## Monday 13 May 2024

### On This Day in Math - May 13

In mathematics you don't understand things.
You just get used to them.

~ John von Neumann

The 133rd day of the year; 133 is a "happy number".  If you sum the squares of the digits and then repeat the process and the sum will eventually come to one. (12 + 32+32= 19 ... ===  82 === 68 === 100 ====1) Some numbers, "unhappy ones", never reach one. (Student's might explore happy numbers to find how many times the process must be iterated for different numbers to reach one, for example I (33) = 5  Alternatively, curious students may wonder what happens to the "unhappy" numbers if they never reach one.)

133 is a repdigit in base 11 (111) and base 18 (77),

133 is the sum of the squares of the first three semi-primes, and is a semi-prime itself. it is the smallest number with this property.  133= 42 + 62 +92 =7*19

133, and 134 were used by Euler in generating birectangular Heronian tetrahedra.  He created a method for deriving them from equal sums of fourth powers  p^4 + q^4 = r^4 + s^4 and used 133 and 134 on one side, and 59 and 158 on the other.  The actual side lengths of the three perpendicular edges created from this quartet were over 332,000,000.

133 is the smallest integer, n, for which 10 n +(1or 3 or 7 or 9) are all composite.  *Prime Curios

The Dewey Decimal system classification for numerology is 133.533, and if you add the first to the reverse of the second 133+335=666....

And Jim Wilder @wilderlab posted this interesting observation about 133 and it's reversal, 331.

### EVENTS

1637 The table knife was created by Cardinal Richelieu in France. Until this time, daggers were used to cut meat, as well as to pick one's teeth. Richelieu had the points rounded off all of the knives to be used at his table *TIS

He was a French statesman and prelate of the Catholic Church. He became known as l'Éminence rouge, or "the Red Eminence", a term derived from the title "Eminence" applied to cardinals and from the red robes that they customarily wear.

1673 Scottish mathematician, physicist and optician James Gregory in a letter to John Collins, remarks on diffraction:

If ye think fit, ye may signify to Mr. Newton a small experiment, which (if he know it not already) may be worthy of his consideration. Let in the sun’s light by a small hole to a darkened house, and at the hole place a feather, (the more delicate and white the better for this purpose,) and it shall direct to a white wall or paper opposite to it a number of small circles and ovals, (if I mistake them not) whereof one is somewhat white, (to wit, the middle, which is opposite to the sun,) and all the rest severally coloured. I would gladly hear his thoughts of it.

"Diffraction was first investigated and described by the Jesuit astronomer, mathematician and physicist Francesco Maria Grimaldi (1618 – 1663) and published posthumously in his Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo in 1665. Grimaldi was one of the prominent products of Clavius’ mathematical education programme who as well as his investigation into light also conducted empirical experiments into the laws of free fall, confirming Galileo’s results, together with another Jesuit scientist Giovanni Battista Riccioli (1598 -1671) with whom he also produced the most accurate map of the moon in the 17th century. On the basis of his optical investigation and in particular his discovery of diffraction, Grimaldi developed a wave theory of light. It was Grimaldi who gave this particular optical phenomenon its name deriving it from the Latin verb diffringere ‘break into pieces’ from ‘dis’ apart and frangere ‘to break’. "

*Thony Christie, The Renaissance Mathematicus

1733 Swedish Astronomer Birger Wassenius reports on the Eclipse and attributes solar prominences to the Moon:

I can tell you is this, that I soon after the sun's total extinction became aware of some small lighter spots UTI the bright ring, or the atmosphere, about 3 or 4, of different temperament and size, which set in towards the moon's periphery , but at no point next to it. As is now not the moon altogether at one time could fall into my eyes through a long tube, so I had particularly esteem of the largest of these spots, which in the tube appeared on the northeast side of the moon. Being that as composed of three reddish cloud drops placed adjacent to one side, with darker colors or stripes in between, such as the figure below shows fairly. "

*Astronomer Guide

After five years of study at Uppsala University, he published his first academic essay in 1717, which was about the planet Mars . It ended with the calculation of the transit of Venus that would occur in 1761. During his studies, he also built several aids to be able to make accurate astronomical studies.

