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may be understood by dividing the formula_78 sequences into two non-overlapping sets where all sequences either begin with 1 or 2:
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Excluding the first element, the remaining terms in each sequence sum to formula_91 or formula_92 and the cardinality of each set is formula_93 or formula_94 giving a total of formula_95 sequences, showing this is equal to formula_78.
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In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the "n"th is equal to the ("n" + 2)-nd Fibonacci number minus 1. In symbols:
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This may be seen by dividing all sequences summing to formula_98 based on the location of the first 2. Specifically, each set consists of those sequences that start formula_99 until the last two sets formula_100 each with cardinality 1.
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A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
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In words, the sum of the first Fibonacci numbers with odd index up to formula_106 is the (2"n")th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to formula_107 is the (2"n" + 1)th Fibonacci number minus 1.
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or in words, the sum of the squares of the first Fibonacci numbers up to formula_78 is the product of the "n"th and ("n" + 1)th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size formula_110 and decompose it into squares of size formula_111; from this the identity follows by comparing areas:
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The sequence formula_112 is also considered using the symbolic method. More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is formula_113. Indeed, as stated above, the formula_114-th Fibonacci number equals the number of combinatorial compositions (ordered ...
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It follows that the ordinary generating function of the Fibonacci sequence, i.e. formula_116, is the complex function formula_117.
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where "L" is the "n"'th Lucas number. The last is an identity for doubling "n"; other identities of this type are
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These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number.
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formula_151 gives the generating function for the negafibonacci numbers, and formula_152 satisfies the functional equation
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Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as
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Every third number of the sequence is even (a multiple of formula_163) and, more generally, every "k"th number of the sequence is a multiple of "F". Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property
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In particular, any three consecutive Fibonacci numbers are pairwise coprime because both formula_165 and formula_166. That is,
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Every prime number "p" divides a Fibonacci number that can be determined by the value of "p" modulo 5. If "p" is congruent to 1 or 4 (mod 5), then "p" divides "F", and if "p" is congruent to 2 or 3 (mod 5), then, "p" divides "F". The remaining case is that "p" = 5, and in this case "p" divides "F".
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where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that "n" is a prime, and if it fails to hold, then "n" is definitely not a prime. If "n" is composite and satisfies the formula, then "n" is a "Fibonacci pseudoprime". When "m" is largesay a 500-bit numberthen we can calculate "F" (...
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Here the matrix power "A" is calculated using modular exponentiation, which can be adapted to matrices.
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Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
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"F" is divisible by "F", so, apart from "F" = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
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The only nontrivial square Fibonacci number is 144. Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.
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1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.
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No Fibonacci number can be a perfect number. More generally, no Fibonacci number other than 1 can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect.
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With the exceptions of 1, 8 and 144 ("F" = "F", "F" and "F") every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). As a result, 8 and 144 ("F" and "F") are the only Fibonacci numbers that are the product of other Fibonacci numbers.
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The divisibility of Fibonacci numbers by a prime "p" is related to the Legendre symbol formula_172 which is evaluated as follows:
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For odd "n", all odd prime divisors of "F" are congruent to 1 modulo 4, implying that all odd divisors of "F" (as the products of odd prime divisors) are congruent to 1 modulo 4.
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All known factors of Fibonacci numbers "F"("i") for all "i" < 50000 are collected at the relevant repositories.
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If the members of the Fibonacci sequence are taken mod "n", the resulting sequence is periodic with period at most "6n". The lengths of the periods for various "n" form the so-called Pisano periods. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instanc...
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The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solut...
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The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):
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These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number as an ordered sum of 1s and 2s (called compositions); there are ways to do this (equivalently, it's also the number of domino tilings of the formula_195 rectang...
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The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.
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The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.
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Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybees. Kepler pointed out the presence of the Fibonacci sequence in n...
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Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.
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A model for the pattern of florets in the head of a sunflower was proposed by in 1979. This has the form
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where is the index number of the floret and is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the cent...
