## Saturday, 31 March 2018

### On This Day in Math - March 31

God runs electromagnetics on Monday, Wednesday, and Friday by the wave theory, and the devil runs it by quantum theory on Tuesday, Thursday, and Saturday.
~Sir Lawrence Bragg  (apparently there is no electricity on Sundays)

The 90th day of the year; 90 is the only number that is the sum of its digits plus the sum of the squares of its digits. (Is there any interesting distinction to the rest of the numbers for which this sum is more (or less) than the original number?)

$\frac{90^3 - 1}{90 - 1}$ is a Mersenne prime.

90 is the smallest number having 6 representations as a sum of four positive squares

90 is the number of degrees in a right angle. Moreover, as a compass direction, 90 degrees corresponds to east.
Which reminds me of a fun math joke:"The number you have dialed is imaginary. Please rotate you phone by 90 degrees and dial again."

And 90 is the sum of the first 9 consecutive even numbers,
the sum of consecutive integers in two different ways,
the sum of two consecutive primes,
and of six consecutive primes,
and the sum of five consecutive squares.
(all proofs left to the reader.)

EVENTS

In 1851, Leon Foucault demonstrated his pendulum experiment at the Pantheon of Paris at the request of Napoleon Bonaparte, who had been informed of Foucault's recent discovery on 6 Jan 1851. He had installed a pendulum in his cellar in the Arras Street of Paris. It was made from 2 m (6.5-ft) long wire supporting a 5-kg weight. He observed a small movement of the oscillation plane of the pendulum - showing that the Earth was rotating underneath the swinging pendulum. A month later, he repeated the experiment at the observatory of Paris, with a 11-m pendulum which gave longer swings and a more clearly visible deviation. His March demonstration at the Pantheon used a 28-kg sphere on a 67-m (220-ft) wire. *TIS (The first date of this demonstration seems to have been on March 28,

1854 The University of Konigsberg awarded Weierstrass an honorary doctorate. Previously he was a Gymnasium teacher without a university degree. *VFR The award was the result of the attention his 1854 paper, Zur Theorie der Abelschen Functionen, which appeared in Crelle's Journal. This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series. With this paper Weierstrass burst from obscurity.*SAU

1899 The EIFFEL TOWER, was built in 26 months and opened in Mar 1889 for the Universal Exposition. it is 320.75 m (1051 ft) high and only weighs 7000 tons – less than the air around it! The tower was inaugurated on 31 March 1889, and opened on 6 May. *VFR

1959 Sof'ja Janovskaja became the first chairperson of the newly created department of mathematical logic at the Moscow State University. *Women of Mathematics

1918 Daylight Savings Time for the USA first applied. Standard time was adopted throughout the United States. 'An Act to preserve daylight and provide standard time for the United States' was enacted on March 19, 1918. It both established standard time zones and set summer DST to begin on March 31, 1918. *WebExhibits.org
I understand that at least three states are trying to repeal daylight savings in their states as of 2014.

In 1921, Professor Albert Einstein arrived in New York to give a lecture on his new theory of relativity. *TIS

1936 The Last day of service of the US Post Office in Eight, West Va. (It Seems the PO in nearby Six, W. Va lasted a little longer, but I can't find it now in Post Office Listings)  Eight was an unincorporated community located in McDowell County, West Virginia.

1939 Harvard and IBM Agree to Build The Mark I "Giant Brain":
Harvard and IBM sign an agreement to build the Mark I, also known as the IBM Automatic Sequence Controlled Calculator (ASCC). Project leader Howard Aiken developed the original concept of the machine: a series of switches, relays, rotating shafts and clutches. The Mark I weighed about five tons and contained more than 750,000 components. It read instructions from paper tape and data from punch cards.*CHM

1981 Time (p. 51) reported that Educational Testing Service had to change the scores on 250,000 PSAT and 19,000 SAT papers because a student had successfully challenged a mathematical question about polyhedrons with no right answer. Mathematics Magazine 54 (1981), pp 152 and 277. *VFR Daniel Lowen, 17, a junior at Cocoa Beach High School in Florida was the first to call the ETS attention to their error. The problem involved putting two pyramids together and determining the number of faces on the new figure. The ETS had failed to allow for the fact that when two faces are joined, other faces meeting at the edges of the union might meld into one face.

