## Sunday, 24 January 2016

### On This Day in Math - January 24

* The art of photography

A mathematician's reputation rests on the number of bad proofs he has given.
~Abram S Besicovitch

The 24th day of the year; 24! (6.2044840173323943936 1023)is almost equal to Avogadro's Number, (6.022141×10^23).
Also 12 + 22 +...+ 242 = 702 the only pyramidal number that is also a square.

There are five regular polyhedra in three space, the Platonic solids. There are six regular 4-space polyhedra. Five of the 4-space polyhedra are analogs of the Platonic solids in 3-space, but there is also a 24-cell, with 24 octahedral faces w/o analog to the Platonic solids. Beyond 4-space, the number of regular polyhedra is always three.

The 24th dimension is the highest dimension for which the exact "kissing number", the number of spheres that can be placed around a central sphere so that they all are touching it, is known. For the 24th dimension, the "kissing number is 196,560. Beyond the fourth dimension, only the eighth and twenty-fourth are known exactly.

EVENTS
1544, a solar eclipse was viewed at Louvain, which was later depicted in the first published book illustration of the camera obscura in use. Dutch mathematician and astronomer Reinerus Gemma-Frisius viewed a solar eclipse using a hole in one wall of a pavillion to project the sun's image upside down onto the opposite wall. He published the first illustrationof a camera obscura, depicting his method of observation of the eclipse in De Radio Astronomica et Geometrica (1545). Several astronomers made use of such a device in the early part of the 16th century. Both Johannes Kepler and Christopher Scheiner used a camera obscura to study the activity of sunspots. The technique was known to Aristotle (Problems, ca 330 BC). *TIS

1672 (14 Jan 1671 OS) In a letter from John Wallis to Oldenberg, he discusses his joint solution with Huygens of the area between the cissoid and it's asymptote, "Let mee know whether what I have inserted from Mr. Hugens ... be to his content." The cissoid (called Cissoid of Diocles) area was found to be 3πa2 , where a is the radius of the circle. *Wallis Corr. (cissoid means "ivy-shaped)

1801 Joseph Piazzi sends letters to Bode in Berlin, Oriani in Milan, and Lelande in Paris regarding a newly sighted "comet without tail and envelope".  To Oriani he admits that :

I have announced this star as a comet, but since it is not accompanied by any nebulosity and, further, since its movement is so slow and rather uniform, it has occurred to me several times that it might be something better than a comet. But I have been careful not to advance this supposition to the public.
*Wik

1870 Spectroscopy was a source of much excitement and interest in science in the last half of the 19th Century, but it created interest in the public sector as well. One example of the degree of the interest was the publication of a lengthy explanation of the topic in a seemingly unusual place, The Baptist Quarterly. The 20+ page article "claimed that the history, processes, achievements and possibilities of spectroscopy constitute "one of the marvels of the nineteenth century, and entitle it to the consideration of every thoughtful mind." *American History, Smithsonian

1889 Charles Darwin elected a Fellow of the Royal Society. *Friends of Darwin ‏@friendsofdarwin

1918 – The Gregorian calendar is introduced in Russia by decree of the Council of People's Commissars effective February 1 *Wik

1895 Karl Pearson first uses the term "Binomial Distribution" in Contributions to the Mathematical Theory of Evolution---II. Skew Variation in Homogeneous Material, Received by the Royal Society of London December 19, 1894, It would be read on January 24, 1895,” Philosophical Transactions of the Royal Society of London for the year MDCCCXCV. A footnote has: “This result seems of considerable importance, and I do not believe it has yet been noticed. It gives the mean square error for any binomial distribution, and we see that for most practical purposes it is identical with the value $\sqrt{npq}$ , hitherto deduced as an approximate result, by assuming the binomial to be approximately a normal curve.” Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics

1925, a motion picture of a solar eclipse was taken by the U.S. Navy from the dirigible Los Angeles. The craft was at an elevation of about 4,500-ft and positioned about 19 miles east of Montauk Point, Long Island, NY. This give a view of a total eclipse of the sun that lasted just over 2-min. Four astronomical cameras and a spectrograph were used as well as two moving picture cameras. This was the first time in the U.S. that a dirigible had been used as a platform for observation of a total eclipse of the sun. The first U.S. attempt to photograph one from an aircraft 10 Sep 1923 was unsuccessful due to cloudy conditions, but on 28 Apr 1930, a flight over California sponsored by the U.S. Naval Observatory recorded a total solar eclipse. *TIS

