Tuesday 5 May 2020

On This Day in Math - May 5

The man ignorant of mathematics 
will be increasingly limited in his grasp
of the main forces of civilization. 
~John Kemeny,

The 126th day of the year; nine points around a circle form the vertices of \( \binom{n}{k} = 126 \)  unique quadrilaterals.   That also means that if you draw all the diagonals of the nonagon, you would be using the same 126 sets of four vertices to get 126 intersections. 

In non-leap  years, there are 126 days in which the day of the month is prime.

The prime gap that covers the first century with no primes (from 1671800 to 1671899) has length 126 (from 1671781 to 1671907). 

There are 9 choose 5, or 126 ways for a random selection to pick the five spaces on a tic tac toe board for the "first player" in a random game.  36 of these configurations are a "win" for both players. They have both three x's and three O's in a line, since they don't have an order of play. Over 58% of those games are a win for the "first player". Geometrically, a student could think of each random game as a pentagon selected from nine points spaced around a circle.


840 A total solar eclipse was recorded over France. Known as Emperor Louis' Eclipse. NASA Eclipse map here. *David Dickinson ‏ @Astroguyz

1642 Théodore Deschamps, a physician from Bergerac, writes to Marin Mersenne that he remembered that in 1609, during his stay at Leiden University, he had not only witnessed a demonstration of a telescope by the mathematics professor, Rudolph Snellius, but had also met a Delft spectacle maker, who in his telescopes had covered up ‘the parts of the convex glass on which the rays coming from the object intersect each other too soon.’ (suggesting an early invention of a diaphragm that would allow a better image from poorer quality lenses) *Huib J. Zuidervaart, The ‘true inventor’ of the telescope. A survey of 400 years of debate,

1777 First use of i for imaginary constant: On May 5, 1777, Euler addressed to the 'Academiae' the paper "De Formulis Differentialibus Angularibus maxime irrationalibus quas tamen per logarithmos et arcus circulares integrare licet," which was published posthumously in his "Institutionum calculi integralis," second ed., vol. 4, pp. 183-194, Impensis Academiae Imperialis Scientiarum, Petropoli, 1794.
Quoniam mihi quidem alia adhuc via non patet istud praestandi nisi per imaginaria procedendo, formulam \( \sqrt{-1}\) ilittera i in posterum designabo, ita ut sit ii = -1 ideoque 1/i = -i.
According to Cajori, the next appearance of i in print is by Gauss in 1801 in the Disquisitiones Arithmeticae. Carl Boyer believes that Gauss' adoption of i made it the standard. By 1821, when Cauchy published Cours d'Analyse, the use of i was rather standard, and Cauchy defines i as "as if \( \sqrt{-1}\) was a real quantity whose square is equal to -1."
*Jeff Miller

1809 Mary Dixon Kies patent is approved and signed by President James Madison. Her patent was for a new technique for weaving straw with silk and thread to make ladies hats. This was the first US patent granted to a woman. *HistoryTime

1834, William Whewell wrote a letter to Michael Faraday concerning names to describe the process of electrolysis which he was investigating. Whewell suggest the names Anode and Cathode. The terms are based on the Greek prefixes "ana-" meaning "up" and "kata-" meaning "down." The chosen prefixes referred to the idea that (as was then applied) that electric current flowed from a battery's positive to a negative pole, in the manner that water would flow down from a hillside to a valley. He suggested a term - ion - for the two together instead of Zetodes or Stechions. Faraday replied that he was "delighted with the facility of expression which the new terms give me and I shall ever be your debtor for the kind assistance you have given me." *TIS Whewell had written on April 25th to Faraday suggesting these terms, but Faraday had been reluctant at first to use them. (PB)

1883 George Cantor writes to Mitag-Leffler that Kronecker had called his work on transfinite set theory "Humbug" in a letter to Hermite. Kronecker reserved his attacks for personal correspondence and student lectures, but said little or nothing publicly against Cantor. *From the Calculus to Set Theory, 1630-1910: An Introductory History
By I. Grattan-Guinness

In 1925, a meeting of local leaders was held in Dayton, Tennessee, to plan a challenge to that state's new law, the Butler Act, which made it illegal to teach Darwin's theory of evolution in a public school. George W. Rappelyea and other local leaders of the small mining town met at Robinson's drug store. The American Civil Liberties Union in New York, concerned by the law's infringement on constitutional rights, had advertised an offer to give legal support to any teacher who would challenge the law. Rappelyea saw the publicity that would accompany such a trial as an opportunity to promote his town. He approached John T. Scopes, a 24-year-old teacher and football coach, who was hesitant at first, to test the legality of the law in court. The infamous “Scopes Monkey Trial” began on 10 Jul 1925.*TIS

