Thursday 13 February 2020

On This Day in Math - February 13

Prejudice for regularity and simplicity is a source of error that has only too often infected philosophy.
~Ruggero Giuseppe (Roger Joseph) Boscovich

The 44th day of the year; there are 44 ways to reorder the numbers 1 through five so that none of the digits is in its natural place. This is called a derangement.  The number of derangements of n items is an interesting study for students.  Some historical notes from here.

If you had five  letters for five different people and five  envelopes  addressed to the five people,  there are 44 ways to put every letter in the wrong envelope.

All even perfect numbers greater than 6 end in 44 in base six, as do all powers of ten greater than 10. *Lord Karl Voldevive ‏@Karl4MarioMugan  (students should be encouraged to understand that the converse of these statements is not true by finding exceptions.

An Euler brick, named after Leonhard Euler, is a cuboid whose edges andface diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.The smallest Euler brick, discovered by Paul Halcke in 1719, has edgess) = (44,117,240) and face diagonals 125, 244, and 267.

1535 During the night of February 12–13 Tartaglia discovered a method of solving cubic equations that enables him to beat Fiore in a contest. *VFR Tartaglia had previously solve cubics of the form, ax2+bx3=n. He had learned that in his contest with del Ferro's Student Fiore, which would occur on Feb 22nd, Fiore had been given del Ferro's method for solving cubics of the type ax + bx3=n. It was this type of problem that Tartaglia leaned to do. On the 22nd, Tartaglia would emerge victorious.

In 1588, Tycho Brahe first outlines his "Tychonic system" idea of the structure of the solar system. The Tychonic system was a hybrid, sharing both the basic idea of the geocentric system of Ptolemy, and the heliocentric idea of Nicholas Copernicus. In his De mundi aethorei recentioribus phaenomenis, Brahe's proposal, retaining Aristotelian physics, kept the the Sun and Moon revolving about Earth in the center of the universe and, at a great distance, the shell of the fixed stars was centered on the Earth. But like Copernicus, he agreed that Mercury, Venus, Mars, Jupiter, and Saturn revolved about the Sun. Thus he could explain the motions of the heavens without "crystal spheres" carrying the planets through complex Ptolemaic epicycles. *TIS

In 1633, Italian astronomer Galileo arrived in Rome for his trial before the Inquisition for professing the belief that the earth revolves around the sun. Enemies of Galileo had convinced Pope Urban VIII that the character Simplicio in the Dialogue ineptly defending the Ptolemaic system, was a thinly veiled caricature of himself. A document was produced alleging that Bellarmine in 1616 forbade Galileo to discuss Copernican ideas in any way. (Modern scholars determined this document is a forgery). He faced two charges: disobeying Bellarmine's order and misleading censors who published his book. Humiliated and threatened with torture, Galileo had no choice but to admit guilt, and "abjure, curse and detest the aforesaid errors and heresies...*TIS

1684 John Evelyn writes in his diary about the interest in a public library in St Martines Parish, which could hopefully be part of the new renovation of St. Paul's. *Lisa Jardine, Ingenious Pursuits, pg 83 Ingenious Pursuits: Building the Scientific Revolution

1809 When the French Academy of Sciences had no winner for the prize competition in Galvanism, it was decided to recommend to Napoleon that the prize money be used to "encourage the mathematical analysis of the experiments made by M. Chladni on the vibration of resonating plates." *Sophie Germain: An Essay in the History of the Theory of Elasticity

1909 Graph Paper proposed for use in Elementary Schools: "The Rochester Section (of the NCTM) held its sixth regular meeting at the University of Rochester on Saturday, February 13, 1909. The program consisted of the following papers :..." The Use of Squared Paper in Elementary Mathematics," by A. S. Gale, the University of Rochester..... The use of cross-section paper in the grammar schools was advocated by Professor Gale. It may well be used in drawing, in connection with the multiplication table, and especially in mensuration. Simple graphs not only interest children, but they are of such general importance as to merit consideration." *The Mathematics Teacher, Vol. 1, No. 2, December, 1908

