On This Day in Math - December 27

At ubi materia, ibi Geometria.
Where there is matter, there is geometry.

~Johannes Kepler


The 361st day of the year, 2³⁶¹ is an apocalyptic number, it contains 666. 2³⁶¹=4697085165547666455778961193578674054751365097816639741414581943064418050229216886927397996769537406063869952 That's 109 digits.
361 = 19² *Lord Karl Voldevive ‏@Karl4MarioMugan

EVENTS
1612 Galileo observed Neptune, but did not recognize it as a planet. Galileo's drawings show that he first observed Neptune on December 28, 1612, and again on January 27, 1613. On both occasions, Galileo mistook Neptune for a fixed star when it appeared very close—in conjunction—to Jupiter in the night sky; hence, he is not credited with Neptune's discovery. (The official discovery is usually cited as September 23, 1846, Neptune was discovered within 1° of where Le Verrier had predicted it to be.) During the period of his first observation in December 1612, Neptune was stationary in the sky because it had just turned retrograde that very day. This apparent backward motion is created when the orbit of the Earth takes it past an outer planet. Since Neptune was only beginning its yearly retrograde cycle, the motion of the planet was far too slight to be detected with Galileo's small telescope.*Wik

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In 1831, Charles Darwin set sail from Plymouth harbour on his voyage of scientific discovery aboard the HMS Beagle, a British Navy ship. The Captain Robert FitzRoy was sailing to the southern coast of South America in order to complete a government survey. Darwin had an unpaid position as the ship's naturalist, at age 22, just out of university. Originally planned to be at sea for two years, the voyage lasted five years, making stops in Brazil, the Galapogos Islands, and New Zealand. From the observations he made and the specimens he collected on that voyage, Darwin developed his theory of biological evolution through natural selection, which he published 28 years after the Beagle left Plymouth. Darwin laid the foundation of modern evolutionary theory. *TIS

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In 1956, the formerly believed "law" of conservation of parity was disproved in the first successful results from an experiment conducted by Madame Chien-Shiung Wu at Columbia University on the beta-decay of cobalt-60. It had been suggested in a paper published by Lee and Yang on 1 Oct 1956. There had been problems to overcome working with the cobalt sample and detectors in a vacuum at a working temperature of one-hundredth of a kelvin. Wu's team repeated the experiment, doing maintenance on the apparatus as necessary, until on 9 Jan 1957 further measurements confirmed the initial results. Leon Lederman performed an independent test of parity with Columbia's cyclotron. They held a press conference on 15 Jan 1957.*TIS

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BIRTHS
1571 Johannes Kepler (27 Dec 1571; 15 Nov 1630) German astronomer who formulated three major laws of planetary motion which enabled Isaac Newton to devise the law of gravitation. Working from the carefully measured positions of the planets recorded by Tycho Brahe, Kepler mathematically deduced three relationships from the data: (1) the planets move in elliptical orbits with the Sun at one focus; (2) the radius vector sweeps out equal areas in equal times; and (3) for two planets the squares of their periods are proportional to the cubes of their mean distances from the sun. Kepler suggested that the tides were caused by the attraction of the moon. He believed that the universe was governed by mathematical rules, but recognized the importance of experimental verification.*TIS

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1654 Jacob Jacques Bernoulli (27 Dec 1654; 16 Aug 1705) was a Swiss mathematician and astronomer who was one of the first to fully utilize differential calculus and introduced the term integral in integral calculus. Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra (1685), work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. He published five treatises on infinite series (1682 - 1704). Jacob was intrigued by the logarithmic spiral and requested it be carved on his tombstone. He was the first of the Bernoulli family of mathematicians. *TIS (see more about the family of Bernoulli's at the Renaissance Mathematicus )

Even as the finite encloses an infinite series

And in the unlimited limits appear,

So the soul of immensity dwells in minutia

And in the narrowest limits no limit in here.

What joy to discern the minute in infinity!

The vast to perceive in the small, what divinity!

