## Tuesday, 5 April 2016

### On This Day in Math - April 5

How dare we speak of the laws of chance? Is not chance the antithesis of all law?
~Joseph Bertrand

The 96th day of the year; 96 is the smallest number that can be written as the difference of 2 squares in 4 ways. *What's So Special About This Number?
(students are encouraged to find them all...Is there a smaller number that can be so expressed in 3 ways?)

The sum of 96 consecutive squared integers is a square number ( $x^2 + (x+1)^2 + (x+2)^2 +(x+3)^2 + \dotsm + (x+95)^2 = y^2$ ) can be solved with eight sets of 96 consecutive year days. One solution is $13^2 + 14^2 + \dotsm + 108^2 = 652^2$ *Ben Vitale

1/4 of the integers less than 96 are prime, The smallest integer n for which (n) = n/4.

96 is also the difference of consecutive factorials, and the difference of two powers of two.

EVENTS
1610 Writing to Galileo, Kepler was impressed by the observation that stars seen through the telescope still sparkled, in contrast to the circular appearance of planets. He asked:
"What other conclusion shall we draw from this difference, Galileo, than that the fixed stars generate their light from within, whereas the planets, being opaque, are illuminated from without; that is, to use Bruno’s terms, the former are suns, the latter, moons, or earths?"
*Steven Soter, Ciclops.org

1752 Taxes are due in England. Previously they were due on March 25, the ﬁrst day of the year, but because the adoption of the Gregorian calendar reform necessitated the dropping of eleven days, the tax date was changed also. Apparently the tax collectors couldn’t do fractions. *VFR

In 1753, the British Museum was founded by an Act of Parliament granting £20,000 to purchase the 50,000 volume library of Sir Hans Sloane and his vast collection of 69,352 items of nature and art. Sloane was a prominent London physician who made the collection available in his will at much below its intrinsic value. Montagu House, Bloomsbury, was purchased in 1754 by the government to house this and other collections. Since it opened, on 15 Jan 1759, the Museum has been collecting, conserving and studying millions of artefacts. The British Museum established its Research Laboratory in 1920 with the appointment of Dr Alexander Scott as its first scientist.*TIS The myth that Sloan had invented the process of Hot Chocolate, which is still strongly promoted in the shops in Chelsea that feature this product, is a myth. See James Delbourgo's Article on Sloan and Cocoa here. Skipping to page 78 for details of the history of Chocolate in use around Europe in the 17th century. It had been used for much longer by the natives of South America with some apparent religious or spiritual relationship.

1792 George Washington cast the ﬁrst presidential veto in the USA. Amazingly, mathematics was involved. It seemed so easy. The 1787 US Constitution laid out simple rules for deciding how many representatives each state shall receive:
"Representatives and direct taxes shall be apportioned among the several States which may be included within this Union, according to their respective numbers, ... The number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative ...". It may have seemed easy, but for the 200+ years of US government, the question of "Who gets how many?" continues to perplex and promote controversy. When congress discussed mathematical methods of applying this constitutional directive there were two methods of prime consideration, Jefferson's method, and Hamilton's method. Congress selected Hamilton's method and in the first use of the Presidential veto . President Washington rejected the bill. Congress submitted and passed another bill using Jefferson's method. The method used has changed frequently over the years with a method by Daniel Webster adopted in 1842, (the original 65 Representatives had grown to 223) and then replaced with Hamilton's method in 1852 (234 Representatives). In a strange "Only in America" moment in 1872, the congress reapportioned without actually adopting an official method and some analysis suggest that the difference caused Rutherford Hayes to win instead of Samuel Tilden who would have won had Hamilton's method been used. Since 1931 the US House has had 435 Representatives with the brief exception of when Alaska and Hawaii became states. Then there was a temporary addition of one seat for each until the new apportionment after the 1960 Census. In 1941 the Huntington-Hill Method was adopted and has remained in continuous (and contentious) use ever since.(Pat B)

Chaotic Elections! A Mathematician Looks at Voting

1800 A UFO sighting near Baton Rouge, Louisiana will be reported to the American Philosophical Society by Thomas Jefferson, President of the society, and (at that time) Vice-President of the United States. The report of a UFO by a Vice-President is still the highest government official to report a UFO. The report itself was written by the naturalist William Dunbar: "A phenomenon was seen to pass Baton Rouge on the night of the 5th April 1800, of which the following is the best description I have been able to obtain. It was first seen in the South West, and moved so rapidly, passing over the heads of the spectators, as to disappear in the North East in about a quarter of a minute. It appeared to be of the size of a large house, 70 or 80 feet long"