1912  Robert Millikan began collecting data from his famous oil drop experiment. On this day he gathered observations on the first of the 58 drops he ultimately published. Millikan used his measurements of the motion of oil drops within an electric field to estimate the fundamental unit of charge. He earned the Nobel Prize for Physics in 1923 for his pioneering measurements of the charge on the electron. *TIS

In 1946, the world's first electronic digital computer, ENIAC (the Electronic Numerical Integrator and Calculator) was first demonstrated at the Moore School of Electrical Engineering at the University of Pennsylvania, by the late John W. Mauchly and J. Presper Eckert. The ENIAC machine occupied a room 30 by 50 feet. Its birth lay in WW II as a classified military project known only as Project PX. The ENIAC is historic because it laid the foundations for the modern electronic computing industry. The ENIAC demonstrated that high-speed digital computing was possible using the vacuum tube technology then available. Built out of some 17,468 electronic vacuum tubes, ENIAC was in its time the largest single electronic apparatus in the world. *TIS

1973 The German Democratic Republic issued a stamp picturing Copernicus and the title page of his De Revolutionibus, which was published 500 years before in 1473. [Scott #1461] *VFR (top)
The USA issued a 500 year commemorative stamp on April 23 of the same year.

1980 Apollo Computer is incorporated in Chelmsford, MA. Apollo helped create the original work stations, small but powerful computer mostly used for engineering. In 1989, Hewlett-Packard Company acquired Apollo in a $476 million deal. *CHM

2012 In the middle of the night on February 13th-14th, something disturbed the animal population of rural Portal, Georgia. Cows started mooing anxiously and local dogs howled at the sky. The cause of the commotion was a rock from space.
"At 1:43 AM Eastern, I witnessed an amazing fireball," reports Portal resident Henry Strickland. "It was very large and lit up half the sky as it fragmented. The event set dogs barking and upset cattle, which began to make excited sounds. I regret I didn't have a camera; it lasted nearly 6 seconds." "These fireballs are particularly slow and penetrating," explains meteor expert Peter Brown, a physics professor at the University of Western Ontario. "They hit the top of the atmosphere moving slower than 15 km/s, decelerate rapidly, and make it to within 50 km of Earth’s surface." So far in February, NASA's All-Sky Fireball Network has photographed about a half a dozen bright meteors that belong to this oddball category. They range in size from basketballs to buses, and all share the same slow entry speed and deep atmospheric penetration. Cooke has analyzed their orbits and come to a surprising conclusion:
February Fireballs "They all hail from the asteroid belt—but not from a single location in the asteroid belt," he says. "There is no common source for these fireballs, which is puzzling." *NASA See the video

2015 The first of three Friday the 13ths this year. They occur in Feb, Mar, and Nov. The last year with three Friday the 13ths was 2012. The 2012 triplet were in a leap year, and each 13 weeks apart (Jan, Apr, and July). This can only happen in a leap year. There seems to be no written evidence of the superstition in English until 1869. Interestingly, the Spanish and Greek Cultures have a similar tradition about Tuesday the 13th. *Wik 2016 only has one, in May.

2017 A plague was erected at the entrance to 2 Colinette Road, Putney, England to commemorate the famous taxicab number, 1729, conversation of Ramanujan and Hardy. In late 1919 Ramanujan was convalescing in a care home at the address when Hardy visited him and the famous conversation took place. The story is so well known that I will not repeat it here, but if it is new to you, read this. Because the story is so popular with mathematicians, Simon Singh began trying to erect a plague in honor of the event at the location, now a private home. Because Ramanujan was at the facility in Putney for such a short period, (He would return to India shortly after) the usual "Blue Plaque" used in England was not permitted, so with the help and funding of the British Society for the History of Mathematics and a group called The Good Thinking Society, and the cooperation of the owner, the plaque below was erected. The bill for the creation and installation of the Plaque was 1729 x 10P.
Home-owner Deborah Gauld and the "Taxicab # Plaque" in Putney, England
It seems strange to me that the actual date of this visit by Hardy does not seem to be known. If it is, and some one is aware of the date, please advise. *The Good Thinking Society
When hearing this story many students assume that Ramanujan was the first to discover this fact, but it seems that it was known as early as 1657 by Bernard Frénicle de Bessy. And because I know you wondered, the smallest number that is the sum of the cubes of two positive numbers in three ways is 87539319.


