Thursday 27 January 2022

On This Day in Math - January 27


Gallleo's Notbook with observation of Neptune?
*Starship Asterisk web site

It troubles me that we are so easily pressured by purveyors of technology into permitting so-called ‘progress’ to alter our lives without attempting to control it—as if technology were an irrepressible force of nature to which we must meekly submit.
~Hyman G. Rickover

The 27th day of the year; 273 = 19,683 which has a digit sum of 27. The sum of the digits of 26^3 also sum to 26.  There is no larger number for which the sum of the digits of the cube is equal to the number but there are three smaller examples, can you find them?

In the 3n+1 or Collatz problem, the sequence for n = 27 takes 111 steps (41 steps through odd numbers), climbing to 9232 before descending to 1. The first starting number to exceed this height is 255.

27 is the smallest composite number which cannot be expressed as the sum of two primes.

I think it is nice that when you add the integers from 2 to 7, you get 27.... 2 + 3 + 4 + 5 + 6 + 7 = 27


1520 Off the Patagonian coast near a small peninsula called Punta Tombo, during Ferdinand Magellan’s voyage around the world, a crewman spied strange creatures swimming in the bay. He called them flightless geese, but scientists believe they were penguins of a sort classified as Spheniscus magellanicus.*VFR

1613 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 A new theory says he may have known it was a planet. Professor David Jamieson, Head of the School of Physics at University of Melbourne is investigating the notebooks of Galileo from 400 years ago and believes that buried in the notations is the evidence that he discovered a new planet that we now know as Neptune. Galileo was observing the moons of Jupiter in the years 1612 and 1613 and recorded his observations in his notebooks. Over several nights he also recorded the position of a nearby star which does not appear in any modern star catalogue. "It has been known for several decades that this unknown star was actually the planet Neptune. Computer simulations show the precision of his observations revealing that Neptune would have looked just like a faint star almost exactly where Galileo observed it," Professor Jamieson says.
In one of his notebooks he noticed the movement of a background star (Neptune) on January 28 and a dot (in Neptune's position) drawn in a different ink suggests that he found it on an earlier sketch, drawn on the night of January 6, suggesting a systematic search among his earlier observations. However, any notification about the discovery hasn't been found. *

1690 Newton returns to Cambridge after spending nearly a year in London serving as an MP from Cambridge University to the Convention Parliament. Declaring the throne vacant after James II escaped to France, the convention offered the throne to William and Mary jointly.According to some reports, his only comments were to complain about a cold draft in the chamber and request that the window be closed

1880 Thomas Edison received U.S. Patent No. 223,898, which was simply titled “Electric Lamp.” Edison did not truly "invent" the electric lightbulb. Prior to Edison’s invention lightbulbs lasted only a few hours and now they could last 50 to 60 days, making them practical. So it is entirely fair to say that Thomas Edison invented the first commercially useful lightbulb. *.ipwatchdog

1921, Albert Einstein suggested the possibility of measuring the universe, which startled the audience, with his address Geometry and Expansion given at the Prussian Academy of Sciences in Berlin. Applying certain results of the relativity theory, he came to the conclusion that if the real velocities of the stars (as could be actually measured) were less than the calculated velocities, then it would prove that real gravitations' great distances were smaller than the gravitational distances demanded by the law of Newton. From such divergence, the finiteness of the universe could be proved indirectly, and it would even permit the estimation of its size. *TIS

1951 Nuclear testing at the Nevada Test Site began with a 1-kiloton-of-TNT (4.2 TJ) bomb dropped on Frenchman Flat. During the 1950s, the mushroom clouds from the 100 atmospheric tests could be seen for almost 100 mi (160 km). The city of Las Vegas experienced noticeable seismic effects, and the distant mushroom clouds, which could be seen from the downtown hotels, became tourist attractions. St. George, Utah, received the brunt of the fallout of above-ground nuclear testing in the Yucca Flats/Nevada Test Site. Winds routinely carried the fallout of these tests directly through St. George and southern Utah. Marked increases in cancers, such as leukemia, lymphoma, thyroid cancer, breast cancer, melanoma, bone cancer, brain tumors, and gastrointestinal tract cancers, were reported from the mid-1950s through 1980. The vast majority—828 of the 928 total nuclear tests—were underground. *Wik