In 1881, Hermann von Helmholtz presented The Faraday Lecture before the Fellows of the Chemical Society in London. His topic was The Modern Development of Faraday's Conception of Electricity. Helmholtz recognized Michael Faraday as being the person who most advanced the general scientific method, saying “His principal aim was to express in his new conceptions only facts, with the least possible use of hypothetical substances and forces.” *TIS

1893, Thomas Corwin Mendenhall, then Superintendent of Weights and Measures, with the approval of the Secretary of the Treasury, decided that the International Meter and Kilogram would in the future be regarded as the fundamental standards of length and mass in the United States, both for metric and customary weights and measures. This decision, which has come to be known as "The Mendenhall Order," was first published as Bulletin No. 26 of the Coast and Geodetic Survey under the title Fundamental Standards of Length and Mass. The Mendenhall Order initiated a departure from the previous policy of attempting to maintain our standards of length and mass to be identical with those of Great Britain.*TIS (And after all this time we have completely converted to metric ;-] )

1955 On the 5th of April, 1955, Nobel laureate Bertrand Russell sent a following letter to Albert Einstein along with a rough draft of what would soon be known as the Russell-Einstein Manifesto - a written warning to the world's population on the dangers of nuclear weapons, and a plea for all leaders to avoid war when faced with conflict - and asked him to be both a signatory and supporter. Einstein's short reply, and in fact the last letter he ever wrote, arrived a week later:

Dear Bertrand Russell,
Thank you for your letter of April 5. I am gladly willing to sign your excellent statement. I also agree with your choice of the prospective signers.

With kind regards,

A. Einstein.

Einstein passed away on the 18th of that month, and the manifesto was released to the public on July 9th.
* Shaun Usher, Letters of Note Web site