1984 Science News reports that Persi Diaconis, a statistician at Stanford, can do a perfect riffle shuffle eight times in a row, thereby returning the 52-card deck to its original order. He has also proved that seven ordinary shuffles is enough to randomize a deck of cards. *VFR

1993 The birth of Spamming, A bug in a program written by Richard Depew sends an article to 200 newsgroups simultaneously. The term spamming is coined by Joel Furr (a writer and software trainer notable as a Usenet personality in the early and mid 1990s.) to describe the incident. *Wik

2011 The first ever "On This Day in Math"... thanks to hundreds of you for all the help.

BIRTHS
1596 René Descartes (31 March 1596 in La Haye (now Descartes),Touraine, France
- 11 Feb 1650 in Stockholm, Sweden)was a French philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. His work had a great influence on both mathematicians and philosophers. La Géométrie is by far the most important part of this work. Scott summarises the importance of this work in four points:

He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes arbitrariness.
Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all.
Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method.
Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all.
*SAU His lifelong habit of laying abed till noon was interrupted by Descartes’ new employer, the athletic, nineteen-year-old Queen Christiana of Sweden, who insisted he tutor her in philosophy in an unheated library early in the morning. This change of lifestyle caused the illness that killed him. [Eves, Circles, 177◦]*VFR

1730 – Étienne Bézout (31 March 1730 in Nemours, France - 27 Sept 1783 in Basses-Loges (near Fontainbleau), France) His most famous and well used book "including an incorrect proof that the quintic was solvable by radicals. In the early nineteenth century some of his in influential textbooks were translated into English. One translator, John Farrah, used
them to teach calculus at Harvard." *VFR
Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his principia, where he claims that two curves have a number of intersection points given by the product of their degrees. The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given.

1795 Louis Paul Emile Richard (31 March 1795 in Rennes, France - 11 March 1849 in Paris, France) Richard perhaps attained his greatest fame as the teacher of Galois and his report on him which stated, "This student works only in the highest realms of mathematics.... "
It is well known. However, he also taught several other mathematicians whose biographies are included in this archive including Le Verrier, Serret and Hermite. He fully realised the significance of Galois' work and so, fifteen years after he left the college, he gave Galois' student exercises to Hermite so that a record of his school-work might be preserved. It is probably fair to say that Richard chose to give them to Hermite since in many ways he saw him as being similar to Galois. Under Richard's guidance, Hermite read papers by Euler, Gauss and Lagrange rather than work for his formal examinations, and he published two mathematics papers while a student at Louis-le-Grand.
Despite being encouraged by his friends to publish books based on the material that he taught so successfully, Richard did not wish to do so and so published nothing. This is indeed rather unfortunate since it would now be very interesting to read textbooks written by the teacher of so many world-class mathematicians.*SAU

1806 Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane as one of ten leading 19th-century British mathematicians. Kirkman's schoolgirl problem, an existence theorem for Steiner triple systems that founded the field of combinatorial design theory, is named after him.
Kirkman's first mathematical publication was in the Cambridge and Dublin Mathematical Journal in 1846, on a problem involving Steiner triple systems that had been published two years earlier in the Lady's and Gentleman's Diary by Wesley S. B. Woolhouse. Despite Kirkman's and Woolhouse's contributions to the problem, Steiner triple systems were named after Jakob Steiner who wrote a later paper in 1853. Kirkman's second research paper paper, in 1848, concerned hypercomplex numbers.
In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in the Lady's and Gentleman's Diary:

Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.

