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20,600 | The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this un... | https://en.wikipedia.org/wiki?curid=26513034 |
20,601 | Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit. According to one legend, Hippasus of Metapontum ("ca." 470 B.C.) was drowned at sea for making known the existe... | https://en.wikipedia.org/wiki?curid=26513034 |
20,602 | Note that "r" is defined to be a positive number or zero but "x" and "y" can be negative as well as positive. Geometrically "r" is the distance of the "z" from zero or the origin "O" in the complex plane. | https://en.wikipedia.org/wiki?curid=26513034 |
20,603 | This can be generalised to find the distance between two points, "z" and "z" say. The required distance is given by | https://en.wikipedia.org/wiki?curid=26513034 |
20,604 | The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If and are points in the plane, then the distance between them, also called the Euclidean distance, is given by | https://en.wikipedia.org/wiki?curid=26513034 |
20,605 | More generally, in Euclidean "n"-space, the Euclidean distance between two points, formula_57 and formula_58, is defined, by generalization of the Pythagorean theorem, as: | https://en.wikipedia.org/wiki?curid=26513034 |
20,606 | If instead of Euclidean distance, the square of this value (the squared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates: | https://en.wikipedia.org/wiki?curid=26513034 |
20,607 | The squared form is a smooth, convex function of both points, and is widely used in optimization theory and statistics, forming the basis of least squares. | https://en.wikipedia.org/wiki?curid=26513034 |
20,608 | If Cartesian coordinates are not used, for example, if polar coordinates are used in two dimensions or, in more general terms, if curvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the st... | https://en.wikipedia.org/wiki?curid=26513034 |
20,609 | Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as: | https://en.wikipedia.org/wiki?curid=26513034 |
20,610 | using the trigonometric product-to-sum formulas. This formula is the law of cosines, sometimes called the generalized Pythagorean theorem. From this result, for the case where the radii to the two locations are at right angles, the enclosed angle and the form corresponding to Pythagoras' theorem is regained: formula_64... | https://en.wikipedia.org/wiki?curid=26513034 |
20,611 | In a right triangle with sides "a", "b" and hypotenuse "c", trigonometry determines the sine and cosine of the angle "θ" between side "a" and the hypotenuse as: | https://en.wikipedia.org/wiki?curid=26513034 |
20,612 | where the last step applies Pythagoras' theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the ... | https://en.wikipedia.org/wiki?curid=26513034 |
20,613 | with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity. | https://en.wikipedia.org/wiki?curid=26513034 |
20,614 | This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions. | https://en.wikipedia.org/wiki?curid=26513034 |
20,615 | The Pythagorean theorem generalizes beyond the areas of squares on the three sides to any similar figures. This was known by Hippocrates of Chios in the 5th century BC, and was included by Euclid in his "Elements": | https://en.wikipedia.org/wiki?curid=26513034 |
20,616 | If one erects similar figures (see Euclidean geometry) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side. | https://en.wikipedia.org/wiki?curid=26513034 |
20,617 | This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are "a:b:c"). While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figure... | https://en.wikipedia.org/wiki?curid=26513034 |
20,618 | The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas "A", "B" and "C" are erected on sides with corresponding lengths "a", "b" and... | https://en.wikipedia.org/wiki?curid=26513034 |
20,619 | Conversely, if we can prove that "A" + "B" = "C" for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle "C" on its hypotenuse, and two similar right triangles ("A"... | https://en.wikipedia.org/wiki?curid=26513034 |
20,620 | The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines: | https://en.wikipedia.org/wiki?curid=26513034 |
20,621 | When formula_73 is formula_77 radians or 90°, then formula_78, and the formula reduces to the usual Pythagorean theorem. | https://en.wikipedia.org/wiki?curid=26513034 |
