Sunday, 4 December 2016

On This Day in Math - December 4

Taking him for all and all, I think it will be conceded that Michael Faraday was the greatest experimental philosopher the world has ever seen.
~John Tyndall

The 339th day of the year; the plane can be divided into 339 regions with 13 hyperbolae.

There are also 339 possible 2x2 matrices with integer entries between zero and 13.

Be amazed, someone checked and found that 339 (repeated 339 times) x 2339 - 1 is prime. (What! You don't believe it, well factor it and prove they're wrong.)


1639 "On this day in 1639 Jeremiah Horrocks and William Crabtree were the first human beings to have recorded a transit of Venus. ...  Moreover, Horrocks predicted the event from his own calculations, improving on Kepler’s ephemeris of Venus and the sun. Horrocks still used the old Julian calendar, which differed then by 10 days with the Gregorian calendar we use today. That is, to Horrocks the transit took place on November 24, (see my blog for that date) while in the rest of Europe it was already December 4."  The image is from the same site . " It’s a mosaic showing Horrocks observing the transit of Venus, and a line from one of his own poems: His mortal eyes to scan the furthest heavens."  *Transit of Venus 
 image by Mark Phillips

1679 Philosopher Thomas Hobbes died, thus ending his 25 year feud with John Wallis over Hobbes’s attempt to square the circle in 1655. It began when Hobbes called Wallis’s Arithmetica Infinitorum a “scab of symbols”. *VFR

1980 Ireland issued a stamp picturing Robert Boyle (1627-1691) and his 1659 Air Pump. [Scott #492]. *VFR

1930 Wolfgang Pauli writes to propose the existence of what would come to be called the neutrino
--in it he thinks very widely of missing stuff, of some of the basic bits of the universe, in a rather open and guarded way, about the ghost of the neutron. He didn't feel very comfortable with his ideas yet, at least for professional consumption--that would have to wait another three years when it was discussed at the 7th Solvay Conference (1933) and another three when it first came into print (1936). The name "neutron" would also be changed to the familiar "neutrino" ("little one") by Enrico Fermi in 1933 to differentiate it from the much larger nuclear particle discovered the year earlier by James Chadwick--Chadwick's paper was published in Nature, which would reject Fermi's paper in 1934 as too radical a leap.
A translation appears here "Dear Radioactive Ladies and Gentlemen" ,*Ptak Science Books

1985 Cray X-MP Supercomputer Begins Operation. The Cray X-MP/48 started operation at the San Diego Supercomputer Center. The X-MP was popular for generating computer graphics, especially for movies. It nearly doubled the operating speed of competing machines with its parallel processing system, which ran at 420 million floating-point operations per second, or megaflops. An even faster speed could be achieved by arranging two Crays to work together on different parts of the same problem. Other applications included the defense industry and scientific research.*CHM

In 1998, the space shuttle Endeavour and a crew of six blasted off on the first mission to begin assembling the international space station.*TIS


1795 Thomas Carlyle (4 Dec 1795 in Ecclefechan, Dumfriesshire, Scotland - 5 Feb 1881 in Chelsea, London, England) was a Scottish writer who was also interested in mathematics. He translated Legendre's work.*SAU

1806 John Thomas Graves (4 December 1806, Dublin, Ireland–29 March 1870, Cheltenham, England) was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions and with personally discovering the octonions, which he called the octaves. He was the brother of both the mathematician Charles Graves and the writer and clergyman Robert Perceval Graves.
In his twentieth year (1826) Graves engaged in researches on the exponential function and the complex logarithm; they were printed in the Philosophical Transactions for 1829 under the title An Attempt to Rectify the Inaccuracy of some Logarithmic Formulæ. M. Vincent of Lille claimed to have arrived in 1825 at similar results, which, however, were not published by him till 1832. The conclusions announced by Graves were not at first accepted by George Peacock, who referred to them in his Report on Algebra, nor by Sir John Herschel. Graves communicated to the British Association in 1834 (Report for that year) on his discovery, and in the same report is a supporting paper by Hamilton, On Conjugate Functions or Algebraic Couples, as tending to illustrate generally the Doctrine of Imaginary Quantities, and as confirming the Results of Mr. Graves respecting the existence of Two independent Integers in the complete expression of an Imaginary Logarithm. It was an anticipation, as far as publication was concerned, of an extended memoir, which had been read by Hamilton before the Royal Irish Academy on 24 November 1833, On Conjugate Functions or Algebraic Couples, and subsequently published in the seventeenth volume of the Transactions of the Royal Irish Academy. To this memoir were prefixed A Preliminary and Elementary Essay on Algebra as the Science of Pure Time, and some General Introductory Remarks. In the concluding paragraphs of each of these three papers Hamilton acknowledges that it was "in reflecting on the important symbolical results of Mr. Graves respecting imaginary logarithms, and in attempting to explain to himself the theoretical meaning of those remarkable symbolisms", that he was conducted to "the theory of conjugate functions, which, leading on to a theory of triplets and sets of moments, steps, and numbers" were foundational for his own work, culminating in the discovery of quaternions.
For many years Graves and Hamilton maintained a correspondence on the interpretation of imaginaries. In 1843 Hamilton discovered the quaternions, and it was to Graves that he made on 17 October his first written communication of the discovery. In his preface to the Lectures on Quaternions and in a prefatory letter to a communication to the Philosophical Magazine for December 1844 are acknowledgments of his indebtedness to Graves for stimulus and suggestion. After the discovery of quaternions, Graves employed himself in extending to eight squares Euler's four-square identity, and went on to conceive a theory of "octaves" (now called octonions) analogous to Hamilton's theory of quaternions, introducing four imaginaries additional to Hamilton's i, j and k, and conforming to "the law of the modulus".
Graves devised also a pure-triplet system founded on the roots of positive unity, simultaneously with his brother Charles Graves, the bishop of Limerick. He afterwards stimulated Hamilton to the study of polyhedra, and was told of the discovery of the icosian calculus. *Wik

