## Tuesday 20 August 2024

### On This Day in Math - August 20

Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfillment in the act but must proclaim and elaborate a poetic form of truth.
~Salomon Bochner

The 232nd Day of the Year
232 is the maximum number of regions that the plane can be divided into with 21 lines (how many of the regions would be of infinite area?)

232 is a palindrome in base ten, but no other base from 2-9

There are 232 bracelets possible with 8 beads of one color and seven of another.

And from Derek Orr
232 is sum of the cubes of the factorials of its digits,
232 = (2!)3 + (3!)3 + (2!)3

and 232 the sum of the first 11 Fibonacci numbers. 232 = 1+1+2+3+5+8+13+21+34+55+89

If you add up all the proper divisors of a number, n, they can be less than n, as 231 is(deficient), equal to n (perfect, like 6 or 28) or abundant. 12 is the smallest abundant number. Nicomachus wrote only of even numbers because he thought all odd numbers were deficient, but he was wrong. The 232nd abundant number is odd, 945.

If you raise 232 to the power of the product of its digits, and then add the sum of its digits, you get a prime. The only other known number with this property is 187. *Prime Curios

232 /4 = 58, so 59^2 - 57^2 = 232. 232 / 8 = 29 so 31^2-27^2 = 232.

Because 58 is a sum of two squares, 232 is also. 58 =7^2 + 3^2, 232 = 14^2 + 6^2. Students might use this to find the sum of two squares for 522 = 9 x 58.

232 is another balanced binary, with four of each ones and zeros, and always the number of ones is greater than, or equal to, the number of zeros.

232 is one less than a Fibonacci Prime.

EVENTS

1601 Pierre Fermat, the child of Claire de Long is baptized in the town of Beaumont de Lomange near Toulouse. almost thirty years later he would marry his mother's distant cousin. *André Weil, Number Theory: An Approach Through History from Hammurabi to Legendre

1638 William Oughtred writes to instrument maker Elias Allen with instructions for the first physical pair of slide rules using his method. Oughtred had invented the rules as early as 1620, and definitely written about them by 1633, but as he says in the opening of his letter, he had never made one before: “I have here sent you directions (as you requested me being at Twickenham) about the making of the two rulers”. He would continue, “would gladly see one of [the two parts of the instrument] when it is finished: wch yet I never have done”. The slide rule that Allen created seems no longer to exist, but a reverse image printed from the rule, perhaps to show to Oughtred still remains.
*Boris Jardine, Cambridge University Library Special Collections

Oughtred's method used two rules with logarithmic scales that were positioned so that one slid against the other. Within two decades, the first slide rule with the scales bound together was created by Robert Bissaker. This is the oldest slide rule still existing today and is in the Science Museum, London.

Oughtred is also known for introducing the X for multiplication, and the double colon, ::, for ratio. His use of a vertical bar to separate decimals from whole numbers was not well accepted.

1699 Newton's name introduced outside the Cambridge area. In 1669 Barrow had brought Newton a copy of Nicholas Mercator's Logarithmotechnia, which included the infinite series for ln(1+x). Newton recognized this as a simple example of his more general work on infinite series during his annus mirabilis in Woolsthorpe. Newton began to share some of his work with Barrow, who talked him into allowing him to send some of it, anonymously, to John Collins, which he did. When Collins highly favorable responses were received, Newton allowed Barrow to identify him to Collins. Barrow's letter to Collins on this date was the first time Newton became known to the mathematical community outside Cambridge. *James Gleick, Isaac Newton

1809 Gauss writes to G. F. Parrot dean of the Philosophical faculty of Tartu University to explain why he could not accept the position of Professor of Mathematics and Astronomy. He chose instead to remain in Gottingen. *PB Notes

1858 The Darwin-Wallace paper was read before the Linnean Society July 1st, 1858 and published in their Proceedings Vol 3 1858. pp 45-62. on this day.
"On the Tendency of Species to form Varieties; and on the Perpetuation of Varieties and Species by Natural Means of Selection" ; By Charles Darwin, Esq., F.R.S., F.L.S.., & F.G.S., and Alfred Wallace​, Esq. Communicated by Sir Charles Lyell, F.R.S., F.L.S., and J.D. Hooker, Esq., M.D., V.P.R.S., F.L.S., &c. *Linnean Society

