## Thursday 30 April 2020

### On This Day in Math - April 30

 Statue of C. F. Gauss in Braunschweig *Wik

I mean the word proof not in the sense of the lawyers,
who set two half proofs equal to a whole one,
but in the sense of a mathematician,
where half a proof is zero,
and it is demanded for proof that
every doubt becomes impossible.
Karl Friedrich Gauss As quoted in Calculus Gems (1992) by George F. Simmons

The 121st day of the year; 121 will be the largest year day of the form n!+1 which is a square number. Brocard conjectured in 1904 that the only solutions of n! + 1 = m2 are n = 4, 5, and 7. There are no other solutions with $n \lt 10^ 9$. 121 is also the only square of the form 1 + n + n2+ n3 + n4. *What's So Special About This Number

121 is also a Smith Number, a composite number for which the sum of its digits is equal to the sum of the digits in its prime factorization. Smith numbers were named by Albert Wilansky of Lehigh University. He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
4937775 = 3 × 5 × 5 × 65837, while 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42.
There are 49 Smith numbers below 1000, collect the whole set.

121 is a palindrome in base ten, and also in base 3 (11111), base 7 (232) and base 8(171). No other year day is a base ten palindrome  and also palindrome in as many other (2-9) bases.

EVENTS

1006 Chinese and Arabic astronomers noted a supernova. The speed of the still-expanding shock wave was measured nearly a millenium later. This was history's brightest "new star" ever recorded, at first seen to be brighter than the planet Venus. It occurred in our Milky Way galaxy, appearing in the southern constellation Lupus, near the star Beta Lupi. It was also recorded by observers in Switzerland, Italy, Japan, Egypt and Iraq. From the careful descriptions of the Chinese astronomers of how the light varied, that it was of apparently yellow color and visible for over a year, it is possible that the supernova reached a magnitude of up to -9. Modern measurements of the speed of the shock wave have been used to estimate its distance. *TIS The associated supernova remnant from this explosion was not identified until 1965, when Doug Milne and Frank Gardner used the Parkes radio telescope to demonstrate that the previously known radio source PKS 1459-41, near the star Beta Lupi, had the appearance of a 30-arcminute circular shell.

1633 Galileo was forced to recant his scientiﬁc ﬁndings(suppositions?) related to the Copernican Theory as “abjured, cursed and detested” by the Inquisition. He was placed under house arrest for the remaining nine years of his life. Legend had it that when Galileo rose from knealing before his inquisitors, he murmured, “e pur, si mouve”—“even so, it does move.” *VFR [church doctrine held that the Earth, God's chosen place, was the center of the universe and everything revolved around it. Copernicans believed that the sun was the center of the solar system, and "the earth moves" around it.]

In 1683, the Boston Philosophical Society held its first meeting.* Rev. Increase Mather, stimulated by a recent comet sighting, and seeking to discuss how God intervenes in the natural order of things, had met earlier in the month with Samuel Willard and a few others to plan the group. Mather's idea was to model their meetings on the Royal Philosophical Society, established in London about 20 years earlier. Each last Monday of most of the following months, the members met and presented papers to emulate the transactions of the London society. However, the few local intellectuals didn't sustain interest in the society beyond about three years. Mather wrote Kometographia, or, A Discourse concerning Comets (1683).*TIS

1695 Bernoulli explains to Leibniz his reasons for the use of the term "integral calculus" for Leibniz's new calculus. Leibniz had used, and tried to get others to use "Sums" but Bernoulli's term had become popular. Bernoulli explained that, "I considered the differential as the infinitesimal part of the whole, or Integral." *VFR

1752 A sealed paper delivered by mathematician/instrument maker James Short to the Royal Society on 30 April 1752 was opened after his death and read publicly on 25 Jan. 1770. It described a method of working object-lenses to a truly spherical form. It seems, from the journal of Lelande that this was done by eye. "from there by water to Surrey Street
to see Mr Short who spoke to me about the difficulty in giving his mirrors a parabolic figure. It is done only by guess-work."
*Richard Watkins

1807 Gauss writes to Sophie Germanin for the first time since being aware she was a woman, (She had formally written using the name Monsieur LeBlanc). In a letter with much praise, he writes:
"The scientific notes with which your letters are so richly
filled have given me a thousand pleasures. I have studied
them with attention and I admire the ease with which you
penetrate all branches of arithmetic, and the wisdom with
which you generalize and perfect."

1837 Massachusetts became the ﬁrst state to establish a board of education. *VFR

1891 Nature magazine publishes Peter Guthrie Tait's "The Role of Quaternions in the Algebra of Vectors."

1895 Georg Cantor, in a letter to Felix Klein, explains the choice of aleph for the cardinality of sets.
In the same letter he comments that, "the usual alphabets seem to me too much used to be fitter for the purpose. On the other hand, I did not want to invent a new symbol, so I chose finally the aleph, which in Hebrew has also the numerical value 1."
*Cantorian Set Theory and Limitation of Size, By Michael Hallett

1897 at the Royal Institution Friday Evening Discourse, Joseph John Thomson (1856-1940) first announced the existence of electrons (as they are now named). Thomson told his audience that earlier in the year, he had made a surprising discovery. He had found a particle of matter a thousand times smaller than the atom. He called it a corpuscle, meaning "small body." Although Thomson was director of the Cavendish Laboratory at the University of Cambridge, and one of the most respected scientists in Great Britain, the scientists present found the news hard to believe. They thought the atom was the smallest and indivisible part of matter that could exist. Nevertheless, the electron was the first elementary particle to be discovered.*TIS

1905 Einstein completed his Doctoral thesis, with Alfred Kleiner, Professor of Experimental Physics, serving as pro-forma advisor. Einstein was awarded a PhD by the University of Zurich. His dissertation was entitled "A New Determination of Molecular Dimensions." This paper included Einstein's initial estimates of Avagadro constant as 2.2×1023 based on diffusion coefficients and viscosities of sugar solutions in water (the error was more in the known estimates of sugar molecules than his method). That same year, which has been called Einstein's annus mirabilis (miracle year), he published four groundbreaking papers, on the photoelectric effect, Brownian motion, special relativity, and the equivalence of mass and energy, which were to bring him to the notice of the academic world. *Wik
Einstein would often say that Kleiner, at first, rejected his thesis for being too short, so he added one more sentence and it was accepted.

1916 Daylight Saving Time has been used in the U.S. and in many European countries since World War I. At that time, in an effort to conserve fuel needed to produce electric power, Germany and Austria took time by the forelock, and began saving daylight at 11:00 p.m. on April 30, 1916, by advancing the hands of the clock one hour until the following October. *WebExhibits.org

1982 Science (pp. 505–506) reported that Stanford magician-statistician Perci Diaconis solved the problem of which arrangements of a deck of cards can occur after repeated perfect rifflee shuffles. The answer involves M12, one of the Mathieu simple groups. Mathematics Magazine 55 (1982),
p. 245].*VFR

1984 30 April-4 May 1984. Teacher Appreciation Week. Celebrated the ﬁrst week of May in Flint, MI. *VFR

1992 The New York Times “in describing the discovery of the new Mersenne prime, felt it necessary to describe the series of primes, which, (according to them) goes: 1, 2, 3, 5, 7, 11, 13, ... . You will notice that they have slipped in what must be another discover (by one of their writers?) of the world’s smallest prime: 1. I’m sure the mathematicians of the world must be tearing their hair out for having missed this one.” [A posting of Ron Rivest to the net.] (In fact it was common prior to the 20th century to consider one as a prime, not that that is an excuse in 1992.)