1994 Silicon Graphics Inc. co-founder Jim Clark leaves the company to start Mosaic Communications, the operation that later became Netscape Communications Corp. With Netscape cofounder Marc Andreesen, Clark helped popularize the World Wide Web by distributing the company's browser for free.*CHM

2003 Pioneer 10 was launched in 1972 . Pioneer 10 crossed the orbit of Saturn in 1976 and the orbit of Uranus in 1979. On June 13, 1983, Pioneer 10 crossed the orbit of Neptune, the outermost planet at the time, and so became the first man-made object to leave the proximity of the major planets of the solar system.

2012 An asteroid, 2012 BX34, passed within about 60,000km of Earth - less than a fifth of the distance to the Moon.The asteroid's path made it the closest space-rock to pass by the Earth since June 2011. The asteroid, estimated to be about 11m (36ft) in diameter, was first detected on Jan 25.*BBC website

2016 This is given as the official date of discovery by the GIMPS (Great Internet Mersenne Prime Search) project for the presently largest known prime number, called M49. The name could change since there may still be smaller Mersenne Primes between this and previously known ones. The previous record was in January of 2013. I am reminded at moments like this of a comment in Barlow's Number Theory (1811) which list all the then known Mersenne Primes up to M31. This number, "is the greatest that will ever be discovered, for, as they are merely curious without being useful it is not likely any person will attempt to find one beyond it." It is strange that we continue the search. It would be terrible if we did not. (Barlow actually listed the Perfect numbers associated with each Mersenne prime.)


1701 Charles-Marie de La Condamine (27 Jan 1701; 4 Feb 1774) French naturalist and mathematician who became particularly interested in geodesy (earth measurement). He was put in charge by the King of France of an expedition to Equador to measure a meridional arc at the equator (1735-43). It was wished to determine whether the Earth was either flattened or elongated at its poles. He then accomplished the first scientific exploration of the Amazon River (1743) on a raft, studying the region, and brought the drug curare to Europe. He also worked on establishment of a universal unit of length, and is credited with developing the idea of vaccination against smallpox, later perfected by Edward Jenner. However, he was almost constantly ill and died in 1773, deaf and completely paralyzed. *TIS

1715 Caspar (or Kaspar) Neumann (14 September 1648 – 27 January 1715) was a German professor and clergyman from Breslau with a special interest in mortality rates.
He first did an apprenticeship as a pharmacist. He finished his higher school education at Breslau's Maria-Magdalen grammar school. In 1667 he became a student of theology at the university of Jena, and on Nov. 30, 1673 was ordained as a priest, having been requested as a traveling chaplain for Prince Christian, the son of Ernest I, Duke of Saxe-Gotha. On his return home, following a two-year journey through west­ern Ger­ma­ny, Switz­er­land, north­ern It­a­ly, and south­ern France, he became a court-chaplain at Altenburg, and married the daughter of J. J. Rabe, physician in ordinary to the prince of Saxe-Friedenstein. In 1678 he was made the deacon of St. Maria-Magdalen in Breslau and became pastor in 1689. *Wik He was a student of Erhard Weigel

1829 Isaac Roberts (27 Jan 1829; 17 Jul 1904) British astronomer who was a pioneer in photography of nebulae. In 1885 he had built an observatory with a 20 inch reflector. Using this instrument Roberts was to make considerable progress in the newly developing science of Astro-photography. He photographed numerous celestial objects including Orion Nebula on 15 Jan 1986 (90 minute exposure) and Pleiades. Undoubtedly his finest work was a photograph showing the spiral structure of the Great Nebula in Andromeda, M31 on 29 Dec 1888. In addition to his contribution to astro-photography, Roberts also devised a machine to be used to engrave stellar positions on copper plates, known as the Stellar Pantograver. He was also a geologist of some considerable note.*TIS