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Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:
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Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level...
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It has been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence. A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as th...
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Thomas Alva Edison (February 11, 1847October 18, 1931) was an American inventor and businessman. He developed many devices in fields such as electric power generation, mass communication, sound recording, and motion pictures. These inventions, which include the phonograph, the motion picture camera, and early versions ...
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Edison was raised in the American Midwest. Early in his career he worked as a telegraph operator, which inspired some of his earliest inventions. In 1876, he established his first laboratory facility in Menlo Park, New Jersey, where many of his early inventions were developed. He later established a botanical laborator...
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Thomas Edison was born in 1847 in Milan, Ohio, but grew up in Port Huron, Michigan, after the family moved there in 1854. He was the seventh and last child of Samuel Ogden Edison Jr. (1804–1896, born in Marshalltown, Nova Scotia) and Nancy Matthews Elliott (1810–1871, born in Chenango County, New York). His patrilineal...
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Edison was taught reading, writing, and arithmetic by his mother, who used to be a school teacher. He attended school for only a few months. However, one biographer described him as a very curious child who learned most things by reading on his own. As a child, he became fascinated with technology and spent hours worki...
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Edison developed hearing problems at the age of 12. The cause of his deafness has been attributed to a bout of scarlet fever during childhood and recurring untreated middle-ear infections. He subsequently concocted elaborate fictitious stories about the cause of his deafness. As he was completely deaf in one ear and ba...
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It is known that early in his career he enrolled in a chemistry course at The Cooper Union for the Advancement of Science and Art to support his work on a new telegraphy system with Charles Batchelor. This appears to have been his only enrollment in courses at an institution of higher learning.
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Thomas Edison began his career as a news butcher, selling newspapers, candy and vegetables on the trains running from Port Huron to Detroit. He turned a $50-a-week profit by age 13, most of which went to buying equipment for electrical and chemical experiments. At age 15, in 1862, he saved 3-year-old Jimmie MacKenzie f...
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Edison obtained the exclusive right to sell newspapers on the road, and, with the aid of four assistants, he set in type and printed the "Grand Trunk Herald", which he sold with his other papers. This began Edison's long streak of entrepreneurial ventures, as he discovered his talents as a businessman. Ultimately, his ...
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In 1866, at the age of 19, Edison moved to Louisville, Kentucky, where, as an employee of Western Union, he worked the Associated Press bureau news wire. Edison requested the night shift, which allowed him plenty of time to spend at his two favorite pastimes—reading and experimenting. Eventually, the latter pre-occupat...
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His first patent was for the electric vote recorder, , which was granted on June 1, 1869. Finding little demand for the machine, Edison moved to New York City shortly thereafter. One of his mentors during those early years was a fellow telegrapher and inventor named Franklin Leonard Pope, who allowed the impoverished y...
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Edison's major innovation was the establishment of an industrial research lab in 1876. It was built in Menlo Park, a part of Raritan Township (now named Edison Township in his honor) in Middlesex County, New Jersey, with the funds from the sale of Edison's quadruplex telegraph. After his demonstration of the telegraph,...
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William Joseph Hammer, a consulting electrical engineer, started working for Edison and began his duties as a laboratory assistant in December 1879. He assisted in experiments on the telephone, phonograph, electric railway, iron ore separator, electric lighting, and other developing inventions. However, Hammer worked p...
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Nearly all of Edison's patents were utility patents, which were protected for 17 years and included inventions or processes that are electrical, mechanical, or chemical in nature. About a dozen were design patents, which protect an ornamental design for up to 14 years. As in most patents, the inventions he described we...
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In just over a decade, Edison's Menlo Park laboratory had expanded to occupy two city blocks. Edison said he wanted the lab to have "a stock of almost every conceivable material". A newspaper article printed in 1887 reveals the seriousness of his claim, stating the lab contained "eight thousand kinds of chemicals, ever...