1948 IBM dedicates the Selective Sequence Electronic Calculator​ (SSEC). Later the SSEC was put on public display near the company's Manhattan headquarters so passers-by could watch its operational speed. Before its decommissioning in 1952, the SSEC produced the moon-position tables used for plotting the course of the 1969 Apollo flight to the moon. *CHM

1986Voyager 2 passes within 81,500 kilometres (50,600 mi) of Uranus. *Wik

BIRTHS
1679 Christian Freiherr von Wolff (24 Jan 1679; 9 Apr 1754) (baron) German philosopher, mathematician, and scientist who worked in many subjects but who is best known as the German spokesman of the Enlightenment, the 18th-century philosophical movement characterized by Rationalism. Wolff's first interest was mathematics. Though he made no original contribution to the discipline, he was important in the teaching of mathematics and instrumental in introducing the new mathematics into German universities. Later, as a philosopher, he developed the most impressive coherent system of his century. Thoroughly eclectic, influenced by Leibniz and Descartes, yet he continued fundamental themes of Aristotle. His system was important in making the discoveries of modern science known in Germany.
*TIS

1798 Karl Georg Christian von Staudt (January 24, 1798 – June 1, 1867) was a German mathematician born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss who was director of the observatory. Staudt provided an ephemeris for the orbits of Mars and the asteroid Pallas. When in 1821 Comet Nicollet-Pons was observed, he provided the elements of its orbit. These accomplishments in astronomy earned him his doctorate from University of Erlangen in 1822.
The book Geometrie der Lage (1847) was a landmark in projective geometry. As Burau (1976) wrote, "Staudt was the first to adopt a fully rigorous approach. Without exception his predecessors still spoke of distances, perpendiculars, angles and other entities that play no role in projective geometry."
Furthermore, this book uses the complete quadrangle to "construct the fourth harmonic associated with three points on a straight line", the projective harmonic conjugate. *Wik (TIS gives birthdate as Jan 23)

1863 August Adler (24 Jan 1863 in Opava, Austrian Silesia (now Czech Republic)-17 Oct 1923 in Vienna, Austria) In 1906 Adler applied the theory of inversion to solve Mascheroni construction problems in his book Theorie der geometrischen Konstruktionen published in Leipzig. In 1797 Mascheroni had shown that all plane construction problems which could be made with ruler and compass could in fact be made with compasses alone. His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass.
Since he was using inversion Adler now had a symmetry between lines and circles which in some sense showed why the constructions needed only compasses. However Adler did not simplify Mascheroni proof. On the contrary, his new methods were not as elegant, either in simplicity or length, as the original proof by Mascheroni.
This 1906 publication was not the first by Adler studying this problem. He had published a paper on the theory of Mascheroni's constructions in 1890, another on the theory of geometrical constructions in 1895, and one on the theory of drawing instruments in 1902. As well as his interest in descriptive geometry, Adler was also interested in mathematical education, particularly in teaching mathematics in secondary schools. His publications on this topic began around 1901 and by the end of his career he was publishing more on mathematical education than on geometry. Most of his papers on mathematical education were directed towards teaching geometry in schools, but in 1907 he wrote on modern methods in mathematical instruction in Austrian middle schools. He produced various teaching materials for teaching geometry in the sixth-form in Austrian schools such as an exercise book which he published in 1908. *SAU

1872 Morris William Travers (24 Jan 1872; 25 Aug 1961)
English chemist who, while working with Sir Willam Ramsay in London, discovered the element krypton (30 May 1898). The name derives from the Greek word for "hidden." It was a fraction separated from liquified air, which when placed in a Plücker tube connected to an induction coil yielded a spectrum with a bright yellow line with a greener tint than the known helium line and a brilliant green line that corresponded to nothing seen before.*TIS