1952 Dummer Proposes Integrated Circuit Concept: G. W. A. Dummer, an English electrical engineer, foresees the fabrication of all electronic components of a circuit or system in a single block of semiconductor material. Several special-function devices were developed at Bell Labs and RCA before Jack Kilby at TI demonstrated a general-purpose concept "integrated circuit" in 1958.*CHM

1961 Alan B. Shepard is the first U.S. astronaut to make a flight into space. His fifteen minute flight in Freedom 7 from Cape Canaveral, Florida, reached an altitude of 115 (116?) miles and ended 302 miles down the Atlantic missile range. [Kane, p. 373; Navy Facts, 204] *VFR

1980 Greece issued a stamp honoring the 2300th anniversary of Aristarchus of Samos, discoverer of the heliocentric theory. [Scott #1350] *VFR

1981 The German Democratic Republic issued a stamp honoring Richard Dedekind. [Scott #2181] *VFR

In 2000, a conjunction of the five bright planets - Mercury, Venus, Mars, Jupiter and Saturn - formed a rough line across the sky with the Sun and Moon. Unfortunately, nothing was visible from the earth, because the the line of planets was behind the Sun and hidden in its brilliance. Such a conjunction last happened in Feb 1962 and will not happen again until Apr 2438. Throughout former history, a conjuction event was regarded with foreboding. However, now science can be dismissive. Donald Olson, an expert on tides at Southwest Texas State University, working with the assistance of a graduate student, Thomas Lytle, calculated the stress on the Earth caused by the Moon and eight planets has often been routinely greater, most recently on 6 Jan 1990. *TIS

2012 The biggest full moon of the year, a so-called "supermoon," will take center stage when it rises this weekend (Saturday, May 5, at 11:35 p.m. EDT ). A supermoon occurs when the moon hits its full phase at the same time it makes closest approach to Earth for the month, a lunar milestone known as perigee. May's full moon timed with the moon's perigee could appear 14 percent bigger and 30 percent brighter than other full moons of 2012. *Huffington Post Science

2016 Meteor Shower from Halley's Comet at it's most active-but meteors will be visible for another few weeks as the Earth passes through the debris trail of Halley's Comet. The eta Aquarid display is one of two meteor showers created by dust from Halley's comet (the Orionid shower in October is the other). It occurs every April and May when the Earth passes through a stream of debris cast off by comet Halley during its 76-year trip around the sun. *PB


1580 Johann Faulhaber (5 May 1580; Ulm, Germany – 10 September 1635; Ulm, Germany) was a German mathematician.
Born in Ulm, Faulhaber was trained as a weaver. However he was taught mathematics in Ulm and showed such promise that the City appointed him city mathematician and surveyor. He opened his own school in Ulm in 1600 but he was in great demand because of his skill in fortification work. He collaborated with Johannes Kepler and Ludolph van Ceulen. Besides his work on the fortifications of cities (notably Basel and Frankfurt), Faulhaber built water wheels in his home town and geometrical instruments for the military. Faulhaber made the first publication of Henry Briggs's Logarithm in Germany.
Faulhaber's major contribution was in calculating the sums of powers of integers, what is now called Faulhaber's formula. Jacob Bernoulli makes references to Faulhaber in his Ars Conjectandi and the Bernouli numbers arise in solving coefficients of Faulhaber's formula.
In Academia Algebra Faulhaber gives ∑ nk as a polynomial in N, for k = 1, 3, 5, ... ,17. He also gives the corresponding polynomials in n. Faulhaber states that such polynomials in N exist for all k, but gave no proof. This was first proved by Jacobi in 1834. It is not known how much Jacobi was influenced by Faulhaber's work, but we do know that Jacobi owned Academia Algebra since his copy of it is now in the University of Cambridge.
At the end of Academia Algebra Faulhaber states that he has calculated polynomials for ∑ nk as far as k = 25. He gives the formulae in the form of a secret code, which was common practice at the time. Donald Knuth suggests he is the first to crack the code: (the task [of cracking the code] is relatively easy with modern computers) and shows that Faulhaber had the correct formulae up to k = 23, but his formulae for k = 24 and k = 25 appear to be wrong.
A nice example of how to calculate sum of powers using Pascal's arithmetic triangle is given at Theorem of the Day.
*SAU *Wik

1811 John William Draper (May 5, 1811 – January 4, 1882) was an American (English-born) scientist, philosopher, physician, chemist, historian and photographer. He is credited with producing the first clear photograph of a female face (1839–40) and the first detailed photograph of the Moon (1840). He was also the first president of the American Chemical Society (1876–77) and a founder of the New York University School of Medicine. One of Draper's books, History of the Conflict between Religion and Science, received worldwide recognition and was translated into several languages, but was banned by the Catholic Church. His son, Henry Draper, and his granddaughter, Antonia Maury, were astronomers, and his eldest son, John Christopher Draper, was a chemist. *Wik