Ars Conjectandi

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1773 Sir George Cayley (27 Dec 1773; 15 Dec 1857)(6th Baronet ) English aeronautical pioneer who built the first successful man-carrying glider (1853). He made extensive anatomical and functional studies of bird flight. By measuring bird and human muscle masses, he realized it would be impossible for humans to strap on a pair of wings and take to the air. His further studies in the principles of lift, drag and thrust founded the science of aerodynamics from which he discovered stabilizing flying craft required both vertical and horizontal tail rudders, that concave wings produced more lift than flat surfaces and that swept-back wings provided greater stability. Cayley also invented the caterpillar tractor (1825), automatic railroad crossing signals, self-righting lifeboats, and an expansion-air (hot-air) engine.
*TIS (He was a distant cousin of the father of mathematician Arthur Cayley)

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1915 Jacob Lionel Bakst Cooper (27 December 1915, Beaufort West, Cape Province, South Africa, 8 August 1979, London, England) was a South African mathematician who worked in operator theory, transform theory, thermodynamics, functional analysis and differential equations.*Wik

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DEATHS
1771 Henri Pitot (3 May 1695, 27 Dec 1771) French hydraulic engineer who invented the Pitot tube (1732), an instrument to measure flow velocity either in liquids or gases. With subsequent improvements by Henri Darcy, its modern form is used to determine the airspeed of aircraft. Although originally a trained mathematician and astronomer, he became involved with an investigation of the velocity of flowing water at different depths, for which purpose he first created the Pitot tube. He disproved the prevailing belief that the velocity of flowing water increased with depth. Pitot became an engineer in charge of maintenance and construction of canals, bridges, drainage projects, and is particularly remembered for his kilometer-long Roman-arched Saint-Clément Aqueduct (1772) at Montpellier, France. *TIS

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1930 Gyula Farkas (28 March 1847 in Sárosd, Fejér County, Hungary - 27 Dec 1930 in Pestszentlorinc, Hungary) He is remembered for Farkas theorem which is used in linear programming and also for his work on linear inequalities. In 1881 Gyula Farkas published a paper on Farkas Bolyai's iterative solution to the trinomial equation, making a careful study of the convergence of the algorithm. In a paper published three years later, Farkas examined the convergence of more general iterative methods. He also made major contributions to applied mathematics and physics, particularly in the areas of mechanical equilibrium, thermodynamics, and electrodynamics.*SAU

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1973 Raymond Woodard Brink (4 Jan 1890 in Newark, New Jersey, USA - 27 Dec 1973 in La Jolla, California, USA) was an American mathematician who studied at Kansas State University, Harvard and Paris. He taught at the University of Minnesota though he spent a year in Edinburgh in 1919. He worked on the convergence of series. *SAU

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1992 Alfred Hoblitzelle Clifford (July 11, 1908 – December 27, 1992) was an American mathematician who is known for Clifford theory and for his work on semigroups. The Alfred H. Clifford Mathematics Research Library at Tulane University is named after him.*Wik

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1995 Boris Vladimirovich Gnedenko (January 1, 1912 - December 27, 1995) was a Soviet mathematician and a student of Andrey Nikolaevich Kolmogorov. He was born in Simbirsk (now Ulyanovsk), Russia, and died in Moscow. He is perhaps best known for his work with Kolmogorov, and his contributions to the study of probability theory. Gnedenko was appointed as Head of the Physics, Mathematics and Chemistry Section of the Ukrainian Academy of Sciences in 1949, and also became Director of the Kiev Institute of Mathematics in the same year.*Wik

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1996 Sister Mary Celine Fasenmyer, R.S.M., (October 4, 1906, Crown, Pennsylvania – December 27, 1996, Erie, Pennsylvania) was a mathematician. She is most noted for her work on hypergeometric functions and linear algebra.*Wik

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2006 Peter L. Hammer (December 23, 1936 - December 27, 2006) was an American mathematician native to Romania. He contributed to the fields of operations research and applied discrete mathematics through the study of pseudo-Boolean functions and their connections to graph theory and data mining.*Wik

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Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

On This Day in Math - December 26

A young man passes from our public schools to the universities, ignorant almost of the elements of every branch of useful knowledge.
~Charles Babbage