1743 Sir Joseph Banks (Baronet)(13 Feb 1743; 19 Jun 1820 at age 77) was an English botanist and explorer who was president of the Royal Society for over 40 years, and known for his promotion of science. As an independent naturalist, Banks participated in a voyage to Newfoundland and Labrador in 1767. He successfully lobbied the Royal Society to be included on what was to be James Cook's first great voyage of discovery, on board the Endeavour (1768-71). King George III appointed Banks adviser to the Royal Botanic Gardens at Kew. Banks established his London home as a scientific base (1776) with natural history collections he made freely available to researchers. In 1819, he was Chairman of committees established by the House of Commons, one to enquire into prevention of banknote forgery, the other to consider systems of weights and measures. *TIS

1766 Thomas Robert Malthus (13 Feb 1766; 23 Dec 1834 at age 68) English economist and demographer who can be regarded as a pioneer sociologist. He was one of the first to systematically analyze human society when he published his theories in An Essay on the Principle of Population. Malthus predicted population would always outrun the food supply and that would result in famine, disease or war to reduce the number of people. As Malthus observed the Industrial Revolution was causing a rapid increase in population, he indicated keeping improved social conditions would require imposing strict limits on reproduction. Reading the book inspired Charles Darwin to reflect upon the survival of the fittest individuals in the process of natural selection in evolving populations of any organism. Alfred Russell Wallace likewise acknowledged his theory was stimulated by the book by Malthus. *TIS

1805 Peter Gustav Lejeune Dirichlet(13 Feb 1805; 5 May 1859 at age 54) German mathematician who made valuable contributions contributions to number theory, analysis, and mechanics. Dirichlet is best known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. He also proposed in 1837 the modern definition of a function. In mechanics he investigated the equilibrium of systems and potential theory. This led him to the Dirichlet problem concerning harmonic functions with given boundary conditions. Dirichlet is considered the founder of the theory of Fourier series, having corrected the earlier mistakes of other workers on Fourier's writings. One of his students was Riemann. In 1855, he succeeded Carl Friedrich Gauss at the University of Göttingen.
*TIS He was precociously interested in mathematics, even using his pocket money to buy mathematical books before the age of 12. Because mathematics in Germany was at a low ebb he studied in Paris from 1822 to 1826. As he earned no doctorate the University of Cologne awarded one honoris causis so that he could teach in Germany. He was an excellent teacher. *VFR

1852 Johan Ludvig Emil Dreyer (13 Feb 1852, 14 Sep 1926) Danish astronomer who compiled the New General Catalogue of Nebulae and Clusters of Stars, (NGC) in 1888. When he became Director of the Armagh Observatory in 1882, financially it was destitute, with no prospect of replacing its aging instruments. Though Dreyer obtained a new 10-inch refractor by Grubb, the lack of funding for an assistant, precluded him from a continuation of traditional positional astronomy. Instead he concentrated on the compilation of observations made earlier. The NGC he listed 7840 objects and in its supplements (1895, 1908) he added a further 5386 objects. It still remains one of the standard reference catalogs.*TIS

1882 Tadeusz Banachiewicz (13 February 1882, Warsaw, Congress Poland, Russian Empire – 17 November 1954, Kraków) was a Polish astronomer, mathematician and geodesist.