1886 Ludwig Georg Elias Moses Bieberbach (4 Dec 1886 in Goddelau, Darmstadt in Hessen, Germany - 1 Sept 1982 in Oberaudorf in Oberbayern, Germany) Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions (German: Theorie der automorphen Funktionen). He began working as a Privatdozent at Königsberg in 1910 and as Professor ordinarius at the University of Basel in 1913. He taught at the University of Frankfurt in 1915 and the University of Berlin from 1921–45.
Bieberbach wrote a habilitation thesis in 1911 about groups of Euclidean motions – identifying conditions under which the group must have a translational subgroup whose vectors span the Euclidean space – that helped solve Hilbert's 18th problem. He worked on complex analysis and its applications to other areas in mathematics. He is known for his work on dynamics in several complex variables, where he obtained results similar to Fatou's. In 1916 he formulated the Bieberbach conjecture, stating a necessary condition for a holomorphic function to map the open unit disc injectively into the complex plane in terms of the function's Taylor series. In 1984 Louis de Branges proved the conjecture (for this reason, the Bieberbach conjecture is sometimes called de Branges' theorem). There is also a Bieberbach theorem on space groups.*Wik

1890 Harry Clyde Carver (December 4, 1890 – January 30, 1977) was an American mathematician and academic, primarily associated with the University of Michigan. He was a major influence in the development of mathematical statistics as an academic discipline.
Born in Waterbury, Connecticut, Carver was educated at the University of Michigan, earning his B.S. degree in 1915, and the next year becoming an instructor in mathematics; he taught statistics in actuarial applications. At the time, the University of Michigan was only the second such institution in the United States to offer this type of course, after the pioneering Iowa State University. Carver was appointed assistant professor at Michigan in 1918, then associate professor (1921) and full professor (1936); during this period the University's program in mathematical statistics and probability underwent significant expansion.
In 1930 Carver founded the journal Annals of Mathematical Statistics, which over time became an important periodical in the field. Financial support, however, was lacking in the midst of the Great Depression; in January 1934 Carver undertook financial responsibility for the Annals and maintained the existence of the journal at his own expense. In 1935 he helped to start the Institute of Mathematical Statistics, which in 1938 assumed control over the journal; Samuel S. Wilks succeeded Carver as editor in the same year. The Institute has named its Harry C. Carver Medal after him.
With the coming of World War II, Carver devoted his energies to solving problems in aerial navigation, an interest he maintained for the remainder of his life. *Wik

1924 Frank Press (4 Dec 1924, )American geophysicist known for his investigations of the structure of the Earth's crust and mantle and the mechanics of earthquakes. Press pioneered the use of seismic waves to explore subsurface geological structures and for his pioneering use of waves to explore Earth's deep interior. In 1950, with William Maurice Ewing, a major innovator in modern geology at Columbia University, he invented an improved seismograph,and they published a landmark paper recognized as beginning a new era in structural seismology. While at Caltech (1955-65) and later MIT, Press became knownin public policy circles for his work on seismic detection of underground nuclear tests and for advocacating a national program for earthquake prediction capabilities. *TIS

1938 George Eyre Andrews (December 4, 1938 in Salem, Oregon) is an American mathematician working in analysis and combinatorics. He is currently an Evan Pugh Professor of Mathematics at Pennsylvania State University. He received his PhD in 1964 at University of Pennsylvania where his advisor was Hans Rademacher.
Andrews's contributions include several monographs and over 250 research and popular articles on q-series, special functions, combinatorics and applications. He is considered to be the world's leading expert in the theory of integer partitions.[citation needed] In 1976 he discovered Ramanujan's Lost Notebook. He is highly interested in mathematical pedagogy, and is a vocal critic of the "calculus reform" movement.*Wik


1131 Omar Khayyam (18 May 1048, 4 Dec 1131) Persian poet, mathematician, and astronomer. Khayyam, who was born at Nishapur (now in Iran), produced a work on algebra that was used as a textbook in Persia until this century. In geometry, he studied generalities of Euclid and contributed to the theory of parallel lines. Around 1074, he set up an observatory and led work on compiling astronomical tables, and also contributed to the reform of the Persian calendar. His contributions to other fields of science included developing methods for the accurate determination of specific gravity. He is known to English-speaking readers for his "quatrains" as The Rubáiyát of Omar Khayyám, published in 1859 by Edward Fitzgerald, though it is now regarded as an anthology of which little or nothing may be by Omar. *TIS   A nice blog with more detail about the Persian Polymath is at Galileo's Pendulum .

1574 Georg Joachim Rheticus (16 Feb 1514, 4 Dec 1574) Austrian-born astronomer and mathematician who was among the first to adopt and spread the heliocentric theory of Nicolaus Copernicus. He was first taught by his father, a physician, who was beheaded for sorcery in 1528, while Rheticus was still a teenager. He is best known as the first disciple of Copernicus. In 1540, Rheticus published the first account of the heliocentric hypothesis which had been elaborated by Copernicus, entitled Narratio prima, which was explicitly authorised by Copernicus, who also asked for his friend's aid in editing the edition of his De revolutionibus orbium coelestium ("On the revolutions of the heavenly spheres"). Rheticus was the first mathematician to regard the trigonometric functions in terms of angles rather than arcs of a circle.*TIS
The First Copernican: Georg Joachim Rheticus and the Rise of the Copernican Revolution

(I have seen his date of death also listed as the Dec 5th)

1798 Luigi Galvani (9 Sep 1737, 4 Dec 1798) Italian physician and physicist studied the structure of organs and the physiology of tissues who is best known for his investigation of the nature and effects of what he conceived to be electricity in animal tissue. He observed how frog muscles twitched when they were touched by metal contacts but he wrongly attributed this to innate "animal electricity" (the current was actually produced by the metal contacts). This was disputed by Alessandro Volta who, in the course of this argument, invented his electrochemical cell. The current produced by this device was for many years called galvanic electricity. The galvanometer was named after him.*TIS