1993 CERN announces World Wide Web protocols will be free. *Wik

2015 Walpurgis night or Witches’ Sabbath is celebrated on the eve of May Day, particularly by university students in northern Europe. *VFR According to the ancient legends, this night was the last chance for witches and their nefarious cohorts to stir up trouble before Spring reawakened the land. They were said to congregate on Brocken, the highest peak in the Harz Mountains - a tradition that comes from Goethe's Faust. *Wik

BIRTHS

1777 Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.
Sometimes referred to as the Princeps mathematicorum (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences".. *Wik
His poorly educated mother couldn’t remember his birthdate, but could relate it to a movable religious feast. To conﬁrm the date of his birth Gauss developed a formula for the date of Easter. *VFR
He transformed nearly all areas of mathematics, for which his talent showed from a very early age. For his contributions to theory in magnetism and electricity, a unit of magnetic field has been named the gauss. He devised the method of least squares in statistics, and his Gaussian error curve remains well-known. He anticipated the SI system in his proposal that physical units should be based on a few absolute units such as length, mass and time. In astronomy, he calculated the orbits of the small planets Ceres and Pallas by a new method. He invented the heliotrope for trigonometric determination of the Earth's shape. With Weber, he developed an electromagnetic telegraph and two magnetometers. *TIS He proved that the heptadecagon (17 gon) was constructable (see April 8) with straight-edge and compass. Dave Renfro has a complete and elementary proof.

1904 George Robert Stibitz (30 Apr 1904, 31 Jan 1995) U.S. mathematician who was regarded by many as the "father of the modern digital computer." While serving as a research mathematician at Bell Telephone Laboratories in New York City, Stibitz worked on relay switching equipment used in telephone networks. In 1937, Stibitz, a scientist at Bell Laboratories built a digital machine based on relays, flashlight bulbs, and metal strips cut from tin-cans. He called it the "Model K" because most of it was constructed on his kitchen table. It worked on the principle that if two relays were activated they caused a third relay to become active, where this third relay represented the sum of the operation. Also, in 1940, he gave a demonstration of the first remote operation of a computer.*TIS

1916 Claude Shannon (30 April 1916 in Petoskey, Michigan, USA - 24 Feb 2001 in Medford, Massachusetts, USA) founded the subject of information theory and he proposed a linear schematic model of a communications system. His Master's thesis was on A Symbolic Analysis of Relay and Switching Circuits on the use of Boole's algebra to analyse and optimize relay switching circuits. *SAU While working with John von Neumann on early computer designs, (John) Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948. Among several statues to Shannon, one is erected in his hometown of Gaylord, Michigan. The statue is located in Shannon Park in the center of downtown Gaylord. Shannon Park is the former site of the Shannon Building, built and owned by Claude Shannon's father. The lady beside the statue, a true mathematical genius in her own right, is Betty, the wife, and closest collaborator of Claude Shannon.
While working with John von Neumann on early computer designs, (John) Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948. In 2016 as his 100th birth anniversary was approaching, the Petoskey News (Shannon's birthplace) described him as the folks who knew him growing up in Gaylord like to recall him, "Who would have guessed the world would be celebrating the birthday of a unicycling, juggling, mathematic academic and engineer from Gaylord? But that is exactly what is happening next week as local historians, youth and others celebrate a special centennial birthday of a local celebrity.

1944 Lee Cecil Fletcher Sallows (April 30, 1944, ) is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares. Sallows has an Erdős number of 2.
Sallows is an expert on the theory of magic squares and has invented several variations on them, including Alphamagic Squares and geomagic squares. The latter invention caught the attention of mathematician Peter Cameron who has said that he believes that "an even deeper structure may lie hidden beyond geomagic squares"
In 1984 Lee Sallows invented the self-enumerating sentence — a sentence that inventories its own letters. Following failure in his attempt to write a computer program to generate such sentences, he constructed a so-called electronic Pangram Machine, among the results of which was the following sentence that appeared in Douglas Hofstadter's Metamagical Themas column in Scientific American in October 1984:
This Pangram contains four as, one b, two cs, one d, thirty es, six fs, five gs, seven hs, eleven is, one j, one k, two ls, two ms, eighteen ns, fifteen os, two ps, one q, five rs, twenty-seven ss, eighteen ts, two us, seven vs, eight ws, two xs, three ys, & one z.
A golygon is a polygon containing only right angles, such that adjacent sides exhibit consecutive integer lengths. Golygons were invented and named by Sallows and introduced by A.K. Dewdney in the Computer Recreations column of the July 1990 issue of Scientific American.
In 2012 Sallows invented and named 'self-tiling tile sets'—a new generalization of rep-tiles
1944*Wik

DEATHS

1865 Robert Fitzroy British naval officer, hydrographer, and meteorologist who commanded the voyage of HMS Beagle, aboard which Charles Darwin sailed around the world as the ship's naturalist. That voyage provided Darwin with much of the material on which he based his theory of evolution. Fitzroy retired from active duty in 1850 and from 1854 devoted himself to meteorology. He devised a storm warning system that was the prototype of the daily weather forecast, invented a barometer, and published The Weather Book (1863). His death was by suicide, during a bout of depression. *TIS
[FitzRoy is buried in the front church yard of All Saints Church in Upper Norwood, London. His memorial was restored by the Meteorological Office in 1981]

1907 Charles Howard Hinton​ (1853, 30 April 1907) was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualizing the geometry of higher dimensions. He also had a strong interest in theosophy.
Hinton created several new words to describe elements in the fourth dimension. According to OED, he first used the word tesseract in 1888 in his book A New Era of Thought. He also invented the words "kata" (from the Greek "down from") and "ana" (from the Greek "up toward") to describe the two opposing fourth-dimensional directions—the 4-D equivalents of left and right, forwards and backwards, and up and down.
Hinton was convicted of bigamy for marrying both Mary Ellen (daughter of Mary Everest Boole and George Boole, the founder of mathematical logic) and Maud Wheldon. He served a single day in prison sentence, then moved with Mary Ellen first to Japan (1886) and later to Princeton University in 1893 as an instructor in mathematics.
In 1897, he designed a gunpowder-powered baseball pitching machine for the Princeton baseball team's batting practice. According to one source it caused several injuries, and may have been in part responsible for Hinton's dismissal from Princeton that year. However, the machine was versatile, capable of variable speeds with an adjustable breech size, and firing curve balls by the use of two rubber-coated steel fingers at the muzzle of the pitcher. He successfully introduced the machine to the University of Minnesota, where Hinton worked as an assistant professor until 1900, when he resigned to move to the U.S. Naval Observatory in Washington, D.C.
At the end of his life, Hinton worked as an examiner of chemical patents for the United States Patent Office. He died unexpectedly of a cerebral hemorrhage on April 30, 1907. One source colorfully suggests that his death came when he died suddenly after being asked to give a toast to "female philosophers" at the Society of Philanthropic Inquiry meeting. *Wik

1977 Charles Fox (17 March 1897 in London, England - 30 April 1977 in Montreal, Canada) Fox's main contributions were on hypergeometric functions, integral transforms, integral equations, the theory of statistical distributions, and the mathematics of navigation. In the theory of special functions he introduced an H-function with a formal definition. It is a type of generalisation of a hypergeometric function and related ideas can be found in the work of Salvatore Pincherle, Hjalmar Mellin, Bill Ferrar, Salomon Bochner and others. He wrote only one book An introduction to the calculus of variations (1950, 2nd edition 1963, reprinted 1987). *SAU

1989 Gottfried Maria Hugo Köthe (25 December 1905 in Graz; 30 April 1989 in Frankfurt) was an Austrian mathematician working in abstract algebra and functional analysis. Köthe received a fellowship to visit the University of Göttingen, where he attended the lectures of Emmy Noether and Bartel van der Waerden on the emerging subject of abstract algebra. He began working in ring theory and in 1930 published the Köthe conjecture stating that a sum of two left nil ideals in an arbitrary ring is a nil ideal. By a recommendation of Emmy Noether, he was appointed an assistant of Otto Toeplitz in Bonn University in 1929–1930. During this time he began transition to functional analysis. He continued scientific collaboration with Toeplitz for several years afterward. Köthe's best known work has been in the theory of topological vector spaces. In 1960, volume 1 of his seminal monograph Topologische lineare Räume was published (the second edition was translated into English in 1969). It was not until 1979 that volume 2 appeared, this time written in English. He also made contributions to the theory of lattices.*WIK

2011 Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. He is renowned for being the "prime architect" of higher algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.
Quillen was a Putnam Fellow in 1959.
Quillen retired at the end of 2006. He died from complications of Alzheimer's disease on April 30, 2011, aged 70, in Florida. *Wik

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

## Wednesday 29 April 2020

### On This Day in Math - April 29

Science is built up of facts, as a house is with stones.
But a collection of facts is no more a science
than a heap of stones is a house.
~Henri Poincare

The 120th day of the year; All primes (except 2 and 3) are of form 6*n +/- 1. Note that 120 = 6*20 is the smallest multiple of six such that neither 6n+1 or 6n-1 is prime. *Prime Curios Can you find the next

120 = 3¹ + 3² + 3³ + 3⁴

(There are only three days of the year that appear in the arithmetic triangle more than five times. What are the other two?)