He was educated at Warsaw University and his thesis was on "reduction constants of the Repsold heliometer". In 1905, after the closure of the University by the Russians, he moved to Göttingen and in 1906 to the Pulkowa Observatory. He also worked at the Engel'gardt Observatory at Kazan University from 1910–1915.
In 1919, after Poland regained her independence, Banachiewicz moved to Kraków, becoming a professor at the Jagiellonian University and the director of Kraków Observatory. He authored approximately 180 research papers and modified the method of determining parabolic orbits. In 1925, he invented a theory of "cracovians" — a special kind of matrix algebra — which brought him international recognition. This theory solved several astronomical, geodesic, mechanical and mathematical problems.
In 1922 he became a member of PAU (Polska Akademia Umiejętności) and from 1932 to 1938 was the vice-president of the International Astronomical Union. He was also the first President of the Polish Astronomical Society, the vice-president of the Geodesic Committee of The Baltic States and, from 1952 to his death, a member of the Polish Academy of Sciences. He was also the founder of the journal Acta Astronomica. He was the recipient of Doctor Honoris Causa titles from the University of Warsaw, the University of Poznań and the University of Sofia in Bulgaria.[citation needed]
Banachiewicz invented a chronocinematograph. The lunar crater Banachiewicz is named after him. He wrote over 230 scientific works. *Wik

1906 Edward Maitland Wright (13 Feb 1906 in Farnley, near Leeds, England - 2 Feb 2005 in Reading, England) was initially self-taught in Mathematics but was able to go and study at Oxford. He spent a year at Göttingen and returned to Oxford. He was appointed to the Char at Aberdeen where he stayed for the rest of his career, eventually becoming Principal and Vice-Chancellor of the University. He is best known for the standard work on Number Theory he wrote with G H Hardy. One of Wright's first papers, published in 1930, was on Bernstein polynomials. Also among his early work was a series of three papers titled Asymptotic partition formulae. The third in the series Asymptotic partition formulae, III. Partitions into kth powers was published by Acta Mathematica in 1934. *SAU

1910 William Bradford Shockley (13 Feb 1910; 12 Aug 1989 at age 79) was an English-American physicist and engineer who shared (with John Bardeen and Walter H. Brattain) the 1956 Nobel Prize for Physics for their development of the transistor, a device that largely replaced the bulkier and less-efficient vacuum tube and ushered in the age of microminiature electronics. *TIS In his later years Shockley became controversial due to his statements related to eugenics. Shockley argued that the higher rate of reproduction among the less intelligent was having a dysgenic effect, and that a drop in average intelligence would ultimately lead to a decline in civilization *Wik

1926 Tonny Albert Springer (February 13, 1926, The Hague – December 7, 2011, Zeist) was a mathematician at Utrecht university who worked on linear algebraic groups, Hecke algebras, complex reflection groups, and who introduced Springer representations and the Springer resolution.*Wik


1237 Jordanus de Nemore, (1225 in Borgentreich (near Warburg), Germany-13 February 1237) along with Leonardo Fibonacci, was the dominant mathematician of the first half of the 13th century. He is best known for his works on mechanics (statics) but he also wrote influential works on arithmetic, geometry and algebra. Little is known about Jordanus' life. His name, de Nemore (literally of the forest or Forester) suggests he was from a wooded area. Some scholars feel Jordanus de Nemore and Jordanus of Saxony, second Grand Master of the Dominican order, are the same person. If this is so, Jordanus was born in the area of Mainz and was educated in Paris. He was elected Grand Master in 1222 and died in a shipwreck on 13 February 1237 while returning from the Holy Land. (His dates of birth is unknown, and his date of death is questionable).
This is page 7 from an early printed edition of the Arithmetica of Jordanus de Nemore from Mathematical Treasures of the MAA

1774 Charles Marie de La Condamine (28 January 1701 – 13 February 1774) was a French explorer, geographer, and mathematician. He spent ten years in present-day Ecuador measuring the length of a degree latitude at the equator and preparing the first map of the Amazon region based on astronomical observations. *Wik