Wassenius Almanack

In 1769 Britain's Board of Longitude awarded 10 Pounds to Israel Lyons, Mathematician for, "Reward for his solution to a problem proposed by the late Dr Halley which the Commissioners of Longitude think will be useful to Navigation."  The problem seemed to be related to "traverse sailing."  In June of 1775 his widow would receive an additional 31.50 Pounds for "some of her husband's Problems & Solutions which have been given up by her..." *Derek Howse, Britain's Board of Longitude: The Finances, 1714-1828

Called Israel Lyons the Younger, he was one of the mathematicians who computed the navigation tables for the first Nautical Almanac (1767).

1829 Charles-Francois Sturm presented his theorem for ﬁnding the number of real roots of a polynomial equation to the French Academy. *VFR  For counting and isolating the real roots, other methods, such as Descartes' rule of signs, are usually preferred, because they are computationally more efficient.

Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate polynomials.

1861 Australian astronomer John Tebbutt discovered C/1861 J1, the Great Comet of 1861.  Tebbutt also discovered Nova Scorpii 1862, a nova visible to the unaided eye.

 *SciHiBlog

In 1890, Nikola Tesla was issued a patent for an electric generator (No. 428,057). *TIS

1940 aviation pioneer Igor Sikorsky made the maiden flight with his newly developed helicopter VS-300 *@yovisto

Designed by Igor Sikorsky and built by the Vought-Sikorsky Aircraft Division of the United Aircraft Corporation, the helicopter was the first to incorporate a single main rotor and tail rotor design. *Wik

2010 The Times reported on 13 May 2010 that Foucault's original Pendulum is damaged, "Historic instrument is irreparably damaged in an accident at a Paris museum. The original pendulum, which was used by French scientist Leon Foucault to demonstrate the rotation of the Earth and which forms an integral part of Eco's novel's labyrinthine plot, has been irreparably damaged in an accident in Paris. The pendulum's cable snapped last month and its sphere crashed to the marble floor of the Musee des Arts et Metiers. In 1851, Foucault used the pendulum to perform a sensational demonstration in the Paris Pantheon, proving to Napoleon III and the Parisian elite that the Earth revolved around its axis. Such was its success that the experiment was replicated throughout Europe.
Thierry Lalande, the museum's ancient scientific instruments curator, said that the pendulum's brass bob had been badly damaged in three places and could not be restored.
"It's not a loss, because the pendulum is still there, but it's a failure because we were unable to protect it," he said. The circumstances surrounding the accident have raised eyebrows in France.
The museum regularly hosts cocktail parties in the chapel that houses the pendulum, and Mr Lalande admitted that several alarming incidents had occurred over the past year. In May 2009, for example, a partygoer grabbed the 28kg instrument and swung it into a security barrier. *Times Higher Education

2013 Peruvian mathematician Harald Andrés Helfgott releases pre-print claiming a completed proof of the weak Goldbach Conjecture. The weak, or ternary, Goldbach conjecture states that every odd integer greater than 5 can be written as the sum of three primes; *The Value of the Variable at Wordpress.com

2013  At a Harvard seminar on May 13, 2013, the first major breakthrough was  produced in solving the twin primes conjecture.  A lecturer from the University of New Hampshire, Yitang Zhang, had proved that there are infinitely many pairs of primes that differ by no ,ore than 70,000,000.  It was a long way from differing by two, but it was an even greater distance from infinity.  He had submitted his paper to the Annals of Mathematics in April, and they had rushed to get reviews, which turned out to be enthusiastically positive. By May 21, 2013, the paper was accepted for publication on the 1st of May 2014.
By the 31st of May 2013, a group led by Scott Morrison and Terry Tao had lowered the gap to 42,342,946; game on!

This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur Fellowship. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015.

