position) are borne in a straight line to masses or aggregates which resemble them, the light toward the heaven, the heavy toward the earth; and all this is very fine language.
But we have an instance of alliance in low comets, which revolve, though far below the heavens; and the fiction of Aristotle, of the comet being fixed to, or necessarily following some star, has been long since exploded; not only because it is improbable in itself, but from the evident fact of the discursive and irregular motion of comets through various parts of the heavens.
Another instance of alliance is that of the motion of air, which appears to revolve from east to west within the tropics, where the circles of revolution are the greatest.
The flow and ebb of the sea would perhaps be another instance, if the water were once found to have a motion of revolution, though slow and hardly perceptible, from east to west, subject, however, to a reaction twice a day. If this be so, it is clear that the motion of revolution is not confined to the celestial bodies, but is shared, also, by air and water.
Again—the supposed peculiar disposition of light bodies to rise is rather shaken; and here we may find an instance of alliance in a water bubble. For if air be placed under water, it rises rapidly toward the surface by that striking motion (as Democritus terms it) with which the descending water strikes the air and raises it, not by any struggle or effort of the air itself; and when it has reached the surface of the water, it is prevented from ascending any further, by the slight resistance it meets with in the water, which does not allow an immediate separation of its parts, so that the tendency of the air to rise must be very slight.
Again, let the required nature be weight. It is certainly a received classification, that dense and solid bodies are borne toward the centre of the earth, and rare and light bodies to the circumference of the heavens, as their appropriate places. As far as relates to places (though these things have much weight in the schools), the notion of there being any determinate place is absurd and puerile. Philosophers trifle, therefore, when they tell you, that if the earth were perforated, heavy bodies would stop on their arrival at the centre. This centre would indeed be an efficacious nothing, or mathematical point, could it affect bodies or be sought by them, for a body is not acted upon except by a body. In fact, this tendency to ascend and descend is either in the conformation of the moving body, or in its harmony and sympathy with another body. But if any dense and solid body be found, which does not, however, tend toward the earth, the classification is at an end. Now, if we allow of Gilbert’s opinion, that the magnetic power of the earth, in attracting heavy bodies, is not extended beyond the limit of its peculiar virtue (which operates always at a fixed distance and no further), and this be proved by some instance, such an instance will be one of alliance in our present subject. The nearest approach to it is that of waterspouts, frequently seen by persons navigating the Atlantic toward either of the Indies. For the force and mass of the water suddenly effused by waterspouts, appears to be so considerable, that the water must have been collected previously, and have remained fixed where it was formed, until it was afterward forced down by some violent cause, rather than made to fall by the natural motion of gravity: so that it may be conjectured that a dense and compact mass, at a great distance from the earth, may be suspended as the earth itself is, and would not fall, unless forced down. We do not, however, affirm this as certain. In the meanwhile, both in this respect and many others, it will readily be seen how deficient we are in natural history, since we are forced to have recourse to suppositions for examples, instead of ascertained instances.
Again, let the required nature be the discursive power of the mind. The classification of human reason and animal instinct appears to be perfectly correct. Yet there are some instances of the actions of brutes which seem to show that they, too, can syllogize. Thus it is related, that a crow, which had nearly perished from thirst in a great drought, saw some water in the hollow trunk of a tree, but as it was too narrow for him to get into it, he continued to throw in pebbles, which made the water rise till he could drink; and it afterward became a proverb.
Again, let the required nature be vision. The classification appears real and certain, which considers light as that which is originally visible, and confers the power of seeing; and color, as being secondarily visible, and not capable of being seen without light, so as to appear a mere image or modification of light. Yet there are instances of alliance in each respect; as in snow when in great quantities, and in the flame of sulphur; the one being a color originally and in itself light, the other a light verging toward color.
XXXVI. In the fourteenth rank of prerogative instances, we will place the instances of the cross, borrowing our metaphor from the crosses erected where two roads meet, to point out the different directions. We are wont also to call them decisive and judicial instances, and in some cases instances of the oracle and of command. Their nature is as follows: When in investigating any nature the understanding is, as it were, balanced, and uncertain to which of two or more natures the cause of the required nature should be assigned, on account of the frequent and usual concurrence of several natures, the instances of the cross show that the union of one nature with the required nature is firm and indissoluble, while that of the other is unsteady and separable; by which means the question is decided, and the first is received as the cause, while the other is dismissed and rejected. Such instances, therefore, afford great light, and are of great weight, so that the course of interpretation sometimes terminates, and is completed in them. Sometimes, however, they are found among the instances already observed, but they are generally new, being expressly and purposely sought for and applied, and brought to light only by attentive and active diligence.
For example: let the required nature be the flow and ebb of the sea, which is repeated twice a day, at intervals of six hours between each advance and retreat, with some little difference, agreeing with the motion of the moon. We have here the following crossways:
This motion must be occasioned either by the advancing and the retiring of the sea, like water shaken in a basin, which leaves one side while it washes the other; or by the rising of the sea from the bottom, and its again subsiding, like boiling water. But a doubt arises, to which of these causes we should assign the flow and ebb. If the first assertion be admitted, it follows, that when there is a flood on one side, there must at the same time be an ebb on another, and the question therefore is reduced to this. Now Acosta, and some others, after a diligent inquiry, have observed that the flood tide takes place on the coast of Florida, and the opposite coasts of Spain and Africa, at the same time, as does also the ebb; and that there is not, on the contrary, a flood tide at Florida when there is an ebb on the coasts of Spain and Africa. Yet if one consider the subject attentively, this does not prove the necessity of a rising motion, nor refute the notion of a progressive motion. For the motion may be progressive, and yet inundate the opposite shores of a channel at the same time; as if the waters be forced and driven together from some other quarter, for instance, which takes place in rivers, for they flow and ebb toward each bank at the same time, yet their motion is clearly progressive, being that of the waters from the sea entering their mouths. So it may happen, that the waters coming in a vast body from the eastern Indian Ocean are driven together, and forced into the channel of the Atlantic, and therefore inundate both coasts at once. We must inquire, therefore, if there be any other channel by which the waters can at the same time sink and ebb; and the Southern Ocean at once suggests itself, which is not less than the Atlantic, but rather broader and more extensive than is requisite for this effect.
We at length arrive, then, at an instance of the cross, which is this. If it be positively discovered, that when the flood sets in toward the opposite coasts of Florida and Spain in the Atlantic, there is at the same time a flood tide on the coasts of Peru and the back part of China, in the Southern Ocean, then assuredly, from this decisive instance, we must reject the assertion, that the flood and ebb of the sea, about which we inquire, takes place by progressive motion; for no other sea or place is left where there can be an ebb. But this may most easily be learned, by inquiring of the inhabitants of Panama and Lima (where the two oceans are separated by a narrow isthmus), whether the flood and ebb takes place on the opposite sides of the isthmus at the same time, or the reverse. This decision or rejection appears certain, if it be granted that the earth is fixed; but if the earth revolves, it may perhaps happen, that from the unequal revolution (as regards velocity) of the earth and the waters of the sea, there may be a violent forcing of the waters into a mass, forming the flood, and a subsequent relaxation of them (when they can no longer bear the accumulation), forming the ebb. A separate inquiry must be made into this. Even with this hypothesis, however, it remains equally true, that there must be an ebb somewhere, at the same time that there is a flood in another quarter.
Again, let the required nature be the latter of the two motions we have supposed; namely, that of a rising and subsiding motion, if it should happen that upon diligent examination the progressive motion be rejected. We have, then, three ways before us, with regard to this nature. The motion, by which the waters raise themselves, and again fall back, in the floods and ebbs, without the addition of any other water rolled toward them, must take place in one of the three following ways: Either the supply of water emanates from the interior of the earth, and returns back again; or there is really no greater quantity of water, but the same water (without any augmentation of its quantity) is extended or rarefied, so as to occupy a greater space and dimension, and again contracts itself; or there is neither an additional supply nor any extension, but the same waters (with regard to quantity, density, or rarity) raise themselves and fall from sympathy, by some magnetic power attracting and calling them up, as it were, from above. Let us then (passing over the first two motions) reduce the investigation to the last, and inquire if there be any such elevation of the water by sympathy or a magnetic force; and it is evident, in the first place, that the whole mass of water being placed in the trench or cavity of the sea, cannot be raised at once, because there would not be enough to cover the bottom, so that if there be any tendency of this kind in the water to raise itself, yet it would be interrupted and checked by the cohesion of things, or (as the common expression is) that there may be no vacuum. The water, therefore, must rise on one side, and for that reason be diminished and ebb on another. But it will again necessarily follow that the magnetic power not being able to operate on the whole, operates most intensely on the centre, so as to raise the waters there, which, when thus raised successively, desert and abandon the sides.
We at length arrive, then, at an instance of the cross, which is this: if it be found that during the ebb the surface of the waters at sea is more curved and round, from the waters rising in the middle, and sinking at the sides or coast, and if, during a flood, it be more even and level, from the waters returning to their former position, then assuredly, by this decisive instance, the raising of them by a magnetic force can be admitted; if otherwise, it must be entirely rejected. It is not difficult to make the experiment (by sounding in straits), whether the sea be deeper toward the middle in ebbs, than in floods. But it must be observed, if this be the case, that (contrary to common opinion) the waters rise in ebbs, and only return to their former position in floods, so as to bathe and inundate the coast.
Again, let the required nature be the spontaneous motion of revolution, and particularly, whether the diurnal motion, by which the sun and stars appear to us to rise and set, be a real motion of revolution in the heavenly bodies, or only apparent in them, and real in the earth. There may be an instance of the cross of the following nature. If there be discovered any motion in the ocean from east to west, though very languid and weak, and if the same motion be discovered rather more swift in the air (particularly within the tropics, where it is more perceptible from the circles being greater). If it be discovered also in the low comets, and be already quick and powerful in them; if it be found also in the planets, but so tempered and regulated as to be slower in those nearest the earth, and quicker in those at the greatest distance, being quickest of all in the heavens, then the diurnal motion should certainly be considered as real in the heavens, and that of the earth must be rejected; for it will be evident that the motion from east to west is part of the system of the world and universal; since it is most rapid in the height of the heavens, and gradually grows weaker, till it stops and is extinguished in rest at the earth.
Again, let the required nature be that other motion of revolution, so celebrated among astronomers, which is contrary to the diurnal, namely, from west to east—and which the ancient astronomers assign to the planets, and even to the starry sphere, but Copernicus and his followers to the earth also—and let it be examined whether any such motion be found in nature, or it be rather a fiction and hypothesis for abridging and facilitating calculation, and for promoting that fine notion of effecting the heavenly motions by perfect circles; for there is nothing which proves such a motion in heavenly objects to be true and real, either in a planet’s not returning in its diurnal motion to the same point of the starry sphere, or in the pole of the zodiac being different from that of the world, which two circumstances have occasioned this notion. For the first phenomenon is well accounted for by the spheres overtaking or falling behind each other, and the second by spiral lines; so that the inaccuracy of the return and declination to the tropics may be rather modifications of the one diurnal motion than contrary motions, or about different poles. And it is most certain, if we consider ourselves for a moment as part of the vulgar (setting aside the fictions of astronomers and the school, who are wont undeservedly to attack the senses in many respects, and to affect obscurity), that the apparent motion is such as we have said, a model of which we have sometimes caused to be represented by wires in a sort of a machine.
We may take the following instances of the cross upon this subject. If it be found in any history worthy of credit, that there has existed any comet, high or low, which has not revolved in manifest harmony (however irregularly) with the diurnal motion, then we may decide so far as to allow such a motion to be possible in nature. But if nothing of the sort be found, it must be suspected, and recourse must be had to other instances of the cross.
Again, let the required nature be weight or gravity. Heavy and ponderous bodies must, either of their own nature, tend toward the centre of the earth by their peculiar formation, or must be attracted and hurried by the corporeal mass of the earth itself, as being an assemblage of similar bodies, and be drawn to it by sympathy. But if the latter be the cause, it follows that the nearer bodies approach to the earth, the more powerfully and rapidly they must be borne toward it, and the further they are distant, the more faintly and slowly (as is the case in magnetic attractions), and that this must happen within a given distance; so that if they be separated at such a distance from the earth that the power of the earth cannot act upon them, they will remain suspended like the earth, and not fall at all.
The following instance of the cross may be adopted. Take a clock moved by leaden weights, and another by a spring, and let them be set well together, so that one be neither quicker nor slower than the other; then let the clock moved by weights be placed on the top of a very high church, and the other be kept below, and let it be well observed, if the former move slower than it did, from the diminished power of the weights. Let the same experiment be made at the bottom of mines worked to a considerable depth, in order to see whether the clock move more quickly from the increased power of the weights. But if this power be found to diminish at a height, and to increase in subterraneous places, the attraction of the corporeal mass of the earth may be taken as the cause of weight.
Again, let the required nature be the polarity of the steel needle when touched with the magnet. We have these two ways with regard to this nature—Either the touch of the magnet must communicate polarity to the steel toward the north and south, or else it may only excite and prepare it, while the actual motion is occasioned by the presence of the earth, which Gilbert considers to be the case, and endeavors to prove with so much labor. The particulars he has inquired into with such ingenious zeal amount to this—1.An iron bolt placed for a long time toward the north and south acquires polarity from this habit, without the touch of the magnet, as if the earth itself operating but weakly from its distance (for the surface or outer crust of the earth does not, in his opinion, possess the magnetic power), yet, by long continued motion, could supply the place of the magnet, excite the iron, and convert and change it when excited. 2.Iron, at a red or white heat, when quenched in a direction parallel to the north and south, also acquires polarity without the touch of the magnet, as if the parts of iron being put in motion by ignition, and afterward recovering themselves, were, at the moment of being quenched, more susceptible and sensitive of the power emanating from the earth, than at other times, and therefore as it were excited. But these points, though well observed, do not completely prove his assertion.
An instance of the cross on this point might be as follows: Let a small magnetic globe be taken, and its poles marked, and placed toward the east and west, not toward the north and south, and let it continue thus. Then let an untouched needle be placed over it, and suffered to remain so for six or seven days. Now, the needle (for this is not disputed), while it remains over the magnet, will leave the poles of the world and turn to those of the magnet, and therefore, as long as it remains in the above position, will turn to the east and west. But if the needle, when removed from the magnet and placed upon a pivot, be found immediately to turn to the north and south, or even by degrees to return thither, then the presence of the earth must be considered as the cause, but if it remains turned as at first, toward the east and west, or lose its polarity, then that cause must be suspected, and further inquiry made.
Again, let the required nature be the corporeal substance of the moon, whether it be rare, fiery, and aërial (as most of the ancient philosophers have thought), or solid and dense (as Gilbert and many of the moderns, with some of the ancients, hold). The reasons for this latter opinion are grounded chiefly upon this, that the moon reflects the sun’s rays, and that light does not appear capable of being reflected except by solids. The instances of the cross will therefore (if any) be such as to exhibit reflection by a rare body, such as flame, if it be but sufficiently dense. Now, certainly, one of the reasons of twilight is the reflection of the rays of the sun by the upper part of the atmosphere. We see the sun’s rays also reflected on fine evenings by streaks of moist clouds, with a splendor not less, but perhaps more bright and glorious than that reflected from the body of the moon, and yet it is not clear that those clouds have formed into a dense body of water. We see, also, that the dark air behind the windows at night reflects the light of a candle in the same manner as a dense body would do. The experiment should also be made of causing the sun’s rays to fall through a hole upon some dark and bluish flame. The unconfined rays of the sun, when falling on faint flames, do certainly appear to deaden them, and render them more like white smoke than flames. These are the only instances which occur at present of the nature of those of the cross, and better perhaps can be found. But it must always be observed that reflection is not to be expected from flame, unless it be of some depth, for otherwise it becomes nearly transparent. This at least may be considered certain, that light is always either received and transmitted or reflected by an even surface.
Again, let the required nature be the motion of projectiles (such as darts, arrows, and balls) through the air. The school, in its usual manner, treats this very carelessly, considering it enough to distinguish it by the name of violent motion, from that which they term natural, and as far as regards the first percussion or impulse, satisfies itself by its axiom, that two bodies cannot exist in one place, or there would be a penetration of dimensions. With regard to this nature we have these two crossways—The motion must arise either from the air carrying the projected body, and collecting behind it, like a stream behind boats, or the wind behind straws; or from the parts of the body itself not supporting the impression, but pushing themselves forward in succession to ease it. Fracastorius, and nearly all those who have entered into any refined inquiry upon the subject, adopt the first. Nor can it be doubted that the air has some effect, yet the other motion is without doubt real, as is clear from a vast number of experiments. Among others we may take this instance of the cross, namely, that a thin plate or wire of iron rather stiff, or even a reed or pen split in two, when drawn up and bent between the finger and thumb, will leap forward; for it is clear that this cannot be attributed to the air’s being collected behind the body, because the source of motion is in the centre of the plate or pen, and not in its extremities.
Again, let the required nature be the rapid and powerful motion of the explosion of gunpowder, by which such vast masses are upheaved, and such weights discharged as we observe in large mines and mortars, there are two crossways before us with regard to this nature. This motion is excited either by the mere effort of the body expanding itself when inflamed, or by the assisting effort of the crude spirit, which escapes rapidly from fire, and bursts violently from the surrounding flame as from a prison. The school, however, and common opinion only consider the first effort; for men think that they are great philosophers when they assert that flame, from the form of the element, is endowed with a kind of necessity of occupying a greater space than the same body had occupied when in the form of powder, and that thence proceeds the motion in question. In the meantime they do not observe, that although this may be true, on the supposition of flame being generated, yet the generation may be impeded by a weight of sufficient force to compress and suffocate it, so that no such necessity exists as they assert. They are right, indeed, in imagining that the expansion and the consequent emission or removal of the opposing body, is necessary if flame be once generated, but such a necessity is avoided if the solid opposing mass suppress the flame before it be generated; and we in fact see that flame, especially at the moment of its generation, is mild and gentle, and requires a hollow space where it can play and try its force. The great violence of the effect, therefore, cannot be attributed to this cause; but the truth is, that the generation of these exploding flames and fiery blasts arises from the conflict of two bodies of a decidedly opposite nature—the one very inflammable, as is the sulphur, the other having an antipathy to flame, namely, the crude spirit of the nitre; so that an extraordinary conflict takes place while the sulphur is becoming inflamed as far as it can (for the third body, the willow charcoal, merely incorporates and conveniently unites the two others), and the spirit of nitre is escaping, as far also as it can, and at the same time expanding itself (for air, and all crude substances, and water are expanded by heat), fanning thus, in every direction, the flame of the sulphur by its escape and violence, just as if by invisible bellows.
Two kinds of instances of the cross might here be used—the one of very inflammable substances, such as sulphur and camphor, naphtha and the like, and their compounds, which take fire more readily and easily than gunpowder if left to themselves (and this shows that the effort to catch fire does not of itself produce such a prodigious effect); the other of substances which avoid and repel flame, such as all salts; for we see that when they are cast into the fire, the aqueous spirit escapes with a crackling noise before flame is produced, which also happens in a less degree in stiff leaves, from the escape of the aqueous part before the oily part has caught fire. This is more particularly observed in quicksilver, which is not improperly called mineral water, and which, without any inflammation, nearly equals the force of gunpowder by simple explosion and expansion, and is said, when mixed with gunpowder, to increase its force.
Again, let the required nature be the transitory nature of flame and its momentaneous extinction; for to us the nature of flame does not appear to be fixed or settled, but to be generated from moment to moment, and to be every instant extinguished; it being clear that those flames which continue and last, do not owe their continuance to the same mass of flame, but to a continued succession of new flame regularly generated, and that the same identical flame does not continue. This is easily shown by removing the food or source of the flame, when it at once goes out. We have the two following crossways with regard to this nature:
This momentary nature either arises from the cessation of the cause which first produced it, as in light, sounds, and violent motions, as they are termed, or flame may be capable, by its own nature, of duration, but is subjected to some violence from the contrary natures which surround it, and is destroyed.
We may therefore adopt the following instance of the cross. We see to what a height the flames rise in great conflagrations; for as the base of the flame becomes more extensive, its vertex is more lofty. It appears, then, that the commencement of the extinction takes place at the sides, where the flame is compressed by the air, and is ill at ease; but the centre of the flame, which is untouched by the air and surrounded by flame, continues the same, and is not extinguished until compressed by degrees by the air attacking it from the sides. All flame, therefore, is pyramidal, having its base near the source, and its vertex pointed from its being resisted by the air, and not supplied from the source. On the contrary, the smoke, which is narrow at the base, expands in its ascent, and resembles an inverted pyramid, because the air admits the smoke, but compresses the flame; for let no one dream that the lighted flame is air, since they are clearly heterogeneous.
The instance of the cross will be more accurate, if the experiment can be made by flames of different colors. Take, therefore, a small metal sconce, and place a lighted taper in it, then put it in a basin, and pour a small quantity of spirits of wine round the sconce, so as not to reach its edge, and light the spirit. Now the flame of the spirit will be blue, and that of the taper yellow; observe, therefore, whether the latter (which can easily be distinguished from the former by its color, for flames do not mix immediately, as liquids do) continue pyramidal, or tend more to a globular figure, since there is nothing to destroy or compress it. If the latter result be observed, it must be considered as settled, that flame continues positively the same, while inclosed within another flame, and not exposed to the resisting force of the air.
Let this suffice for the instances of the cross. We have dwelt the longer upon them in order gradually to teach and accustom mankind to judge of nature by these instances, and enlightening experiments, and not by probable reasons.
XXXVII. We will treat of the instances of divorce as the fifteenth of our prerogative instances. They indicate the separation of natures of the most common occurrence. They differ, however, from those subjoined to the accompanying instances; for the instances of divorce point out the separation of a particular nature from some concrete substance with which it is usually found in conjunction, while the hostile instances point out the total separation of one nature from another. They differ, also, from the instances of the cross, because they decide nothing, but only inform us that the one nature is capable of being separated from the other. They are of use in exposing false forms, and dissipating hasty theories derived from obvious facts; so that they add ballast and weight, as it were, to the understanding.
For instance, let the acquired natures be those four which Telesius terms associates, and of the same family, namely, heat, light, rarity, and mobility, or promptitude to motion; yet many instances of divorce can be discovered between them. Air is rare and easily moved, but neither hot nor light; the moon is light but not hot; boiling water is warm but not light; the motion of the needle in the compass is swift and active, and yet its substance is cold, dense, and opaque; and there are many similar examples.
Again, let the required natures be corporeal nature and natural action. The latter appears incapable of subsisting without some body, yet may we, perhaps, even here find an instance of divorce, as in the magnetic motion, which draws the iron to the magnet, and heavy bodies to the globe of the earth; to which we may add other actions which operate at a distance. For such action takes place in time, by distinct moments, not in an instant; and in space, by regular degrees and distances. There is, therefore, some one moment of time and some interval of space, in which the power or action is suspended between the two bodies creating the motion. Our consideration, then, is reduced to this, whether the bodies which are the extremes of motion prepare or alter the intermediate bodies, so that the power advances from one extreme to the other by succession and actual contact, and in the meantime exists in some intermediate body; or whether there exists in reality nothing but the bodies, the power, and the space? In the case of the rays of light, sounds, and heat, and some other objects which operate at a distance, it is indeed probable that the intermediate bodies are prepared and altered, the more so because a qualified medium is required for their operation. But the magnetic or attractive power admits of an indifferent medium, and it is not impeded in any. But if that power or action is independent of the intermediate body, it follows that it is a natural power or action existing in a certain time and space without any body, since it exists neither in the extreme nor in the intermediate bodies. Hence the magnetic action may be taken as an instance of divorce of corporeal nature and natural action; to which we may add, as a corollary and an advantage not to be neglected, that it may be taken as a proof of essence and substance being separate and incorporeal, even by those who philosophize according to the senses. For if natural power and action emanating from a body can exist at any time and place entirely without any body, it is nearly a proof that it can also emanate originally from an incorporeal substance; for a corporeal nature appears to be no less necessary for supporting and conveying, than for exciting or generating natural action.
XXXVIII. Next follow five classes of instances which we are wont to call by the general term of instances of the lamp, or of immediate information. They are such as assist the senses; for since every interpretation of nature sets out from the senses, and leads, by a regular fixed and well-established road, from the perceptions of the senses to those of the understanding (which are true notions and axioms), it necessarily follows, that in proportion as the representatives or ministerings of the senses are more abundant and accurate, everything else must be more easy and successful.
The first of these five sets of instances of the lamp, strengthen, enlarge, and correct the immediate operations of the senses; the second reduce to the sphere of the senses such matters as are beyond it; the third indicate the continued process or series of such things and motions, as for the most part are only observed in their termination, or in periods; the fourth supply the absolute wants of the senses; the fifth excite their attention and observation, and at the same time limit the subtilty of things. We will now proceed to speak of them singly.
XXXIX. In the sixteenth rank, then, of prerogative instances, we will place the instances of the door or gate, by which name we designate such as assist the immediate action of the senses. It is obvious, that sight holds the first rank among the senses, with regard to information, for which reason we must seek principally helps for that sense. These helps appear to be threefold, either to enable it to perceive objects not naturally seen, or to see them from a greater distance, or to see them more accurately and distinctly.
We have an example of the first (not to speak of spectacles and the like, which only correct and remove the infirmity of a deficient sight, and therefore give no further information) in the lately invented microscopes, which exhibit the latent and invisible minutiæ of substances, and their hidden formation and motion, by wonderfully increasing their apparent magnitude. By their assistance we behold with astonishment the accurate form and outline of a flea, moss, and animalculæ, as well as their previously invisible color and motion. It is said, also, that an apparently straight line, drawn with a pen or pencil, is discovered by such a microscope to be very uneven and curved, because neither the motion of the hand, when assisted by a ruler, nor the impression of ink or color, are really regular, although the irregularities are so minute as not to be perceptible without the assistance of the microscope. Men have (as is usual in new and wonderful discoveries) added a superstitious remark, that the microscope sheds a lustre on the works of nature, and dishonor on those of art, which only means that the tissue of nature is much more delicate than that of art. For the microscope is only of use for minute objects, and Democritus, perhaps, if he had seen it, would have exulted in the thought of a means being discovered for seeing his atom, which he affirmed to be entirely invisible. But the inadequacy of these microscopes, for the observation of any but the most minute bodies, and even of those if parts of a larger body, destroys their utility; for if the invention could be extended to greater bodies, or the minute parts of greater bodies, so that a piece of cloth would appear like a net, and the latent minutiæ and irregularities of gems, liquids, urine, blood, wounds, and many other things could be rendered visible, the greatest advantage would, without doubt, be derived.
We have an instance of the second kind in the telescope, discovered by the wonderful exertions of Galileo; by the assistance of which a nearer intercourse may be opened (as by boats or vessels) between ourselves and the heavenly objects. For by its aid we are assured that the Milky Way is but a knot or constellation of small stars, clearly defined and separate, which the ancients only conjectured to be the case; whence it appears to be capable of demonstration, that the spaces of the planetary orbits (as they are termed) are not quite destitute of other stars, but that the heaven begins to glitter with stars before we arrive at the starry sphere, although they may be too small to be visible without the telescope. By the telescope, also, we can behold the revolutions of smaller stars round Jupiter, whence it may be conjectured that there are several centres of motion among the stars. By its assistance, also, the irregularity of light and shade on the moon’s surface is more clearly observed and determined, so as to allow of a sort of selenography. By the telescope we see the spots in the sun, and other similar phenomena; all of which are most noble discoveries, as far as credit can be safely given to demonstrations of this nature, which are on this account very suspicious, namely, that experiment stops at these few, and nothing further has yet been discovered by the same method, among objects equally worthy of consideration.
We have instances of the third kind in measuring-rods, astrolabes, and the like, which do not enlarge, but correct and guide the sight. If there be other instances which assist the other senses in their immediate and individual action, yet if they add nothing further to their information they are not apposite to our present purpose, and we have therefore said nothing of them.
XL. In the seventeenth rank of prerogative instances we will place citing instances (to borrow a term from the tribunals), because they cite those things to appear, which have not yet appeared. We are wont also to call them invoking instances, and their property is that of reducing to the sphere of the senses objects which do not immediately fall within it.
Objects escape the senses either from their distance, or the intervention of other bodies, or because they are not calculated to make an impression upon the senses, or because they are not in sufficient quantity to strike the senses, or because there is not sufficient time for their acting upon the senses, or because the impression is too violent, or because the senses are previously filled and possessed by the object, so as to leave no room for any new motion. These remarks apply principally to sight, and next to touch, which two senses act extensively in giving information, and that too upon general objects, while the remaining three inform us only, as it were, by their immediate action, and as to specific objects.
There can be no reduction to the sphere of the senses in the first case, unless in the place of the object, which cannot be perceived on account of the distance, there be added or substituted some other object, which can excite and strike the sense from a greater distance, as in the communication of intelligence by fires, bells, and the like.
In the second case we effect this reduction by rendering those things which are concealed by the interposition of other bodies, and which cannot easily be laid open, evident to the senses by means of that which lies at the surface, or proceeds from the interior; thus the state of the body is judged of by the pulse, urine, etc.
The third and fourth cases apply to many subjects, and the reduction to the sphere of the senses must be obtained from every quarter in the investigation of things. There are many examples. It is obvious that air, and spirit, and the like, whose whole substance is extremely rare and delicate, can neither be seen nor touched—a reduction, therefore, to the senses becomes necessary in every investigation relating to such bodies.
Let the required nature, therefore, be the action and motion of the spirit inclosed in tangible bodies; for every tangible body with which we are acquainted contains an invisible and intangible spirit, over which it is drawn, and which it seems to clothe. This spirit being emitted from a tangible substance, leaves the body contracted and dry; when retained, it softens and melts it; when neither wholly emitted nor retained, it models it, endows it with limbs, assimilates, manifests, organizes it, and the like. All these points are reduced to the sphere of the senses by manifest effects.
For in every tangible and inanimate body the inclosed spirit at first increases, and as it were feeds on the tangible parts which are most open and prepared for it; and when it has digested and modified them, and turned them into spirit, it escapes with them. This formation and increase of spirit is rendered sensible by the diminution of weight; for in every desiccation something is lost in quantity, not only of the spirit previously existing in the body, but of the body itself, which was previously tangible, and has been recently changed, for the spirit itself has no weight. The departure or emission of spirit is rendered sensible in the rust of metals, and other putrefactions of a like nature, which stop before they arrive at the rudiments of life, which belong to the third species of process. In compact bodies the spirit does not find pores and passages for its escape, and is therefore obliged to force out, and drive before it, the tangible parts also, which consequently protrude, whence arises rust and the like. The contraction of the tangible parts, occasioned by the emission of part of the spirit (whence arises desiccation), is rendered sensible by the increased hardness of the substance, and still more by the fissures, contractions, shrivelling, and folds of the bodies thus produced. For the parts of wood split and contract, skins become shrivelled, and not only that, but, if the spirit be emitted suddenly by the heat of the fire, become so hastily contracted as to twist and roll themselves up.
On the contrary, when the spirit is retained, and yet expanded and excited by heat or the like (which happens in solid and tenacious bodies), then the bodies are softened, as in hot iron; or flow, as in metals; or melt, as in gums, wax, and the like. The contrary effects of heat, therefore (hardening some substances and melting others), are easily reconciled, because the spirit is emitted in the former, and agitated and retained in the latter; the latter action is that of heat and the spirit, the former that of the tangible parts themselves, after the spirit’s emission.
But when the spirit is neither entirely retained nor emitted, but only strives and exercises itself, within its limits, and meets with tangible parts, which obey and readily follow it wherever it leads them, then follows the formation of an organic body, and of limbs, and the other vital actions of vegetables and animals. These are rendered sensible chiefly by diligent observation of the first beginnings, and rudiments or effects of life in animalculæ sprung from putrefaction, as in the eggs of ants, worms, mosses, frogs after rain, etc. Both a mild heat and a pliant substance, however, are necessary for the production of life, in order that the spirit may neither hastily escape, nor be restrained by the obstinacy of the parts, so as not to be able to bend and model them like wax.
Again, the difference of spirit which is important and of effect in many points (as unconnected spirit, branching spirit, branching and cellular spirit, the first of which is that of all inanimate substances, the second of vegetables, and the third of animals), is placed, as it were, before the eyes by many reducing instances.
Again, it is clear that the more refined tissue and conformation of things (though forming the whole body of visible or tangible objects) are neither visible nor tangible. Our information, therefore, must here also be derived from reduction to the sphere of the senses. But the most radical and primary difference of formation depends on the abundance or scarcity of matter within the same space or dimensions. For the other formations which regard the dissimilarity of the parts contained in the same body, and their collocation and position, are secondary in comparison with the former.
Let the required nature then be the expansion or coherence of matter in different bodies, or the quantity of matter relative to the dimensions of each. For there is nothing in nature more true than the twofold proposition—that nothing proceeds from nothing and that nothing is reduced to nothing, but that the quantum, or sum total of matter, is constant, and is neither increased nor diminished. Nor is it less true, that out of this given quantity of matter, there is a greater or less quantity, contained within the same space or dimensions according to the difference of bodies; as, for instance, water contains more than air. So that if any one were to assert that a given content of water can be changed into an equal content of air, it is the same as if he were to assert that something can be reduced into nothing. On the contrary, if any one were to assert that a given content of air can be changed into an equal content of water, it is the same as if he were to assert that something can proceed from nothing. From this abundance or scarcity of matter are properly derived the notions of density and rarity, which are taken in various and promiscuous senses.
This third assertion may be considered as being also sufficiently certain; namely, that the greater or less quantity of matter in this or that body, may, by comparison, be reduced to calculation, and exact, or nearly exact, proportion. Thus, if one should say that there is such an accumulation of matter in a given quantity of gold, that it would require twenty-one times the quantity in dimension of spirits of wine, to make up the same quantity of matter, it would not be far from the truth.
The accumulation of matter, however, and its relative quantity, are rendered sensible by weight; for weight is proportionate to the quantity of matter, as regards the parts of a tangible substance, but spirit and its quantity of matter are not to be computed by weight, which spirit rather diminishes than augments.
We have made a tolerably accurate table of weight, in which we have selected the weights and size of all the metals, the principal minerals, stones, liquids, oils, and many other natural and artificial bodies: a very useful proceeding both as regards theory and practice, and which is capable of revealing many unexpected results. Nor is this of little consequence, that it serves to demonstrate that the whole range of the variety of tangible bodies with which we are acquainted (we mean tolerably close, and not spongy, hollow bodies, which are for a considerable part filled with air), does not exceed the ratio of one to twenty-one. So limited is nature, or at least that part of it to which we are most habituated.
We have also thought it deserving our industry, to try if we could arrive at the ratio of intangible or pneumatic bodies to tangible bodies, which we attempted by the following contrivance. We took a vial capable of containing about an ounce, using a small vessel in order to effect the subsequent evaporation with less heat. We filled this vial, almost to the neck, with spirits of wine, selecting it as the tangible body which, by our table, was the rarest, and contained a less quantity of matter in a given space than all other tangible bodies which are compact and not hollow. Then we noted exactly the weight of the liquid and vial. We next took a bladder, containing about two pints, and squeezed all the air out of it, as completely as possible, and until the sides of the bladder met. We first, however, rubbed the bladder gently with oil, so as to make it air-tight, by closing its pores with the oil. We tied the bladder tightly round the mouth of the vial, which we had inserted in it, and with a piece of waxed thread to make it fit better and more tightly, and then placed the vial on some hot coals in a brazier. The vapor or steam of the spirit, dilated and become aëriform by the heat, gradually swelled out the bladder, and stretched it in every direction like a sail. As soon as that was accomplished, we removed the vial from the fire and placed it on a carpet, that it might not be cracked by the cold; we also pricked the bladder immediately, that the steam might not return to a liquid state by the cessation of heat, and confound the proportions. We then removed the bladder, and again took the weight of the spirit which remained; and so calculated the quantity which had been converted into vapor, or an aëriform shape, and then examined how much space had been occupied by the body in its form of spirits of wine in the vial, and how much, on the other hand, had been occupied by it in its aëriform shape in the bladder, and subtracted the results; from which it was clear that the body, thus converted and changed, acquired an expansion of one hundred times beyond its former bulk.
Again, let the required nature be heat or cold, of such a degree as not to be sensible from its weakness. They are rendered sensible by the thermometer, as we described it above; for the cold and heat are not actually perceived by the touch, but heat expands and cold contracts the air. Nor, again, is that expansion or contraction of the air in itself visible, but the air when expanded depresses the water, and when contracted raises it, which is the first reduction to sight.
Again, let the required nature be the mixture of bodies; namely, how much aqueous, oleaginous or spirituous, ashy or salt parts they contain; or, as a particular example, how much butter, cheese, and whey there is in milk, and the like. These things are rendered sensible by artificial and skilful separations in tangible substances; and the nature of the spirit in them, though not immediately perceptible, is nevertheless discovered by the various motions and efforts of bodies. And, indeed, in this branch men have labored hard in distillations and artificial separations, but with little more success than in their other experiments now in use; their methods being mere guesses and blind attempts, and more industrious than intelligent; and what is worst of all, without any imitation or rivalry of nature, but rather by violent heats and too energetic agents, to the destruction of any delicate conformation, in which principally consist the hidden virtues and sympathies. Nor do men in these separations ever attend to or observe what we have before pointed out; namely, that in attacking bodies by fire, or other methods, many qualities are superinduced by the fire itself, and the other bodies used to effect the separation, which were not originally in the compound. Hence arise most extraordinary fallacies; for the mass of vapor which is emitted from water by fire, for instance, did not exist as vapor or air in the water, but is chiefly created by the expansion of the water by the heat of the fire.
So, in general, all delicate experiments on natural or artificial bodies, by which the genuine are distinguished from the adulterated, and the better from the more common, should be referred to this division; for they bring that which is not the object of the senses within their sphere. They are therefore to be everywhere diligently sought after.
With regard to the fifth cause of objects escaping our senses, it is clear that the action of the sense takes place by motion, and this motion is time. If, therefore, the motion of any body be either so slow or so swift as not to be proportioned to the necessary momentum which operates on the senses, the object is not perceived at all; as in the motion of the hour hand, and that, again, of a musket-ball. The motion which is imperceptible by the senses from its slowness, is readily and usually rendered sensible by the accumulation of motion; that which is imperceptible from its velocity, has not as yet been well measured; it is necessary, however, that this should be done in some cases, with a view to a proper investigation of nature.
The sixth case, where the sense is impeded by the power of the object, admits of a reduction to the sensible sphere, either by removing the object to a greater distance, or by deadening its effects by the interposition of a medium, which may weaken and not destroy the object; or by the admission of its reflection where the direct impression is too strong, as that of the sun in a basin of water.
The seventh case, where the senses are so overcharged with the object as to leave no further room, scarcely occurs except in the smell or taste, and is not of much consequence as regards our present subject. Let what we have said, therefore, suffice with regard to the reduction to the sensible sphere of objects not naturally within its compass.
Sometimes, however, this reduction is not extended to the senses of man, but to those of some other animal, whose senses, in some points, exceed those of man; as (with regard to some scents) to that of the dog, and with regard to light existing imperceptibly in the air, when not illuminated from any extraneous source, to the sense of the cat, the owl, and other animals which see by night. For Telesius has well observed, that there appears to be an original portion of light even in the air itself, although but slight and meagre, and of no use for the most part to the eyes of men, and those of the generality of animals; because those animals to whose senses this light is proportioned can see by night, which does not, in all probability, proceed from their seeing either without light or by any internal light.
Here, too, we would observe, that we at present discuss only the wants of the senses, and their remedies; for their deceptions must be referred to the inquiries appropriated to the senses, and sensible objects; except that important deception, which makes them define objects in their relation to man, and not in their relation to the universe, and which is only corrected by universal reasoning and philosophy.
XLI. In the eighteenth rank of prerogative instances we will class the instances of the road, which we are also wont to call itinerant and jointed instances. They are such as indicate the gradually continued motions of nature. This species of instances escapes rather our observation than our senses; for men are wonderfully indolent upon this subject, consulting nature in a desultory manner, and at periodic intervals, when bodies have been regularly finished and completed, and not during her work. But if any one were desirous of examining and contemplating the talents and industry of an artificer, he would not merely wish to see the rude materials of his art, and then his work when finished, but rather to be present while he is at labor, and proceeding with his work. Something of the same kind should be done with regard to nature. For instance, if any one investigate the vegetation of plants, he should observe from the first sowing of any seed (which can easily be done, by pulling up every day seeds which have been two, three, or four days in the ground, and examining them diligently), how and when the seed begins to swell and break, and be filled, as it were, with spirit; then how it begins to burst the bark and push out fibres, raising itself a little at the same time, unless the ground be very stiff; then how it pushes out these fibres, some downward for roots, others upward for the stem, sometimes also creeping laterally, if it find the earth open and more yielding on one side, and the like. The same should be done in observing the hatching of eggs, where we may easily see the process of animation and organization, and what parts are formed of the yolk, and what of the white of the egg, and the like. The same may be said of the inquiry into the formation of animals from putrefaction; for it would not be so humane to inquire into perfect and terrestrial animals, by cutting the fœtus from the womb; but opportunities may perhaps be offered of abortions, animals killed in hunting, and the like. Nature, therefore, must, as it were, be watched, as being more easily observed by night than by day: for contemplations of this kind may be considered as carried on by night, from the minuteness and perpetual burning of our watch-light.
The same must be attempted with inanimate objects, which we have ourselves done by inquiring into the opening of liquids by fire. For the mode in which water expands is different from that observed in wine, vinegar, or verjuice, and very different, again, from that observed in milk and oil, and the like; and this was easily seen by boiling them with slow heat, in a glass vessel, through which the whole may be clearly perceived. But we merely mention this, intending to treat of it more at large and more closely when we come to the discovery of the latent process; for it should always be remembered that we do not here treat of things themselves, but merely propose examples.
XLII. In the nineteenth rank of prerogative instances we will class supplementary or substitutive instances, which we are also wont to call instances of refuge. They are such as supply information, where the senses are entirely deficient, and we therefore have recourse to them when appropriate instances cannot be obtained. This substitution is twofold, either by approximation or by analogy. For instance, there is no known medium which entirely prevents the effect of the magnet in attracting iron—neither gold, nor silver, nor stone, nor glass, wood, water, oil, cloth, or fibrous bodies, air, flame, or the like. Yet by accurate experiment, a medium may perhaps be found which would deaden its effect, more than another comparatively and in degree; as, for instance, the magnet would not perhaps attract iron through the same thickness of gold as of air, or the same quantity of ignited as of cold silver, and so on; for we have not ourselves made the experiment, but it will suffice as an example. Again, there is no known body which is not susceptible of heat, when brought near the fire; yet air becomes warm much sooner than stone. These are examples of substitution by approximation.
Substitution by analogy is useful, but less sure, and therefore to be adopted with some judgment. It serves to reduce that which is not the object of the senses to their sphere, not by the perceptible operations of the imperceptible body, but by the consideration of some similar perceptible body. For instance, let the subject for inquiry be the mixture of spirits, which are invisible bodies. There appears to be some relation between bodies and their sources or support. Now, the source of flame seems to be oil and fat; that of air, water, and watery substances; for flame increases over the exhalation of oil, and air over that of water. One must therefore consider the mixture of oil and water, which is manifest to the senses, since that of air and flame in general escapes the senses. But oil and water mix very imperfectly by composition or stirring, while they are exactly and nicely mixed in herbs, blood, and the parts of animals. Something similar, therefore, may take place in the mixture of flame and air in spirituous substances, not bearing mixture very well by simple collision, while they appear, however, to be well mixed in the spirits of plants and animals.
Again, if the inquiry do not relate to perfect mixtures of spirits, but merely to their composition, as whether they easily incorporate with each other, or there be rather (as an example) certain winds and exhalations, or other spiritual bodies, which do not mix with common air, but only adhere to and float in it in globules and drops, and are rather broken and pounded by the air, than received into, and incorporated with it; this cannot be perceived in common air, and other aëriform substances, on account of the rarity of the bodies, but an image, as it were, of this process may be conceived in such liquids as quicksilver, oil, water, and even air, when broken and dissipated it ascends in small portions through water, and also in the thicker kinds of smoke; lastly, in dust, raised and remaining in the air, in all of which there is no incorporation: and the above representation in this respect is not a bad one, if it be first diligently investigated, whether there can be such a difference of nature between spirituous substances, as between liquids, for then these images might conveniently be substituted by analogy.
And although we have observed of these supplementary instances, that information is to be derived from them, when appropriate instances are wanting, by way of refuge, yet we would have it understood, that they are also of great use, when the appropriate instances are at hand, in order to confirm the information afforded by them; of which we will speak more at length, when our subject leads us, in due course, to the support of induction.
XLIII. In the twentieth rank of prerogative instances we will place lancing instances, which we are also wont (but for a different reason) to call twitching instances. We adopt the latter name, because they twitch the understanding, and the former because they pierce nature, whence we style them occasionally the instances of Democritus. They are such as warn the understanding of the admirable and exquisite subtilty of nature, so that it becomes roused and awakened to attention, observation, and proper inquiry; as, for instance, that a little drop of ink should be drawn out into so many letters; that silver merely gilt on its surface should be stretched to such a length of gilt wire; that a little worm, such as you may find on the skin, should possess both a spirit and a varied conformation of its parts; that a little saffron should imbue a whole tub of water with its color; that a little musk or aroma should imbue a much greater extent of air with its perfume; that a cloud of smoke should be raised by a little incense; that such accurate differences of sounds as articulate words should be conveyed in all directions through the air, and even penetrate the pores of wood and water (though they become much weakened), that they should be, moreover, reflected, and that with such distinctness and velocity; that light and color should for such an extent and so rapidly pass through solid bodies, such as glass and water, with so great and so exquisite a variety of images, and should be refracted and reflected; that the magnet should attract through every description of body, even the most compact; but (what is still more wonderful) that in all these cases the action of one should not impede that of another in a common medium, such as air; and that there should be borne through the air, at the same time, so many images of visible objects, so many impulses of articulation, so many different perfumes, as of the violet, rose, etc., besides cold and heat, and magnetic attractions; all of them, I say, at once, without any impediment from each other, as if each had its paths and peculiar passage set apart for it, without infringing against or meeting each other.
To these lancing instances, however, we are wont, not without some advantage, to add those which we call the limits of such instances. Thus, in the cases we have pointed out, one action does not disturb or impede another of a different nature, yet those of a similar nature subdue and extinguish each other; as the light of the sun does that of the candle, the sound of a cannon that of the voice, a strong perfume a more delicate one, a powerful heat a more gentle one, a plate of iron between the magnet and other iron the effect of the magnet. But the proper place for mentioning these will be also among the supports of induction.
XLIV. We have now spoken of the instances which assist the senses, and which are principally of service as regards information; for information begins from the senses. But our whole labor terminates in practice, and as the former is the beginning, so is the latter the end of our subject. The following instances, therefore, will be those which are chiefly useful in practice. They are comprehended in two classes, and are seven in number. We call them all by the general name of practical instances. Now there are two defects in practice, and as many divisions of important instances. Practice is either deceptive or too laborious. It is generally deceptive (especially after a diligent examination of natures), on account of the power and actions of bodies being ill defined and determined. Now the powers and actions of bodies are defined and determined either by space or by time, or by the quantity at a given period, or by the predominance of energy; and if these four circumstances be not well and diligently considered, the sciences may indeed be beautiful in theory, but are of no effect in practice. We call the four instances referred to this class, mathematical instances and instances of measure.
Practice is laborious either from the multitude of instruments, or the bulk of matter and substances requisite for any given work. Those instances, therefore, are valuable, which either direct practice to that which is of most consequence to mankind, or lessen the number of instruments or of matter to be worked upon. We assign to the three instances relating to this class, the common name of propitious or benevolent instances. We will now separately discuss these seven instances, and conclude with them that part of our work which relates to the prerogative or illustrious instances.
XLV. In the twenty-first rank of prerogative instances we will place the instances of the rod or rule, which we are also wont to call the instances of completion or non ultrà. For the powers and motions of bodies do not act and take effect through indefinite and accidental, but through limited and certain spaces; and it is of great importance to practice that these should be understood and noted in every nature which is investigated, not only to prevent deception, but to render practice more extensive and efficient. For it is sometimes possible to extend these powers, and bring the distance, as it were, nearer, as in the example of telescopes.
Many powers act and take effect only by actual touch, as in the percussion of bodies, where the one does not remove the other, unless the impelling touch the impelled body. External applications in medicine, as ointment and plasters, do not exercise their efficacy except when in contact with the body. Lastly, the objects of touch and taste only strike those senses when in contact with their organs.
Other powers act at a distance, though it be very small, of which but few have as yet been noted, although there be more than men suspect; this happens (to take everyday instances) when amber or jet attracts straws, bubbles dissolve bubbles, some purgative medicines draw humors from above, and the like. The magnetic power by which iron and the magnet, or two magnets, are attracted together, acts within a definite and narrow sphere, but if there be any magnetic power emanating from the earth a little below its surface, and affecting the needle in its polarity, it must act at a great distance.
Again, if there be any magnetic force which acts by sympathy between the globe of the earth and heavy bodies, or between that of the moon and the waters of the sea (as seems most probable from the particular floods and ebbs which occur twice in the month), or between the starry sphere and the planets, by which they are summoned and raised to their apogees, these must all operate at very great distances.
Again, some conflagrations and the kindling of flames take place at very considerable distances with particular substances, as they report of the naphtha of Babylon. Heat, too, insinuates itself at wide distances, as does also cold, so that the masses of ice which are broken off and float upon the Northern Ocean, and are borne through the Atlantic to the coast of Canada, become perceptible by the inhabitants, and strike them with cold from a distance. Perfumes also (though here there appears to be always some corporeal emission) act at remarkable distances, as is experienced by persons sailing by the coast of Florida, or parts of Spain, where there are whole woods of lemons, oranges, and other odoriferous plants, or rosemary and marjoram bushes, and the like. Lastly, the rays of light and the impressions of sound act at extensive distances.
Yet all these powers, whether acting at a small or great distance, certainly act within definite distances, which are well ascertained by nature, so that there is a limit depending either on the mass or quantity of the bodies, the vigor or faintness of the powers, or the favorable or impeding nature of the medium, all of which should be taken into account and observed. We must also note the boundaries of violent motions, such as missiles, projectiles, wheels and the like, since they are also manifestly confined to certain limits.
Some motions and virtues are to be found of a directly contrary nature to these, which act in contact but not at a distance; namely, such as operate at a distance and not in contact, and again act with less force at a less distance, and the reverse. Sight, for instance, is not easily effective in contact, but requires a medium and distance; although I remember having heard from a person deserving of credit, that in being cured of a cataract (which was done by putting a small silver needle within the first coat of the eye, to remove the thin pellicle of the cataract, and force it into a corner of the eye), he had distinctly seen the needle moving across the pupil. Still, though this may be true, it is clear that large bodies cannot be seen well or distinctly, unless at the vertex of a cone, where the rays from the object meet at some distance from the eye. In old persons the eye sees better if the object be moved a little further, and not nearer. Again, it is certain that in projectiles the impact is not so violent at too short a distance as a little afterward. Such are the observations to be made on the measure of motions as regards distance.
There is another measure of motion in space which must not be passed over, not relating to progressive but spherical motion—that is, the expansion of bodies into a greater, or their contraction into a lesser sphere. For in our measure of this motion we must inquire what degree of compression or extension bodies easily and readily admit of, according to their nature, and at what point they begin to resist it, so as at last to bear it no further—as when an inflated bladder is compressed, it allows a certain compression of the air, but if this be increased, the air does not suffer it, and the bladder is burst.
We have proved this by a more delicate experiment. We took a metal bell, of a light and thin sort, such as is used for salt-cellars, and immersed it in a basin of water, so as to carry the air contained in its interior down with it to the bottom of the basin. We had first, however, placed a small globe at the bottom of the basin, over which we placed the bell. The result was, that if the globe were small compared with the interior of the bell, the air would contract itself, and be compressed without being forced out, but if it were too large for the air readily to yield to it, the latter became impatient of the pressure, raised the bell partly up, and ascended in bubbles.
To prove, also, the extension (as well as the compression) which air admits of, we adopted the following method:—We took a glass egg, with a small hole at one end; we drew out the air by violent suction at this hole, and then closed the hole with the finger, immersed the egg in water, and then removed the finger. The air being constrained by the effort made in suction, and dilated beyond its natural state, and therefore striving to recover and contract itself (so that if the egg had not been immersed in water, it would have drawn in the air with a hissing sound), now drew in a sufficient quantity of water to allow the air to recover its former dimensions.
It is well ascertained that rare bodies (such as air) admit of considerable contraction, as has been before observed; but tangible bodies (such as water) admit of it much less readily, and to a less extent. We investigated the latter point by the following experiment:
We had a leaden globe made, capable of containing about two pints, wine measure, and of tolerable thickness, so as to support considerable pressure. We poured water into it through an aperture, which we afterward closed with melted lead, as soon as the globe was filled with water, so that the whole became perfectly solid. We next flattened the two opposite sides with a heavy hammer, which necessarily caused the water to occupy a less space, since the sphere is the solid of greatest content; and when hammering failed from the resistance of the water, we made use of a mill or press, till at last the water, refusing to submit to a greater pressure, exuded like a fine dew through the solid lead. We then computed the extent to which the original space had been reduced, and concluded that water admitted such a degree of compression when constrained by great violence.
The more solid, dry or compact bodies, such as stones, wood and metals, admit of much less, and indeed scarcely any perceptible compression or expansion, but escape by breaking, slipping forward, or other efforts; as appears in bending wood, or steel for watch-springs, in projectiles, hammering and many other motions, all of which, together with their degrees, are to be observed and examined in the investigation of nature, either to a certainty, or by estimation, or comparison, as opportunity permits.
XLVI. In the twenty-second rank of prerogative instances we will place the instances of the course, which we are also wont to call water instances, borrowing our expression from the water hour-glasses employed by the ancients instead of those with sand. They are such as measure nature by the moments of time, as the last instances do by the degrees of space. For all motion or natural action takes place in time, more or less rapidly, but still in determined moments well ascertained by nature. Even those actions which appear to take effect suddenly, and in the twinkling of an eye (as we express it), are found to admit of greater or less rapidity.
In the first place, then, we see that the return of the heavenly bodies to the same place takes place in regular times, as does the flood and ebb of the sea. The descent of heavy bodies toward the earth, and the ascent of light bodies toward the heavenly sphere, take place in definite times, according to the nature of the body, and of the medium through which it moves. The sailing of ships, the motions of animals, the transmission of projectiles, all take place in times the sums of which can be computed. With regard to heat, we see that boys in winter bathe their hands in the flame without being burned; and conjurers, by quick and regular movements, overturn vessels filled with wine or water, and replace them without spilling the liquid, with several similar instances. The compression, expansion and eruption of several bodies, take place more or less rapidly, according to the nature of the body and its motion, but still in definite moments.
In the explosion of several cannon at once (which are sometimes heard at the distance of thirty miles), the sound of those nearest to the spot is heard before that of the most distant. Even in sight (whose action is most rapid), it is clear that a definite time is necessary for its exertion, which is proved by certain objects being invisible from the velocity of their motion, such as a musket-ball; for the flight of the ball is too swift to allow an impression of its figure to be conveyed to the sight.
This last instance, and others of a like nature, have sometimes excited in us a most marvellous doubt, no less than whether the image of the sky and stars is perceived as at the actual moment of its existence, or rather a little after, and whether there is not (with regard to the visible appearance of the heavenly bodies) a true and apparent time, as well as a true and apparent place, which is observed by astronomers in parallaxes. It appeared so incredible to us, that the images or radiations of heavenly bodies could suddenly be conveyed through such immense spaces to the sight, and it seemed that they ought rather to be transmitted in a definite time. That doubt, however (as far as regards any great difference between the true and apparent time), was subsequently completely set at rest, when we considered the infinite loss and diminution of size as regards the real and apparent magnitude of a star, occasioned by its distance, and at the same time observed at how great a distance (at least sixty miles) bodies which are merely white can be suddenly seen by us. For there is no doubt, that the light of the heavenly bodies not only far surpasses the vivid appearance of white, but even the light of any flame (with which we are acquainted) in the vigor of its radiation. The immense velocity of the bodies themselves, which is perceived in their diurnal motion, and has so astonished thinking men, that they have been more ready to believe in the motion of the earth, renders the motion of radiation from them (marvellous as it is in its rapidity) more worthy of belief. That which has weighed most with us, however, is, that if there were any considerable interval of time between the reality and the appearance, the images would often be interrupted and confused by clouds formed in the meantime, and similar disturbances of the medium. Let this suffice with regard to the simple measures of time.
It is not merely the absolute, but still more the relative measure of motions and actions which must be inquired into, for this latter is of great use and application. We perceive that the flame of firearms is seen sooner than the sound is heard, although the ball must have struck the air before the flame, which was behind it, could escape: the reason of which is, that light moves with greater velocity than sound. We perceive, also, that visible images are received by the sight with greater rapidity than they are dismissed, and for this reason, a violin string touched with the finger appears double or triple, because the new image is received before the former one is dismissed. Hence, also, rings when spinning appear globular, and a lighted torch, borne rapidly along at night, appears to have a tail. Upon the principle of the inequality of motion, also, Galileo attempted an explanation of the flood and ebb of the sea, supposing the earth to move rapidly, and the water slowly, by which means the water, after accumulating, would at intervals fall back, as is shown in a vessel of water made to move rapidly. He has, however, imagined this on data which cannot be granted (namely, the earth’s motion), and besides, does not satisfactorily account for the tide taking place every six hours.
An example of our present point (the relative measure of motion), and, at the same time, of its remarkable use of which we have spoken, is conspicuous in mines filled with gunpowder, where immense weights of earth, buildings, and the like, are overthrown and prostrated by a small quantity of powder; the reason of which is decidedly this, that the motion of the expansion of the gunpowder is much more rapid than that of gravity, which would resist it, so that the former has terminated before the latter has commenced. Hence, also, in missiles, a strong blow will not carry them so far as a sharp and rapid one. Nor could a small portion of animal spirit in animals, especially in such vast bodies as those of the whale and elephant, have ever bent or directed such a mass of body, were it not owing to the velocity of the former, and the slowness of the latter in resisting its motion.
In short, this point is one of the principal foundations of the magic experiments (of which we shall presently speak), where a small mass of matter overcomes and regulates a much larger, if there but be an anticipation of motion, by the velocity of one before the other is prepared to act.
Finally, the point of the first and last should be observed in all natural actions. Thus, in an infusion of rhubarb the purgative property is first extracted, and then the astringent; we have experienced something of the same kind in steeping violets in vinegar, which first extracts the sweet and delicate odor of the flower, and then the more earthy part, which disturbs the perfume; so that if the violets be steeped a whole day, a much fainter perfume is extracted than if they were steeped for a quarter of an hour only, and then taken out; and since the odoriferous spirit in the violet is not abundant, let other and fresh violets be steeped in the vinegar every quarter of an hour, as many as six times, when the infusion becomes so strengthened, that although the violets have not altogether remained there for more than one hour and a half, there remains a most pleasing perfume, not inferior to the flower itself, for a whole year. It must be observed, however, that the perfume does not acquire its full strength till about a month after the infusion. In the distillation of aromatic plants macerated in spirits of wine, it is well known that an aqueous and useless phlegm rises first, then water containing more of the spirit, and, lastly, water containing more of the aroma; and many observations of the like kind, well worthy of notice, are to be made in distillations. But let these suffice as examples.
XLVII. In the twenty-third rank of prerogative instances we will place instances of quantity, which we are also wont to call the doses of nature (borrowing a word from medicine). They are such as measure the powers by the quantity of bodies, and point out the effect of the quantity in the degree of power. And in the first place, some powers only subsist in the universal quantity, or such as bears a relation to the confirmation and fabric of the universe. Thus the earth is fixed, its parts fall. The waters in the sea flow and ebb, but not in the rivers, except by the admission of the sea. Then, again, almost all particular powers act according to the greater or less quantity of the body. Large masses of water are not easily rendered foul, small are. New wine and beer become ripe and drinkable in small skins much more readily than in large casks. If a herb be placed in a considerable quantity of liquid, infusion takes place rather than impregnation; if in less, the reverse. A bath, therefore, and a light sprinkling, produce different effects on the human body. Light dew, again, never falls, but is dissipated and incorporated with the air; thus we see that in breathing on gems, the slight quantity of moisture, like a small cloud in the air, is immediately dissolved. Again, a piece of the same magnet does not attract so much iron as the whole magnet did. There are some powers where the smallness of the quantity is of more avail; as in boring, a sharp point pierces more readily than a blunt one; the diamond, when pointed, makes an impression on glass, and the like.
Here, too, we must not rest contented with a vague result, but inquire into the exact proportion of quantity requisite for a particular exertion of power; for one would be apt to suppose that the power bears an exact proportion to the quantity; that if a leaden bullet of one ounce, for instance, would fall in a given time, one of two ounces ought to fall twice as rapidly, which is most erroneous. Nor does the same ratio prevail in every kind of power, their difference being considerable. The measure, therefore, must be determined by experiment, and not by probability or conjecture.
Lastly, we must in all our investigations of nature observe what quantity, or dose, of the body is requisite for a given effect, and must at the same time be guarded against estimating it at too much or too little.
XLVIII. In the twenty-fourth rank of prerogative instances we will place wrestling instances, which we are also wont to call instances of predominance. They are such as point out the predominance and submission of powers compared with each other, and which of them is the more energetic and superior, or more weak and inferior. For the motions and effects of bodies are compounded, decomposed, and combined, no less than the bodies themselves. We will exhibit, therefore, the principal kinds of motions or active powers, in order that their comparative strength, and thence a demonstration and definition of the instances in question, may be rendered more clear.
Let the first motion be that of the resistance of matter, which exists in every particle, and completely prevents its annihilation; so that no conflagration, weight, pressure, violence, or length of time can reduce even the smallest portion of matter to nothing, or prevent it from being something, and occupying some space, and delivering itself (whatever straits it be put to), by changing its form or place, or, if that be impossible, remaining as it is; nor can it ever happen that it should either be nothing or nowhere. This motion is designated by the schools (which generally name and define everything by its effects and inconveniences rather than by its inherent cause) by the axiom, that two bodies cannot exist in the same place, or they call it a motion to prevent the penetration of dimensions. It is useless to give examples of this motion, since it exists in every body.
Let the second motion be that which we term the motion of connection, by which bodies do not allow themselves to be separated at any point from the contact of another body, delighting, as it were, in the mutual connection and contact. This is called by the schools a motion to prevent a vacuum. It takes place when water is drawn up by suction or a syringe, the flesh by cupping, or when the water remains without escaping from perforated jars, unless the mouth be opened to admit the air, and innumerable instances of a like nature.
Let the third be that which we term the motion of liberty, by which bodies strive to deliver themselves from any unnatural pressure or tension, and to restore themselves to the dimensions suited to their mass; and of which, also, there are innumerable examples. Thus, we have examples of their escaping from pressure, in the water in swimming, in the air in flying, in the water again in rowing, and in the air in the undulation of the winds, and in springs of watches. An exact instance of the motion of compressed air is seen in children’s popguns, which they make by scooping out elder-branches or some such matter, and forcing in a piece of some pulpy root or the like, at each end; then they force the root or other pellet with a ramrod to the opposite end, from which the lower pellet is emitted and projected with a report, and that before it is touched by the other piece of root or pellet, or by the ramrod. We have examples of their escape from tension, in the motion of the air that remains in glass eggs after suction, in strings, leather, and cloth, which recoil after tension, unless it be long continued. The schools define this by the term of motion from the form of the element; injudiciously enough, since this motion is to be found not only in air, water, or fire, but in every species of solid, as wood, iron, lead, cloth, parchment, etc., each of which has its own proper size, and is with difficulty stretched to any other. Since, however, this motion of liberty is the most obvious of all, and to be seen in an infinite number of cases, it will be as well to distinguish it correctly and clearly; for some most carelessly confound this with the two others of resistance and connection; namely, the freedom from pressure with the former, and that from tension with the latter, as if bodies when compressed yielded or expanded to prevent a penetration of dimensions, and when stretched rebounded and contracted themselves to prevent a vacuum. But if the air, when compressed, could be brought to the density of water, or wood to that of stone, there would be no need of any penetration of dimensions, and yet the compression would be much greater than they actually admit of. So if water could be expanded till it became as rare as air, or stone as rare as wood, there would be no need of a vacuum, and yet the expansion would be much greater than they actually admit of.
We do not, therefore, arrive at a penetration of dimensions or a vacuum before the extremes of condensation and rarefaction, while the motion we speak of stops and exerts itself much within them, and is nothing more than a desire of bodies to preserve their specific density (or, if it be preferred, their form), and not to desert them suddenly, but only to change by degrees, and of their own accord. It is, however, much more necessary to intimate to mankind (because many other points depend upon this), that the violent motion which we call mechanical, and Democritus (who, in explaining his primary motions, is to be ranked even below the middling class of philosophers) termed the motion of a blow, is nothing else than this motion of liberty, namely, a tendency to relaxation from compression. For in all simple impulsion or flight through the air, the body is not displaced or moved in space, until its parts are placed in an unnatural state, and compressed by the impelling force. When that takes place, the different parts urging the other in succession, the whole is moved, and that with a rotatory as well as progressive motion, in order that the parts may, by this means also, set themselves at liberty, or more readily submit. Let this suffice for the motion in question.
Let the fourth be that which we term the motion of matter, and which is opposed to the last; for in the motion of liberty, bodies abhor, reject, and avoid, a new size or volume, or any new expansion or contraction (for these different terms have the same meaning), and strive, with all their power, to rebound and resume their former density; on the contrary, in the motion of matter, they are anxious to acquire a new volume or dimension, and attempt it willingly and rapidly, and occasionally by a most vigorous effort, as in the example of gunpowder. The most powerful, or at least most frequent, though not the only instruments of this motion, are heat and cold. For instance, the air, if expanded by tension (as by suction in the glass egg), struggles anxiously to restore itself; but if heat be applied, it strives, on the contrary, to dilate itself, and longs for a larger volume, regularly passing and migrating into it, as into a new form (as it is termed); nor after a certain degree of expansion is it anxious to return, unless it be invited to do so by the application of cold, which is not indeed a return, but a fresh change. So also water, when confined by compression, resists, and wishes to become as it was before, namely, more expanded; but if there happen an intense and continued cold, it changes itself readily, and of its own accord, into the condensed state of ice; and if the cold be long continued, without any intervening warmth (as in grottoes and deep caves), it is changed into crystal or similar matter, and never resumes its form.
Let the fifth be that which we term the motion of continuity. We do not understand by this simple and primary continuity with any other body (for that is the motion of connection), but the continuity of a particular body in itself; for it is most certain that all bodies abhor a solution of continuity, some more and some less, but all partially. In hard bodies (such as steel and glass) the resistance to an interruption of continuity is most powerful and efficacious, while although in liquids it appears to be faint and languid, yet it is not altogether null, but exists in the lowest degree, and shows itself in many experiments, such as bubbles, the round form of drops, the thin threads which drip from roofs, the cohesion of glutinous substances, and the like. It is most conspicuous, however, if an attempt be made to push this separation to still smaller particles. Thus, in mortars, the pestle produces no effect after a certain degree of contusion, water does not penetrate small fissures, and the air itself, notwithstanding its subtilty, does not penetrate the pores of solid vessels at once, but only by long-continued insinuation.