6 and 28 are prefect numbers because the sum of their proper divisors is equal to the number.  120 is the only year date that is a multi-perfect number.  The sum of its proper divisors is 2 * 120. (known since antiquity, the second smallest was discovered by Fermat in 1636 is 672. Fermat actually showed a method to find an infinite number of  such "sous-doubles".)

120 is the largest number of spheres that can contract a central sphere in eight dimensions. Beyond the fourth dimension, this "kissing number" is only known for the eighth and 24th dimensions.

EVENTS

In 1699, the French Academy of Sciences held its first public meeting, in the Louvre. *TIS

1756 Benjamin Franklin was elected a Fellow of the Royal Society on April 29, 1756. Under the rules candidates had to be recommended in writing by three or more Fellows acquainted with him “either in person or by his Works,” the recommendation had to be approved by the Council, and the certificate publicly displayed at “ten several ordinary meetings” before balloting. Nothing more was required of foreign fellows. British (including colonial) fellows, however, had to pay an admission fee (five guineas after 1752) and a sum of £21 “for the use of the Society in lieu of Contributions,” or give bond for that amount. Only then was a British subject deemed to be a fellow and entitled to be registered in the Journal-Book and be included in the printed List of Fellows. To attend meetings and vote in elections British fellows had also to sign the obligation to “endeavour to promote the Good of the Royall Society … and to pursue the Ends for which the same was formed.” *Franklin Papers, Natl. Archives

1831 Weber is offered the position of full professor of Physics at Gottingen to fill the position of Tobias Mayer, partially on the recommendation of Gauss.

1832 Evariste Galois released from prison. On (1831)Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. For this, he was again arrested and this time sentenced to six months in prison for illegally wearing a uniform. He was released on April 29, 1832. During his imprisonment, he continued developing his mathematical ideas.*Wik (He will be shot on the morning of May 30, and die the next day, 1832)

1854 Lincoln University, the ﬁrst university for Blacks, is incorporated. Lincoln University of the Commonwealth of Pennsylvania was chartered in April 1854 as Ashmun Institute. As Horace Mann Bond, '23, the eighth president of Lincoln University, so eloquently cites in the opening chapter of his book, Education for Freedom, this was "the first institution found anywhere in the world to provide a higher education in the arts and sciences for male youth of African descent." The story of Lincoln University goes back to the early years of the 19th century and to the ancestors of its founder, John Miller Dickey, and his wife, Sarah Emlen Cresson. The Institute was re-named Lincoln University in 1866 after President Abraham Lincoln. *Lincoln University web site

1901 Math Blunder succeeds, "But a more recent, a veritably shocking, example is at hand. On April 29, 1901, a Mr. Israel Euclid Eabinovitch submitted to the Board of University Studies of the Johns Hopkins University, in conformity with the requirements for the degree of doctor of philosophy, a dissertation in which, after an introduction full of the most palpable blunders, he proceeds to persuade himself that he proves Euclid's parallel postulate by using the worn-out device of attacking it from space of three dimensions, a device already squeezed dry and discarded by the very creator of non-Euclidean geometry, John Bolyai. And his dissertation was accepted by the referees. (Science Monthly, Vol 67, page 642)

In 1878, “a monument, in memory of the great physicist, Alessandro Volta, was unveiled at Pavia. Most of the Italian Universities, and several foreign scientific societies had sent deputies to Pavia University for this event. The monument is a masterpiece of the sculptor Tantardini of Milan. The ceremony of unveiling was followed by a dignified celebration at the University, and upon that occasion the following gentlemen were elected honorary doctors of the scientific faculty: Professors Clerk Maxwell (Cambridge) and Sir W. Thomson (Glasgow); M. Dumas (Paris), Dr. W. E. Weber (Leipzig); Professors Bunsen (Heidelberg) and Helmholtz (Berlin), Dr. F. H. Neumann (Koenigsberg), and Dr. P. Riess (Berlin).”*TIS

1925 The ﬁrst woman, F. R. Sabin, is elected to the National Academy of Sciences (Kane, p. 945). *VFR She was a histology professor at Johns Hopkins University. When (who) was the ﬁrst woman mathematician elected?

1931 Robert Lee Moore elected to the National Academy of Sciences. *VFR

BIRTHS

1667 John Arbuthnot (baptised April 29, 1667 – February 27, 1735), fellow of the Royal College of Physicians. In 1710, his paper “An argument for divine providence taken form the constant regularity observ’s in the bith of both sexes” gave the ﬁrst example of statistical inference. In his day he was famous for his political satires, from which we still know the character John Bull. *VFR
He inspired both Jonathan Swift's Gulliver's Travels book III and Alexander Pope's Peri Bathous, Or the Art of Sinking in Poetry, Memoirs of Martin Scriblerus,m (Wikipedia) He also translated Huygens' "De ratiociniis in ludo aleae " in 1692 and extended it by adding a few further games of chance. This was the first work on probability published in English.*SAU

1850 William Edward Story (April 29, 1850 in Boston, Massachusetts, U.S. - April 10, 1930 in Worcester, Massachusetts, U.S.) He taught at Johns Hopkins with Sylvester and then moved on to Clark University which was, during the early 1890’s, the strongest mathematics department in the country. In the 1890’s he edited the short lived Mathematical Reviews.*VFR

1854 Jules Henri Poincare (29 April 1854 – 17 July 1912) born in Nancy, France. He did important work in function theory, alge­braic geometry, number theory, algebra, celestial mechanics, diﬀerential equations, mathemat­ical physics, algebraic topology, and philosophy of mathematics. There may never be another universal mathematician like Poincar´e. *VFR His Poincaré Conjecture holds that if any loop in a given three-dimensional space can be shrunk to a point, the space is equivalent to a sphere. Its proof remains an unsolved problem in topology. He influenced cosmogony, relativity, and topology. In applied mathematics he also studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, and cosmology. He is often described as the last universalist in mathematics. He studied the three-body-problem in celestial mechanics, and theories of light and electromagnetic waves. He was a co-discoverer (with Albert Einstein and Hendrik Lorentz) of the special theory of relativity. *TIS

1872 Forest Ray Moulton (29 Apr 1872 (in a log cabin near the small town of Leroy, Michigan); 7 Dec 1952 at age 80) American astronomer who collaborated with Thomas Chamberlin in advancing the planetesimal theory of the origin of the solar system (1904). They suggested filaments of matter were ejected when a star passed close to the Sun, which cooled into tiny solid fragments, “planetesimals.” Over a very long period, grains collided and stuck together. Continued accretion created pebbles, boulders, and eventually larger bodies whose gravitational force of attraction accelerated the formation of protoplanets. (This formation by accretion is still accepted, but not the stellar origin of the planetesimals.) Moulton was first to suggest that the smaller satellites of Jupiter discovered by Nicholson and others in the early 20th century were captured asteroids, now widely accepted. *TIS