The 360th day of the year; Bryant Tuckerman found the Mersenne prime M19937 using an IBM360. *Prime Curios 
360 is also the number of degrees in a full circle, and there is a (rather new) word for two angles that sum to 360 degrees.  They are called "explementary" .
360 is a highly composite number, it has 24 divisors, more than any other number that is below twice its size.
 It is the smallest number that is divisible by nine of the ten numbers 1-10 (not divisible by 7) What is next, students?
There are 360 possible rook moves on a 6x6 chess board.*Derek Orr
The 359th, 360th,  361st digits of \( \pi \) are 3, 6, 0. (Also from Derek)

EVENTS
1837 Charles Babbage completed his “Calculating Engine” manuscript. *VFR

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1843 John Graves write to William Rowan Hamilton that he has invented an eight-dimension normed division algebra he called "Octaves" Within a few months, Hamilton would realize that the octonions were not associative. This would lead to the first use of the term "associative" by Hamilton in 1844. (Except for matrices, which were not generally considered as "numbers", there were no common non-associative systems at that time) *Joan Baez Rankin Lecture of September 17, 2008 Glascow

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1898 Radium discovered by Pierre and Marie Curie. *VFR Actually, it seems this was the date of their announcement of the discovery(which must have occurred a few days earlier. They created the name radium for their element. This was their second discovery in the first year of her research on her thesis. They had also discovered Polonium earlier in the year.

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 In 1906, the world's first full-length feature film, the 70-min Story of the Kelly Gang was presented in the Town Hall at Melbourne, Australia, where it had been filmed at a cost of £450. It preceded D.W. Griffith's The Birth of a Nation by nine years. The subject of the Australian movie was Ned Kelly, a bandit who lived 1855 to 1880. The film toured through Australia for over 20 years, and abroad in New Zealand and Britain. Since some people, including politicians and police viewed the content of the film as glorifying the criminals, the movie was banned (1907) in Benalla and Wangaratta and also in Victoria (1912). Only fragments totalling about 10 minutes of the original nitrate film have survived to the present.*TIS

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1951 Kurt Godel delivered the Gibbs Lecture, “Some Basic Theorems on the Foundations of Mathe-matics and their Philosophical Implications,” to the annual AMS meeting at Brown University. *VFR

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1982 TIME Names a Non-Human “Man of the Year”
TIME magazine's editors selected the Personal Computer for “Machine of the Year,” in lieu of their well-known “Man of the Year” award. The computer beat out U.S. President Ronald Reagan, U.K. prime minister Margaret Thatcher and Prime Minister of Israel​, Menachem Begin. The planet Earth became the second non-human recipient for the award in 1988. The awards have been given since 1927. The magazine's essay reported that in 1982, 80% of Americans expected that "in the fairly near future, home computers will be as commonplace as television sets or dishwashers.” In 1980, 724,000 personal computers were sold in the United States, according to Time. The following year, that number doubled to 1.4 million. *CHM

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BIRTHS 1532 Wilhelm Xylander (born Wilhelm Holtzman, graecized to Xylander) (December 26, 1532 – February 10, 1576) was a German classical scholar and humanist.
Xylander was the author of a number of important works. He translated the first six books of Euclid into German with notes, the Arithmetica of Diophantus, and the De quattuor mathematicis scientiis of Michael Psellus into Latin. *Wik

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1780 Mary Fairfax Greig Somerville (26 Dec 1780 in Jedburgh, Roxburghshire, Scotland - 29 Nov 1872 in Naples, Italy) Somerville wrote many works which influenced Maxwell. Her discussion of a hypothetical planet perturbing Uranus led Adams to his investigation. Somerville College in Oxford was named after her.*SAU

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1791 Charles Babbage born. *VFR (26 Dec 1791; 18 Oct 1871) English mathematician and pioneer of mechanical computation, which he pursued to eliminate inaccuracies in mathematical tables. By 1822, he had a small calculating machine able to compute squares. He produced prototypes of portions of a larger Difference Engine. (Georg and Edvard Schuetz later constructed the first working devices to the same design which were successful in limited applications.) In 1833 he began his programmable Analytical Machine, a forerunner of modern computers. His other inventions include the cowcatcher, dynamometer, standard railroad gauge, uniform postal rates, occulting lights for lighthouses, Greenwich time signals, heliograph opthalmoscope. He also had an interest in cyphers and lock-picking.*TIS