Let the sixth be that which we term the motion of acquisition, or the motion of need. It is that by which bodies placed among others of a heterogeneous and, as it were, hostile nature, if they meet with the means or opportunity of avoiding them, and uniting themselves with others of a more analogous nature, even when these latter are not closely allied to them, immediately seize and, as it were, select them, and appear to consider it as something acquired (whence we derive the name), and to have need of these latter bodies. For instance, gold, or any other metal in leaf, does not like the neighborhood of air; if, therefore, they meet with any tangible and thick substance (such as the finger, paper, or the like), they immediately adhere to it, and are not easily torn from it. Paper, too, and cloth, and the like, do not agree with the air, which is inherent and mixed in their pores. They readily, therefore, imbibe water or other liquids, and get rid of the air. Sugar, or a sponge, dipped in water or wine, and though part of it be out of the water or wine, and at some height above it, will yet gradually absorb them.
Hence an excellent rule is derived for the opening and dissolution of bodies; for (not to mention corrosive and strong waters, which force their way) if a body can be found which is more adapted, suited, and friendly to a given solid, than that with which it is by some necessity united, the given solid immediately opens and dissolves itself to receive the former, and excludes or removes the latter. Nor is the effect or power of this motion confined to contact, for the electric energy (of which Gilbert and others after him have told so many fables) is only the energy excited in a body by gentle friction, and which does not endure the air, but prefers some tangible substance if there be any at hand.
Let the seventh be that which we term the motion of greater congregation, by which bodies are borne toward masses of a similar nature, for instance, heavy bodies toward the earth, light to the sphere of heaven. The schools termed this natural motion, by a superficial consideration of it, because produced by no external visible agent, which made them consider it innate in the substances; or perhaps because it does not cease, which is little to be wondered at, since heaven and earth are always present, while the causes and sources of many other motions are sometimes absent and sometimes present. They therefore called this perpetual and proper, because it is never interrupted, but instantly takes place when the others are interrupted, and they called the others adscititious. The former, however, is in reality weak and slow, since it yields, and is inferior to the others as long as they act, unless the mass of the body be great; and although this motion have so filled men’s minds, as almost to have obscured all others, yet they know but little about it, and commit many errors in its estimate.
Let the eighth be that which we term the motion of lesser congregation, by which the homogeneous parts in any body separate themselves from the heterogeneous and unite together, and whole bodies of a similar substance coalesce and tend toward each other, and are sometimes congregated, attracted, and meet, from some distance; thus in milk the cream rises after a certain time, and in wine the dregs and tartar sink; which effects are not to be attributed to gravity and levity only, so as to account for the rising of some parts and the sinking of others, but much more to the desire of the homogeneous bodies to meet and unite. This motion differs from that of need in two points: 1st, because the latter is the stimulus of a malignant and contrary nature, while in this of which we treat (if there be no impediment or restraint), the parts are united by their affinity, although there be no foreign nature to create a struggle; 2dly, because the union is closer and more select. For in the other motion, bodies which have no great affinity unite, if they can but avoid the hostile body, while in this, substances which are connected by a decided kindred resemblance come together and are molded into one. It is a motion existing in all compound bodies, and would be readily seen in each, if it were not confined and checked by the other affections and necessities of bodies which disturb the union.
This motion is usually confined in the three following manners: by the torpor of the bodies; by the power of the predominating body; by external motion. With regard to the first, it is certain that there is more or less sluggishness in tangible bodies, and an abhorrence of locomotion; so that unless excited they prefer remaining contented with their actual state, to placing themselves in a better position. There are three means of breaking through this sluggishness—heat; the active power of a similar body; vivid and powerful motion. With regard to the first, heat is, on this account, defined as that which separates heterogeneous, and draws together homogeneous substances; a definition of the Peripatetics which is justly ridiculed by Gilbert, who says it is as if one were to define man to be that which sows wheat and plants vineyards; being only a definition deduced from effects, and those but partial. But it is still more to be blamed, because those effects, such as they are, are not a peculiar property of heat, but a mere accident (for cold, as we shall afterward show, does the same), arising from the desire of the homogeneous parts to unite; the heat then assists them in breaking through that sluggishness which before restrained their desire. With regard to the assistance derived from the power of a similar body, it is most conspicuous in the magnet when armed with steel, for it excites in the steel a power of adhering to steel, as a homogeneous substance, the power of the magnet breaking through the sluggishness of the steel. With regard to the assistance of motion, it is seen in wooden arrows or points, which penetrate more deeply into wood than if they were tipped with iron, from the similarity of the substance, the swiftness of the motion breaking through the sluggishness of the wood; of which two last experiments we have spoken above in the aphorism on clandestine instances.
The confinement of the motion of lesser congregation, which arises from the power of the predominant body, is shown in the decomposition of blood and urine by cold. For as long as these substances are filled with the active spirit, which regulates and restrains each of their component parts, as the predominant ruler of the whole, the several different parts do not collect themselves separately on account of the check; but as soon as that spirit has evaporated, or has been choked by the cold, then the decomposed parts unite, according to their natural desire. Hence it happens, that all bodies which contain a sharp spirit (as salts and the like), last without decomposition, owing to the permanent and durable power of the predominating and imperious spirit.
The confinement of the motion of lesser congregation, which arises from external motion, is very evident in that agitation of bodies which preserves them from putrefaction. For all putrefaction depends on the congregation of the homogeneous parts, whence, by degrees, there ensues a corruption of the first form (as it is called), and the generation of another. For the decomposition of the original form, which is itself the union of the homogeneous parts, precedes the putrefaction, which prepares the way for the generation of another. This decomposition, if not interrupted, is simple; but if there be various obstacles, putrefactions ensue, which are the rudiments of a new generation. But if (to come to our present point) a frequent agitation be excited by external motion, the motion toward union (which is delicate and gentle, and requires to be free from all external influence) is disturbed, and ceases; which we perceive to be the case in innumerable instances. Thus, the daily agitation or flowing of water prevents putrefaction; winds prevent the air from being pestilent; corn turned about and shaken in granaries continues clean: in short, everything which is externally agitated will with difficulty rot internally.
We must not omit that union of the parts of bodies which is the principal cause of induration and desiccation. When the spirit or moisture, which has evaporated into spirit, has escaped from a porous body (such as wood, bone, parchment, and the like), the thicker parts are drawn together, and united with a greater effort, and induration or desiccation is the consequence; and this we attribute not so much to the motion of connection (in order to prevent a vacuum), as to this motion of friendship and union.
Union from a distance is rare, and yet is to be met with in more instances than are generally observed. We perceive it when one bubble dissolves another, when medicines attract humors from a similarity of substance, when one string moves another in unison with it on different instruments, and the like. We are of opinion that this motion is very prevalent also in animal spirits, but are quite ignorant of the fact. It is, however, conspicuous in the magnet, and magnetized iron. While speaking of the motions of the magnet, we must plainly distinguish them, for there are four distinct powers or effects of the magnet which should not be confounded, although the wonder and astonishment of mankind has classed them together. 1.The attraction of the magnet to the magnet, or of iron to the magnet, or of magnetized iron to iron. 2.Its polarity toward the north and south, and its variation. 3.Its penetration through gold, glass, stone, and all other substances. 4.The communication of power from the mineral to iron, and from iron to iron, without any communication of the substances. Here, however, we only speak of the first. There is also a singular motion of attraction between quicksilver and gold, so that the gold attracts quicksilver even when made use of in ointment; and those who work surrounded by the vapors of quicksilver, are wont to hold a piece of gold in their mouths, to collect the exhalations, which would otherwise attack their heads and bones, and this piece soon grows white. Let this suffice for the motion of lesser congregation.
Let the ninth be the magnetic motion, which, although of the nature of that last mentioned, yet, when operating at great distances, and on great masses, deserves a separate inquiry, especially if it neither begin in contact, as most motions of congregation do, nor end by bringing the substances into contact, as all do, but only raise them, and make them swell without any further effect. For if the moon raise the waters, or cause moist substances to swell, or if the starry sphere attract the planets toward their apogees, or the sun confine the planets Mercury and Venus to within a certain distance of his mass; these motions do not appear capable of being classed under either of those of congregation, but to be, as it were, intermediately and imperfectly congregative, and thus to form a distinct species.
Let the tenth motion be that of avoidance, or that which is opposed to the motion of lesser congregation, by which bodies, with a kind of antipathy, avoid and disperse, and separate themselves from, or refuse to unite themselves with others of a hostile nature. For although this may sometimes appear to be an accidental motion, necessarily attendant upon that of the lesser congregation, because the homogeneous parts cannot unite, unless the heterogeneous be first removed and excluded, yet it is still to be classed separately, and considered as a distinct species, because, in many cases, the desire of avoidance appears to be more marked than that of union.
It is very conspicuous in the excrements of animals, nor less, perhaps, in objects odious to particular senses, especially the smell and taste; for a fetid smell is rejected by the nose, so as to produce a sympathetic motion of expulsion at the mouth of the stomach; a bitter and rough taste is rejected by the palate or throat, so as to produce a sympathetic concussion and shivering of the head. This motion is visible also in other cases. Thus it is observed in some kinds of antiperistasis, as in the middle region of the air, the cold of which appears to be occasioned by the rejection of cold from the regions of the heavenly bodies; and also in the heat and combustion observed in subterranean spots, which appear to be owing to the rejection of heat from the centre of the earth. For heat and cold, when in small quantities, mutually destroy each other, while in larger quantities, like armies equally matched, they remove and eject each other in open conflict. It is said, also that cinnamon and other perfumes retain their odor longer when placed near privies and foul places, because they will not unite and mix with stinks. It is well known that quicksilver, which would otherwise reunite into a complete mass, is prevented from so doing by man’s spittle, pork lard, turpentine and the like, from the little affinity of its parts with those substances, so that when surrounded by them it draws itself back, and its avoidance of these intervening obstacles is greater than its desire of reuniting itself to its homogeneous parts; which is what they term the mortification of quicksilver. Again, the difference in weight of oil and water is not the only reason for their refusing to mix, but it is also owing to the little affinity of the two; for spirits of wine, which are lighter than oil, mix very well with water. A very remarkable instance of the motion in question is seen in nitre, and crude bodies of a like nature, which abhor flame, as may be observed in gunpowder, quicksilver and gold. The avoidance of one pole of the magnet by iron is not (as Gilbert has well observed), strictly speaking, an avoidance, but a conformity, or attraction to a more convenient situation.
Let the eleventh motion be that of assimilation, or self-multiplication, or simple generation, by which latter term we do not mean the simple generation of integral bodies, such as plants or animals, but of homogeneous bodies. By this motion homogeneous bodies convert those which are allied to them, or at least well disposed and prepared, into their own substance and nature. Thus flame multiplies itself over vapors and oily substances and generates fresh flame; the air over water and watery substances multiplies itself and generates fresh air; the vegetable and animal spirit, over the thin particles of a watery or oleaginous spirit contained in its food, multiplies itself and generates fresh spirit; the solid parts of plants and animals, as the leaf, flower, the flesh, bone and the like, each of them assimilate some part of the juices contained in their food, and generate a successive and daily substance. For let none rave with Paracelsus, who (blinded by his distillations) would have it, that nutrition takes place by mere separation, and that the eye, nose, brain and liver lie concealed in bread and meat, the root, leaf and flower, in the juice of the earth; asserting that just as the artist brings out a leaf, flower, eye, nose, hand, foot and the like, from a rude mass of stone or wood by the separation and rejection of what is superfluous; so the great artist within us brings out our several limbs and parts by separation and rejection. But to leave such trifling, it is most certain that all the parts of vegetables and animals, as well the homogeneous as organic, first of all attract those juices contained in their food, which are nearly common, or at least not very different, and then assimilate and convert them into their own nature. Nor does this assimilation, or simple generation, take place in animated bodies only, but the inanimate also participate in the same property (as we have observed of flame and air), and that languid spirit, which is contained in every tangible animated substance, is perpetually working upon the coarser parts, and converting them into spirit, which afterward is exhaled, whence ensues a diminution of weight, and a desiccation of which we have spoken elsewhere.
Nor should we, in speaking of assimilation, neglect to mention the accretion which is usually distinguished from aliment, and which is observed when mud grows into a mass between stones, and is converted into a stony substance, and the scaly substance round the teeth is converted into one no less hard than the teeth themselves; for we are of opinion that there exists in all bodies a desire of assimilation, as well as of uniting with homogeneous masses. Each of these powers, however, is confined, although in different manners, and should be diligently investigated, because they are connected with the revival of old age. Lastly, it is worthy of observation, that in the nine preceding motions, bodies appear to aim at the mere preservation of their nature, while in this they attempt its propagation.
Let the twelfth motion be that of excitement, which appears to be a species of the last, and is sometimes mentioned by us under that name. It is, like that, a diffusive, communicative, transitive and multiplying motion; and they agree remarkably in their effect, although they differ in their mode of action, and in their subject matter. The former proceeds imperiously and with authority; it orders and compels the assimilated to be converted and changed into the assimilating body. The latter proceeds by art, insinuation and stealth, inviting and disposing the excited toward the nature of the exciting body. The former both multiplies and transforms bodies and substances; thus a greater quantity of flame, air, spirit and flesh is formed; but in the latter, the powers only are multiplied and changed, and heat, the magnetic power, and putrefaction, in the above instances, are increased. Heat does not diffuse itself when heating other bodies by any communication of the original heat, but only by exciting the parts of the heated body to that motion which is the form of heat, and of which we spoke in the first vintage of the nature of heat. Heat, therefore, is excited much less rapidly and readily in stone or metal than in air, on account of the inaptitude and sluggishness of those bodies in acquiring that motion, so that it is probable, that there may be some substances, toward the centre of the earth, quite incapable of being heated, on account of their density, which may deprive them of the spirit by which the motion of excitement is usually commenced. Thus also the magnet creates in the iron a new disposition of its parts, and a conformable motion, without losing any of its virtue. So the leaven of bread, yeast, rennet and some poisons, excite and invite successive and continued motion in dough, beer, cheese or the human body; not so much from the power of the exciting, as the predisposition and yielding of the excited body.
Let the thirteenth motion be that of impression, which is also a species of motion of assimilation, and the most subtile of diffusive motions. We have thought it right, however, to consider it as a distinct species, on account of its remarkable difference from the last two; for the simple motion of assimilation transforms the bodies themselves, so that if you remove the first agent, you diminish not the effect of those which succeed; thus, neither the first lighting of flame, nor the first conversion into air, are of any importance to the flame or air next generated. So, also, the motion of excitement still continues for a considerable time after the removal of the first agent, as in a heated body on the removal of the original heat, in the excited iron on the removal of the magnet, and in the dough on the removal of the leaven. But the motion of impression, although diffusive and transitive, appears, nevertheless, to depend on the first agent, so that upon the removal of the latter the former immediately fails and perishes; for which reason also it takes effect in a moment, or at least a very short space of time. We are wont to call the two former motions the motions of the generation of Jupiter, because when born they continue to exist; and the latter, the motion of the generation of Saturn, because it is immediately devoured and absorbed. It may be seen in three instances: 1, in the rays of light; 2, in the percussions of sounds; 3, in magnetic attractions as regards communication. For, on the removal of light, colors and all its other images disappear, as on the cessation of the first percussion and the vibration of the body, sound soon fails, and although sounds are agitated by the wind, like waves, yet it is to be observed, that the same sound does not last during the whole time of the reverberation. Thus, when a bell is struck, the sound appears to be continued for a considerable time, and one might easily be led into the mistake of supposing it to float and remain in the air during the whole time, which is most erroneous. For the reverberation is not one identical sound, but the repetition of sounds, which is made manifest by stopping and confining the sonorous body; thus, if a bell be stopped and held tightly, so as to be immovable, the sound fails, and there is no further reverberation, and if a musical string be touched after the first vibration, either with the finger (as in the harp), or a quill (as in the harpsichord), the sound immediately ceases. If the magnet be removed the iron falls. The moon, however, cannot be removed from the sea, nor the earth from a heavy falling body, and we can, therefore, make no experiment upon them; but the case is the same.
Let the fourteenth motion be that configuration or position, by which bodies appear to desire a peculiar situation, collocation, and configuration with others, rather than union or separation. This is a very abstruse notion, and has not been well investigated; and, in some instances, appears to occur almost without any cause, although we be mistaken in supposing this to be really the case. For if it be asked, why the heavens revolve from east to west, rather than from west to east, or why they turn on poles situate near the Bears, rather than round Orion or any other part of the heaven, such a question appears to be unreasonable, since these phenomena should be received as determinate and the objects of our experience. There are, indeed, some ultimate and self-existing phenomena in nature, but those which we have just mentioned are not to be referred to that class: for we attribute them to a certain harmony and consent of the universe, which has not yet been properly observed. But if the motion of the earth from west to east be allowed, the same question may be put, for it must also revolve round certain poles, and why should they be placed where they are, rather than elsewhere? The polarity and variation of the needle come under our present head. There is also observed in both natural and artificial bodies, especially solids rather than fluids, a particular collocation and position of parts, resembling hairs or fibres, which should be diligently investigated, since, without a discovery of them, bodies cannot be conveniently controlled or wrought upon. The eddies observable in liquids by which, when compressed, they successively raise different parts of their mass before they can escape, so as to equalize the pressure, is more correctly assigned to the motion of liberty.
Let the fifteenth motion be that of transmission or of passage, by which the powers of bodies are more or less impeded or advanced by the medium, according to the nature of the bodies and their effective powers, and also according to that of the medium. For one medium is adapted to light, another to sound, another to heat and cold, another to magnetic action, and so on with regard to the other actions.
Let the sixteenth be that which we term the royal or political motion, by which the predominant and governing parts of any body check, subdue, reduce, and regulate the others, and force them to unite, separate, stand still, move, or assume a certain position, not from any inclination of their own, but according to a certain order, and as best suits the convenience of the governing part, so that there is a sort of dominion and civil government exercised by the ruling part over its subjects. The motion is very conspicuous in the spirits of animals, where, as long as it is in force, it tempers all the motions of the other parts. It is found in a less degree in other bodies, as we have observed in blood and urine, which are not decomposed until the spirit, which mixed and retained their parts, has been emitted or extinguished. Nor is this motion peculiar to spirits only, although in most bodies the spirit predominates, owing to its rapid motion and penetration; for the grosser parts predominate in denser bodies, which are not filled with a quick and active spirit (such as exists in quicksilver or vitriol), so that unless this check or yoke be thrown off by some contrivance, there is no hope of any transformation of such bodies. And let not any one suppose that we have forgotten our subject, because we speak of predominance in this classification of motions, which is made entirely with the view of assisting the investigation of wrestling instances, or instances of predominance. For we do not now treat of the general predominance of motions or powers, but of that of parts in whole bodies, which constitutes the particular species here considered.
Let the seventeenth motion be the spontaneous motion of revolution, by which bodies having a tendency to move, and placed in a favorable situation, enjoy their peculiar nature, pursuing themselves and nothing else, and seeking, as it were, to embrace themselves. For bodies seem either to move without any limit, or to tend toward a limit, arrived at which they either revolve according to their peculiar nature, or rest. Those which are favorably situated, and have a tendency to motion, move in a circle with an eternal and unlimited motion; those which are favorably situated and abhor motion, rest. Those which are not favorably situated move in a straight line (as their shortest path), in order to unite with others of a congenial nature. This motion of revolution admits of nine differences: 1, with regard to the centre about which the bodies move; 2, the poles round which they move; 3, the circumference or orbit relatively to its distance from the centre; 4, the velocity, or greater or less speed with which they revolve; 5, the direction of the motion as from east to west, or the reverse; 6, the deviation from a perfect circle, by spiral lines at a greater or less distance from the centre; 7, the deviation from the circle, by spiral lines at a greater or less distance from the poles; 8, the greater or less distance of these spirals from each other; 9, and lastly, the variation of the poles if they be movable; which, however, only affects revolution when circular. The motion in question is, according to common and long-received opinion, considered to be that of the heavenly bodies. There exists, however, with regard to this, a considerable dispute between some of the ancients as well as moderns, who have attributed a motion of revolution to the earth. A much more reasonable controversy, perhaps, exists (if it be not a matter beyond dispute), whether the motion in question (on the hypothesis of the earth’s being fixed) is confined to the heavens, or rather descends and is communicated to the air and water. The rotation of missiles, as in darts, musket-balls, and the like, we refer entirely to the motion of liberty.
Let the eighteenth motion be that of trepidation, to which (in the sense assigned to it by astronomers) we do not give much credit; but in our serious and general search after the tendencies of natural bodies, this motion occurs, and appears worthy of forming a distinct species. It is the motion of an (as it were) eternal captivity; when bodies, for instance, being placed not altogether according to their nature, and yet not exactly ill, constantly tremble, and are restless, not contented with their position, and yet not daring to advance. Such is the motion of the heart and pulse of animals, and it must necessarily occur in all bodies which are situated in a mean state, between conveniences and inconveniences; so that being removed from their proper position, they strive to escape, are repulsed, and again continue to make the attempt.
Let the nineteenth and last motion be one which can scarcely be termed a motion, and yet is one; and which we may call the motion of repose, or of abhorrence of motion. It is by this motion that the earth stands by its own weight, while its extremes move toward the middle, not to an imaginary centre, but in order to unite. It is owing to the same tendency, that all bodies of considerable density abhor motion, and their only tendency is not to move, which nature they preserve, although excited and urged in a variety of ways to motion. But if they be compelled to move, yet do they always appear anxious to recover their former state, and to cease from motion, in which respect they certainly appear active, and attempt it with sufficient swiftness and rapidity, as if fatigued, and impatient of delay. We can only have a partial representation of this tendency, because with us every tangible substance is not only not condensed to the utmost, but even some spirit is added, owing to the action and concocting influence of the heavenly bodies.
We have now, therefore, exhibited the species, or simple elements of the motions, tendencies, and active powers, which are most universal in nature; and no small portion of natural science has been thus sketched out. We do not, however, deny that other instances can perhaps be added, and our divisions changed according to some more natural order of things, and also reduced to a less number; in which respect we do not allude to any abstract classification, as if one were to say, that bodies desire the preservation, exaltation, propagation, or fruition of their nature; or, that motion tends to the preservation and benefit either of the universe (as in the case of those of resistance and connection), or of extensive wholes, as in the case of those of the greater congregation, revolution, and abhorrence of motion, or of particular forms, as in the case of the others. For although such remarks be just, yet, unless they terminate in matter and construction, according to true definitions, they are speculative, and of little use. In the meantime, our classification will suffice, and be of much use in the consideration of the predominance of powers, and examining the wrestling instances which constitute our present subject.
For of the motions here laid down, some are quite invincible, some more powerful than others, which they confine, check, and modify; others extend to a greater distance, others are more immediate and swift, others strengthen, increase, and accelerate the rest.
The motion of resistance is most adamantine and invincible. We are yet in doubt whether such be the nature of that of connection; for we cannot with certainty determine whether there be a vacuum, either extensive or intermixed with matter. Of one thing, however, we are satisfied, that the reason assigned by Leucippus and Democritus for the introduction of a vacuum (namely, that the same bodies could not otherwise comprehend, and fill greater and less spaces) is false. For there is clearly a folding of matter, by which it wraps and unwraps itself in space within certain limits, without the intervention of a vacuum. Nor is there two thousand times more of vacuum in air than in gold, as there should be on this hypothesis; a fact demonstrated by the very powerful energies of fluids (which would otherwise float like fine dust in vacuo), and many other proofs. The other motions direct, and are directed by each other, according to their strength, quantity, excitement, emission, or the assistance or impediments they meet with.
For instance; some armed magnets hold and support iron of sixty times their own weight; so far does the motion of lesser congregation predominate over that of the greater; but if the weight be increased, it yields. A lever of a certain strength will raise a given weight, and so far the motion of liberty predominates over that of the greater congregation, but if the weight be greater, the former motion yields. A piece of leather stretched to a certain point does not break, and so far the motion of continuity predominates over that of tension, but if the tension be greater, the leather breaks, and the motion of continuity yields. A certain quantity of water flows through a chink, and so far the motion of greater congregation predominates over that of continuity, but if the chink be smaller it yields. If a musket be charged with ball and powdered sulphur alone, and fire be applied, the ball is not discharged, in which case the motion of greater congregation overcomes that of matter; but when gunpowder is used, the motion of matter in the sulphur predominates, being assisted by that motion, and the motion of avoidance in the nitre; and so of the rest. For wrestling instances (which show the predominance of powers, and in what manner and proportion they predominate and yield) must be searched for with active and industrious diligence.
The methods and nature of this yielding must also be diligently examined, as for instance, whether the motions completely cease, or exert themselves, but are constrained. For in the bodies with which we are acquainted, there is no real but an apparent rest, either in the whole or in parts. This apparent rest is occasioned either by equilibrium, or the absolute predominance of motions. By equilibrium, as in the scales of the balance, which rest if the weights be equal. By predominance, as in perforated jars, in which the water rests, and is prevented from falling by the predominance of the motion of connection. It is, however, to be observed (as we have said before), how far the yielding motions exert themselves. For if a man be held stretched out on the ground against his will, with arms and legs bound down, or otherwise confined, and yet strive with all his power to get up, the struggle is not the less, although ineffectual. The real state of the case (namely, whether the yielding motion be, as it were, annihilated by the predominance, or there be rather a continued, although an invisible effort) will, perhaps, appear in the concurrence of motions, although it escape our notice in their conflict. For instance: let an experiment be made with muskets; whether a musket-ball, at its utmost range in a straight line, or (as it is commonly called) point-blank, strike with less force when projected upward, where the motion of the blow is simple, than when projected downward, where the motion of gravity concurs with the blow.
The rules of such instances of predominance as occur should be collected: such as the following; the more general the desired advantage is, the stronger will be the motion; the motion of connection, for instance, which relates to the intercourse of the parts of the universe, is more powerful than that of gravity, which relates to the intercourse of dense bodies only. Again, the desire of a private good does not in general prevail against that of a public one, except where the quantities are small. Would that such were the case in civil matters!
XLIX. In the twenty-fifth rank of prerogative instances we will place suggesting instances; such as suggest, or point out, that which is advantageous to mankind; for bare power and knowledge in themselves exalt rather than enrich human nature. We must, therefore, select from the general store such things as are most useful to mankind. We shall have a better opportunity of discussing these when we treat of the application to practice; besides, in the work of interpretation, we leave room, on every subject, for the human or optative chart; for it is a part of science to make judicious inquiries and wishes.
L. In the twenty-sixth rank of prerogative instances we will place the generally useful instances. They are such as relate to various points, and frequently occur, sparing by that means considerable labor and new trials. The proper place for treating of instruments and contrivances, will be that in which we speak of the application to practice, and the methods of experiment. All that has hitherto been ascertained, and made use of, will be described in the particular history of each art. At present, we will subjoin a few general examples of the instances in question.
Man acts, then, upon natural bodies (besides merely bringing them together or removing them) by seven principal methods: 1, by the exclusion of all that impedes and disturbs; 2, by compression, extension, agitation, and the like; 3, by heat and cold; 4, by detention in a suitable place; 5, by checking or directing motion; 6, by peculiar harmonies; 7, by a seasonable and proper alternation, series, and succession of all these, or, at least, of some of them.
With regard to the first—common air, which is always at hand, and forces its admission, as also the rays of the heavenly bodies, create much disturbance. Whatever, therefore, tends to exclude them may well be considered as generally useful. The substance and thickness of vessels in which bodies are placed when prepared for operations may be referred to this head. So also may the accurate methods of closing vessels by consolidation, or the lutum sapientiæ, as the chemists call it. The exclusion of air by means of liquids at the extremity is also very useful, as when they pour oil on wine, or the juices of herbs, which by spreading itself upon the top like a cover, preserves them uninjured from the air. Powders, also, are serviceable, for although they contain air mixed up in them, yet they ward off the power of the mass of circumambient air, which is seen in the preservation of grapes and other fruits in sand or flour. Wax, honey, pitch, and other resinous bodies, are well used in order to make the exclusion more perfect, and to remove the air and celestial influence. We have sometimes made an experiment by placing a vessel or other bodies in quicksilver, the most dense of all substances capable of being poured round others. Grottoes and subterraneous caves are of great use in keeping off the effects of the sun, and the predatory action of air, and in the north of Germany are used for granaries. The depositing of bodies at the bottom of water may be also mentioned here; and I remember having heard of some bottles of wine being let down into a deep well in order to cool them, but left there by chance, carelessness, and forgetfulness for several years, and then taken out; by which means the wine not only escaped becoming flat or dead, but was much more excellent in flavor, arising (as it appears) from a more complete mixture of its parts. But if the case require that bodies should be sunk to the bottom of water, as in rivers or the sea, and yet should not touch the water, nor be inclosed in sealed vessels, but surrounded only by air, it would be right to use that vessel which has been sometimes employed under water above ships that have sunk, in order to enable the divers to remain below and breathe occasionally by turns. It was of the following nature: A hollow tub of metal was formed, and sunk so as to have its bottom parallel with the surface of the water; it thus carried down with it to the bottom of the sea all the air contained in the tub. It stood upon three feet (like a tripod), being of rather less height than a man, so that, when the diver was in want of breath, he could put his head into the hollow of the tub, breathe, and then continue his work. We hear that some sort of boat or vessel has now been invented, capable of carrying men some distance under water. Any bodies, however, can easily be suspended under some such vessel as we have mentioned, which has occasioned our remarks upon the experiment.
Another advantage of the careful and hermetical closing of bodies is this—not only the admission of external air is prevented (of which we have treated), but the spirit of bodies also is prevented from making its escape, which is an internal operation. For any one operating on natural bodies must be certain as to their quantity, and that nothing has evaporated or escaped, since profound alterations take place in bodies, when art prevents the loss or escape of any portion, while nature prevents their annihilation. With regard to this circumstance, a false idea has prevailed (which if true would make us despair of preserving quantity without diminution), namely, that the spirit of bodies, and air when rarefied by a great degree of heat, cannot be so kept in by being inclosed in any vessel as not to escape by the small pores. Men are led into this idea by the common experiments of a cup inverted over water, with a candle or piece of lighted paper in it, by which the water is drawn up, and of those cups which, when heated, draw up the flesh. For they think that in each experiment the rarefied air escapes, and that its quantity is therefore diminished, by which means the water or flesh rises by the motion of connection. This is, however, most incorrect. For the air is not diminished in quantity, but contracted in dimensions, nor does this motion of the rising of the water begin till the flame is extinguished, or the air cooled, so that physicians place cold sponges, moistened with water, on the cups, in order to increase their attraction. There is, therefore, no reason why men should fear much from the ready escape of air: for although it be true that the most solid bodies have their pores, yet neither air, nor spirit, readily suffers itself to be rarefied to such an extreme degree; just as water will not escape by a small chink.
With regard to the second of the seven above-mentioned methods, we must especially observe, that compression and similar violence have a most powerful effect either in producing locomotion, and other motions of the same nature, as may be observed in engines and projectiles, or in destroying the organic body, and those qualities, which consist entirely in motion (for all life, and every description of flame and ignition are destroyed by compression, which also injures and deranges every machine); or in destroying those qualities which consist in position and a coarse difference of parts, as in colors; for the color of a flower when whole, differs from that it presents when bruised, and the same may be observed of whole and powdered amber; or in tastes, for the taste of a pear before it is ripe, and of the same pear when bruised and softened, is different, since it becomes perceptibly more sweet. But such violence is of little avail in the more noble transformations and changes of homogeneous bodies, for they do not, by such means, acquire any constantly and permanently new state, but one that is transitory, and always struggling to return to its former habit and freedom. It would not, however, be useless to make some more diligent experiments with regard to this; whether, for instance, the condensation of a perfectly homogeneous body (such as air, water, oil, and the like) or their rarefaction, when effected by violence, can become permanent, fixed, and, as it were, so changed, as to become a nature. This might at first be tried by simple perseverance, and then by means of helps and harmonies. It might readily have been attempted (if we had but thought of it), when we condensed water (as was mentioned above), by hammering and compression, until it burst out. For we ought to have left the flattened globe untouched for some days, and then to have drawn off the water, in order to try whether it would have immediately occupied the same dimensions as it did before the condensation. If it had not done so, either immediately, or soon afterward, the condensation would have appeared to have been rendered constant; if not, it would have appeared that a restitution took place, and that the condensation had been transitory. Something of the same kind might have been tried with the glass eggs; the egg should have been sealed up suddenly and firmly, after a complete exhaustion of the air, and should have been allowed to remain so for some days, and it might then have been tried whether, on opening the aperture, the air would be drawn in with a hissing noise, or whether as much water would be drawn into it when immersed, as would have been drawn into it at first, if it had not continued sealed. For it is probable (or, at least, worth making the experiment) that this might have happened, or might happen, because perseverance has a similar effect upon bodies which are a little less homogeneous. A stick bent together for some time does not rebound, which is not owing to any loss of quantity in the wood during the time, for the same would occur (after a larger time) in a plate of steel, which does not evaporate. If the experiment of simple perseverance should fail, the matter should not be given up, but other means should be employed. For it would be no small advantage, if bodies could be endued with fixed and constant natures by violence. Air could then be converted into water by condensation, with other similar effects; for man is more the master of violent motions than of any other means.
The third of our seven methods is referred to that great practical engine of nature, as well as of art, cold and heat. Here, man’s power limps, as it were, with one leg. For we possess the heat of fire, which is infinitely more powerful and intense than that of the sun (as it reaches us), and that of animals. But we want cold, except such as we can obtain in winter, in caverns, or by surrounding objects with snow and ice, which, perhaps, may be compared in degree with the noontide heat of the sun in tropical countries, increased by the reflection of mountains and walls. For this degree of heat and cold can be borne for a short period only by animals, yet it is nothing compared with the heat of a burning furnace, or the corresponding degree of cold. Everything with us has a tendency to become rarefied, dry and wasted, and nothing to become condensed or soft, except by mixtures, and, as it were, spurious methods. Instances of cold, therefore, should be searched for most diligently, such as may be found by exposing bodies upon buildings in a hard frost, in subterraneous caverns, by surrounding bodies with snow and ice in deep places excavated for that purpose, by letting bodies down into wells, by burying bodies in quicksilver and metals, by immersing them in streams which petrify wood, by burying them in the earth (which the Chinese are reported to do with their china, masses of which, made for that purpose, are said to remain in the ground for forty or fifty years, and to be transmitted to their heirs as a sort of artificial mine) and the like. The condensations which take place in nature, by means of cold, should also be investigated, that by learning their causes, they may be introduced into the arts; such as are observed in the exudation of marble and stones, in the dew upon the panes of glass in a room toward morning after a frosty night, in the formation and the gathering of vapors under the earth into water, whence spring fountains and the like.
Besides the substances which are cold to the touch, there are others which have also the effect of cold, and condense; they appear, however, to act only upon the bodies of animals, and scarcely any further. Of these we have many instances, in medicines and plasters. Some condense the flesh and tangible parts, such as astringent and inspissating medicines, others the spirits, such as soporifics. There are two modes of condensing the spirits, by soporifics or provocatives to sleep; the one by calming the motion, the other by expelling the spirit. The violet, dried roses, lettuces, and other benign or mild remedies, by their friendly and gently cooling vapors, invite the spirits to unite, and restrain their violent and perturbed motion. Rose-water, for instance, applied to the nostrils in fainting fits, causes the resolved and relaxed spirits to recover themselves, and, as it were, cherishes them. But opiates, and the like, banish the spirits by their malignant and hostile quality. If they be applied, therefore, externally, the spirits immediately quit the part and no longer readily flow into it; but if they be taken internally, their vapor, mounting to the head, expels, in all directions, the spirits contained in the ventricles of the brain, and since these spirits retreat, but cannot escape, they consequently meet and are condensed, and are sometimes completely extinguished and suffocated; although the same opiates, when taken in moderation, by a secondary accident (the condensation which succeeds their union), strengthen the spirits, render them more robust, and check their useless and inflammatory motion, by which means they contribute not a little to the cure of diseases, and the prolongation of life.
The preparations of bodies, also, for the reception of cold should not be omitted, such as that water a little warmed is more easily frozen than that which is quite cold, and the like.
Moreover, since nature supplies cold so sparingly, we must act like the apothecaries, who, when they cannot obtain any simple ingredient, take a succedaneum, or quid pro quo, as they term it, such as aloes for xylobalsamum, cassia for cinnamon. In the same manner we should look diligently about us, to ascertain whether there may be any substitutes for cold, that is to say, in what other manner condensation can be effected, which is the peculiar operation of cold. Such condensations appear hitherto to be of four kinds only. 1.By simple compression, which is of little avail toward permanent condensation, on account of the elasticity of substances, but may still, however, be of some assistance. 2.By the contraction of the coarser, after the escape or departure of the finer parts of a given body; as is exemplified in induration by fire, and the repeated heating and extinguishing of metals, and the like. 3.By the cohesion of the most solid homogeneous parts of a given body, which were previously separated, and mixed with others less solid, as in the return of sublimated mercury to its simple state, in which it occupies much less space than it did in powder, and the same may be observed of the cleansing of all metals from their dross. 4.By harmony, or the application of substances which condense by some latent power. These harmonies are as yet but rarely observed, at which we cannot be surprised, since there is little to hope for from their investigation, unless the discovery of forms and confirmation be attained. With regard to animal bodies, it is not to be questioned that there are many internal and external medicines which condense by harmony, as we have before observed, but this action is rare in inanimate bodies. Written accounts, as well as report, have certainly spoken of a tree in one of the Tercera or Canary Islands (for I do not exactly recollect which) that drips perpetually, so as to supply the inhabitants, in some degree, with water; and Paracelsus says that the herb called ros solis is filled with dew at noon, while the sun gives out its greatest heat, and all other herbs around it are dry. We treat both these accounts as fables; they would, however, if true, be of the most important service, and most worthy of examination. As to the honey-dew, resembling manna, which is found in May on the leaves of the oak, we are of opinion that it is not condensed by any harmony or peculiarity of the oak leaf, but that while it falls equally upon other leaves it is retained and continues on those of the oak, because their texture is closer, and not so porous as that of most of the other leaves.
With regard to heat, man possesses abundant means and power; but his observation and inquiry are defective in some respects, and those of the greatest importance, notwithstanding the boasting of quacks. For the effects of intense heat are examined and observed, while those of a more gentle degree of heat, being of the most frequent occurrence in the paths of nature, are, on that very account, least known. We see, therefore, the furnaces, which are most esteemed, employed in increasing the spirits of bodies to a great extent, as in the strong acids, and some chemical oils; while the tangible parts are hardened, and, when the volatile part has escaped, become sometimes fixed; the homogeneous parts are separated, and the heterogeneous incorporated and agglomerated in a coarse lump; and (what is chiefly worthy of remark) the junction of compound bodies, and the more delicate conformations are destroyed and confounded. But the operation of a less violent heat should be tried and investigated, by which more delicate mixtures and regular conformations may be produced and elicited, according to the example of nature, and in imitation of the effect of the sun, which we have alluded to in the aphorism on the instances of alliance. For the works of nature are carried on in much smaller portions, and in more delicate and varied positions than those of fire, as we now employ it. But man will then appear to have really augmented his power, when the works of nature can be imitated in species, perfected in power, and varied in quantity; to which should be added the acceleration in point of time. Rust, for instance, is the result of a long process, but crocus martis is obtained immediately; and the same may be observed of natural verdigris and ceruse. Crystal is formed slowly, while glass is blown immediately: stones increase slowly, while bricks are baked immediately, etc. In the meantime (with regard to our present subject) every different species of heat should, with its peculiar effects, be diligently collected and inquired into; that of the heavenly bodies, whether their rays be direct, reflected, or refracted, or condensed by a burning-glass; that of lightning, flame, and ignited charcoal; that of fire of different materials, either open or confined, straitened or overflowing, qualified by the different forms of the furnaces, excited by the bellows, or quiescent, removed to a greater or less distance, or passing through different media; moist heats, such as the balneum Mariæ, and the dunghill; the external and internal heat of animals; dry heats, such as the heat of ashes, lime, warm sand; in short, the nature of every kind of heat, and its degrees.
We should, however, particularly attend to the investigation and discovery of the effects and operations of heat, when made to approach and retire by degrees, regularly, periodically, and by proper intervals of space and time. For this systematical inequality is in truth the daughter of heaven and mother of generation, nor can any great result be expected from a vehement, precipitate, or desultory heat. For this is not only most evident in vegetables, but in the wombs of animals also there arises a great inequality of heat, from the motion, sleep, food, and passions of the female. The same inequality prevails in those subterraneous beds where metals and fossils are perpetually forming, which renders yet more remarkable the ignorance of some of the reformed alchemists, who imagined they could attain their object by the equable heat of lamps, or the like, burning uniformly. Let this suffice concerning the operation and effects of heat; nor is it time for us to investigate them thoroughly before the forms and conformations of bodies have been further examined and brought to light. When we have determined upon our models, we may seek, apply, and arrange our instruments.
The fourth mode of action is by continuance, the very steward and almoner, as it were, of nature. We apply the term continuance to the abandonment of a body to itself for an observable time, guarded and protected in the meanwhile from all external force. For the internal motion then commences to betray and exert itself when the external and adventitious is removed. The effects of time, however, are far more delicate than those of fire. Wine, for instance, cannot be clarified by fire as it is by continuance. Nor are the ashes produced by combustion so fine as the particles dissolved or wasted by the lapse of ages. The incorporations and mixtures, which are hurried by fire, are very inferior to those obtained by continuance; and the various conformations assumed by bodies left to themselves, such as mouldiness, etc., are put a stop to by fire or a strong heat. It is not, in the meantime, unimportant to remark that there is a certain degree of violence in the motion of bodies entirely confined; for the confinement impedes the proper motion of the body. Continuance in an open vessel, therefore, is useful for separations, and in one hermetically sealed for mixtures, that in a vessel partly closed, but admitting the air, for putrefaction. But instances of the operation and effect of continuance must be collected diligently from every quarter.
The direction of motion (which is the fifth method of action) is of no small use. We adopt this term, when speaking of a body which, meeting with another, either arrests, repels, allows, or directs its original motion. This is the case principally in the figure and position of vessels. An upright cone, for instance, promotes the condensation of vapor in alembics, but when reversed, as in inverted vessels, it assists the refining of sugar. Sometimes a curved form, or one alternately contracted and dilated, is required. Strainers may be ranged under this head, where the opposed body opens a way for one portion of another substance and impedes the rest. Nor is this process or any other direction of motion carried on externally only, but sometimes by one body within another. Thus, pebbles are thrown into water to collect the muddy particles, and syrups are refined by the white of an egg, which glues the grosser particles together so as to facilitate their removal. Telesius, indeed, rashly and ignorantly enough attributes the formation of animals to this cause, by means of the channels and folds of the womb. He ought to have observed a similar formation of the young in eggs which have no wrinkles or inequalities. One may observe a real result of this direction of motion in casting and modelling.
The effects produced by harmony and aversion (which is the sixth method) are frequently buried in obscurity; for these occult and specific properties (as they are termed), the sympathies and antipathies, are for the most part but a corruption of philosophy. Nor can we form any great expectation of the discovery of the harmony which exists between natural objects, before that of their forms and simple conformations, for it is nothing more than the symmetry between these forms and conformations.
The greater and more universal species of harmony are not, however, so wholly obscure, and with them, therefore, we must commence. The first and principal distinction between them is this; that some bodies differ considerably in the abundance and rarity of their substance, but correspond in their conformation; others, on the contrary, correspond in the former and differ in the latter. Thus the chemists have well observed, that in their trial of first principles sulphur and mercury, as it were, pervade the universe; their reasoning about salt, however, is absurd, and merely introduced to comprise earthy dry fixed bodies. In the other two, indeed, one of the most universal species of natural harmony manifests itself. Thus there is a correspondence between sulphur, oil, greasy exhalations, flame, and, perhaps, the substance of the stars. On the other hand, there is a like correspondence between mercury, water, aqueous vapor, air, and, perhaps, pure inter-sidereal ether. Yet do these two quaternions, or great natural tribes (each within its own limits), differ immensely in quantity and density of substance, while they generally agree in conformation, as is manifest in many instances. On the other hand, the metals agree in such quantity and density (especially when compared with vegetables, etc.), but differ in many respects in conformation. Animals and vegetables, in like manner, vary in their almost infinite modes of conformation, but range within very limited degrees of quantity and density of substance.
The next most general correspondence is that between individual bodies and those which supply them by way of menstruum or support. Inquiry, therefore, must be made as to the climate, soil, and depth at which each metal is generated, and the same of gems, whether produced in rocks or mines, also as to the soil in which particular trees, shrubs, and herbs, mostly grow and, as it were, delight; and as to the best species of manure, whether dung, chalk, sea sand, or ashes, etc., and their different propriety and advantage according to the variety of soils. So also the grafting and setting of trees and plants (as regards the readiness of grafting one particular species on another) depends very much upon harmony, and it would be amusing to try an experiment I have lately heard of, in grafting forest trees (garden trees alone having hitherto been adopted), by which means the leaves and fruit are enlarged, and the trees produce more shade. The specific food of animals again should be observed, as well as that which cannot be used. Thus the carnivorous cannot be fed on herbs, for which reason the order of feuilletans, the experiment having been made, has nearly vanished; human nature being incapable of supporting their regimen, although the human will has more power over the bodily frame than that of other animals. The different kinds of putrefaction from which animals are generated should be noted.
The harmony of principal bodies with those subordinate to them (such indeed may be deemed those we have alluded to above) are sufficiently manifest, to which may be added those that exist between different bodies and their objects, and, since these latter are more apparent, they may throw great light when well observed and diligently examined upon those which are more latent.
The more internal harmony and aversion, or friendship and enmity (for superstition and folly have rendered the terms of sympathy and antipathy almost disgusting), have been either falsely assigned, or mixed with fable, or most rarely discovered from neglect. For if one were to allege that there is an enmity between the vine and the cabbage, because they will not come up well when sown together, there is a sufficient reason for it in the succulent and absorbent nature of each plant, so that the one defrauds the other. Again, if one were to say that there is a harmony and friendship between the corn and the corn-flower, or the wild poppy, because the latter seldom grow anywhere but in cultivated soils, he ought rather to say, there is an enmity between them, for the poppy and the corn-flower are produced and created by those juices which the corn has left and rejected, so that the sowing of the corn prepares the ground for their production. And there are a vast number of similar false assertions. As for fables, they must be totally exterminated. There remains, then, but a scanty supply of such species of harmony as has borne the test of experiment, such as that between the magnet and iron, gold and quicksilver, and the like. In chemical experiments on metals, however, there are some others worthy of notice, but the greatest abundance (where the whole are so few in numbers) is discovered in certain medicines, which, from their occult and specific qualities (as they are termed), affect particular limbs, humors, diseases, or constitutions. Nor should we omit the harmony between the motion and phenomena of the moon, and their effects on lower bodies, which may be brought together by an accurate and honest selection from the experiments of agriculture, navigation, and medicine, or of other sciences. By as much as these general instances, however, of more latent harmony, are rare, with so much the more diligence are they to be inquired after, through tradition, and faithful and honest reports, but without rashness and credulity, with an anxious and, as it were, hesitating degree of reliance. There remains one species of harmony which, though simple in its mode of action, is yet most valuable in its use, and must by no means be omitted, but rather diligently investigated. It is the ready or difficult coition or union of bodies in composition, or simple juxtaposition. For some bodies readily and willingly mix, and are incorporated, others tardily and perversely; thus powders mix best with water, chalk and ashes with oils, and the like. Nor are these instances of readiness and aversion to mixture to be alone collected, but others, also, of the collocation, distribution, and digestion of the parts when mingled, and the predominance after the mixture is complete.
Lastly, there remains the seventh, and last of the seven, modes of action; namely, that by the alternation and interchange of the other six; but of this, it will not be the right time to offer any examples, until some deeper investigation shall have taken place of each of the others. The series, or chain of this alternation, in its mode of application to separate effects, is no less powerful in its operation than difficult to be traced. But men are possessed with the most extreme impatience, both of such inquiries, and their practical application, although it be the clew of the labyrinth in all greater works. Thus far of the generally useful instances.
LI. The twenty-seventh and last place we will assign to the magical instances, a term which we apply to those where the matter or efficient agent is scanty or small, in comparison with the grandeur of the work or effect produced; so that even when common they appear miraculous, some at first sight, others even upon more attentive observation. Nature, however, of herself, supplies these but sparingly. What she will do when her whole store is thrown open, and after the discovery of forms, processes, and conformation, will appear hereafter. As far as we can yet conjecture, these magic effects are produced in three ways, either by self-multiplication, as in fire, and the poisons termed specific, and the motions transferred and multiplied from wheel to wheel; or by the excitement, or, as it were, invitation of another substance, as in the magnet, which excites innumerable needles without losing or diminishing its power; and again in leaven, and the like; or by the excess of rapidity of one species of motion over another, as has been observed in the case of gunpowder, cannon, and mines. The two former require an investigation of harmonies, the latter of a measure of motion. Whether there be any mode of changing bodies per minima (as it is termed), and transferring the delicate conformations of matter, which is of importance in all transformations of bodies, so as to enable art to effect, in a short time, that which nature works out by divers expedients, is a point of which we have as yet no indication. But, as we aspire to the extremest and highest results in that which is solid and true, so do we ever detest, and, as far as in us lies, expel all that is empty and vain.
LII. Let this suffice as to the respective dignity of prerogatives of instances. But it must be noted, that in this our organ, we treat of logic, and not of philosophy. Seeing, however, that our logic instructs and informs the understanding, in order that it may not, with the small hooks, as it were, of the mind, catch at, and grasp mere abstractions, but rather actually penetrate nature, and discover the properties and effects of bodies, and the determinate laws of their substance (so that this science of ours springs from the nature of things, as well as from that of the mind); it is not to be wondered at, if it have been continually interspersed and illustrated with natural observations and experiments, as instances of our method. The prerogative instances are, as appears from what has preceded, twenty-seven in number, and are termed, solitary instances, migrating instances, conspicuous instances, clandestine instances, constitutive instances, similar instances, singular instances, deviating instances, bordering instances, instances of power, accompanying and hostile instances, subjunctive instances, instances of alliance, instances of the cross, instances of divorce, instances of the gate, citing instances, instances of the road, supplementary instances, lancing instances, instances of the rod, instances of the course, doses of nature, wrestling instances, suggesting instances, generally useful instances, and magical instances. The advantage, by which these instances excel the more ordinary, regards specifically either theory or practice, or both. With regard to theory, they assist either the senses or the understanding; the senses, as in the five instances of the lamp; the understanding, either by expediting the exclusive mode of arriving at the form, as in solitary instances, or by confining, and more immediately indicating the affirmative, as in the migrating, conspicuous, accompanying, and subjunctive instances; or by elevating the understanding, and leading it to general and common natures, and that either immediately, as in the clandestine and singular instances, and those of alliance; or very nearly so, as in the constitutive; or still less so, as in the similar instances; or by correcting the understanding of its habits, as in the deviating instances; or by leading to the grand form or fabric of the universe, as in the bordering instances; or by guarding it from false forms and causes, as in those of the cross and of divorce. With regard to practice, they either point it out, or measure, or elevate it. They point it out, either by showing where we must commence in order not to repeat the labors of others, as in the instances of power; or by inducing us to aspire to that which may be possible, as in the suggesting instances; the four mathematical instances measure it. The generally useful and the magical elevate it.
Again, out of these twenty-seven instances, some must be collected immediately, without waiting for a particular investigation of properties. Such are the similar, singular, deviating, and bordering instances, those of power, and of the gate, and suggesting, generally useful, and magical instances; for these either assist and cure the understanding and senses, or furnish our general practice. The remainder are to be collected when we finish our synoptical tables for the work of the interpreter, upon any particular nature; for these instances, honored and gifted with such prerogatives, are like the soul amid the vulgar crowd of instances, and (as we from the first observed) a few of them are worth a multitude of the others. When, therefore, we are forming our tables they must be searched out with the greatest zeal, and placed in the table. And, since mention must be made of them in what follows, a treatise upon their nature has necessarily been prefixed. We must next, however, proceed to the supports and corrections of induction, and thence to concretes, the latent process, and latent conformations, and the other matters, which we have enumerated in their order in the twenty-first aphorism, in order that, like good and faithful guardians, we may yield up their fortune to mankind upon the emancipation and majority of their understanding; from which must necessarily follow an improvement of their estate, and an increase of their power over nature. For man, by the fall, lost at once his state of innocence, and his empire over creation, both of which can be partially recovered even in this life, the first by religion and faith, the second by the arts and sciences. For creation did not become entirely and utterly rebellious by the curse, but in consequence of the Divine decree, “in the sweat of thy brow shalt thou eat bread,” she is compelled by our labors (not assuredly by our disputes or magical ceremonies), at length, to afford mankind in some degree his bread, that is to say, to supply man’s daily wants.
END OF “NOVUM ORGANUM”
FOOTNOTES
Τὸ τὶ ἦν εἶναι, or ἦν οὐσία of Aristotle.—See lib.iii. Metap.
These divisions are from Aristotle’s Metaphysics, where they are termed, 1.ὓλη ἢ τὸ ὑποκείμενον. 2.τὸ τὶ ἦν εἶναι. 3.ὅθεν ἡ ἀρχὴ τῆς κινήσεως. 4.τὸ οὗ ἕνεκεν—καὶ τὸ ἀγαθόν.
See Aphorism li. and second paragraph of Aphorism lxv. in the first book.
Bacon means, that although there exist in nature only individualities, yet a certain number of these may have common properties, and be controlled by the same laws. Now, these homogeneous qualities which distinguish them from other individuals, lead us to class them under one expression, and sometimes under a single term. Yet these classes are only pure conceptions in Bacon’s opinion, and cannot be taken for distinct substances. He evidently here aims a blow at the Realists, who concluded that the essence which united individualities in a class was the only real and immutable existence in nature, inasmuch as it entered into their ideas of individual substances as a distinct and essential property, and continued in the mind as the mold, type or pattern of the class, while its individual forms were undergoing perpetual renovation and decay.—Ed.
Bacon’s definition is obscure. All the idea we have of a law of nature consists in invariable sequence between certain classes of phenomena; but this cannot be the complete sense attached by Bacon to the term form, as he employs it in the fourth aphorism as convertible with the nature of any object; and again, in the first aphorism, as the natura naturans, or general law or condition in any substance or quality—natura naturata—which is whatever its form is, or that particular combination of forces which impresses a certain nature upon matter subject to its influence. Thus, in the Newtonian sense, the form of whiteness would be that combination of the seven primitive rays of light which give rise to that color. In combination with this word, and affording a still further insight into its meaning, we have the phrases, latens processus ad formam, et latens schematismus corporum. Now, the latens schematismus signifies the internal texture, structure, or configuration of bodies, or the result of the respective situation of all the parts of a body; while the latens processus ad formam points out the gradation of movements which takes place among the molecula of bodies when they either conserve or change their figure. Hence we may consider the form of any quality in body as something convertible with that quality, i.e., when it exists the quality is present, and vice versâ. In this sense, the form of a thing differs only from its efficient cause in being permanent, whereas we apply cause to that which exists in order of time. The latens processus and latens schematismus are subordinate to form, as concrete exemplifications of its essence. The former is the secret and invisible process by which change is effected, and involves the principle since called the law of continuity. Thus, the succession of events between the application of the match to the expulsion of the bullet is an instance of latent progress which we can now trace with some degree of accuracy. It also more directly refers to the operation by which one form or condition of being is induced upon another. For example, when the surface of iron becomes rusty, or when water is converted into steam, some change has taken place, or latent process from one form to another. Mechanics afford many exemplifications of the first latent process we have denoted, and chemistry of the second. The latens schematismus is that visible structure of bodies on which so many of their properties depend. When we inquire into the constitution of crystals, and into the internal structure of plants, we are examining into their latent schematism.—Ed.
By the recent discoveries in electric magnetism, copper wires, or, indeed, wires of any metal, may be transformed into magnets; the magnetic law, or form, having been to that extent discovered.
Haller has pursued this investigation in his “Physiology,” and has left his successors little else to do than repeat his discoveries.—Ed.
Bacon here first seems pregnant with the important development of the higher calculus, which, in the hands of Newton and Descartes, was to effect as great a revolution in philosophy as his method.—Ed.
By spirit, Bacon here plainly implies material fluid too fine to be grasped by the unassisted sense, which rather operates than reasons. We sometimes adopt the same mode of expression, as in the words spirits of nitre, spirits of wine. Some such agency has been assumed by nearly all the modern physicists, a few of whom, along with Bacon, would leave us to gather from their expressions, that they believe such bodies endowed with the sentient powers of perception. As another specimen of his sentiment on this subject, we may refer to a paragraph on the decomposition of compounds, in his essay on death, beginning—“The spirit which exists in all living bodies, keeps all the parts in due subjection; when it escapes, the body decomposes, or the similar parts unite.”—Ed.
The theory of the Epicureans and others. The atoms are supposed to be invisible, unalterable particles, endued with all the properties of the given body, and forming that body by their union. They must be separated, of course, which either takes a vacuum for granted, or introduces a tertium quid into the composition of the body.
Compare the three following aphorisms with the last three chapters of the third book of the “De Augmentis Scientiarum.”
Bacon gives this unfortunate term its proper signification; μετα, in composition, with the Greeks signifying change or mutation. Most of our readers, no doubt, are aware that the obtrusion of this word into technical philosophy was purely capricious, and is of no older date than the publication of Aristotle’s works by Andronicus of Rhodes, one of the learned men into whose hands the manuscripts of that philosopher fell, after they were brought by Sylla from Athens to Rome. To fourteen books in these MSS. with no distinguishing title, Andronicus is said to have prefixed the words τα μετα τα φυσικα, to denote the place which they ought to hold either in the order of Aristotle’s arrangement, or in that of study. These books treat first of those subjects which are common to matter and mind; secondly, of things separate from matter, i.e. of God, and of the subordinate spirits, which were supposed by the Peripatetics to watch over particular portions of the universe. The followers of Aristotle accepted the whimsical title of Andronicus, and in their usual manner allowed a word to unite things into one science which were plainly heterogeneous. Their error was adopted by the Peripatetics of the Christian Church. The schoolmen added to the notion of ontology, the science of the mind, or pneumatology, and as that genus of being has since become extinct with the schools, metaphysics thus in modern parlance comes to be synonymous with psychology. It were to be wished that Bacon’s definition of the term had been accepted, and mental science delivered from one of the greatest monstrosities in its nomenclature, yet Bacon whimsically enough in his De Augmentis includes mathematics in metaphysics.—Ed.
“Ne tenues pluviæ, rapidive potentia solis
Acrior, aut Boreæ penetrabile frigus adurat.”
—Virg.Georg. i.92, 93.
This notion, which he repeats again, and particularizes in the 18th aph. of this book, is borrowed from the ancients, and we need not say is as wise as their other astronomical conjectures. The sun also approaches stars quite as large in other quarters of the zodiac, when it looks down upon the earth through the murky clouds of winter. When that luminary is in Leo, the heat of the earth is certainly greater than at any other period, but this arises from the accumulation of heat after the solstice, for the same reason that the maximum heat of the day is at two o’clock instead of noon.—Ed.
Bouguer, employed by Louis XIV. in philosophical researches, ascended the Andes to discover the globular form of the earth, and published an account of his passage, which verifies the statement of Bacon.
Montanari asserts in his book against the astrologers that he had satisfied himself by numerous and oft-repeated experiments, that the lunar rays gathered to a focus produced a sensible degree of heat. Muschenbröck, however, adopts the opposite opinion, and asserts that himself, De la Hire, Villet, and Tschirnhausen had tried with that view the strongest burning-glasses in vain. (Opera de Igne.) De la Lande makes a similar confession in his Astronomy (vol.ii. vii. §1413). Bouguer, whom we have just quoted, demonstrated that the light of the moon was 300,000 degrees less than that of the sun; it would consequently be necessary to invent a glass with an absorbing power 300,000 degrees greater than those ordinarily in use, to try the experiment Bacon speaks of.—Ed.
In this thermometer, mercury was not dilated by heat or contracted by cold, as the one now in use, but a mass of air employed instead, which filled the cavity of the bulb. This being placed in an inverted position to ours, that is to say, with the bulb uppermost, pressed down the liquor when the air became dilated by heat, as ours press it upward; and when the heat diminished, the liquor rose to occupy the place vacated by the air, as the one now in use descends. It consequently was liable to be affected by a change in the temperature, as by the weight of air, and could afford only a rude standard of accuracy in scientific investigations. This thermometer was not Bacon’s own contrivance, as is commonly supposed, but that of Drebbel.—Ed.
La Lande is indignant that the Chaldeans should have more correct notions of the nature of comets than the modern physicists, and charges Bacon with entertaining the idea that they were the mere effects of vapor and heat. This passage, with two others more positive, in the “De Aug.” (cap.xl.) and the “Descript. Globi Intellect.” (cap.vi.) certainly afford ground for the assertion; but if Bacon erred, he erred with Galileo, and with the foremost spirits of the times. It is true that Pythagoras and Seneca had asserted their belief in the solidity of these bodies, but the wide dominion which Aristotle subsequently exercised, threw their opinions into the shade, and made the opposite doctrine everywhere paramount.—Ed.
Was it a silk apron which exhibited electric sparks? Silk was then scarce.
The Italian fire-fly.
This last is found to be the real reason, air not being a good conductor, and therefore not allowing the escape of heat. The confined air is disengaged when these substances are placed under an exhausted receiver.
This is erroneous. Air, in fact, is one of the worst, and metals are the best conductors of heat.
See No.28 in the table of the degrees of heat.
Bacon here mistakes sensation confined to ourselves for an internal property of distinct substances. Metals are denser than wood, and our bodies consequently coming into contact with more particles of matter when we touch them, lose a greater quantity of heat than in the case of lighter substances.—Ed.
This was the ancient opinion, but the moderns incline to the belief that these insects are produced by generation or fecundity from seeds deposited by their tribes in bodies on the verge of putrefaction.—Ed.
The correct measure of the activity of flame may be obtained by multiplying its natural force into the square of its velocity. On this account the flame of vivid lightning mentioned in No.23 contains so much vigor, its velocity being greater than that arising from other heat.—Ed.
The fires supply fresh heat, the water has only a certain quantity of heat, which being diffused over a fresh supply of cooler water, must be on the whole lowered.
If condensation were the cause of the greater heat, Bacon concludes the centre of the flame would be the hotter part, and vice versâ. The fact is, neither of the causes assigned by Bacon is the true one; for the fire burns more quickly only because the draught of air is more rapid, the cold dense air pressing rapidly into the heated room and toward the chimney.—Ed.
Bacon appears to have confounded combustibility and fusibility with susceptibility of heat; for though the metals will certainly neither dissolve as soon as ice or butter, nor be consumed as soon as wood, that only shows that different degrees of heat are required to produce similar effects on different bodies; but metals much more readily acquire and transmit the same degree of heat than any of the above substances. The rapid transmission renders them generally cold to the touch. The convenience of fixing wooden handles to vessels containing hot water illustrates these observations.
Another singular error, the truth being, that solid bodies are the best conductors; but of course where heat is diffused over a large mass, it is less in each part, than if that part alone absorbed the whole quantum of heat.—Ed.
This general law or form has been well illustrated by Newton’s discovery of the decomposition of colors.
I.e., the common link or form which connects the various kinds of natures, such as the different hot or red natures enumerated above.—See Aphorism iii. part 2.
This is erroneous—all metals expand considerably when heated.
“Quid ipsum,” the τὸ τὶ ἦν εἶναι of Aristotle.
To show the error of the text, we need only mention the case of water, which, when confined in corked vases, and exposed to the action of a freezing atmosphere, is sure to swell out and break those vessels which are not sufficiently large to contain its expanded volume. Megalotti narrates a hundred other instances of a similar character.—Ed.
Bacon’s inquisition into the nature of heat, as an example of the mode of interpreting nature, cannot be looked upon otherwise than as a complete failure. Though the exact nature of this phenomenon is still an obscure and controverted matter, the science of thermotics now consists of many important truths, and to none of these truths is there so much as an approximation in Bacon’s process. The steps by which this science really advanced were the discovery of a measure of a heat or temperature, the establishment of the laws of conduction and radiation, of the laws of specific heat, latent heat, and the like. Such advances have led to Ampère’s hypothesis, that heat consists in the vibrations of an imponderable fluid; and to Laplace’s theory, that temperature consists in the internal radiation of a similar medium. These hypotheses cannot yet be said to be even probable, but at least they are so modified as to include some of the preceding laws which are firmly established, whereas Bacon’s “form,” or true definition of heat, as stated in the text, includes no laws of phenomena, explains no process, and is indeed itself an example of illicit generalization.
In all the details of his example of heat he is unfortunate. He includes in his collection of instances, the hot tastes of aromatic plants, the caustic effects of acids, and many other facts which cannot be ascribed to heat without a studious laxity in the use of the word.—Ed.
By this term Bacon understands general phenomena, taken in order from the great mass of indiscriminative facts, which, as they lie in nature, are apt to generate confusion by their number, indistinctness and complication. Such classes of phenomena, as being peculiarly suggestive of causation, he quaintly classes under the title of prerogative inquiries, either seduced by the fanciful analogy, which such instances bore to the prerogativa centuria in the Roman Comitia, or justly considering them as Herschel supposes to hold a kind of prerogative dignity from being peculiarly suggestive of causation.
Two high authorities in physical science (v. Herschel, Nat. Phil., art.192; Whewell’s Philosophy of the Inductive Sciences, vol.ii. p.243) pronounce these instances of little service in the task of induction, being for the most part classed not according to the ideas which they involve, or to any obvious circumstance in the facts of which they consist, but according to the extent and manner of their influence upon the inquiry in which they are employed. Thus we have solitary instances, migrating instances, ostensive instances, clandestine instances, so termed according to the degree in which they exhibit, or seem to exhibit, the property, whose nature we would examine. We have guide-post instances, crucial instances, instances of the parted road, of the doorway, of the lamp, according to the guidance they supply to our advance. Whewell remarks that such a classification is much of the same nature as if, having to teach the art of building, we were to describe tools with reference to the amount and place of the work which they must do, instead of pointing out their construction and use; as if we were to inform the pupil that we must have tools for lifting a stone up, tools for moving it sidewise, tools for laying it square, and tools for cementing it firmly. The means are thus lost in the end, and we reap the fruits of unmethodical arrangement in the confusion of cross division. In addition, all the instances are leavened with the error of confounding the laws with the causes of phenomena, and we are urged to adopt the fundamental error of seeking therein the universal agents, or general causes of phenomena, without ascending the gradual steps of intermediate laws.—Ed.
Of these nine general heads no more than the first is prosecuted by the author.
This very nearly approaches to Sir I. Newton’s discovery of the decomposition of light by the prism.
The mineral kingdom, as displaying the same nature in all its gradations, from the shells so perfect in structure in limestone to the finer marbles in which their nature gradually disappears, is the great theatre for instances of migration.—Ed.
Bacon was not aware of the fact since brought to light by Römer, that down to fourteen fathoms from the earth’s mean level the thermometer remains fixed at the tenth degree, but that as the thermometer descends below that depth the heat increases in a ratio proportionate to the descent, which happens with little variation in all climates. Buffon considers this a proof of a central fire in our planet.—Ed.
All the diversities of bodies depend upon two principles, i.e., the quantity and the position of the elements that enter into their composition. The primary difference is not that which depends on the greatest or least quantity of material elements, but that which depends on their position. It was the quick perception of this truth that made Leibnitz say that to complete mathematics it was necessary to join to the analysis of quantity the analysis of position.—Ed.
Query?
The real cause of this phenomenon is the attraction of the surface-water in the vessel by the sides of the bubbles. When the bubbles approach, the sides nearest each other both tend to raise the small space of water between them, and consequently less water is raised by each of these nearer sides than by the exterior part of the bubble, and the greater weight of the water raised on the exterior parts pushes the bubbles together. In the same manner a bubble near the side of a vessel is pushed toward it; the vessel and bubble both drawing the water that is between them. The latter phenomenon cannot be explained on Bacon’s hypothesis.