1906 Eugène Ehrhart (29 April 1906 Guebwiller – 17 January 2000 Strasbourg) was a French mathematician who introduced Ehrhart polynomials in the 1960s. Ehrhart received his high school diploma at the age of 22. He was a mathematics teacher in several high schools, and did mathematics research on his own time. He started publishing in mathematics in his 40s, and finished his PhD thesis at the age of 60. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem. *Wik

1926 Vera Nikolaevna Maslennikova (29 April 1926, Priluki, Russia - 14 August 2000) Gelfond supervised her diploma work at Moscow and Sobolev directed her Ph.D. at the Steklov Mathematical Institute. She has published more than 80 papers in the theory of partial diﬀerential equations, the mathematical hydrodynamics of rotating ﬂuids, and in function spaces.*VFR She has worked in the field of partial differential equations, the mathematical hydrodynamics of rotating fluids, and in function spaces, having published more than one hundred and forty research papers. *Wik

1928 Laszlo Belady,( April 29, 1928 in Budapest - ) creator of the Belady algorithm (used in optimizing the performance of computers), is born. Belady worked at IBM for 23 years in software engineering before joining the Mitsubishi Electronics Research Laboratory in the mid-1980s. He wins numerous awards, including the J.D. Warnier Prize for Excellence in Information and an IEEE fellowship. *CHM

1930 Yuan Wang (29 April 1930 in Lanhsi, Zhejiang province, China - )Most of Wang Yuan's research has been in the area of number theory. He looked at sieve methods and applied them to the Goldbach Conjecture. He also applied circle methods to the Goldbach Conjecture. In 1956 he published (in Chinese) On the representation of large even integer as a sum of a prime and a product of at most 4 primes in which he assumed the truth of the Riemann hypothesis and with that assumption proved that every large even integer is the sum of a prime and of a product of at most 4 primes. He also proved that there are infinitely many primes p such that p + 2 is a product of at most 4 primes. In 1957 Wang Yuan published four papers: On sieve methods and some of their applications; On some properties of integral valued polynomials; On the representation of large even number as a sum of two almost-primes; and On sieve methods and some of the related problems.*SAU

1936 Volker Strassen
(April 29, 1936 - ) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz. Strassen began his researches as a probabilist; his 1964 paper An Invariance Principle for the Law of the Iterated Logarithm defined a functional form of the law of the iterated logarithm, showing a form of scale invariance in random walks. This result, now known as Strassen's invariance principle or as Strassen's law of the iterated logarithm, has been highly cited and led to a 1966 presentation at the International Congress of Mathematicians.
In 1969, Strassen shifted his research efforts towards the analysis of algorithms with a paper on Gaussian elimination, introducing Strassen's algorithm, the first algorithm for performing matrix multiplication faster than the O(n3) time bound that would result from a naive algorithm. In the same paper he also presented an asymptotically-fast algorithm to perform matrix inversion, based on the fast matrix multiplication algorithm. This result was an important theoretical breakthrough, leading to much additional research on fast matrix multiplication, and despite later theoretical improvements it remains a practical method for multiplication of dense matrices of moderate to large sizes. In 1971 Strassen published another paper together with Arnold Schönhage on asymptotically-fast integer multiplication based on the fast Fourier transform; see the Schönhage–Strassen algorithm. Strassen is also known for his 1977 work with Robert M. Solovay on the Solovay–Strassen primality test, the first method to show that testing whether a number is prime can be performed in randomized polynomial time and one of the first results to show the power of randomized algorithms more generally.*Wik

DEATHS

1713 Francis Hauksbee the elder (baptized on 27 May 1660 in Colchester–buried in St Dunstan's-in-the-West, London on 29 April 1713.), also known as Francis Hawksbee, was an 18th-century English scientist best known for his work on electricity and electrostatic repulsion.
Initially apprenticed in 1678 to his elder brother as a draper, Hauksbee became Isaac Newton’s lab assistant. In 1703 he was appointed curator, instrument maker and experimentalist of the Royal Society by Newton, who had recently become president of the society and wished to resurrect the Royal Society’s weekly demonstrations.
Until 1705, most of these experiments were air pump experiments of a mundane nature, but Hauksbee then turned to investigating the luminosity of mercury which was known to emit a glow under barometric vacuum conditions.
By 1705, Hauksbee had discovered that if he placed a small amount of mercury in the glass of his modified version of Otto von Guericke's generator, evacuated the air from it to create a mild vacuum and rubbed the ball in order to build up a charge, a glow was visible if he placed his hand on the outside of the ball. This glow was bright enough to read by. It seemed to be similar to St. Elmo's Fire. This effect later became the basis of the gas-discharge lamp, which led to neon lighting and mercury vapor lamps. In 1706 he produced an 'Influence machine' to generate this effect. He was elected a Fellow of the Royal Society the same year.
Hauksbee continued to experiment with electricity, making numerous observations and developing machines to generate and demonstrate various electrical phenomena. In 1709 he published Physico-Mechanical Experiments on Various Subjects which summarized much of his scientific work.
In 1708, Hauksbee independently discovered Charles' law of gases, which states that, for a given mass of gas at a constant pressure, the volume of the gas is proportional to its temperature.
The Royal Society Hauksbee Awards, awarded in 2010, were given by the Royal Society to the “unsung heroes of science, technology, engineering and mathematics.” *Wik

1862 John Edward Campbell (27 May 1862, Lisburn, Ireland – 1 October 1924, Oxford, Oxfordshire, England) is remembered for the Campbell-Baker-Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras. *SAU His 1903 book, Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups, popularized the ideas of Sophus Lie among British mathematicians.
He was elected a Fellow of the Royal Society in 1905, and served as President of the London Mathematical Society from 1918 to 1920. *Wik  *Renaissance Mathematicus

1864 Charles-Julien Brianchon (19 Dec 1783, 29 Apr 1864 at age 80) French mathematician who published a geometrical theorem (named as Brianchon's theorem) while a student (1806). He showed that in any hexagon formed of six tangents to a conic, the three diagonals meet at a point. (Conics include circles, ellipses, parabolas, and hyperbolas.) In fact, this theorem is simply the dual of Pascal's theorem which was proved in 1639. After graduation, Brianchon became a lieutenant in artillery fighting in Napoleon's army until he left active service in 1813 due to ill health. His last work in mathematics made the first use of the term "nine-point circle." By 1823, Brianchon's interests turned to teaching and to chemistry. *TIS

1872 Jean-Marie-Constant Duhamel (5 Feb 1797, 29 Apr 1872 at age 75) French mathematician and physicist who proposed a theory dealing with the transmission of heat in crystal structures based on the work of the French mathematicians Jean-Baptiste-Joseph Fourier and Siméon-Denis Poisson. *TIS

1894 Giuseppe Battaglini (11 Jan 1826 in Naples, Kingdom of Naples and Sicily (now Italy) - 29 Apr 1894 in Naples, Italy ) Some of Battaglini's results have proved significant. For example, in his doctoral dissertation of 1868, Klein introduced a classification scheme for second-degree line complexes based on Battaglini's earlier work. However, his main importance is his modern approach to mathematics which played a major role in invigorating the Italian university system, particularly in his efforts to bring the non-Euclidean geometry of Lobachevsky and Bolyai to the Italian speaking world. Jules Hoüel played a similar role for non-Euclidean geometry in the French speaking world and the correspondence between the two (see [6]) provides a vivid picture of the reactions of both the French and the Italian mathematical communities against the non-Euclidean geometries. Battaglini and Hoüel also exchanged ideas relating to mathematical education in various European countries. In particular they debated the use of Euclid's Elements as a textbook for teaching elementary geometry in schools. *SAU

1916 – Jørgen Pedersen Gram (June 27, 1850 – April 29, 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark.
Important papers of his include On series expansions determined by the methods of least squares, and Investigations of the number of primes less than a given number. The mathematical method that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. The Gramian matrix is also named after him.
For number theorists his main fame is the series for the Riemann zeta function (the leading function in Riemann's exact prime-counting function). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function.
Gram was the first mathematician to provide a systematic theory of the development of skew frequency curves, showing that the normal symmetric Gaussian error curve was but one special case of a more general class of frequency curves.
He died after being struck by a bicycle.*Wik

1951 Ludwig Josef Johann Wittgenstein (26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947. In his lifetime he published just one book review, one article, a children's dictionary, and the 75-page Tractatus Logico-Philosophicus (1921). In 1999 his posthumously published Philosophical Investigations (1953) was ranked as the most important book of 20th-century philosophy, standing out as "...the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations". Bertrand Russell described him as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating". *Wik He died three days after his birthday. He is buried in a cemetery off Huntington Road in Cambridge, UK.