1850 William Sturgeon (22 May 1783, 4 Dec 1850) English electrical engineer who devised the first electromagnet capable of supporting more than its own weight (1825). The 7-oz (200-g) magnet supported 9-lb (4-kg) of iron with a single cell's current. He built an electric motor (1832) and invented the commutator, now part of most modern electric motors. In 1836, he invented the first suspended coil galvanometer, a device for measuring current. Sturgeon also worked on improving the voltaic battery, developing a theory of thermoelectricity, and even atmospheric charge conditions. From 500 kite flights made in calm weather, he found the atmosphere is consistently charged positively with respect to the Earth, and increasingly so at increased height.*TIS

1893 John Tyndall (2 Aug 1820, 4 Dec 1893)British physicist who demonstrated why the sky is blue. His initial scientific reputation was based on a study of diamagnetism. He carried out research on radiant heat, studied spontaneous generation and the germ theory of disease, glacier motion, sound, the diffusion of light in the atmosphere and a host of related topics. He showed that ozone was an oxygen cluster rather than a hydrogen compound, and invented the firemans respirator and made other less well-known inventions including better fog-horns. One of his most important inventions, the light pipe, has led to the development of fibre optics. The modern light instrument is known as the gastroscope, which enables internal observations of a patient's stomach without surgery. Tyndall was a very popular lecturer. *TIS

1934 Sir Horace Lamb (27 Nov 1849, 4 Dec 1934) English mathematician who contributed to the field of mathematical physics. Topics he worked on include wave propagation, electrical induction, earthquakes, and the theory of tides. He wrote important papers on the oscillations of a viscous spheroid, the vibrations of elastic spheres, waves in elastic solids, electric waves and the absorption of light. In a famous paper in the Proceedings of the London Mathematical Society he showed how Rayleigh's results on the vibrations of thin plates fitted with the general equations of the theory. Another paper reported on his study of the propagation of waves on the surface of an elastic solid where he tried to understand the way that earthquake tremors are transmitted around the surface of the Earth.*TIS

1948 Frank Albert Benford, Jr., (1883 Johnstown, Pennsylvania – December 4, 1948) was an American electrical engineer and physicist best known for rediscovering and generalizing Benford's Law, a statistical statement about the occurrence of digits in lists of data.
Benford is also known for having devised, in 1937, an instrument for measuring the refractive index of glass. An expert in optical measurements, he published 109 papers in the fields of optics and mathematics and was granted 20 patents on optical devices.
His date of birth is given variously as May 29 or July 10, 1883. After graduating from the University of Michigan in 1910, Benford worked for General Electric, first in the Illuminating Engineering Laboratory for 18 years, then the Research Laboratory for 20 years until retiring in July 1948. He died suddenly at his home on December 4, 1948. *Wik

1978 Samuel Abraham Goudsmit (11 Jul 1902, 4 Dec 1978) Dutch-born U.S. physicist who, with George E. Uhlenbeck, a fellow graduate student at the University of Leiden, Neth., formulated (1925) the concept of electron spin. It led to recognition that spin was a property of protons, neutrons, and most elementary particles and to a fundamental change in the mathematical structure of quantum mechanics. Goudsmit also made the first measurement of nuclear spin and its Zeeman effect with Ernst Back (1926-27), developed a theory of hyperfine structure of spectral lines, made the first spectroscopic determination of nuclear magnetic moments (1931-33), contributed to the theory of complex atoms and the theory of multiple scattering of electrons, and invented the magnetic time-of-flight mass spectrometer (1948).*TIS

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

Saturday, 3 December 2016

On This Day in Math - December 3

Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.
~Hermann Weyl

The 338th day of the year; the last day of the year which will be twice a perfect square.

338 is the arithmetic mean of two triangular numbers.

338 is the smallest number for which the number of divisors (6) and the sum of its prime factors (28) are both perfect numbers.


1610 Galileo dedicates his Sidereus nuncius to Grandduke Cosmos II. *VFR I am not sure what event Professor Rickey is referring here. According to Albert Van Helden in his introduction to his translation, "The Dedicatory letter of Sidereus nuncius is dated 12 March 1610, and on the next day Galileo sent an advance, unbound copy, accompanied by a letter, to the Tuscan court."
Thony Christie sent this translation from page 33 of the same book, "Written in Padua on the fourth day before the Ides of March 1610. Your Highnesses's most loyal servant, Galileo Galilei."

1833 Oberlin College, the first truly coeducational institution of higher education in the U.S., opens with 29 men, 15 women. *VFR

1836 Adolphe Quetelet presents the average number of meteors per hour.
The 1833 Leonid storm had galvanized interest in meteors, and the time was ripe. Adolphe Quetelet, a Belgian statistician and founder and director of the Brussels Observatory, had mentioned mid-August meteors very tentatively six months earlier. His attention had been called to meteors by François Arago of France, who dominated European science at the time with his skill in discerning important scientific problems and suggesting experiments to solve them. What, asked Arago in the wake of the 1833 display, constituted a shower of meteors, and what was the rate of the ordinary, everynight drizzle?
The problem was ideal for Quetelet, whose passion was statistics. In a speech to the Royal Academy of Sciences and Arts of Brussels on December 3, 1836, Quetelet gave his answer: averaged over the night and year, a single observer should expect to see eight sporadic (nonshower) meteors per hour. That figure is still good today. After his speech Quetelet made a brief mention of unusual August meteors, and in his 1836 annual report of the Brussels Observatory he presented the idea timidly and almost in passing: "I thought I also noticed a greater frequency of these meteors in the month of August (from the 8th to the 15th)."
By the following year, Quetelet had accidentally found records in his observatory of exceptional meteor displays on August 10th of 1834 and 1835 to accompany the increase he had seen in 1836. He called for scientists at the March 4, 1837, session of the Royal Academy of Brussel to watch the sky on August 10, 1837. *Sky and Telescope