Modern discoveries appear to bear out the sagacity of Bacon’s remark, and the experiments of Baron Cagnard may be regarded as a first step toward its full demonstration. After the new facts elicited by that philosopher, there can be little doubt that the solid, liquid and aëriform state of bodies are merely stages in a progress of gradual transition from one extreme to the other, and that however strongly marked the distinctions between them may appear, they will ultimately turn out to be separated by no sudden or violent line of demarcation, but slide into each other by imperceptible gradations. Bacon’s suggestion, however, is as old as Pythagoras, and perhaps simultaneous with the first dawn of philosophic reason. The doctrine of the reciprocal transmutation of the elements underlies all the physical systems of the ancients, and was adopted by the Epicureans as well as the Stoics. Ovid opens his last book of the Metamorphoses with the poetry of the subject, where he expressly points to the hint of Bacon:—
——“Tenuatus in auras
Aëraque humor abit, etc., etc.
Inde retro redeunt, idemque retexitur ordo.”—xv. 246–249.
and Seneca, in the third book of his Natural Philosophy, quest.iv., states the opinion in more precise language than either the ancient bard or the modern philosopher.—Ed.
The author’s own system of Memoria Technica may be found in the De Augmentis, chap.xv. We may add that, notwithstanding Bacon’s assertion that he intended his method to apply to religion, politics, and morals, this is the only lengthy illustration he has adduced of any subject out of the domain of physical science.—Ed.
The collective instances here meant are no other than general facts or laws of some degree of generality, and are themselves the result of induction. For example, the system of Jupiter, or Saturn with its satellites, is a collective instance, and materially assisted in securing the admission of the Copernican system. We have here in miniature, and displayed at one view, a system analogous to that of the planets about the sun, of which, from the circumstance of our being involved in it, and unfavorably situated for seeing it otherwise than in detail, we are incapacitated from forming a general idea, but by slow and progressive efforts of reason.
But there is a species of collective instance which Bacon does not seem to have contemplated, in which particular phenomena are presented in such numbers at once, as to make the induction of their law a matter of ocular inspection. For example, the parabolic form assumed by a jet of water spouted out of a hole is a collective instance of the velocities and directions of the motions of all the particles which compose it seen together, and which thus leads us without trouble to recognize the law of the motion of a projectile. Again, the beautiful figures exhibited by sand strewed on regular plates of glass or metal set in vibration, are collective instances of an infinite number of points which remain at rest while the remainder of the plate vibrates, and in consequence afford us an insight into the law which regulates their arrangement and sequence throughout the whole surface. The richly colored lemniscates seen around the optic axis of crystals exposed to polarized light afford a striking instance of the same kind, pointing at once to the general mathematical expression of the law which regulates their production. Such collective instances as these lead us to a general law by an induction which offers itself spontaneously, and thus furnish advanced posts in philosophical exploration. The laws of Kepler, which Bacon ignored on account of his want of mathematical taste, may be cited as a collective instance. The first is, that the planets move in elliptical orbits, having the sun for their common focus. The second, that about this focus the radius vector of each planet describes equal areas in equal times. The third, that the squares of the periodic times of the planets are as the cubes of their mean distance from the sun. This collective instance “opened the way” to the discovery of the Newtonian law of gravitation.—Ed.
Is not this very hasty generalization? Do serpents move with four folds only? Observe also the motion of centipedes and other insects.
Shaw states another point of difference between the objects cited in the text—animals having their roots within, while plants have theirs without; for their lacteals nearly correspond with the fibres of the roots in plants; so that animals seem nourished within themselves as plants are without.—Ed.
Bacon falls into an error here in regarding the syllogism as something distinct from the reasoning faculty, and only one of its forms. It is not generally true that the syllogism is only a form of reasoning by which we unite ideas which accord with the middle term. This agreement is not even essential to accurate syllogisms; when the relation of the two things compared to the third is one of equality or similitude, it of course follows that the two things compared may be pronounced equal, or like to each other. But if the relation between these terms exist in a different form, then it is not true that the two extremes stand in the same relation to each other as to the middle term. For instance, if A is double of B, and B double of C, then A is quadruple of C. But then the relation of A to C is different from that of A to B and of B to C.—Ed.
Comparative anatomy is full of analogies of this kind. Those between natural and artificial productions are well worthy of attention, and sometimes lead to important discoveries. By observing an analogy of this kind between the plan used in hydraulic engines for preventing the counter-current of a fluid, and a similar contrivance in the blood vessels, Harvey was led to the discovery of the circulation of the blood.—Ed.
This is well illustrated in plants, for the gardener can produce endless varieties of any known species, but can never produce a new species itself.
The discoveries of Tournefort have placed moss in the class of plants. The fish alluded to below are to be found only in the tropics.—Ed.
There is, however, no real approximation to birds in either the flying fish or bat, any more than a man approximates to a fish because he can swim. The wings of the flying fish and bat are mere expansions of skin, bearing no resemblance whatever to those of birds.—Ed.
Seneca was a sounder astronomer than Bacon. He ridiculed the idea of the motion of any heavenly bodies being irregular, and predicted that the day would come, when the laws which guided the revolution of these bodies would be proved to be identical with those which controlled the motions of the planets. The anticipation, was realized by Newton.—Ed.
But see Bacon’s own corollary at the end of the Instances of Divorce, Aphorism xxxvii. If Bacon’s remark be accepted, the censure will fall upon Newton and the system so generally received at the present day. It is, however, unjust, as the centre of which Newton so often speaks is not a point with an active inherent force, but only the result of all the particular and reciprocal attractions of the different parts of the planet acting upon one spot. It is evident, that if all these forces were united in this centre, that the sum would be equal to all their partial effects.—Ed.
Since Newton’s discovery of the law of gravitation, we find that the attractive force of the earth must extend to an infinite distance. Bacon himself alludes to the operation of this attractive force at great distances in the Instances of the Rod, Aphorism xlv.
Snow reflects light, but is not a source of light.
Bacon’s sagacity here foreshadows Newton’s theory of the tides.
The error in the text arose from Bacon’s impression that the earth was immovable. It is evident, since gravitation acts at an infinite distance, that no such point could be found; and even supposing the impossible point of equilibrium discovered, the body could not maintain its position an instant, but would be hurried, at the first movement of the heavenly bodies, in the direction of the dominant gravitating power.—Ed.
Fly clocks are referred to in the text, not pendulum clocks, which were not known in England till 1662. The former, though clumsy and rude in their construction, still embodied sound mechanical principles. The comparison of the effect of a spring with that of a weight in producing certain motions in certain times on altitudes and in mines, has recently been tried by Professors Airy and Whewell in Dalcoath mine, by means of a pendulum, which is only a weight moved by gravity, and a chronometer balance moved and regulated by a spring. In his thirty-seventh Aphorism, Bacon also speaks of gravity as an incorporeal power, acting at a distance, and requiring time for its transmission; a consideration which occurred at a later period to Laplace in one of his most delicate investigations.
Crucial instances, as Herschel remarks, afford the readiest and securest means of eliminating extraneous causes, and deciding between the claims of rival hypotheses; especially when these, running parallel to each other, in the explanation of great classes of phenomena, at length come to be placed at issue upon a single fact. A curious example is given by M.Fresnel, as decisive in his mind of the question between the two great theories on the nature of light, which, since the time of Newton and Huyghens, have divided philosophers. When two very clean glasses are laid one on the other, if they be not perfectly flat, but one or both, in an almost imperceptible degree, convex or prominent, beautiful and vivid colors will be seen between them; and if these be viewed through a red glass, their appearance will be that of alternate dark and bright stripes. These stripes are formed between the two surfaces in apparent contact, and being applicable on both theories, are appealed to by their respective supporters as strong confirmatory facts; but there is a difference in one circumstance, according as one or other theory is employed to explain them. In the case of the Huyghenian theory, the intervals between the bright stripes ought to appear absolutely black, when a prism is used for the upper glass, in the other half bright. This curious case of difference was tried, as soon as the opposing consequences of the two theories were noted by M.Fresnel, and the result is stated by him to be decisive in favor of that theory which makes light to consist in the vibrations of an elastic medium.—Ed.
Bacon plainly, from this passage, was inclined to believe that the moon, like the comets, was nothing more than illuminated vapor. The Newtonian law, however, has not only established its solidity, but its density and weight. A sufficient proof of the former is afforded by the attraction of the sea, and the moon’s motion round the earth.—Ed.
Rather the refraction; the sky or air, however, reflects the blue rays of light.
The polished surface of the glass causes the reflection in this case, and not the air; and a hat or other black surface put behind the window in the daytime will enable the glass to reflect distinctly for the same reason, namely, that the reflected rays are not mixed and confused with those transmitted from the other side of the window.
These instances, which Bacon seems to consider as a great discovery, are nothing more than disjunctive propositions combined with dilemmas. In proposing to explain an effect, we commence with the enumeration of the different causes which seem connected with its production; then with the aid of one or more dilemmas, we eliminate each of the phenomena accidental to its composition, and conclude with attributing the effect to the residue. For instance, a certain phenomenon(a) is produced either by phenomenon(B) or phenomenon(C); but C cannot be the cause of a, for it is found in D, E, F, neither of which are connected with a. Then the true cause of phenomenon(a) must be phenomenon(B).
This species of reasoning is liable to several paralogisms, against which Bacon has not guarded his readers, from the very fact that he stumbled into them unwittingly himself. The two principal ones are false exclusions and defective enumerations. Bacon, in his survey of the causes which are able to concur in producing the phenomena of the tides, takes no account of the periodic melting of the Polar ice, or the expansion of water by the solar heat; nor does he fare better in his exclusions. For the attraction of the planets and the progression and retrograde motion communicated by the earth’s diurnal revolution, can plainly affect the sea together, and have a simultaneous influence on its surface.
Bacon is hardly just or consistent in his censure of Ramus; the end of whose dichotomy was only to render reasoning by dilemma, and crucial instances, more certain in their results, by reducing the divisions which composed their parts to two sets of contradictory propositions. The affirmative or negative of one would then necessarily have led to the acceptance or rejection of the other.—Ed.
Père Shenier first pointed out the spots on the sun’s disk, and by the marks which they afforded him, computed its revolution to be performed in twenty-five days and some hours.—Ed.
Rust is now well known to be a chemical combination of oxygen with the metal, and the metal when rusty acquires additional weight. His theory as to the generation of animals, is deduced from the erroneous notion of the possibility of spontaneous generation (as it was termed). See the next paragraph but one.
“Limus ut hic durescit, et hæc ut cera liquescit
Uno eodemque igni.”—Virg.Ecl. viii.
See Table of Degrees, No.38.
Riccati, and all modern physicists, discover some portion of light in every body, which seems to confirm the passage in Genesis that assigns to this substance priority in creation.—Ed.
As instances of this kind, which the progress of science since the time of Bacon affords, we may cite the air-pump and the barometer, for manifesting the weight and elasticity of air: the measurement of the velocity of light, by means of the occultation of Jupiter’s satellites and the aberration of the fixed stars: the experiments in electricity and galvanism, and in the greater part of pneumatic chemistry. In all these cases scientific facts are elicited, which sense could never have revealed to us.—Ed.
The itinerant instances, as well as frontier instances, are cases in which we are enabled to trace the general law of continuity which seems to pervade all nature, and which has been aptly embodied in the sentence, “natura non agit per saltum.” The pursuit of this law into phenomena where its application is not at first sight obvious, has opened a mine of physical discovery, and led us to perceive an intimate connection between facts which at first seemed hostile to each other. For example, the transparency of gold-leaf, which permits a bluish-green light to pass through it, is a frontier instance between transparent and opaque bodies, by exhibiting a body of the glass generally regarded the most opaque in nature, as still possessed of some slight degree of transparency. It thus proves that the quality of opacity is not a contrary or antagonistic quality to that of transparency, but only its extreme lowest degree.
Alluding to his theory of atoms.
Observe the approximation to Newton’s theory. The same notion repeated still more clearly in the ninth motion. Newton believed that the planets might so conspire as to derange the earth’s annual revolution, and to elongate the line of the apsides and ellipsis that the earth describes in its annual revolution round the sun. In the supposition that all the planets meet on the same straight line, Venus and Mercury on one side of the sun, and the earth, moon, Mars, Jupiter and Saturn on the side diametrically opposite; then Saturn would attract Jupiter, Jupiter Mars, Mars the moon, which must in its turn attract the earth in proportion to the force with which it was drawn out of its orbit. The result of this combined action on our planet would elongate its ecliptic orbit, and so far draw it from the source of heat, as to produce an intensity of cold destructive to animal life. But this movement would immediately cease with the planetary concurrence which produced it, and the earth, like a compressed spring, bound almost as near to the sun as she had been drawn from it, the reaction of the heat on its surface being about as intense as the cold caused by the first removal was severe. The earth, until it gained its regular track, would thus alternately vibrate between each side of its orbit, with successive changes in its atmosphere, proportional to the square of the variation of its distance from the sun. In no place is Bacon’s genius more conspicuous than in these repeated guesses at truth. He would have been a strong Copernican, had not Gilbert defended the system.—Ed.
This is not true except when the projectile acquires greater velocity at every successive instant of its course, which is never the case except with falling bodies. Bacon appears to have been led into the opinion from observing that gunshots pierce many objects at a distance from which they rebound when brought within a certain proximity of contact. This apparent inconsistency, however, arises from the resistance of the parts of the object, which velocity combined with force is necessary to overcome.—Ed.
This passage shows that the pressure of the external atmosphere, which forces the water into the egg, was not in Bacon’s time understood.—Ed.
We have already alluded, in a note prefixed to the same aphorism of the first book, to Newton’s error of the absolute lightness of bodies. In speaking again of the volatile or spiritual substances (Aph.xl. b.ii.) which he supposed with the Platonists and some of the schoolmen to enter into the composition of every body, he ascribes to them a power of lessening the weight of the material coating in which he supposes them inclosed. It would appear from these passages and the text that Bacon had no idea of the relative density of bodies, and the capability which some have to diminish the specific gravity of the heavier substances by the dilation of their parts; or if he had, the reveries in which Aristotle indulged in treating of the soul, about the appetency of bodies to fly to kindred substances—flame and spirit to the sky, and solid opaque substances to the earth, must have vitiated his mind.—Ed.
Römer, a Danish astronomer, was the first to demonstrate, by connecting the irregularities of the eclipses of Jupiter’s satellites with their distances from the earth, the necessity of time for the propagation of light. The idea occurred to Dominic Cassini as well as Bacon, but both allowed the discovery to slip out of their hands.—Ed.
The author in the text confounds inertness, which is a simple indifference of bodies to action, with gravity, which is a force acting always in proportion to their density. He falls into the same error further on.—Ed.
The experiments of the last two classes of instances are considered only in relation to practice, and Bacon does not so much as mention their infinitely greater importance in the theoretical part of induction. The important law of gravitation in physical astronomy could never have been demonstrated but by such observations and experiments as assigned accurate geometrical measures to the quantities compared. It was necessary to determine with precision the demi-diameter of the earth, the velocity of falling bodies at its surface, the distance of the moon, and the speed with which she describes her orbit, before the relation could be discovered between the force which draws a stone to the ground and that which retains the moon in her sphere.
In many cases the result of a number of particular facts, or the collective instances rising out of them, can only be discovered by geometry, which so far becomes necessary to complete the work of induction. For instance, in the case of optics, when light passes from one transparent medium to another, it is refracted, and the angle which the ray of incidence makes with the superficies which bounds the two media determines that which the refracted ray makes with the same superficies. Now, all experiment can do for us in this case is, to determine for any particular angle of incidence the corresponding angle of refraction. But with respect to the general rule which in every possible case deduces one of these angles from the other, or expresses the constant and invariable relation which subsists between them, experiment gives no direct information. Geometry must, consequently, be called in, which, when a constant though unknown relation subsists between two angles, or two variable qualities of any kind, and when an indefinite number of values of those quantities are assigned, furnishes infallible means of discovering that unknown relation either accurately or by approximation. In this way it has been found, when the two media remain the same, the cosines of the above-mentioned angles have a constant ratio to each other. Hence, when the relations of the simple elements of phenomena are discovered to afford a general rule which will apply to any concrete case, the deductive method must be applied, and the elementary principles made through its agency to account for the laws of their more complex combinations. The reflection and refraction of light by the rain falling from a cloud opposite to the sun was thought, even before Newton’s day, to contain the form of the rainbow. This philosopher transformed a probable conjecture into a certain fact when he deduced from the known laws of reflection and refraction the breadth of the colored arch, the diameter of the circle of which it is a part, and the relation of the latter to the place of the spectator and the sun. Doubt was at once silenced when there came out of his calculus a combination of the same laws of the simple elements of optics answering to the phenomena in nature.—Ed.
As far as this motion results from attraction and repulsion, it is only a simple consequence of the last two.—Ed.
These two cases are now resolved into the property of the capillary tubes and present only another feature of the law of attraction.—Ed.
This is one of the most useful practical methods in chemistry at the present day.
See Aphorism xxv.
Query?
Observe this approximation to Newton’s theory.
Those differences which are generated by the masses and respective distances of bodies are only differences of quantity, and not specific; consequently those three classes are only one.—Ed.
See the citing instances, Aphorism xl.
Aristotle’s doctrine, that sound takes place when bodies strike the air, which the modern science of acoustics has completely established, was rejected by Bacon in a treatise upon the same subject: “The collision or thrusting of air,” he says, “which they will have to be the cause of sound, neither denotes the form nor the latent process of sound, but is a term of ignorance and of superficial contemplation.” To get out of the difficulty, he betook himself to his theory of spirits, a species of phenomena which he constantly introduces to give himself the air of explaining things he could not understand, or would not admit upon the hypothesis of his opponents.—Ed.
The motion of trepidation, as Bacon calls it, was attributed by the ancient astronomers to the eight spheres, relative to the precession of the equinoxes. Galileo was the first to observe this kind of lunar motion.—Ed.
Part of the air is expanded and escapes, and part is consumed by the flame. When condensed, therefore, by the cold application, it cannot offer sufficient resistance to the external atmosphere to prevent the liquid or flesh from being forced into the glass.
Heat can now be abstracted by a very simple process, till the degree of cold be of almost any required intensity.—Ed.
It is impossible to compare a degree of heat with a degree of cold, without the assumption of some arbitrary test, to which the degrees are to be referred. In the next sentence Bacon appears to have taken the power of animal life to support heat or cold as the test, and then the comparison can only be between the degree of heat or of cold that will produce death.
The zero must be arbitrary which divides equally a certain degree of heat from a certain degree of cold.—Ed.
It may often be observed on the leaves of the lime and other trees.
Transcriber’s Notes
Some page numbers do not appear due to removed blank pages.
The list of Contents was added for reader’s convenience.
Punctuation errors were corrected.
Inconsistent hyphenation was retained.
“De la Lande” and “La Lande” both appear in text and were retained.
“Shenier”, editor’s spelling of “Scheiner”, was retained.
On p. 37, a paragraph break was inserted before "There is a clear example ...".
On p. 44, “the” was changed from “The” (the usual method).
On p. 85, “that” was changed from “That” (that a species).
On p. 86, “that” was changed from “That” (that an instrument).
On p. 118, “aëriform” was changed from “aeriform”.
On p. 178, “borrow” was changed from “brorrow”.
On p. 204, “sufficiently” was changed from “sufficietly”.
On p. 219, “quantity” was changed from “quanity” (quantity of gold).
In footnote , “psychological” was changed from “pyschological”.
In footnote , “that” was changed from “tha”.
In footnote , "72" was changed from “22”.
In footnote , “ix.” was changed from “x.”.
In footnote , “οὐσία” was changed from “οὐδία”.
In footnote , “necessary” was changed from “necesary”.
In footnote , “liquor” was changed from “liqour” (the liquor rose).
In footnote , “the” was changed from “th” (by the flame).
END OF THE PROJECT GUTENBERG EBOOK 45988 |
12060 | https://people.cs.nycu.edu.tw/~tsaiwn/introcs/03_Labs/Lab04/pascal.htm | Pascal's triangle
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The first five rows of Pascal's triangle
In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after Blaise Pascal in much of the western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row zero, and the numbers in odd rows are usually staggered relative to the numbers in even rows. A simple construction of the triangle proceeds in the following manner. On the zeroth row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.
Each number in the triangle is the sum of the two directly above it.
This construction is related to the binomial coefficients by Pascal's rule, which states that if
is the kth binomial coefficient in the binomial expansion of (x + y)n, where n! is the factorial of n, then
for any nonnegative integer n and any integer k between 0 and n.
Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices �X see also pyramid, tetrahedron, and simplex.
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| Contents 1 History 2 The triangle 3 Pascal's triangle and binomial expansions 4 Patterns and properties + 4.1 The diagonals + 4.2 Other patterns and properties + 4.3 More subtle patterns + 4.4 The matrix exponential + 4.5 Geometric properties + 4.6 Calculating an individual row 5 References 6 See also 7 External links |
History
Yang Hui (Pascal's) triangle, as depicted by the Chinese using rod numerals.
Blaise Pascal's version of the triangle
The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an ancient Indian book on Sanskrit prosody written by Pingala between the 5th�V2nd centuries BC. While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers. The Indian mathematician Bhattotpala and his pupil Dhruv Ragunathan (c. 1068) later gives rows 0-16 of the triangle.
At around the same time, it was discussed in Persia (Iran) by the mathematician Al-Karaji (953�V1029) and the poet-astronomer-mathematician Omar Khayy��m (1048-1131); thus the triangle is referred to as the "Khayyam triangle" in Iran. Several theorems related to the triangle were known, including the binomial theorem. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
In 13th century, Yang Hui (1238-1298) presented the arithmetic triangle, which was the same as Pascal's Triangle. Today Pascal's triangle is called "Yang Hui's triangle" in China.
Petrus Apianus (1495-1552) published the triangle on the frontispiece of his book on business calculations 1531/32 and an earlier version in 1527 the first record of it in Europe.
In Italy, it is referred to as "Tartaglia's triangle", named for the Italian algebraist Niccol�� Fontana Tartaglia (1500-1577); Tartaglia is credited with the general formula for solving cubic polynomials (which may be really from Scipione del Ferro but was published by Gerolamo Cardano 1545).
Finally, in 1655, Blaise Pascal wrote a Trait�� du triangle arithm��tique (Treatise on arithmetical triangle), wherein he collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) and Abraham de Moivre (1730).
The triangle
Below are rows zero to sixteen of Pascal's triangle:
Pascal's triangle and binomial expansions
Pascal's triangle determines the coefficients which arise in binomial expansions. For an example, consider the expansion
: (x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2.
Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like x + y is raised to a positive integer power we have:
: (x + y)n = a0xn + a1xn?1y + a2xn?2y2 + �K + an?1xyn?1 + anyn,
where the coefficients ai in this expansion are precisely the numbers on row n of Pascal's triangle. In other words,
This is the binomial theorem.
Notice that entire right diagonal of Pascal's triangle corresponds to the coefficient of yn in these binomial expansions, while the next diagonal corresponds to the coefficient of xyn-1 and so on.
To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of (x + 1)n+1 in terms of the corresponding coefficients of (x + 1)n (setting y = 1 for simplicity). Suppose then that
Now
The two summations can be reorganized as follows:
(because of how raising a polynomial to a power works, a0 = an = 1).
We now have an expression for the polynomial (x + 1)n+1 in terms of the coefficients of (x + 1)n (these are the ais), which is what we need if we want to express a line in terms of the line above it. Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of x, and that the a-terms are the coefficients of the polynomial (x + 1)n, and we are determining the coefficients of (x + 1)n+1. Now, for any given i not 0 or n + 1, the coefficient of the xi term in the polynomial (x + 1)n+1 is equal to ai (the figure above and to the left of the figure to be determined, since it is on the same diagonal) + ai?1 (the figure to the immediate right of the first figure). This is indeed the simple rule for constructing Pascal's triangle row-by-row.
It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem.
An interesting consequence of the binomial theorem is obtained by setting both variables x and y equal to one. In this case, we know that (1 + 1)n = 2n, and so
In other words, the sum of the entries in the nth row of Pascal's triangle is the nth power of 2.
Patterns and properties
Pascal's triangle has many properties and contains many patterns of numbers.
The diagonals
Some simple patterns are immediately apparent in the diagonals of Pascal's triangle:
The diagonals going along the left and right edges contain only 1's.
The diagonals next to the edge diagonals contain the natural numbers in order.
Moving inwards, the next pair of diagonals contain the triangular numbers in order.
The next pair of diagonals contain the tetrahedral numbers in order, and the next pair give pentatope numbers. In general, each next pair of diagonals contains the next higher dimensional "d-simplex" numbers, which can be defined as
An alternative formula is as follows:
The geometric meaning of a function trid is: trid(1) = 1 for all d. Construct a d-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to trid(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find trid(x), have a total of x dots composing the target shape. trid(x) then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore tri1(x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to trid ? 1(x).
Sierpinski triangle
Other patterns and properties
The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called Sierpinski triangle, and this resemblance becomes more and more accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern is the Sierpinski triangle. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other patterns and combinations.
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| Pascal's triangle with odd numbers shaded | Pascal's triangle with numbers not divisible by 3 shaded |
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| Pascal's triangle with numbers not divisible by 4 shaded | Pascal's triangle with numbers not divisible by 5 shaded |
Imagine each number in the triangle is a node in a grid which is connected to the adjacent numbers above and below it. Now for any node in the grid, count the number of paths there are in the grid (without backtracking) which connect this node to the top node (1) of the triangle. The answer is the Pascal number associated to that node. The interpretation of the number in Pascal's Triangle as the number of paths to that number from the tip means that on a Plinko game board shaped like a triangle, the probability of winning prizes nearer the center will be higher than winning prizes on the edges.
The value of each row, if each number in it is considered as a decimal place and numbers larger than 9 are carried over accordingly, is a power of 11 (specifically, 11n, where n is the number of the row). For example, row two reads '1, 2, 1', which is 112 (121). In row five, '1, 5, 10, 10, 5, 1' is translated to 161051 after carrying the values over, which is 115. This property is easily explained by setting x = 10 in the binomial expansion of (x + 1)row number, and adjusting the values to fit in the decimal number system.
More subtle patterns
There are also more surprising, subtle patterns. From a single element of the triangle, a more shallow diagonal line can be formed by continually moving one element to the right, then one element to the bottom-right, or by going in the opposite direction. An example is the line with elements 1, 6, 5, 1, which starts from the row 1, 3, 3, 1 and ends three rows down. Such a "diagonal" has a sum that is a Fibonacci number. In the case of the example, the Fibonacci number is 13:
1 1 1 1 2 1 1 �� 3 �� 3 1 1 4 ��6 �� 4 �� 1 1 5 10 10 ��5 �� 1 �� 1 �� 6 �� 15 20 15 6 ��1 1 7 ��21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1
The second highlighted diagonal has a sum of 233. The numbers 'skipped over' between the move right and the move down-right also sum to Fibonacci numbers, being the numbers 'between' the sums formed by the first construction. For example, the numbers skipped over in the first highlighted diagonal are 3, 4 and 1, making 8.
In addition, if row m is taken to indicate row (n + 1), the sum of the squares of the elements of row m equals the middle element of row (2m ? 1). For example, 12 + 42 + 62 + 42 + 12 = 70. In general form:
Another interesting pattern is that on any row m, where m is odd, the middle term minus the term two spots to the left equals a Catalan number, specifically the (m + 1)/2 Catalan number. For example: on row 5, 6 ? 1 = 5, which is the 3rd Catalan number, and (5 + 1)/2 = 3.
Also, the sum of the elements of row m is equal to 2m?1. For example, the sum of the elements of row 5 is 1 + 4 + 6 + 4 + 1 = 16, which is equal to 24 = 16. This follows from the binomial theorem proved above, applied to (1 + 1)m?1.
Some of the numbers in Pascal's triangle correlate to numbers in Lozani?'s triangle.
Another interesting property of Pascal's triangle is that in rows where the second number (the 1st number following 1) is prime, all the terms in that row except the 1s are multiples of that prime.
Binomial matrix as matrix exponential (illustration for 5��5 matrices). All the dots represent 0.
The matrix exponential
: See also: Pascal matrix
Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the matrix exponential can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, �K on its subdiagonal and zero everywhere else.
Geometric properties
Pascal's triangle can be used as a lookup table for the number of arbitrarily dimensioned elements within a single arbitrarily dimensioned version of a triangle (known as a simplex). For example, consider the 3rd line of the triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements (vertices, or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a tetrahedron has one 3-dimensional element (itself), four 2-dimensional elements (faces), six 1-dimensional elements (edges), and four 0-dimensional elements (vertices). Adding the final 1 again, these values correspond to the 4th row of the triangle (1, 4, 6, 4, 1). Line 1 corresponds to a point, and Line 2 corresponds to a line segment (dyad). This pattern continues to arbitrarily high-dimensioned hyper-tetrahedrons (simplices).
To understand why this pattern exists, one must first understand that the process of building an n-simplex from an (n ? 1)-simplex consists of simply adding a new vertex to the latter, positioned such that this new vertex lies outside of the space of the original simplex, and connecting it to all original vertices. As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 1 face, 3 edges, and 3 vertices (the meaning of the final 1 will be explained shortly). To build a tetrahedron from a triangle, we position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle.
The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements, each of which is built upon elements of one fewer dimension from the original triangle. Thus, in the tetrahedron, the number of cells (polyhedral elements) is 0 (the original triangle possesses none) + 1 (built upon the single face of the original triangle) = 1; the number of faces is 1 (the original triangle itself) + 3 (the new faces, each built upon an edge of the original triangle) = 4; the number of edges is 3 (from the original triangle) + 3 (the new edges, each built upon a vertex of the original triangle) = 6; the number of new vertices is 3 (from the original triangle) + 1 (the new vertex that was added to create the tetrahedron from the triangle) = 4. This process of summing the number of elements of a given dimension to those of one fewer dimension to arrive at the number of the former found in the next higher simplex is equivalent to the process of summing two adjacent numbers in a row of Pascal's triangle to yield the number below. Thus, the meaning of the final number (1) in a row of Pascal's triangle becomes understood as representing the new vertex that is to be added to the simplex represented by that row to yield the next higher simplex represented by the next row. This new vertex is joined to every element in the original simplex to yield a new element of one higher dimension in the new simplex, and this is the origin of the pattern found to be identical to that seen in Pascal's triangle.
A similar pattern is observed relating to squares, as opposed to triangles. To find the pattern, one must construct an analog to Pascal's triangle, whose entries are the coefficients of (x + 2)Row Number, instead of (x + 1)Row Number. There are a couple ways to do this. The simpler is to begin with Row 0 = 1 and Row 1 = 1, 2. Proceed to construct the analog triangles according to the following rule:
That is, choose a pair of numbers according to the rules of Pascal's triangle, but double the one on the left before adding. This results in:
1 1 2 1 4 4 1 6 12 8 1 8 24 32 16 1 10 40 80 80 32 1 12 60 160 240 192 64 1 14 84 280 560 672 448 128
The other way of manufacturing this triangle is to start with Pascal's triangle and multiply each entry by 2k, where k is the position in the row of the given number. For example, the 2nd value in row 4 of Pascal's triangle is 6 (the slope of 1s corresponds to the zeroth entry in each row). To get the value that resides in the corresponding position in the analog triangle, multiply 6 by 2Position Number = 6 �� 22 = 6 �� 4 = 24. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned cube (called a hypercube) can be read from the table in a way analogous to Pascal's triangle. For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to the next line of the analog triangle (1, 6, 12, 8). This pattern continues indefinitely.
To understand why this pattern exists, first recognize that the construction of an n-cube from an (n ? 1)-cube is done by simply duplicating the original figure and displacing it some distance (for a regular n-cube, the edge length) orthogonal to the space of the original figure, then connecting each vertex of the new figure to its corresponding vertex of the original. This initial duplication process is the reason why, to enumerate the dimensional elements of an n-cube, one must double the first of a pair of numbers in a row of this analog of Pascal's triangle before summing to yield the number below. The initial doubling thus yields the number of "original" elements to be found in the next higher n-cube and, as before, new elements are built upon those of one fewer dimension (edges upon vertices, faces upon edges, etc.). Again, the last number of a row represents the number of new vertices to be added to generate the next higher n-cube.
In this triangle, the sum of the elements of row m is equal to 3m ? 1. Again, to use the elements of row 5 as an example: 1 + 8 + 24 + 32 + 16 = 81, which is equal to 34 = 81.
Calculating an individual row
This algorithm is an alternative to the standard method of calculating individual cells with factorials. Starting at the left, the first cell's value is 1. For each cell after, the value is determined by multiplying the value to the left by a slowly changing fraction:
where r = row + 1, starting with 0 at the top, and c = the column, starting with 0 on the left. For example, to calculate row 5, r=6. The first value is 1. The next value is 1 x 5/1 = 5. The numerator decreases by one, and the denominator increases by one with each step. So 5 x 4/2 = 10. Then 10 x 3/3 = 10. Then 10 x 2/4 = 5. Then 5 x 1/5 = 1. Notice that the last cell always equals 1, the final multiplication is included for completeness of the series.
A similar pattern exists on a downward diagonal. Starting with the one and the natural number in the next cell, form a fraction. To determine the next cell, increase the numerator and denominator each by one, and then multiply the previous result by the fraction. For example, the row starting with 1 and 7 form a fraction of 7/1. The next cell is 7 x 8/2 = 28. The next cell is 28 x 9/3 = 84.
Note that for any individual row you only need to calculate half (rounded up) the number of values in the row. This is because the row is symmetrical.
References
^ The binomial coefficient is conventionally set to zero if k is either less than zero or greater than n.
See also
bean machine, Francis Galton's "quincunx"
Euler triangle
Floyd's triangle
Leibniz harmonic triangle
Multiplicities of entries in Pascal's triangle (Singmaster's conjecture)
Pascal matrix
Pascal's tetrahedron
External links
Wikimedia Commons has media related to:
Pascal's triangle
Eric W. Weisstein, Pascal's triangle at MathWorld.
The Old Method Chart of the Seven Multiplying Squares (from the Ssu Yuan Y�� Chien of Chu Shi-Chieh, 1303, depicting the first nine rows of Pascal's triangle)
Pascal's Treatise on the Arithmetic Triangle (page images of Pascal's treatise, 1655; summary: )
Earliest Known Uses of Some of the Words of Mathematics (P)
Leibniz and Pascal triangles
Dot Patterns, Pascal's Triangle, and Lucas' Theorem
Pascal's Triangle From Top to Bottom
Omar Khayyam the mathematician
Info on Pascal's Triangle
Explanation of Pascal's Triangle and common occurrences, including link to interactive version specifying # of rows to view
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Categories: Factorial and binomial topics | Eponyms | Blaise Pascal | Triangles of numbers
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In how many ways do 4 boys and 4 girls be arranged in a row such that no two boys and no two girls sit together?
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1y
To arrange 4 boys and 4 girls in a row such that no two boys and no two girls sit together, we can follow these steps:
Arrange the Boys and Girls Separately:
The boys can be arranged among themselves in 4!4! ways.
The girls can also be arranged among themselves in 4!4! ways.
Choose the Arrangement Pattern:
Since we want to avoid having two boys or two girls sitting together, we can start with either a boy or a girl. The two valid arrangements are:
Boy-Girl-Boy-Girl-Boy-Girl-Boy-Girl (BGBGBGBG)
Girl-Boy-Girl-Boy-Girl-Boy-Girl-Boy (GBGBGBGB)
Calculate the Total Arrangements:
Each arra
Continue Reading
To arrange 4 boys and 4 girls in a row such that no two boys and no two girls sit together, we can follow these steps:
Arrange the Boys and Girls Separately:
The boys can be arranged among themselves in 4!4! ways.
The girls can also be arranged among themselves in 4!4! ways.
Choose the Arrangement Pattern:
Since we want to avoid having two boys or two girls sitting together, we can start with either a boy or a girl. The two valid arrangements are:
Boy-Girl-Boy-Girl-Boy-Girl-Boy-Girl (BGBGBGBG)
Girl-Boy-Girl-Boy-Girl-Boy-Girl-Boy (GBGBGBGB)
Calculate the Total Arrangements:
Each arrangement pattern (BGBGBGBG or GBGBGBGB) allows for the boys and girls to be arranged independently.
Thus, the total number of arrangements is given by:
Total Arrangements=2×(4!)×(4!)Total Arrangements=2×(4!)×(4!)
Now, let's calculate 4!4!:
4!=24 4!=24
Thus, the total arrangements are:
Total Arrangements=2×24×24=2×576=1152 Total Arrangements=2×24×24=2×576=1152
Therefore, the total number of ways to arrange 4 boys and 4 girls in a row such that no two boys and no two girls sit together is 1152 1152.
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Related questions
More answers below
There are 4 boys and 4 girls. In how many ways can they be seated in a row so that all the girls do not sit together?
What are the number of ways in which 5 boys and 5 girls can be arranged in a row so that no two girls and no two boys are together?
In how many ways can 4 boys and 5 girls sit in a row?
In how many ways can six girls and two boys sit together on a bench so that there are always four girls sitting in between two boys?
In how many ways may 4 girls and 4 boys be arranged such that all girls are sitting together?
Divya Keshamoni
MS in Statistics (academic discipline), Texas Tech University (Graduated 2012)
·8y
Originally Answered: In how many ways can 4 boys and 3 girls arranged in a row if no two girls are together? ·
Hey there,
We have 4 boys and 3 girls arranged in a row such that no two girls are adjacent.
I would first start off with arranging/seating the boys . 4 boys can be seated in 4!4!ways. Thus 4 boys can be seated in 4!=4∗3∗2∗1=24 4!=4∗3∗2∗1=24 ways.
{Note: Order matters here. i.e. Seating arrangement of B1 B2 B3 B4 is different from B1 B3 B2 B4, where, B1,B2,B3,B4 are four different boys.}
So, this is a permutation and so 4 boys can be arranged in 4 places in 4 P 4 4 P 4 ways= 4!(4−4)!=4!0!=24.4!(4−4)!=4!0!=24. Thus the boys can be arranged in 4! = 24 ways. {Note: 0!=1 0!=1}
Now we have to place/arrange the girls such tha
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Hey there,
We have 4 boys and 3 girls arranged in a row such that no two girls are adjacent.
I would first start off with arranging/seating the boys . 4 boys can be seated in 4!4!ways. Thus 4 boys can be seated in 4!=4∗3∗2∗1=24 4!=4∗3∗2∗1=24 ways.
{Note: Order matters here. i.e. Seating arrangement of B1 B2 B3 B4 is different from B1 B3 B2 B4, where, B1,B2,B3,B4 are four different boys.}
So, this is a permutation and so 4 boys can be arranged in 4 places in 4 P 4 4 P 4 ways= 4!(4−4)!=4!0!=24.4!(4−4)!=4!0!=24. Thus the boys can be arranged in 4! = 24 ways. {Note: 0!=1 0!=1}
Now we have to place/arrange the girls such that no two girls are adjacent. So we can arrange them in 5 different places as shown below.
B_B_B_B
The first girl can be seated in any of the 5 5 places.
The second girl can be seated in any of the remaining 4 4 places.
The third girl can be seated in any of the remaining 3 3 places.
So, the number of ways that the girls can be seated: 5∗4∗3=60 5∗4∗3=60
This can also be done as follows: There are 5 places available for 3 girls to be seated and the order of seating matters. Thus the 3 girls can be seated in 5 places in 5 P 3 5 P 3 ways = 5!(5−3)!=5!2!=60 5!(5−3)!=5!2!=60
The total number of ways that 4 boys and 3 girls can be arranged in a row such that no two girls are adjacent : 4 P 4∗5 P 3=24∗60=1440 4 P 4∗5 P 3=24∗60=1440 ways.
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Sourav Behera
Studied at Ispat English Medium School, Rourkela
·7y
Originally Answered: There are 4 boys and 4 girls. In how many ways can they be seated in a row so that all the girls do not sit together? ·
There are 8 persons in total (4 boys + 4 girls)
So the number of ways in which 8 persons can be arranged is 8! ways.
Let us consider the 4 girls as a single person so there are now 5 persons in total (4 boys + 4 girls as one group or one person). So the number of ways in which 5 persons can be arranged is 5! ways and also the 4 girls can sit among themselves in 4! way so the total number of ways of seating this group of 5 is 5!×4! ways .
So the total number of ways in which 4 boys and 4 girls can be seated so that not all the girls sit together is given by= (Total no. of ways in which all 8 can s
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There are 8 persons in total (4 boys + 4 girls)
So the number of ways in which 8 persons can be arranged is 8! ways.
Let us consider the 4 girls as a single person so there are now 5 persons in total (4 boys + 4 girls as one group or one person). So the number of ways in which 5 persons can be arranged is 5! ways and also the 4 girls can sit among themselves in 4! way so the total number of ways of seating this group of 5 is 5!×4! ways .
So the total number of ways in which 4 boys and 4 girls can be seated so that not all the girls sit together is given by= (Total no. of ways in which all 8 can sit)-(No. of ways the group of 5 can sit where the 4 girls all together is taken as a single person)
=> 8! - 5!×4! ways
=>37440 ways.
Thanks for reading the sollution!!!
Upvote ·
99 31
Einar S Arason
Studied mathematics at junior college in Reykjavik. · Author has 142 answers and 246.2K answer views
·4y
Originally Answered: There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together? ·
Bisrat Teklu, maybe you want to clarify the question a little. Now it is: “There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together?”
First of all, is it intentional to say “so that NO two boys are NEVER together?” It is a double negative, like - (-2) = +2; “no two … never” = “every pair … must be together”. For a while, I thought you were giving us a trick question and I imagined you meant we must find some combinations of the row which, when added together by rearranging through an event, meant that no two boys are never together (through the
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Bisrat Teklu, maybe you want to clarify the question a little. Now it is: “There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together?”
First of all, is it intentional to say “so that NO two boys are NEVER together?” It is a double negative, like - (-2) = +2; “no two … never” = “every pair … must be together”. For a while, I thought you were giving us a trick question and I imagined you meant we must find some combinations of the row which, when added together by rearranging through an event, meant that no two boys are never together (through the whole event). (See my original answer farther below.)
Or did you simply mean “so that no two boys are together” (in this possible combination of the row)? I find that more likely, as we all sometimes make “a slip of the tongue”. In that case, you are asking for all the combinations where no two boys are together in the row.
Secondly, are you asking of how many ways can you arrange the persons or is the focus just on how many combinations are there? I mean, one combination is B - G - G - B - G - G - B, another would be G - B - G - B - G - B - G, and there are some others. But within each combination you have different way of arranging the persons. Like the combination B - G - G - B - G - G - B could be B1 - G1 - G2 - B2 - G3 - G4 - B3 or it could be B2 - G2 - G1 - B1 - G3 - G4 - B3 et cetera.
– – –
Below is my original answer, which explains how I was imagining the possibilities of solving the “trick question” I thought you meant. [ It’s not completely original; I was always editing it because I was always seeing new possibilities of what you meant.]
– – –
Tom Almy has brought my attention to one thing about how the question is phrased:
“There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together?”
So NO two boys are NEVER together ?!?!?
Even if you put all the three boys together in the row, one of them will be in the middle and the other two will never be together. Unless you rearrange the row.
Wait a moment. Is that what we are asked to do?
To rearrange the row a few times so during the event no two boys … are never together?
That should be possible.
But it is still a bit vague in my opinion what the question is asking us to do. Is it asking us to find the maximum possibilities for rearranging these seven persons, or is it asking us to find the minimum possibilities for rearranging them so no two boys are never together.
If it is asking for the minimum, we need to find two possibilities with all of them grouped together. Boy 1 - boy 2 - boy 3. Then two of them switch, for instance: Boy 1 - boy 3 - boy 2. In the first possibility, boy 1 and boy 3 are not together, but they are together in the second possibility, so we have solved the problem. No two boys are NEVER together.
But the question could also be asking us to let them rearrange in all possible ways during the event, as it says “In how many ways CAN they stand in row …”
And if we let them rearrange in all possible ways during the event, well, no two boys will never be together because some of the arrangements will bring them together.
To solve this, you don’t need to think of Girl - boy - girl - boy … etc. It can be girl - girl - boy - boy … doesn’t matter. We are asked “In how many ways can they stand in row so that no two boys are never together?”
So it is basically asking how many ways to put seven people in a row. I’m sure there are others better at calculating that than I am.
The only thing is that during the event all three boys have to meet. At some point during the event boy 1 needs to be next to boy 2, at some point boy 2 needs to be next to boy 3, and at some point boy 1 needs to be next to boy 3. That way “no two boys are never together”.
Still, the question is odd. If the author intended what they said, why didn’t they rather ask “how many times do they need to rearrange the row so no two boys are never together”? It kind of makes you wonder if they just meant “In how many ways can they stand in row so that no two boys are together?”
– – –
EDIT
This latter possibility is what I find most likely. We all make sometimes an occasional slip of the tongue when we are thinking in a hurry. We mean one thing, but say another. Happens to all.
If this is the case, the answer depends on whether we should just count the possible combinations, like
B - G - G - B - G - G - B
G - B - G - B - G - B - G
etc.
or count every possible way of arranging the persons, for example the combination
B - G - G - B - G - G - B
could be
B1 - G1 - G2 - B2 - G3 - G4 - B3
or
B2 - G2 - G1 - B1 - G3 - G4 - B3
etc.
Upvote ·
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Related questions
More answers below
In how many ways 4 boys and 4 girls be seated such that 2 girls are together?
In how many ways can 4 boys and 3 girls arranged in a row if no two girls are together?
In how many ways can 4 boys and 5 girls be seated alternately?
In how many ways 10 boys and 5 girls can sit around a circular table so that no two girls sit together?
A) in how many ways can 3 boys and 2 girls sit in a row? (b) in how many ways can they sit in a row if the boys and girls are each to sit together? (c) in how many ways can they sit in a row if just the girls are to sit together
Kumar Abhishek
Author has 59 answers and 195.1K answer views
·7y
Originally Answered: In how many ways can 6 boys and 6 girls be arranged in a row so that no two boys and no two girls are seated together? ·
2 6! 6! is the answer. The workings are as below:
There can be two following ways of sitting for the given condition (B= Boy, G=Girl)
BGBGBGBGBGBG
GBGBGBGBGBGB
Now 6 boys can be made to sit on six seats in 6! ways. Similarly 6 girls can be made to sit on six seats in 6! ways.
Hence the total number of seating arrangements will be
2 6! 6!
Hence the answer, hope this helps.
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Ronak Shah
Trainer and Director at Proton Training Solutions (2013–present)
·5y
Originally Answered: There are 6 boys and 5 girls. In how many ways can they be arranged in a row such that no two girls are together and no two boys are together? ·
In such questions, first arrange the boys on 5 chairs in 5! ways. Then there will be six positions abound boys, as shown in the following diagram. On these six positions, six girls can be arranged in 6! ways. So, Ans = 5! x 6!
(n! = n x (n-1) x (n-2) x …. x 1 )
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In such questions, first arrange the boys on 5 chairs in 5! ways. Then there will be six positions abound boys, as shown in the following diagram. On these six positions, six girls can be arranged in 6! ways. So, Ans = 5! x 6!
(n! = n x (n-1) x (n-2) x …. x 1 )
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Satish Gs
Former Student at National Institute of Technology, Warangal (2019–2023) · Author has 127 answers and 868.4K answer views
·5y
Originally Answered: There are 6 boys and 5 girls. In how many ways can they be arranged in a row such that no two girls are together and no two boys are together? ·
Excuse me, but I didn't read complete question. But no problem, I'll give an extended answer
See the pic, in that, only in right most order no two girls or boys were together
So no of ways = 1 x 6! x 5!
Since boys can be arranged 6! Ways and girls can be arranged in 5! Ways
You might not be surprised if I say that the same will be answer if they only give condition that no two boys are together
But if the condition is only that no two girs should be together, then the three arrangements in the pic are accepted
So then the answer would be
3x6!x5!
Thanks for reading:)
Continue Reading
Excuse me, but I didn't read complete question. But no problem, I'll give an extended answer
See the pic, in that, only in right most order no two girls or boys were together
So no of ways = 1 x 6! x 5!
Since boys can be arranged 6! Ways and girls can be arranged in 5! Ways
You might not be surprised if I say that the same will be answer if they only give condition that no two boys are together
But if the condition is only that no two girs should be together, then the three arrangements in the pic are accepted
So then the answer would be
3x6!x5!
Thanks for reading:)
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Tom Almy
Retired Electrical Engineer (after 43 years) · Author has 4.5K answers and 6.6M answer views
·4y
Originally Answered: There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together? ·
Question: There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together.
Quick answer: anywhere from an unlimited number of ways to no way at all.
I saw this question and though they could be arranged G B G B G B G or B G B G B G G. But then I realized it could also be B G G B G B G or even B G G B G G B. And then does it matter left to right or right to left (i.e. is there a front end of the row?).
Then it hit me. I remembered an exercise done with groups of people where each person has to walk across the room in a different manner. It seems like a b
Continue Reading
Question: There are 3 boys and 4 girls. In how many ways can they stand in row so that no two boys are never together.
Quick answer: anywhere from an unlimited number of ways to no way at all.
I saw this question and though they could be arranged G B G B G B G or B G B G B G G. But then I realized it could also be B G G B G B G or even B G G B G G B. And then does it matter left to right or right to left (i.e. is there a front end of the row?).
Then it hit me. I remembered an exercise done with groups of people where each person has to walk across the room in a different manner. It seems like a big disadvantage to be toward the end, however everyone eventually realizes that there are really an unlimited number of ways to walk across the room.
Likewise there there an unlimited number of ways each person can stand — head turned up/down/left/right, eyes open/closed, arms raised, out, facing front, back… So there are unlimited ways they can stand in a row so that no two boys are together.
Then it hit me again — the question asks how many ways they can stand in a row so that no two boys are never together. Looked at it this way, the first and third boys in the row are never together. So the answer is zero.
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Alvin Chen
Author has 124 answers and 500K answer views
·8y
Originally Answered: In how many ways can 5 girls and 3 boys be seated in a row so that no 2 boys are together? ·
No two boys must sit together.
BGBGBGGG
BGBGGBGG
BGBGGGBG
BGBGGGGB
BGGBGBGG
BGGBGGBG
BGGBGGGB
BGGGBGBG
BGGGBGGB
BGGGGBGB
GBGBGBGG
GBGBGGBG
GBGBGGGB
GBGGBGBG
GBGGBGGB
GBGGGBGB
GGBGBGBG
GGBGBGGB
GGBGGBGB
GGGBGBGB
There are 20 of these cases. For each of these, there are 3!=6 3!=6 ways to arrange the boys and 5!=120 5!=120 ways to arrange the girls. This gives us a total of 20⋅6⋅120=14400 20⋅6⋅120=14400 total ways.
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Anil Bapat
Have studied Mathematics up to pre-Degree Level · Author has 2.8K answers and 3.8M answer views
·5y
Originally Answered: There are 6 boys and 5 girls. In how many ways can they be arranged in a row such that no two girls are together and no two boys are together? ·
There are 6 boys and 5 girls. In how many ways can they be arranged in a row such that no two girls are together and no two boys are together?
The only way to achieve this is to have a boy sitting first followed by a girl to be followed by a boy and so on until all the boys are girls have been seated.
A boy could be chosen in 6 ways and then a girl could be chosen in 5 ways to be followed by a boy that could be chosen in 5 ways and then the girl could be chosen in 4 ways until you come to 1 and 1. So, in reality, you need to multiply:
6 5 5 4 4 3 3 2 2 1 1
This gives you 86,400
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There are 6 boys and 5 girls. In how many ways can they be arranged in a row such that no two girls are together and no two boys are together?
The only way to achieve this is to have a boy sitting first followed by a girl to be followed by a boy and so on until all the boys are girls have been seated.
A boy could be chosen in 6 ways and then a girl could be chosen in 5 ways to be followed by a boy that could be chosen in 5 ways and then the girl could be chosen in 4 ways until you come to 1 and 1. So, in reality, you need to multiply:
6 5 5 4 4 3 3 2 2 1 1
This gives you 86,400.
Thus the required answer is, there are 86,400 ways for 6 boys and 5 girls to be arranged such that no two girls are together and no 2 boys are together.
Upvote ·
9 2
John Chadwick
Former Head of Electrical, Mechanical & MV Engineering at South Cheshire College (1972–2007) · Author has 1K answers and 609.9K answer views
·Updated 2y
Originally Answered: In how many ways can 4 boys and 3 girls arranged in a row if no two girls are together? ·
There are 4 boys {b1, b2, b3, b4} and 3 girls {g1, g2, g3} to be arranged in a row such that there are never two girls adjacent:-
_ b1 _ b2 _ b3 _ b4 _ ==> 5 slots available for the 3 girls to fill.
The boys {b1, b2, b3, b4} can be partitioned (permuted) in 4P4 = 24 ways and the girls {g1, g2, g3} can be partitioned (permuted) to fill 3 of the 5 slots available to them in 5P3 = 60 ways
This gives a total of 4P45P3 = 2460 = 1440 possible ways for 3 girls to be partitioned in a row between 4 boys so that there are NEVER any two girls adjacent.
Upvote ·
9 3
Zachary Taylor
International Relations, Economics, and Math major. University of Rochester '15 · Author has 1.7K answers and 5.4M answer views
·8y
Originally Answered: In how many ways can 4 boys and three girls arranged in a row if no two girls are together? ·
Note: There are two ways to interpret what a unique combination is. We can interpret a combination as an order of genders (boy, girl, boy, boy…) or we can interpret a combination as an order of individual people (Bob, Sally, Joe, Greg…). I’ll do it both ways, in that order.
First interpretation:
This can be thought of as a “stars and bars problem” in which the three girls are the bars:
| | |
We need to arrange the first two boys so that there is at least one boy in between each girl:
| | |
This leaves two remaining boys, each of which can be put in one of four positions (to the left of the girls
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Note: There are two ways to interpret what a unique combination is. We can interpret a combination as an order of genders (boy, girl, boy, boy…) or we can interpret a combination as an order of individual people (Bob, Sally, Joe, Greg…). I’ll do it both ways, in that order.
First interpretation:
This can be thought of as a “stars and bars problem” in which the three girls are the bars:
| | |
We need to arrange the first two boys so that there is at least one boy in between each girl:
| | |
This leaves two remaining boys, each of which can be put in one of four positions (to the left of the girls, to the right, or one of the two middle positions.) Putting the third boy in any position doesn’t change where we can put the second boy, so the final answer is 4 4 = 16.
Second interpretation:
We’ve already found that there are 16 combinations of stars and bars that work. For each such stars-and-bars combination, we can reorder the boys and reorder the girls to create a unique combination of people. For example:
| | |
might correspond to:
Sally, Bob, Louisa, Greg, Laura, Bill, Joe.
Any allowable ordering of people can be given by: 1. A stars-and-bars combination 2. An ordering of boys and 3. An ordering of women.
There are 4! = 4321 ways to order the boys and 3! ways to order the women. So, we have:
(16)(4!)(3!) = 2304 unique arrangements of people. If there was no rule against girls sitting together, we would have 7! = 5040 possible arrangements. So, a randomly-selected arrangement has about a 46% chance of satisfying a rule.
Upvote ·
9 4
9 3
Harry Kanigel
Former software executive in finance · Author has 161 answers and 373.9K answer views
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Originally Answered: In how many ways can 3 boys and 3 girls be arranged in a line so that no two boys are next to each other? ·
144
First let’s establish how many arrangements of indistinguishable girls and boys would meet the conditions. If we imagine the three girls standing together we can imagine four potential placement points to place a boy on either side of each girl. I’ll designate the possible placement points with a ‘p’ and the indistinguishable girls with a ‘g’ . Like so:
p,g,p,g,p,g,p
Our first task is to compute the ways of choosing 3 placement points from 4 possible placement points. This is often referred to as 4C3. This can be computed as 4!/3! which is just 4.
Here are the four possible placeholder arrange
Continue Reading
144
First let’s establish how many arrangements of indistinguishable girls and boys would meet the conditions. If we imagine the three girls standing together we can imagine four potential placement points to place a boy on either side of each girl. I’ll designate the possible placement points with a ‘p’ and the indistinguishable girls with a ‘g’ . Like so:
p,g,p,g,p,g,p
Our first task is to compute the ways of choosing 3 placement points from 4 possible placement points. This is often referred to as 4C3. This can be computed as 4!/3! which is just 4.
Here are the four possible placeholder arrangements:
g,p,g,p,g,p / p,g,g,p,g,p / p,g,p,g,g,p / p,g,p,g,p,g
We should think of the placement points now as locations of indistinguishable boys. Like so:
g,b,g,b,g,b / b,g,g,b,g,b / b,g,b,g,g,b / b,g,b,g,b,g
Notice that all these placeholder arrangements meet the criterion of not having any two boys next to each other.
Now let’s compute the number of combinations of distinguishable boys and girls that could slot into any of these arrangements. For the b placeholders it would be computed as 3! = 6; same for the g placeholders.
So our final calculation is:
4 6 6 = 144
Upvote ·
9 1
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12063 | https://en.wikipedia.org/wiki/Substring_index | Jump to content
Substring index
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From Wikipedia, the free encyclopedia
Data structure
In computer science, a substring index is a data structure which gives substring search in a text or text collection in sublinear time. Once constructed from a document or set of documents, a substring index can be used to locate all occurrences of a pattern in time linear or near-linear in the pattern size, with no dependence or only logarithmic dependence on the document size.
The phrase full-text index is often used for substring indexes. But this is ambiguous, as it is also used for regular word indexes such as inverted files and document retrieval. See full text search.
General considerations
[edit]
These data structures typically treat their text and pattern as strings over a fixed alphabet, and search for locations where the pattern occurs as a substring of the text. The symbols of the alphabet may be characters (for instance in Unicode) but in practical applications for text retrieval it may be preferable to treat the (stemmed) words of a document as the symbols of its alphabet, because doing this reduces the lengths of both the text and pattern as measured in numbers of symbols.
Examples
[edit]
Specific data structures that can be used as substring indexes include:
The suffix tree, a radix tree of the suffixes of the string, allowing substring search to be performed symbol-by-symbol
The suffix automaton, the minimal deterministic finite automaton that recognizes substrings of a given text, closely related to the suffix tree and constructable by variants of the same algorithms.
The suffix array, a sorted array of the starting positions of suffixes of the string, allowing substring search to be performed by binary search Augmenting a suffix array with an LCP array of the lengths of common prefixes of consecutive suffixes allows the search to be performed symbol-by-symbol, matching the search time of the suffix tree.
The compressed suffix array, a data structure that combines data compression with the suffix array, allowing the structure to be stored in space sublinear in the text length
The FM-index, another compressed substring index based on the Burrows–Wheeler transform and closely related to the suffix array
References
[edit]
^ Jump up to: a b c d Barsky, Marina; Stege, Ulrike; Thomo, Alex (2012), "Chapter 1: Structures for Indexing Substrings", Full-Text (Substring) Indexes in External Memory, Synthesis Lectures on Data Management, Springer International Publishing, pp. 1–15, doi:10.1007/978-3-031-01885-5_1, ISBN 9783031018855
^ Risvik, Knut Magne (1998), "Approximate word sequence matching over sparse suffix trees", in Farach-Colton, Martin (ed.), Combinatorial Pattern Matching, 9th Annual Symposium, CPM 98, Piscataway, New Jersey, USA, July 20–22, 1998, Proceedings, Lecture Notes in Computer Science, vol. 1448, Springer, pp. 65–79, doi:10.1007/BFB0030781, ISBN 978-3-540-64739-3
^ Jump up to: a b c Grossi, Roberto; Vitter, Jeffrey Scott (2005), "Compressed suffix arrays and suffix trees with applications to text indexing and string matching" (PDF), SIAM Journal on Computing, 35 (2): 378–407, doi:10.1137/S0097539702402354, hdl:1808/18962, MR 2191449
^ Blumer, Anselm; Blumer, J.; Ehrenfeucht, Andrzej; Haussler, David; McConnell, Ross M. (1984), "Building the minimal DFA for the set of all subwords of a word on-line in linear time", in Paredaens, Jan (ed.), Automata, Languages and Programming, 11th Colloquium, Antwerp, Belgium, July 16–20, 1984, Proceedings, Lecture Notes in Computer Science, vol. 172, Springer, pp. 109–118, doi:10.1007/3-540-13345-3_9, ISBN 978-3-540-13345-2
^ Manber, Udi; Myers, Gene (1993), "Suffix arrays: a new method for on-line string searches", SIAM Journal on Computing, 22 (5): 935–948, doi:10.1137/0222058, MR 1237156
^ Ferragina, Paolo; Manzini, Giovanni (2005), "Indexing compressed text", Journal of the ACM, 52 (4): 552–581, doi:10.1145/1082036.1082039, MR 2164632
| v t e Strings | |
--- |
| String metric | Approximate string matching Bitap algorithm Damerau–Levenshtein distance Edit distance Gestalt pattern matching Hamming distance Jaro–Winkler distance Lee distance Levenshtein automaton Levenshtein distance Wagner–Fischer algorithm |
| String-searching algorithm | Apostolico–Giancarlo algorithm Boyer–Moore string-search algorithm Boyer–Moore–Horspool algorithm Knuth–Morris–Pratt algorithm Rabin–Karp algorithm Raita algorithm Trigram search Two-way string-matching algorithm Zhu–Takaoka string matching algorithm |
| Multiple string searching | Aho–Corasick Commentz-Walter algorithm |
| Regular expression | Comparison of regular-expression engines Regular grammar Thompson's construction Nondeterministic finite automaton |
| Sequence alignment | BLAST Hirschberg's algorithm Needleman–Wunsch algorithm Smith–Waterman algorithm |
| Data structure | DAFSA Substring index + Suffix array + Suffix automaton + Suffix tree + Compressed suffix array + LCP array + FM-index Generalized suffix tree Rope Ternary search tree Trie |
| Other | Parsing Pattern matching Compressed pattern matching Longest common subsequence Longest common substring Sequential pattern mining Sorting String rewriting systems String operations |
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On the definition of cofinite.
I am having some difficulty comprehending the definition of a cofinite set. I am seeking confirmation of whether my understanding is correct and some clarification on the definition.
Wikipedia provides the following definition:
Definition: A cofinite subset of a set $X$ is a subset $A$ whose complement in $X$ is a finite set. In other words, $A$ contains all but finitely many elements of $X$.
Based upon my understanding, I have come up with a definition I believe is equivalent.
Definition': Suppose there are two sets, $X$ and $A$ where $A \subset X$. $A$ is cofinite in $X$ if $A^c \cap X$ is finite, where the $c$ superscript denotes the complement.
1 Answer 1
Yes. It's fine.
Note that $A^c\cap X=X\setminus A$, which may be simpler to read.
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12065 | https://digitalcommons.uri.edu/cgi/viewcontent.cgi?article=1037&context=phy204-slides | University of Rhode Island University of Rhode Island
DigitalCommons@URI DigitalCommons@URI
PHY 204: Elementary Physics II -- Slides PHY 204: Elementary Physics II (2021)
2020
13. Effects of magnetic field: torque. Magnetic dipole 13. Effects of magnetic field: torque. Magnetic dipole
Gerhard Müller University of Rhode Island , gmuller@uri.edu
Robert Coyne University of Rhode Island , robcoyne@uri.edu
Follow this and additional works at:
Recommended Citation Recommended Citation
Müller, Gerhard and Coyne, Robert, "13. Effects of magnetic field: torque. Magnetic dipole" (2020).
PHY
204: Elementary Physics II -- Slides. Paper 38.
This Course Material is brought to you by the University of Rhode Island. It has been accepted for inclusion in PHY
204: Elementary Physics II -- Slides by an authorized administrator of DigitalCommons@URI. For more information,
please contact digitalcommons-group@uri.edu . For permission to reuse copyrighted content, contact the author
directly. Dot Product Between Vectors
Consider two vectors ~A = Axˆi + Ayˆj + Az ˆk and ~B = Bxˆi + Byˆj + Bz ˆk.
• ~A ·~B = AB cos φ = AB A = BA B.
• ~A ·~B = ~B · ~A.
• ~A ·~B = AB if ~A ‖ ~B.
• ~A ·~B = 0 if ~A ⊥ ~B.
• ~A ·~B = ( Axˆi + Ayˆj + Az ˆk) · (Bxˆi + Byˆj + Bz ˆk)= AxBx(ˆi · ˆi) + AxBy(ˆi · ˆj) + AxBz(ˆi · ˆk)+AyBx(ˆj · ˆi) + AyBy(ˆj · ˆj) + AyBz(ˆj · ˆk)+AzBx(ˆk · ˆi) + AzBy(ˆk · ˆj) + AzBz(ˆk · ˆk).
• Use ˆi · ˆi = ˆj · ˆj = ˆk · ˆk = 1,
ˆi · ˆj = ˆj · ˆk = ˆk · ˆi = 0.
• ⇒ ~A ·~B = AxBx + AyBy + AzBz.
tsl229
Cross Product Between Vectors
Consider two vectors ~A = Axˆi + Ayˆj + Az ˆk and ~B = Bxˆi + Byˆj + Bz ˆk.
• ~A ×~B = AB sin φ ˆn.
• ~A ×~B = −~B × ~A.
• ~A × ~A = 0.
• ~A ×~B = AB ˆn if ~A ⊥ ~B.
• ~A ×~B = 0 if ~A ‖ ~B.
• ~A ×~B = ( Axˆi + Ayˆj + Az ˆk) × (Bxˆi + Byˆj + Bz ˆk)= AxBx(ˆi × ˆi) + AxBy(ˆi × ˆj) + AxBz(ˆi × ˆk)+AyBx(ˆj × ˆi) + AyBy(ˆj × ˆj) + AyBz(ˆj × ˆk)+AzBx(ˆk × ˆi) + AzBy(ˆk × ˆj) + AzBz(ˆk × ˆk).
• Use ˆi × ˆi = ˆj × ˆj = ˆk × ˆk = 0,
ˆi × ˆj = ˆk, ˆ j × ˆk = ˆi, ˆk × ˆi = ˆj.
• ⇒ ~A ×~B = ( AyBz − AzBy)ˆi + ( AzBx − AxBz)ˆj + ( AxBy − AyBx)ˆk.
tsl230
Magnetic Dipole Moment of Current Loop
N: number of turns
I: current through wire
A: area of loop
ˆn: unit vector perpendicular to plane of loop
~μ = NIA ˆn: magnetic dipole moment
~B: magnetic field
~τ = ~μ ×~B: torque acting on current loop
tsl475
Torque on Current Loop
• magnetic field: ~B (horizontal)
• area of loop: A = ab
• unit vector ⊥ to plane of loop: ˆn
• right-hand rule: ˆn points up.
• forces on sides a: F = IaB (vertical)
• forces on sides b: F = IbB (horizontal, not shown)
• torque: τ = Fb sin θ = IAB sin θ
• magnetic moment: ~μ = IA ˆn
• torque (vector): ~τ = ~μ ×~B
tsl196
Magnetic Moment of a Rotating Disk
Consider a nonconducting disk of radius R with a uniform surface charge density σ. The disk rotates with angular velocity ~ω.Calculation of the magnetic moment ~μ:
• Total charge on disk: Q = σ(πR2).
• Divide the disk into concentric rings of width dr .
• Period of rotation: T = 2πω .
• Current within ring: dI = dQ T = σ(2πrdr ) ω
2π = σω rdr .
• Magnetic moment of ring: dμ = dI (πr2) = πσω r3dr .