1831 Charles Lutwidge Dodgson, pen-name Lewis Carroll (27 Jan 1832, 14 Jan 1898), was an English logician, mathematician, photographer, and novelist, remembered for Alice's Adventures in Wonderland (1865) and its sequel. After graduating from Christ Church College, Oxford in 1854, Dodgson remained there, lecturing on mathematics and writing treatises until 1881. As a mathematician, Dodgson was conservative. He was the author of a fair number of mathematics books, for instance A syllabus of plane algebraical geometry (1860). His mathematics books have not proved of enduring importance except Euclid and his modern rivals (1879) which is of historical interest. As a logician, he was more interested in logic as a game than as an instrument for testing reason.*TIS (I once read that if Dodgson had not written "Alice", he would be remembered today for his photography, and if he had not done either of those, then, if he was remembered at all, it would be for his logic book. One of my favorite Lewis Carroll stories is about his gift of a book to Queen Victoria. Here is the version as it is told on the Mathworld page):
Several accounts state that Lewis Carroll (Charles Dodgson ) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants. Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland, expressed a desire to receive the author's next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author's other books, and was sent one of Dodgson's mathematical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worth while to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson)
And then, I learned that "Lewis Carroll coined 'chortle' in Through the Looking-Glass, in 1871." @OEDonline, Twitter

1870 Jules-Émile Verschaffelt (27 January 1870, Ghent – 22 December 1955) was a Belgian physicist. He worked at Kamerlingh Onnes’s laboratory in Leiden from 1894 to 1906 and once again from 1914 to 1923. From 1906 to 1914 he worked at the Vrije Universiteit Brussel and from 1923 to 1940 at the Ghent University. *Wik

1885 Franciszek Leja (January 27, 1885 in Grodzisko Górne near Przeworsk – October 11, 1979 in Kraków, Poland) Polish mathematician who greatly influenced Polish Mathematics in the period between the two World Wars.
He was born to a poor peasant family in the southeastern Poland. After graduating from the University of Lwów he was a teacher of mathematics and physics in high schools from 1910 until 1923, among others in Kraków. From 1924 until 1926 he was a professor at the Warsaw University of Technology and from 1936 until 1960 in the Jagiellonian University.
During the Second World War he lectured on the underground universities in Łańcut and Lezajsk. But after the German invasion of Poland in 1939 life there became extremely difficult. There was a strategy by the Germans to wipe out the intellectual life of Poland. To achieve this Germans sent many academics to concentration camps and murdered others. In one of such actions he was sent to the Sachsenhausen concentration camp which he fortunately survived.
Since 1948 he worked for the Institute of Mathematics of the Polish Academy of Sciences. He was a co-founder of the Polish Mathematics Society in 1919 and from 1963 until 1965 the chairman. Since 1931 he was a member of the Warsaw Science Society (TNW).
His main scientific interests concentrated on analytic functions, in particular the method of extremal points and transfinite diameters. *Wik

1900 Hyman George Rickover (27 Jan 1900; 8 Jul 1986) was a Polish-American naval officer who immigrated to the US (1906) and graduated from the Naval Academy in 1922. He eventually became an Admiral. He is known as the “Father of the Nuclear Navy” for his leadership to build the atomic-powered submarine, USS Nautilus (1954). He served on active duty with the United States Navy for more than 63 years, receiving exemptions from the mandatory retirement age due to his critical service in the building of the United States Navy's nuclear surface and submarine force. *TIS

1903 Howard Percy Robertson (27 Jan 1903 in Hoquiam, Washington, USA - 26 Aug 1961) made outstanding contributions to differential geometry, quantum theory, the theory of general relativity, and cosmology. He was interested in the foundations of physical theories, differential geometry, the theory of continuous groups, and group representations. He was particularly interested in the application of the latter three subjects to physical problems.
His contributions to differential geometry came in papers such as: The absolute differential calculus of a non-Pythagorean non-Riemannian space (1924); Transformation of Einstein space (1925); Dynamical space-times which contain a conformal Euclidean 3-space (1927); Note on projective coordinates (1928); (with H Weyl) On a problem in the theory of groups arising in the foundations of differential geometry (1929); Hypertensors (1930); and Groups of motion in space admitting absolute parallelism (1932). *SAU