1958 Germany issued a stamp to commemorate the 500th anniversary of the Cusanus Hospice at Kues, founded by Cardinal Nicolaus (1401-1464), Nicolaus Cusanus (Nickolaus Krebs). *VFR

1968 CDC Announces 7600 Supercomputer: Control Data Corporation announces its 7600 model, considered by some to be the first true supercomputer. The CDC 7600 calculated at a speed of nearly 40 megaflops. Seymour Cray designed this computer, as well as its predecessor, the 6600 that was popular with scientific researchers, and a successor, the 8600, which the company never marketed. *CHM


1616 John Wallis (3 Dec 1616; 8 Nov 1703) English mathematician. Wallis was skilled in cryptography and decoded Royalist messages for the Parliamentarians during the Civil War. Wallis was part of a group interested in natural and experimental science who started to meet in London. This group became the Royal Society (1663), with Wallis as a founder member and one of its first Fellows. He contributed substantially to the origins of calculus and was the most influential English mathematician before Newton. Wallis introduced our symbol for infinity (1656), and exponents using negative or fractional numbers (such as 1/x2 = x-2 or square root of x = x-1/2). In 1668, he was the first to suggest the law of conservation of momentum for colliding bodies, the first of all-important conservation laws.*TIS

1903 Sydney Goldstein (3 Dec 1903 in Hull, England - 22 Jan 1989 in Belmont, Massachusetts, USA) Goldstein's work in fluid dynamics is of major importance. He is described as, "... one of those who most influenced progress in fluid dynamics during the 20th century." He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930. In 1935 he published work on the turbulent resistance to rotation of a disk in a fluid. His work was important in aerodynamics, a subject in which Goldstein was extremely knowledgeable. *SAU

1938 Cleveland Abbe (3 Dec 1838; 28 Oct 1916) U.S. astronomer and first meteorologist, born in New York City, the "father of the U.S. Weather Bureau," which was later renamed the National Weather Service. Abbe inaugurated a private weather reporting and warning service at Cincinnati. His weather reports or bulletins began to be issued on Sept. 1, 1869. The Weather Service of the United States was authorized by Congress on 9 Feb 1870, and placed under the direction of the Signal Service. Abbe was the only person in the country who was already experienced in drawing weather maps from telegraphic reports and forecasting from them. Naturally, he was offered an important position in this new service which he accepted, beginning 3 Jan 1871, and was often the official forecaster of the weather.*TIS

1924 John Backus (3 Dec 1924; 28 Oct 1988) American computer scientist who invented the FORTRAN (FORmula TRANslation) programming language in the mid 1950s. He had previously developed an assembly language for IBM's 701 computer when he suggested the development of a compiler and higher level language for the IBM 704. As the first high-level computer programming language, FORTRAN was able to convert standard mathematical formulas and expressions into the binary code used by computers. Thus a non-specialist could write a program in familiar words and symbols, and different computers could use programs generated in the same language. This paved the way for other computer languages such as COBOL, ALGOL and BASIC. *TIS

1942 Joseph Ivor Silk FRS (3 December 1942-) was the Savilian Chair of Astronomy at the University of Oxford from 1999 to September 2011. He was educated at Tottenham County School (1954-1960) and went on to study Mathematics at the University of Cambridge (1960-1963). He gained his PhD in Astronomy from Harvard in 1968. Silk took up his first post at Berkeley in 1970, and the Chair in Astronomy in 1978. Following a career of nearly 30 years there, Silk returned to the UK in 1999 to take up the Savilian Chair at the University of Oxford. He is currently Professor of Physics at the Institut d’Astrophysique de Paris, Université Pierre et Marie Curie, and he joined Johns Hopkins University in 2010 as Homewood Professor of Physics and Astronomy.

He is an Emeritus Fellow of New College, Oxford and a Fellow of the Royal Society (elected May 1999). He was awarded the 2011 Balzan Prize for his works on the early Universe. Silk has given more than two hundred invited conference lectures, primarily on galaxy formation and cosmology.
In 2015 he was selected the Gresham Professor of Astronomy. *Wik


1882 James Challis (12 Dec 1803, 3 Dec 1882) British clergyman and astronomer, famous in the history of astronomy for his failure to discover the planet Neptune. Astronomer and mathematician John Couch Adams had studied the known deviations in the orbit of the planet Uranus which indicated a planet even further out. In 1845, Adams gave Astronomer Royal George Airy a calculated orbital path for the unknown planet. But Airy was more interested in the primary job of navigation and timekeeping observations. Airy informed Challis, who did not begin until July 1846, and actually sighted the new planet four times without recognizing it. On 23 Sep 1845, the new planet was instead discovered from Berlin Observatory. Challis admitted that Adam's prediction was within 2° of the planet's position.*TIS

1956 Felix Bernstein (24 Feb 1878 in Halle, Germany - 3 Dec 1956 in Zurich, Switzerland) established his famous theorem on the equivalence of sets while in Cantor's seminar at Halle in 1897. He also worked on transfinite ordinal numbers.Bernstein is best remembered by mathematicians for the Schröder-Bernstein Theorem. This theorem states:
If each of two sets A and B are equivalent to a subset of the other, then A is equivalent to B. *SAU

1983 Elliott Waters Montroll (May 4, 1916 in Pittsburgh, Pennsylvania, USA - December 3, 1983 in Chevy Chase, Maryland, USA) was an American scientist and mathematician.Montroll had an exceptionally varied career: was a Sterling Research Fellow at Yale University where his work on the Ising model of a ferromagnet led him to solve certain Markov chain problems. Following this he was a Research Associate at Cornell University in 1941-42 where he began his studies of the problem of finding the frequency spectrum of elastic vibrations in crystal lattices. He was elected to the National Academy of Sciences (United States) in 1969, and to the American Academy of Arts and Sciences in 1973. His work on traffic flow led to him winning (jointly) the Lanchester Prize of the Operations Research Society of America in 1959. *Wik