1833 Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a gifted analyst whose works form a bridge between the fundamental researches of Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincaré, Painlevé, and Émile Picard. *SAU

1860 Charles Chree (5 May 1860 – 12 August 1928) studied in Aberdeen and Cambridge. He became Superintendent of Kew Observatory and worked on terrestrial magnetism. *SAU

1861 Peter Cooper Hewitt (May 5, 1861 – August 25, 1921) was an American electrical engineer and inventor, who invented the first mercury-vapor lamp in 1901. Hewitt was issued U.S. patent #682692 on September 17, 1901.
In 1902 Hewitt developed the mercury arc rectifier, the first rectifier which could convert alternating current power to direct current without mechanical means. It was widely used in electric railways, industry, electroplating, and high-voltage direct current (HVDC) power transmission. Although it was largely replaced by power semiconductor devices in the 1970s and 80s, it is still used in some high power applications.
In 1907 he developed and tested an early hydrofoil. In 1916, Hewitt joined Elmer Sperry to develop the Hewitt-Sperry Automatic Airplane, one of the first successful precursors of the UAV. *wik

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

1883 Anna Johnson Pell Wheeler (5 May 1883 in Calliope (now Hawarden), Iowa, USA - 26 March 1966 in Bryn Mawr, Pennsylvania, USA) In 1899 she entered the University of South Dakota where she showed great promise in mathematics. The professor of mathematics, Alexander Pell, recognised her talents and helped persuade Anna Johnson that she should follow a career in mathematics. She received an A.B. degree in 1903.
After winning a scholarship to study for her master's degrees at the University of Iowa, she was awarded the degree for a thesis The extension of Galois theory to linear differential equations in 1904. A second master's degree from Radcliffe was awarded in 1905 and she remained there to study under Bôcher and Osgood.
Anna Johnson was awarded the Alice Freeman Palmer Fellowship from Wellesley College to study for a year at Göttingen University. There she attended lectures by Hilbert, Klein, Minkowski, Herglotz and Schwarzschild. She worked for her doctorate at Göttingen. While there Alexander Pell, her former mathematics professor came to Göttingen so that they could marry.
After returning to the United States, where her husband was by now Dean of Engineering, she taught courses in the theory of functions and differential equations. In 1908 Anna Pell returned to Göttingen where she completed the work for her doctorate but, after a disagreement with Hilbert, she returned to Chicago, where her husband was now on the university staff, without the degree being awarded.
At Chicago she became a student of Eliakim Moore and received her Ph.D. in 1909, her thesis Biorthogonal Systems of Functions with Applications to the Theory of Integral Equations being the one written originally at Göttingen. From 1911 Anna Pell taught at Mount Holyoke College and then at Bryn Mawr from 1918. Anna Pell's husband Alexander, who was 25 years older than she was, died in 1920. In 1924 Anna Pell became head of mathematics when Scott retired, becoming a full professor in 1925.
After a short second marriage to Arthur Wheeler, during which time they lived at Princeton and she taught only part-time, her second husband died in 1932. After this Anna Wheeler returned to full time work at Bryn Mawr where Emmy Noether joined her in 1933. However Emmy Noether died in 1935. The period from 1920 until 1935 certainly must have been one with much unhappiness for Anna Wheeler since during those years her father, mother, two husbands and close friend and colleague Emmy Noether died. Anna Wheeler remained at Bryn Mawr until her retirement in 1948.
The direction of Anna Wheeler's work was much influenced by Hilbert. Under his guidance she worked on integral equations studying infinite dimensional linear spaces. This work was done in the days when functional analysis was in its infancy and much of her work has lessened in importance as it became part of the more general theory.
Perhaps the most important honour she received was becoming the first woman to give the Colloquium Lectures at the American Mathematical Society meetings in 1927.

1897 Francesco Giacomo Tricomi (5 May 1897 – 21 November 1978)  studied differential equations which became very important in the theory of supersonic flight. *SAU

1921 Arthur Leonard Schawlow (May 5, 1921 – April 28, 1999) was an American physicist. He is best remembered for his work on lasers, for which he shared the 1981 Nobel Prize in Physics with Nicolaas Bloembergen and Kai Siegbahn.
In 1991 the NEC Corporation and the American Physical Society established a prize: the Arthur L. Schawlow Prize in Laser Science. The prize is awarded annually to "candidates who have made outstanding contributions to basic research using lasers."
In 1951, he married Aurelia Townes, younger sister to physicist Charles Hard Townes, and together they had three children; Arthur Jr., Helen, and Edith. Arthur Jr. was autistic, with very little speech ability.
Schawlow and Professor Robert Hofstadter at Stanford, who also had an autistic child, teamed up to help each other find solutions to the condition. Arthur Jr. was put in a special center for autistic individuals, and later Schawlow put together an institution to care for people with autism in Paradise, California. It was later named the Arthur Schawlow Center in 1999, shortly before his death on the 29th of April 1999.
Schawlow died of leukemia in Palo Alto, California. *Wik