This problem became known as Kirkman's schoolgirl problem, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years. Kirkman also studied the Pascal lines determined by the intersection points of opposite sides of a hexagon inscribed within a conic section. Any six points on a conic may be joined into a hexagon in 60 different ways, forming 60 different Pascal lines. Extending previous work of Steiner, Kirkman showed that these lines intersect in triples to form 60 points (now known as the Kirkman points), so that each line contains three of the points and each point lies on three of the lines. *Wik

1847 – Yegor Ivanovich Zolotarev, (March 31, 1847, Saint Petersburg – July 19, 1878, Saint Petersburg) In 1874, Zolotarev become a member of the university staff as a lecturer and in the same year he defended his doctoral thesis “Theory of Complex Numbers with an Application to Integral Calculus”. The problem Zolotarev solved there was based on a problem Chebyshev had posed earlier. His steep career ended abruptly with his early death. He was on his way to his dacha when he was run over by a train in the Tsarskoe Selo station. On July 19, 1878 he died from blood poisoning. *Wik

1890 Sir William Lawrence Bragg (31 Mar 1890; 1 Jul 1971 at age 81) was an Australian-English physicist and X-ray crystallographer who at the early age of 25, shared the Nobel Prize for Physics in 1915 (with his father, Sir William Bragg). Lawrence Bragg formulated the Bragg law of X-ray diffraction, which is basic for the determination of crystal structure: nλ = 2dsinθ which relates the wavelength of x-rays, λ, the angle of incidence on a crystal, θ, and the spacing of crystal planes, d, for x-ray diffraction, where n is an integer (1, 2, 3, etc.). Together, the Braggs worked out the crystal structures of a number of substances. Early in this work, they showed that sodium chloride does not have individual molecules in the solid, but is an array of sodium and chloride ions. *TIS

1906 Shin'ichiro Tomonaga (31 Mar 1906; 8 Jul 1979 at age 73)Japanese physicist who shared the Nobel Prize for Physics in 1965 (with Richard P. Feynman and Julian S. Schwinger of the U.S.) for independently developing basic principles of quantum electrodynamics. He was one of the first to apply quantum theory to subatomic particles with very high energies. Tomonaga began with an analysis of intermediate coupling - the idea that interactions between two particles take place through the exchange of a third (virtual particle), like one ship affecting another by firing a cannonball. He used this concept to develop a quantum field theory (1941-43) that was consistent with the theory of special relativity. WW II delayed news of his work. Meanwhile, Feynman and Schwinger published their own independent solutions.*TIS

DEATHS
1624 Joao Baptista Lavanha (1550 in Portugal - 31 March 1624 in Madrid, Spain)Lavanha is said to have studied in Rome. He was appointed by Philip II of Spain to be professor of mathematics in Madrid in 1582.
Philip had sent the Duke of Alba with an army to conquer Portugal in 1580 and soon realized that Portugal was more advanced in studies of navigation than Spain. In an attempt to correct this, Philip founded an Academy of Mathematics in Madrid with Lavanha as its first professor.
From 1587 Lavanha became chief engineer to Philip II. He was appointed cosmographer to the king in 1596 and about the same time he moved to Lisbon where he taught mathematics to sailors and navigators.
Lavanha is best known for his contributions to navigation. His book Regimento nautico gives rules for determining latitude and tables of declination of the Sun. He also worked on maps, producing some interesting new ideas. He produced a map of Aragon in about 1615. Among his publications was a translation of Euclid.
Lavanha also studied instruments used in navigation, constructing astrolabes, quadrants and compasses. *SAU

1726/7 Isaac Newton (25 December 1642 – 20 March 1727 [NS: 4 January 1643 – 31 March 1727) English physicist and mathematician, who made seminal discoveries in several areas of science, and was the leading scientist of his era. His study of optics included using a prism to show white light could be split into a spectrum of colors. The statement of his three laws of motion are fundamental in the study of mechanics. He was the first to describe the moon as falling (in a circle around the earth) under the same influence of gravity as a falling apple, embodied in his law of universal gravitation. As a mathematician, he devised infinitesimal calculus to make the calculations needed in his studies, which he published in Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, 1687)*TIS
Newton died intestate. Immediately his relatives began to quarrel over the division of his estate, which amounted to a considerable fortune. Thomas Pellet examined Newton’s manuscript holdings in hopes of turning a quick proﬁt. His “thick clumsy annotations ‘Not ﬁt to be printed,’ now seem at once pitiful and ludicrous.” See Whiteside, Newton Works, I, xvii ﬀ for details. *VFR

1776 John Bird (1709 – March 31, 1776), the well known mathematical instrument maker, was born at Bishop Auckland. He worked in London for Jonathan Sisson, and by 1745 he had his own business in the Strand. Bird was commissioned to make a brass quadrant 8 feet across for the Royal Observatory at Greenwich, where it is still preserved. Soon after, duplicates were ordered for France, Spain and Russia.