1970 Paul Finsler (born 11 April 1894, in Heilbronn, Germany,- 29 April 1970 in Zurich, Switzerland)Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory. He joined the faculty of the University of Zurich in 1927, and was promoted to ordinary professor there in 1944.
Finsler's thesis work concerned differential geometry, and Finsler spaces were named after him by Élie Cartan in 1934. The Hadwiger–Finsler inequality, a relation between the side lengths and area of a triangle in the Euclidean plane, is named after Finsler and his co-author Hugo Hadwiger. Finsler is also known for his work on the foundations of mathematics, developing a non-well-founded set theory with which he hoped to resolve the contradictions implied by Russell's paradox.
In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler. It states that if a triangle in the plane has side lengths a, b and c and area A, then
$a^{2} + b^{2} + c^{2} \geq (a - b)^{2} + (b - c)^{2} + (c - a)^{2} + 4 \sqrt{3} A \quad \mbox{(HF)}.$
Weitzenböck's inequality is a straightforward corollary of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths a, b and c and area A, then
$a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} A \quad \mbox{(W)}.$
Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. a = b = c.
*Wik

2008 Mary Golda Ross (August 9, 1908 – April 29, 2008) was the first known Native American female engineer, and the first female engineer in the history of Lockheed. She was one of the 40 founding engineers of the renowned and highly secretive Skunk Works project at Lockheed Corporation. She worked at Lockheed from 1942 until her retirement in 1973, where she was best remembered for her work on aerospace design – including the Agena Rocket program – as well as numerous "design concepts for interplanetary space travel, crewed and uncrewed Earth-orbiting flights, the earliest studies of orbiting satellites for both defense and civilian purposes." In 2018, she was chosen to be depicted on the 2019 Native American \$1 Coin by the U.S. Mint celebrating American Indians in the space program. *Wik

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 - April 28

One of the principal objects of theoretical research
in my department of knowledge
is to find the point of view from which
the subject appears in its greatest simplicity.
Willard Gibbs (1839 - 1903)

The 119th day of the year; the largest amount of US money one can have in coins without being able to make change for a dollar is 119 cents. *Tanya Khovanova, Number Gossip

119 is the product of the first two primes ending with 7

119 is the sum of five consecutive primes (17 + 19 + 23 + 29 + 31).

119 is the order of the largest cyclic subgroups of the Monster group.

EVENTS

1664 Trinity College, Cambridge awards a scholarship to Isaac Newton to study for his Master's Degree, thus ending his period as a lowly sizar earning his tuition by cleaning up after wealthier students. Within months his formal education would be put on hold as the college closed under the assault of the plague.

1673 Leeuwenhoeck writes his first letter to the Royal Society, which would be published the next month, May 19, in Philosophical Transactions number 94, "A Specimen of Some Observations Made by a Microscope, Contrived by M. Leewenhoeck in Holland, Lately Communicated by Dr. Regnerus de Graaf." Constantijn Huygens, who lived not far from Delft, visited Leeuwenhoek and read the letter. A week before Leeuwenhoek sent it, Huygens sent his own letter to Robert Hooke that acted as a cover letter and recommendation similar to de Graaf's letter in April. Over the rest of Leeuwenhoeck's life, the Society would publish 116 articles containing excerpts from 113 letters. *lensonleeuwenhoek

1686 Newton shows the handwritten copy of his Principia to the Royal Society. *VFR
28 April 1686 "Dr. Vincent presented a manuscript treatise entitled Philosophiae Naturalis principia mathematica, and dedicated to the Society by Mr. Isaac Newton,..." Minutes of the RS written by Halley clerk to the Society. (It was actually only the manuscript of Book I) *Thony Christie

1693 Leibniz, in a letter to L’Hopital, explains his discovery of determinants. This work was ﬁfty years before that of Cramer who was the real driving force in the development of determinants. Leibniz’s work had no inﬂuence because it was not published until 1850 in his Mathematische Schriften. [Smith, Source Book, p. 267] *VFR
Leibniz was convinced that good mathematical notation was the key to progress so he experimented with different notation for coefficient systems. His unpublished manuscripts contain more than 50 different ways of writing coefficient systems which he worked on during a period of 50 years beginning in 1678. Only two publications (1700 and 1710) contain results on coefficient systems and these use the same notation as in his letter to de l'Hôpital mentioned above.
Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant. He proved various results on resultants including what is essentially Cramer's rule. He also knew that a determinant could be expanded using any column - what is now called the Laplace expansion. As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory. In the 1730's Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death. It contains the first published results on determinants proving Cramer's rule for 2 X 2 and 3X 3 systems and indicating how the 4 X 4 case would work. Cramer gave the general rule for n X  n systems in his book Introduction to the analysis of algebraic curves (1750). It arose out of a desire to find the equation of a plane curve passing through a number of given points. The rule appears in an Appendix to the book but no proof is given] *SAU (edited and corrected with suggestions by Dave Renfro)
Dave adds:   a 715 page book (xxiii + 680 + xii pages), which is freely available on the internet. Cramer's rule itself appears in Appendix 2 (pp. 657-676). Cramer's book itself was motivated by Newton's work in classifying cubic curves, and I believe he was one of three mathematicians that devoted an extensive study to Newton's classification in the 1700s. (I don't remember who the other two were, but I believe one of them was Euler.) There is an excellent annotated and translation of Newton's work published in 1860 and freely available on the internet:

"Sir Isaac Newton's Enumeration of Lines of the Third Order, Generation of
Curves by Shadows, Organic Description of Curves, and Construction of
Equations by Curves", Translated from the Latin, with notes and examples,
by C.R.M. Talbot, 1860.

http://name.umdl.umich.edu/ABQ9451.0001.001
Thanks again to Dave for the corrections.

1817 Gauss wrote the astronomer H. W. M. Olbers, “I am becoming more and more convinced that the necessity of our [Euclidean] geometry cannot be proved, at least not by human intellect nor for the human intellect.” [G. E. Martin, Foundations of Geometry and the Non-Euclidean Plane, p. 306] *VFR

1983 Greece issued a stamp portraying Archimedes and his Hydrostatic Principle

1897 In a letter to Fuchs, Dedekind expressed skepticism of a tale about Gauss attempting to light his pipe with a copy of his DA
Schering in Gottingen in response to a note from Fuchs that he had found materials related to Guass' Disquisitiones Arithmetica in the papers of Dirichlet had described a story that he had shared with Kronecker a decade before,
"The piece of Guass's Disquisitiones Arithmeticiae, which is found among Dirichlet's papers, is probably that portion which, as Dirichlet told me himself, he saved from the hand of Gauss when the latter lit his pipe with his manuscript of the Disquisitiones Arithmeticae on the day of his doctoral jubilee."
Dedekind reasoned, if Guass had saved the paper for fifty years he obviously valued it, and that if the anecdote were true, Dirichlet surely would have shared it with him as well.
*Uta Merzbach, An Early Version of Gauss's Disquisitiones Arithmeticae, Mathematical Perspectives, 1981

2004 At 11:50 AM a paper was submitted electronically to the American Mathematical Monthly which purports to be the shortest journal entry ever, essentially two words," n2 + 2 can". After some correspondence back and forth, (the journal suggested, "a line or two of explanatin might help") the paper was accepted as a "filler" in the January 2005 issue. *wfnmc.org