1936 Samuel C.C. Ting(27 Jan 1936, ) Samuel Chao Chung Ting is an American physicist who shared, with Burton Richter, the Nobel Prize for Physics in 1976 for his discovery of a new subatomic particle, the J/psi particle.*TIS

1941 Beatrice Muriel Hill Tinsley (27 January 1941 – 23 March 1981) was a British-born New Zealand astronomer and cosmologist whose research made fundamental contributions to the astronomical understanding of how galaxies evolve, grow and die.
Tinsley completed pioneering theoretical studies of how populations of stars age and affect the observable qualities of galaxies. She also collaborated on basic research into models investigating whether the universe is closed or open. Her galaxy models led to the first approximation of what protogalaxies should look like.
In 1974 she received the American Astronomical Society's Annie J. Cannon Award in Astronomy, awarded for "outstanding research and promise for future research by a postdoctoral woman researcher", in recognition of her work on galaxy evolution.
In 1977, Tinsley, with Richard Larson of Yale, organized a conference on 'The Evolution of Galaxies and Stellar Populations'.
Shortly after, in 1978, she became the first female professor of astronomy at Yale University. Her last scientific paper, submitted to the Astrophysical Journal ten days before her death, was published posthumously that November, without revision. *Wik


1667 Gregorius Saint Vincent (8 Sept 1584 in Bruges, Belgium - 27 Jan 1667 in Ghent, Belgium). His Opus geometricum (1647) contains the most beautiful frontispiece of any mathematics text. In this work, Gregorius was the first to develop the theory of the geometric series and also the first to show that the area under a hyperbola is a logarithm. *VFR (in the frontispiece he claims to have squared the circle) The engraved frontispiece shows sunrays inscribed in a square frame being arranged by graceful angels to produce a circle on the ground: 'mutat quadrata rotundis'. There was uneasiness in the learned world because no one in that world still believed that under the specific Greek rules the quadrature of a circle could possibly be effected, and few relished the thought of trying to locate an error, or errors, in 1200 pages of text. Four years later, in 1651, Christiaan Huygens found a serious defect in the last book of 'Opus geometricum', namely in Proposition 39 of Book X, on page 1121. This gave the book a bad reputation.*SAU
Bob Mrotek wrote to point out that "The picture shows an angel holding a square frame and the light ray that passes through it forms a circle on the ground. This is ALWAYS the case no matter what the shape of the hole that the light passes through as long as there is enough focal length between the hole (depending on its size) and the ground. When you walk in the woods you will notice that the light passing through the odd shaped spaces between the leaves forms perfect circles on the ground. This is the camera obscura effect and most people never realize it."

1823 Charles Hutton (14 Aug 1737 in Newcastle-upon-Tyne, England - 27 Jan 1823 in London, England) was an English mathematician who wrote arithmetic textbooks. A textbook he wrote while at the Royal Military Academy, Woolwich was later adopted as the first math text by the USMA in West Point, NY and served as the principal math text for two decades. *Wik

1860 János Bolyai (15 Dec 1802; 27 Jan 1860) Hungarian mathematician and one of the founders of non-Euclidean geometry - geometry that does not include Euclid's axiom that only one line can be drawn parallel to a given line through a point not on the given line. His father, Farkas Bolyai, had devoted his life to trying to prove Euclid's famous parallel postulate. Despite his father's warnings that it would ruin his health and peace of mind, János followed in working on this axiom until, in about 1820, he came to the conclusion that it could not be proved. He went on to develop a consistent geometry (published 1882) in which the parallel postulate is not used, thus establishing the independence of this axiom from the others. He also did valuable work in the theory of complex numbers. *TIS
He once accepted a challenge to duel thirteen of his fellow cavalry officers on the condition that after each duel he would serenade the loser with a piece on his violin. He won all thirteen duels.
János had his own funeral card printed with a blank date and built his own coffin. Still alive six years later, he printed a new funeral card to replace the unused one. In his will he left instructions that an apple tree be planted on his grave in remembrance of Eve, Paris, and Newton. In 1911 his ashes were exhumed and laid into his father’s tomb. *robertnowlan