2004 Shiing-shen Chern (26 Oct 1911, 3 Dec 2004) Chinese-American mathematician and educator whose researches in differential geometry include the development of the Chern characteristic classes in fibre spaces, which play a major role in mathematics and in mathematical physics. "When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point. Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people. Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern." *TIS

2008 Oliver Gordon Selfridge (May 10, 1926 – December 3, 2008), grandson of Harry Gordon Selfridge, the founder of Selfridges' department stores, was a pioneer of artificial intelligence. He has been called the "Father of Machine Perception."
Selfridge was born in England, educated at Malvern College and Middlesex School and then earned an S.B. from MIT in mathematics in 1945. He then became a graduate student of Norbert Wiener's at MIT, but did not write up his doctoral research and never earned a Ph.D. While at MIT, he acted as one of the earlier reviewers for Wiener's Cybernetics book in 1949. He was also technically a supervisor of Marvin Minsky, and helped organize the first ever public meeting on Artificial Intelligence (AI) with Minsky in 1955.
Selfridge wrote important early papers on neural networks and pattern recognition and machine learning, and his "Pandemonium" paper (1959) is generally recognized as a classic in artificial intelligence. In it, Selfridge introduced the notion of "demons" that record events as they occur, recognize patterns in those events, and may trigger subsequent events according to patterns they recognize. Over time, this idea gave rise to Aspect-oriented programming.
In 1968, in their formative paper "The Computer as a Communication Device", J. C. R. Licklider and Robert Taylor introduced a concept known as an OLIVER (Online Interactive Expediter and Responder) which was named in honor of Selfridge.
Selfridge spent his career at Lincoln Laboratory, MIT (where he was Associate Director of Project MAC), Bolt, Beranek and Newman, and GTE Laboratories where he became Chief Scientist. He served on the NSA Advisory Board for 20 years, chairing the Data Processing Panel. Selfridge retired in 1993.
Selfridge also authored four children's books, "Sticks", "Fingers Come In Fives", "All About Mud", and "Trouble With Dragons". *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

Friday, 2 December 2016

On This Day in Math - December 2

The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom.
~Isaac Asimov

The 337th day of the year, 337 is a Pythagorean prime number, and when its digits are reversed, that is also prime. (A Pythagorean prime is a prime number of the form 4n + 1. Pythagorean primes are exactly the primes that are the sum of two squares (and from this derives the name in reference to the famous Pythagorean theorem.)

The mean of the first 337 square numbers is itself a square. This is the smallest number for which this is true.

The famous Fibonacci area paradox shows a 13x13 square converted to an 8x21 rectangle. The areas of the two figures, 13x13 + 8x21 = 337 (this illusion works with any Fibonacci number F(n) squared and a rectangle that is F(n-1) by F(n+1)  )

1697 St Paul's Cathedral, reconstructed after the Great Fire of 1666 as redesigned by Christopher Wren, was officially opened on December 2nd, 1697. *History Today 

In 1895, James Dewar exhibited his new apparatus for the production of liquid air at the Royal Institution.*TIS

In 1934, the molten glass was poured in the Corning, N.Y. for the first 200-inch diameter telescope mirror. Pyrex glass at 2,700 degrees Fahrenheit was poured into a ceramic mold. The mold had been constructed over a period of several months. The temperature of the glass was lowered during 11 months, a degree or two a day. It was then allowed to cool to room temperature. The 20-ton disk was shipped 26 Mar 1936 for grinding and polishing at the California Institute of Technology, which spanned 11 years, completed on 3 Oct 1947. It was installed in a telescope at the Mount Palomar Observatory on Palomar Mountain, San Diego County, California, which was named the Hale telescope in honour of Dr George Hale who had conceived and promoted it.*TIS

1942 At 3:36 p.m. in a squash court (Actually it was a Racketball court *James Zug Squash, A History of the Game pgs. 135–136.) under the West Stands of Stagg Field (the abandoned football stadium) at the University of Chicago, the first self-sustaining nuclear (fission) reaction took place. Enrico Fermi (1901–1954) was leader of the Manhattan Project. [DSB 4, 582]. *VFR This first run of the nuclear pile produced a single watt of power, Just enough to show that the process was feasible. One of the “about 40” people who watched was Leo Szilard who had conceived the idea of a chain reaction leading to power while stopped at a red light on Southampton Row in London only four years before. *Frederik Pohl, Chasing Science, Pg 20
One of Fermi's assistants ran from the test to the telephone to notify Openheimer, "The Italian navigator has just landed in the New World." *Brody & Brody, The Science Class You Wished You Had

1954 The U.S. Navy dedicates its Naval Ordnance Research Calculator (NORC) at the Naval Surface Weapons Center in Dahlgren, Virginia. John von Neumann was the keynote speaker. The machine was built at the Watson Scientific Computing Laboratory under the direction of Wallace Eckert.
This computer was in demand by many organizations, including two different Navy facilities and Lawrence Livermore National Laboratory in California. Physicist Edward Teller had been trying to receive NORC arguing that the LLNL's nuclear calculations were more important than Dahlgren's ballistic calculations. The Navy won and NORC was delivered to Dahlgren, following the Mark II (1948) and the Mark III (1951).*CHM

1967 Italy issued a postage stamp to commemorate the 25th anniversary of the first atomic chain reaction. Pictured is Enrico Fermi at Los Alamos and a model of the first Atomic Reactor. *VFR

1978 Science News reports, p. 390, that 221,701 − 1 is prime.


1831 Paul David Gustav du Bois-Reymond (2 Dec 1831 in Berlin, Germany - 7 April 1889 in Freiburg, Germany) Du Bois-Reymond's work is almost exclusively on calculus, in particular partial differential equations and functions of a real variable. The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function. In 1873 du Bois-Reymond was the first person to give an example of a continuous function whose Fourier series diverges at a point. Perhaps what was even more surprising, the Fourier series of du Bois-Reymond function diverged at a dense set of points. The important work Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern ("A new theory of convergence and divergence of series with positive terms") led to an increasing understanding of the whole concept of a function.
Du Bois-Reymond published an example of a continuous function which is nowhere differentiable in 1875. It was inspired by a similar function found by Weierstrass in 1872 but not published by him until much later. This example contradicted most mathematicians' intuition, for it was generally believed that a continuous function was differentiable everywhere except in special points. *SAU