20,622 | At any selected angle of a general triangle of sides "a, b, c", inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled "c". Inscribing the isosceles triangle forms triangle "CAD" with angle θ opposite side "b"... | https://en.wikipedia.org/wiki?curid=26513034 |
20,623 | As the angle θ approaches /2, the base of the isosceles triangle narrows, and lengths "r" and "s" overlap less and less. When θ = /2, "ADB" becomes a right triangle, "r" + "s" = "c", and the original Pythagorean theorem is regained. | https://en.wikipedia.org/wiki?curid=26513034 |
20,624 | One proof observes that triangle "ABC" has the same angles as triangle "CAD", but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, "ABC" is similar to the reflection of "CAD", the triangle "DAC" i... | https://en.wikipedia.org/wiki?curid=26513034 |
20,625 | The theorem remains valid if the angle formula_83 is obtuse so the lengths "r" and "s" are non-overlapping. | https://en.wikipedia.org/wiki?curid=26513034 |
20,626 | Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum... | https://en.wikipedia.org/wiki?curid=26513034 |
20,627 | The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base "b" and height "h". However, the left green parallelogram also has the same area as the left green ... | https://en.wikipedia.org/wiki?curid=26513034 |
20,628 | In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal "BD" is found from Pythagoras' theorem as: | https://en.wikipedia.org/wiki?curid=26513034 |
20,629 | where these three sides form a right triangle. Using horizontal diagonal "BD" and the vertical edge "AB", the length of diagonal "AD" then is found by a second application of Pythagoras' theorem as: | https://en.wikipedia.org/wiki?curid=26513034 |
20,630 | This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components {v} (the three mutually perpendicular sides): | https://en.wikipedia.org/wiki?curid=26513034 |
20,631 | This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes. | https://en.wikipedia.org/wiki?curid=26513034 |
20,632 | A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of th... | https://en.wikipedia.org/wiki?curid=26513034 |
20,633 | This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increa... | https://en.wikipedia.org/wiki?curid=26513034 |
20,634 | The Pythagorean theorem can be generalized to inner product spaces, which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces. For example, a function may be considered as a vector with infinitely many components in an inner product space, as in functional analysis. | https://en.wikipedia.org/wiki?curid=26513034 |
20,635 | In an inner product space, the concept of perpendicularity is replaced by the concept of orthogonality: two vectors v and w are orthogonal if their inner product formula_88 is zero. The inner product is a generalization of the dot product of vectors. The dot product is called the "standard" inner product or the "Euclid... | https://en.wikipedia.org/wiki?curid=26513034 |
20,636 | In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have | https://en.wikipedia.org/wiki?curid=26513034 |
20,637 | Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum v + w. This form of the Pythagorean theorem is a consequence of the properties of the inner product: | https://en.wikipedia.org/wiki?curid=26513034 |
20,638 | A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the "parallelogram law" : | https://en.wikipedia.org/wiki?curid=26513034 |
20,639 | which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is "ipso facto" a norm corresponding to an inner product. | https://en.wikipedia.org/wiki?curid=26513034 |
20,640 | The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If v, v, ..., v are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on solid geometry) results in th... | https://en.wikipedia.org/wiki?curid=26513034 |
20,641 | Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an "m"-dimensional set of objects in one or more parallel "m"-dimensional flats in "n"-dimensional Euclidean space is equal to the sum of the squares o... | https://en.wikipedia.org/wiki?curid=26513034 |
20,642 | The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postu... | https://en.wikipedia.org/wiki?curid=26513034 |
20,643 | do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say "a", "b", and "c") bounding an octant of the unit sphere have length equal to /2, and all its angles are right angles, which violates the Pythagorean theorem because | https://en.wikipedia.org/wiki?curid=26513034 |