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1861 Frederick Engle born in Germany. He became the closest student of the Norwegian mathe¬matician Sophus Lie. Engle was also the first to translate Lobachevsky’s work into a Western language (German). *VFR

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1900 Antoni Zygmund (26 Dec 1900; 30 May 1992) Polish-born mathematician who created a major analysis research centre at Chicago, and recognized in 1986 for this with the National Medal for Science. In 1940, he escaped with his wife and son from German controlled Poland to the USA. He did much work in harmonic analysis, a statistical method for determining the amplitude and period of certain harmonic or wave components in a set of data with the aid of Fourier series. Such technique can be applied in various fields of science and technology, including natural phenomena such as sea tides. He also did major work in Fourier analysis and its application to partial differential equations. Zygmund's book Trigonometric Series (1935) is a classic, definitive work on the subject*TIS

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1903 Lancelot Stephen Bosanquet (26 Dec 1903 in St. Stephen's-by-Saltash, Cornwall, England - 10 Jan 1984 in Cambridge, Cambridgeshire, England) Bosanquet wrote many papers on the convergence and summability of Fourier series. He also wrote on the convergence and summability of Dirichlet series and studied specific kinds of summability such as summability factors for Cesàro means. His later work on integrals include two major papers on the Laplace-Stieltjes integral published in 1953 and 1961. Other topics he studied included inequalities, mean-value theorems, Tauberian theorems, and convexity theorems. *SAU

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1937 John Horton Conway (born 26 December 1937, ) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life.
Conway is currently Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics at Princeton University. He studied at Cambridge, where he started research under Harold Davenport. He received the Berwick Prize (1971),[1] was elected a Fellow of the Royal Society (1981),[2] was the first recipient of the Pólya Prize (LMS) (1987),[1] won the Nemmers Prize in Mathematics (1998) and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society. He has an Erdős number of one.*Wik Conway is known for his sense of humor, and the last proof in his "On Numbers and Games" is this:
Theorem 100; This is the last Theorem in this book.
The Proof is Obvious.

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DEATHS
1624 Simon Marius (10 Jan 1573, 26 Dec 1624) (Also known as Simon Mayr) German astronomer, pupil of Tycho Brahe, one of the earliest users of the telescope and the first in print to make mention the Andromeda nebula (1612). He studied and named the four largest moons of Jupiter as then known: Io, Europa, Ganymede and Callisto (1609) after mythological figures closely involved in love with Jupiter. Although he may have made his discovery independently of Galileo, when Marius claimed to have discovered these satellites of Jupiter (1609), in a dispute over priority, it was Galileo who was credited by other astronomers. However, Marius was the first to prepare tables of the mean periodic motions of these moons. He also observed sunspots in 1611 *TIS You can find a nice blog about the conflict with Galileo by the Renaissance Mathematicus.

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1931 Melvil Dewey (10 Dec 1851, 26 Dec 1931) American librarian who developed library science in the U.S., especially with his system of classification, the Dewey Decimal Classification (1876), for library cataloging. His system of classification (1876) uses numbers from 000 to 999 to cover the general fields of knowledge and designating more specific subjects by the use of decimal points. He was an activist in the spelling reform and metric system movements. Dewey invented the vertical office file, winning a gold medal at the 1893 World's Fair. It was essentially an enlarged version of a card catalogue, where paper documents hung vertically in long drawers. *TIS