1910 Florence Nightingale was buried on 20 August in the family plot at East Wellow, Hampshire. An offer of burial in Westminster Abbey was refused by her relatives. She died one week earlier. *Victorian Web Org

1955 Observances were held on the Island of Samos commemorating the 2500th anniversary of the founding of the ﬁrst school of philosophy by Pythagoras. Four postage stamps were issued by Greece. Naturally one of them illustrated the celebrated 47th proposition of Euclid, the Pythagorean Theorem, by a 3–4–5 triangle with squares erected on its sides.

1960 Two mongrel dogs, Belka(Little Squirrel) and Strelka(Little Arrow) became the first living creatures to perform a space flight and return safely to Earth. Korabl-Sputnik-2 (Spaceship Satellite-2), also known as Sputnik 5, was launched on August 19, 1960. Also on board were 40 mice, 2 rats and a variety of plants.
After a day in orbit, the spacecraft's retrorocket was fired and the landing capsule and the dogs were safely recovered. They were the first living animals to survive orbital flight. *Space Today Online

BIRTHS

1710 Thomas Simpson born (20 August 1710 – 14 May 1761). Best known to elementary calculus students for Simpson's rule, a method to approximate definite integrals. (this rule had been found 100 years earlier by Johannes Kepler, and in German is the so-called Keplersche Fassregel, "Kepler's barrel rule") The method actually uses a method of fitting parabolas to the function (a word not in use when Simpson lived) but is exact for polynomials up to a cubic. Apparently, the method that became known as Simpson's rule was well known and used earlier by Bonaventura Cavalieri (a student of Galileo) in 1639, later rediscovered by James Gregory (who Simpson succeeded as Regius Professor of Mathematics at the University of St Andrews) and was only attributed to Simpson because of the popularity of his math books in which it was included. *Wik

1779  Jöns Jakob Berzelius, (20 August 1779 – 7 August 1848) a Swedish chemist, was born Aug. 20, 1779.  Berzelius was part of the second generation of chemists who accepted the reform in chemical nomenclature proposed by Antoine Lavoisier and his confrères and set out to improve the nomenclature and flesh it out with newly discovered elements.  Berzelius did his share in the discovery department, being the first to isolate silicon (which he called silicium), selenium, and thorium, but he is best known for suggesting a new system of chemical symbols for the elements. John Dalton, the father of chemical atomism, had suggested using geometric symbols to represent each element: oxygen was a white circle, carbon a black circle, hydrogen a white circle with a dot in the center, etc.  A graphic display of Dalton's chemical symbols can be seen as the first image on our Scientist of the Day entry for Dalton. Berzelius proposed instead a symbolism based on the first, or the first and second, letters of the Latin names of the elements; in his system, oxygen was O, hydrogen was H, and carbon was C, while calcium was Ca, copper was Cu, and cobalt was Co. In the table above (second image), we see 31 of the 46 known elements and their symbols, along with their specific weights.

Berzelius proposed to represent compounds by combining these symbols; so water was 2H + O, while iron oxide (the red kind) was Fe + 3 0, as we see on another page, showing the remaining 15 chemical symbols and some proposed ways of representing compounds.  We have made some changes since Berzelius, but basically it is his system that we still use. When you look at a chart of the periodic table, the language you are reading is the nomenclature Berzelius initiated back in 1814.

 *Linda Hall Org

1862 Paul Gustav Samuel Stäckel (20 August 1862, Berlin — 12 December 1919, Heidelberg) was a German mathematician, active in the areas of differential geometry, number theory, and non-Euclidean geometry. In the area of prime number theory, he used the term twin prime (Primzahlzwillinge i.e. "prime number twins")for the first time. *Wik

1863 Corrado Segre (20 August 1863, 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry.
Segre spent his entire career at the University of Turin, first as a student of Enrico D'Ovidio. In 1883 he published a dissertation on quadrics in projective space and was named as assistant to professors in algebra and analytic geometry. In 1885 he also assisted in descriptive geometry. He began to instruct in projective geometry, as stand-in for Giuseppe Bruno, from 1885 to 88. Then for 36 years he had the chair in higher geometry following D'Ovidio. Segre and Giuseppe Peano made Turin known in geometry.*Wik