1787 Ruggero Giuseppe (Roger Joseph) Boscovich (18 May 1711, 13 Feb 1787 at age 75) or in native language Ruđer Josip Bošković was a Serbo-Croatian astronomer and mathematician who gave the first geometric procedure for determining the equator of a rotating planet from three observations of a surface feature and for computing the orbit of a planet from three observations of its position. Boscovich was one of the first in continental Europe to accept Newton's gravitational theories and he wrote 70 papers on optics, astronomy, gravitation, meteorology and trigonometry. Boscovich also showed much ability in dealing with practical problems. He suggested and directed the draining of the Pontine marshes near Rome, and recommended the use of iron bands to control the spread of cracks in the dome of St. Peter's basilica. *TIS A slightly enlarged description of his life is here.

1874 Franz Taurinus (15 Nov 1794 in Bad König, German - 13 Feb 1874 in Cologne, Germany) was a German mathematician best known for his work on non-Euclidean geometry. In 1697 Girolamo Saccheri assumed the fifth postulate is false and attempted to derive a contradiction. Of course, although he did not intend it to be so, he was then studying non-euclidean geometry. In 1766 Lambert followed a similar line to Saccheri. Lambert noticed that, in this new geometry where the sum of the angles of a triangle was less than 180 degrees, the angle sum of a triangle increased as the area of the triangle decreased. Schweikart himself is famed for investigating this new geometry which he called astral geometry.
Taurinus not only corresponded on mathematical topics with his uncle but he also corresponded with Gauss about his ideas on geometry. At first Taurinus tried to prove that Euclidean geometry was the only geometry but, in 1826, he accepted the lack of contradiction in other geometries. He published Theorie der Parallellinien in Cologne in 1825 and in the following year he published Geometriae prima elementa also in Cologne.
In this last mentioned publication Taurinus accepts that a third system of geometry exists in which the sum of the angles of a triangle is less than 180 degrees. He called this geometry "logarithmic-spherical geometry" and he recognized the lack of a contradiction in this geometry as meaning that it was internally consistent. He had developed a non-euclidean trigonometry which he applied to a number of elementary problems.
Taurinus came up with the important idea that elliptic geometry could be realized on the surface of a sphere, an idea taken up by Riemann. He also realized that there were an infinite number of non-euclidean geometries and this, Taurinus claimed, was highly significant. It showed that euclidean geometry held a unique dominating role. This is an interesting sideways move since his original aim had been to prove that euclidean geometry was the unique geometry. Finding that this was not so, he still wanted to demonstrate that euclidean geometry was "the" geometry. *SAU

1914 Alphonse Bertillon (23 Apr 1853, 13 Feb 1914 at age 60) French criminologist who was chief of criminal identification for the Paris police from 1880. He developed an identification system known as anthropometry, or the Bertillon system, that came into wide use in France and other countries. The system records physical characteristics (eye colour, scars, deformities, etc.) and specified measurements (height, fingertip reach, head length and width, ear, foot, arm and finger length, etc) These are recorded on cards and classified according to the length of the head. After two decades this system was replaced by fingerprinting in the early 1900s because Bertillon measurements were difficult to take with uniform exactness, and could change later due to growth or surgery. *TIS

1926 Francis Ysidro Edgeworth FBA (8 February 1845, Edgeworthstown – 13 February 1926, Oxford) was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. Edgeworth was a highly influential figure in the development of neo-classical economics. He was the first to apply certain formal mathematical techniques to individual decision making in economics. He developed utility theory, introducing the indifference curve and the famous Edgeworth box, which is now familiar to undergraduate students of microeconomics. He is also known for the Edgeworth conjecture which states that the core of an economy shrinks to the set of competitive equilibria as the number of agents in the economy gets large. In statistics Edgeworth is most prominently remembered by having his name on the Edgeworth series. *Wik In 1881 he published Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. This work, really on economics, looks at the Economical Calculus and the Utilitarian Calculus. In fact most of his work could be said to be applications of mathematical psychics which Edgeworth saw as analogous to mathematical physics. They were applied to the measure of utility, the measure of ethical value, the measure of evidence, the measure of probability, the measure of economic value, and the determination of economic equilibria. He formulated mathematically a capacity for happiness and a capacity for work. His conclusions that women have less capacity for pleasure and for work than do men would not be popular today. *SAU