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2006 Martin David Kruskal (September 28, 1925 – December 26, 2006) was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His single most celebrated contribution was the discovery and theory of solitons. His Ph.D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic "The Bridge Theorem For Minimal Surfaces." He received his Ph.D. in 1952.
In the 1950s and early 1960s, he worked largely on plasma physics, developing many ideas that are now fundamental in the field. His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include the Kruskal–Shafranov instability and the Bernstein–Greene–Kruskal (BGK) modes. With I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or magnetohydrodynamic) Energy Principle. His interests extended to plasma astrophysics as well as laboratory plasmas. Martin Kruskal's work in plasma physics is considered by some to be his most outstanding.
In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in General Relativity. A spherically symmetric black hole can be described by the Schwarzschild solution, which was discovered in the early days of General Relativity. However, in its original form, this solution only describes the region exterior to the horizon of the black hole. Kruskal (in parallel with George Szekeres) discovered the maximal analytic continuation of the Schwarzschild solution, which he exhibited elegantly using what are now called Kruskal–Szekeres coordinates.
This led Kruskal to the astonishing discovery that the interior of the black hole looks like a "wormhole" connecting two identical, asymptotically flat universes. This was the first real example of a wormhole solution in General Relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other. This is now believed to be the general fate of wormholes in General Relativity.
Martin Kruskal was married to Laura Kruskal, his wife of 56 years. Laura is well known as a lecturer and writer about origami and originator of many new models.[3] Martin, who had a great love of games, puzzles, and word play of all kinds, also invented several quite unusual origami models including an envelope for sending secret messages (anyone who unfolded the envelope to read the message would have great difficulty refolding it to conceal the deed).
His Mother, Lillian Rose Vorhaus Kruskal Oppenheimer was an American origami pioneer. She popularized origami in the West starting in the 1950s, and is credited with popularizing the Japanese term origami in English-speaking circles, which gradually supplanted the literal translation paper folding that had been used earlier. In the 1960s she co-wrote several popular books on origami with Shari Lewis.*wik

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Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

On This Day in Math - December 25

God Bless Us, Everyone

Me, and Tiny Tim

The 359th day of the year; 359 is a Sophie Germain prime. If you start with n=89 and iterate 2n+1 you will get a string of primes that includes 359. (How many in all?)


EVENTS
1843 Sir Henry Cole British industrial designer, museum director and writer who produced the first commercial Christmas card.* In 1843, wishing to save much handwriting of seasonal correspondance, Cole introduced the world's first commercial Christmas card. He commissioned artist John Callcott Horsley to make the artwork for 1000 hand-coloured lithographs. (Individuals' homemade Christmas cards had existed earlier.) (See Image at top)

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1656 Huygens (who was a bachelor) spent Christmas Day making the first model of a pendulum clock. *VFR

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In 1741, the Centigrade temperature scale was devised by astronomer Anders Celsius (1701-44) and incorporated into a Delisle thermometer at Uppsala in Sweden. Celsius divided the fixed-point range of the Fahrenheit scale (the freezing and boiling temperatures of water) into 100 equal divisions, but curiously set the freezing point at 100 and the boiling point at 0. This reverse scaling was changed to match the sense of the other temperature scales after Celsius's death.*TIS

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1758 Halley’s comet first sighted after he predicted its return. After Newton explained planetary motion, he suggested that comets could have elongated elliptical orbits. Halley’s comet has eccentricity 0.9675. [UMAP Journal, 4(1983), p. 162] *VFR
In 1758, the predicted return of Halley's comet was first sighted by German farmer and amateur astronomer, Johann Georg Palitzsch, as a faint object in Pisces. Edmund Halley had predicted in 1705 the return of the comet to the Earth's vicinity every 75.5 years. For the first time the scientific prediction had been proven. Halley himself had died 16 years before this new event. Palitzsch also observed the 6 Jun 1761 transit of Venus, when he saw a black band linking Venus and the Sun near the beginning and end of the transit ("black drop effect") and correctly interpreted this as evidence that Venus possessed an atmosphere. He also measured the period of the variation of the brightness of the star Algol. *TIS
The BAYEUX TAPESTRY (Tapisserie de la Reine Mathilde) includes a clear picture of Halley's Comet, as shown on the stamp below.

and here is a Blog regarding the return in 1758.