In 1963, the U.S. Atomic Energy Commission gave the Fermi Award to J. Robert Oppenheimer for research in nuclear energy. Oppenheimer was the chief scientist of the Manhattan Project during WWII that created the atomic bomb. Later, he opposed the more destructive hydrogen bomb development and his security clearance was revoked (1954). Nine years later, a wiser U.S. government awarded Oppenheimer the prestigious Fermi Award, "For contributions to theoretical physics as a teacher and originator of ideas, and for leadership of the Los Alamos Laboratory and the atomic energy program during critical years." The actual presentation of the medal and $50,000 was made 2 Dec 1963 by President Lyndon B. Johnson. *TIS American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer BIRTHS 1588- Thomas Hobbes(5 April 1588 – 4 December 1679) was an English scholar and amateur mathematician who wrote on optics and on geometry. He attacked the 'new' methods of mathematical analysis. Hobbes was caught in a series of conflicts from the time of publishing his De Corpore in 1655. In Leviathan he had assailed the system of the original universities. Because Hobbes was so evidently opposed to the existing academic arrangements, and because De Corpore contained not only tendentious views on mathematics, but an unacceptable proof of the squaring of the circle (which was apparently an afterthought), mathematicians took him to be a target for polemics. John Wallis was not the first such opponent, but he tenaciously pursued Hobbes. The resulting controversy continued well into the 1670s. *Wik 1607 Honor´e Fabri, or Honoratus Fabrius,(1607 in Ain, France; 8 March 1688 at Rome,) He developed the inﬁnitesimal methods of Cavalieri and Torricelli and his quadrature of the cycloid inspired Leibniz. Some of his geometrical work boils down to special cases of xn sin x dx, sinn x dx and arcsin x dx dy. [DSB 4, 506] *VFR In his treatise on man he claims to have discovered the circulation of the blood, prior to William Harvey, but after having investigated this question, Father Auguste Bellynck arrives at the conclusion that, at best, Father Fabri may have made the discovery independently of Harvey. *Wik 1622 – Vincenzo Viviani,(April 5, 1622 – September 22, 1703) Italian mathematician In 1639, at the age of 17, he was an assistant of Galileo Galilei in Arcetri. He remained a disciple until Galileo's death in 1642. From 1655 to 1656, Viviani edited the first edition of Galileo's collected works. He was a leader in his field and founded the Accademia del Cimento. As one of the first important scientific societies, this organization came before England's Royal Society. In 1639, at age 17, he became the student, secretary and assistant of Galileo (now blind) in Arcetri, until Galileo died in 1642. A note from Thony Christie informed me that after Galileo's death, his papers were being used by the local butcher to wrap his meat and sausages until Viviani rescued what was left of them. During his long career, Viviani published a number of books on mathematical and scientific subjects. He edited the first edition of Galileo's collected works (1655-1656), and worked tirelessly to have his master's memory rehabilitated. In 1660, together with Borelli, he measured the velocity of sound by timing the difference between the flash and the sound of a cannon. They obtained the value of 350 metres per second.*TIS Viviani's Theorem is named for him. The theorem states that in an equilateral triangle, the sum of the perpendicular distances to the sides is equal to the altitude of the triangle. In the figure h=PE+PF+PG. If the point is outside the triangle, the relationship will still hold if one or more of the perpendiculars is treated as a negative value. The theorem can be generalized to a regular n-gon to state, for any point P interior to a regular n-gon, the sum of the perpendicular distances to the n sides is n times the apothem of the figure. 1877 Georg Faber (5 April 1877 in Kaiserslautern, Germany - 7 March 1966 in Munich, Germany) Faber's most important work was on the polynomial expansion of functions. This is the problem of expanding an analytical function in an area bounded by a smooth curve as a sum of polynomials, where the polynomials are determined by the area. These polynomials are now known as 'Faber polynomials' and first appear in Faber's 1903 paper Über polynomische Entwickelungen published in Mathematische Annalen. Another important paper which he also published in Mathematische Annalen, this time in 1909, was Über stetige Funktionen. In this paper he introduced the 'hierarchical basis' and explicitly used it for the representation of functions. In fact Faber was building on the idea of Archimedes who computed approximately using a hierarchy of polygonal approximations of a circle. Only in the 1980s was Faber's idea seen to be an important ingredient for the efficient solution of partial differential equations. One further achievement of Faber is worthy of mention. In 1894 Lord Rayleigh made the following claim:" ... given a fixed area of ox-hide to make a drum, the ground tone is lowest if you make your drum circular. " Two mathematicians independently verified Rayleigh's conjecture, Faber and Edgar Krahn. *SAU 1901 Subbayya Sivasankaranarayana Pillai (April 5, 1901 Nagercoil, Tamil Nadu - 31 August 1950, Cairo, Egypt) was an Nagercoil native Indian mathematician specializing in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan". In number theory, a Pillai prime, named for him, is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, $n! \equiv -1 \mod p but p \not\equiv 1 \mod n$. The first few Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... (sequence A063980 in OEIS). *Wik 1911 Computer Pioneer Cuthbert Hurd Is Born: The mathematician son of an itinerant preacher, IBM President Thomas Watson Sr. hired Hurd in early 1949 as IBM's second Ph.D. A figure generally unknown to history, Hurd quietly encouraged IBM upper management to enter into the computer field, convincing them in the early 1950s that a market for scientific computers existed after a cross-country sales trip revealed pent-up demand. At the time, IBM enjoyed large profits from its traditional punch card accounting business so the change was difficult for IBM to make internally. Hurd's first great success was in selling 10 of IBM's 701 computers, its first commercial scientific machine which rented for$18,000 a month. Shortly thereafter, he became manager of the IBM team that invented and developed the FORTRAN programming language under John Backus. Hurd died on May 22, 1996 in Portola Valley, California.*CHM

1911 Walter Warwick Sawyer (or W. W. Sawyer) (April 5,1911–February 15, 2008) was a mathematician, mathematics educator and author, who taught on several continents.
Born in London, England , he attended Highgate School and was an undergraduate at St. John's College, Cambridge, obtaining a BA in 1933. He was an assistant lecturer in mathematics from 1933 to 1937 at University College, Dundee and from 1937 to 1944 at Manchester University. From 1945 to 1947, he was the head of mathematics at Leicester College of Technology.
In 1948 W. W. Sawyer became the first head of the mathematics department of what is now the University of Ghana. From 1951 to 1956, he was at Canterbury College (now the University of Canterbury in New Zealand). He left Canterbury College to become an associate professor at the University of Illinois, where he worked from winter 1957 through June 1958. While there, he criticized the New Math movement, which included the people who had hired him. From 1958 to 1965, he was a professor of mathematics at Wesleyan University. In the fall of 1965 he became a professor at the University of Toronto, appointed to both the College of Education and the Department of Mathematics. He retired in 1976.
W. W. Sawyer was the author of some 11 books. He is probably best known for his semi-popular works Mathematicians Delight and Prelude to Mathematics. Both of these have been translated into many languages. Mathematician's Delight was still in print 65 years after it was written. Some mathematicians have credited these books with helping to inspire their choice of a career.
W.W. Sawyer died on February 15, 2008, at the age of 96. He was survived by a daughter, Anne. *Wik
His first book "Mathematician’s Delight" (1943), was written with the aim "to dispel the fear of mathematics." It is one of the most successful math book ever written, going through numerous editions, translations into 10 languages, and selling more than 500,000 copies.