2016 Friday the 13th.  The thirteenth of the month is more likely to occur on Friday than on any other day of the week.
Each Gregorian 400-year cycle contains 146,097 days (365 × 400 = 146,000 normal days, plus 97 leap days) and they equal 146,097 days, total. 146,097 ÷ 7 = 20,871 weeks. Thus, each cycle contains the same pattern of days of the week (and thus the same pattern of Fridays that are on the 13th). The 13th day of the month is slightly more likely to be a Friday than any other day of the week.   On average, there is a Friday the 13th once every 212.35 days (compared to Thursday the 13th, which occurs only once every 213.59 days).
According to the Stress Management Center and Phobia Institute in Asheville, North Carolina, an estimated 17 to 21 million people in the United States are affected by a fear of this day. Some people are so paralyzed by fear that they avoid their normal routines in doing business, taking flights or even getting out of bed. "It's been estimated that [US]$800 or$900 million is lost in business on this day". Despite this, representatives for both Delta and Continental Airlines say that their airlines do not suffer from any noticeable drop in travel on those Fridays.
According to folklorists, there is no written evidence for a "Friday the 13th" superstition before the 19th century. The earliest known documented reference in English occurs in Henry Sutherland Edwards' 1869 biography of Gioachino Rossini.

In both Greek and Spanish populations, Tuesday the 13th is considered an "unlucky" day, and Italians have a similar tradition for Friday the 17th.

Don't worry, if something terrible doesn't happen to you today, you have another chance for disaster next January. *PB (Or, for 2024, in September.)

BIRTHS

1713 Alexis Clairaut (sometimes Clairault)( 13 May 1713 – 17 May 1765) a French mathematician who worked to confirm the Newton-Huygens belief that the Earth was flattened at the poles. He was a child prodigy was studying calculus at age 10 and was admitted to the Academy of Sciences at age 18. He was the first person to estimate the mass of Venus to a close value. He also calculated the return date of Halley's comet. In about 1737, Pierre de Maupertuis led an expedition (including Clairaut) to measure a degree along a meridian in Lapland, while Bouguer and La Condamine went to Peru. The results, even before the Peru expedition had returned, showed that Newton was correct in predicting that the earth was flattened at the poles. He published the results in Théorie de la figure de la Terre in 1743.(various)
A nice brief summary of Clairaut's life and works is here.

As a child prodigy, at age ten he was studying calculus, tutored by his father. Clairaut read his first paper, Quatre problèmes sur de nouvelles courbes to the Paris Academy (1726) at the age of 13. He accompanied Maupertuis on an expedition to Lapland to measure the length of the meridian. From this experience, he began a book (1743) on the shape of the rotating earth under the influences of gravity and centrifugal forces. Further, he showed how to measure the shape by use of measurements of the effect of gravity at different location on the swing of a pendulum. He also determined the first reasonable value for the mass of Venus, an improved value for the mass of the moon, and predicted the timing of the return of Halley's Comet.

1750 Lorenzo Mascheroni (May 13, 1750 – July 14, 1800) was a geometer who proved in 1797 that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed.*SAU He is also known for the Euler–Mascheroni constant which gives the limit of the difference between ln(n) and the sum of the harmonic series for the first n terms. The constant first appeared in a 1735 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835. *Wikipedia,

He was also a founder of the science of mechanics, asserting that the velocity of a falling body was independent of its weight.

1753 Lazare-Nicolas-Marguerite Carnot,  (13 May 1753 – 2 August 1823)  who published his "Reﬂections on the Metaphysics of the Inﬁnitesimal Calculus" in 1797. It was written in 1784 for a competition of the Berlin academy seeking a “clear and precise” foundation for the calculus. *VFR  His son Sadi Carnot was a founder of the field of thermodynamics and the theory of heat engines .  Lazare is better known outside of mathematics as a military tactician and politician.

1804 Janet Taylor was (born Jane Ann Ionn, 13 May 1804 – 25 January 1870the sixth child of the Reverend Peter Ionn and Jane Deighton, the daughter of a country gentleman.