• Magnetic moment of disk: μ =
∫R
0
πσω r3dr = π
4 σR4ω.
• Vector relation: ~μ = π
4 σR4 ~ω = 14 QR 2 ~ω.
tsl199
Direct-Current Motor
tsl408
Galvanometer
Measuring direct currents.
• magnetic moment ~μ (along needle)
• magnetic field ~B (toward right)
• torque ~τ = ~μ ×~B (into plane)
tsl409
Magnetic Dipole in Uniform Magnetic Field
• Magnetic dipole moment: ~μ = IA ˆn
• Torque exerted by magnetic field: ~τ = ~μ ×~B
• Potential energy: U = −~μ ·~BU(θ) = −
∫ θπ/2
τ(θ)dθ = μB
∫ θπ/2
sin θdθ = −μB cos θ
Note: τ(θ) and dθ have opposite sign. I θ
μ = IAn ^
n^
τ
B
μ
.
tsl198
Electric Dipole in Uniform Electric Field
• Electric dipole moment: ~p = q~L
• Torque exerted by electric field: ~τ = ~p ×~E
• Potential energy: U = −~p ·~EU(θ) = −
∫ θπ/2
τ(θ)dθ = pE
∫ θπ/2
sin θdθ = −pE cos θ
Note: τ(θ) and dθ have opposite sign. L
τ θ
E
p−q +q
p = qL
tsl197
Magnetic Force Application (11)
If the magnetic moment of the current loop (1) is μ1 = 1Am 2, what are the magnetic moments μ2, μ3, μ4 of the current loops (2), (3), (4), respectively? (1) (2) (3) (4)
tsl206
Unit Exam III: Problem #1 (Spring ’12)
In a region of uniform magnetic field B = 5mTˆ i, a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) is launched with velocity v0 = 4000m/s ˆ k.(a) Calculate the magnitude F of the magnetic force that keeps the proton on a circular path. (b) Calculate the radius r of the circular path. (c) Calculate the time T it takes the proton to go around that circle once. (d) Sketch the circular path of the proton in the graph. xm, q Bv0
zy
tsl435
Unit Exam III: Problem #1 (Spring ’12)
In a region of uniform magnetic field B = 5mTˆ i, a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) is launched with velocity v0 = 4000m/s ˆ k.(a) Calculate the magnitude F of the magnetic force that keeps the proton on a circular path. (b) Calculate the radius r of the circular path. (c) Calculate the time T it takes the proton to go around that circle once. (d) Sketch the circular path of the proton in the graph. xm, q Bv0
zy
Solution:
(a) F = qv 0B = 3.2 × 10 −18 N.
tsl435
Unit Exam III: Problem #1 (Spring ’12)
In a region of uniform magnetic field B = 5mTˆ i, a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) is launched with velocity v0 = 4000m/s ˆ k.(a) Calculate the magnitude F of the magnetic force that keeps the proton on a circular path. (b) Calculate the radius r of the circular path. (c) Calculate the time T it takes the proton to go around that circle once. (d) Sketch the circular path of the proton in the graph. xm, q Bv0
zy
Solution:
(a) F = qv 0B = 3.2 × 10 −18 N.(b) mv 20
r = qv 0B ⇒ r = mv 0
qB = 8.35mm .
tsl435
Unit Exam III: Problem #1 (Spring ’12)
In a region of uniform magnetic field B = 5mTˆ i, a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) is launched with velocity v0 = 4000m/s ˆ k.(a) Calculate the magnitude F of the magnetic force that keeps the proton on a circular path. (b) Calculate the radius r of the circular path. (c) Calculate the time T it takes the proton to go around that circle once. (d) Sketch the circular path of the proton in the graph. xm, q Bv0
zy
Solution:
(a) F = qv 0B = 3.2 × 10 −18 N.(b) mv 20
r = qv 0B ⇒ r = mv 0
qB = 8.35mm .(c) T = 2πrv0
= 2πmqB = 13.1 μs.
tsl435
Unit Exam III: Problem #1 (Spring ’12)
In a region of uniform magnetic field B = 5mTˆ i, a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) is launched with velocity v0 = 4000m/s ˆ k.(a) Calculate the magnitude F of the magnetic force that keeps the proton on a circular path. (b) Calculate the radius r of the circular path. (c) Calculate the time T it takes the proton to go around that circle once. (d) Sketch the circular path of the proton in the graph. xm, q Bv0
zy
Solution:
(a) F = qv 0B = 3.2 × 10 −18 N.(b) mv 20
r = qv 0B ⇒ r = mv 0
qB = 8.35mm .(c) T = 2πrv0
= 2πmqB = 13.1 μs.(d) Center of circle to the right of proton’s initial position (cw motion).
tsl435
Magnetic Force Application (6)
An electric current flows through each of the letter-shaped wires in a region of uniform magnetic field pointing into the plane.
• Find the direction of the resultant magnetic force on each letter. NW ESNW NE SE SW
B
tsl193
Magnetic Force Application (7)
The rectangular 20-turn loop of wire is 10cm high and 5cm wide. It carries a current I = 0.1 A and is hinged along one long side. It is mounted with its plane at an angle of 30 ◦ to the direction of a uniform magnetic field of magnitude
B = 0.50 T.
• Calculate the magnetic moment μ of the loop.
• Calculate the torque τ acting on the loop about the hinge line. 30 o
a = 10cm
I B
b = 5cm
tsl202
Magnetic Force Application (10)
A triangular current loop is free to rotate around the vertical axis PQ .If a uniform magnetic field ~B is switched on, will the corner R of the triangle start to move out of the plane, into the plane, or will it not move at all? Find the answer for a field ~B pointing (a) up, (b) to the right, (c) into the plane. I (a) (b) (c) BBBPQR
tsl205
Intermediate Exam III: Problem #1 (Spring ’07)
Consider a rectangular conducting loop in the xy -plane with a counterclockwise current I = 7A in a uniform magnetic field ~B = 3Tˆ i.(a) Find the magnetic moment ~μ (magnitude and direction) of the loop. (b) Find the force ~F (magnitude and direction) acting on the side ab of the rectangle. (c) Find the torque ~τ (magnitude and direction) acting on the loop. 5m 9m BIabzyx
tsl365
Intermediate Exam III: Problem #1 (Spring ’07)
Consider a rectangular conducting loop in the xy -plane with a counterclockwise current I = 7A in a uniform magnetic field ~B = 3Tˆ i.(a) Find the magnetic moment ~μ (magnitude and direction) of the loop. (b) Find the force ~F (magnitude and direction) acting on the side ab of the rectangle. (c) Find the torque ~τ (magnitude and direction) acting on the loop. 5m 9m BIabzyx
Solution:
(a) ~μ = ( 7A )( 45m 2)ˆk = 315Am 2 ˆk.
tsl365
Intermediate Exam III: Problem #1 (Spring ’07)
Consider a rectangular conducting loop in the xy -plane with a counterclockwise current I = 7A in a uniform magnetic field ~B = 3Tˆ i.(a) Find the magnetic moment ~μ (magnitude and direction) of the loop. (b) Find the force ~F (magnitude and direction) acting on the side ab of the rectangle. (c) Find the torque ~τ (magnitude and direction) acting on the loop. 5m 9m BIabzyx
Solution:
(a) ~μ = ( 7A )( 45m 2)ˆk = 315Am 2 ˆk.
(b) ~F = I~L ×~B = ( 7A )( 5mˆ j) × (3Tˆ i) = −105N ˆk.
tsl365
Intermediate Exam III: Problem #1 (Spring ’07)
Consider a rectangular conducting loop in the xy -plane with a counterclockwise current I = 7A in a uniform magnetic field ~B = 3Tˆ i.(a) Find the magnetic moment ~μ (magnitude and direction) of the loop. (b) Find the force ~F (magnitude and direction) acting on the side ab of the rectangle. (c) Find the torque ~τ (magnitude and direction) acting on the loop. 5m 9m BIabzyx
Solution:
(a) ~μ = ( 7A )( 45m 2)ˆk = 315Am 2 ˆk.
(b) ~F = I~L ×~B = ( 7A )( 5mˆ j) × (3Tˆ i) = −105N ˆk.
(c) ~τ = ~μ ×~B = ( 315Am 2 ˆk) × (3Tˆ i) = 945Nmˆ j
tsl365
Unit Exam III: Problem #1 (Spring ’08)
Consider two circular currents I1 = 3A at radius r1 = 2m and I2 = 5A at radius r2 = 4m in the directions shown. (a) Find magnitude B and direction ( , ⊗) of the resultant magnetic field at the center. (b) Find magnitude μ and direction ( , ⊗) of the magnetic dipole moment generated by the two currents. rrII
1122
tsl381
Unit Exam III: Problem #1 (Spring ’08)
Consider two circular currents I1 = 3A at radius r1 = 2m and I2 = 5A at radius r2 = 4m in the directions shown. (a) Find magnitude B and direction ( , ⊗) of the resultant magnetic field at the center. (b) Find magnitude μ and direction ( , ⊗) of the magnetic dipole moment generated by the two currents. rrII
1122
Solution:
(a) B = μ0(3A )
2(2m ) − μ0(5A )
2(4m ) = ( 9.42 − 7.85 ) × 10 −7T
⇒ B = 1.57 × 10 −7T ⊗
tsl381
Unit Exam III: Problem #1 (Spring ’08)
Consider two circular currents I1 = 3A at radius r1 = 2m and I2 = 5A at radius r2 = 4m in the directions shown. (a) Find magnitude B and direction ( , ⊗) of the resultant magnetic field at the center. (b) Find magnitude μ and direction ( , ⊗) of the magnetic dipole moment generated by the two currents. rrII
1122
Solution:
(a) B = μ0(3A )
2(2m ) − μ0(5A )
2(4m ) = ( 9.42 − 7.85 ) × 10 −7T
⇒ B = 1.57 × 10 −7T ⊗
(b) μ = π(4m )2(5A ) − π(2m )2(3A ) = ( 251 − 38 )Am 2
⇒ μ = 213Am 2
tsl381
Unit Exam III: Problem #1 (Spring ’09)
A triangular conducting loop in the yz -plane with a counterclockwise current I = 3A is free to rotate about the axis PQ . A uniform magnetic field ~B = 0.5T ˆk is present. (a) Find the magnetic moment ~μ (magnitude and direction) of the triangle. (b) Find the magnetic torque ~τ (magnitude and direction) acting on the triangle. (c) Find the magnetic force ~FH (magnitude and direction) acting on the long side (hypotenuse) of the triangle. (d) Find the force ~FR (magnitude and direction) that must be applied to the corner R to keep the triangle from rotating. IQPRB8m 8m xzy
tsl395
Unit Exam III: Problem #1 (Spring ’09)
A triangular conducting loop in the yz -plane with a counterclockwise current I = 3A is free to rotate about the axis PQ . A uniform magnetic field ~B = 0.5T ˆk is present. (a) Find the magnetic moment ~μ (magnitude and direction) of the triangle. (b) Find the magnetic torque ~τ (magnitude and direction) acting on the triangle. (c) Find the magnetic force ~FH (magnitude and direction) acting on the long side (hypotenuse) of the triangle. (d) Find the force ~FR (magnitude and direction) that must be applied to the corner R to keep the triangle from rotating. IQPRB8m 8m xzy
Solution:
(a) ~μ = ( 3A )( 32m 2)ˆi = 96Am 2ˆi.
tsl395
Unit Exam III: Problem #1 (Spring ’09)
A triangular conducting loop in the yz -plane with a counterclockwise current I = 3A is free to rotate about the axis PQ . A uniform magnetic field ~B = 0.5T ˆk is present. (a) Find the magnetic moment ~μ (magnitude and direction) of the triangle. (b) Find the magnetic torque ~τ (magnitude and direction) acting on the triangle. (c) Find the magnetic force ~FH (magnitude and direction) acting on the long side (hypotenuse) of the triangle. (d) Find the force ~FR (magnitude and direction) that must be applied to the corner R to keep the triangle from rotating. IQPRB8m 8m xzy
Solution:
(a) ~μ = ( 3A )( 32m 2)ˆi = 96Am 2ˆi.
(b) ~τ = ~μ ×~B = ( 96Am 2ˆi) × (0.5T ˆk) = −48Nmˆ j.
tsl395
Unit Exam III: Problem #1 (Spring ’09)
A triangular conducting loop in the yz -plane with a counterclockwise current I = 3A is free to rotate about the axis PQ . A uniform magnetic field ~B = 0.5T ˆk is present. (a) Find the magnetic moment ~μ (magnitude and direction) of the triangle. (b) Find the magnetic torque ~τ (magnitude and direction) acting on the triangle. (c) Find the magnetic force ~FH (magnitude and direction) acting on the long side (hypotenuse) of the triangle. (d) Find the force ~FR (magnitude and direction) that must be applied to the corner R to keep the triangle from rotating. IQPRB8m 8m xzy
Solution:
(a) ~μ = ( 3A )( 32m 2)ˆi = 96Am 2ˆi.
(b) ~τ = ~μ ×~B = ( 96Am 2ˆi) × (0.5T ˆk) = −48Nmˆ j.(c) FH = ( 3A )( 8√2m )( 0.5T )( sin 45 ◦ ) = 12N .
tsl395
Unit Exam III: Problem #1 (Spring ’09)
A triangular conducting loop in the yz -plane with a counterclockwise current I = 3A is free to rotate about the axis PQ . A uniform magnetic field ~B = 0.5T ˆk is present. (a) Find the magnetic moment ~μ (magnitude and direction) of the triangle. (b) Find the magnetic torque ~τ (magnitude and direction) acting on the triangle. (c) Find the magnetic force ~FH (magnitude and direction) acting on the long side (hypotenuse) of the triangle. (d) Find the force ~FR (magnitude and direction) that must be applied to the corner R to keep the triangle from rotating. IQPRB8m 8m xzy
Solution:
(a) ~μ = ( 3A )( 32m 2)ˆi = 96Am 2ˆi.
(b) ~τ = ~μ ×~B = ( 96Am 2ˆi) × (0.5T ˆk) = −48Nmˆ j.(c) FH = ( 3A )( 8√2m )( 0.5T )( sin 45 ◦ ) = 12N .(d) (−8m ˆk) ×~FR = −~τ = 48Nmˆ j ⇒ ~FR = −6Nˆ i.
tsl395
Unit Exam III: Problem #1 (Spring ’13)
In a region of uniform magnetic field B a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) experiences a force
F = 8.0 × 10 −19 N ˆ i as it passes through point P with velocity v0 = 2000m/s ˆ k on a circular path. (a) Find the magnetic field B (magnitude and direction). (b) Calculate the radius r of the circular path. (c) Locate the center C of the circular path in the coordinate system on the page. 1 2 3 4 5
x [cm] y [cm] z [cm]
5432154321
P
tsl462
Unit Exam III: Problem #1 (Spring ’13)
In a region of uniform magnetic field B a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) experiences a force
F = 8.0 × 10 −19 N ˆ i as it passes through point P with velocity v0 = 2000m/s ˆ k on a circular path. (a) Find the magnetic field B (magnitude and direction). (b) Calculate the radius r of the circular path. (c) Locate the center C of the circular path in the coordinate system on the page. 1 2 3 4 5
x [cm] y [cm] z [cm]
5432154321
P
Solution:
(a) B = Fqv 0
= 2.50 × 10 −3T, ˆi = ˆk × (−ˆj)
⇒ B = −2.50 × 10 −3T ˆ j.
tsl462
Unit Exam III: Problem #1 (Spring ’13)
In a region of uniform magnetic field B a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) experiences a force
F = 8.0 × 10 −19 N ˆ i as it passes through point P with velocity v0 = 2000m/s ˆ k on a circular path. (a) Find the magnetic field B (magnitude and direction). (b) Calculate the radius r of the circular path. (c) Locate the center C of the circular path in the coordinate system on the page. 1 2 3 4 5
x [cm] y [cm] z [cm]
5432154321
P
Solution:
(a) B = Fqv 0
= 2.50 × 10 −3T, ˆi = ˆk × (−ˆj)
⇒ B = −2.50 × 10 −3T ˆ j.(b) F = mv 20
r = qv 0B
⇒ r = mv 20
F = mv 0
qB = 0.835cm .
tsl462
Unit Exam III: Problem #1 (Spring ’13)
In a region of uniform magnetic field B a proton (m = 1.67 × 10 −27 kg, q = 1.60 × 10 −19 C) experiences a force
F = 8.0 × 10 −19 N ˆ i as it passes through point P with velocity v0 = 2000m/s ˆ k on a circular path. (a) Find the magnetic field B (magnitude and direction). (b) Calculate the radius r of the circular path. (c) Locate the center C of the circular path in the coordinate system on the page. 1 2 3 4 5
x [cm] y [cm] z [cm]
5432154321
P
Solution:
(a) B = Fqv 0
= 2.50 × 10 −3T, ˆi = ˆk × (−ˆj)
⇒ B = −2.50 × 10 −3T ˆ j.(b) F = mv 20
r = qv 0B
⇒ r = mv 20
F = mv 0
qB = 0.835cm .(c) C = 3.84cm ˆ i + 3.00cm ˆ k.
tsl462
Charged Particle in Crossed Electric and Magnetic Fields (1)
• Release particle from rest.
• Force: ~F = q(~E +~v ×~B)
• (1) Fx = m dv x
dt = −qv yB ⇒ dv x
dt = − qB m vy
• (2) Fy = m dv y
dt = qv xB + qE ⇒ dv y
dt = qB m vx + qE m
• Ansatz: vx(t) = wx cos (ω0t) + ux, vy(t) = wy sin (ω0t) + uy
• Substitute ansatz into (1) and (2) to find wx, wy, ux, uy, ω0.
• (1) −ω0wx sin (ω0t) = − qB m wy sin (ω0t) − qB m uy
• (2) ω0wy cos (ω0t) = qB m wx cos (ω0t) + qB m ux + qE m
• ⇒ uy = 0, ux = − EB , ω0 = qB m , wx = wy ≡ w
• Initial condition: vx(0) = vy(0) = 0 ⇒ w = EBzqm
E
yx
B
tsl208
Charged Particle in Crossed Electric and Magnetic Fields (2)
• Solution for velocity of particle:
vx(t) = EB
[
cos
( qBt m
)
− 1
]
, vy(t) = EB sin
( qBt m
)
• Solution for position of particle:
x(t) = EB
∫t
0
[
cos
( qBt m
)
− 1
]
dt = Em qB 2 sin
( qBt m
)
− Et By(t) = EB
∫t
0
sin
( qBt m
)
dt = Em qB 2
[
1 − cos
( qBt m
)]
• Path of particle in (x, y)-plane: cycloid qB 2 qm
E
yx
B
2mE mE π2
qB 2
t = 2πmqB
tsl209
Hall Effect
Method for dermining whether charge carriers are positively or negatively charged.
• Magnetic field ~B pulls charge carriers to one side of conducting strip.
• Accumulation of charge carriers on that side and depletion on opposite side produce transverse electric field ~E.
• Transverse forces on charge carrier: FE = qE and FB = qv dB.
• In steady state forces are balanced: ~FE = −~FB.
• Hall voltage in steady state: VH = Ew = vdBw .
positive charge carriers negative charge carriers
tsl201
Magnetic Force Application (9)
Two charged particles are released in different uniform fields. Ignore gravity. (a) Find the the horizontal velocity components vEx , vBx and the vertical velocity components vEy , vBy at the instant each particle hits the wall. (b) Find the times tE, tB it takes each particle to reach the wall. E = 1N/C B = 1T
v = 1m/s 0
1m m = 1kg q = 1C q = 1C m = 1kg v = 1/m/s 0
+
tsl204
Magnetic Force Application (12)
An electric current I = 1A flows through the M-shaped wire in the direction indicated. The wire is placed in a magnetic field B = 1T pointing into the plane. (a) Find the magnitude of the magnetic forces F1, F2, F3, F4 acting on each part of the wire. (b) Find the direction of the resultant force ~F = ~F1 +~F2 +~F3 +~F4 acting on the wire. 412 31m 1m 1m 1m
B NW ESNW NE SE SW
tsl207
Magnetic Force Application (4)
A negatively charged basketball is thrown vertically up against the gravitational field ~g.Which direction of (a) a uniform electric field ~E,(b) a uniform magnetic field ~B
will give the ball a chance to find its way into the basket? (up/down/left/right/back/front) g
V |
12066 | https://static.hlt.bme.hu/semantics/external/pages/kérdésmegválaszolás/en.wikipedia.org/wiki/Text_corpus.html | Text corpus - Wikipedia
Text corpus
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In linguistics, a corpus (plural corpora) or text corpus is a large and structured set of texts (nowadays usually electronically stored and processed). In corpus linguistics, they are used to do statistical analysis and hypothesis testing, checking occurrences or validating linguistic rules within a specific language territory.
[x]
Contents
1 Overview
2 Archaeological corpora
3 Some notable text corpora
4 See also
5 References
6 External links
Overview[edit]
A corpus may contain texts in a single language (monolingual corpus) or text data in multiple languages (multilingual corpus).
Multilingual corpora that have been specially formatted for side-by-side comparison are called aligned parallel corpora. There are two main types of parallel corpora which contain texts in two languages. In a translation corpus, the texts in one language are translations of texts in the other language. In a comparable corpus, the texts are of the same kind and cover the same content, but they are not translations of each other. To exploit a parallel text, some kind of text alignment identifying equivalent text segments (phrases or sentences) is a prerequisite for analysis. Machine translation algorithms for translating between two languages are often trained using parallel fragments comprising a first language corpus and a second language corpus which is an element-for-element translation of the first language corpus.
In order to make the corpora more useful for doing linguistic research, they are often subjected to a process known as annotation. An example of annotating a corpus is part-of-speech tagging, or POS-tagging, in which information about each word's part of speech (verb, noun, adjective, etc.) is added to the corpus in the form of tags. Another example is indicating the lemma (base) form of each word. When the language of the corpus is not a working language of the researchers who use it, interlinear glossing is used to make the annotation bilingual.
Some corpora have further structured levels of analysis applied. In particular, a number of smaller corpora may be fully parsed. Such corpora are usually called Treebanks or Parsed Corpora. The difficulty of ensuring that the entire corpus is completely and consistently annotated means that these corpora are usually smaller, containing around one to three million words. Other levels of linguistic structured analysis are possible, including annotations for morphology, semantics and pragmatics.
Corpora are the main knowledge base in corpus linguistics. The analysis and processing of various types of corpora are also the subject of much work in computational linguistics, speech recognition and machine translation, where they are often used to create hidden Markov models for part of speech tagging and other purposes. Corpora and frequency lists derived from them are useful for language teaching. Corpora can be considered as a type of foreign language writing aid as the contextualised grammatical knowledge acquired by non-native language users through exposure to authentic texts in corpora allows learners to grasp the manner of sentence formation in the target language, enabling effective writing.
Archaeological corpora[edit]
Text corpora are also used in the study of historical documents, for example in attempts to decipher ancient scripts, or in Biblical scholarship. Some archaeological corpora can be of such short duration that they provide a snapshot in time. One of the shortest corpora in time, may be the 15–30 year Amarna letters texts (1350 BC). The corpus of an ancient city, (for example the "Kültepe Texts" of Turkey), may go through a series of corpora, determined by their find site dates.
Some notable text corpora[edit]
Main article: List of text corpora
See also[edit]
Concordance
Corpus linguistics
Linguistic Data Consortium
Natural language processing
Natural Language Toolkit
Parallel text alignment
Search engines: they access the "web corpus".
Speech corpus
Translation memory
Treebank
Zipf's Law
References[edit]
^Wołk, K.; Marasek, K. "A Sentence Meaning Based Alignment Method for Parallel Text Corpora Preparation". Advances in Intelligent Systems and Computing. Springer. 275: 107–114. ISBN978-3-319-05950-1. ISSN2194-5357.
^Wołk, K.; Marasek, K. (2015). "Tuned and GPU-accelerated Parallel Data Mining from Comparable Corpora". Lecture Notes in Artificial Intelligence. Springer: 32–40. ISBN978-3-319-24032-9.
^Yoon, H., & Hirvela, A. (2004). ESL Student Attitudes toward Corpus Use in L2 Writing. Journal of Second Language Writing, 13(4), 257–283. Retrieved 21 March 2012.
External links[edit]
ACL SIGLEX Resource Links: Text Corpora
Developing Linguistic Corpora: a Guide to Good Practice
Free samples (not free), web-based corpora (45-425 million words each): American (COCA, COHA, TIME), British (BNC), Spanish, Portuguese
Intercorp Building synchronous parallel corpora of the languages taught at the Faculty of Arts of Charles University.
Sketch Engine: Open corpora with free access
TS Corpus - A Turkish Corpus freely available for academic research.
Turkish National Corpus - A general-purpose corpus for contemporary Turkish
Corpus of Political Speeches, publicly accessible with speeches from United States, Hong Kong, Taiwan, and China, provided by Hong Kong Baptist University Library
| v t e Natural language processing |
| General terms | Natural language understanding Text corpus Speech corpus Stopwords Bag-of-words AI-complete n-gram (Bigram, Trigram) |
| Text analysis | Text segmentation Part-of-speech tagging Text chunking Compound term processing Collocation extraction Stemming Lemmatisation Named-entity recognition Coreference resolution Sentiment analysis Concept mining Parsing Word-sense disambiguation Ontology learning Terminology extraction Truecasing |
| Automatic summarization | Multi-document summarization Sentence extraction Text simplification |
| Machine translation | Computer-assisted Example-based Rule-based Neural |
| Automatic identification and data capture | Speech recognition Speech synthesis Optical character recognition Natural language generation |
| Topic model | Pachinko allocation Latent Dirichlet allocation Latent semantic analysis |
| Computer-assisted reviewing | Automated essay scoring Concordancer Grammar checker Predictive text Spell checker Syntax guessing |
| Natural language user interface | Automated online assistant Chatbot Interactive fiction Question answering Voice user interface |
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12067 | https://www.webqc.org/molecular-weight-of-argon.html | Printed from
Molar Mass, Molecular Weight and Elemental Composition Calculator
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| Molar mass of Argon (Ar) is 39.9480 g/mol Convert between Ar weight and moles | Compound | Moles | Weight, g | --- | Ar | | | Elemental composition of Ar | Element | Symbol | Atomic weight | Atoms | Mass percent | --- --- | Argon | Ar | 39.948 | 1 | 100.0000 | | Computing molar mass step by step | | First, compute the number of each atom in Ar:Ar: 1 Then, lookup atomic weights for each element in periodic table: Ar: 39.948 Now, compute the sum of products of number of atoms to the atomic weight:Molar mass (Ar) = ∑ Counti Weighti = Count(Ar) Weight(Ar) = 1 39.948 = 39.9480 g/mol | | Mass Percent Composition | Atomic Percent Composition | --- | | Ar Argon (100.00%) | Ar Argon (100.00%) | | Mass Percent Composition | | Ar Argon (100.00%) | | Atomic Percent Composition | | Ar Argon (100.00%) | | Appearance | | Argon is a colorless, odorless, and tasteless gas that is typically found in its natural state as a monatomic molecule. | Related compounds | Formula | Compound name | --- | | Ar2 | Diargon | | Related | | Oxidation state calculator | | Compound properties | |
| Computing molar mass (molar weight)To calculate molar mass of a chemical compound enter its formula and click 'Compute'. In chemical formula you may use: Any chemical element. Capitalize the first letter in chemical symbol and use lower case for the remaining letters: Ca, Fe, Mg, Mn, S, O, H, C, N, Na, K, Cl, Al. Functional groups: D, T, Ph, Me, Et, Bu, AcAc, For, Tos, Bz, TMS, tBu, Bzl, Bn, Dmg parenthesis () or brackets []. Common compound names. Examples of molar mass computations: NaCl, Ca(OH)2, K4[Fe(CN)6], CuSO45H2O, nitric acid, potassium permanganate, ethanol, fructose, caffeine, water. Molar mass calculator also displays common compound name, Hill formula, elemental composition, mass percent composition, atomic percent compositions and allows to convert from weight to number of moles and vice versa.Computing molecular weight (molecular mass)To calculate molecular weight of a chemical compound enter it's formula, specify its isotope mass number after each element in square brackets. Examples of molecular weight computations: CO2, SO2.Definitions Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). (1 u is equal to 1/12 the mass of one atom of carbon-12) Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. Mole is a standard scientific unit for measuring large quantities of very small entities such as atoms and molecules. One mole contains exactly 6.022 ×1023 particles (Avogadro's number) Steps to calculate molar mass 1. Identify the compound: write down the chemical formula of the compound. For example, water is H2O, meaning it contains two hydrogen atoms and one oxygen atom. 2. Find atomic masses: look up the atomic masses of each element present in the compound. The atomic mass is usually found on the periodic table and is given in atomic mass units (amu). 3. Calculate molar mass of each element: multiply the atomic mass of each element by the number of atoms of that element in the compound. 4. Add them together: add the results from step 3 to get the total molar mass of the compound. Example: calculating molar mass Let's calculate the molar mass of carbon dioxide (CO2): Carbon (C) has an atomic mass of about 12.01 amu. Oxygen (O) has an atomic mass of about 16.00 amu. CO2 has one carbon atom and two oxygen atoms. The molar mass of carbon dioxide is 12.01 + (2 × 16.00) = 44.01 g/mol. Lesson on computing molar massPractice what you learned: Practice calculating molar mass Weights of atoms and isotopes are from NIST article. Related: Molecular weights of amino acids |
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12068 | https://stackoverflow.com/questions/10568081/partitioning-a-sequence-into-sets-of-unique-pairs | Stack Overflow
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Partitioning a sequence into sets of unique pairs
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4
I'm in need a of function which can split a sequence into pairs, and then combine them such that all elements in a combination is unique. I have tried a number of approaches using python's itertools, but have not found a solution.
To illustrate i would like a function which would take this sequence: [1, 2, 3, 4]
and split it into the following 3 combinations:
it should also work for longer sequences, but does not have to handle sequences of odd length. eg.
[1,2,3,4,5,6]
splits into the following 15 combinations:
... and so on.
The CAS called Maple has this function implemented under the name setpartition.
Edit: fixed a critical late night typing error pointed out by wks, and clarified the outputs.
python
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edited May 13, 2012 at 8:41
FrederikNSFrederikNS
asked May 12, 2012 at 23:07
FrederikNSFrederikNS
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1
1
Yes, I must have been very tired.
FrederikNS
– FrederikNS
2012-05-13 08:23:38 +00:00
Commented May 13, 2012 at 8:23
Add a comment |
3 Answers 3
Reset to default
5
itertoolsis indeed your friend:
from itertools import permutations def group(iterable, n=2): return zip(([iter(iterable)] n)) for each in permutations([1, 2, 3, 4, 5, 6]): print map(list, group(each))
Result:
...
[EDIT] @FrederikNS: After you clarified your question and found an answer yourself, here is my solution:
from itertools import combinations def setpartition(iterable, n=2): iterable = list(iterable) partitions = combinations(combinations(iterable, r=n), r=len(iterable) / n) for partition in partitions: seen = set() for group in partition: if seen.intersection(group): break seen.update(group) else: yield partition for each in setpartition([1, 2, 3, 4]): print each print for each in setpartition([1, 2, 3, 4, 5, 6]): print each
Result:
((1, 2), (3, 4)) ((1, 3), (2, 4)) ((1, 4), (2, 3)) ((1, 2), (3, 4), (5, 6)) ((1, 2), (3, 5), (4, 6)) ((1, 2), (3, 6), (4, 5)) ((1, 3), (2, 4), (5, 6)) ((1, 3), (2, 5), (4, 6)) ((1, 3), (2, 6), (4, 5)) ((1, 4), (2, 3), (5, 6)) ((1, 4), (2, 5), (3, 6)) ((1, 4), (2, 6), (3, 5)) ((1, 5), (2, 3), (4, 6)) ((1, 5), (2, 4), (3, 6)) ((1, 5), (2, 6), (3, 4)) ((1, 6), (2, 3), (4, 5)) ((1, 6), (2, 4), (3, 5)) ((1, 6), (2, 5), (3, 4))
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edited May 23, 2017 at 12:04
CommunityBot
111 silver badge
answered May 12, 2012 at 23:50
pillmuncherpillmuncher
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2 Comments
jamylak
jamylak
+1 this is great but i don't like the use of each as a name since it can be confused as a language construct from other languages.
FrederikNS
FrederikNS
This is cool, but unfortunately not what I was going for. I have clarified my question further.
1
I finally got it my self (pillmuncher's answer really gave me a nudge in the right direction, and the group function is entirely his):
def group(iterable, n=2): return zip(([iter(iterable)] n)) def set_partition(iterable, n=2): set_partitions = set() for permutation in itertools.permutations(iterable): grouped = group(list(permutation), n) sorted_group = tuple(sorted([tuple(sorted(partition)) for partition in grouped])) set_partitions.add(sorted_group) return set_partitions partitions = set_partition([1,2,3,4], 2) for part in partitions: print(part)
this prints :
((1, 4), (2, 3)) ((1, 3), (2, 4)) ((1, 2), (3, 4))
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answered May 13, 2012 at 9:54
FrederikNSFrederikNS
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-1
Try this:
def function(list): combinations = [] for i in list: for i2 in list: if not [i2, i] in combinations: combinations.append([i, i2]) return combinations
This returns every possible combination.
Hope this helps!
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edited May 13, 2012 at 0:01
answered May 12, 2012 at 23:47
HarrisonHarrison
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Nick Craig-Wood
Nick Craig-Wood
That isn't what the question wanted - see the extra grouping in the example. You've re-written itertools.combinations
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12069 | https://www.vedantu.com/content-files-downloadable/ncert-solutions/ncert-solutions-class-11-chemistry-chapter-10-the-s-block-elements.pdf | Class XI Chemistry www.vedantu.com 1 Chemistry Chapter – 10 The s-Block Elements NCERT Exercise: 1. What are the common physical and chemical features of alkali metals? Ans: Physical properties of alkali metals: ● Alkali metals have low ionization enthalpies. ● Alkali metals are highly electropositive in nature ● Alkali metals exhibit +1 oxidation states in their compounds. ● Alkali metals impart characteristic colours to the flame. Chemical properties of alkali metals: ● Alkali metals are highly reactive in nature. ● Alkali metals hydroxides are highly basic in nature. ● Alkali metals dissolve in liquid ammonia to form blue and conducting solution. 2. Discuss the general characteristics and gradation in properties of alkaline earth metals. Ans: ● Atomic size goes on increasing down the group. ● Ionization energy goes on decreasing down the group. ● They are harder than alkali metals. ● They are less electropositive than alkali metals. ● Electropositive character increases on going down the group. 3. Why are alkali metals not found in nature? Ans: Alkali metals are highly reactive in nature. That’s why they always exist in a combined state in nature. 4. Find out the oxidation state of sodium in 2 2 Na O . Ans: let x be the oxidation of sodium in 2 2 Na O 2x 2( 1) 0 2x 2 0 x 1 5. Explain why sodium is less reactive than potassium. Ans: it is because ionization enthalpy i H of potassium = 419 kJ/mol Ionization enthalpy of sodium = 496 kJ/mol Since, ionization enthalpy of potassium is less than that of sodium, potassium is NCERT solutions for Class 11 Class XI Chemistry www.vedantu.com 2 more reactive than sodium 6. Compare the alkali metals and alkaline earth metals with respect to a. Ionization enthalpy Ans: Because of high nuclear charge the ionization enthalpy of alkaline earth metals are higher than those of the corresponding alkali metals. b. Basicity of oxides Ans: Basicity of oxides of alkali metals are higher than that of alkaline earth metals. c. Solubility of hydroxides Ans: Solubility of hydroxides of alkali metals are higher than that of alkaline earth metals. Alkali metals due to lower ionization enthalpy are more electropositive than the corresponding group 2 elements. 7. In what ways lithium shows similarities to magnesium in its chemical behavior? Ans: ● Both react with nitrogen to form nitrides ● Both react with 2 O to form monoxides. ● Both the elements have the tendency to form covalent compounds ● Both can form complex compounds 8. Explain why alkali and alkaline earth metals can not be obtained by chemical reduction methods. Ans: Alkali and alkaline earth metals are better reducing agents, and reducing agents better than alkali metals are not available. That is why these metals are not obtained by chemical reduction methods. 9. Why are potassium and caesium, rather than lithium used in photoelectric cells? Ans: Potassium and caesium have much lower ionization enthalpy than that of lithium. As a result, these metals easily emit electrons on exposure to light. Due to this, K and Cs are used in photoelectric cells rather than lithium. 10. When alkali metal dissolves in liquid ammonia, the solution can acquire different colors. Explain the reason for this type of color change. Ans: Alkali metals dissolve in liquid ammonia and give deep blue solutions which are conducting in nature because ammoniated electrons absorb energy in the visible region of light and impart blue colour. Class XI Chemistry www.vedantu.com 3 3 3 x 3 y Ammoniated electrons M (x y)NH [M(NH ) ] e (NH ) 11. Beryllium and magnesium do not give colour to flame whereas other alkaline earth metals do so. Why? Ans: Due to the small size, the ionization enthalpies of Be and Mg are much higher than those of other alkaline earth metals. Therefore, a large amount of energy is needed to excite their valence electrons, and that’s why they do not impart colour to the flame. 12. Discuss the various reactions that occur in the Solvay process. Ans: Heat 3 2 CaCO (s) CaO CO 3 2 4 NH H O NH OH 4 2 3 4 NaCl NH OH CO NaHCO NH Cl heat 3 2 3 2 2 2NaHCO (s) Na CO CO H O 2 3 2 2 3 2 Na CO 10H O Na CO .10H O 13. Potassium Carbonate cannot be prepared by solvay process. Why? Ans: Potassium carbonate being more soluble than sodium bicarbonate does not get precipitated when 2 CO is passed through a concentrated solution of KCl saturated with ammonia. 14. Why is 2 3 Li CO decomposed at a lower temperature whereas 2 3 Na Co at high temperature? Ans: 2 3 Li CO is a covalent compound whereas 2 3 Na Co is an ionic compound. Therefore, Lattice energy of 2 3 Na Co is higher than that of 2 3 Li CO . Thus, 2 3 Li CO is decomposed at a lower temperature. 15. Compare the solubility and thermal stability of the following compounds of the alkali metals with those of the alkaline earth metals. a. Nitrates Ans: Nitrates of both group 1 and group 2 elements are soluble in water because Hydration energy is more than the lattice energy. Nitrates of both group 1 and group 2 elements are thermally unstable but they decompose differently except 3 LiCO e.g. heat 3 2 2 2NaNO 2NaNO O 3 2 2 2KNO 2KNO O Class XI Chemistry www.vedantu.com 4 3 2 2 2 4LiNO 2LiO 4NO O 3 2 2 2 2Mg(NO ) 2MgO 4NO O b. Carbonates Ans: carbonates of group 1 elements are soluble in water except 2 3 Li CO . 2 3 2 2 Li CO Li O CO Group 2 carbonates are insoluble in water because their Lattice energy are higher than hydration energy.Thermal stability of carbonates of group 2 increases down the group because Lattice energy goes on increasing due to increase in ionic character. c. Sulphates Ans: Sulphates of group 1 are soluble in water except Li2SO4. They are thermally stable. Solubility of sulphates of group 2 decreases down the group because Lattice energy dominates over hydration energy. Sulphates of group 2 elements are thermally stable and increasing down the group due to increases in Lattice energy 16. Starting with sodium chloride how could you proceed to prepare a. Sodium metal Ans: sodium metal is manufactured by electrolysis of a fused mass of NaCl 40% and 2 CaCl 60% in Down’s cell at 873K, using iron as cathode and graphite as anode. Na is liberated at the cathode. At cathode: Na e Na(l) At anode: 2 2Cl (melt) Cl (g) 2e b. Sodium hydroxide Ans: sodium hydroxide is manufactured by electrolysis of an aqueous solution of NaCl (brine) in Castner- Kellner cell At cathode: Na e Na 2 2Na Hg Na Hg 2H O 2 2 2Na Hg 2H O 2NaOH H Hg At anode: Cl e Cl 2 Cl Cl Cl c. Sodium peroxide Ans: 2 2 2 4Na 2O 2Na O O d. Sodium carbonate Class XI Chemistry www.vedantu.com 5 Ans: sodium carbonate is obtained by Solvay ammonia process 3 2 2 3 4 NaCl NH CO H O NaHCO NH Cl 3 2 3 2 2 2NaHCO Na CO CO H O 17. What happens when (i) magnesium is burnt in air, (ii) Quick lime is heated with silica (iii) chlorine reacts with slaked lime (iv) calcium nitrate is heated? Ans: a. 2 2Mg(s) O (g) 2MgO(s) b. 2 3 CaO(s) SiO (s) CaSiO (s) c. 2 2 2 2 2 2Ca(OH) 2Cl CaCl Ca(OCl) 2H O d. 3 2 2 2 2Ca(NO ) (s) 2CaO(s) 4NO (g) O (g) 18. Describe the two important uses of the following. a. Caustic soda Ans: i. It is used in the manufacturing of soap paper , artificial silk etc. ii. It is used in the textile industries b. Sodium carbonate Ans: i. Used in the softening of water, for laundry and cleaning purposes. ii. It is used in glass manufacturing c. Quick lime Ans: i. It is used in the preparation of bleaching powder ii. Used in the purification of sugar and in the manufacturing of cement. 19. Draw the structure of the following: a. 2 BeCl (vapour) Ans: in the vapor site , it exists as a chloro bridged dimer b. 2 BeCl (solid) Class XI Chemistry www.vedantu.com 6 20. The hydroxides and carbonates of sodium and potassium are easily soluble in water while the corresponding salts of magnesium and calcium are sparingly soluble in water. Explain Ans: Since group 1 hydroxides and carbonates due to large size contain higher hydration energy than the lattice energy so, they are easily soluble in water. Whereas, in magnesium and calcium due to small size, their lattice energy dominates over hydration energy. Hence they are sparingly soluble in water. 21. Describe the importance of the following. a. Limestone Ans: ● Extensively used in manufacturing of high quality paper ● Used in mild abrasive in toothpaste ● As a filler in cosmetics ● Used as an antacid b. Cement Ans: ● An important building material ● Used in concrete and reinforced cement c. Plaster of Paris Ans: ● Used in plasters ● In dentistry, in ornamental work for making status 22. Why are lithium salts commonly hydrated and those of the other alkali metal ions usually anhydrous? Ans: Due to smallest size, Li can polarize water molecules easily than the other alkali metal ions 23. Why is LiF almost insoluble in water whereas LiCl soluble not only in water but also in acetone? Ans: It is due to high lattice energy of LiF as compared to LiCl. LiCl is soluble in water because its hydration energy is higher than its lattice energy Class XI Chemistry www.vedantu.com 7 24. Explain the significance of sodium, potassium, magnesium and calcium in biological fluids. Ans: Sodium ions: ● Na ions participate in the transmission of nerve signals, in regulating the flow of water across cell membranes. ● In the transport of sugars and amino acids into cell Potassium ions: ● They active many enzymes ● Participate in the oxidation of glucose to produce ATP. Magnesium ions: ● All enzymes that utilize ATP in phosphate transfer require magnesium as a cofactor. ● Mg is the main pigment for the absorption of light in plants. Calcium ions: ● 2 Ca ions are present in bones ● Plays important roles in neuromuscular function 25. What happens when a. Sodium metal is dropped in water? Ans: 2 2 2Na 2H O 2NaOH H b. Sodium metal is heated in a free supply of air? Ans: 2 2 2 2Na O Na O c. Sodium peroxide dissolves in water? Ans: 2 2 2 2 2 Na O 2H O 2NaOH H O 26. Comment on each of the following observation: a. The mobility of the alkali metal ions in aqueous solutions are Li Na K Rb Cs Ans: Smaller the size of the ion, more highly it is hydrated and hence greater is the mass of the hydrated ion and thus the ionic mobility becomes lesser. The extent of hydration decreases in the order. Li Na K Rb Cs Thus, the mobility of Cs will be the highest b. Lithium is the only alkali metal to form a nitride directly. Ans: due to its smaller size lithium can form nitride directly Class XI Chemistry www.vedantu.com 8 c. E for 2 M (aq) 2e M(s) ( where M= Ca, Sr, or Ba) Ans: it is because reduction potential depends upon sublimation energy, ionization energy and hydration energy. Their resultant is almost constant for these ions. 27. State as to why a. A solution of 2 3 Na CO is alkaline? Ans: 2 3 Na CO is a salt of a weak acid, carbonic acid 2 3 (H CO ) and a strong base NaOH. Thus it undergoes hydrolysis to produce a strong base NaOH and its aqueous solution is alkaline in nature. 2 3 2 Na CO (s) H O(l) 2NaOH b. Alkali metals are prepared by electrolysis of their fused chlorides? Ans: Because the discharge potential of alkali metals is much higher than that of hydrogen, therefore when the aqueous solution of any alkali metal chlorides is subjected to electrolysis, 2 H , instead of the alkali metal, is produced at the cathode. Therefore alkali metals are prepared by electrolysis of their fused chlorides. c. Sodium is found to be more useful than potassium? Ans: since potassium is more reactive than sodium and it is found in nature to a less extent than Na, sodium is found to be more useful. 28. Write balanced equations for reactions between. a. 2 2 Na O and water Ans: 2 2 2 2 2 Na O 2H O 2NaOH H O b. 2 KO and water Ans: 2 2 2 2 2 2KO 2H O 2KOH O H O c. 2 Na O and 2 CO Ans: 2 2 2 3 Na O CO Na CO 29. How would you explain the following observations? BeO is almost insoluble 4 BeSO is soluble in water. Ans: Lattice energy of BeO is comparatively higher than the hydration energy. Therefore, it is almost insoluble in water. Whereas 4 BeSO is ionic in nature and its hydration energy dominates the lattice energy. BaO is soluble but 4 BaSO is insoluble in water. Class XI Chemistry www.vedantu.com 9 Ans: Both BaO and 4 BaSO are ionic compounds but the hydration energy of BaO is higher than the lattice energy therefore it is soluble in water. LiI is more soluble than KI in ethanol. Ans: Since the size of Li ion is very small in comparison to K ion, it polarizes the electron cloud I- ion to great extent. Thus LiI dissolves in ethanol more easily than the KI. 30. Which of the following alkali metals has the least melting point? a. Na b. K c. Rb d. Cs Ans: size of Cs is the biggest thus, its melting point is the lowest. Option (d) is correct 31. Which one of the following alkali metals gives hydrated salts? a. Li b. Na c. K d. Cs Ans: Li is the smallest. Thus it has the highest charge density and hence attracts the water molecules more strongly. 32. Which one of the following alkaline earth metal carbonates is thermally most stable? a. 3 MgCO b. 3 CaCO c. 3 SrCO d. 3 BaCO Ans: 3 BaCO I. Very Short Answer Type Questions 1. Name the alkali metal which shows diagonal relationship with magnesium? Ans: Lithium (Li) reduction method? 2. Why alkali and alkaline earth metals cannot be obtained by chemical Ans: Because alkali and alkaline earth metals are themselves stronger reducing Class XI Chemistry www.vedantu.com 10 agents than the majority of other reducing agents. 3. Name the compounds used for the manufacture of washing soda by Solvay process. Ans: 3 3 NaCl,CaCO & NH 4. Which electrolyte is used to obtain sodium in Castner’s process? Ans: Fused NaOH 5. What happens when crystals of washing soda are exposed to air? Ans: Monohydrate ( 2 3 2 Na CO H O ) is formed as a result of efflorescence. 6. Name the alkaline earth metals whose salt do not impart colour to a non-luminous flame. Ans: beryllium does not impart colour to a non-luminous flame. 7. What is dead burnt plaster? Ans: it is anhydrous calcium sulphate ( 4 CaSO ) 8. What is Quicklime? What happens when it is added to water? Ans: CaO is quicklime. When it is added to water, 2 Ca(OH) is formed. 9. Arrange the following in the increasing order of solubility in water? 2 2 2 2 MgCl ,CaCl ,SrCl ,BaCl Ans: 2 2 2 2 BaCl SrCl CaCl MgCl 10. Give the chemical formula of Epsom salt. Ans: 4 2 MgSO .7H O 11. How would you prepare sodium silicate from silica? Ans: Fuse 2 3 2 2 3 2 Sod. Silicate Na CO SiO Na SiO CO 12. What happens when sodium metal is heated in a free supply of air? Ans: sodium peroxide is formed 2 2 2 2Na O Na O 13. What is the general name for elements of group 1? Ans: alkali metals Class XI Chemistry www.vedantu.com 11 14. Why are alkali metals soft? Ans: since the atoms of alkali metals have bigger kernels and smaller number of valence electrons, the metallic bonds in them are very weak and hence are soft. 15. What do you mean by diagonal relationship in periodic table? Ans: The resemblance of the first element of the second period with diagonally situated elements of neighboring elements is called diagonal relationship. 16. Why is 2 BeCl soluble in organic solvent? Ans: since, 2 BeCl is a covalent compound that is soluble in organic solvent. 17. Why do alkali metals give characteristic flame coloration? Ans: Alkali metals due to low ionization energy absorb energy from the visible region to radiate complementary color. 18. Why is the solution of alkali metals in liquid ammonia conducting in nature? Ans: Due to ammoniated electrons and cations. 19. Which is more basic NaOH or 2 Mg(OH) ? Ans: NaOH is more basic 20. Which alkaline earth metals do not impart color to the flame? Ans: Be and Mg 21. What is soda ash? Ans: soda ash is anhydrous sodium carbonate ( 2 3 Na CO ) II. Short Answer Type Questions 1. Why are alkali metals always univalent? Which alkali metal ion forms the largest hydrated ion in aqueous solution? Ans: they are always univalent because after losing one electron, they acquire the nearest inert gas configuration. Li forms largest hydrated cations because it has the highest hydration energy. 2. What is the effect of heat on the following compounds (give equations for the reactions) a. 3 CaCO Ans: 3 2 CaCO CaO CO Class XI Chemistry www.vedantu.com 12 b. 4 2 CaSO .2H O Ans: 4 2 4 2 Dead burnt plaster CaSO .2H O CaSO 2H O 3. Explain the following: i. Lithium iodide is more covalent than lithium fluoride Ans: According to Fazan’s rule, Li ion can polarize I ion more than the F ion due to the bigger size of the anion. Thus LiI has more covalent character than LiF. ii. Lattice enthalpy of LiF is maximum among all the alkali metal halides. Ans: Smaller the size (internuclear distance), more is the value of Lattice enthalpy since internuclear distance is expected to be least in the LiF. 4. Write the chemical formula of the following compounds. i. Chile saltpetre Ans: 3 NaNO ii.Marble Ans: 3 CaCO iii. Brine Ans: NaCl 5. Explain the following: a. Why is Cs considered as the most electropositive element? Ans: Due to its lowest ionization energy, Cs is considered as the most electropositive element. b. Lithium cannot be used in making photoelectric cells Ans: lithium cannot be used in making photoelectric cells because out of all alkali metals it has highest ionization energy and thus cannot emit electrons when exposed to light. c. Lithium does not form alums Ans: due to small size, lithium does not form alums. 6. a. What makes lithium show properties uncommon to the rest of the alkali metals? Ans: The unusual properties of lithium as compared to other alkali metals are due to its exceptionally small size of atom and its ion and its high polarizing power. Class XI Chemistry www.vedantu.com 13 b. When is cation highly polarizing? Which alkali metal cation has the highest polarizing power? Ans: A cation is highly polarizing if its charge/size ratio is very high. Li ion has the highest polarizing power. 7. Why are only ionic hydrides of only alkali metals and alkaline earth metals known? Give two examples. Ans: Alkali metals and alkaline earth metals are most electropositive due to low ionization enthalpy therefore they form ionic hydrides, e.g. NaH, KH and 2 CaH . 8. Why does the solution of alkali metals become blue in liquid ammonia? Give the chemical equation also Ans: The blue colour of the solution is due to an ammoniated electron which absorbs energy in the visible region of light and imparts blue colour. 3 2 2 1 Na(am) e (am) NH (l) NaNH (am) ( )H (g) 2 9. Give the important uses of the following: i. 3 NaHCO Ans: ● It is used in fire extinguisher ● It is mild antiseptic for skin infections ● It is used as an antacid. ii. NaOH Ans: ● It is used in soap industry ● It is used in textile industry ● It is used as reagent in laboratory ● It is used in absorbing poisonous gases 10. What is the mixture of 2 CaC and 2 N called? How is it prepared? Ans: it is called Nitro lime. It is prepared by heating 2 CaC with 2 N at high temperature. 2 2 2 CaC N CaCN C III. Long Answer Type Questions 1. Compare a) Compare four properties of alkali and alkaline earth metals Class XI Chemistry www.vedantu.com 14 Alkali metals Alkaline earth metals (i) They are soft metals (i) they are harder than alkali metals (ii) Alkali metals show +1 oxidation s t state (ii) Alkaline earth metals show +2 oxidation s state (iii) Their carbonates are soluble in in awater except 2 3 Li CO (iii) their carbonates are insoluble in water (iv) except Li, alkali metals do not c form complex compounds (iv) they can form complex compounds b) What happens when alkali metals are dissolved in ammonia? Ans: they form a blue colored solution. The solution is paramagnetic in nature. c) 2 MgCl is electrolyzed. Ans: electrolysis 2 2 MgCl Mg 2Cl At cathode: electrolysis 2 Mg (aq) 2e Mg(s) At anode: 2 2Cl (aq) 2e Cl (g) 2. State as to why a) Alkali metals show only +1 oxidation state. Ans: Alkali metals have low ionization enthalpies. They have a strong tendency to lose 1 electron to form unipositive ions. Thus they show an oxidation state of +1 and are strongly electropositive b) Na and K impart colour to the flame but Mg does not. Ans: Valence electrons of alkali metals like Na and K easily absorb energy from the flame and are excited to higher energy levels. When these electrons return to the ground state, the energy is emitted in the form of light. Magnesium atoms have small size so electrons are strongly bound to the nucleus. [Thus they need large amount of energy for excitation of electrons to higher energy levels which is not possible in Bunsen flame] c) Lithium on being heated in air mainly forms the monoxide and not the peroxide. Ans: Due to the small size of Li it has a strong positive field which attracts the negative charge so strongly that it does not permit the oxide ion, 02-to combine with another oxygen atom to form peroxide ion d) Li is the best reducing agent in aqueous solution Class XI Chemistry www.vedantu.com 15 Ans: Since, among alkali metals, lithium has the most negative electrode potential ( 0 E 3.04V ) so, it is the strongest reducing agent in the aqueous solution. IV. Multiple Choice Questions 1. The reducing property of alkali metals follows that order a) Na K Rb Cs Li b) K Na Rb Cs Li c) Li Cs Rb K Na d) Rb Cs K Na Li Ans: a) Na K Rb Cs Li 2. Which of the following is least thermally stable? a) 3 MgCO b) 3 CaCO c) 3 SrCO d) 3 BeCO Ans: 3 BeCO 3. When heated to o 800 C , 3 NaNO gives a) 2 2 Na N O b) 2 2 NaNO O c) 2 2 2 Na O O N d) 3 2 NaN O Ans: 2 2 2 Na O O N 4. Lithium shows a diagonal relationship with a) Sodium b) Silicon c) Nitrogen d) Magnesium Ans: magnesium 5. In the Solvay process a) An ammonical brine solution is bicarbonate with 2 CO , forming 3 NaHCO which on decomposition at 0 150 C produces 2 3 Na CO b) A sodium amalgam reacts with water to produce NaOH which gives 2 3 Na CO on reacting with 2 CO c) A brine solution is made react with 3 BaCO to produce 2 3 Na CO Class XI Chemistry www.vedantu.com 16 d) All of the above Ans: An ammonical brine solution is bicarbonate with 2 CO , forming 3 NaHCO which on decomposition at 0 150 C produces 2 3 Na CO 6. Which oxide of the following is amphoteric? a) Pb b) Mg c) Ca d) Al Ans: a) Pb and d) Al 7. Alkaline earth metals are a) More reactive b) Less reducing c) More oxidizing d) Less basic than alkali metals Ans: b) less reducing and d) less basic than alkali metals 8. Which of the following is not peroxides? a) 2 KO b) 5 CrO c) 2 2 Na O d) 2 BaO Ans: a) 2 KO 9. Hydrides as well as halides of alkaline earth metals tend to polymerize a) Sr b) Ca c) Be d) Mg Ans: c) Be 10. Which of the following is used in photovoltaic cells? a) Na b) K c) Li d) Cs Ans: Cs V. Hots Questions: Class XI Chemistry www.vedantu.com 17 1. Why are alkali metals soft and have low melting points? Ans: Alkali metals have only one valence electron per metal atom. As a result, the binding energy of alkali metal ions in the close-packed metal lattices are weak. Therefore, these are soft and have low melting point. 2. Which out of the following can be used to store an alkali metal? 2 2 5 6 6 H O,C H OH&C H Ans: benzene can be used to store an alkali metal because other substances react with alkali metal as: 2 2 1 Na H O NaOH H 2 2 5 2 5 2 1 Na C H OH C H ONa H 2 3. Potassium carbonate cannot be prepared by Solvay process. Why? Ans: This is due to the reason that potassium bicarbonate ( 3 KHCO ) formed as an intermediate (when 2 CO gas is passed through an ammoniated solution of potassium chloride) is highly soluble in water and cannot be separated by filtration. 4. The hydroxides and carbonates of sodium and potassium are easily soluble in water while the corresponding salts of magnesium and calcium are sparingly soluble in water. Explain. Ans: All the compounds are crystalline solids and their solubility in water is guided by both lattice enthalpy and hydration enthalpy. In case of sodium and potassium compounds, the magnitude of lattice enthalpy is quite small as compared to sodium and potassium that are mentioned, readily dissolve in water. However, in case of corresponding magnesium and calcium compounds, the cations have smaller sizes and more magnitude of positive charge. This means that their lattice enthalpies are more as compared to the compounds of sodium and potassium. Therefore, the hydroxides and carbonates of these metals are only sparingly soluble in water. 5. Why is LiF almost insoluble in water whereas LiCl soluble not only in water but also in acetone? Ans: The low solubility of LiF in water is due to its very high lattice enthalpy ( F ion is very small in size). On the other hand, in lithium chloride (LiCl) the lattice enthalpy is comparatively very small. This means that the magnitude of hydration enthalpy is quite large. Therefore lithium chloride dissolves in water. It is also soluble in acetone due to dipolar attraction. (Acetone is polar in nature). |
12070 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3153074/ | Statistical Inference: The Big Picture - PMC
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Published in final edited form as: Stat Sci. 2011 Feb 1;26(1):1–9. doi: 10.1214/10-STS337
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Statistical Inference: The Big Picture
Robert E Kass
Robert E Kass
Department of Statistics, Center for the Neural Basis of Cognition, and Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, 15217
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Department of Statistics, Center for the Neural Basis of Cognition, and Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, 15217
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Robert E Kass: Professor
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PMCID: PMC3153074 NIHMSID: NIHMS274606 PMID: 21841892
The publisher's version of this article is available at Stat Sci
Abstract
Statistics has moved beyond the frequentist-Bayesian controversies of the past. Where does this leave our ability to interpret results? I suggest that a philosophy compatible with statistical practice, labelled here statistical pragmatism, serves as a foundation for inference. Statistical pragmatism is inclusive and emphasizes the assumptions that connect statistical models with observed data. I argue that introductory courses often mis-characterize the process of statistical inference and I propose an alternative “big picture” depiction.
Keywords: Bayesian, confidence, frequentist, statistical education, statistical pragmatism, statistical significance
1 INTRODUCTION
The protracted battle for the foundations of statistics, joined vociferously by Fisher, Jeffreys, Neyman, Savage, and many disciples, has been deeply illuminating, but it has left statistics without a philosophy that matches contemporary attitudes. Because each camp took as its goal exclusive ownership of inference, each was doomed to failure. We have all, or nearly all, moved past these old debates, yet our textbook explanations have not caught up with the eclecticism of statistical practice.
The difficulties go both ways. Bayesians have denied the utility of confidence and statistical significance, attempting to sweep aside the obvious success of these concepts in applied work. Meanwhile, for their part, frequentists have ignored the possibility of inference about unique events despite their ubiquitous occurrence throughout science. Furthermore, intepretations of posterior probability in terms of subjective belief, or confidence in terms of long-run frequency, give students a limited and sometimes confusing view of the nature of statistical inference. When used to introduce the expression of uncertainty based on a random sample, these caricatures forfeit an opportunity to articulate a fundamental attitude of statistical practice.
Most modern practitioners have, I think, an open-minded view about alternative modes of inference, but are accutely aware of theoretical assumptions and the many ways they may be mistaken. I would suggest that it makes more sense to place in the center of our logical framework the match or mis-match of theoretical assumptions with the real world of data. This, it 3 seems to me, is the common ground that Bayesian and frequentist statistics share; it is more fundamental than either paradigm taken separately; and as we strive to foster widespread understanding of statistical reasoning, it is more important for beginning students to appreciate the role of theoretical assumptions than for them to recite correctly the long-run interpretation of confidence intervals. With the hope of prodding our discipline to right a lingering imbalance, I attempt here to describe the dominant contemporary philosophy of statistics.
2 STATISTICAL PRAGMATISM
I propose to call this modern philosophy statistical pragmatism. I think it is based on the following attitudes:
Confidence, statistical significance, and posterior probability are all valuable inferential tools.
Simple chance situations, where counting arguments may be based on symmetries that generate equally-likely outcomes (6 faces on a fair die; 52 cards in a shuffled deck), supply basic intuitions about probability. Probability may be built up to important but less immediately intuitive situations using abstract mathematics, much the way real numbers are defined abstractly based on intuitions coming from fractions. Probability is usefully calibrated in terms of fair bets: another way to say the probability of rolling a 3 with a fair die is 1/6 is that 5 to 1 odds 4 against rolling a 3 would be a fair bet.
Long-run frequencies are important mathematically, interpretively, and pedagogically. However, it is possible to assign probabilities to unique events, including rolling a 3 with a fair die or having a confidence interval cover the true mean, without considering long-run frequency. Long-run frequencies may be regarded as consequences of the law of large numbers rather than as part of the definition of probability or confidence.
Similarly, the subjective interpretation of posterior probability is important as a way of understanding Bayesian inference, but it is not fundamental to its use: in reporting a 95% posterior interval one need not make a statement such as, “My personal probability of this interval covering the mean is .95.”
Statistical inferences of all kinds use statistical models, which embody theoretical assumptions. As illustrated in Figure 1, like scientific models, statistical models exist in an abstract framework; to distinguish this framework from the real world inhabited by data we may call it a “theoretical world.” Random variables, confidence intervals, and posterior probabilities all live in this theoretical world. When we use a statistical model to make a statistical inference we implicitly assert that the variation exhibited by data is captured reasonably well by the statistical model, so that the theoretical world corresponds reasonably well to the real world. Conclusions are drawn by applying a statistical inference technique, which is a theoretical construct, to some real data. Figure 1 depicts the conclusions as straddling the theoretical and real worlds. Statistical inferences may have implications for the real world of new observable phenomena, but in scientific contexts, conclusions most often concern scientific models (or theories), so that their “real world” implications (involving new data) are somewhat indirect (the new data will involve new, and different experiments).
Figure 1.
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The big picture of statistical inference. Statistical procedures are abstractly defined in terms of mathematics but are used, in conjunction with scientific models and methods, to explain observable phenomena. This picture emphasizes the hypothetical link between variation in data and its description using statistical models.
The statistical models in Figure 1 could involve large function spaces or other relatively weak probabilistic assumptions. Careful consideration of the connection between models and data is a core component of both the art of statistical practice and the science of statistical methodology. The purpose of Figure 1 is to shift the grounds for discussion.
Note, in particular, that data should not be confused with random variables. Random variables live in the theoretical world. When we say things like, “Let us assume the data are normally distributed” and we proceed to make a statistical inference, we do not need to take these words literally as asserting that the data form a random sample. Instead, this kind of language is a convenient and familiar shorthand for the much weaker assertion that, for our specified purposes, the variability of the data are adequately consistent with variability that would occur in a random sample. This linguistic amenity is used routinely in both frequentist and Bayesian frameworks. Historically, the distinction between data and random variables, the match of the model to the data, was set aside, to be treated as a separate topic apart from the foundations of inference. But once the data themselves were considered random variables, the frequentist-Bayesian debate moved into the theoretical world: it became a debate about the best way to reason from random variables to inferences about parameters. This was consistent with developments elsewhere. In other parts of science, the distinction between quantities to be measured and their theoretical counterparts within a mathematical theory can be relegated to a different subject—to a theory of errors. In statistics, we do not have that luxury, and it seems to me important, from a pragmatic viewpoint, to bring to center stage the identification of models with data. The purpose of doing so is that it provides different interpretations of both frequentist and Bayesian inference, interpretations which, I believe, are closer to the attitude of modern statistical practitioners.
A familiar practical situation where these issues arise is binary regression. A classic example comes from a psychophysical experiment conducted by Hecht et al. (1942), who investigated the sensitivity of the human visual system by constructing an apparatus that would emit flashes of light at very low intensity in a darkened room. Those authors presented light of varying intensities repeatedly to several subjects and determined, for each intensity, the proportion of times each subject would respond that he or she had seen a flash of light. For each subject the resulting data are repeated binary observations (“yes” perceived vs. “no” did not perceive) at each of many intensities and, these days, the standard statistical tool to analyze such data is logistic regression. We might, for instance, use maximum likelihood to find a 95% confidence interval for the intensity of light at which the subject would report perception with probability p = .5. Because the data reported by Hecht et al. involved fairly large samples, we would obtain essentially the same answer if instead we applied Bayesian methods to get an interval having 95% posterior probability. But how should such an interval be interpreted?
A more recent example comes from DiMatteo, Genovese, and Kass (2001), who illustrated a new nonparametric regression method called Bayesian adaptive regression splines (BARS) by analyzing neural firing rate data from inferotemporal cortex of a macaque monkey. The data came from a study ultimately reported by Rollenhagen and Olson (2005), which investigated the differential response of individual neurons under two experimental conditions. Figure 2 displays BARS fits under the two conditions. One way to quantify the discrepancy between the fits is to estimate the drop in firing rate from peak (the maximal firing rate) to the trough immediately following the peak in each condition. Let us call these peak minus trough differences, under the two conditions, φ 1 and φ 2. Using BARS, DiMatteo, Genovese, and Kass, reported a posterior mean of φ̂ 1 − φ̂ 2 = 50.0 with posterior standard deviation (±20.8). In follow-up work, Wallstrom, Liebner, and Kass (2008) reported very good frequentist coverage probability of 95% posterior probability intervals based on BARS for similar quantities under simulation conditions chosen to mimic such experimental data. Thus, a BARS-based posterior interval could be considered from either a Bayesian or frequentist point of view. Again we may ask how such an inferential interval should be interpreted.
Figure 2.
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(A) BARS fits to a pair of peri-stimulus time histograms displaying neural firing rate of a particular neuron under two alternative experimental conditions. (B) The two BARS fits are overlaid for ease of comparison.
3 INTERPRETATIONS
Statistical pragmatism involves mildly altered interpretations of frequentist and Bayesian inference. For definiteness I will discuss the paradigm case of confidence and posterior intervals for a normal mean based on a sample of size n, with the standard deviation being known. Suppose that we have n = 49 observations that have a sample mean equal to 10.2.
Frequentist assumptions
Suppose X 1, X 2, …, X n are i.i.d. random variables from a normal distribution with mean μ and standard deviation σ = 1. In other words, suppose X 1, X 2, …, X n form a random sample from a N(μ, 1) distribution.
Noting that x̄ = 10.2 and 49=7 we define the inferential interval
I=(10.2−2 7,10.2+2 7).