1947 Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician. He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students.
Hecke was born in Buk, Posen, Germany (now Poznań, Poland), and died in Copenhagen, Denmark. His early work included establishing the functional equation for the Dedekind zeta function, with a proof based on theta functions. The method extended to the L-functions associated to a class of characters now known as Hecke characters or idele class characters: such L-functions are now known as Hecke L-functions. He devoted most of his research to the theory of modular forms, creating the general theory of cusp forms (holomorphic, for GL(2)), as it is now understood in the classical setting.*Wik

1956 Jan Łukasiewicz (Polish pronunciation: [ˈjan wukaˈɕɛvʲitʂ]) (21 December 1878 – 13 February 1956) was a Polish logician and philosopher born in Lwów (Lemberg in German), Galicia, Austria–Hungary (now Lviv, Ukraine). His work centred on analytical philosophy and mathematical logic. He thought innovatively about traditional propositional logic, the principle of non-contradiction and the law of excluded middle.*Wik He created a notation he called Polish Notation (after his homeland) that use the argument of a function before the actual notation for the function to eliminate the need for parenthetical enclosures. This notation is the root of the idea of the recursive stack, a last-in, first-out computer memory store proposed by several researchers including Turing, Bauer and Hamblin, and first implemented in 1957. In 1960, Łukasiewicz notation concepts and stacks were used as the basis of the Burroughs B5000 computer designed by Robert S. Barton and his team at Burroughs Corporation in Pasadena, California. The concepts also led to the design of the English Electric multi-programmed KDF9 computer system of 1963, which had two such hardware register stacks. A similar concept underlies the reverse Polish notation (RPN, a postfix notation) of the Friden EC-130 calculator and its successors, many Hewlett Packard calculators. *Wik

1980 Marian Adam Rejewski (16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in 1932 solved the plugboard-equipped Enigma machine, the main cipher device used by Germany. The success of Rejewski and his colleagues Jerzy Różycki and Henryk Zygalski jump-started British reading of Enigma in World War II; the intelligence so gained, code-named "Ultra", contributed, perhaps decisively, to the defeat of Nazi Germany.
While studying mathematics at Poznań University, Rejewski had attended a secret cryptology course conducted by the Polish General Staff's Biuro Szyfrów (Cipher Bureau), which he joined full-time in 1932. The Bureau had achieved little success reading Enigma and in late 1932 set Rejewski to work on the problem. After only a few weeks, he deduced the secret internal wiring of the Enigma. Rejewski and his two mathematician colleagues then developed an assortment of techniques for the regular decryption of Enigma messages. Rejewski's contributions included devising the cryptologic "card catalog," derived using his "cyclometer," and the "cryptologic bomb."
Five weeks before the German invasion of Poland in 1939, Rejewski and his colleagues presented their results on Enigma decryption to French and British intelligence representatives. Shortly after the outbreak of war, the Polish cryptologists were evacuated to France, where they continued their work in collaboration with the British and French. They were again compelled to evacuate after the fall of France in June 1940, but within months returned to work undercover in Vichy France. After the country was fully occupied by Germany in November 1942, Rejewski and fellow mathematician Henryk Zygalski fled, via Spain, Portugal and Gibraltar, to Britain. There they worked at a Polish Army unit, solving low-level German ciphers. In 1946 Rejewski returned to his family in Poland and worked as an accountant, remaining silent about his cryptologic work until 1967. *Wik

1997 Mark Alexandrovich Krasnosel'skii (April 27, 1920, Starokostiantyniv – February 13, 1997, Moscow) was a Soviet, Russian and Ukrainian mathematician renowned for his work on nonlinear functional analysis and its applications. *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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