1923 Cathleen Synge Morawetz (May 5, 1923; Toronto, Canada - ) is a mathematician. Morawetz's research was mainly in the study of the partial differential equations governing fluid flow, particularly those of mixed type occurring in transonic flow. She is Professor Emerita at the Courant Institute of Mathematical Sciences at the New York University, where she has also served as director from 1984 to 1988.
Morawetz's father, John Lighton Synge was an Irish mathematician, specializing in the geometry of general relativity and her mother also studied mathematics for a time. Her childhood was split between Ireland and Canada. Both her parents were supportive of her interest in mathematics and science, and it was a woman mathematician, Cecilia Krieger, who had been a family friend for many years who later encouraged Morawetz to pursue a PhD in mathematics. Morawetz says her father was influential in stimulating her interest in mathematics, but he wondered whether her studying mathematics would be wise (suggesting they might fight like the Bernoulli brothers)
In 1981, she became the first woman to deliver the Gibbs Lecture of The American Mathematical Society, and in 1982 presented an Invited Address at a meeting of the Society for Industrial and Applied Mathematics. She was named Outstanding Woman Scientist for 1993 by the Association for Women in Science. In 1995, she became the second woman elected to the office of president of the American Mathematical Society. In 1998 she was awarded the National Medal of Science; she was the first woman to receive the medal for work in mathematics. In 2004 she received the Leroy P. Steele Prize for Lifetime Achievement. In 2006 she won the George David Birkhoff Prize in Applied Mathematics. In 2012 she became a fellow of the American Mathematical Society.*Wik

1961 G. L. Honaker, Jr., (May 5, 1961- ) in the strobogrammatic year 1961 and lives in Bristol, Virginia. Number art and design has interested him since an early age. He became fascinated when an elementary school teacher (J. N. Ely, Jr.) in his nearby birthplace of Pennington Gap drew a factor tree on the blackboard. "I saw great beauty in this 'numerical fingerprint' and was hooked." After a tour in the US Navy he became a K-12 math/science educator and created 'Prime Curios!' (primes.utm.edu/curios/) with the assistance of Chris K. Caldwell, a well-known mathematics professor and technical editor of the site at the University of Tennessee at Martin. Here, people from all over the world submit facts, curiosities, oddities, etc., about anything related to prime numbers.

BookAuthority (https://bookauthority.org/books/best-prime-numbers-books) includes it in their list of 53 Best Prime Numbers Books of All Time. 



1859 Peter Gustav Lejeune Dirichlet  (13 Feb 1805, 5 May 1859 at age 54)   is credited with the modern formal definition of a function.   After his death, Dirichlet's lectures and other results in number theory were collected, edited and published by his friend and fellow mathematician Richard Dedekind under the title Vorlesungen über Zahlentheorie (Lectures on Number Theory).  Dirichlet's brain is preserved in the department of physiology at the University of Göttingen, along with the brain of Gauss.(Wikipedia) (Dirichlet proved in 1826 that in any arithmetic progression with first term coprime to the difference there are infinitely many primes.) *SAU [Dirichlet is buried in Bartholomaus cemetery in Gottingen]

1957 Leopold Löwenheim (26 June 1878 in Krefeld, Germany  – 5 May 1957 in Berlin) was a German mathematician who worked on mathematical logic and is best-known for the Löwenheim-Skolem paradox. *SAU  [Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence which intuitively says (or which precisely says in the standard model of the theory) that there exist sets that are not countable. Thus the seeming contradiction is that a model which is itself countable, and which contains only countable sets, satisfies the first order sentence that intuitively states "there are uncountable sets".] *Wik

1989 Stefan E Warschawski (April 18, 1904 – May 5, 1989) With careful scholarship, he made lasting contributions to the theory of complex analysis, particularly to the theory of conformal mappings. With keen judgment, he guided two mathematics departments to eminence. With modest gratitude, he cemented many friendships along the way.*SAU  [He is buried in El Camino Memorial Park, San Diego, California.]

2001Theodore Harold "Ted" Maiman (July 11, 1927 – May 5, 2007) was an American Engineer and physicist credited with the invention of the first working laser.[ Maiman’s laser led to the subsequent development of many other types of lasers.[9][10] The laser was successfully fired on May 16, 1960. In a July 7, 1960 press conference in Manhattan,[1] Maiman and his employer, Hughes Aircraft Company, announced the laser to the world. Maiman was granted a patent for his invention, and he received many awards and honors for his work. Maiman's experiences in developing the first laser and subsequent related events are described in his book, The Laser Odyssey. *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

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