The interval I may be regarded as a 95% confidence interval. I now contrast the standard frequentist interpretation with the pragmatic interepretation.
Frequentist interpretation of confidence interval
Under the assumptions above, if we were to draw infinitely many random samples from a N(μ, 1) distribution, 95% of the corresponding confidence intervals ( X¯−2 7,X¯+2 7) would cover μ.
Pragmatic interpretation of confidence interval
If we were to draw a random sample according to the assumptions above, the 10 resulting confidence interval ( X¯−2 7,X¯+2 7) would have probability .95 of covering μ. Because the random sample lives in the theoretical world, this is a theoretical statement. Nonetheless, substituting
X¯=x¯(1)
together with
x¯=10.2(2)
we obtain the interval I, and are able to draw useful conclusions as long as our theoretical world is aligned well with the real world that produced the data.
The main point here is that we do not need a long-run interpretation of probability, but we do have to be reminded that the unique-event probability of .95 remains a theoretical statement because it applies to random variables rather than data. Let us turn to the Bayesian case.
Bayesian assumptions
Suppose X 1, X 2, …, X n form a random sample from a N(μ, 1) distribution and the prior distribution of μ is N(μ 0, τ 2), with τ 2≫1 49 and 49 τ 2 ≫ |μ 0|.
The posterior distribution of μ is normal, the posterior mean becomes
μ¯=τ 2 1 49+τ 2 10.2+1 49 1 49+τ 2 μ 0
and the posterior variance is
v=(49+1 τ 2)−1
but because τ 2≫1 49 and 49 τ 2 ≫ |μ 0| we have
μ¯≈10.2
and
v≈1 49.
Therefore, the inferential interval I defined above has posterior probability .95.
Bayesian interpretation of posterior interval: Under the assumptions above, the probability that μ is in the interval I is .95. Pragmatic interpretation of posterior interval: If the data were a random sample for which (2) holds, i.e., x̄ = 10.2, and if the assumptions above were to hold, then the probability that μ is in the interval I would be .95. This refers to a hypothetical value x̄ of the random variable X̄, and because X̄ lives in the theoretical world the statement remains theoretical. Nonetheless, we are able to draw useful conclusions from the data as long as our theoretical world is aligned well with the real world that produced the data.
Here, although the Bayesian approach escapes the indirectness of confidence within the theoretical world, it can not escape it in the world of data analysis because there remains the additional layer of identifying data with random variables. According to the pragmatic interpretation, the posterior is not, literally, a statement about the way the observed data relate to the unknown parameter μ because those objects live in different worlds. The language of Bayesian inference, like the language of frequentist inference, takes a convenient shortcut by blurring the distinction between data and random variables.
The commonality between frequentist and Bayesian inferences is the use of theoretical assumptions, together with a subjunctive statement. In both approaches a statistical model is introduced—in the Bayesian case the prior distributions become part of what I am here calling the model—and we may say that the inference is based on what would happen if the data were to be random variables distributed according to the statistical model. This modeling assumption would be reasonable if the model were to describe accurately the variation in the data.
4 IMPLICATIONS FOR TEACHING
It is important for students in introductory statistics courses to see the subject as a coherent, principled whole. Instructors, and textbook authors, may try to help by providing some notion of a “big picture.” Often this is done literally, with an illustration such as Figure 3 (e.g., Lovett, Meyer, and Thille, 2008). This kind of illustration can be extremely useful if referenced repeatedly throughout a course.
Figure 3.
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The big picture of statistical inference according to the standard conception. Here, a random sample is pictured as a sample from a finite population.
Figure 3 represents a standard story about statistical inference. Fisher introduced the idea of a random sample drawn from a hypothetical infinite population, and Neyman and Pearson’s work encouraged subsequent mathematical statisticians to drop the word “hypothetical” and instead describe statistical inference as analogous to simple random sampling from a finite population. This is the concept that Figure 3 tries to get across. My complaint is that it is not a good general description of statistical inference, and my claim is that Figure 1 is more accurate. For instance, in the psychophysical example of Hecht et al. discussed in Section 2, there is no population of “yes” or “no” replies from which a random sample is drawn. We do not need to struggle to make an analogy with a simple random sample. Furthermore, any thoughts along these lines may draw attention away from the most important theoretical assumptions, such as independence among the responses. Figure 1 is supposed to remind students to look for the important assumptions, and ask whether they describe the variation in the data reasonably accurately..
One of the reasons the population and sample picture in Figure 3 is so attractive pedagogically is that it reinforces the fundamental distinction between parameters and statistics through the terms population mean and sample mean. To my way of thinking, this terminology, inherited from Fisher, is unfortunate. Instead of “population mean” I would much prefer theoretical mean, because it captures better the notion that a theoretical distribution is being introduced, a notion that is reinforced by Figure 1.
I have found Figure 1 helpful in teaching basic statistics. For instance, when talking about random variables I like to begin with a set of data, where variation is displayed in a histogram, and then say that probability may be used to describe such variation. I then tell the students we must introduce mathematical objects called random variables, and in defining them and applying the concept to the data at hand, I immediately acknowledge that this is an abstraction, while also stating that—as the students will see repeatedly in many examples—it can be an extraordinarily useful abstraction whenever the theoretical world of random variables is aligned well with the real world of the data.
I have also used Figure 1 in my classes when describing attitudes toward data analysis that statistical training aims to instill. Specifically, I define statistical thinking, as in Brown and Kass (2009), to involve two principles:
Statistical models of regularity and variability in data may be used to express knowledge and uncertainty about a signal in the presence of noise, via inductive reasoning.
Statistical methods may be analyzed to determine how well they are likely to perform.
Principle 1 identifies the source of statistical inference to be the hypothesized link between data and statistical models. In explaining, I explicitly distinguish the use of probability to describe variation and to express knowledge. A probabilistic description of variation would be “The probability of rolling a 3 with a fair die is 1/6” while an expression of knowledge would be “I’m 90% sure the capital of Wyoming is Cheyenne.” These two sorts of statements, which use probability in different ways, are sometimes considered to involve two different kinds of probability, which have been called “aleatory probability” and “epistemic probability.” Bayesians merge these, applying the laws of probability to go from quantitative description to quantified belief, but in every form of statistical inference alleatory probability is used, somehow, to make epistemic statements. This is Principle 1. Principle 2 is that the same sorts of statistical models may be used to evaluate statistical procedures—though in the classroom I also explain that performance of procedures is usually investigated under varying circumstances.
For somewhat more advanced audiences it is possible to elaborate, describing in more detail the process trained statisticians follow when reasoning from data. A big picture of the overall process is given in Figure 4. That figure indicates the hypothetical connection between data and random variables, between key features of unobserved mechansims and parameters, and between real-world and theoretical conclusions. It further indicates that data display both regularity (which is often described in theoretical terms as a “signal,” sometimes conforming to simple mathematical descriptions or “laws”) and unexplained variability, which is usually taken to be “noise.” The figure also includes the components exploratory data analysis—EDA—and algorithms, but the main message of Figure 4, given by the labels of the two big boxes, is the same as that in Figure 1.
Figure 4.
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A more elaborate big picture, reflecting in greater detail the process of statistical inference. As in Figure 1, there is a hypothetical link between data and statistical models but here the data are connected more specifically to their representation as random variables.
5 DISCUSSION
According to my understanding, laid out above, statistical pragmatism has two main features: it is eclectic and it emphasizes the assumptions that connect statistical models with observed data. The pragmatic view acknowledges that both sides of the frequentist-Bayesian debate made important points. Bayesians scoffed at the artificiality in using sampling from a finite population to motivate all of inference, and in using long-run behavior to define characteristics of procedures. Within the theoretical world, posterior probabilities are more direct, and therefore seemed to offer much stronger inferences. Frequentists bristled, pointing to the subjectivity of prior distributions. Bayesians responded by treating subjectivity as a virtue on the grounds that all inferences are subjective yet, while there is a kernel of truth in this observation—we are all human beings, making our own judgments—subjectivism was never satisfying as a logical framework: an important purpose of the scientific enterprise is to go beyond personal decision-making. Nonetheless, from a pragmatic perspective, while the selection of prior probabilities is important, their use is not so problematic as to disqualify Bayesian methods, and in looking back on history the introduction of prior distributions may not have been the central bothersome issue it was made out to be. Instead, it seems to me, the really troubling point for frequentists has been the Bayesian claim to a philosophical high ground, where compelling inferences could be delivered at negligible logical cost. Frequentists have always felt that no such thing should be possible. The difficulty begins not with the introduction of prior distributions but with the gap between models and data, which is neither frequentist nor Bayesian. Statistical pragmatism avoids this irritation by acknowledging explicitly the tenuous connection between the real and theoretical worlds. As a result, its inferences are necessarily subjunctive. We speak of what would be inferred if our assumptions were to hold. The inferential bridge is traversed, by both frequentist and Bayesian methods, when we act as if the data were generated by random variables. In the normal mean example discussed in Section 4, the key step involves the conjunction of the two equations (1) and (2). Strictly speaking, according to statistical pragmatism, Equation (1) lives in the theoretical world while Equation (2) lives in the real world; the bridge is built by allowing x̄ to refer to both the theoretical value of the random variable and the observed data value.
In pondering the nature of statistical inference I am, like others, guided partly by past and present sages (for an overview see Barnett, 1999), but also by my own experience and by watching many colleagues in action. Many of the sharpest, and most vicious Bayes-frequentist debates took place during the dominance of pure theory in academia. Statisticians are now more inclined to argue about the extent to which a method succeeds in solving a data analytic problem. Much statistical practice revolves around getting good estimates and standard errors in complicated settings where statistical uncertainty is smaller than the unquantified aggregate of many other uncertainties in scientific investigation. In such contexts, the distinction between frequentist and Bayesian logic becomes unimportant and contemporary practioners move freely between frequentist and Bayesian techniques using one or the other depending on the problem. Thus, in a review of statistical methods in neurophysiology in which my colleagues and I discussed both frequentist and Bayesian methods (Kass, Ventura, and Brown, 2005), not only did we not emphasize this dichotomy but we didn’t even mention the distinction between the approaches or their inferential interpretations.
In fact, in my first publication involving analysis of neural data (Olson, et al., 2001) we reported more than a dozen different statistical analyses, some frequentist, some Bayesian. Futhermore, methods from the two approaches are sometimes glued together in a single analysis. For example, to examine several neural firing-rate intensity functions λ 1(t), …, λ p(t), assumed to be smooth functions of time t, Behseta et al. (2007) developed a frequentist approach to testing the hypothesis H 0 : λ 1(t) = · · · = λ p(t), for all t, that incorporated BARS smoothing. Such hybrids are not uncommon, and they do not force a practitioner to walk around with mutually inconsistent interpretations of statistical inference. Figure 1 provides a general framework that encompasses both of the major approaches to methodology while emphasizing the inherent gap between data and modeling assumptions, a gap that is bridged through subjunctive statements. The advantage of the pragmatic framework is that it considers frequentist and Bayesian inference to be equally respectable and allows us to have a consistent interpretation, without feeling as if we must have split personalities in order to be competent statisticians. More to the point, this framework seems to me to resemble more closely what we do in practice: statisticians offer inferences couched in a cautionary attitude. Perhaps we might even say that most practitioners are subjunctivists.
I have emphasized subjunctive statements partly because, on the frequentist side, they eliminate any need for long-run interpretation. For Bayesian methods they eliminate reliance on subjectivism. The Bayesian point of view was articulated admirably by Jeffreys (see Robert, Chopin, and Rousseau, 2009, and accompanying discussion) but it became clear, especially from the arguments of Savage and subsequent investigations in the 1970s, that the only solid foundation for Bayesianism is subjective (see Kass and Wasserman, 1995, and Kass, 2008). Statistical pragmatism pulls us out of that solopsistic quagmire. On the other hand, I do not mean to imply that it really doesn’t matter what approach is taken in a particular instance. Current attention frequently focuses on challenging, high-dimensional data sets where frequentist and Bayesian methods may differ. Statistical pragmatism is agnostic on this. Instead, procedures should be judged according to their performance under theoretical conditions thought to capture relevant real-world variation in a particular applied setting. This is where our juxtaposition of the theoretical world with the real world earns its keep.
I called the story about statistical inference told by Figure 3 “standard” because it is imbedded in many introductory texts, such as the path-breaking book by Freedman, Pisani, and Purves (1978) and the excellent and very popular book by Moore and McCabe (1989). My criticism is that the standard story misrepresents the way statistical inference is commonly understood by trained statisticians, portraying it as analogous to simple random sampling from a finite population. As I noted, the population vs. sampling terminology comes from Fisher, but I believe the conception in Figure 1 is closer to Fisher’s conception of the relationship between theory and data. Fisher spoke pointedly of a hypothetical infinite population, but in the standard story of Figure 3 the “hypothetical” part of this notion—which is crucial to the concept—gets dropped (confer also Lenhard, 2006). I understand Fisher’s “hypothetical” to connote what I have here called “theoretical.” Fisher did not anticipate the co-option of his framework and was, in large part for this reason, horrified by subsequent developments by Neyman and Pearson. The terminology “theoretical” avoids this confusion and thus may offer a clearer representation of Fisher’s idea.1
We now recognize Neyman and Pearson to have made permanent, important contributions to statistical inference through their introduction of hypothesis testing and confidence. From today’s vantage point, however, their behavioral interpretation seems quaint, especially when represented by their famous dictum, “We are inclined to think that as far as a particular hypothesis is concerned, no test based upon the theory of probability can by itself provide any valuable evidence of the truth or falsehood of that hypothesis.” Nonetheless, that interpretation seems to have inspired the attitude behind Figure 3. In the extreme, one may be led to insist that statistical inferences are valid only when some chance mechanism has generated the data. The problem with the chance-mechanism conception is that it applies to a rather small part of the real world, where there is either actual random sampling or situations described by statistical or quantum physics. I believe the chance-mechanism conception errs in declaring that data are assumed to be random variables, rather than allowing the gap of Figure 1 to be bridged2 by statements such as (2). In saying this I am trying to listen carefully to the voice in my head that comes from the late David Freedman (see Freedman and Ziesel, 1988). I imagine he might call crossing this bridge, in the absence of an explicit chance mechanism, a leap of faith. In a strict sense I am inclined to agree. It seems to me, however, that it is precisely this leap of faith that makes statistical reasoning possible in the vast majority of applications.
Statistical models that go beyond chance mechanisms have been central to statistical inference since Fisher and Jeffreys, and their role in reasoning has been considered by many authors (e.g., Cox, 1990, Lehmann, 1990). An outstanding issue is the extent to which statistical models are like the theoretical models used throughout science (see Stanford, 2006). I would argue, on the one hand, that they are similar: the most fundamental belief of any scientist is that the theoretical and real worlds are aligned. On the other hand, as observed in Section 2, statistics is unique in having to face the gap between theoretical and real worlds every time a model is applied and, it seems to me, this is a big part of what we offer our scientific collaborators. Statistical pragmatism recognizes that all forms of statisical inference make assumptions, assumptions which can only be tested very crudely (with such things as goodness-of-fit methods) and can almost never be verified. This is not only at the heart of statistical inference, it is also the great wisdom of our field.
Acknowledgments
This work was supported in part by NIH grant MH064537. The author is grateful for comments on an earlier draft by Brian Junker, Nancy Reid, Steven Stigler, Larry Wasserman, and Gordon Weinberg.
Footnotes
1
Fisher also introduced populations partly because he used long-run frequency as a foundation for probability, which statistical pragmatism considers unnecessary.
2
Because probability is introduced with the goal of drawing conclusions via statistical inference, it is, in a philosophical sense, “instrumental.” See Glymour (2001).
References
Barnett V. Comparative Statistical Inference. 3. Wiley; 1999. [Google Scholar]
Behseta S, Kass RE, Moorman D, Olson CR. Testing equality of several functions: Analysis of single-unit firing rate curves across multiple experimental conditions. Statist Med. 2007;26:3958–3975. doi: 10.1002/sim.2940. [DOI] [PubMed] [Google Scholar]
Brown EN, Kass RE. What is statistics? (With discussion) Amer Statist. 2009;63:105–110. [Google Scholar]
Cox DR. Role of models in statistical analysis. Statist Sci. 1990;5:169–174. [Google Scholar]
DiMatteo I, Genovese CR, Kass RE. Bayesian curve-fitting with free-knot splines. Biometrika. 2001;88:1055–1071. [Google Scholar]
Freedman D, Pisani R, Purves R. Statistics. 4. W.W. Norton; 1978. 2007. [Google Scholar]
Glymour C. Instrumental probability. Monist. 2001;84:284–300. [Google Scholar]
Hecht, Schlaer, Pirenne J Gen Physiol. 1942;25:819–840. doi: 10.1085/jgp.25.6.819. [DOI] [PMC free article] [PubMed] [Google Scholar]
Kass RE. Kinds of Bayesians (Comment on articles by Berger and by Goldstein) Bayesian Anal. 2006;1:437–440. [Google Scholar]
Kass RE, Ventura V, Brown EN. Statistical issues in the analysis of neuronal data. J Neurophysiol. 2005;94:8–25. doi: 10.1152/jn.00648.2004. [DOI] [PubMed] [Google Scholar]
Kass RE, Wasserman LA. The selection of prior distributions by formal rules. J Amer Statist Assoc. 1996;91:1343–1370. [Google Scholar]
Lehmann EL. Model specification: The views of Fisher and Neyman, and later developments. Statist Sci. 1990;5:160–168. [Google Scholar]
Lenhard J. Models and statistical inference: The controversy between Fisher and Neyman-Pearson. Brit J Philo Sci. 2006;57:69–91. [Google Scholar]
Lovett M, Meyer O, Thille C. The open learning initiative: Measuring the effectiveness of the OLI statistics course in accelerating student learning. J Interactive Media in Education. 2008:14. [Google Scholar]
Moore DS, McCabe G. Introduction to the Practice of Statistics. 5. W.H. Freeman; 1989. 2005. [Google Scholar]
Robert CP, Chopin N, Rousseau J. Harold Jeffreys’ Theory of Probability, revisited (with discussion) Statistical Science. 2010 to appear. [Google Scholar]
Stanford PK. Exceeding Our Grasp. Oxford; 2006. [Google Scholar]
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12071 | https://possiblywrong.wordpress.com/2019/12/24/a-different-sock-matching-problem/ | A different sock matching problem | Possibly Wrong
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A different sock matching problem
Posted on December 24, 2019 by possiblywrong
When I take a load of laundry from the dryer, there are socks mixed in with all of the other clothes. Suppose that there are pairs of socks, socks in total, that I match and fold by randomly drawing one sock at a time from the basket. If the sock matches one that I have already drawn, I fold the matching pair and put it away in the sock drawer. Otherwise, I set the unmatched sock aside, anticipating matching it later. How much space does this take? That is, let be the maximum number of unmatched socks set aside at any point during this process. What is the distribution of ?
There are at least a couple of different possible problems here, depending on what constitutes a matching pair of socks. Arguably the most natural setup is that all pairs are distinct (e.g., each pair of my dress socks is a different color), so that each individual sock has exactly one mate. This is what has been described as the sock matching problem in the literature; see the references below.
My athletic socks, on the other hand, are essentially identical pairs, with each individual sock being distinguished only by a “left” or “right” label stitched into it, so that each sock may be matched with any of the other “differently-handed” socks. In this case, it’s a nice problem to show that
and thus
But what I found most interesting about this problem is that appears to be very well approximated by , with an error that I conjecture is always less than 1/2, and approaches zero in the limit as grows large. I don’t see how to prove this, though.
[Edit 2021-02-20: It turns out that the above conjecture is incorrect; thanks to Michael Earnest, who in the comments shows that the expected value is in fact , referencing an interesting paper by de Bruijn, Knuth, and S. O. Rice. Interestingly, that constant multiplier in the term is approximately 1.22857, slightly larger than my incorrect guess of which is about 1.22474. Thanks, Michael!]
References:
Gilliand, S., Johnson, C., Rush, S., and Wood, D., The sock matching problem, Involve, 7(5) 2014, p. 691-697. [PDF]
Pantic, B. and Bodroza-Pantic, O., A brief overview of the sock matching problem. [arXiv]
OEIS Foundation Inc. (2019), The On-Line Encyclopedia of Integer Sequences [A225177]
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17 thoughts on “A different sock matching problem”
Wilmington on December 24, 2019 at 2:43 pmsaid: Folding laundry must be quite interesting at your house!
I don’t know how to derive your distribution functions (hopefully you’ll show how later), but I did mock it up in Excel and could calculate up to 16 pairs of socks. No idea how to prove it.
For the conjecture, I’d put it slightly differently: It looks like E[U] approaches SQRT(3u/2) – 1/2 with an error in the limit of zero and the largest error of 1 of 1/2 – SQRT(3/2) occurring at n = 1.
Here are the Excel functions if anyone want to play with them:
Option Explicit
Private Function PrSpaceGreaterOrEqualGuts(u As Long, n As Long) As Long
Dim k As Long
Dim s As Long: s = 0
Dim c As Long
If u = 0 Then
PrSpaceGreaterOrEqualGuts = 1
Exit Function
End If
Dim limit As Long: limit = Int(n / u)
For k = 1 To limit
c = WorksheetFunction.Combin(n + n, n – k u)
If k Mod 2 = 1 Then
s = s + c
Else
s = s – c
End If
Next
PrSpaceGreaterOrEqualGuts = s
End Function
Private Function NormalizationFactor(n As Long) As Double
NormalizationFactor = 2# / WorksheetFunction.Combin(n + n, n)
End Function
Public Function PrSpaceGreaterOrEqual(u As Long, n As Long) As Double
PrSpaceGreaterOrEqual = NormalizationFactor(n) PrSpaceGreaterOrEqualGuts(u, n)
End Function
Public Function E_SockSpace(n As Long) As Double
Dim u As Long
Dim s As Long: s = 0
For u = 1 To n
s = s + PrSpaceGreaterOrEqualGuts(u, n)
Next
E_SockSpace = NormalizationFactor(n) s
End Function Reply ↓
wilmington on December 24, 2019 at 2:46 pmsaid: typo: largest error of 1/2 – SQRT(3/2) occurring at n = 1. This is approximately 0.275 Reply ↓
wilmington on December 24, 2019 at 6:51 pmsaid: another typo: It looks like E[U] approaches SQRT(3n/2) – 1/2 with an error in the limit of zero and the largest error of 1/2 – SQRT(3/2) occurring at n = 1. The error at 1 is approximately 0.275.
Image 20: possiblywrong's avatarpossiblywrong on December 26, 2019 at 1:21 pmsaid: Your code looks right, but I may misunderstand your description of the approximation error. If we use sqrt(3n/2)-1/2 as the approximation, then its error at n=1 (where E[U]=1) is sqrt(3/2)-3/2, or about -0.275. I’m guessing this is just a typo, since although your expression sqrt(3/2)-1/2 is off by 1, the decimal figure looks right (mod the sign)?
At any rate, although you are correct that offsetting the approximation by 1/2 like this does indeed reduce the magnitude of its error for small n, the resulting trend in the magnitude of that error is increasing for large n. See the plot linked in the comment replying to Chris. Reply ↓
Chris Wellons on December 25, 2019 at 10:50 pmsaid: Passes a Monte Carlo sniff test, though your estimate very consistently overestimates:
Code:
Output: Reply ↓
possiblywrong on December 26, 2019 at 1:24 pmsaid: Right; see this plot showing the error between the approximation and the exact expected value. It appears to approach zero from above… or at least that’s my guess. Reply ↓
Chris Wellons on December 26, 2019 at 1:48 pmsaid: Oh, duh, that makes sense. Now I understand what you meant by “approaches zero in the limit”. (Plus I wasn’t running enough simulations to see the pattern.)
possiblywrong on December 26, 2019 at 3:16 pmsaid: Sorry, that was confusing; I have edited the last paragraph of the post to try to clarify.
Re the simulation results, since we have an exact expression for the cumulative distribution function (e.g., contrast with the recent Skittles experiment, where we can only efficiently-but-accurately approximate the expected number of packs before first duplicate, but computing the distribution requires resorting to Monte Carlo estimation), we can compute the variance exactly as well. This plot shows the exact expected value in blue as before, with 2-sigma (roughly 95% containment) bounds for 65536 samples in red (the sample size that you used), and for one million samples in gray. Reply ↓
4. Wilmington on December 30, 2019 at 5:20 pmsaid: Ah! A hump for low n, then a downward trend. As I mentioned, my Excel code only worked up to n=16 or so. I was lulled by the initial trend.
What have you tried so far to prove it?
Suppose we convert it to the continuous domain? I don’t know how to finish it, but here’s my attempt to use the gamma function:
Would it be possible to apply L’Hopital’s rule?
In any case, if you can prove it, it would be tombstone worthy, (though maybe hard to express pithily) especially since it involves the numbers 3 and 2 like Archimedes had inscribed on his tombstone: Reply ↓
5. Michael Earnest on November 21, 2020 at 12:28 amsaid: This is a super cool problem! After seeing this, I asked on math stack exchange to see if anyone had a proof of the result, but no dice so far.
One thing I figured out is that
which can be found by applying a roots-of-unity filter to evaluate the sums of evenly spaced binomial coefficients. Since this is a summation of terms, I think this is bit faster to compute when . Using this, I computed up to . It seems that becomes negative at around , and continues to decrease thereafter. Perhaps it oscillates, or perhaps it approaches some constant.
Finally, a related problem to this is finding the expected value of the maximum height of a randomly chosen Dyck path. This is discussed in , where they show this expected value is . I think that similar method can solve this problem, but I do not have the skills to replicate the work of Knuth and de Bruijn!
Bruijn, de, N. G., Knuth, D. E., & Rice, S. O. (1972). The average height of planted plane trees. In R. C. Read (Ed.), Graph Theory and Computing (pp. 15-22). Academic Press Inc. Reply ↓
possiblywrong on November 21, 2020 at 10:48 amsaid: Very interesting! I like the transformation to reduce the number of terms involved in computing . (Note that I edited the RHS of the formula in your comment, where it looked like you were using in place of the parameter .)
Thanks for the pointer to the paper, I will take a look… although I won't be surprised if it's over my head. I struggle with asymptotics, where the discrete of combinatorics meets the continuous of analysis, so to speak :), and I'm much less comfortable with the latter. Reply ↓
Pingback: Expected maximum deviation of lattice path ~ Mathematics ~ mathubs.com
Michael Earnest on February 20, 2021 at 3:28 amsaid: Okay, I finally dove in to that paper and figured it out. It turns out that $\sqrt{3/2 n}$ is not the best approximation! Instead,
Note that , while , so you would have to use pretty large to distinguish the two with data alone.
Here is code my code I used to confirm the approximation: Reply ↓
8. Michael Earnest on February 20, 2021 at 6:52 amsaid: aaand, I just realized that simplifies to . Reply ↓
possiblywrong on February 20, 2021 at 1:21 pmsaid: Very interesting, great work! I have edited the post to refer to your comments and point out this result. Thanks! Reply ↓
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12072 | https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/11%3A_Coordination_Chemistry_III_-_Electronic_Spectra/11.01%3A_Absorption_of_Light/11.1.01%3A_Beer-Lambert_Absorption_Law | Skip to main content
11.1.1: Beer-Lambert Absorption Law
Last updated
: Aug 12, 2022
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11.1: Absorption of Light
11.2: Quantum Numbers of Multielectron Atoms
Page ID
: 281609
Kathryn Haas
Duke University
( \newcommand{\kernel}{\mathrm{null}\,})
When light passes through a solution that absorbs light, it enters the solution with an initial intensity () at a given wavelength, and it emerges with an intensity, .
The Beer-Lambert Law defines the relationship between absorbance at a given wavelength and the concentration of the solution.
The absorbance (A) is a unitless number because is unitless. The absorbance depends on the concentration () and the path length (). The concentration of the sample solution is measured in molarity (M) and the length of the light path in centimeters (cm). The Greek letter epsilon () in these equations is called the molar absorptivity (also called the molar absorption coefficient). The units of are or .
Chemists most often measure and report absorbed light in terms of wavelength () in units of nanometers (nm). But the wavelength scale is inconvenient for measuring energy because it is inversely proportional to both frequency and energy. In other disciplines, like physics for example, absorption spectra are more often reported in terms of frequency () using units of inverse centimeters (). The relationships between energy (E), , and are given by the equation below:
11.1: Absorption of Light
11.2: Quantum Numbers of Multielectron Atoms |
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12074 | https://www-personal.umd.umich.edu/~jameshet/IntroLabs/IntroLabDocuments/150-03%20Projectile%20Motion/Projectile%20Motion%207.0.pdf | Team: __ __ Projectile Motion So far you have focused on motion in one dimension: x(t). In this lab, you will study motion in two dimensions: x(t) , y(t). This 2D motion, called “projectile motion”, consists of a ball projected with an initial velocity in the earth’s gravitational field.
Basic Principles Consider launching a ball with an initial velocity vo near the surface of the earth where the acceleration of gravity is g .
The position of the ball is given by the coordinates (x , y). The position of the ball depends on time t. The motion of the ball is defined by the motion functions: x(t) , y(t). Note that at time t = 0 , the ball is launched from the point (x , y) = (0 , yo) with the velocity vo . The initial velocity vector vo has magnitude vo and direction θo . Given the position and velocity of the ball at time zero, can we figure out the position of the ball for all future time? In short, here is the fundamental “Kinematic Quest” of projectile motion: Given: The Initial Conditions: yo , vo , θo . Find: The Motion Functions: x(t), y(t) .
The quest to find how x(t) and y(t) depend on t is greatly simplified by the following facts, first discovered by Galileo: The horizontal x(t) and vertical y(t) motions are completely independent of each other. x(t) = constant-velocity motion. y(t) = constant-acceleration motion.
Analyzing the whole complicated motion as a superposition of manageable parts is a paradigm of modern theoretical physics. (x , y) vo θo yo g y 0 x Projectile Motion - Version 7.0 - University of Michigan-Dearborn 2 The theory of projectile motion goes as follows. The general motion functions for any kind of uniformly-accelerated motion in two dimensions are x(t) = xo + vox t + ½ ax t2 .
y(t) = yo + voy t + ½ ay t2 .
Projectile motion is a special case of uniformly-accelerated motion. Near the surface of the earth, the acceleration of gravity points downward and has magnitude 9.8 m/s2 and therefore (ax , ay ) = (0 , − 9.8 m/s2). Substituting these special “earth gravity values” of ax and ay into the general motion functions x(t) and y(t) displayed above, and also setting xo equal to zero for convenience, gives the following motion functions for any object projected in the earth’s gravitational field: The Projectile Motion Equations These equations tell you everything about the motion of a projectile (neglecting air resistance).
If you know the conditions (yo , vox , voy ) at t = 0 , then these equations tell you the position (x(t) , y(t)) of the projectile for all future time t > 0. Make sure you understand The Projectile Motion Equations. They will be used in all future parts of this lab.
Note: In terms of the initial launch angle θo , the components (vox , voy) of the initial velocity vector vo are vox = vo cos θo and voy = vo sin θo .
Exercise The initial (t = 0) launch parameters of a projectile are yo = 3.6 m, vo = 8.9 m/s, θo = 54o. Where is the projectile at time t = 1.2 seconds ? x = __ m. y = __ m. x(t) = vox t y(t) = yo + voy t − 4.9 t2 vo vo sin θo θo vo cos θo 3 Part I. An Illustration of the Independence of x(t) and y(t) Roll the plastic ball off the edge of the table. At the instant the ball leaves the table, drop a coin from the edge of the table. Listen for the ball and the coin to hit the floor. Try rolling the ball faster off the table. Summarize your observations: The picture below shows the position of the dropped coin at six different times. The time interval between consecutive positions is the same. The point at which the ball hits the floor is also shown. Use a ruler to draw horizontal and vertical grid lines – spaced just right – that will allow you to plot the precise location of the ball at six different times. Neglect air friction. Use your six points to sketch the path of the ball as it moves through the air from table to floor.
Table Floor Point where ball hits floor 4 Part II. The Ball Launcher. Finding the Initial Speed.
In this lab, you will use a “projectile machine” to give the ball an initial velocity, i.e. to project the ball in a certain direction with a certain speed. This ball launcher is a spring-loaded “cannon”. Note that you can change the angle θo of the launch (the direction of the vo vector) by tilting the launcher (loosen and tighten the nut). Note that the value of θo is measured relative to the horizontal. A protractor on the side of the launcher specifies the numerical value θo . Tilt the launcher so that it is set for a θo = 30o launch. Place the plastic ball inside the barrel of the launcher. Use the plunger rod to slowly push down on the ball until you hear and feel the first “click”. At this first setting, the spring is locked into a state of minimum compression.
CAUTION: DO NOT compress the spring beyond the first setting.
CAUTION: Always make sure that the launcher is aimed in a SAFE direction, away from people and objects.
When it is safe, pull the lever that launches the ball. Observe the projectile motion.
What is the Initial Speed of the Projectile?
Here you will find the value of vo = the speed of the ball as it exits the launcher. This is known as the “muzzle velocity” of the cannon. First note that the point at which the ball exits the launcher (leaves the spring) is marked as a dot at the center of the small circle that appears on the side of the launcher. This dot represents the “center of mass” of the ball (circle). Since the initial speed of the ball is determined by the force of the spring (gravity has a negligible effect), the value of v0 is constant, independent of the tilt of the launcher.
Tilt the launcher so that it points upward in the vertical direction (θ0 = 90o). Launch the ball and observe how high the ball rises. By measuring the maximum height H, you can determine the launch speed v0.
Here is an experimental technique to determine H. On your table is a vertical rod assembly with a small metal plate attached to the rod. Position the plate directly above the launcher near the point where the ball reaches its maximum height. Adjust the plate up or down so that the launched ball barely hits or barely misses the plate. Launch the ball several times, each time “fine tuning” the vertical position of the plate (slightly up/down) until you are confident (± 1 cm) in the location of the maximum height. Measure the distance from the top of the ball at the launch point to the bottom of the plate. plate H initial v0 final v = 0 5 Maximum Height: H = __ m.
From this measured value of H, compute the initial speed v0 of the ball. Show your calculation. Initial Speed: v0 = __ m/s.
Part III. Discovering the Parabola Geometric trademark of projectile motion: Projectile Path = Parabolic Curve.
Launch the ball at some angle and observe the curved path traced out by the ball as it moves through space. It is difficult to map out the exact shape of the path when you only have about one second to make the observation! The path is definitely some kind of curve, but how do you know it is a parabola? Can you rule other shapes: semi-circular arc, or an oval curve, or a cubic curve, or an exponential curve, or a piece of a sine curve ?
Let’s explore the special case of a horizontal launch. Set the launch angle to be θo = 0o.
The Initial Conditions Measure the initial height yo of the ball. Note: measure from the table surface to the bottom of the ball (see picture). Record your previously measured value of the initial speed vo of the ball. yo = ___ m . vo = __ m/s .
These initial parameters, which specify how you start the projectile motion, uniquely determine the shape of the projectile path. y vo ( x(t) , y(t) ) yo Table x 6 Mapping Out the Parabola First find the motion functions x(t) and y(t) that specify the position of the ball at any time t between the initial launch time t = 0 and the final hit-the-table time. Hint: Substitute the numerical values of your measured initial parameters yo and vo into The Projectile Motion Equations. x(t) = _ t . y(t) = _ + _ t − 4.9 t2 .
Compute x(t) at t = 0 , 0.05 s , 0.10 s , 0.15 s , 0.20 s and mark these five values of x by placing five dots at the appropriate locations on the x(m) axis shown below. Compute y(t) at the same five times and mark these five values of y on the y(m) axis below.
Place five dots within the xy plane pictured above that represent the actual five positions of the projectile at the five times considered above. Label each point with the corresponding time.
Use these five points as a guide to draw the entire smooth path of the projectile as it flies through the air.
Find the equation y(x) that describes this curved path. Hint: Eliminate t from your x(t) and y(t) equations by solving the x equation for t and then substituting this t expression into the y equation. You will be left with “y as a function of x” which should be a quadratic function of the form y(x) = Ax2 + C. Show your derivation of y(x) in the space below. The underlined coefficients that you fill in for the parabola equation y(x) are the special “parabolic parameters” of your projectile motion. y(x) = ___ + __ x2 . 0.3 0.2 0.1 y (m) x (m) 0.6 0.7 0.4 0.5 0.2 0.3 0.1 0 7 Where Does the Projectile Land ?
In theory, the landing point is defined by the coordinate point (x , y) = (L , 0) .
Use your parabola equation y(x) to compute the horizontal landing distance L of your projectile.
Show your calculation. L (theory) = ___ m .
Launch the ball five times. Arrange for the ball to land on a piece of carbon paper, which is placed on top of copy paper taped to the table. The scatter of landing points (dots) recorded on the paper provides a nice visual display of the uncertainty in L. Find the average value of L and the uncertainty in L (half-width spread around the average).
L (m) L (experiment) = __ ± __ m .
Does your value of L (theory) fall within the range of the values of L (experiment)?
% difference between L (theory) and L (experiment) is __ % . y Launch Point Landing Point x 0 x = L 8 Part IV. Range, Altitude, Flight Time There are three important properties of projectile motion: Range R ≡ Maximum horizontal distance. Maximum Height H ≡ Maximum vertical distance. Time of Flight tf ≡ Time in air between launch and land.
Theory The Motion Functions Find the motion functions x(t) and y(t) that describe the motion of the ball projected at an angle of θo = 60o from your launcher. Use your measured values of yo and vo as the initial parameters in the motion equations. Remember: vox = vocosθo , voy = vosinθo . So as not to clutter the equations, do not include the units of the numbers that you write in the blanks below. But make sure that all your numbers are expressed in the metric units of meters and seconds. x(t) = __ t . y(t) = __ + __ t − 4.9 t2 .
The Parabola Use your motion functions x(t) and y(t) to compute the position (x, y) of the ball every tenth of a second (t = 0.0 , 0.1 , 0.2 , 0.3 , ...) until the ball hits the table. Note that the ball hits the table when y = 0. Record the position coordinates (x, y) in the data table below. Launch R y H x Land time 0 0 time tf 9 Position Coordinates (x, y) of the Projectile t (s) x (m) y (m) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 The x and y columns of your data table describe the parabolic path of the ball through the air.
Use the program Graphical Analysis (on the Desktop) to graph y versus x. Include at least one negative value of y on your graph. Fit the points (x, y) on your graph with a quadratic (parabolic) function: y = Ax2 +Bx + C. PRINT the graph showing the parabolic fit. Report the equation of your parabola: y = __ x2 + __ x + __ .
Finding R , H , tf Find the horizontal range R and the maximum height H of the parabolic path directly on your printed graph. DO NOT solve any equations. Simply look at your graph of the parabola. Draw the horizontal line representing the table surface (the x-axis) on your graph. Write “Table Surface” on this line. Mark the value of R on this x-axis. Mark the value of H on the y-axis.
Report your marked values of R and H here. Note: You can only read numbers on the x-axis and y-axis to two significant figures. R (theory) = __ m. H (theory) = __ m.
Use your x(t) equation and value of R to calculate the time of flight tf . Show your calculation. tf (theory) = __ s .
Note: Negative values of y denote the vertical position of the ball below the table surface.
The ball does not have access to this region of space unless you cut a hole in the table for the ball to whiz through !
10 Experiment Set the launcher for a θo = 60o launch. Launch the ball and measure tf , H , and R. One person should measure tf − the “hang time”of the ball through the air − with a stopwatch. Another person should measure H − the maximum altitude − with a meter stick and a simple visible inspection of where the trajectory of the ball peaks. Use carbon paper on top of copy paper to record the landing point. Measure the range R − the horizontal distance between the launch point and the landing point − with a meter stick.
Repeat this 60o launch four more times. Fill in the table. Find the average values of your measured parameters R , H , and tf .
Experimental Values of the Projectile Motion Parameters Average R (exp) m H (exp) m tf (exp) s Compare Theory and Experiment % diff between R (theory) = _ and R (exp) = _ is __ % .
% diff between H (theory) = _ and H (exp) = _ is __ % .
% diff between tf (theory) = _ and tf (exp) = _ is __ % .
If any percent difference is greater than 10%, then consult your instructor.
11 Part V. Design Problem: Human Cannonball.
A circus performer is launched from a cannon and lands in a net. Your tabletop ball launcher represents a small-scale working model of this human cannonball system. Your goal is to figure out what angle to launch the ball and where to place the net so that the flight time of the ball is equal to 0.50 seconds and the ball lands in the net.
The Theory Write the motion equations x(t) and y(t) for this problem. Use your previously measured values of the “cannon” parameters, yo and vo , as the initial parameters in the equations. x(t) = _ cosθo t . y(t) = _ + _ sinθo t − _ t2 .
Substitute the specified value of t = tf = 0.50 s into x(t) and y(t). Remember: At the time t = tf , the ball is at the landing point (x , y) = (R , 0). Solve your two equations for the two unknowns θo and R. Show all your algebra in the space below. θo = _ o . R = __ m.
Ask your instructor to visit your table with the “net” in order to check your theory and run the experiment with your team.
The Experiment Set the launch angle of the cannon at your theoretical (predicted) value of θo. Place the landing net on the table at your theoretical (predicted) value of R. Note that the radius of the net defines the allowed uncertainty in R. Launch the ball. Observe where the ball lands. Predicted R = _ m and Actual R = _ m differ by __ %.
Now launch the ball again and measure the time of flight. Specified tf = 0.50 s and Actual tf = _ s differ by ______ %. |
12075 | https://fiveable.me/key-terms/principles-physics-ii/cyclotron-motion | printables
🎢principles of physics ii review
key term - Cyclotron motion
Citation:
MLA
Definition
Cyclotron motion refers to the circular motion of charged particles, such as electrons or ions, when they move through a uniform magnetic field. This phenomenon occurs because the magnetic field exerts a force perpendicular to the velocity of the charged particle, causing it to change direction continuously and move in a circular path. The frequency of this motion is constant and is determined by the charge and mass of the particle as well as the strength of the magnetic field.
5 Must Know Facts For Your Next Test
Cyclotron motion occurs due to the Lorentz force acting on charged particles moving through a magnetic field, resulting in a continuous change in direction without a change in speed.
The radius of the circular path in cyclotron motion is directly proportional to the momentum of the particle and inversely proportional to the magnetic field strength.
Cyclotron motion can be observed in various applications, including cyclotrons used for accelerating particles in nuclear physics and medical imaging techniques like MRI.
The energy of a charged particle in cyclotron motion increases as it moves further away from the center of its circular path due to conservation of angular momentum.
If a charged particle enters a region with a varying magnetic field, it can lose its cyclotron motion and spiral outwards or become trapped depending on the conditions.
Review Questions
How does the Lorentz force affect cyclotron motion for charged particles?
The Lorentz force plays a crucial role in cyclotron motion as it provides the necessary centripetal force that keeps charged particles moving in a circular path within a magnetic field. This force acts perpendicular to both the velocity of the particle and the direction of the magnetic field, continuously changing the particle's direction while maintaining its speed. As a result, the charged particle undergoes uniform circular motion, resulting in what we observe as cyclotron motion.
Explain how changing either the charge or mass of a particle affects its cyclotron frequency.
Cyclotron frequency is determined by both the charge and mass of the particle as described by the formula $$f = \frac{qB}{2\pi m}$$. If you increase the charge $q$, while keeping mass $m$ and magnetic field strength $B$ constant, the frequency increases, meaning that the particle orbits faster. Conversely, if you increase mass $m$, while keeping charge and magnetic field strength constant, the frequency decreases, resulting in slower orbital motion. Therefore, both charge and mass have inverse relationships with cyclotron frequency.
Evaluate how cyclotron motion contributes to advancements in technology such as medical imaging and particle accelerators.
Cyclotron motion is fundamental to advancements in technology like medical imaging techniques, such as MRI, and in particle accelerators used in nuclear physics. In MRI, understanding how protons behave in magnetic fields allows for detailed imaging of soft tissues in the body. Meanwhile, in particle accelerators like cyclotrons, controlling charged particles' cyclotron motion enables scientists to accelerate particles to high energies for experiments. These applications highlight how principles of cyclotron motion translate into real-world technological innovations that enhance our understanding of biology and physics.
Related terms
Lorentz force:
The force experienced by a charged particle moving through a magnetic field, calculated as the cross product of the velocity vector and the magnetic field vector.
magnetic field: A region around a magnetic material or moving electric charge within which the force of magnetism acts.
cyclotron frequency:
The frequency at which a charged particle orbits in a magnetic field, given by the formula $$f = \frac{qB}{2\pi m}$$, where $q$ is the charge, $B$ is the magnetic field strength, and $m$ is the mass of the particle. |
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IOPscience - Identification of Penicillium Species of Fruits Using Morphology and Spectroscopic Methods
National Center for Biotechnology Information - PubMed Central - Identification and nomenclature of the genus Penicillium
BMC - IMA Fungus - Comparative genomic study of the Penicillium genus elucidates a diverse pangenome and 15 lateral gene transfer events
ASM Journals - Journal of Clinical Microbiology - Clinical, Morphological, and Molecular Characterization of Penicillium canis sp. nov., Isolated from a Dog with Osteomyelitis
University of Adelaide - Penicillium
Nature - Scientific Reports - The diversity and ecological roles of Penicillium in intertidal zones
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Related Topics:
: penicillin
: Penicillium chrysogenum
: Penicillium notatum
: Penicillium roqueforti
: Penicillium glaucum
See all related content
Penicillium, genus of blue or green mold fungi (kingdom Fungi) that exists as asexual forms (anamorphs, or deuteromycetes). Those species for which the sexual phase is known are placed in the Eurotiales. Found on foodstuffs, leather, and fabrics, they are of economic importance in the production of antibiotics (penicillin, q.v.), organic acids, and cheeses.
This article was most recently revised and updated by Kara Rogers. |
12077 | https://www.sparkl.me/learn/collegeboard-ap/precalculus/analyzing-periodic-properties-of-sine-and-cosine/revision-notes/668 | 1.1.1
Graphing parametric functions in x-y planes
1.1.2
Comparing parametric and Cartesian forms
1.1.3
Identifying parametric equations for simple curves
1.2.1
Calculating derivatives of parametric equations
1.2.2
Understanding tangent lines in parametric graphs
1.2.3
Exploring relationships between x and y changes
1.3.1
Adding and subtracting vectors
1.3.2
Computing magnitudes and directions
1.3.3
Representing motion using vector components
1.4.1
Graphing 2D vector fields
1.4.2
Understanding velocity and acceleration in vector form
1.4.3
Combining vectors with scalar functions
1.5.1
Basics of matrix addition and multiplication
1.5.2
Representing linear equations with matrices
1.5.3
Analyzing matrix properties algebraically
1.6.1
Understanding matrix-based transformations
1.6.2
Verifying transformations using input-output pairs
1.6.3
Connecting linear transformations to geometry
1.7.1
Solving systems of equations with inverse matrices
1.7.2
Verifying solutions using determinants
1.7.3
Simplifying complex systems algebraically
2.1.1
Identifying transformations of logarithmic functions
2.1.2
Solving logarithmic equations symbolically
2.1.3
Defining logarithmic bases and their properties
2.2.1
Expanding and simplifying logarithmic terms
2.2.2
Combining terms with the same base
2.2.3
Factoring logarithms for solving equations
2.3.1
Solving inequalities involving logarithms
2.3.2
Validating solutions using substitution
2.3.3
Using exponentiation to verify logarithmic solutions
2.4.1
Representing exponential data with semi-log scales
2.4.2
Identifying linear trends in transformed data
2.4.3
Comparing semi-log plots to regular plots
2.5.1
Recognizing differences in arithmetic sequences
2.5.2
Identifying ratios in geometric sequences
2.5.3
Connecting sequences to linear and exponential functions
2.6.1
Identifying linear growth or decay rates
2.6.2
Comparing linear vs. exponential rates of increase
2.6.3
Understanding when exponential models are appropriate
2.7.1
Understanding domain and range of exponential functions
2.7.2
Analyzing transformations: vertical and horizontal shifts
2.7.3
Exploring asymptotic behavior in exponential graphs
2.8.1
Simplifying exponential expressions using rules of exponents
2.8.2
Factoring exponential terms in equations
2.8.3
Expanding powers for simplification
2.9.1
Representing patterns algebraically
2.9.2
Validating assumptions in exponential equations
2.9.3
Generalizing growth models without specific real-world scenarios
2.10.1
Testing linear vs. exponential models for fit
2.10.2
Comparing residuals to determine accuracy
2.10.3
Analyzing errors in model assumptions
2.11.1
Combining two exponential functions
2.11.2
Using compositions for recursive sequences
2.11.3
Exploring commutativity in function compositions
2.12.1
Identifying conditions for inverses
2.12.2
Verifying inverse relationships algebraically
2.12.3
Sketching inverse functions on a graph
2.13.1
Expanding logarithmic terms using rules of logs
2.13.2
Simplifying nested logarithms
2.13.3
Converting between logarithmic and exponential forms
2.14.1
Analyzing inverse symmetry in graphs
2.14.2
Understanding logarithms as inverses of exponents
2.14.3
Exploring how inverse transformations affect function shapes
3.
Polynomial and Rational Functions
3.1.1
Simplifying rational functions to reveal holes
3.1.2
Testing function continuity at holes
3.1.3
Identifying removable discontinuities
3.2.1
Factoring and simplifying expressions
3.2.2
Expanding rational expressions
3.2.3
Verifying equivalency through algebraic manipulation
3.3.1
Horizontal and vertical shifts
3.3.2
Stretching and compressing functions
3.3.3
Reflecting across axes
3.4.1
Choosing appropriate polynomial or rational models
3.4.2
Testing assumptions through equation constraints
3.4.3
Analyzing limitations of chosen models
3.5.1
Writing polynomial and rational models
3.5.2
Simplifying derived models for analysis
3.5.3
Interpreting model parameters algebraically
3.6.1
Identifying relationships between variables
3.6.2
Representing co-variation with graphs and tables
3.6.3
Recognizing proportional relationships
3.7.1
Definition and calculation of average rate of change
3.7.2
Interpreting instantaneous rate of change
3.7.3
Slope as a measure of rate in linear functions
3.8.1
Identifying constant rates in linear models
3.8.2
Analyzing variable rates in quadratic functions
3.8.3
Determining intervals of increase and decrease
3.9.1
Graphical representation of polynomial behaviors
3.9.2
Understanding the degree's role in rate analysis
3.9.3
Identifying turning points and critical points
3.10.1
Finding complex zeros using the quadratic formula
3.10.2
Factoring polynomials with real and complex roots
3.10.3
Verifying solutions with synthetic division
3.11.1
Predicting end behavior using leading coefficients
3.11.2
Determining symmetry in polynomial graphs
3.11.3
Understanding multiplicities of roots
3.12.1
Determining horizontal asymptotes
3.12.2
Analyzing degrees of numerator and denominator
3.12.3
Identifying oblique asymptotes
3.13.1
Finding zeros in rational functions
3.13.2
Multiplicity and their impact on graph shape
3.13.3
Using synthetic and long division
3.14.1
Defining vertical asymptotes algebraically
3.14.2
Testing continuity near asymptotes
3.14.3
Identifying behaviors approaching vertical asymptotes
4.
Trigonometric and Polar Functions
4.1.1
Understanding cycles in functions
4.1.2
Representing periodic intervals
4.1.3
Exploring periodicity in graphs
4.2.1
Computing trigonometric ratios using the unit circle
4.2.2
Verifying identities involving sine, cosine and tangent
4.2.3
Exploring co-functions and their relationships
4.3.1
Computing exact values for common angles
4.3.2
Understanding reflection and symmetry properties
4.3.3
Analyzing periodic properties of sine and cosine
4.4.1
Graphing sine and cosine with amplitude and period
4.4.2
Identifying shifts in sine and cosine graphs
4.4.3
Exploring relationships between graphs of sine and cosine
4.5.1
Graphing tangent over its domain
4.5.2
Identifying vertical asymptotes in tangent graphs
4.5.3
Analyzing tangent transformations
4.6.1
Determining restricted domains of inverses
4.6.2
Using inverse functions to solve equations
4.6.3
Verifying identities with inverses
4.7.1
Solving multi-angle equations
4.7.2
Exploring periodic solutions in equations
4.7.3
Analyzing domain-specific inequalities
4.8.1
Graphing reciprocal functions of sine, cosine and tangent
4.8.2
Identifying domains and asymptotes
4.8.3
Verifying reciprocal relationships in identities
4.9.1
Using Pythagorean identities to simplify
4.9.2
Verifying equivalences through substitution
4.9.3
Rewriting expressions for computational ease
4.10.1
Converting Cartesian equations to polar form
4.10.2
Graphing polar coordinates
4.10.3
Analyzing polar symmetry and rotations
4.11.1
Graphing cardioids, roses and spirals
4.11.2
Analyzing periodicity and intersections in polar graphs
4.11.3
Using technology to explore complex polar shapes
Analyzing periodic properties of sine and cosine
Topic 2/3
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Analyzing Periodic Properties of Sine and Cosine
Introduction
Understanding the periodic properties of sine and cosine functions is fundamental in precalculus, particularly for students preparing for the Collegeboard AP examinations. These trigonometric functions not only describe oscillatory motions but also play a crucial role in modeling real-world phenomena such as sound waves, electrical currents, and circular motion. This article delves into the periodic nature of sine and cosine functions, exploring their key properties, applications, and comparative aspects to provide a comprehensive understanding tailored for precalculus learners.
Key Concepts
1. Definition of Periodic Functions
A function is considered periodic if it repeats its values at regular intervals over its domain. Mathematically, a function f(x) is periodic with period P if:
f(x+P)=f(x)
for all x in the domain of f. The smallest positive value of P for which this equation holds is called the fundamental period of the function.
2. Periodicity of Sine and Cosine Functions
The sine and cosine functions are quintessential examples of periodic functions with a fundamental period of 2π. This means that:
sin(x+2π)=sin(x)andcos(x+2π)=cos(x)
for all real numbers x. This periodicity arises from the unit circle representation, where rotating an angle by 2π radians brings a point back to its initial position.
3. Amplitude, Frequency, and Phase Shift
Besides periodicity, sine and cosine functions are characterized by their amplitude, frequency, and phase shift:
Amplitude: The amplitude determines the peak value of the function, represented by A in the general form y=Asin(Bx−C)+D or y=Acos(Bx−C)+D. It indicates the maximum displacement from the equilibrium position.
Frequency: Frequency relates to how often the function completes one full cycle within a specific interval. It is inversely proportional to the period (P=B2π), where B affects the number of cycles within 2π.
Phase Shift: Phase shift represents the horizontal displacement of the function along the x-axis, determined by C in the equations. A positive C shifts the graph to the right, while a negative C shifts it to the left.
4. Graphical Representation
Graphing sine and cosine functions involves plotting their periodic oscillations over intervals of their periods. Key points to plot include:
Intercepts: Points where the function crosses the x-axis.
Maximum and Minimum Points: Peaks and troughs of the wave corresponding to the amplitude.
Points of Inflection: Where the concavity of the graph changes.
For example, the graph of y=sin(x) starts at the origin, reaches a maximum of 1 at 2π, crosses zero at π, reaches a minimum of -1 at 23π, and returns to zero at 2π, completing one full cycle.
5. Phase Relationships
Sine and cosine functions are phase-shifted counterparts of each other. Specifically:
cos(x)=sin(x+2π)
This relationship implies that the cosine function leads the sine function by 2π radians. Understanding this phase shift is crucial when analyzing systems where these functions interact, such as in wave interference and alternating current circuits.
6. Applications of Periodic Properties
The periodic properties of sine and cosine are employed in various applications:
Signal Processing: Modeling and analyzing periodic signals like sound and electromagnetic waves.
Engineering: Designing components that rely on oscillatory motion, such as springs and pendulums.
Architecture: Calculating forces and stresses in structures subjected to cyclical loads.
Medicine: Understanding biological rhythms and designing medical imaging techniques.
By leveraging the inherent periodicity, engineers and scientists can predict and manipulate patterns in diverse fields.
7. Mathematical Derivation of Periodicity
The periodicity of sine and cosine functions can be derived using Euler's formula, which relates complex exponentials to trigonometric functions:
eix=cos(x)+isin(x)
Since ei(x+2π)=eixei2π=eix (because ei2π=1), it follows that:
cos(x+2π)+isin(x+2π)=cos(x)+isin(x)
Equating the real and imaginary parts, we obtain the periodicity of cosine and sine functions respectively.
8. Amplitude and Energy
In physical systems, the amplitude of sine and cosine functions is directly related to the energy of the system. For instance, in harmonic oscillators, the amplitude determines the maximum potential energy stored, while the frequency relates to the kinetic energy. Understanding the interplay between amplitude and frequency helps in analyzing the stability and behavior of oscillatory systems.
9. Composite Functions and Superposition
When multiple sine and cosine functions with different amplitudes, frequencies, and phase shifts are combined, the resulting function exhibits complex periodic behavior. This principle of superposition is fundamental in fields like acoustics and signal processing, where overlapping waves must be analyzed to determine the resultant waveforms.
For example, the sum of two sine functions can be expressed as:
Asin(Bx−C)+Dsin(Ex−F)
Analyzing such composite functions involves using trigonometric identities and Fourier analysis to decompose and understand the individual contributions to the overall periodic behavior.
10. Real-World Examples
Several real-world phenomena can be modeled using the periodic properties of sine and cosine functions:
Sound Waves: Described by sine functions where amplitude corresponds to loudness and frequency to pitch.
Tidal Movements: Regular fluctuations in sea levels modeled by sinusoidal functions.
Electrical Alternating Current (AC): Voltage and current variations in AC circuits follow sine wave patterns.
Pendulum Motion: The swinging of a pendulum approximates simple harmonic motion described by sine or cosine functions.
These examples highlight the versatility and applicability of sine and cosine functions in explaining and predicting cyclical behaviors in nature and technology.
Comparison Table
| | | |
---
| Aspect | Sine Function | Cosine Function |
| Starting Point | Origin (0,0) | Maximum at (0,1) |
| Phase Shift Relationship | Lagging by 2π radians | Leading by 2π radians |
| Graph Symmetry | Odd function: sin(−x)=−sin(x) | Even function: cos(−x)=cos(x) |
| Applications | Modeling oscillations with zero baseline, such as simple harmonic motion starting at equilibrium | Modeling oscillations with maximum displacement at origin, such as spring compression |
| Derivative | cos(x) | −sin(x) |
| Integral | −cos(x)+C | sin(x)+C |
Summary and Key Takeaways
Sine and cosine functions are inherently periodic with a fundamental period of 2π.
Key properties include amplitude, frequency, and phase shift, which dictate their behavior and applications.
Graphical analysis involves identifying intercepts, maxima, minima, and points of inflection.
Understanding the phase relationship between sine and cosine is essential for analyzing oscillatory systems.
Applications of these functions span across various real-world phenomena, highlighting their versatility in modeling cyclic behaviors.
Coming Soon!
Examiner Tip
Tips
To master periodic functions, remember the mnemonic "All Students Take Calculus" for the sign of sine and cosine derivatives in each quadrant. Practice sketching graphs by identifying key features like amplitude, period, and phase shift. Utilize unit circle diagrams to visualize phase relationships. For AP exam success, solve various problems involving transformations of sine and cosine functions to build confidence and accuracy.
Did You Know
Did You Know
Did you know that the ancient Greeks used the sine function to calculate the heights of stars? Additionally, sine and cosine waves are fundamental in modern technology, enabling the transmission of information through radio, television, and internet signals. Another fascinating fact is that Beethoven's Fifth Symphony exhibits periodic properties similar to sine waves, showcasing the intersection of mathematics and art.
Common Mistakes
Common Mistakes
Students often confuse the amplitude with the period of sine and cosine functions. For example, mistakenly identifying the period as the height of the wave can lead to incorrect graphing. Another common error is neglecting the phase shift, which results in misplacing the function horizontally. Lastly, forgetting to apply the negative sign when dealing with cosine derivatives can lead to wrong answers in calculus problems.
FAQ
What is the fundamental period of the sine function?
The fundamental period of the sine function is 2π radians, meaning it repeats every 2π units.
How do you determine the amplitude of a cosine function?
The amplitude of a cosine function y=Acos(Bx−C)+D is the absolute value of A, representing the peak deviation from the equilibrium position.
What causes a phase shift in sine and cosine functions?
A phase shift is caused by the constant C in the function's equation y=Asin(Bx−C)+D or y=Acos(Bx−C)+D, which shifts the graph horizontally.
How are sine and cosine functions related?
Sine and cosine functions are phase-shifted versions of each other. Specifically, cos(x)=sin(x+2π), meaning cosine leads sine by 2π radians.
Why are sine and cosine functions important in real-world applications?
Their periodic nature makes them ideal for modeling oscillatory and wave-like phenomena, such as sound waves, electrical currents, and mechanical vibrations, which are prevalent in various scientific and engineering fields.
1.
Functions Involving Parameters, Vectors and Matrices
1.1.1
Graphing parametric functions in x-y planes
1.1.2
Comparing parametric and Cartesian forms
1.1.3
Identifying parametric equations for simple curves
1.2.1
Calculating derivatives of parametric equations
1.2.2
Understanding tangent lines in parametric graphs
1.2.3
Exploring relationships between x and y changes
1.3.1
Adding and subtracting vectors
1.3.2
Computing magnitudes and directions
1.3.3
Representing motion using vector components
1.4.1
Graphing 2D vector fields
1.4.2
Understanding velocity and acceleration in vector form
1.4.3
Combining vectors with scalar functions
1.5.1
Basics of matrix addition and multiplication
1.5.2
Representing linear equations with matrices
1.5.3
Analyzing matrix properties algebraically
1.6.1
Understanding matrix-based transformations
1.6.2
Verifying transformations using input-output pairs
1.6.3
Connecting linear transformations to geometry
1.7.1
Solving systems of equations with inverse matrices
1.7.2
Verifying solutions using determinants
1.7.3
Simplifying complex systems algebraically
2.
Exponential and Logarithmic Functions
2.1.1
Identifying transformations of logarithmic functions
2.1.2
Solving logarithmic equations symbolically
2.1.3
Defining logarithmic bases and their properties
2.2.1
Expanding and simplifying logarithmic terms
2.2.2
Combining terms with the same base
2.2.3
Factoring logarithms for solving equations
2.3.1
Solving inequalities involving logarithms
2.3.2
Validating solutions using substitution
2.3.3
Using exponentiation to verify logarithmic solutions
2.4.1
Representing exponential data with semi-log scales
2.4.2
Identifying linear trends in transformed data
2.4.3
Comparing semi-log plots to regular plots
2.5.1
Recognizing differences in arithmetic sequences
2.5.2
Identifying ratios in geometric sequences
2.5.3
Connecting sequences to linear and exponential functions
2.6.1
Identifying linear growth or decay rates
2.6.2
Comparing linear vs. exponential rates of increase
2.6.3
Understanding when exponential models are appropriate
2.7.1
Understanding domain and range of exponential functions
2.7.2
Analyzing transformations: vertical and horizontal shifts
2.7.3
Exploring asymptotic behavior in exponential graphs
2.8.1
Simplifying exponential expressions using rules of exponents
2.8.2
Factoring exponential terms in equations
2.8.3
Expanding powers for simplification
2.9.1
Representing patterns algebraically
2.9.2
Validating assumptions in exponential equations
2.9.3
Generalizing growth models without specific real-world scenarios
2.10.1
Testing linear vs. exponential models for fit
2.10.2
Comparing residuals to determine accuracy
2.10.3
Analyzing errors in model assumptions
2.11.1
Combining two exponential functions
2.11.2
Using compositions for recursive sequences
2.11.3
Exploring commutativity in function compositions
2.12.1
Identifying conditions for inverses
2.12.2
Verifying inverse relationships algebraically
2.12.3
Sketching inverse functions on a graph
2.13.1
Expanding logarithmic terms using rules of logs
2.13.2
Simplifying nested logarithms
2.13.3
Converting between logarithmic and exponential forms
2.14.1
Analyzing inverse symmetry in graphs
2.14.2
Understanding logarithms as inverses of exponents
2.14.3
Exploring how inverse transformations affect function shapes
3.
Polynomial and Rational Functions
3.1.1
Simplifying rational functions to reveal holes
3.1.2
Testing function continuity at holes
3.1.3
Identifying removable discontinuities
3.2.1
Factoring and simplifying expressions
3.2.2
Expanding rational expressions
3.2.3
Verifying equivalency through algebraic manipulation
3.3.1
Horizontal and vertical shifts
3.3.2
Stretching and compressing functions
3.3.3
Reflecting across axes
3.4.1
Choosing appropriate polynomial or rational models
3.4.2
Testing assumptions through equation constraints
3.4.3
Analyzing limitations of chosen models
3.5.1
Writing polynomial and rational models
3.5.2
Simplifying derived models for analysis
3.5.3
Interpreting model parameters algebraically
3.6.1
Identifying relationships between variables
3.6.2
Representing co-variation with graphs and tables
3.6.3
Recognizing proportional relationships
3.7.1
Definition and calculation of average rate of change
3.7.2
Interpreting instantaneous rate of change
3.7.3
Slope as a measure of rate in linear functions
3.8.1
Identifying constant rates in linear models
3.8.2
Analyzing variable rates in quadratic functions
3.8.3
Determining intervals of increase and decrease
3.9.1
Graphical representation of polynomial behaviors
3.9.2
Understanding the degree's role in rate analysis
3.9.3
Identifying turning points and critical points
3.10.1
Finding complex zeros using the quadratic formula
3.10.2
Factoring polynomials with real and complex roots
3.10.3
Verifying solutions with synthetic division
3.11.1
Predicting end behavior using leading coefficients
3.11.2
Determining symmetry in polynomial graphs
3.11.3
Understanding multiplicities of roots
3.12.1
Determining horizontal asymptotes
3.12.2
Analyzing degrees of numerator and denominator
3.12.3
Identifying oblique asymptotes
3.13.1
Finding zeros in rational functions
3.13.2
Multiplicity and their impact on graph shape
3.13.3
Using synthetic and long division
3.14.1
Defining vertical asymptotes algebraically
3.14.2
Testing continuity near asymptotes
3.14.3
Identifying behaviors approaching vertical asymptotes
4.
Trigonometric and Polar Functions
4.1.1
Understanding cycles in functions
4.1.2
Representing periodic intervals
4.1.3
Exploring periodicity in graphs
4.2.1
Computing trigonometric ratios using the unit circle
4.2.2
Verifying identities involving sine, cosine and tangent
4.2.3
Exploring co-functions and their relationships
4.3.1
Computing exact values for common angles
4.3.2
Understanding reflection and symmetry properties
4.3.3
Analyzing periodic properties of sine and cosine
4.4.1
Graphing sine and cosine with amplitude and period
4.4.2
Identifying shifts in sine and cosine graphs
4.4.3
Exploring relationships between graphs of sine and cosine
4.5.1
Graphing tangent over its domain
4.5.2
Identifying vertical asymptotes in tangent graphs
4.5.3
Analyzing tangent transformations
4.6.1
Determining restricted domains of inverses
4.6.2
Using inverse functions to solve equations
4.6.3
Verifying identities with inverses
4.7.1
Solving multi-angle equations
4.7.2
Exploring periodic solutions in equations
4.7.3
Analyzing domain-specific inequalities
4.8.1
Graphing reciprocal functions of sine, cosine and tangent
4.8.2
Identifying domains and asymptotes
4.8.3
Verifying reciprocal relationships in identities
4.9.1
Using Pythagorean identities to simplify
4.9.2
Verifying equivalences through substitution
4.9.3
Rewriting expressions for computational ease
4.10.1
Converting Cartesian equations to polar form
4.10.2
Graphing polar coordinates
4.10.3
Analyzing polar symmetry and rotations
4.11.1
Graphing cardioids, roses and spirals
4.11.2
Analyzing periodicity and intersections in polar graphs
4.11.3
Using technology to explore complex polar shapes
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12078 | https://matthey.com/documents/161599/3948762/Orobraze%2C+Pallabraze+-+EN.pdf/99c04a99-e688-b205-d088-06f3461942c1?t=1740158178695 | GOLD AND PALLADIUM BASED BRAZING FILLER METALS
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can exhibit strength at elevated temperatures making them ideal for components that will see these conditions in service. Gold and palladium in the filler metals can impart improved resistance to oxidation at elevated temperatures. Orobraze TM
and Pallabraze TM
products can also offer improved resistance to chemical attack in certain corrosive conditions. Some of the products have low volatile elements which makes them ideal for use in vacuum tube / thermionic valve devices that operate in vacuum at elevated temperatures. Selected products have no readily-oxidising elements making them suitable for reducing atmosphere brazing conditions. Within the range there are ductile filler metals which are suitable for use on ceramic to metal joints where there is a need to balance differences in coefficients of thermal expansion.