Bird supplied the astronomer James Bradley with further instruments of such quality that the commissioners of longitude paid him £500 (a huge sum) on condition that he take on a 7-year apprentice and produce in writing upon oath, a full account of his working methods. This was the origin of Bird's two treatises The Method of Dividing Mathematical Instruments (1767) and The Method of Constructing Mural Quadrants (1768). Both had a foreword from the astronomer-royal Nevil Maskelyne. When the Houses of Parliament burned down in 1834, the standard yards of 1758 and 1760, both constructed by Bird, were destroyed.

Bird was an early influence in the life of Jerimiah Dixon, and in all probability it was he who recommended Dixon as a suitable companion to accompany Mason. *Wik

1841 George Green (14 Jul 1793, 31 Mar 1841 at age 47) was an English mathematician, born near Nottingham, who was first to attempt to formulate a mathematical theory of electricity and magnetism. He was a baker while, remarkably, he became a self-taught mathematician. He became an undergraduate at Cambridge in October 1833 at the age of 40. Lord Kelvin (William Thomson) subsequently saw, was excited by the Essay. Through Thomson, Maxwell, and others, the general mathematical theory of potential developed by an obscure, self-taught miller's son heralded the beginning of modern mathematical theories of electricity.*TIS His most famous work, An Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnetism was published, by subscription, in March 1828. Most of the ﬁfty-two subscribers were friends and patrons. The work lay unnoticed until William Thomson rediscovered it and showed it to Liouville and Sturm in Paris in 1845. The Theory of Potential it developed led to the modern mathematical theory of electicity. *VFR

1877 Antoine-Augustin Cournot (28 Aug 1801; 31 Mar 1877) French economist and mathematician, who was the first economist who applied mathematics to the treatment of economic questions. In 1838, he published Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth) which was a treatment of mathematical economics. In particular, he considered the supply-and-demand functions. Further, he studied the conditions for equilibrium with monopoly, duopoly and perfect competition. He included the effect of taxes, treated as changes in production costs, and discussed problems of international trade. His definition of a market is still the basis for that presently used in economics. In other work, he applied probability to legal statistics *TIS

1997 Friedrich (Hermann) Hund (4 Feb 1896 - 31 Mar 1997) was a German physicist known for his work on the electronic structure of atoms and molecules. He introduced a method of using molecular orbitals to determine the electronic structure of molecules and chemical bond formation. His empirical Hund's Rules (1925) for atomic spectra determine the lowest energy level for two electrons having the same n and l quantum numbers in a many-electron atom. The lowest energy state has the maximum multiplicity consistent with the Pauli exclusion principle. The lowest energy state has the maximum total electron orbital angular momentum quantum number, consistent with rule. They are explained by the quantum theory of atoms by calculations involving the repulsion between two electrons. *TIS

Harold Scott MacDonald Coxeter (9 Feb 1907 in London, England - 31 March 2003 in Toronto, Canada) graduated from Cambridge and worked most of his life in Canada. His work was mainly in geometry. In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics. Among his most famous geometry books are The real projective plane (1955), Introduction to geometry (1961), Regular polytopes (1963), Non-euclidean geometry (1965) and, written jointly with S L Greitzer, Geometry revisited (1967). He also published a famous work on group presentations, which was written jointly with his first doctoral student W O J Moser, Generators and relations for discrete groups.
His 12 books and 167 published articles cover more than mathematical research. Coxeter met Escher in 1954 and the two became lifelong friends. Another friend, R Buckminister Fuller, used Coxeter's ideas in his architecture. In 1938 Coxeter revised and updated Rouse Ball's Mathematical recreations and essays, a book which Rouse Ball first published in 1892. *SAU

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 & Campbel