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Over his desk Edison displayed a placard with Sir Joshua Reynolds' famous quotation: "There is no expedient to which a man will not resort to avoid the real labor of thinking." This slogan was reputedly posted at several other locations throughout the facility.
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In Menlo Park, Edison had created the first industrial laboratory concerned with creating knowledge and then controlling its application. Edison's name is registered on 1,093 patents.
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Edison began his career as an inventor in Newark, New Jersey, with the automatic repeater and his other improved telegraphic devices, but the invention that first gained him wider notice was the phonograph in 1877. This accomplishment was so unexpected by the public at large as to appear almost magical. Edison became k...
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His first phonograph recorded on tinfoil around a grooved cylinder. Despite its limited sound quality and that the recordings could be played only a few times, the phonograph made Edison a celebrity. Joseph Henry, president of the National Academy of Sciences and one of the most renowned electrical scientists in the US...
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In 1876, Edison began work to improve the microphone for telephones (at that time called a "transmitter") by developing a carbon microphone, which consists of two metal plates separated by granules of carbon that would change resistance with the pressure of sound waves. A steady direct current is passed between the pla...
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Up to that point, microphones, such as the ones developed by Johann Philipp Reis and Alexander Graham Bell, worked by generating a weak current. The carbon microphone works by modulating a direct current and, subsequently, using a transformer to transfer the signal so generated to the telephone line. Edison was one of ...
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Edison used the carbon microphone concept in 1877 to create an improved telephone for Western Union. In 1886, Edison found a way to improve a Bell Telephone microphone, one that used loose-contact ground carbon, with his discovery that it worked far better if the carbon was roasted. This type was put in use in 1890 and...
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In 1878, Edison began working on a system of electrical illumination, something he hoped could compete with gas and oil-based lighting. He began by tackling the problem of creating a long-lasting incandescent lamp, something that would be needed for indoor use. However, Thomas Edison did not invent the light bulb. In 1...
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These early bulbs all had flaws such as an extremely short life and requiring a high electric current to operate which made them difficult to apply on a large scale commercially. In his first attempts to solve these problems, Edison tried using a filament made of cardboard, carbonized with compressed lampblack. This bu...
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The patent described several ways of creating the carbon filament including "cotton and linen thread, wood splints, papers coiled in various ways". It was not until several months after the patent was granted that Edison and his team discovered that a carbonized bamboo filament could last over 1,200 hours.
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In 1878, Edison formed the Edison Electric Light Company in New York City with several financiers, including J. P. Morgan, Spencer Trask, and the members of the Vanderbilt family. Edison made the first public demonstration of his incandescent light bulb on December 31, 1879, in Menlo Park. It was during this time that ...
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Henry Villard, president of the Oregon Railroad and Navigation Company, attended Edison's 1879 demonstration. Villard was impressed and requested Edison install his electric lighting system aboard Villard's company's new steamer, the "Columbia". Although hesitant at first, Edison agreed to Villard's request. Most of th...
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In 1880, Lewis Latimer, a draftsman and an expert witness in patent litigation, began working for the United States Electric Lighting Company run by Edison's rival Hiram S. Maxim. While working for Maxim, Latimer invented a process for making carbon filaments for light bulbs and helped install broad-scale lighting syst...
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On October 8, 1883, the US patent office ruled that Edison's patent was based on the work of William E. Sawyer and was, therefore, invalid. Litigation continued for nearly six years. In 1885, Latimer switched camps and started working with Edison. On October 6, 1889, a judge ruled that Edison's electric light improveme...
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The incandescent light bulb patented by Edison also began to gain widespread popularity in Europe as well. Mahen Theatre in Brno (in what is now the Czech Republic), opened in 1882, and was the first public building in the world to use Edison's electric lamps. Francis Jehl, Edison's assistant in the invention of the la...
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After devising a commercially viable electric light bulb on October 21, 1879, Edison developed an electric "utility" to compete with the existing gas light utilities. On December 17, 1880, he founded the Edison Illuminating Company, and during the 1880s, he patented a system for electricity distribution. The company es...