1898 Leopold Infeld born (20 August 1898, Kraków – 15 January 1968, Warsaw), He was a Polish theoretical physicist. In 1948 he published Whom the Gods Love, a biographical novel about Evariste Galois. *VFR
He was awarded a doctorate at the Jagiellonian University (1921), worked as an assistant and a docent at the University of Lwów (1930–1933) and then as a professor at the University of Toronto between 1939 and 1950. In 1939 he married Helen Schlauch, an American mathematician and a graduate of Cornell.
He worked together with Albert Einstein at Princeton University (1936–1938). The two scientists co-formulated the equation describing star movements as well as co-wrote a popular science book The Evolution of Physics.
Infeld was one of the 11 signatories to the Russell–Einstein Manifesto in 1955, and is the only signatory never to receive a Nobel Prize. *Wik

1899 Salomon Bochner (20 Aug 1899; 2 May 1982) Galician-born American mathematician and educator responsible for the development of the Bochner theorem of positive-definite functions and the Bochner integral.*TIS

1905 Anna Adelaide Stafford Henriques (August 20, 1905 – November 28, 2004) was an American mathematician known for her pioneering role as a female researcher at the Institute for Advanced Study.

Anna was born on August 20, 1905, in Chicago, the first of five children in her family. Her father was a factory manager, and both her parents were children of immigrants. The family moved from Chicago to Wisconsin and Minnesota; her parents died in 1919 and the children moved again to a relative's home in Missouri.

Stafford graduated from high school in 1922 and, with a scholarship from the American Association of University Women, did her undergraduate studies at the Western College for Women, graduating in 1926 with a double major in Greek and mathematics and a minor in French. She became a school mathematics and science teacher in New York, Pennsylvania, and New Jersey, while attending summer classes at the University of Chicago.

Stafford completed a master's degree at Chicago in 1931, with a thesis on An Application of the Dihedral Group. She became interested in topology after seeing a talk by Raymond Louis Wilder, and became a student of Mayme Logsdon at Chicago, completing her doctorate in 1933 with a dissertation on Knotted Varieties.

In preparation for her postdoctoral studies, Stafford had applied to Princeton University to work with James Waddell Alexander II and Oswald Veblen, but was rejected because she was female. She then wrote directly to Veblen, who had been newly appointed to the Institute for Advanced Study (also in Princeton, New Jersey, but separate from the university), and after talking to him when he visited Chicago, she was accepted there.

Stafford became one of two women, with Mabel Schmeiser, in the first group of postdoctoral researchers to visit the Institute. She worked at the Institute from 1933 to 1935, and in order to support herself she also held a teaching position at a school in Princeton. She worked mornings at the school, freeing her afternoons to attend seminars at the Institute.

After her time at the Institute, Stafford decided to aim for a career teaching mathematics rather than one as a researcher. She became an instructor at the University of Nebraska, and then in 1937 moved to the University of Utah. At Utah, her students included Tom M. Apostol, who remembered her as his "best mathematics teacher". She also served there as president of the Utah branch of the National Council of Teachers of Mathematics.

In 1942, in Salt Lake City, she married (as his second wife) Douglas Emmanuel Henriques, an administrative judge. Although Stafford and her husband had no children together, they raised Henriques' teenage son and fostered two Navajo girls.

In 1956, a change of position for Stafford's husband caused their family to move to New Mexico. She gave up what was then an associate professorship in Utah and became a lecturer at St. Michael's College in Santa Fe, New Mexico, and soon after also at the University of New Mexico. She became a full professor at St. Michael's in 1962, and gave up her second position at the University of New Mexico. She eventually became department chair at St. Michael's (later the Santa Fe University of Art and Design), and retired as chair emeritus in 1971.

Stafford and her husband lived in their retirement in Falls Church, Virginia. Her husband died in 1987, and she died on November 28, 2004, in Bailey's Crossroads, Virginia.