Brazing Characteristics
Both Orobraze TM
and Pallabraze TM
filler metals have short melting ranges and free-flowing characteristics making them ideal for furnace brazing applications without the problem of liquation. Both ranges exhibit well-defined melting ranges, which make them suitable for step brazing operations. Orobraze TM
and Pallabraze TM
filler metals produce small exceptionally smooth fillets at the joint edges. Gold-nickel Orobraze™ filler metals exhibit flow characteristics superior to the gold-copper filler metals on nimonics, super-alloys and stainless steel. Gold-copper Orobraze TM
filler metals produce ductile joints without excessive inter-alloying / erosion of the parent metals. Pallabraze TM
filler metals have excellent flow and penetration qualities but some are also capable of filling joint gaps up to 0.5mm. Pallabraze TM
filler metals also exhibit minimal erosion on ferrous and nickel bearing alloys during brazing. These filler metals do not cause liquid metal induced stress cracking and are recommended on iron-nickel-cobalt alloys.
High Purity Vacuum Grade
Most Orobraze TM
and Pallabraze TM
filler metals are manufactured to contain only minimum levels of volatile elements. As a result they have low vapour pressures under elevated temperature and ultra-high vacuum conditions. This makes them suitable for brazing components that operate under these conditions such as vacuum tube devices. In particular Pallabraze TM
filler metals are widely used in this application. Products are in general supplied to conform to the impurity limits specified in ISO17672 for vacuum grade 1 (V1 Grade) materials.
Elevated Temperature Properties
In general Orobraze TM
filler metals exhibit excellent high temperature strength up to 500˚C and oxidation resistance up to 800˚C. And the nickel-bearing Orobraze TM
filler metals exhibit improved high temperature strength and resistance to oxidation at elevated temperatures compared to gold-copper alloys. Pallabraze TM
filler metals can be used to produce joints for elevated-temperature service conditions between 300˚C and 400˚C. Under these conditions they exhibit good oxidation resistance and mechanical properties. Pallabraze TM
810 shows oxidation resistance up to 500˚C and Pallabraze TM
1237 up to 700˚C.
Corrosion Resistance
Orobraze TM
and Pallabraze TM
filler metals also offer good corrosion resistance in many chemical environments and joints are resistant to interfacial corrosion on stainless steels in aqueous service environments. It is recommended that joints that will be subjected to chemical environments in service be tested prior to adoption of a particular brazing filler metal. 4
PRODUCTS
GOLD-BASED BRAZING FILLER METALS
These products are gold-based filler metals manufactured to the highest purity ‘vacuum grade’ to meet the needs of aerospace and electronics applications.
Orobraze TM
910 Orobraze TM
940 Orobraze TM
950 Orobraze TM
970 Orobraze TM
990
Specification
Melting Range ˚C EN 1044 ISO 17672
890 Au 800
Melting Range ˚C EN 1044 ISO 17672
930-940 AU102 Au 625 V1
Melting Range ˚C EN 1044 ISO 17672
950 AU105 Au 827 V1
Melting Range ˚C EN 1044 ISO 17672
955-970 Au 503
Melting Range ˚C EN 1044 ISO 17672
950-990 AU106 Au 752 V1
Melting Range ˚C EN 1044 ISO 17672
980-998 AU103 Au 375 V1
Melting Range ˚C EN 1044 ISO 17672
970-1005 Au 354
Melting Range ˚C EN 1044 ISO 17672
996-1018 AU104 Au 295 V1
Melting Range ˚C EN 1044 ISO 17672
1000-1030 AU101 Au 351
Melting Range ˚C EN 1044 ISO 17672
1030-1040
Melting Range ˚C EN 1044 ISO 17672
1005-1045 Au 700
Au Cu
Composition
Au
Au
Au
Au
Au
Au
Au
Au
Au
Au
Cu
Ni
Cu
Cu
Cu
Ni
Ni
Cu
Cu
Ag
Ni
Pd
Orobraze TM
998 Orobraze TM
1005 Orobraze TM
1018 Orobraze TM
1030 Orobraze TM
1040 Orobraze TM
1045
80 20 35 62 370 22 862.5 37.5 82 18 50 50 75 25 37.5 62.5 35 65 30 70 70 30 5
Metal Joining
PRODUCTS
PALLADIUM-BEARING BRAZING FILLER METALS
These palladium-bearing filler metals are manufactured to the highest purity ‘vacuum grade’ to meet the needs of aerospace and electronics applications.
Melting Range ˚C EN 1044 ISO 17672
807-810 PD106 Pd 287 V1
Melting Range ˚C EN 1044 ISO 17672
834-840 PD104 Pd 388 V1
Melting Range ˚C EN 1044 ISO 17672
825-850 PD105 Pd 387 V1
Melting Range ˚C EN 1044 ISO 17672
856-880 PD103 Pd 481 V1
Melting Range ˚C EN 1044 ISO 17672
876-900 PD102 Pd 484 V1
Melting Range ˚C EN 1044 ISO 17672
901-950 PD101 Pd 587 V1
Melting Range ˚C EN 1044 ISO 17672
970-1010 PD204 Pd 288 V1
Melting Range ˚C EN 1044 ISO 17672
1080-1090 PD203 Pd 483 V1
Melting Range ˚C EN 1044 ISO 17672
1000-1120 - Pd 485 V1
Melting Range ˚C EN 1044 ISO 17672
1150-1225
Melting Range ˚C EN 1044 ISO 17672
1237 PD201 Pd 647 V1
Specification Composition
Pd
Pd
Pd
Pd
Pd
Pd
Pd
Ag
Ag
Ag
Ag
Ag
Ag
Ag
Cu
Cu
Cu
Cu
Cu
Cu
Pd Cu
Pd Ag Mn
Pd Cu
Pd Ni
Pallabraze TM
810 Pallabraze TM
840 Pallabraze TM
850 Pallabraze TM
880 Pallabraze TM
900 Pallabraze TM
950 Pallabraze TM
1010 Pallabraze TM
1090 Pallabraze TM
1120 Pallabraze TM
1225 Pallabraze TM
1237
5 68.5 26.5 10 67.5 22.5 10 58 32 15 65 20 20 52 28 25 54 21 5 95 18 82 20 75 530 70 60 40 6
In Ag Cu Sn Ni
Au Ag Cu In Pd Ni Si Cr Other
Au Cu In Pd Ni
Au Ag Cu Mn
Mn
Pd Ni Co
PRODUCTS
MISCELLANEOUS PRECIOUS METAL BRAZING FILLER METALS
These products are less commonly specified and are included for reference purposes. Availability may vary and it is recommended to consult Johnson Matthey.
MELTING UP TO 1000˚C
Melting Range °C
Pallabraze TM
851 46.7 47.2 6.1 810-851
Pallabraze TM
880Ga 9 82 9 Ga 845-880
Orobraze TM
895 75 5 20 885-895
Orobraze TM
900 60 37 3 860-900
Pallabraze TM
960 36 50 0.5 10.5 3 B 820-960
Pallabraze TM
977 30 57.1 10.5 2.4 B 941-977
Orobraze TM
1000Cr 72 22 6 975-1000
Orobraze TM
1000 40 60 980-1000
MELTING BETWEEN 1000-1050˚C MELTING ABOVE 1050˚C ARGO-BRAZE ™ RANGE FOR VACUUM TUBES, VACUUM BRAZING AND ELECTRONIC COMPONENTS
Melting Range °C AMS AWS A5.8 EN1044:1999 ISO 17672:2016
Argo-braze TM
72NiV 71.5 28 0.5 780-795 BAg-8b
Argo-braze TM
72V 72 28 778 BAg-8 AG401 Ag 272 V1
Argo-braze TM
63V 63 27 10 685-730
Argo-braze TM
61V 61.5 24 14.5 630-705 BAg-29
Argo-braze TM
60V 60 30 10 602-718 4773 BAg-18 AG402 Ag 160
ASSOCIATED PRODUCTS
This range of products can be supplied to a vacuum-grade specification, with lower impurity limits, which can be required for brazing in vacuum or service in vacuum (e.g. Thermionic Valves). A ‘V’ added as a suffix after the product name shows that a vacuum-grade product, according to ISO17672: 2016 Vacuum Grade 1, will be supplied. This range of products can also be supplied in a non-vacuum grade where ultra-high purity of the alloy is not required but freedom from Zinc is. Argo-braze TM
72NiV is a modified version of Argo-braze TM
72V that contains nickel for improved wetting on ferrous and nickel-based parent materials. Argo-braze TM
72V (formerly Silver-Copper Eutectic TM
) is ideal for flux-less brazing of copper, nickel and metalised ceramics.
Argo-braze TM
63V and 61V are low temperature, indium-bearing filler metals often used in step brazing operations.
Argo-braze TM
60V (formerly RTSN TM
) can meet a variety of niche requirements. It is also used in air brazing operations (flame/induction) in its non-vacuum specification.
Melting Range °C
Orobraze TM
1004 35 10 14 31.5 9.5 971-1004
Orobraze TM
1010 73.8 26.2 980-1010
Orobraze TM
1013 25 15 10 37 13 970-1013
Orobraze TM
1025 20 78 2 975-1025
Orobraze TM
1037 70 8 22 1005-1037
Orobraze TM
1050 70 30 960-1050
Melting Range °C AMS/AWS A5.8 ISO 17672:2016
Orobraze TM
1052 25 15 18 31 11 1017-1052
Pallabraze TM
1070 10 90 1025-1070
Orobraze TM
1121 50 25 25 1102-1121 4784 Au 507
Pallabraze TM
1169 30 34 36 1135-1169 4785 Au 300
Pallabraze TM
1197 22.5 48.5 10 19 910-1179
Pallabraze TM
1219 65 35 1219 BPd-1
Orobraze TM
1270 92 8 1200-1270 BAu-8 7
Metal Joining
Standards
ISO 17672
Products will be supplied according to the current version at time of supply i.e. 2010/2016
EN 1044 Refers to EN 1044:1999 which was superseded by ISO 17672:2010
Key
Elements
Ag Silver
Cu Copper
In Indium
Mn Manganese
Ni Nickel
Zn Zinc Metal Joining
Europe and Worldwide Distribution
Johnson Matthey Metal Joining York Way, Royston, Hertfordshire, SG8 5HJ, UK. Tel. +44 (0)1763 253200, Fax. +44 (0)1763 253168 Email: mj@matthey.com
www.jm-metaljoining.com Switzerland, Germany and Austria
Johnson Matthey & Brandenberger AG Glattalstrasse 18, Postfach 485 CH-8052 Zürich Telefon +41 (0) 44 307 19 30, Fax +41 (0) 44 307 19 20 Email: info@johnson-matthey.ch
www.johnson-matthey.ch
Johnson Matthey Plc cannot anticipate all conditions under which this information and our products or the products of other manufacturers in combination with our products will be used. This information relates only to the specific material designated and may not be valid for such material used in combination with any other materials or in any process. Such information is given in good faith, being based on the latest information available to Johnson Matthey Plc and is, to the best of Johnson Matthey Plc’s knowledge and belief, accurate and reliable at the time of preparation. However, no representation, warranty or guarantee is made as to the accuracy or completeness of the information and Johnson Matthey Plc assumes no responsibility therefore and disclaims any liability for any loss, damage or injury howsoever arising (including in respect of any claim brought by any third party) incurred using this information. The product is supplied on the condition that the user accepts responsibility to satisfy himself as to the suitability and completeness of such information for his own particular use. Freedom from patent or any other proprietary rights of any third party must not be assumed. The text and images on this document are Copyright and property of Johnson Matthey. This datasheet may only be reproduced as information, for use with or for resale of Johnson Matthey products. The JM logo©, Johnson Matthey name© and product names referred to in this document are trademarks of Johnson Matthey Plc, Royston, United Kingdom.
JMMJ Orobraze - Pallabraze/En/v01
SILVER BRAZING FILLER METALS
Metal Joining
SILVER BRAZING FILLER METALS FOR TUNGSTEN CARBIDE AND PCD
Metal Joining Metal Joining
FOR BRAZING OF STAINLESS STEEL JOINTS FOR WET ENVIRONMENTS
BRAZING FLUXES
Metal Joining Metal Joining
GOLD AND PALLADIUM BASED BRAZING FILLER METALS
Metal Joining
BASE METAL BRAZING FILLER METALS
SILVER/COPPER-PHOSPHORUS BRAZING FILLER METALS
Metal Joining Metal Joining
• • • • •lnspırıng scıence, enhancıng lıfe
SILVER BRAZING FILLER METALS
Metal Joining
SILVER BRAZING FILLER METALS FOR TUNGSTEN CARBIDE AND PCD
Metal Joining
FOR BRAZING OF STAINLESS STEEL JOINTS FOR WET ENVIRONMENTS
Metal Joining
BRAZING FLUXES
Metal Joining Metal Joining
GOLD AND PALLADIUM BASED BRAZING FILLER METALS
Metal Joining
BASE METAL BRAZING FILLER METALS
SILVER/COPPER-PHOSPHORUS BRAZING FILLER METALS
Metal Joining Metal Joining
FIILLER METALS & FLUXES FOR SOFT SOLDERING
Europe and Worldwide Distribution
Johnson Matthey Metal Joining York Way, Royston, Hertfordshire, SG8 5HJ, UK. Tel. +44 (0)1763 253200, Fax. +44 (0)1763 253168 Email: mj@matthey.com
www.jm-metaljoining.com Switzerland, Germany and Austria
Johnson Matthey & Brandenberger AG Glattalstrasse 18, Postfach 485 CH-8052 Zürich Telefon +41 (0) 44 307 19 30, Fax +41 (0) 44 307 19 20 Email: info@johnson-matthey.ch
www.johnson-matthey.ch
Metal Joining
Johnson Matthey Plc cannot anticipate all conditions under which this information and our products or the products of other manufacturers in combination with our products will be used. This information relates only to the specific material designated and may not be valid for such material used in combination with any other materials or in any process. Such information is given in good faith, being based on the latest information available to Johnson Matthey Plc and is, to the best of Johnson Matthey Plc’s knowledge and belief, accurate and reliable at the time of preparation. However, no representation, warranty or guarantee is made as to the accuracy or completeness of the information and Johnson Matthey Plc assumes no responsibility therefore and disclaims any liability for any loss, damage or injury howsoever arising (including in respect of any claim brought by any third party) incurred using this information. The product is supplied on the condition that the user accepts responsibility to satisfy himself as to the suitability and completeness of such information for his own particular use. Freedom from patent or any other proprietary rights of any third party must not be assumed. The text and images on this document are Copyright and property of Johnson Matthey. This datasheet may only be reproduced as information, for use with or for resale of Johnson Matthey products. The JM logo©, Johnson Matthey name© and product names referred to in this document are trademarks of Johnson Matthey Plc, Royston, United Kingdom. JMMJ Silver-flo/En/v01
Metal Joining
Metal Joining
Die in dieser Information genannten technischen Produkteigenschaften basieren auf Johnson Matthey PLC Qualitätsstandards und Verarbeitungsrichtlin oder Verwendung kann die Eigenschaften und Ergebnisse beeinflussen. Der Nutzer handelt stets eigenverantwortlich, auch wenn er die in dieser In
Europe and W
Johnson Yo Hertfo Tel. +44 (0)1763 2 Emai
www.j
SILVER BRAZING FILLER METALS
Metal Joining
SILVER BRAZING FILLER METALS FOR TUNGSTEN CARBIDE AND PCD
Metal Joining Metal Joining
FOR BRAZING OF STAINLESS STEEL JOINTS FOR WET ENVIRONMENTS
BRAZING FLUXES
Metal Joining Metal Joining
GOLD AN BASED BRAZING FI
SILVER/COPPER-PHOSPHORUS BRAZING FILLER METALS
Metal Joining
•••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe
•••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe •••••lnspırıng scıence, enhancıng lıfe
Johnson Matthey Platinum Group Metal Services Orchard Road, Royston, SG8 5HE United Kingdom Email: mj@matthey.com
www.matthey.com |
12079 | https://www.mathsteacher.com.au/year7/ch06_decimals/13_div_dec/dec.htm | Year 7 Interactive Maths - Second Edition
Division of a Decimal by another Decimal
To divide a decimal by another decimal:
Move the decimal point in the divisor to the right until it is a whole number.
Move the decimal point in the dividend to the right by the same number of places as the decimal point was moved to make the divisor a whole number.
Then divide the new dividend by the new divisor
Example 36
Solution:
Alternatively, we can divide as follows:
Note:
Example 37
Solution:
Alternatively, we can divide as follows:
Note:
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12080 | https://pubmed.ncbi.nlm.nih.gov/34097295/ | How should we define cure after parathyroidectomy for normocalcemic primary hyperparathyroidism? A retrospective cohort study - PubMed
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. 2021 Dec;73(6):2293-2299.
doi: 10.1007/s13304-021-01108-1. Epub 2021 Jun 7.
How should we define cure after parathyroidectomy for normocalcemic primary hyperparathyroidism? A retrospective cohort study
Oscar Cano-Valderrama123,Santiago Ochagavía45,Concepción Sanabria6,Cristina Familiar6,Jesús Díaz4,Sara Picazo4,Patricia Sáez-Carlin47,Antonio J Torres457
Affiliations Expand
Affiliations
1 Department of Surgery, Hospital Clínico San Carlos, C/ Profesor Martín Lagos SN, 28040, Madrid, Spain. oscarcanovalderrama@hotmail.com.
2 Department of Surgery, Universidad Complutense de Madrid, Madrid, Spain. oscarcanovalderrama@hotmail.com.
3 Instituto de Investigación San Carlos, Madrid, Spain. oscarcanovalderrama@hotmail.com.
4 Department of Surgery, Hospital Clínico San Carlos, C/ Profesor Martín Lagos SN, 28040, Madrid, Spain.
5 Department of Surgery, Universidad Complutense de Madrid, Madrid, Spain.
6 Department of Endocrinology, Hospital Clínico San Carlos, Madrid, Spain.
7 Instituto de Investigación San Carlos, Madrid, Spain.
PMID: 34097295
DOI: 10.1007/s13304-021-01108-1
Item in Clipboard
How should we define cure after parathyroidectomy for normocalcemic primary hyperparathyroidism? A retrospective cohort study
Oscar Cano-Valderrama et al. Updates Surg.2021 Dec.
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. 2021 Dec;73(6):2293-2299.
doi: 10.1007/s13304-021-01108-1. Epub 2021 Jun 7.
Authors
Oscar Cano-Valderrama123,Santiago Ochagavía45,Concepción Sanabria6,Cristina Familiar6,Jesús Díaz4,Sara Picazo4,Patricia Sáez-Carlin47,Antonio J Torres457
Affiliations
1 Department of Surgery, Hospital Clínico San Carlos, C/ Profesor Martín Lagos SN, 28040, Madrid, Spain. oscarcanovalderrama@hotmail.com.
2 Department of Surgery, Universidad Complutense de Madrid, Madrid, Spain. oscarcanovalderrama@hotmail.com.
3 Instituto de Investigación San Carlos, Madrid, Spain. oscarcanovalderrama@hotmail.com.
4 Department of Surgery, Hospital Clínico San Carlos, C/ Profesor Martín Lagos SN, 28040, Madrid, Spain.
5 Department of Surgery, Universidad Complutense de Madrid, Madrid, Spain.
6 Department of Endocrinology, Hospital Clínico San Carlos, Madrid, Spain.
7 Instituto de Investigación San Carlos, Madrid, Spain.
PMID: 34097295
DOI: 10.1007/s13304-021-01108-1
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Abstract
Cure after surgery for normocalcemic primary hyperparathyroidism (NHPT) is defined as parathyroid hormone (PTH) normalization. However, an increase of PTH is frequently observed in cured patients with hypercalcemic primary hyperparathyroidism (HHPT). Therefore, this criterion must be redefined. A single-center retrospective study was performed including all patients who underwent surgery for Primary Hyperparathyroidism from 2013 to 2019. Cure rates of different types of hyperparathyroidism were analyzed. PTH reduction was studied as a possible criterion to define cure in patients with NHPT. One-hundred and eighty-six patients were included: 173 with HHPT and 13 with NHPT. After a mean follow-up of 33.4 months, 174 (93.6%) patients were considered cured. Cure was more frequent in the group of patients with HHPT (97.1% vs. 46.2%, p < 0.001). In the multivariate analysis, surgical failure was associated with NHPT and multiglandular disease. Forty-nine (30.1%) cured patients with HHPT had an increased PTH during the follow-up. When decline of PTH levels was studied in patients with HHPT to define cure, the area under curve was 0.92. A cut-off value of 40% in PTH reduction achieved a sensitivity and specificity of 83.4% and 80.0%. If cure was defined as a 40% reduction of PTH, cure rate in the group of patients with NHPT would increase to 69.2%. Patients with NHPT had a lower cure rate than patients with HHPT. A significant number of cured patients with HHPT had an increased PTH during follow-up. A 40% reduction in PTH levels is proposed as an alternative definition for cure in patients with NHPT.
Keywords: Cure; Hypercalcemic primary hyperparathyroidism; Normocalcemic primary hyperparathyroidism; Parathyroid gland; Parathyroidectomy; Primary hyperparathyroidism; Remission; Surgery; morbidity.
© 2021. Italian Society of Surgery (SIC).
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Is parathyroidectomy safe and effective in patients with normocalcemic primary hyperparathyroidism?Traini E, Bellantone R, Tempera SE, Russo S, De Crea C, Lombardi CP, Raffaelli M.Traini E, et al.Langenbecks Arch Surg. 2018 May;403(3):317-323. doi: 10.1007/s00423-018-1659-0. Epub 2018 Mar 14.Langenbecks Arch Surg. 2018.PMID: 29541851
Normocalcemic Primary Hyperparathyroidism: Need for a Standardized Clinical Approach.Zavatta G, Clarke BL.Zavatta G, et al.Endocrinol Metab (Seoul). 2021 Jun;36(3):525-535. doi: 10.3803/EnM.2021.1061. Epub 2021 Jun 1.Endocrinol Metab (Seoul). 2021.PMID: 34107603 Free PMC article.Review.
Elevated parathyroid hormone levels after successful parathyroidectomy for primary hyperparathyroidism: a clinical review.de la Plaza Llamas R, Ramia Ángel JM, Arteaga Peralta V, García Amador C, López Marcano AJ, Medina Velasco AA, González Sierra B, Manuel Vázquez A, Latorre Fragua RA.de la Plaza Llamas R, et al.Eur Arch Otorhinolaryngol. 2018 Mar;275(3):659-669. doi: 10.1007/s00405-017-4836-9. Epub 2017 Dec 5.Eur Arch Otorhinolaryngol. 2018.PMID: 29209851 Review.
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References
Wills MR, Pak CY, Hammond WG, Bartter FC (1969) Normocalcemic primary hyperparathyroidism. Am J Med 47(3):384–391. - DOI - PubMed
Bilezikian JP, Khan AA, Potts JT Jr, Third International Workshop on the Management of Asymptomatic Primary H (2009) Guidelines for the management of asymptomatic primary hyperparathyroidism: summary statement from the third international workshop. J Clin Endocrinol Metab 94(2):335–339. - DOI - PubMed - PMC
Maruani G, Hertig A, Paillard M, Houillier P (2003) Normocalcemic primary hyperparathyroidism: evidence for a generalized target-tissue resistance to parathyroid hormone. J Clin Endocrinol Metab 88(10):4641–4648. - DOI - PubMed
Gomez-Ramirez J, Mihai R (2017) Normocalcaemic primary hyperparathyroidism: a diagnostic and therapeutic algorithm. Langenbecks Arch Surg 402(7):1103–1108. - DOI - PubMed
Gomez-Ramirez J, Gomez-Valdazo A, Luengo P et al (2020) Comparative prospective study on the presentation of normocalcemic primary hyperparathyroidism. Is it more aggressive than the hypercalcemic form? Am J Surg 219(1):150–153. - DOI - PubMed
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12081 | https://www.youtube.com/watch?v=kdMep5GUOBw | Formal definition of partial derivatives
Khan Academy
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Partial derivatives are formally defined using a limit, much like ordinary derivatives.
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12082 | https://www.thedataschool.co.uk/naledi-hollbruegge/compound-growth-rate-quick-table-calculation-tableau/ | Compound Growth Rate quick table calculation in Tableau - The Data School
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Compound Growth Rate quick table calculation in Tableau
One of the options amongst your quick table calculations in to compute the ‘compound growth rate’ (CGR). The CGR is a measure of growth over multiple time periods. It denotes the growth rate from the initial value to the final value, assuming that the investment has compoundinggrowth over the time period.
A compunding growth rate is calculated with the following formula.
This translates to Tableau as:
POWER(ZN(SUM([Sales]))/LOOKUP(ZN(SUM([Sales])),-[N Years]), ZN(1/[N Years])) – 1
or
((End Value/Beginning Value)^(1/n)) – 1
Luckily, we don’t have to create a calculated field manually every time we want to figure out the CGR. We just right click our measure pill in the view > select ‘Quick table calculation’ > ‘Compoung growth Rate’. Note that this option will only be available once you have at least one date field in the view.
In the example below the sum of sales, year over year growth (relative to 2010) and CGR (compute using ‘Order Date’) have been plotted. You can see how the CGR becomes less jagged over time and comes closer to predicting a steady growth rate, as it takes into account the values of each month across the years.
Note that measures that contain both positive and negative values across the chosen time period are problematic in this context. This is because the calculation is taking the root of a number and if you have a negative start value and a positive end value then this would mean taking the root of a negative number. ( When using profit for instance, the first sum of sales value is negative and subsequently only those values that are also negative are plotted. Those calculations involving positive profit result in null values.
Looking at your CGR may be useful when you have highly fluctuating values or data that spans a long period of time, as it allows you to tease out a general trend.
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Naledi Hollbruegge
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12083 | https://www.youtube.com/watch?v=0YqZIHFmVzg | Proof by Contrapositive | Method & First Example
Dr. Trefor Bazett
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Learning objective: prove an implication by showing the contrapositive is true.
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if you recall when we talked about a normal implication P implies Q we've seen before that P implies Q is logically equivalent to not Q implies not P so I'm going to make use of that logical equivalency when I try to go out and prove things in particular my first goal is going to be to try to prove an implication that's what I'm trying to do I'm trying to show prove an assumption leads to a conclusion but what I'm going to do is treat this as the other way around I'm know that P implies Q is the same thing as not Q implies not P that is I'm instead going to go and try to prove the Contra positive I'm going to try to prove not Q implies not p and that's the way I'm going to demonstrate that P implies Q so let's see an example in this theorem I'm saying if the square is even then n itself is also going to be even now I want to not carefully this is different from the theorem that we've seen in a previous video this is not not saying if n is even then n s is even it's the other way around it's saying if n squar is even then n is going to be even I'm going to just try to use a direct proof and show you why it might be challenging so my assumption might be that N2 is even which is going to be that N2 is going to be twice some k for some K I'm not going to write all my symbols down because this is my plain around step I don't have to be all that precise all right and I want to say something about n I've got this Square here so maybe what I should be doing is taking a square root of both sides that might give me my n if I took the positive square root for instance but then what I have is it being like equal to < tk2 < TK K which doesn't really look anything at all like being even so assuming that my n squar was even and taking square roots just didn't leave anywhere it's got all this messy rot2 stuff floating around so proving it directly seems a little bit challenging so let's let's get rid of that since proving this directly wasn't all that useful let's try to prove it by the contrapositive we're we're going to do it the other way around with negations so when I write it as proof by contradiction by Contra positive rather note what happens first of all the the N squar is even and the N is even they flip orders but they also get two knots in that's what happens when I do Contra positive so in other words I'm assuming that n is not even and I'm trying to deduce that n^2 is not even by the way the Assumption n is not even is the same thing as n is odd and the conclusion n squ is not even is the same as the conclusion n s is odd but I've actually seen that particular proof before I've seen a proof that n is odd implies N squared is odd and so we can sort of bring that one in and this is the proof that we've seen before indeed this claim n^2 even implies n is even is the same claim it's logically equivalent to n is odd implies n squ is odd it's just that when it's written this way in its contrapositive form it's quite tractable and we were able to do its proof almost directly it sort of just fell out by our assumptions in a way that just wasn't true when I tried to directly prove the original so it's a wonderful illustration of how a a proof that's challenging and hard if you try to go and prove its condra positive the condra positive while logically equivalent might be much easier to prove |
12084 | https://en.wikipedia.org/wiki/Invasion_of_the_Spice_Islands | Jump to content
Invasion of the Spice Islands
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Coordinates: 3°9′S 129°23′E / 3.150°S 129.383°E / -3.150; 129.383
From Wikipedia, the free encyclopedia
1810 British military campaign in the Dutch East Indies
| Invasion of the Spice Islands | |
--- |
| Part of the Napoleonic Wars | |
| View of the Island of Banda Neira – captured by a force landed from a squadron under the Command of Captain Cole in the morning of 9 August 1810 | |
| | | | --- | | Date | 15 February – 31 August 1810 | | Location | Maluku Islands, Dutch East Indies 3°9′S 129°23′E / 3.150°S 129.383°E / -3.150; 129.383 | | Result | British victory | | Territorial changes | Spice Islands occupied by Britain until 1814 | | |
| Belligerents | |
| United Kingdom East India Company | France |
| Commanders and leaders | |
| Christopher Cole Edward Tucker | Herman Willem Daendels R. Coop à Groen |
| Strength | |
| 7 ships 1,000 soldiers and Marines | Various forts and coastal defences |
| Casualties and losses | |
| Light | All islands, fortifications and military stores captured |
| v t e Dutch colonial conflicts | |
--- |
| 17th century Bantam (1601) Amboina (1605) Malacca (1606) Cape Rachado (1606) Pulo Buton (1606) Mozambique (1607) Mozambique (1608) Banda Islands (1609–21) Johor (1613) Jayakarta (1618–19) Jayakarta (1619) Macau (1622) Pescadores (1622–24) Salvador (1624) Luanda (1624) Persian Gulf (1625) Salvador (1625) Elmina (1625) Cuba (1628) Batavia (1628–29) Recife (1630) Jambi (1630) Abrolhos (1631) Liaoluo Bay (1633) Taiwan (1635–36) Brazil (1636) Liuqiu Island (1636) Porto Calvo (1637) Elmina (1637) Vietnam (1637–43) Goa (1638) Salvador (1638) Mormugão (1639) Itamaracá (1640) Ceylon (1640) Malacca (1641) Luanda (1641) Taiwan (1641) Taiwan (1642) Chile (1643) Cambodia (1643–44) New Netherland (1643–45) India (1644-45) Tabocas (1645) Brazil (1645) Philippines (1646) Kombi (1647) Guararapes (1648) Guararapes (1649) Taiwan (1652) 2nd Recife (1652-1654) 2nd Colombo (1654) Mannar (1658) Malabar (1658-1663) New Netherland (1659–63) Cape Colony (1659-1677) Formosa (1661–62) Sulawesi (1666–69) Ceylon (1670-1670) India (1673) Java (1674–80) 18th century Java (1704–07) Java (1719–23) India (1739–41) Java (1741–43) Penfui (1749) Java (1749–57) Ceylon (1764-1766) Persian gulf (1765) India (1781) Sumatra (1781) Ceylon (1782) Gold Coast (1782) Cape Colony (1795) 19th century Surinam (1804) Cape Colony (1806) Java (1806–07) Moluccas (1810) Java (1811) Algiers (1816) Ambon (1817) Palembang (1819) Palembang (1821) Sumatra (1821–37) Borneo (1823) Bone (1824–25) Java (1825–30) Aceh (1831) Ahanta (1837–39) Bali (1846) Bali (1848) Bali (1849) Palembang (1851–59) Montrado (1854–55) Nias (1855–64) Bali (1858) Bone (1858–59) Borneo (1859–63) Japan (1863–64) Pasoemah (1864–68) Gold Coast (1869–70) Aceh (1873–1904) Mandor (1884–85) Jambi (1885) Edi (1890) Lombok and Karangasem (1894) Pedir (1897–98) 20th century Kerinci (1903) Bone (1905–06) Bali (1906) Bali (1908) Venezuela (1908) Indonesia (1941–45) Indonesia (1946–49) | |
| v t e Napoleonic Wars | |
--- |
| Prelude French Revolutionary Wars + First Coalition + Second Coalition Russo-Turkish War --- Major wars Anglo-Spanish War Russo-Persian War Third Coalition Franco-Swedish War Fourth Coalition Russo-Turkish War English Wars + Gunboat War Dano-Swedish War Anglo-Turkish War Peninsular War Finnish War Anglo-Russian War Fifth Coalition + Austro-Polish War Anglo-Swedish War French invasion of Russia Sixth Coalition + Dano-Swedish War Swedish-Norwegian War Hundred Days + Neapolitan War + Waterloo campaign + Minor campaigns of 1815 --- Rebellions Anti-French rebellions in Italy [it] + Calabria [it] Irish rebellion of 1803 Castle Hill convict rebellion Swiss uprising [de] Greater Poland Uprising Spanish revolts + Spanish American revolts [fr] Portuguese revolts [pt] Dörnberg insurrection [de] Tyrolean Rebellion 1809 Gottscheer rebellion Vorarlberg insurrection [de] Galician Uprising Russian partisan movement [ru] Revolution of 1813 [nl] Vendean and Chouan revolts [fr] --- Overseas and naval conflicts Haitian Revolution + Saint-Domingue expedition Linois's expedition Caribbean campaign Senyavin's expedition Franco-Haitian War Missiessy's expedition Allemand's expedition L'Hermite's expedition Atlantic campaign Cape Colony Lamellerie's expedition River Plate 1st Java Adriatic campaign Santo Domingo Mauritius campaign Troude's expedition Roquebert's expedition Spice Islands Dutch East Indies Banda Oriental 2nd Java Anglo-Haitian conflict | |
The invasion of the Spice Islands was a military invasion by British forces that took place between February and August 1810 on and around the Dutch owned Maluku Islands (or Moluccas) also known as the Spice Islands in the Dutch East Indies during the Napoleonic wars.
By 1810 the Kingdom of Holland was a vassal of Napoleonic France and Great Britain along with the East India Company sought to control the rich Dutch spice islands in the East Indies. Two British forces were allocated; one to the island of Ambon and Ternate, then another force would capture the more heavily defended islands of Banda Neira, following which any other island that was defended.
In a campaign that lasted seven months British forces took all of the islands in the region; Ambon was captured in February, Banda Neira in August and Ternate and all other islands in the region later that same month.
The British held on to the islands until the end of the war. After the Anglo-Dutch Treaty of 1814 the islands were handed back to the Dutch, but in the meantime the East India Company had uprooted a lot of the spice trees for transplantation throughout the British Empire.
Background
[edit]
The Moluccas were known as the Spice Islands because of the nutmeg, mace and cloves that were exclusively found there. The presence of these sparked European colonial interest in the sixteenth century, starting with Portugal who virtually held a monopoly on the spice trade. The Dutch East India Company (VOC) arrived in the islands in 1599 and eventually ousted the Portuguese.
The English East India Company arrived soon after who in turn competed with the Dutch and had claimed the island of Ambon and the small island of Run. The competition soon came to a head with the Amboyna massacre in 1623 which influenced Anglo-Dutch relations for decades. After 1667 under the Treaty of Breda, both agreed to maintain the colonial status quo and relinquish their respective claims, which continued well in the eighteenth century.
After the financially disastrous Fourth Anglo-Dutch War the Dutch Republic became the Batavian Republic and allied with Revolutionary France. The VOC was then nationalised in 1796. During the war that followed Prince William V of Orange ordered the Dutch East India Company to hand over their colonies to the British to stop trade falling into French hands. The VOC was officially dissolved in 1799; the overseas possessions then became Dutch government colonies (the Moluccas became part of the Dutch East Indies). The islands were captured by Vice Admiral Peter Rainier – Ternate was later viciously contested by the Dutch. These were subsequently returned as a result of the Treaty of Amiens seven years later. Peace however did not last long and thus began the Napoleonic Wars. By 1808, most of the Dutch colonies had been neutralised in a series of brief but successful campaigns; the Cape was invaded and captured by Sir Home Riggs Popham in January 1806 and the Dutch navy's East Indies squadron was destroyed by Sir Edward Pellew in the Java campaign of 1806–1807. The French and British were each seeking to control the lucrative Indian Ocean trade routes. The British had started by invading the French Indian Ocean islands of Ile de France and Île Bonaparte in 1809. The Dutch East Indies had to be taken by the British for a number of reasons; firstly it was necessary to subvert French power there before it entrenched itself too firmly to be dislodged easily by the British. This was the primary concern of the East India Company who felt that their China trade would be threatened.
Preparations
[edit]
| Maluku Island spices |
| Clove Nutmeg Mace |
In 1806, Herman Willem Daendels became Governor-General of the Dutch East Indies and sought to defend the region against the British. Using forced labour Daendels reinforced the garrisons, improved defences and built the Great Post Road in Java to counter a potential British threat.
In the middle of 1809, the Colonial Governor of India, 1st Earl of Minto wanted to set up two squadrons to conquer the Moluccas. This fell under Rear Admiral William O'Brien Drury who was resolved to seize the Dutch settlements. The first was set up in February 1810 – Captain Edward Tucker commanded a small squadron comprising HMS Dover, the frigate HMS Cornwallis under Captain William Augustus Montagu and the sloop HMS Samarang. They carried two companies of troops numbering around 400 men of the Madras European Regiment and the Madras Artillery. Their main objective were the islands of Amboyna and Ternate.
The second force intended to capture the Banda Islands, the heart of the Spice trade – notably the strongly defended island of Banda Neira. The force comprised the 36-gun frigate Caroline, the former French frigate HMS Piedmontaise, the 18-gun sloop HMS Barracouta, and a 12-gun transport, the captured Dutch vessel Mandarin, which was serving as a tender to Caroline. The frigates and sloop carried a hundred officers and men of the Madras European Regiment, as well as sailors and Royal Marines, and twenty men and two guns from the Madras Artillery. The squadron was commanded by Captain Christopher Cole, with Captain Charles Foote on Piedmontaise and Captain Richard Kenah aboard Barracouta. They departed from Madras and sailed via Singapore, where Captain Richard Spencer informed Cole that over 700 regular Dutch troops may be located in the Bandas.
A number of East India Company agents were in tow partially to look at the idea of uprooting all of the spice trees namely nutmeg and clove which they had done on a small scale during the islands previous occupation in the 1790s. The primary consideration was the commercial advantage – the occupation of the Spice Islands meant not only a curtailment of the Dutch trade and power in the East Indies but also an equivalent gain to the company of the rich trade in spice.
By 1810, the Kingdom of Holland was a vassal of Napoleonic France after being annexed under orders from Napoleon Bonaparte.
Invasion of the Moluccas
[edit]
Amboyna
[edit]
The British force allocated to take Ambon left Madras on 9 October 1809. By the middle of the following February they arrived off the island the most considerable of the Dutch Spice islands and seat of government. On 6 February Dover captured the Dutch brig-of-war Rambang. Both Dover and Cornwallis anchored off the town of Ambon situated at the bottom of a small bay beneath a line of low hills. These were defended by batteries along the beach as well as on some of the neighbouring heights and by Fort Victoria the main fort mounting a number of heavy guns. As the elevations on the left and in the rear of the town commanded its defences, the British intended to assault them.
The British launched their attack on 16 February – the squadron at the same time occupied the attention of the Dutch by a vigorous cannonade. The troops aided by seamen and marines led by a Captain Court were landed on the right of the bay unnoticed by the Dutch, capturing two batteries that overlooked the port and Fort Victoria. During the night, Samarang landed forty men, who were joined by two field pieces from Dover. These joined in the bombardment of Fort Victoria as well as from the two captured batteries.
British cannon fire from the ships and shore guns proved effective at showing considerable force – three ships were sunk in the harbour. A summons was then given to the Dutch Governor Colonel Filz for the surrender of the island, and after a few hours the articles of capitulation were agreed upon. On 18 February the town capitulated; British casualties were extremely light, with only three dead, one of whom was a marine from Samarang. The entire island defended by 250 Europeans and around 1,000 Javanese and Madurese people laid down their arms.
During the campaign the British captured several Dutch vessels. One was the Dutch brig Mandurese which had twelve guns. She was one of three vessels sunk in the inner harbour of Amboyna. However, the British raised her after the island surrendered. They took her into service as Mandarin. From Amboyna, the squadron went on to capture the islands of Saparua, Haruku, Nusa Laut, Buru, and Manipa.
After the attack on Amboyna, Spencer sailed Samarang to the island of Pulo Ay (or Pulo Ai), in the Banda Islands. There he conducted a successful and bloodless attack on Fort Revenge. Spencer disguised Samarang to look like a Dutch merchant vessel, which fooled the fort's commander, enabling Spencer to take the fort by surprise. The Dutch commander committed suicide by taking poison after he realised that he had surrendered to what was a relatively weak British force.
Next, Samarang captured the Dutch brig Recruiter on 28 March, when she arrived off the island of Pulau Ai. She was armed with twelve guns and had a crew of fifty men under the command of Captain Hegenheard. She had on board 10,000 dollars, the payroll of which were for the Dutch garrison at Banda Neira as well as provisions, a doctor, nurse, and twenty infants, on their way to conduct a vaccination campaign. Samarang shared the prize money by agreement with Dover and Cornwallis. Between 29 April and 18 May, Dover, Cornwallis, and Samarang captured the Dutch ships Engelina and Koukiko. Both Pulau Ai and Run were captured without a fight.
After sending all the Dutch officers and troops from Amboyna to Java, Captain Tucker sailed for the Dutch port of Gorontello, in the Bay of Tommine, on the north-east part of the island of Celebes in June 1810. Although the Dutch flag was flying over Fort Nassau, the settlement was governed by a Sultan and his two sons on behalf of the Dutch. He persuaded the Sultan to allow the British to replace the Dutch – which he agreed to.
Finally on 26 June Dover captured the island of Manado, where Fort Amsterdam was protected by two heavy batteries. The fort surrendered without opposition when Captain Tucker pointed out to the Dutch Governor that an English frigate, with guns ready to fire and volunteers waiting in her boats, was waiting to storm the Dutch position. Manado had a garrison of 113, including officers, and the fort and the batteries mounted fifty guns.[Note 1]
Banda Neira
[edit]
The British force destined for the Banda Islands appeared at Banda Neira on 9 August. The main defence on the islands was Fort Belgica which is a fairly significant position built in the stereotypical Vauban pentagon and surrounded by a ditch. The older Fort Nassau lay further down to Belgica. The defences had been strengthened since the British had left the islands. In addition there were ten batteries (exclusive of the two forts) and as Spencer had predicted, there were 700 regular Dutch troops and 800 native militia.
The operational concept was to approach Banda Neira after dark on 8 August and disembark the landing force of about 400 sailors, marines, and European infantry in small boats. These boats would run into the harbour before dawn and take Fort Belgica and other strongpoints by surprise.
In the event, not much went according to plan. They were taken under fire from a battery during the night from the small island of Rosensgan as the Dutch were on alert, then the weather worsened dispersing the fleet of small boats. About 100 yards from the shore and directly opposite a Dutch battery consisting of ten 18-pounders the boats grounded on a coral reef. The men leapt into the water, then after an hour and a half before daylight the force were able to land in a sandy cove. When Captain Cole reached the assembly area for the attack there were less than 200 seaman, marines and soldiers remaining. He made the decision to press ahead with the attack – Commander Richard Kenah of Barracouta attacked the Dutch battery in front of them from the rear. Using boarding pikes they killed one sentry and captured the remainder of the garrison, some sixty men, without firing a shot.
Leaving a small guard in the battery – twenty minutes later they attempted to storm Fort Belgica. As dawn was breaking they were helped by a native guide to the outer ramparts of the fort. Despite the British being spotted and the alarm called, the heavy rain worked to the advantage of the attackers. The defenders visibility was reduced and their firearms rendered useless. Having erected scaling ladders the attackers stormed over the outer walls and despite coming under desultory musket fire this part of the fort was stormed and taken. The inner walls were then attempted but the ladders were too short. However, the main gate had been opened to admit the commandant who lived outside. The British took the opportunity, made a rush, and by 5:30am the fort was in their possession. The Dutch side the fort commandant and ten men were killed, two officers and thirty men were taken prisoner, along with fifty two cannon. The British loss was trifling with only a few men wounded.
Captain Cole sent Commander Kenah to demand the surrender of the Dutch governor. As negotiations were taking place the Caroline, Piedmontaise, and Barracouda attempted to enter the harbour but were fired on by Dutch batteries. The British used the guns of Fort Belgica to return fire and threatened to destroy the nearby town if the governor did not surrender but he complied.
Map and elevation of Fort Belgica on Banda Neira
Fort Belgica today
Ternate
[edit]
The final part of the campaign involved the capture of the island of Ternate, the last remaining Dutch possession of any consequence in the Moluccas; after capturing Manado, Tucker and 174 men from HMS Dover arrived there on 25 August. The plan was to capture Fort Kalamata a small fortress which lies near the main town which was made up Fort Oranje; a bigger fort – this contained 92 heavy calibre guns and a garrison of 500 men of which 150 were Dutch. Many of the natives soldiers however had been in near mutiny over their pay.
After having found an amphibious landing difficult at night Tucker and his men resorted in daylight and succeeded at Sasa, a village screened by a point of land from the fort by 7am on 28 August. They managed to ascend a hill, occupied it to place a field gun to command the position. The area on top was covered in thick forest and so Tucker attempted a night march. They were met by a roadblock, so a detour along a stream led to a sharp firefight against a strong Dutch detachment. After driving the Dutch away with a bayonet charge, they came across a beach which was only within a hundred yards of Fort Kalamata. The Dutch opened fire, but even so the attackers decided to make an assault. After crossing a ditch, the British stormed the fort using scaling ladders on the flank of the bastion, and carried it after some sharp fighting. Owing to the darkness and rapidity of the advance the casualties were moderate with three killed and fourteen wounded.
Tucker attempted to demand the Dutch governor Colonel Jon van Mithman to surrender but in response guns were fired at the British ships from Fort Kota Baro a small fort which lay in between Kalamata and Oranje. Dover having placed her herself in front of it then used hers guns to silence the battery, following which a small detachment took possession of after a surprise attack. Dover then faced off against Fort Oranje pounding it with considerable effect. The Dutch fought back for a few hours, but after experiencing severe damage to the fort and rising losses it was then assaulted by a surprise attack from the rear by a Royal Marine detachment from Kalamata led by Lieutenant Cursham. The British then turned both fort guns on the town itself, and with Dover then pounded the town. By 5am the Dutch had had enough and the town finally surrendered.
A day after the town's surrender Mithman then surrendered the island to the British who promptly took possession. By August 31 the campaign in the Moluccas had ended.
Aftermath
[edit]
After the Dutch surrender, Captain Charles Foote (of Piedmontaise) was appointed Lieutenant-Governor of the Banda Islands. This action was a prelude to Britain's invasion of Java in 1811 which Cole also took a leading role in planning and executing. This was successfully completed under Rear-Admiral Robert Stopford. For his services, Cole was knighted in May 1812, awarded a specially minted medal, and given an honorary doctorate by the University of Oxford.
Before the Dutch regained control of the islands, the East India Company used their occupation of the Spice Islands to gather spice seedlings. Thus began a transplantation on an almost industrial scale – most were sent to Bencoolen and Penang, as well as Ceylon and other British colonies. In the previous occupation in the 1790s the EIC had established spice gardens in Penang; by 1805 these contained 5,100 nutmeg trees and 15,000 clove trees. After the occupation in 1815 that number had jumped to 13,000 nutmeg trees and as many as 20,000 clove trees. From these locations the trees were then transplanted to other British colonies elsewhere, notably Grenada and later Zanzibar. As a result, this competition largely destroyed the value of the Banda Islands to the Dutch.
The Anglo-Dutch Treaty of 1814 restored the islands as well as Java to the Dutch. The islands then remained part of the Dutch East Indies, a colony of the Kingdom of the Netherlands, until Indonesia's independence in 1945.
See also
[edit]
Pattimura
References
[edit]
Notes
^ A first-class share of the prize money was worth £270 9s; a sixth-class share was worth £2 8s 7½d.
Citations
^ Jump up to: a b c Woodman pp. 104–06
^ "Sir Christopher Cole, K.C.B.". The Annual biography and obituary. Vol. 21. London: Longman, Hurst, Rees, Orme, and Brown. 1837. pp. 114–123. Retrieved 22 December 2011.
^ Jump up to: a b Milne, Peter (16 January 2011). "Banda, the nutmeg treasure islands". Jakarta Post. Jakarta. pp. 10–11. Retrieved 22 December 2011. But the economic importance of the Bandas was only fleeting. With the Napoleonic wars raging across Europe, the British returned to the Bandas in the early 19th century, temporarily taking over control from the Dutch. The English uprooted hundreds of valuable nutmeg seedlings and transport them to their own colonies in Ceylon and Singapore, breaking forever the Dutch monopoly and consigning the Bandas to economic decline and irrelevance.
^ Wolter, John Amadeus (1999). he Napoleonic War in the Dutch East Indies: An Essay and Cartobibliography of the Minto Collection Issue 2 of Occasional paper series // Philip Lee Phillips Society Issue 2. Geography and Map Division, Library of Congress. p. 5.
^ Gardiner 2001, p. 92
^ Das p. 194
^ Moore & van Nierop p. 121
^ Jump up to: a b c d e James pp 191–94
^ Jump up to: a b c d e f g Paget, William Henry (1907). Frontier and overseas expeditions from India. Government Monotype Press. pp. 319–24.
^ Jump up to: a b c d e f Burnett pp 201–03
^ "102nd Regiment of Foot (Royal Madras Fusiliers): Locations". Regiments.org. Archived from the original on 22 February 2006.
^ Jump up to: a b Tracy p 86
^ Jump up to: a b Adkins, Roy; Adkins, Lesley (2008). The War for All the Oceans: From Nelson at the Nile to Napoleon at Waterloo. Penguin. pp. 344–355. ISBN 978-0-14-311392-8. Retrieved 22 December 2011.
^ Das pp 191–92
^ Jump up to: a b Clarke & McArthur 337-41
^ Winfield (2008), p. 350.
^ Jump up to: a b Chessell (2005), pp. 51–4.
^ "No. 17081". The London Gazette. 18 November 1815. p. 2308.
^ "No. 17100". The London Gazette. 16 January 1816. p. 93.
^ van de Wall, V.I (1928). De Nederlandsche Oudheden in de Molukken [Dutch Antiquities in the Moluccans] (in Dutch). 's-Gravenhage: Martinus Nijhoff.
^ Jump up to: a b c Wilson, Horace Hayman (1845). The History of British India: From 1805–1835 Volume 1 of The History of British India from 1805 to 1835. J. Madden & Company. pp. 341–48.
^ Jump up to: a b Hanna, Willard Anderson; Alwi, Des (1990). Turbulent Times Past in Ternate and Tidore. Yayasan Warisan dan Budaya Banda Naira. p. 233.
^ Thomas p. 223
^ Wilson, William John (1883). History of the Madras Army, Volume 3. E. Keys at the Government Press. pp. 316–18.
^ Milton p. 380
Bibliography
[edit]
Burnett, Ian (2013). East Indies. Rosenberg Publishing. ISBN 978-1-922013-87-3.
Chessell, Gwen S. J (2005). Richard Spencer: Napoleonic war naval hero and Australian pioneer. UWA Publishing. ISBN 978-1-920694-40-1.
Clarke, James Stanier; McArthur, John, eds. (2010). The Naval Chronicle: Volume 24, July-December 1810: Containing a General and Biographical History of the Royal Navy of the United Kingdom with a Variety of Original Papers on Nautical Subjects, Volume 24 Cambridge Library Collection – Naval Chronicle. Cambridge University Press. ISBN 978-1-108-01863-0.
Das, Aditya, ed. (2016). Defending British India Against Napoleon: The Foreign Policy of Governor-General Lord Minto, 1807–13 Volume 13 of Worlds of the East India Company. Boydell & Brewer. ISBN 978-1-78327-129-0.
Gardiner, Robert, ed. (2001) . The Victory of Seapower. Caxton Editions. ISBN 1-84067-359-1.
James, William (2011). The Naval History of Great Britain: A New Edition, with Additions and Notes, and an Account of the Burmese War and the Battle of Navarino. Vol. 5. Cambridge University Press London. ISBN 978-1-108-02169-2.
Milton, Giles (2012). Nathaniel's Nutmeg: How One Man's Courage Changed the Course of History. Hachette UK. ISBN 978-1-4447-1771-6.
Moore, Bob; van Nierop, Henk (2017). Colonial Empires Compared: Britain and the Netherlands, 1750–1850. Routledge. ISBN 978-1-351-95050-3.
Thomas, David (1998). Battles and Honours of the Royal Navy. Pen and Sword. ISBN 978-0-85052-623-3.
Tracy, Nicholas (1998). Who's Who in Nelson's Navy; 200 Naval Heroes. Chatham Publishing. ISBN 1-86176-244-5.
Winfield, Rif (2008). British Warships in the Age of Sail 1793–1817: Design, Construction, Careers and Fates. Seaforth. ISBN 978-1-86176-246-7.
Woodman, Richard (2005). The Victory of Seapower: Winning the Napoleonic War 1806–1814. Mercury Books. ISBN 978-1-86176-038-8.
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12085 | https://arxiv.org/abs/2212.02639 | We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
math > arXiv:2212.02639
Mathematics > Number Theory
arXiv:2212.02639 (math)
[Submitted on 5 Dec 2022 (v1), last revised 1 Feb 2023 (this version, v2)]
Title:Sums of Reciprocals of Recurrence Relations
Authors:Hao Cui, Xiaoyu Cui, Sophia C. Davis, Irfan DurmiÄ, Qingcheng Hu, Lisa Liu, Steven J. Miller, Fengping Ren, Alicia Smith Reina, Eliel Sosis
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Abstract:There is a growing literature on sums of reciprocals of polynomial functions of recurrence relations with constant coefficients and fixed depth, such as Fibonacci and Tribonacci numbers, products of such numbers, and balancing numbers (numbers $n$ such that the sum of the integers less than $n$ equals the sum of the $r$ integers immediately after, for some $r$ which is called the balancer of $n$; If $n$ is included in the summation, we have the cobalancing numbers, and $r$ is called the cobalancer of $n$). We generalize previous work to reciprocal sums of depth two recurrence sequences with arbitrary coefficients and the Tribonacci numbers, and show our method provides an alternative proof of some existing results. We define $(a,b)$ balancing and cobalancing numbers, where $a$ and $b$ are constants that multiply the left-hand side and right-hand side respectively, and derive recurrence relations describing these sequences. We show that for balancing numbers, the coefficients $(3,1)$ is unique such that every integer is a $(3,1)$ balancing number, and proved there does not exist an analogous set of coefficients for cobalancing numbers. We also found patterns for certain coefficients that have no balancing or cobalancing numbers.
| | |
--- |
| Comments: | 31 pages, 2 figures, 3 tables |
| Subjects: | Number Theory (math.NT) |
| MSC classes: | 11B39 (primary), 33C05 (secondary) |
| Cite as: | arXiv:2212.02639 [math.NT] |
| | (or arXiv:2212.02639v2 [math.NT] for this version) |
| | arXiv-issued DOI via DataCite |
Submission history
From: Irfan DurmiÄ [view email] [v1] Mon, 5 Dec 2022 22:50:52 UTC (406 KB) [v2] Wed, 1 Feb 2023 06:58:57 UTC (293 KB)
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12086 | https://www.merriam-webster.com/thesaurus/obese | Est. 1828
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12087 | https://study.com/academy/lesson/energy-consumption-of-the-world-the-differences-in-consumption-between-developing-and-developed-nations.html | Energy Consumption by Country & Per Capita | Overview & Examples - Lesson | Study.com
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Environmental Science 101: Environment and Humanity
Energy Consumption by Country & Per Capita | Overview & Examples
Contributors: Melissa Bialowas, Rebecca Gillaspy
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Learn about energy consumption per capita and energy consumption by country. See the top few countries based on these two measures of energy consumption. Updated: 11/21/2023
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Energy Consumption
Energy Consumption by Country
Energy Consumption per Capita
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Which country uses the most electricity per person?
As of 2019, the country of Gibraltar uses the most energy per capita. It uses 1.52 million kWh on average.
How much energy is consumed in the world?
As of 2019, the world's energy consumption was 173,340 terawatt-hours (TWh). World energy consumption has increased all but four years in the last 50 years.
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Energy Consumption ------------------
Energy consumption is the amount of energy consumed, or used, by a person, organization, state, country, or even in a specific process. It is important to note that energy doesn't always mean electricity. For example, the typical household energy consumption would include electricity, gas, water, and other energy sources. The energy consumption of a car would be the amount of gasoline, diesel, or electricity used to run it. In order to calculate energy consumption, simply add the total of all energy bills. This figure is used to understand where savings can be implemented and if certain upgrades or changes would reduce energy consumption. The energy consumption of a home can vary greatly as can the energy consumption of a country.
Developing countries tend to use more energy than developed countries. Countries without stable electricity or a modern road system tend to use less energy when compared with countries that are in the process of building those systems. Once the systems are built, the energy consumption decreases. The weather and sources of energy also vary greatly between countries. Some countries require certain energy efficiency requirements on appliances, cars, and factors. All of these things combined affect the energy consumption of an individual country.
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Energy Consumption by Country -----------------------------
In 2019, the world's energy consumption was measured at 173,340 terawatt-hours (TWh). As populations grow and access to technologies increases, more energy is consumed. In fact, over the last 50 years, there were only four years (1980, 1981, 1982, 2009) when the amount of energy consumed didn't increase worldwide. As expected, larger and more populated countries have the highest energy consumption. These are also the countries in the best position to use public policy to increase energy efficiency and lower global usage. The top ten countries for energy consumption, as of 2019, include:
| Country | Usage | Sources |
---
| China | 39,361 TWh | 58% coal, 20% petroleum, 8% hydroelectric, 8% natural gas, 6% other |
| United States | 26,291 TWh | 37% petroleum, 29% natural gas, 14% coal, 9% nuclear, 11% renewable |
| India | 9,461 TWh | 55% coal, 30% crude oil, 8% natural gas, 7% other |
| Russia | 8,279 TWh | 54% natural gas, 22% oil, 12% coal, 7% nuclear, 5% other |
| Japan | 5,187 TWh | 40% petroleum, 26% coal, 21% natural gas, 10% renewable, 3% other |
| Canada | 3,948 TWh | 32% oil, 30% gas, 24% hydro, 6% nuclear, 8% other |
| Germany | 2,650 TWh | 36% oil, 24% gas, 18% coal, 9% wind, 5% nuclear, 8% other |
| Brazil | 3,445 TWh | 38% oil, 29% hydro, 10% gas, 5% coal, 9% other renewable, 9% other |
| South Korea | 3,436 TWh | 42% oil, 28% coal, 17% natural gas, 11% nuclear, 2% other |
| Iran | 3,429 TWh | 65% gas, 32% oil, 3% other |
It is important to note that different countries are approaching energy consumption with different methodologies. When comparing the energy consumption of China and the United States over the last 50 years, this is easily observed. China's energy usage increases each year on a sharp slope, increasing the annual usage by more than 2,038% over 50 years. In the last 20 years, this slope is at a faster pace than ever before. Over the same 50 year period, the United States has increased only 48%. This has led to criticism of the countries with large percentage increases year after year. Some smaller countries have a higher percentage increase as they have developed the technology to provide energy sources to their countries. For example, Bhutan increased 74,909% over 50 years, but the actual measure of energy increased is only 25 TWh.
There is also criticism of the types of energy sources being used. While most countries rely heavily on non-renewable resources, some are making adjustments to their sources to be more environmentally friendly. There is also the consideration of natural resources versus imported. Iran has a large supply of oil and gas, but countries like South Korea must import their main energy sources, thus using additional energy in transportation.
This graph from 2017 predicts the sources of energy consumption worldwide in the future.
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Energy Consumption per Capita -----------------------------
Of course, the largest and most populated countries have the highest energy consumption. The best way to compare energy consumption is by determining how much energy is used on average based on the population. The energy consumption per capita is when the total energy consumption is divided by the total population. The top ten countries for energy consumption per capita look completely different than the top countries for energy consumption.
| Country | Annual Per Capita Energy Consumption |
--- |
| Gibraltar | 1.52 million kWh |
| Qatar | 198,418 kWh |
| Iceland | 179,963 kWh |
| Singapore | 169,886 kWh |
| Bahrain | 142,159 kWh |
| Trinidad and Tobago | 142,119 kWh |
| United Arab Emirates | 137,318 kWh |
| Brunei | 123,823 kWh |
| Kuwait | 108,127 kWh |
| Canada | 105,540 kWh |
Canada is actually the only country on both top ten lists. This list includes countries that are either in very hot or very cold climates. Temperature control is one of the top uses for energy whether it is heating in a cold climate or air conditioning in a hot climate.
Energy Sources
There are many different ways to produce energy. While most of the world relies on non-renewable resources, there is a push for more sustainable energy production. The three largest non-renewable energy sources include coal, oil, and natural gas. These sources of energy are limited and cause pollution. Renewable energy sources include hydropower (dams), wind, solar, biofuels, and other unlimited energy sources.
50 years ago, only 5% of energy production was from renewable energy sources and that was mostly hydropower. 20 years ago this was up to 7% and included hydropower, solar, wind, biofuels, and others. By 2019 this was up to 11% and included mostly hydropower and wind power in addition to solar, biofuels, and others. So, during the same time that overall energy production is increasing, the fastest rates of increase are in renewable energy. The top ten countries for renewable energy production include:
| Rank | Country | Total Renewable Energy (GWh) |
---
| 1 | China | 1,398,207 |
| 2 | United States | 572,409 |
| 3 | Brazil | 426,638 |
| 4 | Canada | 418,679 |
| 5 | India | 195,242 |
| 6 | Germany | 193,735 |
| 7 | Russia | 170,077 |
| 8 | Japan | 169,660 |
| 9 | Norway | 140,240 |
| 10 | Italy | 109,962 |
Hydropower is the largest source of renewable energy around the world. However, wind and solar power are the most popular options for increasing renewable energy. They are both effective and affordable. Denmark was the first country to generate more electricity from wind power in one day than the country used. On July 9th, 2015, Denmark generated 140% of the electricity demand.
This graph displays the potential sources of energy from renewable and non-renewable sources. Clearly solar and wind power have the most potential.
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Lesson Summary --------------
Energy consumption is the amount of energy consumed by a country, community, household, or person. It includes all forms of energy, not just electricity. Energy can come from either non-renewable resources like oil and coal or renewable resources like hydropower, wind, and solar. The countries that have the highest energy consumption are the largest and most populated countries.
In order to compare energy consumption more fairly, the energy consumption per capita is compared. This is figured by taking the amount of energy consumed and dividing it by the number of people living in the country. The highest uses by this figure are typically countries in very hot or cold regions as a great deal of energy is spent on keeping homes and buildings at a comfortable temperature.
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Video Transcript
Developed and Developing Countries
Back in the year 1800, there were about one billion people living on our planet. One hundred years later, in the year 1900, the world population grew a modest amount to about 1.6 billion people. But, over the next 100 years, the world's population exploded, and today, we have nearly eight billion people walking around planet Earth. That's a lot of people! And, you might think that the rapid growing population is the reason we see such a rise in the energy consumption around the world, but this is only half-right.
What really makes a difference in the world's energy consumption is how much energy each person consumes. When we consider this fact, we see that there is a big difference between the energy consumption of people living in developed nations as compared to those living in developing countries.
So, what do we mean by a developed country? A developed country is a term used to describe an industrialized country with a highly developed economy. Examples of developed countries include the United States, Canada, Australia, Austria, France, Germany, Italy, Japan and the United Kingdom. Each of these countries are in contrast to a developing country, which can be defined as a non-industrialized, poor country that is seeking to develop its resources. There are many developing countries, with some of the more notable being China and India.
Energy Consumption
Energy consumption is highest among developed countries. In fact, Americans make up less than 5% of the world's population and yet consume as much as 25% of its energy. Because of America's extravagant use of energy, the United States often gets singled out in discussions on global energy consumption. Some might say the U.S. needs to go on an 'energy diet.'
When we use the term 'energy consumption,' we mean consumption of the sources of energy that generate power, including fossil fuels and renewable energy. The main fossil fuels are oil, coal and natural gas. The main renewable energy sources include wind, solar, hydroelectric power and biofuels. Currently, the world's energy consumption is about 15 terawatts of power. So, you might be wondering, what is a terawatt?
Well, the term 'tera' refers to one trillion, so a terawatt is a unit of measuring electrical power that is equal to one trillion watts. If we think of a 'watt' as a unit of power used in a common household object, such as a 100-watt light bulb, then we see that a terawatt could power about 10 billion 100-watt light bulbs at the same time. In fact, we use so much energy that the light produced by developed countries can be seen from space.
If we lump all of the developing countries together, we see that combined they only use about 30 percent of the planet's energy. However, this may be changing as the economies of some of the developing countries grow. For example, China tops the list of developing countries that are consuming more energy each year. China has nearly doubled its energy usage within the past 15 years and has seen auto sales climb. Because economic growth leads to increased energy use and because cars consume fossil fuels, prosperity in developing countries (like China), leads to more global energy consumption.
Energy Sources
Most of the world's population relies on oil as their primary source of energy, and this may be due to the fact that we use oil for many different things. Most of us think of oil products being used for transportation fuels, such as gasoline, diesel and jet fuel. However, we use oil for many other applications, including asphalt for roads, fertilizer for our farmer's fields, heat for our homes and the production of plastics. Natural gas is the second most consumed power source. This is followed by combustible renewables and waste, such as biomass and industrial waste, which can be converted into energy. Coal and alternative energies, such as geothermal, hydro, wind and solar power make up the remaining energy sources.
The world is hungry for energy. However, many environmental scientists warn that the burning of fossil fuels is contributing to global climate change, which is the term used to describe the long-term change in the Earth's climate. Despite these warnings, the use of fossil fuels in both developed and developing countries continues to rise. The link between energy demands and global climate change has some experts encouraging developed countries to conserve energy, while encouraging developing nations to adopt alternative and renewable energy sources of power.
Lesson Summary
Let's review. Energy consumption is the term used to describe the consumption of the sources of energy that generate power. Energy is generated from nonrenewable fossil fuels, such as oil, coal and natural gas. It can also be generated from renewable energy sources, such as wind, solar, hydroelectric power and biofuels.