2012 Mountain View, Ca—January 19, 2012—
The Computer History Museum (CHM), the world’s leading institution exploring the history of computing and its ongoing impact on society, today announced its 2012 Fellow Award honorees: Edward A. Feigenbaum, pioneer of artificial intelligence and expert systems; Steve Furber and Sophie Wilson, chief architects of the ARM processor architecture; and Fernando J. Corbató, pioneer of timesharing and the Multics operating system. The four Fellows will be inducted into the Museum’s Hall of Fellows on Saturday, April 28, 2012, at a formal ceremony where Silicon Valley insiders, technology leaders, and Museum supporters will gather to celebrate the accomplishments of the Fellows and their impact on society. This year’s celebration commemorates the 25th Anniversary of the Fellow Awards and will reunite pioneers from more than two decades. *CHM

BIRTHS
1765 Sylvestre François Lacroix (April 28, 1765, Paris – May 24, 1843, Paris) was a French mathematician. He displayed a particular talent for mathematics, calculating the motions of the planets by the age of 14. In 1782 at the age of 17 he became an instructor in mathematics at the École Gardes de Marine in Rochefort, France. He returned to Paris and taught courses in astronomy and mathematics at the Lycée. In 1787 he was the co-winner of that year's Grand Prix of the French Académie des Sciences, but was never awarded the prize. The same year the Lycée was abolished and he again moved to the provinces.
In Besançon he taught course in mathematics, physics, and chemistry at the École d'Artillerie. In 1793 he became examiner of the Artillery Corps, replacing Pierre-Simon Laplace in the post. By 1794 he was aiding his old instructor, Gaspard Monge, in creating material for a course on descriptive geometry. In 1799 he was appointed professor at the École Polytechnique. Lacroix produced most of his texts for the sake of improving his courses. The same year he was voted into the newly formed Institut National des Sciences et des Arts. In 1812 he began teaching at the Collège de France, and was appointed chair of mathematics in 1815.
During his career he produced a number of important textbooks in mathematics. Translations of these books into the English language were used in British universities, and the books remained in circulation for nearly 50 years. In 1812 Babbage set up The Analytical Society for the translation of Differential and Integral Calculus and the book was translated into English in 1816 by George Peacock. *Wik He coined the term “analytic geometry.” *VFR

1773 Robert Woodhouse (28 April 1773 – 23 December 1827) was an English mathematician. He was born at Norwich and educated at Caius College, Cambridge, (BA 1795) of which society he was subsequently a fellow. He was elected a Fellow of the Royal Society in December 1802.
His earliest work, entitled the Principles of Analytical Calculation, was published at Cambridge in 1803. In this he explained the differential notation and strongly pressed the employment of it; but he severely criticized the methods used by continental writers, and their constant assumption of non-evident principles. This was followed in 1809 by a trigonometry (plane and spherical), and in 1810 by a historical treatise on the calculus of variations and isoperimetrical problems. He next produced an astronomy; of which the first book (usually bound in two volumes), on practical and descriptive astronomy, was issued in 1812, and the second book, containing an account of the treatment of physical astronomy by Pierre-Simon Laplace and other continental writers, was issued in 1818.
He became the Lucasian Professor of Mathematics in 1820, and subsequently the Plumian professor in the university. As Plumian Professor he was responsible for installing and adjusting the transit instruments and clocks at the Cambridge Observatory. He held that position until his death in 1827.
On his death in Cambridge he was buried in Caius College Chapel.*Wik He was interested in the “metaphysics of the calculus,” i.e., questions such as the proper theoretical foundations of the calculus, the role of geometric and analytic methods, and the importance of notation. *VFR

1774 Francis Baily (28 April 1774 – 30 August 1844) was an English astronomer. He is most famous for his observations of 'Baily's beads' during an eclipse of the Sun. Bailey was also a major figure in the early history of the Royal Astronomical Society, as one of the founders and president four times.
Baily was born at Newbury in Berkshire in 1774 to Richard Baily. After a tour in the unsettled parts of North America in 1796–1797, his journal of which was edited by Augustus de Morgan in 1856, Baily entered the London Stock Exchange in 1799. The successive publication of Tables for the Purchasing and Renewing of Leases (1802), of The Doctrine of Interest and Annuities (1808), and The Doctrine of Life-Annuities and Assurances (1810), earned him a high reputation as a writer on life-contingencies; he amassed a fortune through diligence and integrity and retired from business in 1825, to devote himself wholly to astronomy.
His observations of "Baily's Beads", during an annular eclipse of the sun on 15 May 1836, at Inch Bonney in Roxburghshire, started the modern series of eclipse expeditions. The phenomenon, which depends upon the irregular shape of the moon's limb, was so vividly described by him as to attract an unprecedented amount of attention to the total eclipse of 8 July 1842, observed by Baily himself at Pavia. *Wik

1831 Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical discipline. His name is known in Graph theory mainly for Tait's conjecture.*Wik (His conjecture was proved wrong by counterexample in 1946 by W. T. Tutte. The problem is related to the four color theorem.) He helped develop quaternions, an advanced algebra that gave rise to vector analysis and was instrumental in the development of modern mathematical physics. *TIS
Below is The First Seven Orders of Knottiness"-table compiled by P.G. Tait in 1884 with a big hat-tip to Ben Gross@bhgross144 .

1854 Phoebe Sarah Hertha Ayrton (28 April 1854 – 23 August 1923), was a British engineer, mathematician, physicist, and inventor. Known in adult life as Hertha Ayrton, born Phoebe Sarah Marks, she was awarded the Hughes Medal by the Royal Society for her work on electric arcs and ripples in sand and water.
In 1880, Ayrton passed the Mathematical Tripos, but Cambridge did not grant her an academic degree because, at the time, Cambridge gave only certificates and not full degrees to women. Ayrton passed an external examination at the University of London, which awarded her a Bachelor of Science degree in 1881.
In 1899, she was the first woman ever to read her own paper before the Institution of Electrical Engineers (IEE). Her paper was entitled "The Hissing of the Electric Arc". Shortly thereafter, Ayrton was elected the first female member of the IEE; the next woman to be admitted to the IEE was in 1958. She petitioned to present a paper before the Royal Society but was not allowed because of her sex and "The Mechanism of the Electric Arc" was read by John Perry in her stead in 1901. Ayrton was also the first woman to win a prize from the Society, the Hughes Medal, awarded to her in 1906 in honour of her research on the motion of ripples in sand and water and her work on the electric arc. By the late nineteenth century, Ayrton's work in the field of electrical engineering was recognised more widely, domestically and internationally. At the International Congress of Women held in London in 1899, she presided over the physical science section. Ayrton also spoke at the International Electrical Congress in Paris in 1900. Her success there led the British Association for the Advancement of Science to allow women to serve on general and sectional committees. *Wik

1868 Georgy Fedoseevich Voronoy (also voronoi)(28 April 1868 – 20 November 1908) introduced what are today called Voronoi diagrams or Voronoi tessellations. Today they have wide applications to the analysis of spatially distributed data, so have become important in topics such as geophysics and meteorology. Although known under different names, the notion occurs in condensed matter physics, and in the study of Lie groups. (Two dimensional diagrams of Voronoi type were considered as early at 1644 by René Descartes and were used by Dirichlet (1850) in the investigation of positive quadratic forms. They were also studied by Voronoi (1907), who extended the investigation of Voronoi diagrams to higher dimensions. They find widespread applications in areas such as computer graphics, epidemiology, geophysics, and meteorology. A particularly notable use of a Voronoi diagram was the analysis of the 1854 cholera epidemic in London, in which physician John Snow determined a strong correlation of deaths with proximity to a particular (and infected) water pump on Broad Street. *Mathworld)

1900 Jan Hendrik Oort (28 April 1900 – 5 November 1992) was a Dutch astronomer who made significant contributions to the understanding of the Milky Way and who was a pioneer in the field of radio astronomy. His New York Times obituary called him “one of the century's foremost explorers of the universe;” the European Space Agency website describes him as, “one of the greatest astronomers of the 20th century,” and states that he “revolutionised astronomy through his ground-breaking discoveries.” In 1955, Oort’s name appeared in Life Magazine’s list of the 100 most famous living people. He has been described as “putting the Netherlands in the forefront of postwar astronomy.”