1923 Tom Mike Apostol ( August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks.

Apostol was born in Helper, Utah. His parents, Emmanouil Apostolopoulos and Efrosini Papathanasopoulos, were Greek immigrants. Apostolopoulos's name was shortened to Mike Apostol when he obtained his United States citizenship, and Tom Apostol inherited this Americanized surname.

Apostol received his Bachelor of Science in chemical engineering in 1944 at the University of Utah. He was particularly enthused by one of his mathematics teachers Anna Stafford Henriques. He said in the interview :-
My best mathematics teacher was Anna Henriques, who taught me college algebra and analytic geometry. She's in her nineties now and lives in a retirement complex in Virginia. I telephoned her recently, and she remembers me very well.
By one of life's strange coincidences Tom and Anna share this date as a birthdate. A brief bio of her is also on this page.
Master's degree in mathematics from the University of Washington in 1946, and a PhD in mathematics from the University of California, Berkeley in 1948. Thereafter Apostol was a faculty member at UC Berkeley, MIT, and Caltech. He was the author of several influential graduate and undergraduate level textbooks.

Apostol was the creator and project director for Project MATHEMATICS! producing videos which explore basic topics in high school mathematics. He helped popularize the visual calculus devised by Mamikon Mnatsakanian with whom he also wrote a number of papers, many of which appeared in the American Mathematical Monthly. Apostol also provided academic content for an acclaimed video lecture series on introductory physics, The Mechanical Universe.

In 2001, Apostol was elected in the Academy of Athens. He received a Lester R. Ford Award in 2005, in 2008, and in 2010. In 2012 he became a fellow of the American Mathematical Society.

 Simply a beautiful book for math lovers

1925
Elza Furtado Gomide (August 20, 1925 – October 26, 2013) was a Brazilian mathematician and the first woman to receive a doctorate in mathematics from the University of São Paulo, in 1950, and the second in Brazil. Gomide was involved in the creation of the Society of Mathematics of São Paulo and was elected head of the department of mathematics of the University of São Paulo in 1968. (The first Brazilian woman to receive a doctorate in mathematics and the first Brazilian woman to join the Brazilian Academy of Sciences was *Wik

1957 Sir Simon Kirwan Donaldson FRS (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. *WikHe
received a Fields Medal for his work on 4 dimensional manifolds. He was knighted in 2012 and awarded the most prestigious mathematical prizes including the Shaw Prize, the Breakthrough Prize in Mathematics, the Oswald Veblen Prize, and the Wolf Prize in Mathematics. *Wik

1957 Sir Simon Kirwan Donaldson FRS (born 20 August 1957 Cambridge, England - ) In 1986 he received a Fields Medal for his work on the topology of four-manifolds. *VFR Remarkably, Donaldson has solved problems of mathematics by using ideas from physics. From the Yang-Mills generalizations of James Clerk Maxwell's electromagnetic equations, Donaldson used special solutions to these equations, called instantons, to look at general four-manifolds. After being awarded the Fields Medal, Donaldson continued his exploitation of ideas from physics with applications to mathematics. *TIS

DEATHS

1622 Baha ad-din Muhammad ibn Husayn al-Amili (20 Mar 1546; 20 Aug 1622 at age 76) A Syrian-Iranian theologian, mathematician and astronomer, a.k.a. Shaykh Baha'i). He became a very learned Muslim whose genius touched every field of knowledge from mathematics and philosophy to architecture and landscape design. He revived the study of mathematics in Iran. His treatise on the subject, Khulasat al-hisab (“The Essentials of Arithmetic”), and translations from the original Arabic was in use as a textbook until the end of the 19th century. His treatise in astronomy, Tashrihu'l-aflak ("Anatomy of the Heavens") summarised the works of earlier masters. He was born within a year of William Gilbert in England and Tycho Brahe in Denmark, and was still a child when his family left Syria to escape religious persecution.*TIS

1672 Jan de Witt (September 24, 1625, Dordrecht - August 20, 1672) murdered by a mob from the (William of) Orange faction. For the previous twenty years he served as grand pensionary in Holland, essentially the prime minister of the Netherlands. Consequently this talented mathematician had little time to devote to mathematics. He wrote the ﬁrst systematic account of the analytic geometry of the straight line and conics. It was published in Van Schooten’s second Latin edition of Descartes’ Geometrie *VFR de Witt and his brother were both killed by a mob which was probably supported by William III of Orange. At the very least, as the Wikipedia articles states, "he protected and rewarded the killers." After a previous attempt on his life, he was lured by a forged letter to the cell where his brother was held, and both were hanged and then their bodies were mutilated. The story of their deaths are a critical element in the plot of Alexander Dumas' "The Black Tulip". *Wik