The world's energy consumption is determined by the energy consumption of individuals. Generally, we see that a person living in a developed country, which has a highly developed economy, will consume more energy than a person living in a developing country that is non-industrialized and poor. If the economy of a developing country improves, the energy consumption for that country rises. This is the case in such developing countries as China.
Fossil fuels are heavily consumed for energy, but they have drawbacks, which include the fact that they are nonrenewable, and their use is contributing to global climate change. These concerns have led some experts to encourage energy conservation in developed countries and the adoption of alternative and renewable energy in developing countries.
Learning Outcomes
After this lesson, you will have the ability to:
Differentiate between a developed country and developing country and the energy consumption in each
Describe the methods used to generate energy
Explain how the world's energy consumption is determined
Summarize negative consequences of energy consumption
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Environmental Science 101: Environment and Humanity
20 chapters
157 lessons
14 flashcard sets
Chapter 1
Introduction to Environmental Science
Environmental Science | Definition & Goals 8:10 min
U.S. Environmental History: Movements & Timeline 6:35 min
Conservation vs. Preservation | Definition & History 8:25 min
Scientific Evaluation of Environmental Problems: Process and Steps 8:25 min
The Scientific Method Applied to Environmental Problems: Definition, Steps and Applications 7:06 min
Chapter 2
Ecosystems and the Effects of Pollution
Ecosystem, Habitat & Niche | Definition, Differences & Examples 10:20 min
The Nitrogen Cycle, Acid Rain and Fossil Fuels 8:37 min
Fossil Fuels, Greenhouse Gases, and Global Warming 10:11 min
Chapter 3
Habitats & Ecosystems
Biodiversity | Definition, Types & Examples 6:58 min
Energy Flow in an Ecosystem | Food Chain & Trophic Levels 9:33 min
Decomposer | Definition, Stages & Examples 4:24 min
Carrying Capacity | Definition, Graph & Examples 7:15 min
Global Ecology | Biogeography, Conservation & Examples 6:55 min
Conservation Biology | Types & Examples 7:34 min
Conservation Biology, Habitat Fragmentation, and Metapopulations 9:46 min
How Introduced and Invasive Species Alter Ecological Balance 8:57 min
Chapter 4
The Evolution of Ecosystems
Biomes of The World | Savannas, Deserts, & Rainforests 10:42 min
Biome Overview & Classifications 11:55 min
Types of Forests | Temperate, Coniferous & Tropical Rainforest 7:03 min
Desert Ecosystems: How Biodiversity Impacts Hot and Cold Deserts 5:49 min
Primary Productivity | Definition & Effects 8:07 min
Primary & Secondary Ecological Succession | Causes & Examples 8:23 min
Theories of Evolution: Lamarck vs. Darwin 7:30 min
Natural Selection vs. Adaptation | What is an Example of Natural Selection? 6:26 min
Chapter 5
Meteorology and the Environment
Solar Energy on Earth | Definition, Temperature & Effects 7:57 min
Greenhouse Gases | Types, Effect & Implications 7:53 min
Climate Change | Definition, Effects & Impact 7:56 min
Factors Involved in Climate Change 10:02 min
Methods for Determining Past Climates 9:50 min
Pollution Overview, Types & Examples 8:02 min
Ozone Layer | Definition, Role & Importance 7:00 min
Acid Deposition | Definition, Causes & Effects 7:04 min
Air's Chemical Composition & Environmental Effects 7:17 min
Air Pollution Definition, Sources & Types 6:45 min
Controlling Air Pollution: How Developed & Developing Nations Differ 6:40 min
Indoor Air Pollution | Pollutants, Examples & Consequences 7:14 min
Chapter 6
Geology and the Environment
Ground and Surface Water: Effects On Landscapes 5:55 min
How a Landform Diagram Describes the Geological Progression of a Landscape 7:59 min
Tectonic Plate Movement | Types, Causes & Effects 7:13 min
Extraction and Processing of Minerals & the Environmental Impacts of Mineral Use 5:21 min
Soil | Definition, Types & Examples 8:30 min
Soil Contamination | Sources, Types & Prevention
Soil Conservation: Fighting Soil Erosion With Sustainable Soil Use 8:47 min
Land Conservation Definition & Methods 8:03 min
Chapter 7
Biological Science
Species Richness | Definition, Equation & Example 5:29 min
Ecosystems of Oceans and Freshwater: Biological Diversity and Water 9:27 min
Extinction | Definition & Types 6:36 min
Threatened vs. Endangered Species | Differences & Levels 7:32 min
Invasive Species | Definition, Causes & Examples 10:41 min
Overexploitation Definition & Examples 4:47 min
Ecological Sampling: Definition & Example 6:16 min
Ecology Restoration Techniques & Examples | What is Ecology Restoration? 4:34 min
The Endangered Species Act & The World Conservation Strategy: Goals & Purposes 10:31 min
Chapter 8
Causes and Effects of Freshwater Pollution
Water Resource Issues: Activities Affecting the Water Supply 6:52 min
Aquifer Depletion and Overdrawing of Surface Waters: Effects on Water Resources 6:21 min
Water Conservation | Ways to Save & Importance 8:31 min
Water Storage | Pros and Cons of Dams & Reservoirs 6:17 min
Water Pollution | Definition, Sources & Examples 7:08 min
Water Treatment | Overview, Systems & Process 7:15 min
Clean Water & Safe Drinking Acts | Difference & Importance 5:43 min
Chapter 9
The Relationship Between Land and the Environment
Sustainable Forest Management | Definition, Benefits & Examples 7:11 min
Deforestation | Definition, Causes & Effects 7:10 min
Rangeland Uses, Importance & Physical Characteristics 6:55 min
Protecting Private Rangelands: Conservation Easements 6:07 min
National Parks in the US: Governing Agencies and Areas of Concern 7:28 min
Wilderness Definition, Classification & Act 7:10 min
Managing Federal Public Lands: Wise-Use vs. Environmental Movements 6:43 min
Using Qualitative & Quantitative Geographic Data 7:04 min
Chapter 10
Effects of Population Growth on the Environment
The History of Human Population Growth and Carrying Capacity 8:50 min
Poverty, Carrying Capacity, Population Growth & Sustainability 8:42 min
Population Size: Impacts on Resource Consumption 5:44 min
Overconsumption & Overpopulation | Definition, Effects & Examples 5:16 min
Energy Consumption History: Global Trends & Implications 6:40 min
Chapter 11
Environmental Impact of Agriculture
Problems in Agriculture | Challenges, Policies & Current Events 6:52 min
Land Degradation and Habitat Fragmentation: The Environmental Impacts of Industrialized Agriculture 6:26 min
Green Revolution | Definition, Impact & Examples 8:00 min
Land Conservation: Criteria For Evaluation of Land and Most Endangered Lands in the US 7:07 min
Sustainable Agriculture | Definition, Importance & Practices 6:33 min
Genetic Engineering | Overview, Benefits & Examples 8:52 min
What Are Pesticides? - Definition and Difference Between Narrow-Spectrum & Broad-Spectrum 6:20 min
Alternative Pest Control Methods for Agricultural Use 7:40 min
Chapter 12
Solid and Hazardous Waste
Solid Waste | Definition, Types & Examples 6:54 min
Sanitary Landfill | Definition, Origin & Examples 7:21 min
Incineration Definition, Benefits & Example 7:12 min
Compost Definition, Methods & Examples 5:47 min
Reduce, Reuse & Recycle | Definition & Examples 7:34 min
What Is Integrated Waste Management? - Definition and Examples 6:23 min
Hazardous Waste Definition, Types & Disposal 9:05 min
Waste Legislation: Resource Conservation and Recovery Act vs. the Superfund Act 6:31 min
Green Chemistry | Overview, Principles & Examples 6:53 min
Chapter 13
Human Impact on the Environment
IPAT | Overview, Equation & Model 7:42 min
Deforestation | Desertification and Declining Biodiversity 7:23 min
The Agricultural Revolution: Impacts on the Environment 6:55 min
Industrial Revolution | Impact, Effects & Consequences 5:23 min
Industrial Development of Emerging Nations and the Environment 6:34 min
Industrial Ecology Definition, Goals & Examples 6:57 min
Pollution and Habitat Destruction: The Human Factors Contributing To Endangerment & Extinction 5:51 min
Eutrophication Definition, Process & Causes 7:14 min
Chapter 14
Renewable Resources and the Environment
Solar Energy Definition, Advantages & Disadvantages 7:01 min
Solar Panels & Photovoltaic Cells | Definition & Overview 6:22 min
Biomass Energy Definition, Pros & Cons 10:05 min
Wind Energy | Definition, Pros & Cons 7:53 min
Hydropower Overview, Pros & Cons 7:39 min
Geothermal Energy | Meaning, Advantages & Disadvantages 9:14 min
Tidal Energy Sources, Advantages & Disadvantages 7:19 min
Energy Conservation and Energy Efficiency: Examples and Differences 5:41 min
Chapter 15
Nonrenewable Resources
Fossil Fuels | Meaning, Advantages & Disadvantages 7:47 min
Viewing now Energy Consumption by Country & Per Capita | Overview & Examples 6:32 min
Up next Coal Mining: Environmental Impacts and Technologies Used To Make Cleaner Coal 7:15 min Watch next lesson
Oil & Natural Gas Reserves| Extraction, Availability & Uses 7:44 min
Hydraulic Fracturing History, Mechanics & Alternatives 4:28 min
Nuclear Energy | Definition, Types & Examples 5:17 min
Risks of Nuclear Power Plants and Radioactive Waste: Safety and Health Concerns 9:25 min
Chapter 16
Environmental Sustainability
Environmental Sustainability | Definition, Objectives & Examples 7:32 min
Human Behaviors that Threaten Environmental Sustainability 8:20 min
Sustainable Development within Environmental Limits 5:18 min
Sustainable Consumption | Definition, Background & Goals 6:08 min
Understanding Voluntary Simplicity as a Form of Sustainable Consumption 6:24 min
Chapter 17
Analyzing Environmental Risks
Environmental Health Definition, Issues & Factors 6:47 min
Understanding & Assessing Threats to Environmental Health 7:42 min
Environmental Toxins | Definition, Types & Examples 6:52 min
Disease-Causing Agent | Overview & Disease-Carrying Organisms 6:16 min
Using Indicator Bacteria to Monitor Public Water Supplies 4:45 min
Emerging Diseases Linked to Environmental Change 6:51 min
Biological Magnification Definition, Processes & Effects 7:37 min
The Stockholm Convention on Persistent Organic Pollutants 5:31 min
The Dose-Response Curve: Determining Health Effects of Environmental Pollutants 7:42 min
Toxicants and Children: Why Children Are More Susceptible 5:19 min
The Precautionary Principle and New Technologies & Products 6:59 min
Chapter 18
Impact of Environmental Policy
Environmental Ethics Overview & Examples | What is Environmental Ethics? 7:05 min
Environmental Worldviews | Western & Deep Ecology 5:21 min
Environmental Justice | Definition, Principles & Examples 6:38 min
Environmental Planning | Definition & Components 5:40 min
Environmental Law Definition & Policies 6:59 min
Environmental Impact Statement | Definition, Purpose & Examples 5:15 min
Natural Capital | Definition & Examples 5:06 min
Air Pollution & Economic Development | Marginal Cost & Benefit 5:55 min
Incentive-Based Regulatory Approaches: Environmental Taxes & Tradable Permits 7:30 min
World Trade Rules & Environmental Policies 5:00 min
Paris Climate Agreement Overview & Goals
Global Environmental Law & International Treaties 7:30 min
Future Environmental Concerns: Population, Food Supply, Energy & Pollution 7:15 min
Chapter 19
Required Assignments for Environmental Science 101
Environmental Science 101 - Assignment 1: Research Paper
Chapter 20
Studying for Environmental Science 101
Ecosystems Flashcards
Effects of Pollution on the Environment Flashcards
Ecology & Environmental Science Flashcards
Atmosphere, Meteorology & the Environment Flashcards
Geology & the Environment Flashcards
Biological Science Flashcards
Land & the Environment Flashcards
Environmental Effects of Population Growth Flashcards
Solid & Hazardous Waste Flashcards
Human Impact on the Environment Flashcards
Renewable & Nonrenewable Resources Flashcards
Energy Resources & Sustainability Flashcards
Analyzing Environmental Risks Flashcards
Environmental Policy Flashcards
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12088 | https://fiveable.me/ap-pre-calc/unit-1/review/study-guide/Px2KiWz3udBhZKep | Unit 1 Overview: Polynomial and Rational Functions | AP Pre-Calculus Class Notes | Fiveable | Fiveable
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📈AP Pre-Calculus Unit 1 Review
Unit 1 Overview: Polynomial and Rational Functions
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AP Pre-Calculus
Unit 1 – Polynomial and Rational Functions
Unit 1 Overview: Polynomial and Rational Functions
📈AP Pre-Calculus Unit 1 Review
Unit 1 Overview: Polynomial and Rational Functions
Written by the Fiveable Content Team • Last updated September 2025
Verified for the 2026 exam
Verified for the 2026 exam•Written by the Fiveable Content Team • Last updated September 2025
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📈AP Pre-Calculus
Unit & Topic Study Guides
AP Precalculus Exam
Unit 1 – Polynomial and Rational Functions
Unit 1 Overview: Polynomial and Rational Functions
1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
Unit 2 – Exponential and Logarithmic Functions
Unit 3 – Trigonometric and Polar Functions
Unit 4 – Functions Involving Parameters, Vectors, and Matrices
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Welcome to Fiveable’s AP Precalculus study guides! 🥳
Before we even jump into the content itself, let’s look at the big, big, big picture of this course:
🧐What’s the Point of AP Pre-Calculus?
AP Precalculus is all about understanding how functions work in real-world situations. It's designed to help you get ready for college-level math and other classes that use math. In this class, we're going to look at a lot of different types of functions that are important for careers in math, science, and other fields. Regardless of whether this is your last math class before college or you’re looking for a refresher of advanced algebra, we’ll make sure you have a really solid understanding of what you’ll be learning! 🌎
We're going to spend a lot of time working on real-world problems and using math tools to solve them. This is called "modeling," allowing us to understand functions better by looking at different examples. We'll also look at functions in different ways, like graphically and numerically, so you can get a good sense of how they work.
Throughout the class, we'll work on math skills like manipulatingequations and solving problems. Similarly, you'll learn how different types of functions and their different parts, like inverses and transformations, can be understood in different ways.
One of the main things we're going to focus on is understanding what functions are and how they work. A function is just a way to show how different inputs (or "domain") are related to different outputs (or "range"). There are different ways to classify functions based on how they work. We're going to look at how functions change over time and how different variables are related. Research shows that understanding functions like this is the best way to prepare for calculus.
We'll also spend time working with data and making models to better understand the functions. We'll learn how to use the models to understand different kinds of information, like how different variables change over time. And we'll learn that all models have their own limitations and assumptions. By looking at functions from different perspectives, you'll develop a deeper understanding of not only the specific types of functions we're studying, but functions in general. This will help you in this class and in other classes in the future.
Phew! Sounds like a lot, huh? Don’t worry—you’ll learn these concepts in the span of four units throughout the year. Let’s start with Unit 1! ⭐
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🔊Unit Breakdown
AP Precalculus Unit 1 covers the topics of polynomial and rational functions:
A polynomialfunction involves a sum of powers in one or more variables. The powers are non-negative integers and the coefficients can be any real number. The degree of a polynomial is the highest power of the variable in the polynomial. 👊
A polynomial function can be represented in the form of f(x)=a n x n+a n−1 x n−1+...+a 1 x+a 0 f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 f(x)=a nx n+a n−1x n−1+...+a 1x+a 0, where n is the degree of the polynomial, x is the variable, and a n a_n a n, a n−1 a_{n-1}a n−1, ..., a 1 a_1 a 1, a 0 a_0 a 0 are the coefficients.
A rationalfunction is a function that can be defined as the ratio of two polynomial functions. It is represented by a polynomial function in the numerator and a non-zero polynomial function in the denominator.
Graph displaying functions. Image Courtesy of Wikiversity
A rational function can be represented in the form of f(x)=p(x)/q(x)f(x) = p(x)/q(x)f(x)=p(x)/q(x), where p(x)p(x)p(x) and q(x)q(x)q(x) are polynomial functions. Rational functions have certain behavior patterns like vertical asymptotes, horizontal asymptotes, and sometimes holes.
Tandem and Rates of Change
The unit starts with learning about the relationship between two or more functions and how changes in one function can affect the other. Then, it moves on to Rates of Change, where you’ll learn about the rate at which a function is changing and how to calculate it.
This is followed by Rates of Change in Linear and Quadratic Functions, where you’ll learn how to calculate the rate of change for linear and quadratic functions and how it relates to their graph. 📈
Polynomial Functions
The unit then covers the properties of polynomial and rational functions.
In Polynomial Functions and Rates of Change, you’ll learn how to calculate the rate of change for polynomial functions, including the use of powerfunctions and the power rule.
In Polynomial Functions and Complex Zeros, students learn about the concept of complexzeros and how they relate to the behavior of polynomial functions. 🕵️♀️
Graph displaying polynomial end behavior
Image Courtesy of Mrs. Hahn
The section also covers Polynomial Functions and End Behavior, where students learn how to determine the leading coefficient and degree of a polynomial function.
Rational Functions
In Rational Functions and End Behavior, students learn about rational functions and how they are different from polynomial functions.
In Rational Functions and Zeros, students learn how to find the zeros of rational functions, including the use of the FactorTheorem and long division.
The section also covers Rational Functions and Vertical Asymptotes, where you’ll learn about the concept of verticalasymptotes and how they relate to the behavior of rational functions.
Finally, the unit goes over Rational Functions and Holes, where you’ll learn about holes in rational functions and how they relate to the behavior of the function. ⛳
Graph displaying rational functions
Image Courtesy of Sciencing
🎁Wrapping Up Unit 1 Overview
Overall, Unit 1 provides a thorough understanding of polynomial and rational functions and their properties, including rates of change, zeros, end behavior, and transformations. It also covers the process of selecting and constructing appropriate function models and applying them to real-world situations. Woo-hoo! 😊
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12089 | https://www.compellingspeaker.com/blog/finding-your-optimal-speaking-pitch | Finding Your Optimal Speaking Pitch
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Finding Your Optimal Speaking Pitch: A Guide to Effective Communication
In every facet of life, effective communication sits at the core, shaping our relationships, influencing our careers, and even determining the trajectory of our personal growth. It is more than just a string of words; it's an art that requires mastery of various elements, one of which is your speaking pitch.
The tone of your voice can say as much, if not more, about your emotions and intentions as the words you utter. It can make or break your credibility, setting the stage for how others perceive you.
Have you ever wondered why some people can captivate an audience effortlessly while others struggle to hold attention? Often, it boils down to understanding and finding their optimal speaking pitch.
This article explores the fascinating concept of optimal speaking pitch. We will explore what it means, why it matters, and most importantly, provide actionable tips to help you discover and refine your own. So, buckle up for an exciting journey toward becoming a more impactful communicator!
Key takeaways:
The role of pitch is crucial in both delivering and understanding communications.
Maintaining the health and proper training of your vocal cords is key to your capacity for clear and effective communication.
Several factors, such as our sex, cultural upbringing, and age, significantly shape our voice.
Table of Contents
Understanding Speaking Pitch
High Pitch vs. Low Pitch
Influence of Pitch on Message Perception and Interpretation
The Science Behind Optimal Pitch
How Your Vocal Cords Generate Speech
Pitch, Resonance, & Voice Quality
The Interplay of Mind and Body in Determining Optimal Voice Pitch
Factors Influencing Optimal Speaking Pitch
The Harmonic Duet of Gender and Voice
Cultural Variations in Pitch Preferences
Age-Related Changes in Pitch Perception
Benefits of Finding Your Optimal Pitch
Enhanced Communication Effectiveness
Improved Credibility and Confidence
Minimized Vocal Strain and Fatigue
Steps to Discover Your Optimal Speaking Pitch
Step #1: Vocal Warm-up and Awareness
Step #2: Simple Vocal Exercises to Increase Awareness of Your Vocal Range
Step #3: Recording and Self-Evaluation
Step #4: Noting Any Inconsistencies or Discomfort
Step #5: Experimentation
Step #6: Gauging the Impact of Each Pitch on Clarity and Emotions Conveyed
Step #7: Seeking Constructive Feedback
Adapting Your Speaking Pitch
Matching the Context and Audience
Using Pitch to Emphasize Key Points and Emotions
Avoiding Monotony by Incorporating Pitch Variations
Overcoming Challenges
How to Deal with Pitch-Related Insecurities
Addressing Vocal Habits and Tendencies
Seeking Professional Guidance for Persistent Difficulties
Maintaining a Healthy Voice
Importance of Vocal Care and Hydration
Avoiding Excessive Strain on Vocal Cords
Balancing Speaking Pitch with Naturalness
Embracing Your Unique Pitch
How to Build Confidence in Your Natural Speaking Pitch
Aligning Pitch with Authentic Self-Expression
Embrace Your Unique Voice and Transform Your Communication
Understanding Speaking Pitch
Just imagine, if you will, the world as a grand piano and your voice as its keys. Your speaking pitch? That's the melody you play with those keys.
With each word you utter, you strike a note – sometimes high-pitched, like the twinkling tones at the piano's right end, resonating with excitement or surprise. Other times, it's low-pitched, echoing the deep, mellow notes on the left, imbued with seriousness or calm.
Put simply, speaking pitch is your voice's highness or lowness when speaking. It's all about the frequency of your vocal vibrations.
These changes in pitch are an essential aspect of verbal communication, subtly conveying your mood, interest level, and even your confidence. You can become a more effective and engrossing communicator by understanding and controlling your speaking pitch.
High Pitch vs. Low Pitch
Pitch is a fundamental concept crucial in perceiving and interpreting sounds around us. Whether it's the chirping of birds, the strumming of a guitar, or the human voice, pitch adds color and depth to the symphony of life.
In the human voice, pitch is determined by the rate at which the vocal cords vibrate. Faster vibrations result in a higher pitch, while slower vibrations produce a lower pitch.
For example, when we are excited or anxious, our vocal cords might stretch and vibrate more quickly, resulting in a higher pitch. Conversely, when calm or serious, our vocal cords may vibrate more slowly, leading to a lower pitch.
A high-pitched voice can convey enthusiasm, urgency, or surprise. It can draw attention to critical points in a speech and help to engage the audience's interest. However, excessive use of high pitch can come across as shrill or nervous, so it's crucial to use this tool judiciously.
On the other hand, a low-pitched voice often suggests authority, calmness, and confidence — qualities that can help a speaker establish credibility. A well-timed drop in pitch can underscore a profound point or signal a transition to a new topic. However, a too-low pitch can be hard to hear or may come across as monotonous, so balance is vital.
Influence of Pitch on Message Perception and Interpretation
Pitch serves as an important tool in conveying and interpreting messages. Whether in public speaking, everyday conversation, or even music and sound design, the pitch can significantly influence how a message is received. Here are three key impacts of pitch on the perception and interpretation of messages:
Conveys Emotion
One of the most powerful impacts of pitch is its ability to convey emotion. For instance, in a presentation, a speaker might raise their pitch to display enthusiasm about a point or lower it to emphasize the gravity of a statement.
Enhances Engagement
Variations in pitch can significantly enhance listener engagement. A monotonous pitch can make a speech uninteresting, causing the audience to lose focus. On the other hand,varying the pitch can make the speech more dynamic and captivating, encouraging the audience to pay more attention and stay engaged.
I ndicates Question or Statement
In many languages, pitch variations at the end of a sentence can indicate whether it is a question or a statement. In English, for example, a rising pitch towards the end of a sentence often signifies a question, while a falling pitch typically indicates a statement. This use of pitch helps listeners understand the speaker's intentions and respond appropriately.
Understanding the impact of pitch on the reception and interpretation of messages is crucial for effective communication. Using pitch strategically, speakers can better convey their emotions, engage their audience, and clarify their intentions.
The Science Behind Optimal Pitch
Think of your vocal cords as the engine of a car. Just like how your vehicle's smooth running depends on its engine's efficiency, your ability to speak clearly and effectively hinges on the health and conditioning of your vocal cords. Let's delve into how your vocal cords work to create speech.
How Your Vocal Cords Generate Speech
Your vocal cords, also known as vocal folds, are twin bands of flexible muscle tissue located in your larynx or voice box. When you're not talking, these cords stay open, allowing air to pass freely to and from your lungs. But when you decide to speak, they close up and vibrate as the air from your lungs rushes past them, creating sound.
The pitch of this sound is determined by how tense and elongated your vocal cords are. If the muscles tighten and the cords stretch thin, you'll produce higher pitched sounds. On the other hand, if the muscles loosen and the cords shorten and thicken, you'll generate sounds of a lower pitch.
Now, imagine if the engine isn't working properly - say there's inadequate fuel, or the parts aren't working together - the car might sputter or stall. Similarly, if your vocal cords aren't working as they should - perhaps they're tired, strained, or lacking enough breath support - your voice may become hoarse, or you might experience a vocal growl.
A vocal growlis a low, gruff, often raspy sound similar to a struggling car engine's rumbling noise. This can happen when your vocal cords aren't sufficiently energized or coordinated, causing irregular vibrations.
Just as you service your car engine regularly and ensure it has enough fuel, you must take care of your vocal cords with proper techniques, rest, and hydration. Regular vocal exercises can help condition your vocal cords and improve their flexibility and stamina. Sufficient rest can help prevent vocal fatigue and strain while staying hydrated keeps your vocal cords moist, reducing friction during vibration.
Pitch, Resonance, & Voice Quality
When you speak, the quality of your voice is determined by a combination of various factors, including pitch, resonance, and other vocal attributes. Understanding these elements can help you improve your speech and communicate more effectively. Let's explore how pitch, resonance, and voice quality are interconnected.
Pitch
Pitch refers to the perceived frequency of sound that you produce. It's determined by the rate at which your vocal cords vibrate. When your vocal cords stretch and become thin, they vibrate faster, creating a higher pitch. Conversely, when the vocal cords relax and thicken, they vibrate slower, resulting in a lower pitch.
Resonance
Resonance, however, is the process that enriches and amplifies the sound produced by your vocal cords. This occurs in the cavities in your throat, mouth, and nose, which act like an echo chamber. The shape and size of these cavities can be altered by moving your tongue, lips, jaw, and soft palate, affecting the resonance and, ultimately, the tone of your voice.
Voice Quality
Voice quality, also known astimbre, makes your voice uniquely yours. The characteristic enables us to distinguish between different voices even when speaking at the same pitch and loudness. Voice quality is influenced by pitch, resonance, volume, and articulation, along with physiological factors like the size and shape of the individual's vocal tract and vocal folds.
The Link Between Pitch, Resonance, and Voice Quality
Pitch and resonance play integral roles in determining voice quality.
The pitch of your voice contributes significantly to how others perceive and interpretyour speech. Resonance affects the richness and fullness of your voice. A voice with good resonance can seem warm, vibrant, and clear, while a voice with poor resonance might sound nasal, muffled, or thin.
Together, pitch and resonance interact to create your unique voice quality. By consciously controlling and modifying your pitch and resonance, you can alter the quality of your voice. Whether to better project your voice in a public speaking scenario, convey specific emotions, or make everyday communication more effective and engaging, they work hand in hand to help enrich your speaking voice.
The Interplay of Mind and Body in Determining Optimal Voice Pitch
Your voice tone, or pitch, is influenced by both your body and your mind. Physically, the size of your voice box and its muscles can determine your voice's natural pitch. This is why men, who generally have larger voice boxes, often have deeper voices than women.
On the other hand, your mental state can also affect your voice. For example, when you're excited or nervous, you might notice that your voice gets higher. This is a physiological reaction to your emotions. If you are physically fatigued, you might notice your voice drop in pitch from the lack of energy.
Factors Influencing Optimal Speaking Pitch
Our voice is the music of our personality, a unique symphony that tells our story to the world. It's influenced by a multitude of factors, including our gender, cultural background, and age. Let's explore these melodies and their impact on our optimal speaking pitch.
The Harmonic Duet of Gender and Voice
Society's orchestra plays a significant role in dictating the pitch of our voices, often setting different tunes for men and women. This musical metaphor applies to how we perceive masculinity and femininity in vocal tones.
Deep, resonant voices are often linked with masculinity - a societal expectation that stems from biological differences but is amplified by cultural norms. This 'masculine melody' is associated with authority, strength, and assertiveness, influencing many men to lower their pitch to adhere to this ideal unconsciously.
On the other hand, higher-pitched voices are typically associated with femininity - a 'feminine harmony' that implies approachability, warmth, and nurturing qualities. As a result, women might unconsciously raise their pitch to sound more aligned with these societal expectations.
However, it's important to remember that these are not rigid rules but evolving societal constructs. With changing perceptions of gender roles, the melodies of masculinity and femininity are continually being rewritten. Some women might adopt a lower pitch to project authority, while some men may embrace a higher pitch to convey empathy.
Cultural Variations in Pitch Preferences
Our voice, the unique musical instrument we all possess, has a pitch influenced by many factors. One of the most fascinating among these is culture. From the bustling streets of New York to the serene temples of Kyoto, our cultural backgrounds shape our perspectives and how we sound.
Studies show that speakers of a tonal language, for instance, have an enhanced ability to process different pitches. This suggests that our cultural and linguistic environments significantly shape our auditory capabilities and preferences.
In the Western world, there seems to be a preference for extremes regarding voice pitch. Men with voices slightly lower than average are often favored, perhaps reflecting a societal association of deep voices with strength and authority.
But that's not a universal standard. In other cultures, lowered-pitch voices were judged by men as being produced by individuals with more remarkable fighting abilities. Furthermore, our cultural environment doesn't just influence our pitch preferences; it also shapes how we perceive music.
Some research suggests that culture significantly impacts how we hear and interpret musical pitches. This finding further underscores the powerful influence of culture on our auditory experiences.
Interestingly, women's preferences for men's voices seem to be influenced by their voice pitch. Women with lower-pitched voices tend to prefer men with lower-pitched voices, suggesting that our vocal characteristics might also shape our cultural preferences.
Age-Related Changes in Pitch Perception
As we age, our bodies and minds undergo numerous transformations. One aspect often overlooked is the change in our auditory system, particularly in pitch perception. Here are three significant age-related modifications:
Presbycusis or Age-Related Hearing Loss
As we grow older, the ability to hear higherfrequencies diminishes. This condition, known as presbycusis, affects pitch perception. High-pitched sounds become harder to distinguish, leading to difficulties in understanding speech, especially when there's background noise. It's a gradual process that starts around the age of 30 and accelerates after 50.
Change in Vocal Pitch
Age doesn't just affect our ability to perceive pitch; it also alters the pitch of our voice. Generally, children and young adults have higher-pitched voices. However, as we age, our vocal cords lose elasticity and bulk, causing our voices to become lower and less resonant. This transformation can be more pronounced in men than women.
Decreased Pitch Discrimination
The ability to distinguish between different pitches, known as pitch discrimination, also declines with age. This can make it challenging for older individuals to appreciate music or understand tonal languages, where pitch changes can alter the meaning of words.
These changes underline the importance of regular auditory check-ups, especially as we age. Understanding these shifts can help us maintain effective communication and enjoy the symphony of life at all stages.
Benefits of Finding Your Optimal Pitch
Our voices are one of our most potent tools for communication. Beyond the words we utter, the pitch of our voice can convey a wealth of information about our emotions, intentions, and identity. Understanding and harnessing our optimal pitch can offer several benefits:
Enhanced Communication Effectiveness
The pitch of your voice isn't just about sounding pleasant—it's a powerful tool for conveying meaning beyond words. By locating and utilizing your natural pitch, you can express your thoughts and feelings more accurately, comprehensibly, and persuasively.
It boosts clarity in your speech, making you more understandable to your listeners. An optimal pitch can captivate your audience's attention, preventing your voice from becoming monotonous or challenging.
Additionally, a well-modulated pitch signifies authority and confidence, enhancing your credibility whether you're giving a presentation, negotiating a deal, or leading a team in an online setup or in-person meeting. Speaking at your natural pitch helps prevent vocal strain, ensuring your voice remains healthy and strong.
Improved Credibility and Confidence
Securing your optimal pitch is not merely about sounding good—it's a strategic move that can significantly enhance your credibility and boost your confidence. When you communicate with your natural pitch, your voice carries a resonance and authenticity that can command respect and attention. It signals to your listeners that you are comfortable in your skin and confident in your message, making you more persuasive and influential.
Furthermore, it can exude authority, enhancing your credibility in personal, professional, or public speaking situations. It can add weight to your words, ensuring they are taken seriously and have the desired impact.
Minimized Vocal Strain and Fatigue
Speaking at your natural pitch minimizes the risk of vocal strain and fatigue. Pushing your voice into an unnatural pitch can cause unnecessary stress on your vocal cords, leading to discomfort and potential long-term damage.
On the other hand, speaking in your optimal pitch allows your vocal cords to work efficiently, reducing the risk of strain and enabling you to talk comfortably for extended periods. One common pitfall for speakers is falling into a pattern of vocal fry—a low, creaky vibration typically at the end of sentences. While this doesn't necessarily harm your voice, it can make you harder to hear and understand.
Moreover, it tends to follow predictable patterns, which can cause your listeners to lose interest over time. By finding and adhering to your optimal pitch, you can avoid these issues, maintain your audience's attention, and ensure the longevity of your most crucial tool as a speaker—your voice.
Steps to Discover Your Optimal Speaking Pitch
Your voice is more than just a means of conveying words—it's a powerful tool that can influence perceptions, command attention, and evoke emotions. One crucial aspect of your voice that can significantly impact your effectiveness as a speaker is your pitch.
However, discovering this perfect pitch can be challenging. It requires self-awareness, experimentation, and feedback.
So, how do you embark on this journey? Here are some practical steps to guide you in discovering your optimal speaking pitch.
Step #1: Vocal Warm-up and Awareness
The journey toward discovering your optimal speaking pitch begins with vocal warm-ups and developing a keen awareness of your voice. Like any other muscle in your body, your vocal cords must be warmed up before you start experimenting with different pitches. This prepares your voice for the exercises ahead and prevents potential strain or damage.
Warming up your voice has significant benefits. Itincreases blood flow, and helps establish a more flexible and responsive vocal range. It also promotes awareness of how your voice feels at different pitches, laying the groundwork for more effective and safe experimentation.
Step #2: Simple Vocal Exercises to Increase Awareness of Your Vocal Range
Simple vocal exercises can heighten your awareness of your vocal range. Try humming, lip trills, or scales to explore the breadth of your voice. Singing exercises, even basic ones, can be profoundly advantageous. They not only help you understand your range but also introduce melody—a change in pitch—which can make your speech more captivating and unpredictable, enhancing your public speaking.
Step #3: Recording and Self-Evaluation
Next, record a sample of your speech. This lets you listen and identify your current pitch, tone, and resonance. Pay attention to how your voicesounds and feels when speaking naturally—this is likely close to your optimal pitch.
Once you have a recording, listencarefully. Identify the pitch at which you naturally speak. Notice whether your voice tends to be lower or higher, softer or louder. This will give you a baseline from which to start your exploration.
Step #4: Noting Any Inconsistencies or Discomfort
As you look over your recording, it may be helpful to make note of any instances of pitch inconsistencies and any moments where you may have felt uncomfortable. These could be signs that you are straying from your optimal pitch. If your voice cracks, feels strained, or you run out of breath quickly, these might indicate that you speak outside of your ideal vocal range.
Step #5: Experimentation
Now, it's time to experiment. Try speaking at different pitches within your comfortable range. Notice how each pitch feels and sounds. Some pitches might feel forced or unnatural, while others might feel relaxed and communicate clearly and resonant.
Assess how each pitch feels in your throat and how it sounds to your ears. You're looking for a comfortable pitch that sounds clear, full, and natural. Remember, the goal is to find a pitch you can sustain withoutstrain, not to achieve a particular high or low pitch.
Step #6: Gauging the Impact of Each Pitch on Clarity and Emotions Conveyed
Consider the impact of each pitch on the clarity of your speech and the emotions it conveys. Different pitches can evoke different feelings and responses from your listeners.
Step #7: Seeking Constructive Feedback
Seeking feedback is a crucial step in finding your optimal speaking pitch. This involves reaching out to people you trust—friends, family members, or colleagues—and asking them to listen as you experiment with different pitches. You could do this in person, over a video call, or by sharing recordings of your speech exercises.
When they provide feedback, encourage them to be honest and constructive. Ask them to comment on the most natural, authentic, and engaging pitch when you speak. They should also consider how well they can hear and understand you at each pitch.
Feedback from others can provide a valuable outside perspective that complements your self-assessment. It can help you identify a pitch that feels comfortable and sustainable for you, resonates with your listeners, and effectively communicates your message. This final input can be instrumental in helping you determine your optimal speaking pitch, enhancing your overall effectiveness as a speaker.
Adapting Your Speaking Pitch
How you deliver a message can be just as important as the content of the directive itself. A successful speaker knows how to use their voice to capture and maintain an audience's attention, and this involves being aware of the different tones that can be used and adjusting them accordingly to fit the situation.
Matching the Context and Audience
Different settings and audiences may require different pitches to achieve the desired impact.
For instance, a professional setting like a business presentation might require a lower, more authoritative pitch. At the same time, an intimate gathering or storytelling session might benefit from a higher, more engaging pitch. It's also important to consider cultural and language nuances, as perceptions of pitch can vary across different societies and linguistic groups.
Using Pitch to Emphasize Key Points and Emotions
Your pitch can be a powerful tool to highlightkeypoints and convey emotions in your speech. A sudden change in pitch can draw attention to a particular word or phrase, adding emphasis and making your message more memorable.
Similarly, adjusting your pitch can help express various emotions—higher pitches can convey excitement or urgency, while lower pitches can evoke calmness or seriousness. This use of pitch can make your speech more dynamic and engaging, helping to connect with your audience on a deeper emotional level.
Avoiding Monotony by Incorporating Pitch Variations
Incorporating pitch variations into your speech can prevent monotony and keep your audience engaged. Speaking at a single pitch throughout your address can make your delivery sound flat and dull, potentially losing your audience's attention. By varying your pitch—raising it for questions, lowering it for statements, or changing it to reflect the mood—you can add interest and expressiveness to your speech.
This unpredictability and variation in pitch, akin to a melody in music, can make your voice more captivating and your message more impactful. Remember, the goal is not to overdo the variations but to use them strategically to enhance your communication and keep your audience engaged.
Overcoming Challenges
In every aspect of life, challenges are inevitable. They often catalyzegrowth, pushing us to step out of our comfort zones and adapt to new circumstances. This applies to our communication skills as well, particularly when it comes to our speaking pitch.
Whether you're dealing with insecurities about the sound of your voice, struggling to match your pitch to different contexts and audiences, or finding it difficult to use pitch effectively to emphasize critical points and emotions, these challenges can feel daunting. However, overcoming these hurdles is entirely within reach with the right strategies and a determined mindset.
The following sections provide practical advice and actionable steps to help you navigate these challenges and enhance your speaking pitch.
How to Deal with Pitch-Related Insecurities
Insecurities related to your speaking pitch can be a significant challenge, but they are not insurmountable. Here are some strategies to help you overcome these insecurities.
Shift Your Perspective:The first step is to change your perspective. Instead of viewing your speaking pitch as a threat or weakness, consider it an opportunity for growth and improvement. Remember that everyone's voice is unique, and what matters most is how effectively you use it to convey your message.
Analyze the Problem:Understand the specific aspects of your pitch that make you insecure. Is it too high? Too low? Monotonous? Once you'veidentified the issue, you can start working on solutions.
Make a Plan:Develop a strategy to improve your pitch. This could involve practicing different pitch levels, seeking feedback, or working with a voice coach. Remember, improvement takes time and consistent effort.
Practice:Finally, practice is vital. Like any other skill, improving your speaking pitch requires regular exercise. The more you practice, the more comfortable and confident you'll become.
Addressing Vocal Habits and Tendencies
Our voices are critical in delivering our messages; certain vocal habits can enhance or hinder this process. Some common vocal tendencies, such as mumbling, speaking too quickly, using filler words, or maintaining a monotonous pitch, can obstruct clear communication and diminish the impact of your speech.
You can significantly improve your public speaking skills by acknowledging and addressing these habits. For instance, if you tend to speak too quickly, practicing pace control can help you deliver your message more clearly and allow your audience to follow along more rapidly. Learning to vary your pitch and tone can make your speech more engaging and expressive if you often talk in a monotone.
Moreover, understanding your vocal tendencies can also assist in managing public speaking anxiety. For example, many people unconsciously increase their speaking speed due to nerves. By recognizing this tendency, you can consciously slow your speech, reducing stress and improving clarity.
Seeking Professional Guidance for Persistent Difficulties
Public speaking coaches, speech therapists, or even experienced mentors in your field can provide expert advice tailored to your specific needs. They can help identify the root causes of your difficulties, whether it's pitch variation, articulation, pacing, or anxiety. Furthermore, they can provide targeted exercises and techniques to address these issues.
This personalized feedback and instruction can fast-track your progress, helping you overcome obstacles you might have struggled with. In addition, professional guidance often includes opportunities for practice and constructive feedback, which are key elements for improvement in public speaking. Therefore, if self-guided methods haven't resolved your difficulties, seeking professional guidance can be a practical step toward overcoming your public speaking challenges.
Maintaining a Healthy Voice
Public speaking is an art that requires not only a command over language and effective communication skills but also an understanding of how to use and care for one's voice. Many people overlook the importance of vocal health in public speaking, focusing more on the content of their speech. However, maintaining a healthy voice is crucial for delivering clear, impactful presentations and speeches.
Let's delve into the importance of vocal care and hydration and strategies to avoidexcessive strain on your vocal cords, providing practical insights on maintaining a healthy voice for successful public speaking.
Importance of Vocal Care and Hydration
The health of your voice plays a significant role in the quality of your public speaking. Like any other body part, your vocal cords need proper care to function optimally.
One key aspect of vocal care is hydration. Drinking plenty of water helps lubricate your vocal cords, reducing friction and strain during speech. This can help prevent vocal fatigue and maintain the clarity and strength of your voice.
In addition, staying hydrated can also improve your overall vocal quality, making your speech sound more fluid and pleasant. Therefore, regular hydration should be integral to your vocal care routine.
Avoiding Excessive Strain on Vocal Cords
Overusing or misusing your voice, such as yelling, screaming, or prolonged speaking without rest, can lead to vocal strain or damage to your vocal cords. This can result in hoarseness, loss of voice, and potentially long-term vocal issues.
To avoid this, using your voice correctly and giving it ample time to rest and recover is essential. This includes utilizing proper breath support when speaking, avoiding shouting or screaming, and taking regular breaks during prolonged speaking sessions. By doing so, you can protect your vocal cords from unnecessary strain and ensure the longevity of your voice for effective public speaking.
Balancing Speaking Pitch with Naturalness
When it comes to sharing ideas, whether in a formal speech or a casual chat, finding the right balance between your message and your delivery is essential. This way, you can ensure your ideas come across clearly and without unnecessary distractions.
By doing this, you'll be able to have more productive and engaging conversations that get your ideas across effectively. Here are some strategies you can employ:
Comfortable Starting Pitch
To avoid straining your vocal cords and maintain control, start with a comfortable pitch and gradually increase as needed. This approach prevents damage or discomfort, allowing confident and effortless speaking.
Practicing Pitch Shifts
Incorporating vocal exercises into your practice routine can be beneficial for maintaining a relaxed feeling while changing pitch. You can improve your control and accuracy when transitioning between different pitches by actively engaging your vocal muscles and focusing on proper technique. Give it a try and see how it can enhance your vocal performance!
Understanding Your Natural Voice
Your natural speaking pitch is the foundation of your vocal identity. It's the pitch at which your vocal cords vibrate comfortably and efficiently, and it plays a crucial role in maintaining vocal health. Discovering and embracing your natural voice is the first step toward effective communication.
When you speak within your comfortable pitch range, you reduce the strain on your vocal cords, helping to prevent voice fatigue and potential long-term damage. To find your optimal speaking pitch, start by paying attention to your voice when you're relaxed and not trying to project or imitate someone else. This self-awareness can be the key to unlocking better vocal health and making your communication more authentic and sustainable.
Embracing Your Unique Pitch
Each person's voice is as unique as a fingerprint, with distinct pitch, tone, and timbre. This individuality in voices adds to the beauty and diversity of human communication. It shapes our identity, expresses our personality, and influences how we connect with others.
The uniqueness of your voice is something to be celebrated rather than altered. Embracing your natural pitch can help you communicate more authentically and effectively. Moreover, it can also foster a more profound sense of self-acceptance and self-appreciation. After all, your voice is integral to who you are and deserves to be heard in its most accurate form.
How to Build Confidence in Your Natural Speaking Pitch
This section will guide you through three practical strategies - regular practice, positive self-talk, and seeking constructive feedback - that can help you become more comfortable with your natural pitch and improve your vocal strengths. So, let's dive into these methods and start your journey toward confident and authentic vocal expression.
Practice Regularly
Regular vocal practice can help you become more comfortable with your natural pitch. This could involve reading aloud, practicing speeches, or even singing. Over time, you'll better understand your voice and gain confidence in using it.
Positive Self-Talk
Recognize the value of your unique voice and reinforce this through positive self-talk. Remind yourself that your voice expresses your individuality and deserves to be heard.
Seek Constructive Feedback
Share your voice with others and seek their feedback. This could be through public speaking groups, voice coaching, or simply conversing with friends or family. Constructive feedback can help you identify strengths in your voice and areas for improvement.
Aligning Pitch with Authentic Self-Expression
Authentic self-expression involves embracing your natural voice to convey thoughts and feelings. Rather than attempting to imitate someone else's voice or pitch, focus on how to use your voice to express yourself most effectively.
Pay attention to the natural variations in your pitch and how they reflect different emotions or emphases. Doing so allows you to use your pitch as a powerful tool for authentic self-expression, enhancing personal and professional communication.
Remember, authenticity resonates with people more deeply than any perfected pitch or tone. When used with confidence and sincerity, your unique voice can truly make an impact.
Embrace Your Unique Voice and Transform Your Communication
In our exploration, we've underscored how the pitch plays a crucial role in speaking, the importance of building confidence in your natural voice, and the power of aligning it with authentic self-expression. We've also highlighted some essential techniques in caring for and understanding our voice to help us harness the power of its uniqueness.
Remember, your optimal speaking pitch isn't about imitation but authenticity. Embrace your unique voice, continually hone your communication skills, and see the transformative impact on your personal and professional relationships.
Ready to elevate your speaking skills? Consider enrolling in the Compelling Speaker Programto master your unique voice and become an even more impactful speaker. Your voice is powerful - let it be heard.
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12090 | https://www.youtube.com/watch?v=jMPOqbPeThE | Basis for the vector space of all Real Polynomials of Degree less than equal to n Linear Algebra
Spectrum of Mathematics
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Posted: 7 Feb 2023
For more information and LIVE classes contact me on conceptbasedmaths@gmail.com
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foreign [Music] Matrix with IJ entry 1 and 0 elsewhere if a is equal to a i j m by n then a is equal to summation on a i j e i j where I is from 1 to M and J is from 1 to n now here it is given to us that e i j is n by n Matrix which means it's having Ambrose and N columns so what kind of Matrix is this suppose this is your let me take this e i j to be E11 where this is your first row and this is your first column that represents suppose I am having many numbers of rows and then I'm having so many columns so this E11 is taking the entry on first row and First Column which means this is your first row and this is your first column and this is your this first entry this entry is they are saying that this entry is one and zero elsewhere and rest of the entries are zero zero zero zero all our zeros only E11 the entry in one wanted entry here this is the element is 1. right and suppose if it is e 1 2 suppose then this means first row and second column right which means this is your first row and this is your second column so this entry will now become 1 and rest of them all becomes zero right so such kind of Matrix is your aij and if a if you take to be equal to a i j m by n then a is submission of a i j into e i j now what is a i j this is the entry a 1 1. right so let me take an example to explain this suppose I take my a to the equal to a 2 by 2 Matrix having the entries a11 this is a12 which is first row and second column then entry present at the second row and First Column then second row and second column suppose this is your a now I have to show that this a is submission of a i j is an e i j right so this is equal to can I write this as I'm taking a a11 and then rest of them as 0 and if I add this Matrix with this 0 a 1 2 0 0 then 0 0 8 2 1 0 0 0 0 8 2 2 is it fine all right now let me take this a11 outside and I'm having here one zero zero zero that is taking a12 outside this is zero one zero zero and then taking a two one outside this is zero zero one zero and at the last a two two a two zero zero zero one now if you look at here this is nothing but your e now this is this entry is at the first row and First Column so this is E11 this is what I have explained earlier and now if the entry present over here which means this is first row and second column so this is E12 E12 first row and second column right here the columns are greater in number this is just an example to show the summation and this is a21 and this is nothing but your e now here the entry is present in the second row and First Column so this is E21 similarly this is e 2 2. so these this so this a is represented as sum of these four terms having a i j and e i j submission where I is from 1 to 2 and J is from 1 to 2. write simple two by two Matrix so this is just an example to show that and please keep this as your remark only that if aij is such kind of Matrix then a can be written as the summation of a i g and e i j right so just take this as your remark only let's move on to the further examples prove that 1X x square X cubed XL these are so many polynomials and we have to prove that this is a basis of the vector space V of all real polynomials of degree less than equal to n together with the zero polynomials over field r so I have to prove that all these set of polynomials forms the basis for V for that I need to show that these set of polynomials are these are the generators of v and all these are linearly independent so uh now since we know that one property that none of these polynomials is linear combinations of the preceding polynomials right we have studied this property and when this happens that if none of the polynomials is a linear combinations of the preceding ones then such polynomials are linearly independent polynomials or vectors right so clearly these are linearly independent as since none of the polynomials 1 x x square up to X N these are linear combinations of the proceeding polynomials so therefore the polynomials are linearly independent so 1 x x square clearly these are linearly independent and let's mark this as one then we have to show that these polynomials forms the generators of V these are the generators of V which means every element of V is a linear combinations of these polynomials so let me take one polynomial from V and let me name that polynomial that f x is equal to a naught A1 X Plus A2 x square up to so on a and x and let this polynomial belongs to V then FX is clearly clearly you will see here that this folder is a linear combinations of 1 x x square up to so on right so but f x was arbitrary so every element of V is a linear combinations of these polynomials so V is generated by them so mark this as 2 so from 1 and 2 we say that these forms base is for V this is a clear example and please mark our note that this set of polynomials is a standard basis for the vector space V of all real polynomials of degree less than equal to n together with zero polynomial over field are right please mark this in your mind that these are the standard basis for the vector space we clearly the result is already proved thank you so much [Music] thank you |
12091 | https://www.quora.com/What-is-the-equation-for-a-parabola-with-focus-0-p-and-directrix-y-p | Something went wrong. Wait a moment and try again.
Graph Representations
Horizontal Parabola
Conic Sections (geometry)
Analytic Curves
Focus (mathematics)
PLANE GEOMETRY
Parabolas (geometry)
5
What is the equation for a parabola with focus (0,p) and directrix y=−p?
Ed Partida
B.S. in Applied Mathematics, California Polytechnic State University, San Luis Obispo (Graduated 1995)
·
Author has 390 answers and 138K answer views
·
3y
We should start by stating the definition of a parabola with respect to the focus and directrix: given a line(the directrix) and a point(the focus) not on the line, the set of points equidistant from both the point and the line form a parabola. Consequently the vertex of the parabola is equidistant from the focus , so the vertex is located at (0,(p+(-p))/2); that is (0,0).
To derive the equation we apply the definition using the distance formula . Let P(x,y) be a point on the parabola and F(0,p) the focus. Then the point on the directrix that lies on the perpendicular from the point( and is hen
We should start by stating the definition of a parabola with respect to the focus and directrix: given a line(the directrix) and a point(the focus) not on the line, the set of points equidistant from both the point and the line form a parabola. Consequently the vertex of the parabola is equidistant from the focus , so the vertex is located at (0,(p+(-p))/2); that is (0,0).
To derive the equation we apply the definition using the distance formula . Let P(x,y) be a point on the parabola and F(0,p) the focus. Then the point on the directrix that lies on the perpendicular from the point( and is hence the shortest distance from the point ) is D(x,-p).
By the definition of the parabola we must have
d(P,F) = d(P,D) ,
where d is the distance formula . Thus
√[(x-0)²+(y-p)²] = √[(x-x)²+(y-(-p))²];
Squaring both sides and simplifying, we get
x²+y²-2py+p² = y²+2py+p²; so
x²-2py = 2py; adding 2py ;
x² = 4py ; dividing by 4p;
(1/(4p))x² = y. By symmetry of =,
y = (1/(4p))x².
the coefficient of x², 1/(4p) , is important in determining the shape of the parabola. When p is small the parabola rises quickly with respect to x and when p is large , it rises slowly with respect to x.
Related questions
Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 2?
P(A) = 0.7, P (A and B) = 0.01, what is P(B)?
P(4,6) is a point on parabola whose focus is at (0,2) and directrix is parallel to x axis. What is the equation of parabolas vertex, directrix, and the distance from p to the directrix?
What’s the equation for the parabola with a vertex at the origin and focus of (7,0)?
What is the equation of the parabola vertex (0, 0) and focus (-6, 0)?
Mohammad Afzaal Butt
B.Sc in Mathematics & Physics, Islamia College Gujranwala (Graduated 1977)
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Author has 24.6K answers and 22.9M answer views
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3y
Focus of the parabola lies on y-axis. The vertex of the parabola is (0 , 0). The parabola opens upward and is symmetrical about x-axis.
Equation of the parabola is
x2 = 4py
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Bot
·
Sep 6
A parabola is the set of points equidistant from a focus and a directrix. For focus (0,p) and directrix y = −p, let (x,y) be a point on the parabola. Distance to focus:
√[(x−0)² + (y−p)²].
Distance to directrix (horizontal line y = −p): |y − (−p)| = |y + p|.
Equate squared distances to avoid the square root:
x² + (y − p)² = (y + p)².
Expand and simplify:
x² + y² − 2py + p² = y² + 2py + p²
x² − 2py = 2py
x² = 4py.
Thus the parabola's equation is:
x² = 4 p y.
(For p > 0 this opens upward with vertex at the origin; for p < 0 it opens downward.)
George Ivey
Former Math Professor at Gallaudet University
·
Upvoted by
BowTangey
, PhD Mathematics, Iowa State University (1988) · Author has 23.7K answers and 2.6M answer views
·
3y
There are infinitely many such parabolas. The fact that focus is at (0, p) and the directrix is y= -p means that the vertex is at (0, 0). Such a parabola has equation y= ax^2 for a any number.
Related questions
What is the equation of a parabola with focus (1,1) and directrix y=-x?
P (4, 6) is a point on a parabola whose focus is at (0, 2) and the directrix is parallel to the x-axis. What is the equation of the Parabola, its vertex and directrix?
Can you find the equation of parabola, focus at (-2,0) directrix the line x=2?
What is the equation of the parabola with V (0,2) and F (0,1)?
What is the equation of the parabola with focus at (0.-4) and directrix y=4?
Mitchell Schoenbrun
MS in Mathematics, San Francisco State University (SFSU) (Graduated 2005)
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Author has 4.5K answers and 6M answer views
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3y
Hmmm,
Well you know the focus and the directrix, and you should know that points on the parabola are equidistant from each, so careful application of the distance formula will give you your answer.
Rick McGeer
CEO at engageLively (2018–present)
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Author has 5.5K answers and 12.1M answer views
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3y
You have
(x−0)2+(y−p)2=(y−(−p))2
x2+y2−2py+p2=y2+2py+p2
x2=4py
y=x24p
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Haresh Sagar
Studied Science & Mathematics (Graduated 1988)
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Author has 6.2K answers and 7M answer views
·
3y
Related
P (4, 6) is a point on a parabola whose focus is at (0, 2) and directrix is parallel to x-axis. What is the equation of parabolas?
P (4, 6) is a point on a parabola whose focus is at (0, 2) and directrix is parallel to x-axis. What is the equation of parabolas?
One of the properties of a parabola is that, distance between any point on parabola to focus is equal to the shortest distance between that point and directrix. Here, distance between P and F is √16+16=4√2.
Directrix is parallel to x axis, so subtracting the distance from y coordinate of P, we get equation of directrix.
y=6−4√2=2(3−2√2)
Standard vertex form of vertical parabola equation is,
(x−h)2=4p(y−k)
where h and k are x and y coordinates of
P (4, 6) is a point on a parabola whose focus is at (0, 2) and directrix is parallel to x-axis. What is the equation of parabolas?
One of the properties of a parabola is that, distance between any point on parabola to focus is equal to the shortest distance between that point and directrix. Here, distance between P and F is √16+16=4√2.
Directrix is parallel to x axis, so subtracting the distance from y coordinate of P, we get equation of directrix.
y=6−4√2=2(3−2√2)
Standard vertex form of vertical parabola equation is,
(x−h)2=4p(y−k)
where h and k are x and y coordinates of vertex respectively and 4p is length of the focal chord whose midpoint is focus point. Note that distance between focus and directrix is 2p and vertex is exactly between focus and directrix. Also remember that in a vertical parabola axis of symmetry passes through x coordinates of focus and vertex. So,
2p=2−(6−4√2)=4(√2−1)
4p=8(√2−1)
V=(0,2+2(3−2√2)2)=(0,4−2√2)
So the equation of parabola is,
x2=8(√2−1)[y−4+2√2]
Robert Paxson
BSME in Mechanical Engineering, Lehigh University (Graduated 1983)
·
Author has 3.9K answers and 4M answer views
·
1y
Related
What is the equation of a parabola with a focus at (0, 4) and a directrix at x = -1?
The distance from the focus, (0,4), to a point, (x,y), on the parabola, is equal to the distance from that same point on the parabola to the directrix, x=−1:
√(x−0)2+(y−4)2=|x−(−1)|
√x2+(y−4)2=√(x+1)2
x2+(y−4)2=(x+1)2
(y−4)2=2x+1
x=12(y−4)2−12
where
x=14f(y−k)2+h
such that f=12 and the vertex is at (h,k)=(−12,4), or:
x2+y2−8y+16=x2+2x+1
y2−2x−8y+15=0
A plot looks like this:
The distance from the focus, (0,4), to a point, (x,y), on the parabola, is equal to the distance from that same point on the parabola to the directrix, x=−1:
√(x−0)2+(y−4)2=|x−(−1)|
√x2+(y−4)2=√(x+1)2
x2+(y−4)2=(x+1)2
(y−4)2=2x+1
x=12(y−4)2−12
where
x=14f(y−k)2+h
such that f=12 and the vertex is at (h,k)=(−12,4), or:
x2+y2−8y+16=x2+2x+1
y2−2x−8y+15=0
A plot looks like this:
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Pradeep Hebbar
Many years of Structural Engineering & Math enthusiasm
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Author has 9.3K answers and 6.2M answer views
·
2y
Related
How do you derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4?
Given parabola has focus at (0,−4) and equation of directrix y=4
Equation of directrix indicates that it is parallel to x-axis. That makes the axis of parabola parallel to y-axis. Parabola either opens up or down.
x-coordinate of focus is 0 and hence the axis of parabola is along y-axis.
Focus is below the directrix and therefore parabola opens down.
Vertex lies on the y-axis and it is at midway between directrix and focus. We note that it is at origin O(0,0).
Equation of such a parabola is given by,
x2=4ay
a is the distance of focus from vertex. For a<0, focus is below x-axis
a=−4
x2=4(−4)y
x2=−
Given parabola has focus at (0,−4) and equation of directrix y=4
Equation of directrix indicates that it is parallel to x-axis. That makes the axis of parabola parallel to y-axis. Parabola either opens up or down.
x-coordinate of focus is 0 and hence the axis of parabola is along y-axis.
Focus is below the directrix and therefore parabola opens down.
Vertex lies on the y-axis and it is at midway between directrix and focus. We note that it is at origin O(0,0).
Equation of such a parabola is given by,
x2=4ay
a is the distance of focus from vertex. For a<0, focus is below x-axis
a=−4
x2=4(−4)y
x2=−16y
Robert Paxson
BSME in Mechanical Engineering, Lehigh University (Graduated 1983)
·
Author has 3.9K answers and 4M answer views
·
1y
Related
What is the equation of a parabola with a focus at (0, -2) and a directrix of y = 15?
For a parabola, the distance from the focus to a point (x,y) is equal to the distance from that same point to the directrix:
√(x−0)2+(y+2)2=|y−15|
√(x−0)2+(y+2)2=√(y−15)2
(x−0)2+(y+2)2=(y−15)2
x2+y2+4y+4=y2−30y+225
y=−134x2+22134
y=−134x2+132
A plot looks like this:
For a parabola, the distance from the focus to a point (x,y) is equal to the distance from that same point to the directrix:
√(x−0)2+(y+2)2=|y−15|
√(x−0)2+(y+2)2=√(y−15)2
(x−0)2+(y+2)2=(y−15)2
x2+y2+4y+4=y2−30y+225
y=−134x2+22134
y=−134x2+132
A plot looks like this:
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Gordon M. Brown
Math Tutor at San Diego City College (2018-Present)
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Upvoted by
BowTangey
, PhD Mathematics, Iowa State University (1988) · Author has 6.2K answers and 4.3M answer views
·
2y
Related
What is the equation of the parabola given directrix x=3, focus (0,8)?
Before I begin, I want to make clear that you have submitted two requests for my answers to two problems pertaining to the parabola, the focus and directrix. I’ve also noted that by posting numerous questions on your profile on this same topic, you’ve made it plain that you want others on Quora to do your homework for you. That’s not what we’re here for!
Accordingly, this will be the first and only question from you that I will address. Afterward, you are expected to learn from the answers already given you, so that you can execute these problems by yourself. It’s high time you learned to think
Before I begin, I want to make clear that you have submitted two requests for my answers to two problems pertaining to the parabola, the focus and directrix. I’ve also noted that by posting numerous questions on your profile on this same topic, you’ve made it plain that you want others on Quora to do your homework for you. That’s not what we’re here for!
Accordingly, this will be the first and only question from you that I will address. Afterward, you are expected to learn from the answers already given you, so that you can execute these problems by yourself. It’s high time you learned to think by yourself, for yourself.
Got that?
Now consider the graph below:
With the directrix being to the right of the focus, we need a parabola whose arms open to the left. Such a parabola is given by the general equation y^2 = -4px, where p is the distance of the vertex from both the focus and directrix. Since the given focus lies 3 units to the left of the directrix, the distance p = 3/2. Consequently, our parabola should have the equation y^2 = -4 3/2 x, or y^2 = -6x. (See the parabola in dashed green above.)
But there is a problem with this parabola: its vertex is located at the origin of the graph. We need to translate this parabola so that its vertex lies midway between the focus and directrix given. This means translating the parabola 8 units upward, and 3/2 units to the right, as shown by the parabola drawn in solid purple above. Now satisfy yourself that every point on the purple curve is equidistant from the focus and the directrix.
P. S. Please be apprised that I never address more than one question per person, per day, for reasons that should be clear by now. But more to my point, if you remain hell-bent to have others do your thinking for you, I strongly suggest that you never submit any further requests for my answers in the future.
Janet Heberling
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What is the equation of the parabola given directrix y=-7/4, focus (0, 7/4)?
Since the directrix is horizontal, the parabola is vertical. The focus lies above the directrix, so the parabola opens upwards.
Apply your data and solve for h, k, p, a:
[math]\text{The vertex $(h,k)$ is halfway between}\ \text{the directrix and focus.} \ (h,k)=(0,0)\ h=k=0[/math]
[math]\text{The focal length $p$ is the magnitude of}\ \text{the distance between vertex and focus.}\ p=\dfra[/math]
Since the directrix is horizontal, the parabola is vertical. The focus lies above the directrix, so the parabola opens upwards.
[math]\text{Vertex-form equation for an up-opening parabola:} \quad y=a(x-h)^2+k, \,a>0\ \quad\text{vertex }(h,k)\ \quad\text{focal length }p=\dfrac{1}{4\left|a\right|}\ \quad\text{focus }(h,k+p)\ \quad\text{directrix } y=k-p\[/math]
Apply your data and solve for h, k, p, a:
[math]\text{The vertex $(h,k)$ is halfway between}\ \text{the directrix and focus.} \ (h,k)=(0,0)\ h=k=0[/math]
[math]\text{The focal length $p$ is the magnitude of}\ \text{the distance between vertex and focus.}\ p=\dfrac{7}{4}\ a=\dfrac{1}{4p}=\dfrac{1}{7}[/math]
[math]\text{Equation of parabola in vertex form:}\ y=\dfrac{1}{7}x^2[/math]
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
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What is the equation of a parabola written in general form predicate focus (0,4) and directrix (y=-4)?
I know it is possible to just find a general equation for what you want but “just putting numbers in a formula” is not my idea of good mathematics.
It is much more fun if you use the idea of what a parabola is!
P(x, y) is a general point on the parabola
if the distance PF is equal to the distance PR
All we have ...
I know it is possible to just find a general equation for what you want but “just putting numbers in a formula” is not my idea of good mathematics.
It is much more fun if you use the idea of what a parabola is!
—————————————————————————————————————————
P(x, y) is a general point on the parabola
if the distance PF is equal to the distance PR
All we have ...
Related questions
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12092 | https://math.stackexchange.com/questions/1854538/combinations-and-permutations-with-constraints | Skip to main content
Combinations and permutations with constraints
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In how many ways can 6 people be seated around a table for 6, if 2 of the group are always:
together? - I solved this by doing 2! x 4!
separate? - I do not know how to do this, the answer is 72
Can you plz explain the second part?
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edited Jul 10, 2016 at 0:07
Joffan
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asked Jul 9, 2016 at 23:55
math help plzzzmath help plzzz
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it does not have to be very detailed just write down ur calculations
– math help plzzz
Commented
Jul 9, 2016 at 23:59
its a round table, and yes just 2 need to be seated together, basically they need to be right next to each other
– math help plzzz
Commented
Jul 10, 2016 at 0:02
@Joffan pleasee i answered ur questions, may u plz help m e
– math help plzzz
Commented
Jul 10, 2016 at 0:04
Try finding all possible permutations, then subtracting the ones in which the two are sitting together. (You already found how many to subtract)
– Jack
Commented
Jul 10, 2016 at 0:09
3
There are 5!=120 arrangements with no restriction. There are 48 with our friends together, as you found, so the number of separated arrangements is 120−48=72.
– André Nicolas
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Jul 10, 2016 at 0:10
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You can use the total arrangement minus the number of arrangements when the two persons sitting together to get the number of the separate arrangements. 5!−2!×4!.
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answered Jul 10, 2016 at 0:13
Xianjin YangXianjin Yang
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There's always this tension between people when answering "round table" questions. Your solution of 2!4! for the first part suggests that you are in the camp that holds that the seat positions are indistinguishable. I would personally multiply your answer by 6, as I have never experienced such a table.
However, in the spirit of your first answer, I would say a similar process should be used. First arbitrarily seat the first person from your restricted pair, then chose one of the three seats not next to that person for other one. Then permute the other four less-fussy dining companions into the remaining four seats. Which is just 32 times your other answer - whichever camp you are in.
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answered Jul 10, 2016 at 0:14
JoffanJoffan
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12093 | https://math.stackexchange.com/questions/1287527/example-2-sec-11-1-in-apostols-calculus-vol-1-how-to-calculate-this-limit | real analysis - Example 2, Sec. 11.1 in Apostol's CALCULUS, vol. 1: How to calculate this limit? - Mathematics Stack Exchange
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Example 2, Sec. 11.1 in Apostol's CALCULUS, vol. 1: How to calculate this limit?