1860 Sir Thomas Makdougall Brisbane, Baronet (23 Jul 1773, 27 Jan 1860) British soldier and astronomical observer for whom the city of Brisbane, Australia, is named. He was Governor of NSW (1821-25). Mainly remembered as a patron of science, he built an astronomical observatory at Parramatta, Australia, made the first extensive observations of the southern stars since Lacaille in (1751-52) and built a combined observatory and magnetic station at Makerstoun, Roxburghshire, Scotland. He also conducted (largely unsuccessful) experiments in growing Virginian tobacco, Georgian cotton, Brazilian coffee and New Zealand flax.*TIS

1895 James Cockle (14 Jan 1819 in Great Oakley, Essex, England - 27 Jan 1895 in Bayswater, London, England) Cockle was remarkably productive as a mathematician publishing over 100 papers. He wrote papers on both pure and applied mathematics, as well as on the history of science. On the former topic he wrote on fluid dynamics and magnetism. Most of his work, however, was in pure mathematics where he studied algebra, the theory of equations, and differential equations. He had a collaborator on mathematical work, a Congregationalist minister named Robert Harley. *SAU

1947 Alexander Brown (5 May 1877 in Dalkeith, near Edinburgh, Scotland - 27 Jan 1947 in Cape Town, South Africa) In 1903 Brown was appointed as Professor of Applied Mathematics in the South African College. In 1911 he married Mary Graham; they had a son and a daughter. He remained in Cape Town until his death in 1947, but his status changed in 1918 when the South African College became the University of Cape Town.
He was a member of the Edinburgh Mathematical Society, joining the Society in December 1898. He contributed papers to meetings of the Society such as On the Ratio of Incommensurables in Geometry to the meeting on Friday 9 June 1905 and Relation between the distances of a point from three vertices of a regular polygon, at the meeting on Friday 11 June 1909, communicated by D C McIntosh.
Brown was elected a Fellow of the Royal Society of South Africa in 1918, was on its Council from 1931 to 1935 and again in 1941, was its Honorary Treasurer from 1936 to 1940, and President from 1942 to 1945. Alexander Brown was elected to the Royal Society of Edinburgh on 20 May 1907. *SAU

1965 Philip Franklin (October 5, 1898 in New York — January 27, 1965 in Belmont, Massachusetts) was an American mathematician and professor whose work was primarily focused in analysis.
His dissertation, The Four Color Problem, was supervised by Oswald Veblen. After teaching for one year at Princeton and two years at Harvard (as the Benjamin Peirce Instructor), Franklin joined the MIT Department of Mathematics, where he stayed until his 1964 retirement.
In 1922, Franklin gave the first proof that all planar graphs with at most 25 vertices can be four-colored.
In 1928, Franklin gave the first description of an orthonormal basis for L²([0,1]) consisting of continuous functions (now known as "Franklin's system").
In 1934, Franklin published a counterexample to the Heawood conjecture, this 12-vertex cubic graph is now known as the Franklin graph.
He was married to Norbert Wiener's sister Constance. *Wik

1972 Richard Courant (8 Jan 1888, 27 Jan 1972) German-American mathematician who, upon joining the faculty of New York University in 1934, began to build the nucleus of a small research group based on the Göttingen model he had experienced as a student of David Hilbert in Germany. Courant's published papers were in variational problems, finite difference methods, minimal surfaces, and partial differential equations. He encouraged the publication of mathematical texts and high quality monographs, such as Methods of Mathematical Physics by Courant and Hilbert. His leadership was commemorated in 1964 when the institute he founded was named the Courant Institute of Mathematical Sciences at New York University *TIS He died at age 84 of a stroke in New Rochelle, NY. Today it is named after him: The Courant Institute. *VFR