Oort determined that the Milky Way rotates and overturned the idea that the Sun was at its center. He also postulated the existence of the mysterious invisible dark matter in 1932, which is believed to make up roughly 84.5% of the total matter in the Universe and whose gravitational pull causes “the clustering of stars into galaxies and galaxies into connecting strings of galaxies.” He discovered the galactic halo, a group of stars orbiting the Milky Way but outside the main disk. Additionally Oort is responsible for a number of important insights about comets, including the realization that their orbits “implied there was a lot more solar system than the region occupied by the planets.”

The Oort cloud, the Oort constants, and the Asteroid, 1691 Oort, were all named after him. *Wik

1906 Kurt Godel (April 28, 1906 – January 14, 1978) Austrian-born US mathematician, logician, and author of Gödel's proof. He is best known for his proof of Gödel's Incompleteness Theorems (1931) He proved fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. This ended a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis.*TIS

1906 Richard Rado FRS(28 April 1906 – 23 December 1989) was a Jewish German mathematician. He earned two Ph.D.s: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He made contributions in combinatorics and graph theory. He wrote 18 papers with Paul Erdős. In 1964, he discovered the Rado graph (The Rado graph contains all finite and countably infinite graphs as induced subgraphs..)
In 1972, he was awarded the Senior Berwick Prize*Wik

1923 Fritz Joseph Ursell FRS (28 April 1923 – 11 May 2012) was a British mathematician noted for his contributions to fluid mechanics, especially in the area of wave-structure interactions.[5] He held the Beyer Chair of Applied Mathematics at the University of Manchester from 1961–1990, was elected Fellow of the Royal Society in 1972 and retired in 1990.
Ursell came to England as a refugee in 1937 from Germany. From 1941 to 1943 he studied at Trinity College, Cambridge, graduating with a bachelor degree in mathematics. *Wik

DEATHS
1843 William Wallace (23 September 1768, Dysart—28 April 1843, Edinburgh) worked on geometry and discovered the (so-called)
Simson line of a triangle.*SAU In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace.*Wik
Mary Sommerville was one of his students.  He succeeded John Playfair as Math Chair in Edinburgh. He also invented a complicated type of pantograph called the eidograph.

1903 Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist and chemist known for contributions to vector analysis and as one of the founders of physical chemistry. In 1863, He was awarded Yale University's first engineering doctorate degree. His major work was in developing thermodynamic theory, which brought physical chemistry from an empirical inquiry to a deductive science. In 1873, he published two papers concerning the fundamental nature of entropy of a system, and established the "thermodynamic surface," a geometrical and graphical method for the analysis of the thermodynamic properties of substances. His famous On the Equilibrium of Homogeneous Substances, published in 1876, established the use of "chemical potential," now an important concept in physical chemistry. *TIS
He is buried at the  Grove Street Cemetery in New Haven Connecticut, USA.

1946 Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 – April 28, 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, which was part of his PhD thesis The Theory of Speculation, (published 1900).
His thesis, which discussed the use of Brownian motion to evaluate stock options, is historically the first paper to use advanced mathematics in the study of finance. Thus, Bachelier is considered a pioneer in the study of financial mathematics and stochastic processes. *Wik Bachelier is now recognised internationally as the father of financial mathematics, but this fame, which he so justly deserved, was a long time coming. The Bachelier Society, named in his honour, is the world-wide financial mathematics society and mathematical finance is now a scientific discipline of its own. The Society held its first World Congress on 2000 in Paris on the hundredth anniversary of Bachelier's celebrated PhD Thesis, Théorie de la Spéculation *SAU

1986 R H Bing (October 20, 1914, Oakwood, Texas – April 28, 1986, Austin, Texas) He wrote papers on general topology, particularly on metrization; planar sets where he examined in particular planar webs, cuttings and planar embeddings. He worked on topological classification of the 2-sphere, the 3-sphere, pseudo arcs, simple closed curves and Hilbert space. He studied partitions and decompositions of locally connected continua. He considered several different aspects of 3-manifolds including decompositions, maps, approximating surfaces, recognizing tameness, triangulation and the Poincaré conjecture. *SAU Oakwood had a population of 471 at the 2000 census.

1991 Paul Ernest Klopsteg (May 30, 1889 – April 28, 1991) was an American physicist. The asteroid 3520 Klopsteg was named after him and the yearly Klopsteg Memorial Award was founded in his memory.
He performed ballistics research during World War I at the US Army's Aberdeen Proving Grounds in Maryland. He applied his knowledge of ballistics to the study of archery.
He was director of research at Northwestern University Technical Institution. From 1951 through 1958 he was an associate director of the National Science Foundation and was president of the American Association for the Advancement of Science from 1958 through 1959.*Wik

1999 Arthur Leonard Schawlow (May 5, 1921 – April 28, 1999) was an American physicist. He is best remembered for his work on lasers, for which he shared the 1981 Nobel Prize in Physics with Nicolaas Bloembergen and Kai Siegbahn.
In 1991 the NEC Corporation and the American Physical Society established a prize: the Arthur L. Schawlow Prize in Laser Science. The prize is awarded annually to "candidates who have made outstanding contributions to basic research using lasers."
In 1951, he married Aurelia Townes, younger sister to physicist Charles Hard Townes, and together they had three children; Arthur Jr., Helen, and Edith. Arthur Jr. was autistic, with very little speech ability.
Schawlow and Professor Robert Hofstadter at Stanford, who also had an autistic child, teamed up to help each other find solutions to the condition. Arthur Jr. was put in a special center for autistic individuals, and later Schawlow put together an institution to care for people with autism in Paradise, California. It was later named the Arthur Schawlow Center in 1999, shortly before his death on the 29th of April 1999.
Schawlow died of leukemia in Palo Alto, California.*Wik

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

## Tuesday 28 April 2020

### Everything's Coming up Polynomials, (and it gets complex)

So I've been doing what I imagine most mathematicians have been doing during the lockdown, prowling through old journals (1770 - 1970 mostly) a few old blogs and web pages I hadn't scanned in a while, and catching up when I could on who's tweeting what in math and history... and it seems that the things leaping out at me are all about polynomials, at least mostly.

So reading through my notes about Leonard of Pisa (I hate to call him Fibonacci, no body did when he was alive, except maybe himself once but I'm feeling doubtful.  And if you want to dive down that rabbit hole, here are my notes to start you off with "Who Was Fibonacci before He Was Fibonacci."  )
Leonardo's most famous book was the 1202 Liber Abaci, first because it had the rabbit problem, and second because it was really important in introducing Arabic base ten numbers to the West.  Perhaps number two on his hit parade was his 1225 Libre Quadratorum, Book of Squares.
Some of what he writes was known to the Pythagoreans, Proclus, and Eulcid, and almost all was known by the time of Diaphontus of Alexandria (about 100 AD). Certianly the proposition VI that I focus on was known to him, and to Brahamagupta, and in 1225 to Leonardo.  Then it shows up in my other reading, but first, Proposition VI.  It is sometimes called the Brahmagupta-Fibonacci identity, because Brahamagupta extend it even farther than this method, but the basic version is this.
If you have four numbers, and in the Pisano's words, "not in proportion, The first less than the second, and the third less than the fourth.." then, in modern terminology, The sum of the squares of the first two, times the sum of the squares of the last two, then the product will be expressible in two different ways as the sum of two squares.  And he tells you how to get those two numbers.  If the first numbers are a, b, c, and d, then (a^2 + b^2) (c^2 + d^2)  =  (ac+bd)^2 + (ad-bc)^2, and don't worry about the signs, because it works if you switch them (ac-bd)^2 + (ad+bc)^2.
Now it doesn't matter much what you start with , so let's do 1, 2, 4, 7 'cause they are easy.  The product of the sums of squares is (1^1 + 2^2)(4^2 +7^2) = 5(65) = 325.  So the first way to get this as a sum of two squares is (1*4+2*7)^2 + (1*7-4*2)^2= 18^2 +(-1)^2 = (Eureka) 325.
Ok, but it has to work the other way, so (1*4-2*7)^2 + (1*7+4*2)^2 = (-10)^2 + 15^2 = 100 + 225 = 325....again.
Immediately you recognize (of course you would, even I did) that if you did this with two numbers that were legs of Pythagorean right triangles, you could find two right triangles that had a common hypotenuse that was the product of the two original hypotenii.  And someone else who did that was the brilliant Francois Viete, around 1570 in his Genesis triagulorum.