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In 1780, Luigi Galvani recorded, "The electric fluid should be considered a means to the nervo-muscular force." He reached this conclusion from work in his laboratory in Bologna, Italy, after a series of experiments and his accidental discovery that muscles are operated by electrical stimulation of nerves. He worked diligently along these lines, but waited for eleven years before he published the results and an ingenious and simple theory. His theory was that of a nervous electric fluid, secreted by the brain, conducted by the nerves, and stored in the muscles. Though his ideas were abandoned by scientists on account of later discoveries, his work opened the way to new research in the physiology of muscle and nerve and pioneered the subject of electrophysiology. *TIS

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In 1999, space shuttle Discovery's astronauts finished their mainenance work on the Hubble Space Telescope, installing correcting optics to repair problems due to a design flaw in the mirror. The first images the Hubble Telescope took after its original launch were disappointingly fuzzy, but after this repair mission the instrument returned crisp images of a clarity never before possible from terrestrial observatories *TIS

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BIRTHS
1642 Isaac Newton born on Julian Calendar. (5 January 1643 New style) *VFR
Born 25 Dec 1642; died 20 Mar 1727. English physicist and mathematician, who made seminal discoveries in several areas of science, and was the leading scientist of his era. His study of optics included using a prism to show white light could be split into a spectrum of colours. The statement of his three laws of motion are fundamental in the study of mechanics. He was the first to describe the moon as falling (in a circle around the earth) under the same influence of gravity as a falling apple, embodied in his law of universal gravitation. As a mathematician, he devised infinitesimal calculus to make the calculations needed in his studies, which he published in Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, 1687)*TIS

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1763 Claude Chappe (25 Dec 1763; 23 Jan 1805) French engineer who invented the semaphore visual telegraph. He began experimenting in 1790, trying various types of telegraph. An early trial used telescopes, synchronised pendulum clocks and a large white board, painted black on the back, with which he succeeded in sending a message a few sentences long across a 16km (10mi) distance. To simplify construction, yet still easily visible to read from far away, he changed to using his semaphore telegraph in 1793. Smaller indicators were pivoted at each end of large horizontal member. The two indicators could each be rotated to stand in any of eight equally spaced positions. By setting them at different orientations, a set of corresponding codes was used to send a message.*TIS

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1900 Antoni Szczepan Zygmund (25 Dec 1900 in Warsaw, Russian Empire (now Poland)
- 30 May 1992 in Chicago, Illinois, USA) Zygmund's work in harmonic analysis has application in the theory of waves and vibrations. He also did major work in Fourier analysis and its application to partial differential equations.*SAU

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1905 Gottfried Maria Hugo Köthe (born 25 December 1905 in Graz; died 30 April 1989 in Frankfurt) was an Austrian mathematician working in abstract algebra and functional analysis.*SAU

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DEATHS
1921 Piers Bohl (October 23, 1865 – December 25, 1921) was a Latvian mathematician, who worked in differential equations, topology and quasi-periodic functions.
He was born in 1865 in Walk, Livonia, in the family of a poor Baltic German merchant. In 1884, after graduating from a German school in Viljandi, he entered the faculty of physics and mathematics at the University of Tartu. In 1893 Bohl was awarded his Master's degree. This was for an investigation of quasi-periodic functions. The notion of quasi-periodic functions was generalised still further by Harald Bohr when he introduced almost-periodic functions. *Wik

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1929 Percy Alexander MacMahon (b. 26 September 1854, Sliema, Malta – 25 December 1929, Bognor Regis, England) was a mathematician, especially noted in connection with the partitions of numbers and enumerative combinatorics. MacMahon was elected a Fellow of the Royal Society in 1890. He received the Royal Society Royal Medal in 1900, the Sylvester Medal in 1919, and the Morgan Medal by the London Mathematical Society in 1923. MacMahon was the President of the London Mathematical Society from 1894 to 1896.
MacMahon is best known for his study of symmetric functions and enumeration of plane partitions; see MacMahon Master theorem. His two volume Combinatory analysis, published in 1915/16, is the first major book in enumerative combinatorics. MacMahon also did pioneering work in recreational mathematics and patented several successful puzzles.*WIK