1865 Niels Nielsen (2 Dec 1865 in Orslev, Denmark - 16 Sept 1931 in Copenhagen, Denmark) was a Danish mathematician who worked on special functions and number theory. *SAU

1901 Dom George Frederick James Temple​ FRS(born 2 December 1901, London; died 30 January 1992, Isle of Wight) was an English mathematician, recipient of the Sylvester Medal in 1969. He was President of the London Mathematical Society in the years 1951-1953.[2]
Temple took his first degree as an evening student at Birkbeck College, London, between 1918 and 1922, and also worked there as a research assistant. In 1924 he moved to Imperial College as a demonstrator, where he worked with Professor Sydney Chapman. After a period spent with Eddington at Cambridge, he returned to Imperial as reader in mathematics. He was appointed professor of mathematics at King's College London in 1932, where he returned after war service with the Royal Aircraft Establishment at Farnborough. In 1953 he was appointed Sedleian Professor of Natural Philosophy at the University of Oxford, a chair which he held until 1968, and in which he succeeded Chapman. He was also an honorary Fellow of Queen's College, Oxford.
After the death of his wife in 1980, Temple, a devout Christian, took monastic vows in the Benedictine order and entered Quarr Abbey on the Isle of Wight, where he remained until his death. *Wik

1914 Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right". He is best known for Dilworth's theorem (Dilworth 1950) relating chains and antichains in partial orders; he was also the first to study antimatroids (Dilworth 1940). Dilworth advised 17 Ph.D. students and as of 2010 has 373 academic descendants listed at the Mathematics Genealogy Project, many through his student Juris Hartmanis, a noted complexity theorist.*Wik


1594 Gerardus Mercator (5 Mar 1512- 2 Dec 1594) Flemish cartographer whose most important innovation was a map, embodying what was later known as the Mercator projection, on which parallels and meridians are rendered as straight lines spaced so as to produce at any point an accurate ratio of latitude to longitude. He also introduced the term atlas for a collection of maps. *TIS

1873 Karl Gräffe Gräffe (7 Nov 1799 in Brunswick, Germany - 2 Dec 1873 in Zurich, Switzerland) is best remembered for his method of numerical solution of algebraic equations, developed to answer a prize question of the Berlin Academy of Sciences. It is particularly suitable for methods developed for using computers to solve mathematical problems. This method is today called the Dandelin-Gräffe method after the two mathematicians who independently investigated it. The history of the Dandelin-Gräffe method is discussed in and . Lobachevsky is also credited with the independent discovery of the method which appears in his little-known book on algebra.*SAU

1966 L(uitzen) E(gbertus) J(an) Brouwer (27 Feb 1881, 2 Dec 1966) was a Dutch mathematician who founded mathematical Intuitionism (a doctrine that views the nature of mathematics as mental constructions governed by self-evident laws). He founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem. (Topology is the study of the most basic properties of geometric surfaces and configurations.) The Brouwer fixed point theorem is named in his honor. He proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, the treatment of general continuous mappings. *TIS

1982 Geoffrey Timms studied at Leeds and Cambridge and then took up a post at St Andrews. During World War II he served at Bletchley Park and Cheltenham and joined the Foreign Office afterwards. He became President of the EMS in 1941.*SAU

2006 Dikran "Dick" Tahta (7 August 1928 – 2 December 2006) was a British-Armenian mathematician, teacher and author.
Dikran Tahta is a descendant of an Ottoman Armenian family who settled in Manchester after the First World War. Much of his childhood, and the influence of his Armenian religious upbringing, is reflected upon in his penultimate book Ararat Associations, in which he notes how his parents were keen for their children to have an English education, yet made sure that they spoke Armenian at home. He was christened by Bishop Tourian in the Armenian Church in Manchester, and his name Dikran was shortened to Dick, but he never forgot his Armenian roots.
From Rossall School, in Fleetwood, Lancashire, he gained a scholarship to Christ Church, Oxford, in 1946. His main subject was Mathematics, but he also read widely in English literature, philosophy and history.
In the 1970s he was involved in the ATV television programme of mathematics for schools entitled 'Leapfrogs' (produced and directed by Paul Martin) and promoted visual approaches to mathematics. His paper "On Geometry" argued that geometrical approaches to mathematics could not be reduced to algebraic approaches. In line with this thinking, he produced the ATM book Geometric Images, and co-authored Images of Infinity with Ray Hemmings. The Leapfrogs group of Tahta and Hemmings, together with David Sturgess, Leo Rogers and Derick Last also produced hands-on teaching materials including workbooks for the polycube. He also drew upon insights into pedagogy in the writings of Mary Boole on mathematics education.
After retirement, he went to teach in the United States and South Africa, and became a tutor for the Open University.
His last book was The Fifteen Schoolgirls about Thomas Kirkman, known for the Kirkman's schoolgirl problem, a problem in combinatorics, which also delved into the byways of Victorian amateur mathematics.
In his obituary, The Guardian newspaper described Dick as "one of the outstanding mathematics teachers of his generation", who was notable for having inspired physicist Stephen Hawking. The Guardian commented on his death that "He was a wise and generous man who inspired love and an increase of intellectual energy in everyone who came within his ambit." *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

Thursday, 1 December 2016

On This Day in Math - December 1

Beauty is the first test:
there is no permanent place in the world for ugly mathematics.

~Godfrey Harold Hardy

The 336th day of the year; there are 336 dimples on an American golf ball. There are 336 ways to partition 41 into primes.

Given eight points around a circle, there are 336 triangles with these points as the vertices.