1677 Pierre Petit (8 Dec 1594 in Montluçon, France - 20 Aug 1677 in Lagny-sur-Marne, France) was a French scientist who had a strong influence on the French government. He was one of Mersenne's collaborators. Petit was an influential figure with important government positions which enabled him to try to influence national science policy. A firm believer, as were the other members of Mersenne's group, of the experimental method rather than the philosophical approach advocated by Descartes, Petit argued strongly for better astronomical facilities in France. He wanted the King to establish a Royal Observatory to allow France to again take a leading role in astronomy. Petit argued that France had fallen behind some other European countries and was relying on observations made in other countries. Petit himself had a fine collection of astronomical instruments and several of these were of his own invention. In particular, late in his life, Petit devised a filar micrometer to measure the diameters of celestial objects such as the Sun, Moon and planets.*SAU

1791 Jacques Charles, (probably 1752, August 20, 1791) Mathematician, born in Cluny, France. He is often confused with the Jacques A. C. Charles who is credited (or mis-credited) with Charles' Law and much of the work of this Jacques Charles. During the Late 18th Century both were active in Paris scientific circles and both were members of the Paris Scientific Academy. They were often distinguished by calling this one Charles the Geometer, and the other Charles the Balloonist since JAC Charles was active in promoting the use of hydrogen balloons and had designed the first balloon that is known to have been used.
This Jacques Charles is also frequently referred to by the historians who are aware of the confusion between them as Charles the Obscure.
Jacques Charles first contact with the Paris Acad of Sci was in a 1770 letter in which he submitted an article on a problem in Algebra at about the age of 18. It was turned down by the academy due to it's elementary level. The address shows that he was living in Cluny at the time. But two years later a second correspondence to the academy is read to the Academy, and Lavosier's minutes list his position as a professor of Mathematics as the school at Nanterre, on the outskirts of Paris. It is suspected that this was a preparatory school for young nobles who were training to become engineers that had been located there since the 1760's.
Between 1779 and 1785 Jacques Charles submitted seven articles to the Paris Academy, all of which were deemed worthy of publication, but only the last seemed to merit his admittance to this esteemed group. Condorcet, who was then perpetual secretary of the Academy said that this, as well as his prior papers certainly warranted his admission. It seems that Laplace, who had a conflict with Charles' mentor/sponsor, Bossut, and had been blocking his entry. With some behind the scenes effort by Lavosier had created a new geometry section, he was voted into the Academy on May 11 (often given as May 12).
By 1792 due to the confusion of their names, much of the mathematical work of Charles the Geometer would be credited to Charles the Balloonist and the "Geometer" would become the "obscure". Even the energetic J. C. Poggendorf would miscredit eight papers by the geometer to the other, and in biographies of J. A. C. Charles written even in the 20th century, you will see him credited as a "mathematician" and statements that suggest that "most of his writings were in mathematics." J. B. Gough, writing in an article in Isis in 1979 describes the ballooning Charles as, "nearly a mathematical illiterate."
The confusion between the two men of common names was exacerbated by the timing of this Charles' death. The year 1791 and the problems related to the Revolution made this the Academy of Sciences did not publish a Memoires, and as a result, no eloge's for the members who died in that year. Strangely, this was still four years before the better remembered Charles was admitted to the Academy.
He was buried (according to an old note to Cvomptes Rendes) at St. Germain l'auxerrois, but this seems hard to confirm in the church records. (*J. B. Gough)
Charles was also the Royal Professor of Hydrodynamics, and as such was also inducted into The Academy of Architecture. *Roger Hahn, More Light on Charles the Obscure, Isis, Vol. 72, No. 1 (Mar., 1981), pp. 83-86