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1930 Eugen Goldstein (5 Sep 1850, 25 Dec 1930) German physicist who discovered and named canal rays (1886) which emerge through holes in the anodes of low-pressure electrical discharge tubes (later shown to be positively charged particles). Earlier, he coined the term "cathode ray" (1876) emitted from a cathode. He was the first to see that they could cast a shadow, and were emitted at right angles to the surface. He also investigated the wavelengths of light emitted by metals and oxides when canal rays impinge on them. When the Berlin Urania, opened in 1889 it had five scientific departments and a "science theatre", it was Goldstein who had recommended the "hall of physics in which the visitor could experiment on his own". Students of his that continued his work included Wien and Stark. *TIS

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1941 Theodor Molien​ or Fedor Eduardovich Molin​ (September 10, 1861 - December 25, 1941) was a Baltic-German mathematician. He was born in Riga, Latvia, which at that time was a part of Russian Empire. Molien studied associative algebras and polynomial invariants of finite groups.*Wik

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1944 Wilhelm Kutta was a German engineer who is best known for his work on the numerical solution of differential equations (the Runge-Kutta method).*SAU

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2000 Willard Van Orman Quine (June 25, 1908 – December 25, 2000) (known to intimates as "Van")[1] was an American philosopher and logician in the analytic tradition. From 1930 until his death 70 years later, Quine was continuously affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of mathematics, and finally as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. A recent poll conducted among philosophers named Quine one of the five most important philosophers of the past two centuries.[2] He won the first Schock Prize in Logic and Philosophy in 1993, for "his systematical and penetrating discussions of how learning of language and communication are based on socially available evidence and of the consequences of this for theories on knowledge and linguistic meaning.*Wik

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Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

A Pretty Little Voting Result in Combinatorics

I was updating the Bio for  Joseph Louis François Bertrand for my On This Day in Math pages, (see Births, 1822) and was reminded of the beautiful simplicity (and symmetry) hiding in this old problem.  Bertrand gave the problem, and a proof, in 1887 in J. Bertrand, Solution d'un problème, Comptes Rendus de l'Académie des Sciences.  

The problem can be simply stated in a couple of ways, but Betrand's question was along the order of, " If at the end of an election, A had m votes and B had n votes, with m greater than n, what is the probability that at any point in the counting A had more votes than B.  
In a second variation one might ask for the probability that A was never behind, thus including sequences that include one or more possible ties in the vote counts. 

For those who want to sit and work on the problem for awhile, I will talk a little about the subsequent history of the problem, and give a solution to two variations much farther down.  

In Bertrand's original paper, he sketches a proof based on a general formula for the number of favorable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method. Such a proof was given by Désiré André based on the observation that the unfavorable sequences can be divided into two equally probable cases, one of which (the case where B receives the first vote) is easily computed; he proves the equality by an explicit bijection. A variation of his method is popularly known as André's reflection method, although André did not use any reflections. The first known use of reflections in the solution was by Percy MacMahon who applied theory of partitions to the problem. The most famous and elegant of these variations is the “reflection method” in which ballot permutations are represented as lattice paths and portions of the bad paths are reflected across a line.


Lost(and Found) in Translation: André’s ActualMethod and its Application to the Generalized Ballot Problem.  This article, by Marc Renault, appeared in the American Mathematical Monthly, April 2008.

The same author has also written the following which gives four different proofs of the general theorem, including some nice graphics.  

Four Proofs of the Ballot Theorem,

Some steps and the Solution:

 Renault gives a proof following these steps

How many ways can the  a + b  votes be arranged so that  A  maintains a lead throughout the counting of the ballots?
Here is the proof by reflection as given by the Wikipedia article on the problem, which also has several other proofs.

For A to be strictly ahead of B throughout the counting of the votes, there can be no ties. Separate the counting sequences according to the first vote. Any sequence that begins with a vote for B must reach a tie at some point, because A eventually wins. For any sequence that begins with A and reaches a tie, reflect the votes up to the point of the first tie (so any A becomes a B, and vice-versa) to obtain a sequence that begins with B. Hence every sequence that begins with A and reaches a tie is in one to one correspondence with a sequence that begins with B, and the probability that a sequence begins with B is , so the probability that A always leads the vote is

the probability of sequences that tie at some point

the probability of sequences that tie at some point and begin with A or B