My favorite Sawyer quote:
Complete success would mean that every individual felt,
"I enjoyed the mathematics that I had time to learn.
If I ever need or want to learn some more,
I shall not be afraid to do so."

DEATHS
1678 - Claude Hardy (1598 in Le Mans, France- 5 April 1678 in Paris, France) was a French lawyer and amateur mathematician who made Latin translations of some of Euclid's work. A translation into French of Viète's book on algebra, originally written in Latin, appeared around 1630 with Antoine Vasset as the translator. It is believed that "Antoine Vasset" was a pseudonym for Claude Hardy. In 1630, under his own name, Hardy published Examen and in 1638 he published Refutation. These works dealt with the problem of the duplication of the cube*SAU

1684 William Brouncker (1620 – 5 April 1684) He was the King’s nominee and ﬁrst president of the Royal Society of London (1666–1677). His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation of the natural logarithm function by infinite series. He was the first European to solve what is now known as Pell's equation. He was the first in England to take interest in generalized continued fractions and, following the work of John Wallis, he provided development in the generalized continued fraction of pi *Wik
In 1656 he gave the continued fraction expansion

. . . and used it to calculate π correct to ten decimal places. *VFR

1861 Ferdinand Joachimsthal (9 March 1818 in Goldberg, Prussian Silesia (now Złotoryja, Poland) - 5 April 1861 in Breslau, Germany (now Wrocław, Poland)) Influenced by the work of Jacobi, Dirichlet and Steiner, Joachimsthal wrote on the theory of surfaces where he made substantial contributions, particularly to the problem of normals to conic sections and second degree surfaces. *SAU

1900 Joseph-Louis-François Bertrand (11 Mar 1822; 5 Apr 1900 at age 78) was a French mathematician and educator and educator remembered for his elegant applications of differential equations to analytical mechanics, particularly in thermodynamics, and for his work on statistical probability and the theory of curves and surfaces. In 1845 Bertrand conjectured that there is at least one prime between n and (2n-2) for every n greater than 3, as proved five years later by Chebyshev. It is not clear to me if he was the one who suggested the jingle
I've told you once and I'll tell you again
There's always a prime between n and 2n.
In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French. He wrote a number of notes on the reduction of data from observations. *TIS At age 11 he started to attend classes at the Ecole Polytechnique, where his Uncle Duhamel was a well-known professor of mathematics. At 17 he received his doctor of science degree. *VFR
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be strictly ahead of B throughout the count?" which Bertrand asked, and proved in 1887 in Comptes Rendus de l'Académie des Sciences.
The answer is   $\frac{p-q}{p+q}.$

2004 – Heiner Zieschang (12 November 1936 in Kiel – 5 April 2004) was a German mathematician. He was a professor at Ruhr University in Bochum from 1968 till 2002. He was a topologist. In 1996 he was an honorary doctor of University of Toulouse and in 1997 he was an honorary professor of Moscow State University.

2009 Irving John ("I.J."; "Jack") Good (9 December 1916 – 5 April 2009) was a British mathematician who worked as a cryptologist at Bletchley Park with Alan Turing. After World War II, Good continued to work with Turing on the design of computers and Bayesian statistics at the University of Manchester. Good moved to the United States where he was professor at Virginia Tech.
He was born Isadore Jacob Gudak to a Polish-Jewish family in London. He later anglicized his name to Irving John Good and signed his publications "I. J. Good."
An originator of the concept now known as "technological singularity," Good served as consultant on supercomputers to Stanley Kubrick, director of the 1968 film 2001: A Space Odyssey. Good's published work ran to over three million words. He was known for his work on Bayesian statistics. He published a number of books on probability theory. In 1958 he published an early version of what later became known as the Fast Fourier Transform but in a journal so obscure that it never became widely known.*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