1882 Harold Delos Babcock (24 Jan 1882(Edgerton,Wisconsin) - 8 Apr 1968) American astronomer who with his son, Horace, invented the solar magnetograph (1951), for detailed observation of the Sun's magnetic field. With their magnetograph the Babcocks measured the distribution of magnetic fields over the solar surface to unprecedented precision and discovered magnetically variable stars. In 1959 Harold Babcock announced that the Sun reverses its magnetic polarity periodically. Babcock's precise laboratory studies of atomic spectra allowed others to identify the first "forbidden" lines in the laboratory and to discover the rare isotopes of oxygen. With C.E. St. John he greatly improved the precision of the wavelengths of some 22,000 lines in the solar spectrum, referring them to newly-determined standards.*TIS

1891 Abram Samoilovitch Besicovitch (24 Jan 1891 in Berdyansk, Russia -2 Nov 1970 in Cambridge, Cambridgeshire, England) Besicovitch left Petrograd for Copenhagen in 1924 and there worked with Harald Bohr. He had been awarded a Rockefeller Fellowship but his applications for permission to work abroad had been refused. He escaped across the border with a colleague J D Tamarkin under the cover of darkness. He managed to reach Copenhagen where he was supported financially for a year with the Rockefeller Fellowship. His interest in almost periodic functions came about through this year spent working with Harald Bohr. After he visited Oxford in 1925 Hardy, who quickly saw the mathematical genius in Besicovitch, found a post for him in Liverpool. At Cambridge Besicovitch lectured on analysis in most years but he also gave an advanced course on a topic which was directly connected with his research interests such as almost periodic functions, Hausdorff measure, or the geometry of plane sets. Besicovitch was famous for his work on almost periodic functions, his interest in which, as we mentioned above, came from his time in Copenhagen with Harald Bohr. In 1932 he wrote an influential text Almost periodic functions covering his work in this area.
One of the achievements, with which he will always be associated, was his solution of the Kakeya problem on minimising areas. The problem had been posed in 1917 by a Japanese mathematician S Kakeya and asked what was the smallest area in which a line segment of unit length could be rotated through 2p. Besicovitch proved in 1925 that given any e, an area of less than e could be found in which the rotation was possible. The figures that resulted from Besicovitch's construction were highly complicated, unbounded figures.
Other areas on which Besicovitch worked included geometric measure theory, Hausdorff measure, real function theory, and complex function theory. In addition to this work on deep mathematical theories, Besicovitch loved problems, particularly those which could be stated in elementary terms but which proved resistant to attack. Often he showed that the "obvious solution" to certain problems is false. An example of such a problem is the Lion and the Man problem posed by Richard Rado in the mid 1920s. *SAU
On his 36th birthday, feeling that his most fertile years were behind him, mathematician Abram Besicovitch said, “I have had four-fifths of my life.”
At age 59 he was elected to the Rouse Ball Chair of Mathematics at Cambridge.
When J.C. Burkill reminded him of his earlier remark, he said, “Numerator was correct.” *Greg Ross, Futility Closet

1902 Oskar Morgenstern (24 Jan 1902; 26 Jul 1977) German-American economist and mathematician who popularized "game theory" which mathematically analyzes behaviour of man or animals in terms of strategies to maximize gains and minimize losses. He coauthored Theory of Games and Economic Behavior (1944), with John von Neumann, which extended Neumann's 1928 theory of games of strategy to competitive business situations. They suggested that often in a business situation ("game'), the outcome depends on several parties ("players"), each estimating what all of the others will do before determining their own strategy. Morgenstern was a professor at Vienna University, Austria, from 1931 until the Nazi occupation in 1938), when he fled to America and joined the faculty at Princeton University. His later publications included works on economic prediction and aspects of U.S. defence.*TIS

1912 Nils Aall Barricelli (January 24, 1912; Rome–January 27, 1993) was a Norwegian-Italian mathematician.
Barricelli's early computer-assisted experiments in symbiogenesis and evolution are considered pioneering in artificial life research. Barricelli, who was independently wealthy, held an unpaid residency at the Institute for Advanced Study in Princeton, New Jersey in 1953, 1954, and 1956. He later worked at the University of California, Los Angeles, at Vanderbilt University (until 1964), in the Department of Genetics of the University of Washington, Seattle (until 1968) and then at the Mathematics Institute of the University of Oslo. Barricelli published in a variety of fields including virus genetics, DNA, theoretical biology, space flight, theoretical physics and mathematical language. *Wik