After the death of her mother when she was just seven years old, Janet gained a scholarship at the precociously young age of nine, to attend Queen Charlotte’s school in Ampthill, Bedfordshire, where the other girls were all aged over 14. Her life thereafter took her into the heart of maritime London.

Her father, the curate of the church of St Mary and St Stephen and schoolmaster of the Free Grammar School at Wolsingham, inspired her in the wonders of navigation. She became a prodigious author of nautical treatises and textbooks, born of a fascination in particular in measuring longitude by the lunar distance method.

She conducted her own Nautical Academy in Minories in the east end of the City, not far from the Tower of London; she was a sub-agent for Admiralty charts; ran a manufacturing business for nautical instruments, many of which she designed herself; and embarked on the business of compass-adjusting at the height of the controversies generated by magnetic deviation and distortions on iron ships.

Through her scientific work, Janet established a respectful correspondence with those in the highest positions in the maritime community: men like the head of the Admiralty’s Hydrographic Office, Captain, later Rear-Admiral Sir Francis Beaufort, and Professor Sir George Biddell Airy, the Astronomer Royal.

Where they were hesitant at first in their engagement with Mrs Taylor, she clearly won their support and respect. Between 1617 and 1852 there 79 patents awarded for nautical instruments – Janet was the only women among them for her Mariner’s Calculator. Dismissed by the Admiralty, it had no commercial future and only one instrument is known to remain in existence.

In 1835, in consideration of ‘services she has extended to seamen’, through her Lunar Tables, the Admiralty awarded her £100 ‘from scientific funds’, a ‘handsome pecuniary award’. She was similarly honoured by the two other members of the ‘big three’ of the 19th Century maritime world in Britain: the Elder Brethren of Trinity House and the East India Company.

Her Mariners compass was displayed on the first page of her Lunar Tables.

She also received international recognition for her contributions: gold medals from the King of Holland and King Friedrich Wilhelm III of Prussia; and, by 1844, a medal from the Pope.

Janet passed away on in January 1870. She was the author of many books, including some that ran to 27 editions and several are still in print today. She was also an inventor of several nautical instruments with some being held in the national maritime Museum in Greenwich.

Sadly, she died in obscurity and bankrupt, estranged from all her children, several of whom lived in Australia. Her death certificate records her occupation simply as ‘Teacher of Navigation’, but she was far more than this.  *Science Focus

1857 Sir Ronald Ross (born 13 May 1857, 16 Sep 1932) English physician, bacteriologist and mathematician whose discovery of the malarial parasite in the gastrointestinal tract of the Anopheles mosquito led to the realization that malaria was transmitted by Anopheles. For this work, he was awarded the 1902 Nobel Prize for Physiology or Medicine, becoming the first British Nobelist. He began studying malaria in 1892. In 1894 he made an experimental investigation in India of the hypothesis of Alphonse Laveran and Patrick Manson that mosquitoes are connected with the propagation of the disease. After two and a half years' failure, Ross succeeded in demonstrating the life-cycle of the parasites of malaria in mosquitoes, thus establishing the hypothesis of Laveran and Manson. Later, in West Africa he found the species of mosquitoes which convey the deadly African fever.*TIS (He is most remembered for his work on malaria, but his greatest influence may have come from his development and publishing of a mathematical theory of epidemiology.)