20,644 | Here two cases of non-Euclidean geometry are considered—spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines. | https://en.wikipedia.org/wiki?curid=26513034 |
20,645 | However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say "A"+"B" = "C". The sides are then related as follows: the sum of the areas of the circles with diameters "a" ... | https://en.wikipedia.org/wiki?curid=26513034 |
20,646 | For any right triangle on a sphere of radius "R" (for example, if γ in the figure is a right angle), with sides "a", "b", "c", the relation between the sides takes the form: | https://en.wikipedia.org/wiki?curid=26513034 |
20,647 | This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles: | https://en.wikipedia.org/wiki?curid=26513034 |
20,648 | By expressing the Maclaurin series for the cosine function as an asymptotic expansion with the remainder term in big O notation, | https://en.wikipedia.org/wiki?curid=26513034 |
20,649 | it can be shown that as the radius "R" approaches infinity and the arguments "a/R", "b/R", and "c/R" tend to zero, the spherical relation between the sides of a right triangle approaches the Euclidean form of the Pythagorean theorem. Substituting the asymptotic expansion for each of the cosines into the spherical relat... | https://en.wikipedia.org/wiki?curid=26513034 |
20,650 | The constants "a", "b", and "c" have been absorbed into the big "O" remainder terms since they are independent of the radius "R". This asymptotic relationship can be further simplified by multiplying out the bracketed quantities, cancelling the ones, multiplying through by −2, and collecting all the error terms togethe... | https://en.wikipedia.org/wiki?curid=26513034 |
20,651 | After multiplying through by "R", the Euclidean Pythagorean relationship "c" = "a" + "b" is recovered in the limit as the radius "R" approaches infinity (since the remainder term tends to zero): | https://en.wikipedia.org/wiki?curid=26513034 |
20,652 | For small right triangles ("a", "b" « "R"), the cosines can be eliminated to avoid loss of significance, giving | https://en.wikipedia.org/wiki?curid=26513034 |
20,653 | In a hyperbolic space with uniform Gaussian curvature −1/"R", for a right triangle with legs "a", "b", and hypotenuse "c", the relation between the sides takes the form: | https://en.wikipedia.org/wiki?curid=26513034 |
20,654 | where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: | https://en.wikipedia.org/wiki?curid=26513034 |
20,655 | By using the Maclaurin series for the hyperbolic cosine, , it can be shown that as a hyperbolic triangle becomes very small (that is, as "a", "b", and "c" all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem. | https://en.wikipedia.org/wiki?curid=26513034 |
20,656 | For small right triangles ("a", "b" « "R"), the hyperbolic cosines can be eliminated to avoid loss of significance, giving | https://en.wikipedia.org/wiki?curid=26513034 |
20,657 | For any uniform curvature "K" (positive, zero, or negative), in very small right triangles (|"K"|"a", |"K"|"b" « 1) with hypotenuse "c", it can be shown that | https://en.wikipedia.org/wiki?curid=26513034 |
20,658 | The Pythagorean theorem applies to infinitesimal triangles seen in differential geometry. In three dimensional space, the distance between two infinitesimally separated points satisfies | https://en.wikipedia.org/wiki?curid=26513034 |
20,659 | with "ds" the element of distance and ("dx", "dy", "dz") the components of the vector separating the two points. Such a space is called a Euclidean space. However, in Riemannian geometry, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) take... | https://en.wikipedia.org/wiki?curid=26513034 |
20,660 | which is called the metric tensor. (Sometimes, by abuse of language, the same term is applied to the set of coefficients .) It may be a function of position, and often describes curved space. A simple example is Euclidean (flat) space expressed in curvilinear coordinates. For example, in polar coordinates: | https://en.wikipedia.org/wiki?curid=26513034 |
20,661 | There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period (20th... | https://en.wikipedia.org/wiki?curid=26513034 |
20,662 | Written between 2000 and 1786 BC, the Egyptian Middle Kingdom "Berlin Papyrus 6619" includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet "Plimpton 322", written between 1790 and 1750 BC during the reign of King Hammurabi the Great, cont... | https://en.wikipedia.org/wiki?curid=26513034 |