1995 Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an American mathematician.
In 1941, Robinson married his former student Julia Bowman. She became his Berkeley colleague and the first woman president of the American Mathematical Society.
He worked on mathematical logic, set theory, geometry, number theory, and combinatorics. Robinson (1937) set out a simpler and more conventional version of John Von Neumann's 1923 axiomatic set theory. Soon after Alfred Tarski joined Berkeley's mathematics department in 1942, Robinson began to do major work on the foundations of mathematics, building on Tarski's concept of "essential undecidability," by proving a number of mathematical theories undecidable. Robinson (1950) proved that an essentially undecidable theory need not have an infinite number of axioms by coming up with a counterexample Robinson's work on undecidability culminated in his coauthoring Tarski et al. (1953), which established, among other things, the undecidability of group theory, lattice theory, abstract projective geometry, and closure algebras.
Robinson worked in number theory, even employing very early computers to obtain results. For example, he coded the Lucas-Lehmer primality test to determine whether 2n − 1 was prime for all prime n less than 2304 on a SWAC. In 1952, he showed that these Mersenne numbers were all composite except for 17 values of n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281. He discovered the last 5 of these Mersenne primes, the largest ones known at the time.
Robinson wrote several papers on tilings of the plane, in particular a clear and remarkable 1971 paper "Undecidability and nonperiodicity for tilings of the plane" simplifying what had been a tangled theory.*Wik

2001 Robert Alexander Rankin (27 Oct 1915 in Garlieston, Wigtownshire, Scotland - 27 Jan 2001 in Glasgow, Scotland) At Cambridge Rankin began to undertake research in number theory on the difference between two successive primes which won him the Rayleigh Prize in 1939. He published four papers on The difference between consecutive prime numbers between this time and 1950. in 1939 he began to work with G H Hardy on the results of Ramanujan. Although Ramanujan had died nearly twenty years earlier, he had left a number of unpublished notebooks filled with theorems that Hardy and other mathematicians continued to study.
After an interruption during WWII, Rankin wrote over 100 research papers, mostly on the theory of numbers and the theory of functions. He wrote The modular group and its subgroups published in 1969 and Modular forms and functions which was published in 1977. The former of these is described by Rankin himself in the Preface, "This short course of lectures was given at the Ramanujan Institute for Advanced Study in Mathematics, in the University of Madras, in September 1968. The object of the course was to study the modular group and some of its subgroups, with help of algebraic rather than analytic or topological methods." He made a number of remarkable contributions to the theory of numbers have played a major part in the modern development of the topic. *SAU

2008 Irene Anne Stegun (February 9, 1919 – January 27, 2008) was a mathematician at the National Bureau of Standards who, with Milton Abramowitz, edited a classic book of mathematical tables called A Handbook of Mathematical Functions, widely known as Abramowitz and Stegun. When Abramowitz died of a heart attack in 1958, Stegun took over management of the project and finished the work by 1964, working under the direction of the NBS Chief of Numerical Analysis Philip J. Davis, who was also a contributor to the book. *Wik

2015 Charles Hard Townes (July 28, 1915 – January 27, 2015) was an American Nobel Prize-winning physicist and educator. Townes was known for his work on the theory and application of the maser, on which he got the fundamental patent, and other work in quantum electronics connected with both maser and laser devices. He shared the Nobel Prize in Physics in 1964 with Nikolay Basov and Alexander Prokhorov.
In a career that spanned six decades, Dr. Townes developed radar bombing systems and navigation devices during World War II, advised presidents and government commissions on lunar landings and the MX missile system, verified Einstein’s cosmological theories, discovered ammonia molecules at the center of the Milky Way, and created an atomic clock that measured time to within one second in 300 years. He died at the age of 99 in Berkeley, California*Wik *NY Times

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

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