1923 Vilfredo Pareto (15 Jul 1848, 20 Aug 1923) Italian economist and sociologist, known for his application of mathematics to economic analysis and for his theory of the 'circulation of elites'. His initial five-year course in civil engineering, graduating in 1870, gave him a grounding in mathematics. While working as an engineer, he studied philosophy and politics and wrote many periodical articles in which he was one of the first to analyse economic problems with mathematical tools. Pareto's first work, Cours d'economie politique (1896-97), included his famous 'law' of income distribution, a complicated mathematical formulation attempting to prove the distribution of incomes and wealth in society is not random and that a consistent pattern appears throughout history, in all parts of the world and in all societies. *TIS Pareto's Law was not created by him, but named in honor of him. The Pareto principle (also known as the 80-20 rule, the law of the vital few, and the principle of factor sparsity) states that, for many events, roughly 80% of the effects come from 20% of the causes.
(From my personal experience using statistics in Quality Engineering, this one rule of thumb is the single most important idea of all the things you learn.)
Business-management consultant Joseph M. Juran suggested the principle and named it after Pareto, who had observed in 1906 that 80% of the land in Italy was owned by 20% of the population; he developed the principle by observing that 20% of the pea pods in his garden contained 80% of the peas. *Wik

1930 Herbert Hall Turner (13 Aug 1861, 20 Aug 1930) English astronomer who pioneered many of the procedures now universally employed in determining stellar positions from astronomical photographs. After serving as chief assistant at the Royal Greenwich Observatory for nine years, he spent most of his career as Savilian professor of astronomy at Oxford University. One of the leaders in the worldwide effort to produce an astrographic chart of the sky, he developed improved methods for obtaining both positions and magnitudes from photographic plates. Most of his later work was in seismology; he compiled and published worldwide earthquake data starting in 1918, and he discovered the existence of deep-focus earthquakes in 1922. *TIS

1972 Carol (Vander Velde)Karp (10 August 1926, Forest Grove, Ottawa County, Michigan – 20 August 1972, Maryland) died of breast cancer. At the time she was at the height of her career in logic. She received her Ph.D. in 1959 from Southern California under the direction of Leon Henkin. She created the ﬁeld of Inﬁnitary Logics which studies logics such as Lω,ω which allowed for the conjunction and disjunction of countably many formulas. This work has become very important in modern logic. *VFR

2001 Sir Fred Hoyle (24 June 1915, 20 Aug 2001) English mathematician and astronomer, best known as the foremost proponent and defender of the steady-state theory of the universe. This theory holds both that the universe is expanding and that matter is being continuously created to keep the mean density of matter in space constant. He became Britain's best-known astronomer in 1950 with his broadcast lectures on The Nature of the Universe, and he recalled coining the term "Big Bang" in the last of those talks. Although over time, belief in a "steady state" universe as Hoyle had proposed was shared by fewer and fewer scientists because of new discoveries, Hoyle never accepted the now most popular "Big Bang" theory for the origin of the universe. *TIS
When  Willy Fowler, who was 72 years old at the time, was told he would share the Nobel  prize with Indian astrophysicist Subrahmanyan Chandrasekhar, who had carried out pioneering work on the structure of stars he was shocked.  Of Fowler's own close collaborator, Fred Hoyle – the British scientist who had led their joint research work – there was no mention.
Scientists' dismay at the refusal to give Hoyle a Nobel prize is understandable, although it should be noted that he could be cantankerous and opinionated and had offended a large number of influential colleagues unused to his Yorkshire bluntness. He had called some of them liars and cheats in public, while his beliefs, in later life, verged on the lunatic. He said Earth was being constantly bombarded by microbes from outer space and that these were responsible for outbreaks of flu and other illnesses. He also claimed that remains of archaeopteryx – the British Museum fossil that demonstrates the early link between dinosaurs and birds – was a fake. Such notions went down badly in scientific circles.

2006 Professor William (Bill) Parry FRS (3 July 1934–20 August 2006) was an English mathematician. During his research career, he was highly active in the study of dynamical systems, and, in particular, ergodic theory, and made significant contributions to these fields. He is considered to have been at the forefront of the introduction of ergodic theory to the United Kingdom. He played a founding role in the study of subshifts of finite type, and his work on nilflows was highly regarded.*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