1914 Vladimir Petrovich Potapov (24 Jan 1914 in Odessa, Ukraine - 21 Dec 1980 in Kharkov, Russia) In 1948 Potapov was invited to the Pedagogical Institute at Odessa. He soon became Head of Mathematics and, from 1952, Dean of the Faculty of Physics and Mathematics. He used his position to invite Livsic and others to the Institute.
During the 1950s Potapov worked on the theory of J-contractive matrix functions and the analysis of matrix functions became his main work. He published papers on the multiplicative theory of analytic matrix functions in the years 1950-55 which contain work from his doctoral thesis. He also worked on interpolation problems.
From 1974 Potapov lectured at Odessa Institute of National Economy, then he went to Kharkov to head the Department of Applied Mathematics at the Institute for low temperature physics. *SAU

1931 Lars V. Hörmander (24 Jan 1931 - ) Swedish mathematician who was awarded the Fields Medal in 1962 for his work on partial differential equations. Spending five years in writing, he produced a text The analysis of linear partial differential operators, in four volumes (1983-85). Between 1987 and 1990 he served as a vice president of the International Mathematical Union. In 1988 Hörmander was awarded the Wolf Prize. Hörmander's text, An Introduction to Complex Analysis in Several Variables, has become a classic dealing with the theory of functions of several complex variables. It developed from lecture notes of a course which he gave in Stanford in 1964 and published in book form two years later, with updates in 1973 and 1990.*TIS

1941 Dan Shechtman (January 24, 1941 in Tel Aviv - ) is the Philip Tobias Professor of Materials Science at the Technion – Israel Institute of Technology, an Associate of the US Department of Energy's Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S. National Bureau of Standards in Washington, D.C., Shechtman discovered the icosahedral phase, which opened the new field of quasiperiodic crystals. He was awarded the 2011 Nobel Prize in Chemistry for "the discovery of quasicrystals". *Wik

1947 Michio Kaku (January 24, 1947 - ) is an American theoretical physicist, the Henry Semat Professor of Theoretical Physics in the City College of New York of City University of New York, the co-founder of string field theory, and a "communicator" and "popularizer" of science. He has written several books about physics and related topics; he has made frequent appearances on radio, television, and film; and he writes extensive online blogs and articles.*Wik

DEATHS
1860 James Pollard Espy (9 May 1785, 24 Jan 1860) American meteorologist who was one of the first to collect meteorological observations by telegraph. He gave apparently the first essentially correct explanation of the thermodynamics of cloud formation and growth. Every great atmospheric disturbance begins with a rising mass of heated, thus less dense air. While rising, the air mass dilates and cools. Then, as water vapour precipitates as clouds, latent heat is liberated so the dilation and rising continues until the moisture of the air forming the upward current is practically exhausted. The heavier air flows in beneath, and, finding a diminished pressure above it, rushes upward with constantly increasing violence. Water vapor precipitated during this atmospheric disturbance results in heavy rains.*TIS

1865 Samuel Hunter Christie (22 March 1784 – 24 January 1865) was a British scientist and mathematician.
He studied mathematics at Trinity College, Cambridge, where he won the Smith's Prize and was second wrangler. It may help to understand the difficulty of this exam by looking at some of the great achievers who did NOT win Senior Wrangler. A short list of Second Wranglers, include Alfred Marshall, James Clerk Maxwell, J. J. Thomson, and Lord Kelvin.
Those who have finished between third and 12th include Karl Pearson and William Henry Bragg (third), George Green and G. H. Hardy (fourth), Adam Sedgwick (fifth), John Venn (sixth), Bertrand Russell and Nevil Maskelyne (seventh), Thomas Malthus (ninth), and John Maynard Keynes (12th).