20,663 | In India, the "Baudhayana Shulba Sutra", the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the "Apastamba Shulba Sutra"... | https://en.wikipedia.org/wiki?curid=26513034 |
20,664 | Byzantine Neoplatonic philosopher and mathematician Proclus, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato, the other to Pythagoras", for generating special Pythagorean triples. The rule attributed to Pythagoras () starts from an odd number and produces a triple with leg... | https://en.wikipedia.org/wiki?curid=26513034 |
20,665 | With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text "Zhoubi Suanjing" (周髀算经), ("The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven") gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Go... | https://en.wikipedia.org/wiki?curid=26513034 |
20,666 | Alexander Graham Bell (, born Alexander Bell; March 3, 1847 – August 2, 1922) was a Scottish-born inventor, scientist and engineer who is credited with patenting the first practical telephone. He also co-founded the American Telephone and Telegraph Company (AT&T) in 1885. | https://en.wikipedia.org/wiki?curid=852 |
20,667 | Bell's father, grandfather, and brother had all been associated with work on elocution and speech, and both his mother and wife were deaf; profoundly influencing Bell's life's work. His research on hearing and speech further led him to experiment with hearing devices which eventually culminated in Bell being awarded th... | https://en.wikipedia.org/wiki?curid=852 |
20,668 | Many other inventions marked Bell's later life, including groundbreaking work in optical telecommunications, hydrofoils, and aeronautics. Bell also had a strong influence on the National Geographic Society and its magazine while serving as the second president from January 7, 1898, until 1903. | https://en.wikipedia.org/wiki?curid=852 |
20,669 | Bell was born in Edinburgh, Scotland, on March 3, 1847. The family home was at South Charlotte Street, and has a stone inscription marking it as Bell's birthplace. He had two brothers: Melville James Bell (1845–1870) and Edward Charles Bell (1848–1867), both of whom would die of tuberculosis. His father was Alexander M... | https://en.wikipedia.org/wiki?curid=852 |
20,670 | As a child, Bell displayed a curiosity about his world; he gathered botanical specimens and ran experiments at an early age. His best friend was Ben Herdman, a neighbour whose family operated a flour mill. At the age of 12, Bell built a homemade device that combined rotating paddles with sets of nail brushes, creating ... | https://en.wikipedia.org/wiki?curid=852 |
20,671 | From his early years, Bell showed a sensitive nature and a talent for art, poetry, and music that was encouraged by his mother. With no formal training, he mastered the piano and became the family's pianist. Despite being normally quiet and introspective, he revelled in mimicry and "voice tricks" akin to ventriloquism ... | https://en.wikipedia.org/wiki?curid=852 |
20,672 | His family was long associated with the teaching of elocution: his grandfather, Alexander Bell, in London, his uncle in Dublin, and his father, in Edinburgh, were all elocutionists. His father published a variety of works on the subject, several of which are still well known, especially his "The Standard Elocutionist" ... | https://en.wikipedia.org/wiki?curid=852 |
20,673 | As a young child, Bell, like his brothers, received his early schooling at home from his father. At an early age, he was enrolled at the Royal High School, Edinburgh, Scotland, which he left at the age of 15, having completed only the first four forms. His school record was undistinguished, marked by absenteeism and la... | https://en.wikipedia.org/wiki?curid=852 |
20,674 | His father encouraged Bell's interest in speech and, in 1863, took his sons to see a unique automaton developed by Sir Charles Wheatstone based on the earlier work of Baron Wolfgang von Kempelen. The rudimentary "mechanical man" simulated a human voice. Bell was fascinated by the machine and after he obtained a copy of... | https://en.wikipedia.org/wiki?curid=852 |
20,675 | Intrigued by the results of the automaton, Bell continued to experiment with a live subject, the family's Skye Terrier, Trouve. After he taught it to growl continuously, Bell would reach into its mouth and manipulate the dog's lips and vocal cords to produce a crude-sounding "Ow ah oo ga ma ma". With little convincing,... | https://en.wikipedia.org/wiki?curid=852 |