336 is the product of three consecutive integers, 6*7*8  = 336


1729 Euler/Goldbach correspondence begins: Goldbach was also a kind of mentor to Leonhard Euler. For over 25 years they exchanged letters, 196 of which survive. These letters give us a window into Euler’s scientific and personal life. In Goldbach’s very first letter to Euler, dated December 1, 1729, Goldbach got Euler interested in number theory. Goldbach added note at the end of the letter: “P. S. Have you noticed the observation of Fermat that all numbers of the form 22x+1, that is 3, 5, 17, etc., are prime numbers, but he did not dare to claim he could demonstrate it, nor, as far as I know, has anyone else been able to prove it.” Three years later, in a five-page paper that now bears the index number E26, Euler shows that the F(5) = 4,294,967,297 = 641× 6,700,417 . That is, Fermat was wrong. *Ed Sandifer, How Euler Did It

1764 Alexander Small writes Benjamin Franklin from England, "My Namesake the Virginian Professor (William Small) is here; and desires to be most particularly remembered to you."
Small is known for being Thomas Jefferson's professor of Natural Philosophy at William and Mary, and for having an influence on the young Jefferson. (I could not determine if Alexander and William Small are related) *Natl. Archives

1783 J. A. C. Charles was the first man to see the sun set twice in one day. He did it by making a flight (to 9000 feet) in a hydrogen balloon. *VFR (Charles is often considered the inventor of the hydrogen balloon.) The first manned voyage of a hydrogen balloon left Paris carrying Professor Jacques Alexander Cesar Charles and Marie-Noel Robert to about 600 m and landed 43 km away after 2 hours in the air. Robert then left the balloon, and Charles continued the flight briefly to 2700 m altitude, measured by a barometer. This hydrogen-filled balloon was generally spherical and used a net, load ring, valve, open appendix and sand ballast, all of which were to be universally adopted later. His hydrogen generator mixed huge quantities of sulfuric acid with iron filings. On 27 Aug 1783, Charles had launched an unmanned hydrogen balloon, just before the Montgolfiers' flight. *TIS (One of these altitudes is obviously wrong. )

1851 On December the first, Louis-Napoleon Bonaparte, who had been instrumental in supporting Foucault in the demonstration of his pendulum, ordered that the pendulum demonstration cease and the Pantheon return to being used as a church (Louis Philippe had secularized the Pantheon in 1830 and stopped burials in the crypt). Why did he stop the popular demonstrations? We do not know, but on the next day citizens of France awoke to find notices posted on the major buildings, “The National Assembly is dissolved… “ Louis-Napoleon had taken his first step to becoming Emperor of France. *Amir D Aczel, Pendulum, pg 174

1890, after regular competition, Peano was named extraordinary
professor of infinitesimal calculus at the University of Turin. *Hubert Kennedy
Eight Mathematical Biographies Pg 23

1896 Frank Broaker of New York City received certificate No. 1 from the New York State Board of Certified Public Account Examiners thus becoming the first CPA in the US. *JN Kane, Famous First Facts,

In 1997 eight planets from our Solar System lined up from West to East beginning with Pluto, followed by Mercury, Mars, Venus, Neptune, Uranus, Jupiter, and Saturn, with a crescent moon alongside, in a rare alignment visible from Earth that lasted until Dec 8. Mercury, Mars, Venus, Jupiter and Saturn were visible to the naked eye, with Venus and Jupiter by far the brightest. A good pair of binoculars is needed to see the small blue dots that are Uranus and Neptune. Pluto is visible only by telescope. The planets also aligned in May 2000, but too close to the sun to be visible from Earth. It will be at least another 100 years before so many planets will be so close and so visible.*TIS


1671 John Keill (1 Dec 1671; 31 Aug 1721) Scottish mathematician and natural philosopher, who was a major proponent of Newton’s theories. He began his university education at Edinburgh under David Gregory, whom he followed to Oxford, where Keill lectured on Newton's work, and eventually became professor of astronomy. In his book, An Examination of Dr. Burnett's Theory of the Earth (1698), Keill applied Newtonian principles challenging Burnett's unsupportable speculations on Earth's formation. In 1701, Keill published Introductio ad Veram Physicam, which was the first series of experimental lectures and provided a clear and influential introduction to Isaac Newton’s Principia. He supported Newton against priority claims by Leibnitz for the invention of calculus. *TIS

1792 Nikolay Ivanovich Lobachevsky (1 Dec 1792; 24 Feb 1856) Russian mathematician who, with János Bolyai of Hungary, is considered the founder of non-Euclidean geometry. Lobachevsky constructed and studied a type of geometry in which Euclid's parallel postulate is false (the postulate states that through a point not on a certain line only one line can be drawn not meeting the first line). This was not well received at first, but his greatest vindication came with the advent of Einstein's theory of relativity when it was demonstrated experimentally that the geometry of space is not described by Euclid's geometry. Apart from geometry, Lobachevsky also did important work in the theory of infinite series, algebraic equations, integral calculus, and probabilty. *TIS William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his Songs by Tom Lehrer album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as "Plagiarize!", as long as one is always careful to call it "research". According to Lehrer, the song is "not intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons".*Wik (The lyrics are here)

1847 Christine Ladd-Franklin (1 Dec 1847; 5 Mar 1930) American scientist and logician known for contributions to the theory of colour vision accounting for the development of man's color sense which countered the established views of Helmholtz, Young, and Hering. Her position was that color-sense developed in stages. Ladd- Franklin's conclusions were particularly useful in accounting for color-blindness in some individuals. In logic, she published an original method for reducing all syllogisms to a single formula *TIS Ladd-Franklin was the first woman to have a published paper in the Analyst (at this time, 1877, it was more of a recreational mathematics publication still edited by the self-educated Ohio farmboy, Joel E Hendricks. The article was simply titled "Quaternions." ). She was also the first woman to receive a Ph.D. in mathematics and logic. The majority of her publications were based on visual processes and logic. Her views on logic influenced Charles S. Peirce’s logic and she was highly praised by Prior. *Wik