1857 Frederick William Sanderson (13 May 1857 – 15 June 1922) was headmaster of Oundle School from 1892 until his death. He was an education reformer, and both at Oundle, and previously at Dulwich College where he had started as assistant master, he introduced innovative programs of education in engineering. Under his headmastership, Oundle saw a reversal of a decline from which it had been suffering in the middle of the 19th century, with school enrolment rising from 92 at the time of his appointment to 500 when he died.
Sanderson was the inspiration for the progressive headmaster character in H. G. Wells' novel Joan and Peter. Wells had sent his own sons to Oundle, and was friendly with Sanderson. After Sanderson's death, which occurred shortly after delivering an address to Wells and others, Wells initially worked on his official biography, entitled Sanderson of Oundle, but later abandoned it in favour of an unofficial biography, The Story of a Great Schoolmaster. *Wik

1888 Inge Lehmann  (13 May 1888 – 21 February 1993) was a Danish seismologist and geophysicist. In 1936, she discovered that the Earth has a solid inner core inside a molten outer core. Before that, seismologists believed Earth's core to be a single molten sphere, being unable, however, to explain careful measurements of seismic waves from earthquakes, which were inconsistent with this idea. Lehmann analysed the seismic wave measurements and concluded that Earth must have a solid inner core and a molten outer core to produce seismic waves that matched the measurements. Other seismologists tested and then accepted Lehmann's explanation. Lehmann was also one of the longest-lived scientists, having lived for over 104 years *Wik

Lehmann Memorial

1931 András Hajnal (May 13, 1931 - July 30, 2016) is an emeritus professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics. Hajnal is the author of over 150 publications. Among the many co-authors of Paul Erdős, he has the second largest number of joint papers, 56. With Peter Hamburger, he wrote a textbook, Set Theory

In 1992, Hajnal was awarded the Officer's Cross of the Order of the Republic of Hungary.[5] In 1999, a conference in honor of his 70th birthday was held at DIMACS, and a second conference honoring the 70th birthdays of both Hajnal and Vera Sós was held in 2001 in Budapest. Hajnal became a fellow of the American Mathematical Society in 2012.*Wik

DEATHS

1826 Christian Kramp(July 8, 1760 – May 13, 1826) As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers. His work on factorials is independent of that of Stirling and Vandermonde. The word factorial is reported to be the creation of Louis François Antoine Arbogast (1759-1803). The symbol now commonly used for factorial seems to have been created by Christian Kramp in 1808. It is referred to as "Kramp's notation" in Chrystal's famous Algebra.

My

1878 Joseph Henry (17 Dec 1797, 13 May 1878 at age 80) One of the first great American scientists after Benjamin Franklin. Although Henry at an early age appeared to be headed for a career in the theater, a chance encounter with a book of lectures on scientific topics turned his interest to science. He aided Samuel F.B. Morse in the development of the telegraph and discovered several important principles of electricity, including self-induction, a phenomenon of primary importance in electronic circuitry. He was the first Secretary (director) of the Smithsonian Institution (1846-1878), where he established the foundation of a national weather service. For more than thirty years, Henry insisted that basic research was of fundamental importance. *TIS

Henry discovered the electromagnetic phenomenon of self-inductance. He also discovered mutual inductance independently of Michael Faraday, though Faraday was the first to make the discovery and publish his results. In his honor, the SI unit of inductance is named the henry.

He may have also been responsible, indirectly for the mention of the cycloid in Moby-Dick, and it's tautochrone property, that "all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time."

It is almost certain that the limited public school education would not include this fact.  Most high school students today would never be introduced to it.  But in Melville's brief time at the Albany Academy it was said that Herman excelled in "ciphering" and won the school prize.  Perhaps his interest in geometry and such was an outstanding teacher, and former alumni of the Albany Academy, young Joseph Henry.

The old Albany Academy building, known officially as Academy Park by the City School District of Albany, its owner (after the park in which it is located), and formerly known as the Joseph Henry Memorial.

1919 Eugen Otto Erwin Netto (30 June 1848 – 13 May 1919) was a German mathematician. He was born in Halle and died in Giessen.

Netto's theorem, on the dimension-preserving properties of continuous bijections, is named for Netto. Netto published this theorem in 1878, in response to Georg Cantor's proof of the existence of discontinuous bijections between the unit interval and unit square. His proof was not fully rigorous, but its errors were later repaired.

Netto made major steps towards abstract group theory when he combined permutation group results and groups in number theory. He also worked on space-filling curves.