Now I don't thing Diophantus or Fibonacci either worked with negative numbers, and neither did Viete, but he did throw a curve ball because well after Robert Record gave that perfectly good, and really long, equal sign in his Whetstone of Witte, 1557, Viete and some continental renegades continued to use the same symbol for the absolute difference of two numbers, so when he wrote a=b he meant what's the difference between the bigger and the smaller.  But he wrote the same rules.
Now I admit when I first used this my mind went right to FOIL (I was lucky to be teaching before the mnemonic was  sin), and the ac +bd chimed "first and last" to me, and ad + bc was outside inside, and that's how I remembered it.
And then reading Viete (translated thank you) it looked different, and for some reason for the first time I saw what everyone may be seeing who is clever, but I never had.
But before you actually think it, remember who we are reading here, Diaphontus, Leonardo de Pisa, Francois Viete, all of them living well before it was understood how to multiply and divide COMPLEX NUMBERS.

That's right, if like me it didn't hit you, let's rewrite our first problem as (1+2i)(4+7i)  Multiply away using the proper distributive property without an unsavory mnemonic, 1*4 + 2i*4 + 1*7i + 2i*7i, and we get  4 + 8i + 7i - 14, or -10 + 15i, and the (modulus, magnitude, absolute value, take your choice is \(\sqrt{(-10)^2 + 15^2} |)  And the other solution is just to mulitply by the conjugate of the second (1+2i)(4-7i)  = (18 + 1i)
Now Viete noticed something else "complex" about those numbers. And he described the two techniques with different names, the second (First and Last negative in my example)  he called synaereseos, from a Greek verb meaning 'taken together'.  The first (First and Last positive),  he referred to as diaeresos, meaning 'taken to pieces.' He stated clearly that if you took the difference between the two smaller angles in the first two triangles, In this case, the arctan(1/2) = 26.56 degrees; and arctan(4/7) = 29.74 degrees.  Not much difference, but then by his diaeresos, or conjugate method, the result is arctan(1/18) = 3.1798 degrees.
Because he focused on the smallest angle, and didn't care whether it was next to the x or the y axis, he didn't notice that had he stayed with the angle next to the x-axis for his reference, in the Synaereseos result, the angle would have been the sum 56.31 degrees of the two original angles.
But then he broke Leonardo of Pisa's first rule.  Remember he said pick four numbers not in proportion.  Veite said what if we make the two sets equal, that is, square the number.  Using (3,4) and (3,4), using his first (taking together) we get (9+16)^2 +(12-12)^2 =25^2 which sort of gets the answer, but the (taken apart) method gives (9-16)^2 + (12+12)^2 = 7^2 + 24^ 2 = 25^2.  If we preserve the angle of reference to be the one between the x axis and the hypotenuse, then the original 3,4,5 triangles have a measure of 36.869 degrees, their sum should be 73.739 degrees,exactly arctan(24/7) and their difference should be zero, perfect for the x=7, y=0.
You get the same answers if you do these by (3+4i)^2; and it works in general for (3+4i)^n of your choice.
Because I was looking at some old blogs of Prof Dan Kalman and he had some really clever notes about.... polynomials. Some of which reminded me of some notes I wrote after reading one of his blogs back in 2013. Some of it I will share here, ignoring all the parts that required some small amount of calculus; but if you want the whole enchilada, about what Professor Kalman calls the "Marden's Theorem":

I once read a description of math as like seeing islands in a great ocean covered by a mist.  As you learn the subject you work around on an island and clear away some of the mist.  Often your education jumps from one island to another at the direction of a teacher and eventually you have mental maps of parts of many separate islands.  But at some point, you clear away a fog on part of an island and see it connects off to another island you had partially explored, and now you know something deeper about both islands and the connectedness of math.

I was recently reminded of one of those kinds of connections that ties together several varied topics from the high school education of most good math students.  It starts with that over-criticized (and under-appreciated,  Algebra I technique of factoring.

Almost ever student in introductory algebra is introduced to a "sum and product" rule that relates the factors of a simple quadratic (with quadratic coefficient of one) to the coefficients.  The rule says that if the roots are at p and q, then the linear coefficient will be the negative of p+q, and the constant term will be their product, pq.  So for example, the simple quadratic with roots at x=2 and x=3 will be x2 - 5x + 6.

I know from experience that if you take a cross section of 100 students who enter calculus classes after two+ year of algebra, very few will know that you can extend that idea out to cubics and higher power polynomials.  An example for a polynomial with four roots will probably suffice for most to understand.  Because the constant terms in linear factors are always the opposite of the roots, {if 3 is a root, (x-3) is a factor} it is easiest to negate all the roots before doing the math involved (at least for me it always was). (addendum: If you're starting without knowing the roots, the rule of thumb is to change the sign of the coefficients that modify an odd power, like the x, x^3 etc.)
So if we wanted to find the simple polynomial with roots at -1, -2, -3, and -4 (chosen so all the multipliers are +, the factors are (x+1)(x+2)(x+3)(x+4) we would find that the fourth degree polynomial will have 10 for the coefficient of x3 because 1+2+3+4 = 10, just as it works in the second term of a quadratic.  After that, the method starts to combine sets of them.  The next coefficient will be the sum of the products of each pair of factor coefficients.  In the example I created we would add 1x2+1x3+1x4+2x3+2x4+3x4 to get 35x2.  The next term sums all triple products of the numbers, 1x2x3 + 1x2x4 + 1x3x4 + 2x3x4 = 50 for the linear coefficient.  And in the constant term, we simply multiply all of them together to get 24.

In a talk in 2011 (I think) Prof. Kelman introduced what I guess I would call a rule of thumb about "reverse polynomials".  Most of us are better at polynomials that have a first term of one, but sometimes you get some with "ugly" leading coefficients and 1 as the constant, Maybe 12x^2 - 7x + 1, and we need to find the roots.  One of the nice rules Prof Kalman points out is a relation between polynomials like that, and their reversal, 1x^2-7x +12, which you suddenly, almost instantly recognize as (x-4)(x-3) with roots of x=3, 4.  So how does that help, well let's walk through the other one. Reversing our thinking, then it must be (4x-1)(3x-1) and so the roots are...aha, 1/4 and 1/3, the inverse of the reversal. Sure enough, the sum of the roots is 7/12 and the product is 1/12.  The professor points out one more way you can get clues about finding, or checking solutions. If  you look at some big long polynomial like 2x^5 + 5x^4 + 3x^3 - 7x^2 + 3x +4, you not only know that the sum of the roots are -5/2, and the product of the roots is 4/2 = 2, you also can imagine the reverse and say the sum of the reciprocals is -4/3, and the product of the reciprocals is 1/2.

He extends this idea that the reverse polynomials have roots that are the inverse of the original to something he calls palindrom Polynomials, and shows how their roots must come in pairs of inverses.  The entire set of notes in PDF are here.