1892 Krishnaswami Ayyangar (1 Dec 1892 in Attipattu, Chingleput district, Tamil Nadu, India - June 1953 in Mysore, India) was an Indian mathematician who worked in Mysore. He produced important work on the history of Hindu mathematics. *SAU

1913 Colossus' Team Member Chandler is Born W.W. Chandler was born in Bridport, England. He obtained his B.Sc. from London University in 1938 by private study while working as a telephone engineer at the British Post Office Research Department. During the war he was responsible for the installation and maintenance of the Colossus at Bletchley Park. The Colossus represented the first electronic computer, however it was programmed by a mechanical switchboard. Its was used to crack the German Fish codes which guarded the highest levels of German communication. Winston Churchill characterized the Bletchley Park team as the geese who laid the golden eggs but never cackled.
After the war Chandler participated in development and installation of the MOSAIC computer and worked on optical character recognition. He died on September 11, 1989. *CHM

1941 Stephen A. Benton (1 Dec 1941; 9 Nov 2003.) American physicist who was a pioneer in medical imaging and fine-arts holography. His fascination with optical phenomena began with the 3-D glasses he used as an 11-year-old to watch te 1953 movie "House of Wax." In 1968, he invented the "rainbow holograms" as seen on credit cards while working for Polaroid Corporation. He turned to academia as an assistant professor at Harvard (1968) and later a professor at Massachusetts Institute of Technology from 1985 where he helped set up the Spatial Imaging Group and headed the M.I.T. media art and sciences program. Benton was a pioneer in natural light holography as a artistic medium, and was a curator at the Museum of Holography in Manhattan until it closed in 1992.*TIS


1750 Johann Doppelmayr (27 Sept 1677 in Nuremberg, Germany - 1 Dec 1750 in Nuremberg, Germany)was a German mathematician who wrote on astronomy, spherical trigonometry, sundials and mathematical instruments. Doppelmayr also wrote a book of tremendous value giving biographical details of 360 mathematicians and instrument makers of Nuremberg from the 15th to the 18th century. This had the lengthy title Historische Nachricht von den Nürnbergischen Mathematicis und Künstlern, welche fast von dreyen Seculis her durch ihre Schriften und Kunst-Bemühungen die Mathematic und mehrere Künste in Nürnberg vor andern trefflich befördert und sich um solche sehr wohl verdient gemacht zu einem guten Exempel, und zur weitern rühmlichen Nachahmung and was published in 1730. *SAU

1866 Sir George Everest (1790, 1 Dec 1866) British military engineer and geodesist, born in Gwernvale, Powys, Wales, UK. He worked on the trigonometrical survey of India (1818-43), providing the accurate mapping of the subcontinent. For more than twenty-five years and despite numerous hardships, he surveyed the longest arc of the meridian ever accomplished at the time. Everest was relentless in his pursuit of accuracy. He made countless adaptations to the surveying equipment, methods, and mathematics in order to minimize problems specific to the Great Survey: immense size and scope, the terrain, weather conditions, and the desired accuracy. Mount Everest, formerly called Peak XV, was renamed in his honour in 1865. *TIS (Mary Boole, self-taught mathematician and wife of George Boole was his niece)

1935 Bernhard Voldemar Schmidt (30 Mar 1879, 1 Dec 1935) Astronomer and optical instrument maker who invented the telescope named for him. In 1929, he devised a new mirror system for reflecting telescopes which overcame previous problems of aberration of the image. He used a vacuum to suck the glass into a mold, polishing it flat, then allowing in to spring back into shape. The Schmidt telescope is now widely used in astronomy to photograph large sections of the sky because of its large field of view and its fine image definition. He lost his arm as a child while experimenting with explosives. Schmidt spent the last year of his life in a mental hospital.*TIS

1947 Godfrey Harold Hardy (1877, 1 Dec 1947)English mathematician known for his work in number theory and mathematical analysis. Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. Although Hardy considered himself a pure mathematician, early in his career, he nevertheless worked in applied mathematics when he formulated a law that describes how proportions of dominant and recessive genetic traits will propagate in a large population (1908). Hardy considered it unimportant but it has proved of major importance in blood group distribution. As it was also independently discovered by Weinberg, it is known as the Hardy-Weinberg principle. *TIS G. H. Hardy died—on the same day that the Copley Medal was to be presented to him by the Royal Society of London. [Collected Papers of G. H. Hardy, vol. 1, p. 8].

1964 J.B.S.(John Burdon Sanderson) Haldane (5 Nov 1892, 1 Dec 1964) was a British geneticist and biometrician who opened new paths of research in population genetics and evolution. He began studying science at the age of eight, as assistant to his father (the noted physiologist John Scott Haldane). J.B.S. Haldane also worked in biochemistry, and on the effects of diving on human physiology. A Marxist from the 1930s, Haldane was well known for his outspoken Marxist views.He resigned from the Communist Party c. 1950 on the issue of Lysenko's claims to have manipulated the genetic structure of plants and "Stalin's interference with science". He became known to a large public as a witty popularizer of science with such works as Daedalus (1924), and Possible Worlds (1927).*TIS

1977 Kenneth O. May (July 8, 1915, Portland, Or. – December 1,1977) was an American mathematician and historian of mathematics, who developed May's theorem. The Kenneth O. May Prize is awarded for outstanding contributions to the history of mathematics. Ken May established Historia Mathematica, and preserved it by separating it from its creator, "The distinguished predecessors of HM were associated with their founders and died with them.  If HM is to avoid this fate, we must prepare and carry through a prompt transfer of editorial responsibility to younger hands." His list of publications numbers above 300.  *Henry S. Tropp, E'loge, Isis 70, Sept 1979, Pgs 419-422

1983 Leon Mirsky (19 Dec 1918 in Russia - 1 Dec 1983 in Sheffield, England)worked in Number Theory, Linear Algebra and Combinatorics.*SAU

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