Christie was particularly interested in magnetism, studying the earth's magnetic field and designing improvements to the magnetic compass. Some of his magnetic research was done in collaboration with Peter Barlow. He became a Fellow of the Royal Society in 1826, delivered their Bakerian Lecture in 1833 and served as their Secretary from 1837 to 1853. In 1833 he published his 'diamond' method, the forerunner of the Wheatstone bridge, in a paper on the magnetic and electrical properties of metals, as a method for comparing the resistances of wires of different thicknesses. However, the method went unrecognized until 1843, when Charles Wheatstone proposed it, in another paper for the Royal Society, for measuring resistance in electrical circuits. Although Wheatstone presented it as Christie's invention, it is his name, rather than Christie's, that is now associated with the device.
Christie taught mathematics at the Royal Military Academy, Woolwich, from 1838 until his retirement in 1854.[1] He died at Twickenham, on 24 January 1865. *Wik

1877 Johann Christian Poggendorff (29 December 1796 – 24 January 1877), was a German physicist and science historian born in Hamburg. By far the greater and more important part of his work related to electricity and magnetism. Poggendorff is known for his electrostatic motor which is analogous to Wilhelm Holtz's electrostatic machine. In 1841 he described the use of the potentiometer for measurement of electrical potentials without current draw.
Even at this early period he had conceived the idea of founding a physical and chemical scientific journal, and the realization of this plan was hastened by the sudden death of Ludwig Wilhelm Gilbert, the editor of Gilbert's Annalen der Physik, in 1824 Poggendorff immediately put himself in communication with the publisher, Barth of Leipzig. He became editor of Annalen der Physik und Chemie, which was to be a continuation of Gilbert's Annalen on a somewhat extended plan. Poggendorff was admirably qualified for the post, and edited the journal for 52 years, until 1876. In 1826, Poggendorff developed the mirror galvanometer, a device for detecting electric currents.
He had an extraordinary memory, well stored with scientific knowledge, both modern and historical, a cool and impartial judgment, and a strong preference for facts as against theory of the speculative kind. He was thus able to throw himself into the spirit of modern experimental science. He possessed in abundant measure the German virtue of orderliness in the arrangement of knowledge and in the conduct of business. Further he had an engaging geniality of manner and much tact in dealing with men. These qualities soon made Poggendorff's Annalen (abbreviation: Pogg. Ann.) the foremost scientific journal in Europe.
In the course of his fifty-two years editorship of the Annalen Poggendorff could not fail to acquire an unusual acquaintance with the labors of modern men of science. This knowledge, joined to what he had gathered by historical reading of equally unusual extent, he carefully digested and gave to the world in his Biographisch-literarisches Handworterbuch zur Geschichte der exacten Wissenschaften, containing notices of the lives and labors of mathematicians, astronomers, physicists, and chemists, of all peoples and all ages. This work contains an astounding collection of facts invaluable to the scientific biographer and historian. The first two volumes were published in 1863; after his death a third volume appeared in 1898, covering the period 1858-1883, and a fourth in 1904, coming down to the beginning of the 20th century.
His literary and scientific reputation speedily brought him honorable recognition. In 1830 he was made royal professor, in 1838 Hon. Ph.D. and extraordinary professor in the University of Berlin, and in 1839 member of the Berlin Academy of Sciences. In 1845, he was elected a foreign member of the Royal Swedish Academy of Sciences.
Many offers of ordinary professorships were made to him, but he declined them all, devoting himself to his duties as editor of the Annalen, and to the pursuit of his scientific researches. He died at Berlin on 24 January 1877.
The Poggendorff Illusion is an optical illusion that involves the brain's perception of the interaction between diagonal lines and horizontal and vertical edges. It is named after Poggendorff, who discovered it in the drawing of Johann Karl Friedrich Zöllner, in which he showed the Zöllner illusion in 1860. In the picture to the right, a straight black line is obscured by a dark gray rectangle. The black line appears disjointed, although it is in fact straight; the second picture illustrates this fact.*Wik