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For each n∈N n∈N, let f n:[0,1]→R f n:[0,1]→R be defined as
f n(x):=n x(1−x 2)n for all x∈[0,1].f n(x):=n x(1−x 2)n for all x∈[0,1].
Then f n(0)=0=f n(1)f n(0)=0=f n(1).
So let 0<x<1 0<x<1.
Then what is
lim n→∞f n(x)?lim n→∞f n(x)?
That is, what is
lim n→∞n x(1−x 2)n lim n→∞n x(1−x 2)n
for a given 0<x<1 0<x<1?
How to calculate this limit?
And how to show rigorously that the value obtained is indeed the desired limit?
My effort:
If 0<x<1 0<x<1, then 0<x 2<1 0<x 2<1 and hence 0<1−x 2<1 0<1−x 2<1; so 0<(1−x 2)n<1 0<(1−x 2)n<1. Thus 0<n x(1−x 2)n<n x 0<n x(1−x 2)n<n x, but this doesn't lead us anywhere as n x→+∞n x→+∞ as n→∞n→∞.
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asked May 18, 2015 at 7:10
Saaqib MahmoodSaaqib Mahmood
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If 0<α<1 0<α<1, then lim y→+∞y⋅α y=0 lim y→+∞y⋅α y=0
Indeed, for instance one can use the argument
ln(y⋅α y)=y ln α+ln y→−∞ln(y⋅α y)=y lnα+lny→−∞
Added: Another way to show the weaker, but sufficient, result 0<α<1⇒lim n n⋅α n=0 0<α<1⇒lim n n⋅α n=0 could be: let b n=n⋅α n b n=n⋅α n.
b n+1 b n=n+1 n α⟶n α b n+1 b n=n+1 n α⟶n α
So ∃N∀n≥N b n+1 b n≤1+α 2<1∃N∀n≥N b n+1 b n≤1+α 2<1
Therefore definitely
0≤b n≤(1+α 2)n−N b N→n 0 0≤b n≤(1+α 2)n−N b N→n 0
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edited May 18, 2015 at 7:31
answered May 18, 2015 at 7:18
user228113 user228113
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how do we know that y ln a+ln y→−∞y lna+lny→−∞ as y→+∞y→+∞?Saaqib Mahmood –Saaqib Mahmood 2015-05-18 17:14:44 +00:00 Commented May 18, 2015 at 17:14
Well, the way I did it back in high school was using l'Hopital rule to calc lim y→+∞y ln a ln y=lim y→+∞ln a 1 y=sgn(ln a)⋅(+∞)lim y→+∞y lna lny=lim y→+∞lna 1 y=sgn(lna)⋅(+∞) but, of course, one may argue that these tools are much harder to be proven from scratch than the actual problem. I actually do not remember if that was the case. @SaaqibMahmuud user228113 –user228113 2015-05-18 17:26:13 +00:00 Commented May 18, 2015 at 17:26
My second approach, on the other hand, I'm 100%100% sure it requires absolutely nothing.user228113 –user228113 2015-05-18 17:32:55 +00:00 Commented May 18, 2015 at 17:32
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Another approach can be this. We have, if 0<x<1 0<x<1,
∑n≥0(1−x 2)n=1 x 2∑n≥0(1−x 2)n=1 x 2
then if we derive
−2 1−x 2∑n≥1 x n(1−x 2)n=−2 x 3−2 1−x 2∑n≥1 x n(1−x 2)n=−2 x 3
hence
∑n≥1 x n(1−x 2)n=1−x 2 x 3∑n≥1 x n(1−x 2)n=1−x 2 x 3
so the series is convergent and so we have
x n(1−x 2)n⟶0 x n(1−x 2)n⟶0
as n→∞n→∞.
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answered May 18, 2015 at 13:01
Marco CantariniMarco Cantarini
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how to derive the second equality in your answer?Saaqib Mahmood –Saaqib Mahmood 2015-05-18 17:12:05 +00:00 Commented May 18, 2015 at 17:12
@SaaqibMahmuud d d x(1−x 2)n=−2 x n(1−x 2)n−1=−2 1−x 2 x n(1−x 2)n d d x(1−x 2)n=−2 x n(1−x 2)n−1=−2 1−x 2 x n(1−x 2)n and d d x 1 x 2=−2 x 3.d d x 1 x 2=−2 x 3.Marco Cantarini –Marco Cantarini 2015-05-18 17:19:54 +00:00 Commented May 18, 2015 at 17:19
can we justify this sort of term by term differentiation?Saaqib Mahmood –Saaqib Mahmood 2015-05-20 04:23:07 +00:00 Commented May 20, 2015 at 4:23
@SaaqibMahmuud See here en.wikipedia.org/wiki/Power_seriesMarco Cantarini –Marco Cantarini 2015-05-20 07:06:53 +00:00 Commented May 20, 2015 at 7:06
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12094 | http://www.personal.soton.ac.uk/ab1u06/teaching/phys3002/course/04_liquiddrop.pdf | Chapter 4 The Liquid Drop Model 4.1 Some Nuclear Nomenclature • Nucleon: A proton or neutron.
• Atomic Number, Z: The number of protons in a nucleus.
• Atomic Mass number, A: The number of nucleons in a nucleus.
• Nuclide: A nucleus with a specified value of A and Z. This is usually written as A Z{Ch} where Ch is the Chemical symbol. e.g. 56 28Ni means Nickel with 28 protons and a further 28 neutrons.
• Isotope: Nucleus with a given atomic number but different atomic mass number, i.e.
different number of neutrons.
Isotopes have very similar atomic and chemical behaviour but may have very different nuclear properties.
• Isotone: Nulceus with a given number of neutrons but a different number of protons (fixed (A-Z)).
• Isobar: Nucleus with a given A but a different Z.
• Mirror Nuclei: Two nuclei with odd A in which the number of protons in one nucleus is equal to the number of neutrons in the other and vice versa.
4.2 Binding Energy The mass of a nuclide is given by mN = Z mp + (A −Z) mn −B(A, Z)/c2, where B(A, Z) is the binding energy of the nucleons and depends on both Z and A. The binding energy is due to the strong short-range nuclear forces that bind the nucleons together.
29 Unlike Coulomb binding these cannot even in principle be calculated analytically as the strong forces are much less well understood than electromagnetism.
Binding energies per nucleon increase sharply as A increases, peaking at iron (Fe) and then decreasing slowly for the more massive nuclei.
The binding energy divided by c2 is sometimes known as the “mass defect”.
4.3 Semi-Empirical Mass Formula For most nuclei (nuclides) with A > 20 the binding energy is well reproduced by a semi-empirical formula based on the idea the the nucleus can be thought of as a liquid drop.
1. Volume term: Each nucleon has a binding energy which binds it to the nucleus.
Therefore we get a term proportional to the volume i.e. proportional to A.
aV A This term reflects the short-range nature of the strong forces. If a nucleon interacted with all other nucleons we would expect an energy term of proportional to A(A −1), but the fact that it turns out to be proportional to A indicates that a nucleon only interact with its nearest neighbours.
30 2. Surface term: The nucleons at the surface of the ‘liquid drop’ only interact with other nucleons inside the nucleus, so that their binding energy is reduced. This leads to a reduction of the binding energy proportional to the surface area of the drop, i.e.
proportional to A2/3 −aS A2/3.
3. Coulomb term: Although the binding energy is mainly due to the strong nuclear force, the binding energy is reduced owing to the Coulomb repulsion between the protons. We expect this to be proportional to the square of the nuclear charge, Z, ( the electromagnetic force is long-range so each proton interact with all the others), and by Coulomb’s law it is expected to be inversely proportional to the nuclear radius, (the Coulomb energy of a charged sphere of radius R and charge Q is 3Q2/(20πǫ0R)) The Coulomb term is therefore proportional to 1/A1/3 −aC Z2 A1/3 4. Asymmetry term: This is a quantum effect arising from the Pauli exclusion principle which only allows two protons or two neutrons (with opposite spin direction) in each energy state. If a nucleus contains the same number of protons and neutrons then for each type the protons and neutrons fill to the same maximum energy level (the ‘fermi level’). If, on the other hand, we exchange one of the neutrons by a proton then that proton would be required by the exclusion principle to occupy a higher energy state, since all the ones below it are already occupied.
The upshot of this is that nuclides with Z = N = (A−Z) have a higher binding energy, whereas for nuclei with different numbers of protons and neutrons (for fixed A) the binding energy decreases as the square of the number difference. The spacing between energy levels is inversely proportional to the volume of the nucleus - this can be seen by treating the nucleus as a three-dimensional potential well- and therefore inversely proportional to A. Thus we get a term −aA (Z −N)2 A 5. Pairing term: It is found experimentally that two protons or two neutrons bind more strongly than one proton and one neutron.
In order to account for this experimentally observed phenomenon we add a term to the binding energy if number of protons and number of neutrons are both even, we subtract 31 the same term if these are both odd, and do nothing if one is odd and the other is even.
Bohr and Mottelson showed that this term was inversely proportional to the square root of the atomic mass number.
We therefore have a term ¡ (−1)Z + (−1)N¢ 2 aP A1/2.
The complete formula is, therefore B(A, Z) = aV A −aS A2/3 −aC Z2 A1/3 −aA (Z −N)2 A + ¡ (−1)Z + (−1)N¢ 2 aP A1/2 From fitting to the measured nuclear binding energies, the values of the parameters aV , aS, aC, aA, aP are aV = 15.56 MeV aS = 17.23 MeV aC = 0.697 MeV aA = 23.285 MeV aP = 12.0 MeV For most nuclei with A > 20 this simple formula does a very good job of determining the binding energies - usually better than 0.5%.
For example we estimate the binding energy per nucleon of 80 35Br (Bromine), for which Z=35, A=80 (N = 80 −35 = 45) and insert into the above formulae to get Volume term: (15.56 × 80) = 1244.8 MeV Surface term: (−17.23 × (80)2/3) = −319.9 MeV Coulomb term: µ0.697 × 352 (80)1/3 ¶ = −198.2 MeV Asymmetry term: µ23.285 × (45 −35)2 80 ¶ = −29.1 MeV Pairing term: µ −12.0 (80)1/2 ¶ = −1.3 MeV Note that we subtract the pairing term since both (A-Z) and Z are odd. This gives a total binding energy of 696.3 MeV. The measured value is 694.2 MeV.
In order to calculate the mass of the nucleus we subtract this binding energy (divided by c2) from the total mass of the protons and neutrons (mp = 938.4MeV/c2, mn = 939.6MeV/c2) mBr = 35mp + 45mn −696.1MeV/c2 = 74417 Mev/c2.
32 Nuclear masses are nowadays usually quoted in MeV/c2 but are still sometimes quoted in atomic mass units, defined to be 1/12 of the atomic mass of 12 6 C (Carbon). The conversion factor is 1 a.u. = 931.5 MeV/c2 Since different isotopes have different atomic mass numbers they will have different bind-ing energies and some isotopes will be more stable than others.
It turns out (and can be seen by looking for the most stable isotopes using the semi-empirical mass formula) that for the lighter nuclei the stable isotopes have approximately the same number of neutrons as protons, but above A ∼20 the number of neutrons required for stability in-creases up to about one and a half times the number of protons for the heaviest nuclei.
Qualitatively, the reason for this arises from the Coulomb term. Protons bind less tightly than neutrons because they have to overcome the Coulomb repulsion between them. It is therefore energetically favourable to have more neutrons than protons. Up to a certain limit this Coulomb effect beats the asymmetry effect which favours equal numbers of protons and neutrons.
33 34 |
12095 | https://people.richland.edu/james/lecture/m170/ch07-clt.html | Stats: Central Limit Theorem
Stats: Central Limit Theorem
Sampling Distribution of the Sample Means
Instead of working with individual scores, statisticians often work with means. What happens is that several samples are taken, the mean is computed for each sample, and then the means are used as the data, rather than individual scores being used. The sample is a sampling distribution of the sample means.
When all of the possible sample means are computed, then the following properties are true:
The mean of the sample means will be the mean of the population
The variance of the sample means will be the variance of the population divided by the sample size.
The standard deviation of the sample means (known as the standard error of the mean) will be smaller than the population mean and will be equal to the standard deviation of the population divided by the square root of the sample size.
If the population has a normal distribution, then the sample means will have a normal distribution.
If the population is not normally distributed, but the sample size is sufficiently large, then the sample means will have an approximately normal distribution. Some books define sufficiently large as at least 30 and others as at least 31.
The formula for a z-score when working with the sample means is:
Finite Population Correction Factor
If the sample size is more than 5% of the population size and the sampling is done without replacement, then a correction needs to be made to the standard error of the means.
In the following, N is the population size and n is the sample size. The adjustment is to multiply the standard error by the square root of the quotient of the difference between the population and sample sizes and one less than the population size.
For the most part, we will be ignoring this in class.
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Number of ordered 4-tuples
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How do we find the number of ordered 4-tuples $(a,b,c,d)$ such that $a,b,c,d \in {1,2,\cdots, 6}$ and $a \leq b \leq c \leq d$?
I tried to make cases, considering first different values of $a$, but they were just too many..
How do we approach such problems? Hints and answers appreciated.
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asked Oct 9, 2014 at 12:36
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Let $A=a,B=b+1,C=c+2,D=d+3$, then we want four numbers from ${1,2,...,9}$ with $A
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answered Oct 9, 2014 at 13:14
Empy2Empy2
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I propose to transform the problem in the following way:How many combinations of $(a,b)$, $(b,c)$ and $(c,d)$ do exist if $a,b,c,d \in {1,2,...,6}$ and $a \leq b \leq c \leq d$? We will mean that e.g. $(1,3)$, $(3,4)$, $(4,6)$ is one combination and it will give us tuple $(1, 3, 4, 6)$ in the terms of initial problem.Let's look at $(a,b)$. Possible values for them are: \begin{matrix} 1 & 1 \ 1 & 2 \ 1 & 3 \ ... & ... \ 2 & 2 \ 2 & 3 \ ... & ... \ 5 & 5 \ 5 & 6 \ 6 & 6 \ \end{matrix}
It gives us total of $\sum_{k=1}^{N}k$, $N = 6$ in our case, so this sum equals $21$.Now look at $(b,c)$. If $(a,b) = (1,1)$, then $(b,c)$ may take all values from the table above, so there are $21$ possible options. If $b = 2$, then $(b,c)$ can't start with $1$, so we need to remove $(1,1)...(1,6)$ from the list of possible values of $(b,c)$. Note that $b=2$ refers to $2$ options of $(a,b)$: $(1,2)$ and $(2,2)$, so there are $2 \sum_{k=1}^{6}k = 2 15$ options of $(b,c)$ for $b=2$. We can continue this reasoning further and denote $\phi_i = \sum_{k=1}^ik$ (so $\phi_1 = 1$, $\phi_2 = 3$, $\phi_3 = 6$ and so on). Then we have $\sum_{j=1}^N{(N-j+1)\phi_j} = 1 21 + 2 15 + 3 10 + 4 6 + 5 3 + 6 1 = 126$ combinations of $(a,b)$ and $(b,c)$.Now add $(c,d)$ into our analysis.If $(a,b) = (1,1)$, then we have $21$ possible options of $(b,c)$ and $(c,d)$ can be any eligible pair. We found above that there are $126$ such pairs. You can see that for any pair $(a,b)$ we can choose $\phi_m$ pairs of $(b,c)$, where $m = N - b + 1 = 7 - b$. So, for example, if $b = 3$, we can choose $\phi_4 = 10$ different pairs of $(b,c)$. Similarly, for any $(b,c)$ we will have $\phi_m$ options of $(c,d)$, where $m = N - c + 1 = 7 - c$.Now denote $\Phi_i = \sum_{k=1}^i{(N-k+1)\sigma_k}$. So $\Phi_6 = 126$, $\Phi_5 = 70$ and so on. $\Phi_m$ shows how many possible $(c,d)$ can we choose based on chosen $(a,b)$, where $m = N - b + 1 = 7 - b$.Thus, looking at the first figure, we have $\Phi_6$ = 126 options for $b=1 \Leftrightarrow (a,b) = (1,1)$, $\Phi_5 = 70$ options for $b = 2 \Leftrightarrow (a,b) = (1,2) \lor (2,2)$ and so on. So the total number of tuples is $$\sum_{i=1}^{N}{(N-i+1)\Phi_i}$$In this case $N=6$, so$$\sum_{i=1}^{6}{(7-i)\Phi_i} = \Phi_6 + 2\Phi_5 + ... + 6\Phi_1 = 457.$$
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answered Oct 9, 2014 at 14:07
Andrei RykhalskiAndrei Rykhalski
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$\begingroup$ @Pkwssis Right. It seems that the last step in my solution was ambigious and it probably solved the same problem for 5-tuples. I should have stopped at calculating $\Phi_6$. $\endgroup$
Andrei Rykhalski
– Andrei Rykhalski
2014-10-09 15:25:14 +00:00
Commented Oct 9, 2014 at 15:25
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12097 | https://simple.wikipedia.org/wiki/Venn_diagram | Jump to content
Venn diagram
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From Simple English Wikipedia, the free encyclopedia
Stained glass window in Cambridge, where John Venn studied. It shows a Venn diagram.
A Venn diagram is a diagram that shows the logical relation between sets. They were popularised by John Venn in the 1880s, and are now widely used. They are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses closed curves drawn on a plane to represent sets. Very often, these curves are circles or ellipses.
Similar ideas had been proposed before Venn. Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler (Letters to a German Princess) in 1768, for instance, came up with similar ideas. The idea was popularised by Venn in Symbolic Logic, Chapter V "Diagrammatic Representation", 1881.
Example
[change | change source]
The following example uses two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged. The blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that can both fly and have two legs—for example, parrots—are then in both sets, and hence correspond to points in the area where the blue and orange circles overlap. That area contains all such (and only) such living creatures.
Humans and penguins are bipedal, and hence are in the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all be represented by points outside both circles.
The combined area of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged, or that can fly (or both).
The area in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles.
Venn diagrams of common operations on sets
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In the illustrations below the left circle shows set and the right circle shows set . Results of operations are shown as colored areas.
Related pages
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Exclusive disjunction
Inclusive disjunction
References
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↑ Jump up to: 1.0 1.1 "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-09-04.
↑ "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". www.probabilitycourse.com. Retrieved 2020-09-04.
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12098 | https://math.stackexchange.com/questions/4814788/is-there-any-square-harmonic-divisor-number-greater-than-1 | Skip to main content
Is there any square harmonic divisor number greater than 1?
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A harmonic divisor number or Ore number is a positive integer whose harmonic mean of its divisors is an integer. In other words, n is a harmonic divisor number if and only if nd(n)σ(n) is an integer, where d(n) is the number of divisors of n and σ(n) the sum of divisors of n. For example, for n=140, we have nd(n)σ(n)=140×12336=5, so 140 is a harmonic divisor number. I was wondering:
Is there any square harmonic divisor number greater than 1?
I check the conjectured list of the first 10000 harmonic divisor numbers and found no squares. In the OEIS page it is conjectured that every harmonic divisor number greater than 1 is a Zumkeller number, one whose divisors can be partitioned into two disjoint sets with equal sum. Of course no square number can be a Zumkeller number because σ(n) is odd for square n, but perhaps proving this conjecture is even harder. So I was wondering if we could prove there is no square harmonic divisor number greater than 1 or, in case where this seems to be equally difficult as finding an odd perfect number, provide any lower bound for a square harmonic divisor number? Any help/reference appreciated.
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asked Nov 26, 2023 at 16:23
Jianing SongJianing Song
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I did not even find yet a harmonic divisor number (except 1) which is powerful that is every prime factor occurs with an exponent larger than 1. I finished the search limit upto n=108
– Peter
Commented
Apr 5, 2024 at 10:58
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In this range , 1 is also the only odd harmonic divisor number.
– Peter
Commented
Apr 5, 2024 at 11:29
@Peter Thanks for the remark!
– Jianing Song
Commented
Apr 5, 2024 at 12:12
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Concerning the entries in OEIS : Apart from 1 and 6 , they are all divisible by 4 or by 18 , so 1 and 6 are the only squarefree values and 1 is the only odd value. Apart from 1 , there is no powerful value. So, my results hold also upto 6.6⋅1019 at least , if we can trust this entry.
– Peter
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Apr 7, 2024 at 7:40
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I found out that all the 104 entries in OEIS can be proven to be Zumkeller-numbers with the greedy-algorithm (always choosing the largest divisor so that the sum is not exceeded). I wonder whether this algorithm works for all Zumkeller numbers or whether there are examples where it fails.
– Peter
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1 Department of Obstetrics and Gynecology, Peking Union Medical College Hospital, Peking Union Medical College, Chinese Academy of Medical Sciences, Beijing 100730, China.
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Correspondence to: Dr. Qi Yu, Department of Obstetrics and Gynecology, Peking Union Medical College Hospital, Peking Union Medical College, Chinese Academy of Medical Sciences, Beijing 100730, China E-Mail: yuqi2008001@sina.com
Received 2019 Aug 27; Issue date 2019 Dec 20.
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PMCID: PMC6964939 PMID: 31809322
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Keywords: Endometriosis, Non-hormonal therapy, Pelvic pain, Subfertility
Abstract
Objective:
Endometriosis is a common gynecologic disease that frequently leading to chronic pelvic pain, severe dysmenorrhea, and subfertility. As first-line hormonal treatment can interfere with ovulation and may cause recurrent pelvic pain, exploration of new non-hormonal therapeutic approaches becomes increasingly necessary. This review aimed to evaluate the pre-clinical and clinical efficacy and safety of non-hormonal treatment for endometriosis
Data sources:
Databases including PubMed, Embase, Cochrane Library, SINOMED, ClinicalTrials.gov, and Google Scholar were searched up to October 2019, using search terms “endometriosis” and “non-hormonal therapy.”
Study selection:
Twenty-four articles were reviewed for analysis, including nine animal studies and 15 human trials; all were published in English.
Results:
Twenty-four articles were identified, including 15 human trials with 861 patients and nine animal studies. Some agents have been evaluated clinically with significant efficacy in endometriosis-related pelvic pain and subfertility, such as rofecoxib, etanercept, pentoxifylline, N-palmitoylethanolamine, resveratrol, everolimus, cabergoline (Cb2), and simvastatin. Other drugs with similar pharmacological properties, like parecoxib, celecoxib, endostatin, rapamycin, quinagolide, and atorvastatin, have only been tested in animal studies.
Conclusions:
Clinical data about most of the non-hormonal agents are not sufficient to support them as options for replacement therapy for endometriosis. In spite of this, a few drugs like pentoxifylline showed strong potential for real clinical application.
Introduction
Endometriosis is one of the most common benign gynecological diseases in pre-menopausal women, since approximately 10% to 15% of reproductive-aged women experience pelvic endometriosis. It is characterized by pelvic pain, dysmenorrhea, dyspareunia, and infertility. The influence of endometriosis on infertility cannot be ignored, as it is found in 33% of infertile women. As the American Society for Reproductive Medicine proposed, endometriosis should be considered a chronic disease that requires lifelong treatment, with the goal of using medical treatment as much as possible and avoiding invasive procedures such as surgery.
The pharmacological therapy consists of hormonal and non-hormonal agents. The well-known hormonal therapy includes combined oral contraceptives (COC) and the levonorgestrel-releasing intra-uterine system. As a second line, desogestrel progestin-only pills, etonogestrel implants, and gonadotrophin-releasing hormone analogs, and dienogest are available. These agents have been reported to ameliorate pain and reduce disease recurrence. However, because these drugs may interfere with ovulation during the post-operative period that is critical for conception, their use for treating endometriosis has been a controversy. Another shortcoming of hormonal therapy is long-term use and high recurrence rate after discontinuation of treatment. Non-hormonal drugs are receiving attention these days because of their protection of ovulation and the potential to eradicate lesions or alleviate symptoms.
Given the complex pathogenesis of endometriosis, non-hormonal drugs could be categorized into different types. A large number of these non-hormonal drugs target angiogenesis, which is a pre-requisite for endometriosis development according to basic studies. As numerous cytokines and inflammatory factors were suggested to facilitate the development of ectopic endometrial tissues, the therapeutic potential of non-steroidal anti-inflammatory drugs (NSAIDs) and tumor necrosis factor (TNF)-α antagonist has been assessed. Additionally, some natural components have been tested for their potential effects in endometriosis.
The safety and efficacy of most non-hormonal drugs have not been thoroughly evaluated through large clinical trials. The lack of high-quality evidence has left clinicians in the dark when choosing the most suitable treatment. This paper aimed to investigate the effect of non-hormonal therapy approaches on endometriosis that have been tested in clinical trials and explore their value for being implemented in clinical practice or the best combination usage with hormonal drugs.
Literature search strategy
PubMed, Embase, Cochrane Library, SINOMED, ClinicalTrials.gov, and Google Scholar were searched. Additionally, references of each article that met the criteria with regard to the target population and therapy were included. We assessed all studies that evaluated any aspect of non-hormonal therapy for endometriosis using the searched terms “endometriosis” and “non-hormonal therapy.” We included in vitro experiment studies in which animal models were treated with new non-hormonal drug and clinical trials recruiting patients who have been diagnosed with endometriosis according to the criteria of the American Fertility Society and received non-hormonal drug therapy. Then, a final selection was made by FYC and ZXG after reading the full text of included articles. We excluded duplicate studies and removed publications that were not in English. Potentially eligible articles were checked by title and abstract by FYC. Then, a final decision was made by FYC and ZXG after reading the full text of the articles. Studies were independently evaluated by two reviewers (FYC, ZXG). Data on study design, study population, and therapy were extracted. Institutional review board approval was not necessary for this review. The quality of each study was assessed independently by two reviewers (FYC, ZXG), with discussion with a third reviewer (XW) in cases of disagreement. For clinical trials, reviewers used a tool adapted from the Cochrane Risk Of Bias Assessment Tool. The clinical studies consist of randomized controlled trials (RCTs), single intervention study, and retrospective case-control study. Quality assessment measures included evaluation of selection of cohorts, assessment of exposure and outcome, and consideration of confounding variables. Study quality was graded as no risk, moderate risk, high risk, and critical risk of bias using the adapted tool. The outcomes measured varied across the studies, such as endometriotic area or volume, pain scores, and birth rate. Many studies only documented a measure of the targeted index but did not document the odd risk or relative risk. Visual analog score (VAS) of pelvic pain within the confined populations was the most common outcome across studies. During the literature search, 49 articles were identified. After the initial screening, three duplicates, four unrelated documents, and four non-English studies were excluded. In addition, 14 articles with non-available full text were excluded. Twenty-four articles met the inclusion criteria for analysis, including nine animal studies and 15 human studies. Figure 1 shows the selection process. The results are presented by drug category and the related analyses are concluded both in animal studies [Table 1] and clinical trials [Table 2]. The risk of bias for clinical trials is shown in Supplementary Table 1.
Figure 1.
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Flow diagram for study selection.
Table 1.
Studies of agents tested in endometriosis induced in animal models but not in humans.
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Table 2.
Studies of agents tested in human trials.
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NSAIDs
Cyclooxygenase (COX), as a synthetic enzyme for pain mediators, plays a positive role in the development and progression of endometriosis. With additional benefits in relatively sparing the gastrointestinal side effects, COX-2 specific inhibitors were used to alleviate pain and inflammation, such as post-surgery pain, headache, and toothache. The reduction in endometriosis-relevant pain and the anti-proliferative capacity of some COX-2 specific inhibitors have been tested in vivo and in vitro.
Parecoxib
Machado et al have found in an experimental rat model with autotransplanted endometriotic tissue that 3 mg·kg−1·d−1 of parecoxib could inhibit the growth of endometriotic tissue with 80% reduction in volume compared to controls. The decreased expression of vascular endothelial growth factor (VEGF) and polyclonal antibody against VEGF receptor 2 (VEGFR2; Flk-1) and the reduced density of von Willebrand factor-positive vessels in endometriotic tissue indicate that the suppression effect of parecoxib could be achieved through its anti-angiogenic activity.
A mouse model of the endometrium induced by grafting uterine tissue samples to the peritoneum was used to test the effect of telmisartan combined with parecoxib on lesions. Significant reduction in the volume of induced endometrium was observed in the combination therapy compared with single treatment with telmisartan or parecoxib, though they also exhibited inhibitory effect on the growth of endometriotic tissue. In immunohistochemical analysis, the combination therapy was associated with less proliferating Ki67-positive cells and more apoptotic cleaved caspase-3-positive stromal cells as well as decreased microvessel density in the lesions. However, to date, there are limited human studies about the application of parecoxib in clinical environment.
Rofecoxib
Rofecoxib resulted in decreased size of the implanted endometriotic tissue in the same degree compared with leuprolide in a rat model (62.4% vs. 64.3%). The regression and atrophy of induced endometriosis were documented in histologic examination, accompanied by decreased VEGF levels of lesions compared with controls.
A pilot study has evaluated the effect of rofecoxib on endometriosis-induced pelvic pain in 28 women who underwent conservative surgery. Pain symptom scores in 28 patients with stage I or stage II endometriosis were recorded before and after 6 months of treatment, through which rofecoxib demonstrated a higher effect on endometriosis (dysmenorrhea 8:3; dyspareunia 7.5:1.5; and chronic pelvic pain 6:1.5). Additionally, no pain recurrence occurred in the rofecoxib group, while two of 12 in the controls had pain recurrence. However, rofecoxib has been withdrawn from the market because of severe cardiovascular effects in long-term users.
Celecoxib
Y-box-binding protein 1 (YB-1) is a member of DNA- and RNA-binding protein superfamily, which is recently reported to be highly expressed in endometrial epithelial cells of endometriosis. OSU-03012, a celecoxib analog, was demonstrated to inhibit the process of endometriosis by affecting YB-1 related signaling pathway in vitro and in vivo. With exposure to different concentrations of OSU-03012, the proliferation of 12Z human endometriotic epithelial cells was hindered. In the mouse model of Silveira et al, reduced volume of implanted endometriotic lesions was recorded in 100 mg/kg OSU-03012-treated mice, showing 56.4% of shrinking tissue after a 2-week treatment, while the lesion size enlarged by 44.7% in controls. Data on the application of OSU-03012 in humans are still lacking.
A mouse model showed that celecoxib and rosiglitazone, combined or separately, have a therapeutic effect on overall endometriotic growth.
TNF-α antagonist
TNF-α is a secretory factor of active macrophages known to have potent inflammatory cytotoxic and angiogenic characteristics. Several studies have demonstrated higher concentration of TNF-α in the peritoneal fluid of women with endometriosis than those without, as well as a direct correlation between TNF-α level and disease severity.[15,16] Thus, TNF-α blockers were suggested as a promising novel therapy for endometriosis.
Etanercept
A rat model with homologous transplanted endometriotic tissue was used to determine the anti-TNF-α activity of etanercept, a group of immunoglobulin coupled with TNF-α receptor. Moreover, a 31.8% decrease in endometriosis volume was observed in etanercept-treated rats with statistically significant reduction in histologic proliferation scores.
Besides, elevation of TNF-α in peritoneal fluids is associated with infertility induced by endometriosis. A retrospective case-control study was conducted in 68 infertile patients who had endometrioma diagnosed by ultrasound imaging and confirmed by cytopathological examination during oocyte retrieval. Of these patients, 19 patients who were administered with 50 mg of etanercept intramuscularly on the second day of their previous menstrual cycle and then treated by assisted reproduction technique demonstrated statistically higher rate of pregnancy. Although those women did not reach statistically significant elevation in live birth rate (LBR) (P = 0.075), a two-fold increase in LBR was recorded. Another problem in this trial was the lack of information regarding the stages of endometrioma and surgery history.
Pentoxifylline
Pentoxifylline is a methylxanthine with anti-inflammatory property, acting as a phosphodiesterase inhibitor, that has been proposed for treating endometriosis. Cerebrovascular and peripheral vascular diseases, as well as other conditions with defective microcirculation, have been the indication of pentoxifylline treatment for many years.
No statistical significance was observed in overall rates and recurrence rate of pain symptoms in a RCT recruiting 88 infertile patients with endometriosis stages III/IV. The participants received 800 mg pentoxifylline daily and were followed for 12 months. The negative results may be related to high stages of most patients as well as the small sample size. Another study recruited 60 endometriosis-related infertile women with stage I or II and documented that the pentoxifylline-treated group has good results, in which 31% of patients conceived during the 1-year follow-up period compared with 18.5% pregnancy rates in the placebo group. However, the data did not reach statistical significance because of a limited sample size. Furthermore, 40% to 45% of them had additional infertile factors; however, after controlling for all these confounding factors, no significant effect on pregnancy rate was recorded.
Another prospective RCT of the effect of pentoxifylline only comprised infertile women with asymptomatic minimal or mild endometriosis. The results are similar (31% vs. 18.5%), which showed no significant difference in the overall pregnancy rate between the intervention group and the placebo group. To confirm the argument, more rigorous trial eliminating confounding infertile causes was performed, showing a two-fold increase of pregnancy rate in the pentoxifylline group as recorded in controls.
On the basis of anti-inflammatory property of pentoxifylline, a prospective RCT was designed to test the effect of oral pentoxifylline combined with laparoscopic surgery on pelvic pain. Reduction in pain scores was documented most significantly in the second and third months since treatment with 400 mg of pentoxifylline twice a day. No evaluation of pain scores in long-term use was the limitation of the study.
N-palmitoylethanolamine
N-palmitoylethanolamine (N-PEA) is a structural analog of anandamide with anti-inflammatory, immunosuppressive, analgesic, neuroprotective, and antioxidant effects. It inhibits mast cell activation, blocking out the crosstalk between mast cells and nociceptive nerve fibers peripherally. It is also able to downregulate microglial cell behaviors and thus reduces associated central pain hypersensitization.[24,25] Besides endometriosis, N-PEA has been successfully tested for reducing pain due to chronic lumbosciatalgia.
Polydatin, a natural glucoside of resveratrol and a phytoalexin polyphenolic compound found in various plants, such as grapes, berries, and peanuts, downregulates the inflammatory response by inhibiting the synthesis and release of proinflammatory mediators, inducing mast cell degranulation, and modifying eicosanoid synthesis. A nude model has proved its anti-inflammatory activity.
An RCT including 61 patients indicates that the association between micronized N-PEA and transpolydatin is effective in the management of pelvic pain (pelvic pain, dysmenorrhea, and dyspareunia) related to endometriosis after laparoscopy (16/21 vs. 8/20), especially in pelvic pain. Meanwhile, celecoxib decreased pelvic pain more effectively than the association with micronized N-PEA and transpolydatin or placebo (P< 0.001).
In another clinical trial, the efficacy of N-PEA plus transpolydatin was tested in 47 women with endometriosis (ovarian cyst <4 cm or rectovaginal septum nodule <2 cm in instrumental evaluation) before surgery. They were divided into recto-vaginal endometriosis group and ovary endometriosis group. In a 3-month follow-up of their symptoms, chronic pelvic pain, dysmenorrhea, and dyschezia were all ameliorated and significantly associated with time. The effect was more apparent in dysmenorrhea of ovary endometriosis and dyschezia in recto-vaginal endometriosis.
Natural agents
Resveratrol
Resveratrol (3,5,4′-trihydroxystilbene) is a natural phytoestrogen in plants, such as dark grapes and blueberries, synthesized after exposure to ultraviolet radiation. Given its known anti-proliferative activity, various studies were designed to test its effect on endometriosis in vitro and in animal models.[27,30,31]
A mouse model with endometriosis induced by intraperitoneal injection of human endometrial tissue showed an 80% decrease of volume when treated with 6 mg of resveratrol for 18 to 20 days. Ectopic implants by homologous transplantation in mice displayed similar results. In a stepwise manner, some studies showed promising outcomes of the potential usage of resveratrol in women with endometriosis. A pilot open-label trial with 42 patients demonstrated statistically more significant reduction in pelvic pain scores when treated by resveratrol (30 mg/d) combined with drospirenone (DRSP)/ethinyl estradiol (EE) (3 mg/3 μg), compared to the same dose of DRSP/EE treatment alone (pain score: 0.2 vs. 2.1, baseline = 3). Besides, the combination therapy led to complete resolution of dysmenorrhea in 82% of patients, while the single COC group had consistent breakthrough bleeding. After a 2-month treatment, the ectopic endometrium was biopsied via laparoscopic surgery and showed significantly lower expression of aromatase and COX-2 activity, confirming the anti-inflammatory property of resveratrol. No findings about the effect of resveratrol on subsequent pregnancy were reported.
However, a rigorous RCT argued against the benefits of resveratrol. The trial recruited 44 women with endometriosis and was followed for 42 days, indicating no superiority of resveratrol (40 mg/d) + COC therapy (DRSP 3 mg/EE 3 μg) on pain relief compared to COC alone. Despite the negative results, the bias related with the small sample size and short medication time should not be neglected. Some side effects of resveratrol have also been pointed out, such as diplopia (one case), headache (six cases), reduced libido (one case), nausea (one case), breast tenderness (one case), hot flushes (one case), increased uterine bleeding (one case), and increased uterine bleeding (one case). Thus, the safety of resveratrol may need further investigation.
Epigallocatechin-3-gallate (EGCG)
Ricci et al stated that in an endometriosis mice model induced by grafting uterine horns into the peritoneum, treatment with resveratrol or EGCG resulted in shrinkage of endometriotic tissue as well as the inhibited proliferation of endometriotic stromal cells immunochemically. In the same study, human endometrial epithelial cells (EECs) were also cultured in vitro, showing that both resveratrol and EGCG could cause the apoptosis of human EECs through the TdT-mediated dUTP nick-end labeling technique.
Green tea EGCG has been indicated to be effective in inhibition of VEGF expression and thus angiogenesis in endometriosis of an experimental endometriosis model in mice, with significant suppression in the growth of implanted endometriotic tissue.[34,35] Furthermore, EGCG would not affect blood vessel development in ovarian follicles.[34,35] Although there are no data of human trials, a phase II double-blind RCT is currently underway to evaluate the efficacy of 3-month pre-surgical administration of green tea extract (400 mg, twice per day) in patients with endometriosis (NCT02832271). However, poor bioavailability prevented the utilization in humans. A new prodrug, EGCG octaacetate, synthesized to increase the serum level and enhance EGCG stability, may shed light on clinical practice, since it is effective in inhibiting the growth of transplanted endometriosis in a mouse model.
Anti-angiogenesis
Endostatin
Among other anti-angiogenic agents, endostatin, a proteolytic fragment of collagen XVIII, significantly suppressed the growth of endometriotic implants in mice possibly through inhibition of VEGF expression.[37–39] Jiang et al used severe combined immunodeficient mice whose peritoneum was transplanted with human eutopic endometriotic tissue and treated with 2 mg·kg−1·d−1 of recombinant human endostatin YH-16 intraperitoneally. The microscopic examination showed decreased volume of the endometriotic tissue in the endostatin-treated group compared to the control group, with consistent results in hematoxylin-eosin staining and immunochemistry showing decreased microvessel density and lower VEGF expression. A Korean study showed a negative correlation of serum endostatin levels with the development of endometriosis, specifically in early-stage endometriosis patients compared with women without endometriosis, which provides support for the performance of human trials in some degree.
Rapamycin
As a drug targeting the mammalian target of rapamycin pathway to control the cell cycle, cell size, translation initiation, and transcription, rapamycin has aroused interest as a new therapeutic approach in many tumors including endometriosis. The activity of rapamycin in the reduction of implanted endometriosis has been tested and backed by some mice models. Besides, the activity of an oral analog of rapamycin, everolimus, has been assessed in an open-labeled phase 2 study. The study enrolled 35 patients with measurable recurrent endometriosis, who had failed at least one and no more than two prior chemotherapeutic regimens. All these patients had prolonged stable disease status, and 21% of them had a confirmed clinical benefit response at 20 weeks of therapy, indicating an encouraging prospect for clinical administration.
VEGF inhibition
The growth of endometriotic tissue in rich vascularized area implies the role of angiogenesis in the pathogenesis of endometriosis. On the basis of a particularly high VEGF expression in endometriomas, therapy targeting VEGF has been evaluated in a series of recent studies.
Cabergoline
One of the suggestions is that dopamine agonists, such as Cb2, have potential in the inhibition of neoangiogenesis because of its explicit preventive activity in VEGF-VEGF receptor binding. Scarpellini et al recruited ten women with relapsing stage IV endometriosis and provided them with 300 mg/d of thalidomide for 6 to 14 months following 6 months of treatment of goserelin combined with 300 mg/d of thalidomide. None of them complained of endometriosis relapse in the follow-up.
A randomized cohort study with 140 patients was carried out to compare the effect of Cb2 on endometriosis with luteinizing hormone releasing hormone (LHRH). The cut-off value was more than 25% of the reduction in the mean endometrioma diameter. A total of 64.7% of patients who were administered Cb2 met the requirement, compared against only 21.7% in the LHRH group (P< 0.05%). Even with the labels open to both clinicians and patients, ultrasound measures ensured outcomes with some degree of objectivity. Side effects of the new drug therapy should be evaluated before it becomes a new first-line therapy. However, one of the most common adverse events, gastrointestinal symptoms, appeared with higher rates in the Cb2-treated group than in the decapeptyl-treated group (12.5% vs. 0). Other side effects of Cb2, such as cardiac valve regurgitation and metabolic disorders, have largely prevented its clinical usage.
Non-ergot-derived dopamine agonists – quinagolide
Quinagolide is a non-ergot-derived selective D2 dopamine receptor agonist with fewer side effects than Cb2 and does not interfere with pregnancy. Thus, the potential use of quinagolide has been tested in a mouse model with ectopic human endometriotic tissue grafted on peritoneum. The authors found that quinagolide was equally effective as Cb2 in decreasing the size of the endometrioma (39.5% vs. 45.8%, P> 0.05) as well as the expression of VEGF and VEGFR2. As quinagolide is not derived from ergot, it does not produce side effects of ergot derivatives, such as cardiac valve regurgitation. Moreover, it has a much shorter life than carbegoline, making it an attractive alternative to Cb2 in women who wish to become pregnant shortly after the treatment. Notwithstanding, the absence of clinical trials prevents the application of quinagolide in real clinical environment. A phase II clinical trial of 72 patients with endometrioma, deep infiltrating endometriosis, and adenomyosis, which compares the effect of quinagolide in a dose of 1080 μg and placebo, will be finished on November 2020.
Statins
Atorvastatin
Besides the known lipid-lowering activity by inhibition of 3-hydroxy-3-methylglutaryl-coenzyme A (HMG-CoA) reductase, statins are also capable of inhibiting angiogenesis and cell proliferation in the development of endometriosis-like tissue, as tested in vitro and in vivo.[48–50] A rat model study induced by ligation of uterine horns demonstrated 68.2% regression of endometriotic implant area with prescription of atorvastatin at 2.5 mg·kg−1·d−1. A randomized cohort trial with an estimated enrollment of 44 participants in 2008, which determined to compare the effect of atorvastatin, oral contraceptive, or COC and atorvastatin in pain relief and inflammatory status in surgically confirmed endometriosis, is still ongoing.
Simvastatin
Simvastatin is a cholesterol-lowering drug that acts by inhibiting HMG-CoA reductase, resulting in a decrease in mevalonate, a precursor of cholesterol and monocyte chemoattractant protein-1 (MCP-1), which is an important peritoneal inflammatory factor and stimulates the process of endometrial adhesion and proliferation, as well as neovascularization. Bruner-Tran et al also found dose-dependent effect of simvastatin in inhibiting the proliferation of endometrial implants in a nude mouse model that was established by giving external estradiol. Evidence is scarce in clinical environment. However, in a randomized study consisting of only 60 patients with surgically confirmed endometriosis, no significant difference was detected in pain scores of simvastatin-treating patients, relevant to dyspareunia, dysmenorrhea, and pelvic pain, and those of gonadotropin-releasing hormone agonists users in a follow-up of 16 weeks. However, none of the patients accepted second-look laparoscopy to evaluate the size and number of endometriotic lesions after the medication. In a more recent RCT including 40 endometriosis patients, simvastatin was given at a dose of 2 weeks before they received the surgery. The MCP-1 gene and CD68 expression were detected with no significant reduction compared to the no-treated group. It might indicate that a higher dose was needed to take effect, but it would increase the risk of myopathy. To explore further the usage of simvastatin, larger and rigorous clinical trials are awaited.
Rosiglitazone
The peroxisome proliferator-activated receptors (PPARs) ligand rosiglitazone, a known drug for diabetes mellitus, has been shown to have anti-angiogenesis ability through ligand-induced apoptosis in breast cancer. Blitek et al demonstrated that the activation of PPARs affected the expression of genes involved in amino acid and glucose transport, prostaglandin and cytokine synthesis, angiogenesis, and apoptosis in the porcine endometrium. The inhibitory effect for the establishment and development of endometriosis has been illustrated in a rat model with autografted lesions. Besides, another mouse model showed that rosiglitazone did not have any negative effects on reproduction and offspring delivery. Moreover, the safety of rosiglitazone on fertility has been demonstrated in a baboon endometriosis model and it has no effect on the menstrual cycle.
However, Pich et al showed that PPARγ agonist rosiglitazone could activate the proliferation of human metastatic melanoma cells by inducing the intracellular paracrine signaling. Thus, its application potential in real clinical environment was doubted.
Discussion and conclusions
The traditional hormonal therapy for endometriosis has been reported to be ineffective in a group of patients as well as interfering with ovulation. For those who are unresponsive or show unacceptable side effects to hormonal treatment, NSAIDs may be an alternative approach or supplemental therapy. In this review, we discussed the latest non-steroidal therapies for endometriosis categorized by their pharmacological principles.
The most promising one would be pentoxifylline, as this supported by four human trials which showed no negative effects on women's fertility. The adverse events are not reported in these studies, while gastroenterological side effects have been widely mentioned in studies on its use in peripheral vascular diseases. In contrast, most of the other drugs are still restricted to pilot studies or case-control trials, and prospective randomized controlled studies are awaited.
For those currently limited to animal models, EGCG may be the most worthwhile option to be tested in clinical trials because of its significant results and harmlessness on follicles. The effect on fertility needs to be evaluated in animal experiments for the rest of the listed drugs.
One limitation of the study lies in the varied animal models in non-human experiments. In human trials, the heterogeneity of outcome indicators makes comparison between included studies difficult. The time recording pain scores following drug usage varied among human trials. Besides, the evaluating standard, VAS, which depends on participants’ own feelings rather than on objective detecting tools, weakens the credibility of evidence. However, most of the included studies set controls and conformed to strict blinding principles, which provided advantages for our analysis in this sort of topic.
In conclusion, the review exposes the effect of several non-hormonal agents in treating endometriosis, most of which have shown promising results but only been tested in animal models or pilot studies. Randomized controlled studies with more rigorous intervention and homogenous outcomes are needed.
Conflicts of interest
None.
Supplementary Material
Supplemental Digital Content
cm9-132-2984-s001.docx (47.4KB, docx)
Footnotes
How to cite this article: Chen FY, Wang X, Tang RY, Guo ZX, Deng YZJ, Yu Q. New therapeutic approaches for endometriosis besides hormonal therapy. Chin Med J 2019;132:2984–2993. doi: 10.1097/CM9.0000000000000569
References
1.Fukunaga M. Uterus-like mass in the uterine cervix: superficial cervical endometriosis with florid smooth muscle metaplasia? Virchows Arch 2001; 438:302–305.. doi: 10.1097/MD.0000000000004961. [DOI] [PubMed] [Google Scholar]
2.Thomas MD, Berek JS, Rinehart RD. Berek & Novak's gynecology. 15th ed.2011; Philadelphia: Lippincott Williams & Wilkins, 518–519. [Google Scholar]
3.Practice Committee of the American Society for Reproductive Medicine. Treatment of pelvic pain associated with endometriosis: a committee opinion. Fertil Steril 2014; 101:927–935.. doi: 10.1016/j.fertnstert.2014.02.012. [DOI] [PubMed] [Google Scholar]
4.Geoffron S, Cohen J, Sauvan M, Legendre G, Wattier JM, Darai E, et al. Endometriosis medical treatment: hormonal treatment for the management of pain and endometriotic lesions recurrence. CNGOF-HAS endometriosis guidelines. Gynecol Obstet Fertil Senol 2018; 46:231–247.. doi: 10.1016/j.gofs.2018.02.011. [DOI] [PubMed] [Google Scholar]
5.Michel C, Jacques GD, David SG, Jouko KH, John AR, Robert SC, et al. Revised American Society for Reproductive Medicine classification of endometriosis: 1996. Fertil Steril 1997; 67:817–821.. doi: 10.1016/s0015-0282(97)81391-x. [DOI] [PubMed] [Google Scholar]
6.Sterne JA, Hernán MA, Reeves BC, Savović J, Berkman ND, Viswanathan M, et al. ROBINS-I: a tool for assessing risk of bias in non-randomised studies of interventions. BMJ 2016; 355:i4919.doi: 10.1136/bmj.i4919. [DOI] [PMC free article] [PubMed] [Google Scholar]
7.Rudzitis-Auth J, Nickels RM, Menger MD, Laschke MW. Inhibition of cyclooxygenase-2 suppresses the recruitment of endothelial progenitor cells in the microvasculature of endometriotic lesions. Am J Pathol 2018; 188:450–460.. doi: 10.1016/j.ajpath.2017.10.013. [DOI] [PubMed] [Google Scholar]
8.Machado DE, Berardo PT, Landgraf RG, Fernandes PD, Palmero C, Alves LM, et al. A selective cyclooxygenase-2 inhibitor suppresses the growth of endometriosis with an antiangiogenic effect in a rat model. Fertil Steril 2010; 93:2674–2679.. doi: 10.1016/j.fertnstert.2009.11.037. [DOI] [PubMed] [Google Scholar]
9.Nenicu A, Gu Y, Körbel C, Menger MD, Laschke MW. Combination therapy with telmisartan and parecoxib induces regression of endometriotic lesions. Br J Pharmacol 2017; 174:2623–2635.. doi: 10.1111/bph.13874. [DOI] [PMC free article] [PubMed] [Google Scholar]
10.Dogan E, Saygili U, Posaci C, Tuna B, Caliskan S, Altunyurt S, et al. Regression of endometrial explants in rats treated with the cyclooxygenase-2 inhibitor rofecoxib. Fertil Steril 2004; 82: Suppl 3: 1115–1120.. doi: 10.1016/j.fertnstert.2004.06.033. [DOI] [PubMed] [Google Scholar]
11.Cobellis L, Razzi S, De Simone S, Sartini A, Fava A, Danero S, et al. The treatment with a COX-2 specific inhibitor is effective in the management of pain related to endometriosis. Eur J Obstet Gynecol Reprod Biol 2004; 116:100–102.. doi: 10.1016/j.ejogrb.2004.02.007. [DOI] [PubMed] [Google Scholar]
12.Burnier M. The safety of rofecoxib. Expert Opin Drug Saf 2005; 4:491–499.. doi: 10.1517/14740338.4.3.491. [DOI] [PubMed] [Google Scholar]
13.Silveira CGT, Marschner G, Canny GO, Klocke S, Hunold P, Köster F, et al. Disrupting Y-box-binding protein 1 function using OSU-03012 prevents endometriosis progression in in vitro and in vivo models. Reprod Sci 2017; 24:67–76.. doi: 10.1177/1933719116649695. [DOI] [PubMed] [Google Scholar]
14.Olivares C, Ricci A, Bilotas M, Barañao RI, Meresman G. The inhibitory effect of celecoxib and rosiglitazone on experimental endometriosis. Fertil Steril 2011; 96:428–433.. doi: 10.1016/j.fertnstert.2011.05.063. [DOI] [PubMed] [Google Scholar]
15.Chae U, Min JY, Kim SH, Ihm HJ, Oh YS, Park SY, et al. Decreased progesterone receptor B/A ratio in endometrial cells by tumor necrosis factor-alpha and peritoneal fluid from patients with endometriosis. Yonsei Med J 2016; 57:1468–1474.. doi: 10.3349/ymj.2016.57.6.1468. [DOI] [PMC free article] [PubMed] [Google Scholar]
16.Salmeri FM, Lagana AS, Sofo V, Triolo O, Sturlese E, Retto G, et al. Behavior of tumor necrosis factor-alpha and tumor necrosis factor receptor 1/tumor necrosis factor receptor 2 system in mononuclear cells recovered from peritoneal fluid of women with endometriosis at different stages. Reprod Sci 2015; 22:165–172.. doi: 10.1177/1933719114536472. [DOI] [PMC free article] [PubMed] [Google Scholar]
17.Yildirim G, Attar R, Ficicioglu C, Karateke A, Ozkan F, Yesildaglar N. Etanercept causes regression of endometriotic implants in a rat model. Arch Gynecol Obstet 2011; 283:1297–1302.. doi: 10.1007/s00404-010-1543-9. [DOI] [PubMed] [Google Scholar]
18.Ozcelik K, Capar M, Gazi Ucar M, Cakiotar T, Ozcelik F, Tuyan Ilhan T. Are cytokine levels in serum, endometrial tissue, and peritoneal fluid a promising predictor to diagnosis of endometriosis-adenomyosis? Clin Exp Obstet Gynecol 2016; 43:569–572.. doi: 10.12891/ceoj.3022.2016. [PubMed] [Google Scholar]
19.Önalan G, Tohma YA, Zeyneloğlu HB. Effect of etanercept on the success of assisted reproductive technology in patients with endometrioma. Gynecol Obstet Invest 2018; 83:358–364.. doi: 10.1159/000484895. [DOI] [PubMed] [Google Scholar]
20.Alborzi S, Ghotbi S, Parsanezhad ME, Dehbashi S, Alborzi S, Alborzi M. Pentoxifylline therapy after laparoscopic surgery for different stages of endometriosis: a prospective, double-blind, randomized, placebo-controlled study. J Minim Invasive Gynecol 2007; 14:54–58.. doi: 10.1016/j.jmig.2006.06.024. [DOI] [PubMed] [Google Scholar]
21.Balasch J, Creus M, Fabregues F, Carmona F, Martinez-Roman S, Manau D, et al. Pentoxifylline versus placebo in the treatment of infertility associated with minimal or mild endometriosis: a pilot randomized clinical trial. Hum Reprod 1997; 12:2046–2050.. doi: 10.1093/humrep/12.9.2046. [DOI] [PubMed] [Google Scholar]
22.Creus M, Fabregues F, Carmona F, del Pino M, Manau D, Balasch J. Combined laparoscopic surgery and pentoxifylline therapy for treatment of endometriosis-associated infertility: a preliminary trial. Hum Reprod 2008; 23:1910–1916.. doi: 10.1093/humrep/den167. [DOI] [PubMed] [Google Scholar]
23.Kamencic H, Thiel JA. Pentoxifylline after conservative surgery for endometriosis: a randomized, controlled trial. J Minim Invasive Gynecol 2008; 15:62–66.. doi: 10.1016/j.jmig.2007.07.018. [DOI] [PubMed] [Google Scholar]
24.Skaper SD, Facci L, Fusco M, Della Valle MF, Zusso M, Costa B, et al. Palmitoylethanolamide, a naturally occurring disease-modifying agent in neuropathic pain. Inflammopharmacology 2014; 22:79–94.. doi: 10.1007/s10787-013-0191-7. [DOI] [PubMed] [Google Scholar]
25.Mattace Raso G, Russo R, Calignano A, Meli R. Palmitoylethanolamide in CNS health and disease. Pharmacol Res 2014; 86:32–41.. doi: 10.1016/j.phrs.2014.05.006. [DOI] [PubMed] [Google Scholar]
26.Jack BD. Aliamides: a new approach to the treatment of inflammation. Drug News Perspect 1996; 9:93–98.. [Google Scholar]
27.Bruner-Tran KL, Osteen KG, Taylor HS, Anna S, Kaitlin H, Duleba AJ. Resveratrol inhibits development of experimental endometriosis in vivo and reduces endometrial stromal cell invasiveness in vitro. Biol Reprod 2011; 84:106–112.. doi: 10.1095/biolreprod.110.086744. [DOI] [PMC free article] [PubMed] [Google Scholar]
28.Cobellis L, Castaldi MA, Giordano V, Trabucco E, De Franciscis P, Torella M, et al. Effectiveness of the association micronized N-palmitoylethanolamine (PEA)-transpolydatin in the treatment of chronic pelvic pain related to endometriosis after laparoscopic assessment: a pilot study. Eur J Obstet Gynecol Reprod Bio 2011; 158:82–86.. doi: 10.1016/j.ejogrb.2011.04.011. [DOI] [PubMed] [Google Scholar]
29.Giugliano E, Cagnazzo E, Soave I, Lo Monte G, Wenger JM, Marci R. The adjuvant use of N-palmitoylethanolamine and transpolydatin in the treatment of endometriotic pain. Eur J Obstet Gynecol Reprod Biol 2013; 168:209–213.. doi: 10.1016/j.ejogrb.2013.01.009. [DOI] [PubMed] [Google Scholar]
30.Ricci AG, Olivares CN, Bilotas MA, Bastón JI, Singla JJ, Meresman GF, et al. Natural therapies assessment for the treatment of endometriosis. Hum Reprod 2013; 28:178–188.. doi: 10.1093/humrep/des369. [DOI] [PubMed] [Google Scholar]
31.Rudzitisauth J, Menger MD, Laschke MW. Resveratrol is a potent inhibitor of vascularization and cell proliferation in experimental endometriosis. Hum Reprod 2013; 28:1339–1347.. doi: 10.1093/humrep/det031. [DOI] [PubMed] [Google Scholar]
32.Maia H, Jr, Haddad C, Pinheiro N, Casoy J. Advantages of the association of resveratrol with oral contraceptives for management of endometriosis-related pain. Int J Womens Health 2012; 4:543–549.. doi: 10.2147/IJWH.S36825. [DOI] [PMC free article] [PubMed] [Google Scholar]
33.da Silva DM, Gross LA, Neto EPG, Lessey BA, Savaris RF. The use of resveratrol as an adjuvant treatment of pain in endometriosis: a randomized clinical trial. J Endocr Soc 2017; 1:359–369.. doi: 10.1210/js.2017-00053. [DOI] [PMC free article] [PubMed] [Google Scholar]
34.Laschke MW, Schwender C, Scheuer C, Vollmar B, Menger MD. Epigallocatechin-3-gallate inhibits estrogen-induced activation of endometrial cells in vitro and causes regression of endometriotic lesions in vivo. Hum Reprod 2008; 23:2308–2318.. doi: 10.1093/humrep/den245. [DOI] [PubMed] [Google Scholar]
35.Wang J, Man GCW, Chan TH, Kwong J, Wang CC. A prodrug of green tea polyphenol (–)-epigallocatechin-3-gallate (Pro-EGCG) serves as a novel angiogenesis inhibitor in endometrial cancer. Cancer Lett 2018; 412:10–20.. doi: 10.1016/j.canlet.2017.09.054. [DOI] [PubMed] [Google Scholar]
36.Wang CC, Xu H, Man GCW, Zhang T, Chu KO, Chu CY, et al. Prodrug of green tea epigallocatechin-3-gallate (Pro-EGCG) as a potent anti-angiogenesis agent for endometriosis in mice. Angiogenesis 2013; 16:59–69.. doi: 10.1007/s10456-012-9299-4. [DOI] [PubMed] [Google Scholar]
37.Jiang HQ, Li YL, Jie Z. Effect of recombinant human endostatin on endometriosis in mice. Chin Med J 2007; 120:1241–1246.. doi: 10.1097/00029330-200707020-00007. [PubMed] [Google Scholar]
38.Becker CM, Sampson DA, Rupnick MA, Rohan RM, Efstathiou JA, Short SM, et al. Endostatin inhibits the growth of endometriotic lesions but does not affect fertility. Fertil Steril 2005; 84:1144–1155.. doi: 10.1016/j.fertnstert.2005.04.040. [DOI] [PubMed] [Google Scholar]
39.Ma Y, He Y. Study of an antiangiogenesis gene therapy with endostatin on endometriosis in the nude mouse model. Clin Exp Obstet Gynecol 2014; 41:328–334.. doi: 10.12891/ceog17172014. [PubMed] [Google Scholar]
40.Kim JG, Kim JY, Jee BC, Suh CS, Kim SH, Choi YM. Association between endometriosis and polymorphisms in endostatin and vascular endothelial growth factor and their serum levels in Korean women. Fertil Steril 2008; 89:243–245.. doi: 10.1016/j.fertnstert.2007.02.023. [DOI] [PubMed] [Google Scholar]
41.Laschke M, Elitzsch A, Scheuer C, Holstein J, Vollmar B, Menger M. Rapamycin induces regression of endometriotic lesions by inhibiting neovascularization and cell proliferation. Br J Pharmacol 2006; 149:137–144.. doi: 10.1038/sj.bjp.0706857. [DOI] [PMC free article] [PubMed] [Google Scholar]
42.Slomovitz BM, Lu KH, Johnston T, Coleman RL, Munsell M, Broaddus RR, et al. A phase 2 study of the oral mammalian target of rapamycin inhibitor, everolimus, in patients with recurrent endometrial carcinoma. Cancer 2010; 116:5415–5419.. doi: 10.1002/cncr.25515. [DOI] [PMC free article] [PubMed] [Google Scholar]
43.Novella-Maestre E, Carda C, Ruiz-Sauri A, Garcia-Velasco JA, Simon C, Pellicer A. Identification and quantification of dopamine receptor 2 in human eutopic and ectopic endometrium: a novel molecular target for endometriosis therapy. Biol Reprod 2010; 83:866–873.. doi: 10.1095/biolreprod.110.084392. [DOI] [PubMed] [Google Scholar]
44.Scarpellini F, Sbracia M, Lecchini S, Scarpellini L. Anti-angiogenesis treatment with thalidomide in endometriosis: a pilot study. Fertil Steril 2002; 78:S87.doi: 10.1016/S0015-0282(02)03609-9. [Google Scholar]
45.Hamid AMSA, Madkour WA, Moawad A, Elzaher MA, Roberts MP. Does cabergoline help in decreasing endometrioma size compared to LHRH agonist? A prospective randomized study. Arch Gynecol Obstet 2014; 290:677–682.. doi: 10.1007/s00404-014-3242-4. [DOI] [PubMed] [Google Scholar]
46.Delgado-Rosas F, Gomez R, Ferrero H, Gaytan F, Garcia-Velasco J, Simon C, et al. The effects of ergot and non-ergot-derived dopamine agonists in an experimental mouse model of endometriosis. Reproduction 2011; 142:745–755.. doi: 10.1530/rep-11-0223. [DOI] [PubMed] [Google Scholar]
Quinagolide Vaginal Ring on Lesion Reduction Assessed by MRI in Women with Andometriosis/Adenomyosis. ClinicalTrials.gov, 2018. Available from: [Last accessed on October 24, 2019] [Google Scholar]
48.Esfandiari N, Khazaei M, Ai J, Bielecki R, Gotlieb L, Ryan E, et al. Effect of a statin on an in vitro model of endometriosis. Fertil Steril 2007; 87:257–262.. doi: 10.1016/j.fertnstert.2006.06.040. [DOI] [PubMed] [Google Scholar]
49.Nasu K, Yuge A, Tsuno A, Narahara H. Simvastatin inhibits the proliferation and the contractility of human endometriotic stromal cells: a promising agent for the treatment of endometriosis. Fertil Steril 2009; 92:2097–2099.. doi: 10.1016/j.fertnstert.2009.06.055. [DOI] [PubMed] [Google Scholar]
50.Piotrowski PC, Kwintkiewicz J, Rzepczynska IJ, Seval Y, Cakmak H, Arici A, et al. Statins inhibit growth of human endometrial stromal cells independently of cholesterol availability. Biol Reprod 2006; 75:107–111.. doi: 10.1095/biolreprod.106.051763. [DOI] [PubMed] [Google Scholar]
51.Oktem M, Esinler I, Eroglu D, Haberal N, Bayraktar N, Zeyneloglu H. High-dose atorvastatin causes regression of endometriotic implants: a rat model. Hum Reprod 2007; 22:1474–1480.. doi: 10.1093/humrep/del505. [DOI] [PubMed] [Google Scholar]
Efficacy Study of Atorvastatin in Pelvic Pain Relief in Women With Endometriosis. ClinicalTrials.gov, 2018. Available from: [Last accessed on October 24, 2019] [Google Scholar]
53.Zhang GQ, Tao YK, Bai YP, Yan ST, Zhao SP. Inhibitory effects of simvastatin on oxidized low-density lipoprotein-induced endoplasmic reticulum stress and apoptosis in vascular endothelial cells. Chin Med J 2018; 131:950–955.. doi: 10.4103/0366-6999.229891. [DOI] [PMC free article] [PubMed] [Google Scholar]
54.Bruner-Tran KL, Osteen KG, Duleba AJ. Simvastatin protects against the development of endometriosis in a nude mouse model. J Clin Endocrinol Metab 2009; 94:2489–2494.. doi: 10.1210/jc.2008-2802. [DOI] [PMC free article] [PubMed] [Google Scholar]
55.Almassinokiani F, Mehdizadeh A, Sariri E, Rezaei M, Almasi A, Akbari H, et al. Effects of simvastatin in prevention of pain recurrences after surgery for endometriosis. Med Sci Monit 2013; 19:534–539.. doi: 10.12659/msm.883967. [DOI] [PMC free article] [PubMed] [Google Scholar]
56.Waiyaput W, Pumipichet S, Weerakiet S, Rattanasiri S, Sophonsritsuk A. Effect of simvastatin on monocyte chemoattractant protein-1 expression in endometriosis patients: a randomized controlled trial. BMC Women Health 2017; 17:89.doi: 10.1186/s12905-017-0446-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
57.Yee LD, Williams N, Wen P, Young DC, Lester J, Johnson MV, et al. Pilot study of rosiglitazone therapy in women with breast cancer: effects of short-term therapy on tumor tissue and serum markers. Clin Cancer Res 2007; 13:246–252.. doi: 10.1158/1078-0432.CCR-06-1947. [DOI] [PubMed] [Google Scholar]
58.Blitek A, Szymanska M. Regulation of expression and role of peroxisome proliferator-activated receptors (PPARs) in luminal epithelial and stromal cells of the porcine endometrium. Theriogenology 2019; 127:88–101.. doi: 10.1016/j.theriogenology.2019.01.002. [DOI] [PubMed] [Google Scholar]
59.Demirturk F, Aytan H, Caliskan AC, Aytan P, Koseoglu DR. Effect of peroxisome proliferator-activated receptor-gamma agonist rosiglitazone on the induction of endometriosis in an experimental rat model. J Soc Gynecol Investig 2006; 13:58–62.. doi: 10.1016/j.jsgi.2005.10.002. [DOI] [PubMed] [Google Scholar]
60.Lebovic DI, Mwenda JM, Chai DC, Mueller MD, Santi A, Fisseha S, et al. PPAR-gamma receptor ligand induces regression of endometrial explants in baboons: a prospective, randomized, placebo-and drug-controlled study. Fertil Steril 2007; 88:1108–1119.. doi: 10.1016/j.fertnstert.2006.12.072. [DOI] [PMC free article] [PubMed] [Google Scholar]
61.Pich C, Meylan P, Mastelic-Gavillet B, Nguyen TN, Loyon R, Trang BK, et al. Induction of paracrine signaling in metastatic melanoma cells by PPARgamma agonist rosiglitazone activates stromal cells and enhances tumor growth. Cancer Res 2018; 78:6447–6461.. doi: 10.1158/0008-5472.can-18-0912. [DOI] [PubMed] [Google Scholar]
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Abstract
Introduction
Literature search strategy
NSAIDs
TNF-α antagonist
Natural agents
Anti-angiogenesis
Statins
Rosiglitazone
Conflicts of interest
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