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In January 1882, Edison switched on the first steam-generating power station at Holborn Viaduct in London. The DC supply system provided electricity supplies to street lamps and several private dwellings within a short distance of the station. On January 19, 1883, the first standardized incandescent electric lighting s...
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As Edison expanded his direct current (DC) power delivery system, he received stiff competition from companies installing alternating current (AC) systems. From the early 1880s, AC arc lighting systems for streets and large spaces had been an expanding business in the US. With the development of transformers in Europe ...
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Edison expressed views that AC was unworkable and the high voltages used were dangerous. As George Westinghouse installed his first AC systems in 1886, Thomas Edison struck out personally against his chief rival stating, ""Just as certain as death, Westinghouse will kill a customer within six months after he puts in a ...
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Parallel to expanding competition between Edison and the AC companies was rising public furor over a series of deaths in the spring of 1888 caused by pole mounted high voltage alternating current lines. This turned into a media frenzy against high voltage alternating current and the seemingly greedy and callous lightin...
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Thomas Edison's staunch anti-AC tactics were not sitting well with his own stockholders. By the early 1890s, Edison's company was generating much smaller profits than its AC rivals, and the War of Currents would come to an end in 1892 with Edison forced out of controlling his own company. That year, the financier J.P. ...
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Edison moved from Menlo Park after the death of his first wife, Mary, in 1884, and purchased a home known as "Glenmont" in 1886 as a wedding gift for his second wife, Mina, in Llewellyn Park in West Orange, New Jersey. In 1885, Thomas Edison bought 13 acres of property in Fort Myers, Florida, for roughly $2,750 () and ...
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Due to the security concerns around World War I, Edison suggested forming a science and industry committee to provide advice and research to the US military, and he headed the Naval Consulting Board in 1915.
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Edison became concerned with America's reliance on foreign supply of rubber and was determined to find a native supply of rubber. Edison's work on rubber took place largely at his research laboratory in Fort Myers, which has been designated as a National Historic Chemical Landmark.
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The laboratory was built after Thomas Edison, Henry Ford, and Harvey Firestone pulled together $75,000 to form the Edison Botanical Research Corporation. Initially, only Ford and Firestone were to contribute funds to the project, while Edison did all the research. Edison, however, wished to contribute $25,000 as well. ...
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During the 1911 New York Electrical show, Edison told representatives of the copper industry it was a shame he didn't have a "chunk of it". The representatives decided to give a cubic foot of solid copper weighing 486 pounds with their gratitude inscribed on it in appreciation for his part in the "continuous stimulatio...
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Edison is credited with designing and producing the first commercially available fluoroscope, a machine that uses X-rays to take radiographs. Until Edison discovered that calcium tungstate fluoroscopy screens produced brighter images than the barium platinocyanide screens originally used by Wilhelm Röntgen, the technol...
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The fundamental design of Edison's fluoroscope is still in use today, although Edison abandoned the project after nearly losing his own eyesight and seriously injuring his assistant, Clarence Dally. Dally made himself an enthusiastic human guinea pig for the fluoroscopy project and was exposed to a poisonous dose of ra...
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In 1903, a shaken Edison said: "Don't talk to me about X-rays, I am afraid of them." Nonetheless, his work was important in the development of a technology still used today.
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Edison invented a highly sensitive device, that he named the tasimeter, which measured infrared radiation. His impetus for its creation was the desire to measure the heat from the solar corona during the total Solar eclipse of July 29, 1878. The device was not patented since Edison could find no practical mass-market a...
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The key to Edison's initial reputation and success was his work in the field of telegraphy. With knowledge gained from years of working as a telegraph operator, he learned the basics of electricity. This, together with his studies in chemistry at the Cooper Union, allowed him to make his early fortune with the stock ti...
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Edison was granted a patent for a motion picture camera, labeled the "Kinetograph". He did the electromechanical design while his employee William Kennedy Dickson, a photographer, worked on the photographic and optical development. Much of the credit for the invention belongs to Dickson. In 1891, Thomas Edison built a ...