20,676 | At age 19, Bell wrote a report on his work and sent it to philologist Alexander Ellis, a colleague of his father. Ellis immediately wrote back indicating that the experiments were similar to existing work in Germany, and also lent Bell a copy of Hermann von Helmholtz's work, "The Sensations of Tone as a Physiological B... | https://en.wikipedia.org/wiki?curid=852 |
20,677 | Dismayed to find that groundbreaking work had already been undertaken by Helmholtz who had conveyed vowel sounds by means of a similar tuning fork "contraption", Bell pored over the German scientist's book. Working from his own erroneous mistranslation of a French edition, Bell fortuitously then made a deduction that w... | https://en.wikipedia.org/wiki?curid=852 |
20,678 | In 1865, when the Bell family moved to London, Bell returned to Weston House as an assistant master and, in his spare hours, continued experiments on sound using a minimum of laboratory equipment. Bell concentrated on experimenting with electricity to convey sound and later installed a telegraph wire from his room in S... | https://en.wikipedia.org/wiki?curid=852 |
20,679 | Helping his father in Visible Speech demonstrations and lectures brought Bell to Susanna E. Hull's private school for the deaf in South Kensington, London. His first two pupils were deaf-mute girls who made remarkable progress under his tutelage. While his older brother seemed to achieve success on many fronts includin... | https://en.wikipedia.org/wiki?curid=852 |
20,680 | In 1870, 23-year-old Bell travelled with his parents and his brother's widow, Caroline Margaret Ottaway, to Paris, Ontario, to stay with Thomas Henderson, a Baptist minister and family friend. The Bell family soon purchased a farm of at Tutelo Heights (now called Tutela Heights), near Brantford, Ontario. The property c... | https://en.wikipedia.org/wiki?curid=852 |
20,681 | At the homestead, Bell set up his own workshop in the converted carriage house near to what he called his "dreaming place", a large hollow nestled in trees at the back of the property above the river. Despite his frail condition upon arriving in Canada, Bell found the climate and environs to his liking, and rapidly imp... | https://en.wikipedia.org/wiki?curid=852 |
20,682 | After setting up his workshop, Bell continued experiments based on Helmholtz's work with electricity and sound. He also modified a melodeon (a type of pump organ) so that it could transmit its music electrically over a distance. Once the family was settled in, both Bell and his father made plans to establish a teaching... | https://en.wikipedia.org/wiki?curid=852 |
20,683 | Bell's father was invited by Sarah Fuller, principal of the Boston School for Deaf Mutes (later to become the public Horace Mann School for the Deaf) to introduce the Visible Speech System by providing training for Fuller's instructors, but he declined the post in favour of his son. Travelling to Boston in April 1871, ... | https://en.wikipedia.org/wiki?curid=852 |
20,684 | Returning home to Brantford after six months abroad, Bell continued his experiments with his "harmonic telegraph". The basic concept behind his device was that messages could be sent through a single wire if each message was transmitted at a different pitch, but work on both the transmitter and receiver was needed. | https://en.wikipedia.org/wiki?curid=852 |
20,685 | Unsure of his future, he contemplated returning to London to complete his studies, but decided to return to Boston as a teacher. His father helped him set up his private practice by contacting Gardiner Greene Hubbard, the president of the Clarke School for the Deaf for a recommendation. Teaching his father's system, in... | https://en.wikipedia.org/wiki?curid=852 |
20,686 | Throughout his lifetime, Bell sought to integrate the deaf and hard of hearing with the hearing world. Bell encouraged speech therapy and lip reading over sign language. He outlined this in a 1898 paper detailing his belief that with resources and effort, the deaf could be taught to read lips and speak (known as oralis... | https://en.wikipedia.org/wiki?curid=852 |
20,687 | In 1872, Bell became professor of Vocal Physiology and Elocution at the Boston University School of Oratory. During this period, he alternated between Boston and Brantford, spending summers in his Canadian home. At Boston University, Bell was "swept up" by the excitement engendered by the many scientists and inventors ... | https://en.wikipedia.org/wiki?curid=852 |