1914 Sir David Gill (12 Jun 1843, 24 Jan 1914) Scottish astronomer known for his measurements of solar and stellar parallax, showing the distances of the Sun and other stars from Earth, and for his early use of photography in mapping the heavens. His early training in timekeeping as a watchmaker led to astronomy and he designed, equipped, and operated a private observatory near Aberdeen. To determine parallaxes, he perfected the use of the heliometer, a telescope that uses a split image to measure the angular separation of celestial bodies. In 1877, Gill and his wife measured the solar parallax by observing Mars from Ascension Island. He was appointed Her Majesty's Astronomer at the Cape of Good Hope (1879-1906). Gill also made geodetic surveys of South Africa. In fact he carried out all of the observations to measure the distances to stars in terms of the standard meter. His precise redetermination of the solar parallax was used for almanacs until 1968. *TIS

1930 Adolf Kneser (19 March 1862 in Grüssow, Mecklenburg, Germany - 24 Jan 1930 in Breslau, Germany (now Wrocław, Poland)) He is remembered most for work mainly in two areas. One of these areas is that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general. He wrote an important text on integral equations. The second main area of his work was the calculus of variations. He published Lehrbruch der Variationsrechnung (Textbook of the calculus of variations) (1900) and he gave the topic many of the terms in common use today including 'extremal' for a resolution curve, 'field' for a family of extremals, 'transversal' and 'strong' and 'weak' extremals *SAU

1955 Percy John Heawood (8 September 1861 Newport, Shropshire, England - 24 January 1955 Durham, England) was a British mathematician. He devoted essentially his whole working life to the four color theorem and in 1890 he exposed a flaw in Alfred Kempe's proof, that had been considered as valid for 11 years. With the four color theorem being open again he established the five color theorem instead. The four color theorem itself was finally established by a computer-based proof in 1976. *Wik

1961 Albert Carlton Gilbert (15 Feb 1884, 24 Jan 1961) was an American inventor who patented the Erector set after he founded the A.C. Gilbert Co. New Haven, Connecticut (1908) to manufacture boxed magic sets. In 1913, he introduced Erector Sets. Similar construction toys then existed, such as Hornby's Meccano set made in England. Meccano sets included pulleys, gears, and several 1/2" wide strips of varying length with holes evenly spaced on them. Gilbert needed something unique for his Erector sets, so he created the square girder, made using several 1" wide strips with triangles cut in them. These had their edges bent over so 4 strips could be screwed together to form a very sturdy square girder. Over the next 40 years, some 30 million Erector Sets were sold.*TIS
Gilbert was also an accomplished athlete, he broke the world record for consecutive chin-ups in 1900 and distance record for running long dive in 1902. He invented the pole vault box and set two world records in the pole vault including a record for 12' 3" (3.66 meters) at the Spring meet of the Irish American Athletic Club, held at Celtic Park, Queens, New York, in 1906. He tied for gold with fellow American Edward Cook at the 1908 Summer Olympics in London for pole vaulting. *Wik

1982 Karol Borsuk (8 May 1905 in Warsaw, - 24 Jan 1982 in Warsaw) Borsuk introduced the important concept of absolute neighbourhood retracts in his doctoral dissertation, published in 1931, which was to lead to new and fruitful ideas in metric differential geometry. In 1936 he introduced the notion of cohomotopy groups, which could be said to mark the beginning of stable homotopy theory.   Shape theory grew up at the same time as infinite-dimensional topology and the interaction between the two fields was of great mutual benefit. He was important for the many deep questions which Borsuk posed which stimulated most of the top mathematicians working in the area. *SAU

1988 Moritz Werner Fenchel (3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for Nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. *Wik

2016 Marvin Minsky (August 9, 1927 - January 24, 2016 (aged 88)) Biochemist and the founder of the MIT Artificial Intelligence Project. Marvin Minsky has made many contributions to AI, cognitive psychology, mathematics, computational linguistics, robotics, and optics. He holds several patents, including those for the first neural-network simulator (SNARC, 1951), the first head-mounted graphical display, the first confocal scanning microscope, and the LOGO "turtle" device. His other inventions include mechanical hands and the "Muse" synthesizer for musical variations (with E. Fredkin). In recent years he has worked chiefly on imparting to machines the human capacity for commonsense reasoning. *TIS He died in Boston of a cerebral hemorrhage .

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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