1939 Stanisław Leśniewski (March 30, 1886, Serpukhov – May 13, 1939, Warsaw) was a Polish mathematician, philosopher and logician. Leśniewski belonged to the first generation of the Lwów-Warsaw School of logic founded by Kazimierz Twardowski. Together with Alfred Tarski and Jan Łukasiewicz, he formed the troika which made the University of Warsaw, during the Interbellum, perhaps the most important research center in the world for formal logic. *Wik

The grand staircase that marks the entrance to the World of Knowledge at the University of Warsaw Library culminates with four statues of famous Polish philosophers and thinkers of the Modern period: Kazimierz Twardowski, Jan Łukasiewicz, Alfred Tarski, and Stanisław Leśniewski. These statues are also a modern interpretation of the ancient propylaea and refer back to the porticos of Ancient Greek and Roman temples of science.

1944 William Edward Hodgson Berwick (11 March 1888 in Dudley Hill, Bradford – 13 May 1944 in Bangor, Gwynedd) was a British mathematician, specializing in algebra, who worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals.*Wik

1983 Otto (Hermann Leopold) Heckmann (23 Jun 1901, 13 May 1983 at age 81) was a German astronomer noted for measuring stellar positions and his studies of relativity and cosmology. He also made notable contributions to statistical mechanics. In 1931, He proved that, under the assumptions that matter is homogeneously distributed throughout the universe and is isotropic (having identical properties in every direction), the theory of general relativity could result in an open, or Euclidean, universe as readily as a closed one. Heckmann organized an international program to photograph and chart the positions of the stars in the Northern Hemisphere, which led to the publication in 1975 of the third German Astronomical Society catalog, Astronomische Gesellschaft Katalog (AGK3). *TIS

1984 Stanislaw Marcin Ulam (13 April 1909 – 13 May 1984)  Polish-American mathematician who played a major role in the development of the hydrogen bomb at Los Alamos. He solved the problem of how to initiate fusion in the hydrogen bomb by suggesting that compression was essential to explosion and that shock waves from a fission bomb could produce the compression needed. He further suggested that careful design could focus mechanical shock waves in such a way that they would promote rapid burning of the fusion fuel. Ulam, with J.C. Everett, also proposed the "Orion" plan for nuclear propulsion of space vehicles. While Ulam was at Los Alamos, he developed "Monte-Carlo method" which searched for solutions to mathematical problems using a statistical sampling method with random numbers. *TIS He is buried in Santa Fe National Cemetery in Santa Fe, New Mexico, USA

“While chatting at the Scottish Caf´e with Borsuk, an outstanding Warsaw topologist, he [Ulam] saw in a ﬂash the truth of what is now called the Borsuk-Ulam theorem. Borsuk had to commandeer all his technical resources to prove it.” For n = 2, this theorem can be interpreted as asserting that some point on the globe has precisely the same weather as its antipodal point. The ‘weather’ has to mean two variables (R2) that vary continuously (f) on the surface (S 2) of the earth. Perhaps temperature and humidity will do? *theoremoftheday.org

2005 George Bernard Dantzig (November 8, 1914 – May 13, 2005) was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics.
Dantzig is known for his development of the simplex algorithm, an algorithm for solving linear programming problems, and his work with linear programming, some years after it was invented by the Soviet mathematician & economist Leonid Kantorovich. In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture of Jerzy Neyman.
Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford. *Wik

2018 Ennackal Chandy George Sudarshan (16 September 1931 – 13 May 2018)  is a prominent Indian-American physicist, author and professor at the University of Texas at Austin. Sudarshan has made significant contributions to several areas of physics. He was the originator (with Robert Marshak) of the V-A theory of the weak force (also done later by Richard Feynman and Murray Gell-Mann), which eventually paved the way for the electroweak theory. Feynman said in 1963: "The V-A theory that was discovered by Sudarshan and Marshak, publicized by Feynman and Gell-Mann".

He also developed a quantum representation of coherent light (for which Glauber was awarded the 2005 Nobel). *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