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In April 1896, Thomas Armat's Vitascope, manufactured by the Edison factory and marketed in Edison's name, was used to project motion pictures in public screenings in New York City. Later, he exhibited motion pictures with voice soundtrack on cylinder recordings, mechanically synchronized with the film.
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Officially the kinetoscope entered Europe when wealthy American businessman Irving T. Bush (1869–1948) bought from the Continental Commerce Company of Frank Z. Maguire and Joseph D. Baucus a dozen machines. Bush placed from October 17, 1894, the first kinetoscopes in London. At the same time, the French company Kinétos...
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The first kinetoscopes arrived in Belgium at the Fairs in early 1895. The Edison's Kinétoscope Français, a Belgian company, was founded in Brussels on January 15, 1895, with the rights to sell the kinetoscopes in Monaco, France and the French colonies. The main investors in this company were Belgian industrialists. On ...
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Edison's film studio made nearly 1,200 films. The majority of the productions were short films showing everything from acrobats to parades to fire calls including titles such as "Fred Ott's Sneeze" (1894), "The Kiss" (1896), "The Great Train Robbery" (1903), "Alice's Adventures in Wonderland" (1910), and the first "Fra...
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As the film business expanded, competing exhibitors routinely copied and exhibited each other's films. To better protect the copyrights on his films, Edison deposited prints of them on long strips of photographic paper with the U.S. copyright office. Many of these paper prints survived longer and in better condition th...
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In 1908, Edison started the Motion Picture Patents Company, which was a conglomerate of nine major film studios (commonly known as the Edison Trust). Thomas Edison was the first honorary fellow of the Acoustical Society of America, which was founded in 1929.
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Edison said his favorite movie was "The Birth of a Nation". He thought that talkies had "spoiled everything" for him. "There isn't any good acting on the screen. They concentrate on the voice now and have forgotten how to act. I can sense it more than you because I am deaf." His favorite stars were Mary Pickford and Cl...
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Starting in the late 1870s, Edison became interested and involved with mining. High-grade iron ore was scarce on the east coast of the United States and Edison tried to mine low-grade ore. Edison developed a process using rollers and crushers that could pulverize rocks up to 10 tons. The dust was then sent between thre...
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In 1901, Edison visited an industrial exhibition in the Sudbury area in Ontario, Canada and thought nickel and cobalt deposits there could be used in his production of electrical equipment. He returned as a mining prospector and is credited with the original discovery of the Falconbridge ore body. His attempts to mine ...
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In the late 1890s, Edison worked on developing a lighter, more efficient rechargeable battery (at that time called an "accumulator"). He looked on them as something customers could use to power their phonographs but saw other uses for an improved battery, including electric automobiles. The then available lead acid rec...
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Edison obtained a US and European patent for his nickel–iron battery in 1901 and founded the Edison Storage Battery Company, and by 1904 it had 450 people working there. The first rechargeable batteries they produced were for electric cars, but there were many defects, with customers complaining about the product. When...
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At the start of World War I, the American chemical industry was primitive. Most chemicals were imported from Europe. The outbreak of war in August 1914 resulted in an immediate shortage of imported chemicals. One of particular importance to Edison was phenol, which was used to make phonograph records—presumably as phen...
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At the time, phenol came from coal as a by-product of coke oven gases or manufactured gas for gas lighting. Phenol could be nitrated to picric acid and converted to ammonium picrate, a shock resistant high explosive suitable for use in artillery shells. A telling of the phenol story is found in "The Aspirin Wars". Most...
https://en.wikipedia.org/wiki?curid=29778
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Edison responded by undertaking production of phenol at his Silver Lake facility using processes developed by his chemists. He built two plants with a capacity of six tons of phenol per day. Production began the first week of September, one month after hostilities began in Europe. He built two plants to produce raw mat...
https://en.wikipedia.org/wiki?curid=29778