20,688 | Deciding to give up his lucrative private Boston practice, Bell retained only two students, six-year-old "Georgie" Sanders, deaf from birth, and 15-year-old Mabel Hubbard. Each pupil would play an important role in the next developments. George's father, Thomas Sanders, a wealthy businessman, offered Bell a place to st... | https://en.wikipedia.org/wiki?curid=852 |
20,689 | By 1874, Bell's initial work on the harmonic telegraph had entered a formative stage, with progress made both at his new Boston "laboratory" (a rented facility) and at his family home in Canada a big success. While working that summer in Brantford, Bell experimented with a "phonautograph", a pen-like machine that could... | https://en.wikipedia.org/wiki?curid=852 |
20,690 | In 1874, telegraph message traffic was rapidly expanding and in the words of Western Union President William Orton, had become "the nervous system of commerce". Orton had contracted with inventors Thomas Edison and Elisha Gray to find a way to send multiple telegraph messages on each telegraph line to avoid the great c... | https://en.wikipedia.org/wiki?curid=852 |
20,691 | In March 1875, Bell and Pollok visited the scientist Joseph Henry, who was then director of the Smithsonian Institution, and asked Henry's advice on the electrical multi-reed apparatus that Bell hoped would transmit the human voice by telegraph. Henry replied that Bell had "the germ of a great invention". When Bell sai... | https://en.wikipedia.org/wiki?curid=852 |
20,692 | With financial support from Sanders and Hubbard, Bell hired Thomas Watson as his assistant, and the two of them experimented with acoustic telegraphy. On June 2, 1875, Watson accidentally plucked one of the reeds and Bell, at the receiving end of the wire, heard the overtones of the reed; overtones that would be necess... | https://en.wikipedia.org/wiki?curid=852 |
20,693 | In 1875, Bell developed an acoustic telegraph and drew up a patent application for it. Since he had agreed to share U.S. profits with his investors Gardiner Hubbard and Thomas Sanders, Bell requested that an associate in Ontario, George Brown, attempt to patent it in Britain, instructing his lawyers to apply for a pate... | https://en.wikipedia.org/wiki?curid=852 |
20,694 | Meanwhile, Elisha Gray was also experimenting with acoustic telegraphy and thought of a way to transmit speech using a water transmitter. On February 14, 1876, Gray filed a caveat with the U.S. Patent Office for a telephone design that used a water transmitter. That same morning, Bell's lawyer filed Bell's application ... | https://en.wikipedia.org/wiki?curid=852 |
20,695 | Bell's patent 174,465, was issued to Bell on March 7, 1876, by the U.S. Patent Office. Bell's patent covered "the method of, and apparatus for, transmitting vocal or other sounds telegraphically ... by causing electrical undulations, similar in form to the vibrations of the air accompanying the said vocal or other soun... | https://en.wikipedia.org/wiki?curid=852 |
20,696 | On March 10, 1876, three days after his patent was issued, Bell succeeded in getting his telephone to work, using a liquid transmitter similar to Gray's design. Vibration of the diaphragm caused a needle to vibrate in the water, varying the electrical resistance in the circuit. When Bell spoke the sentence "Mr. Watson—... | https://en.wikipedia.org/wiki?curid=852 |
20,697 | Although Bell was, and still is, accused of stealing the telephone from Gray, Bell used Gray's water transmitter design only after Bell's patent had been granted, and only as a proof of concept scientific experiment, to prove to his own satisfaction that intelligible "articulate speech" (Bell's words) could be electric... | https://en.wikipedia.org/wiki?curid=852 |
20,698 | The question of priority for the variable resistance feature of the telephone was raised by the examiner before he approved Bell's patent application. He told Bell that his claim for the variable resistance feature was also described in Gray's caveat. Bell pointed to a variable resistance device in his previous applica... | https://en.wikipedia.org/wiki?curid=852 |
20,699 | The patent examiner, Zenas Fisk Wilber, later stated in an affidavit that he was an alcoholic who was much in debt to Bell's lawyer, Marcellus Bailey, with whom he had served in the Civil War. He claimed he showed Gray's patent caveat to Bailey. Wilber also claimed (after Bell arrived in Washington D.C. from Boston) th... | https://en.wikipedia.org/wiki?curid=852 |
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