Saturday, 31 May 2014

On THis Day in Math - May 31

Geometry is the science of 
correct reasoning on incorrect figures.
~George Polya

The 151st day of the year; The smallest prime that begins a 3-run of sums of 5 consecutive primes: 151 + 157 + 163 + 167 + 173 = 811; and 811 + 821 + 823 + 827 + 829 = 4111; and 4111 + 4127 + 4129 + 4133 + 4139 = 20639. *Prime Curios... Can you find the smallest 4-run example?
151 is also the mean (and median) of the first five three digit palindromic primes, 101, 131, 151, 181, 191
Since this is also 5/31 I should point out that 5!+31 = 151,  Thanks to Derek Orr, who also pointed out that any day in May (in non-leap year) 5/d is such that 5! + d = year day

1503 Copernicus received a doctoral degree in canon law from the University of Ferrara. *VFR

1676 Antonie van Leeuwenhoek describes the little animals he sees through a microscope. "The 31th of May, I perceived in the same water more of those Animals, as also some that were somewhat bigger. And I imagine, that [ten hundred thousand] of these little Creatures do not equal an ordinary grain of Sand in bigness: And comparing them with a Cheese-mite (which may be seen to move with the naked eye) I make the proportion of one of these small Water-creatures to a Cheese-mite, to be like that of a Bee to a Horse: For, the circumference of one of these little Animals in water, is not so big as the thickness of a hair in a Cheese-mite." *The Collected Letters of Antoni van Leeuwenhoek (1957), Vol. 2, 75.

1753 A View of the Relation between the Celebrated. Dr. Halley's Tables, and the Notions of Mr. De Buffon, for Establishing a Rule for the Probable Duration of the Life of Man; By Mr. William Kersseboom, of the Hague. Translated from the French, by James Parsons, M. D. and F. R. S. read by the Royal Society on May 31.  

1764 “I went this far with him: ‘Sir, allow me to ask you one question. If the Church should say to you, ‘two and three make ten,’ what would you do? ‘Sir,’ said he, ‘I should believe it, and I should count like this: one, two, three, four, ten.’ I was now fully satisfied.” From Boswell’s Journal as quoted by J. Gallian, Contemporary Abstract Algebra, p. 43. *VFR  (Now you know, It was Boswell who invented Base Five... )

1790 US Copyright law passed. *VFR

1796 Gauss records in his diary a prime number theorem conjecture. Clifford Pickover, in “The Math Book”, points out that 1796 was “an auspicious year for Gauss, when his ideas gushed like a fountain from a fire hose.” In addition to the construction of the 17-gon in March, and the prime number theorem conjecture, he proved that every positive number could be expressed as the sum of (at most) three triangular numbers in July, and another about solutions of polynomials in October.
On May 31 he conjectured that π(n), the number of primes less than n is approximated (for large n) by the area under the logarithmic integral (from 2 to n I assume).
Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in the same year that π(x) is approximated by the function x/(ln(x)-1.08),. Gauss considered the same question and he came up with his own approximating function, the logarithmic integral li(x), although he did not publish his results. Both Legendre's and Gauss' formulas imply the same conjectured asymptotic equivalence of π(x) = x / ln(x), although Gauss' approximation is closer in terms of the differences instead of quotients.
Most teachers tell the story of Gauss as a nine-year old summing the digits from 1 to 100 in his head. Here is another nice Gauss anecdote about his ability to do mental calculations: Once, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, "I used logarithms." The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. "Look them up?" Gauss responded. "Who needs to look them up? I just calculate them in my head!"

1813 Louis Poinsot elected to the mathematics section of of the French Acad´emie des Sciences, replacing Lagrange. [DSB 11, 61] *VFR Although little known today, he was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple. When Gustave Eiffel built the famous tower, he included the names of 72 prominent French scientists on plaques around the first stage, Poinsot included.*Wik

1823 In a letter to a cousin, William Rowan Hamilton disclosed that he had made a “very curious discovery.” It is believed that he was referring to the characteristic function. [Thanks to Howard Eves] *VFR

1868 During the eclipse of 18 August 1868 from the Red Sea through India to Malaysia and New
Guinea, prominences are first studied with spectroscopes and shown to be composed primarily of hydrogen by James Francis Tennant, John Herschel, George Rayet, Norman Pogson
and others. *NSEC

1975 “I had today my virginal experience with the HP [Hewlett-Packard 65 calculator] as a celestial triangle-breaker ... it worked! But I’ll keep plotting the sun to make sure.” William F. Buckley Jr. discussing celestial navigation in his delightful book, Airborn, a Sentimental Journey, about sailing. His caution was justified, for later he learned that the prepackaged program contained errors. *VFR

1985 Marion Tinsley retains the world checker championship by defeating Asa Long 6–1. The one game Long won was the first time in nearly 25 years that anyone has beaten Tinsley in a checkers game. But then perhaps Tinsley had an unfair advantage—a Ph.D. in mathematics from Ohio State with a dissertation in combinatorics directed by Herbert Ryser. [Clipping of June 2, 1985] *VFR
He is considered the greatest checkers player who ever lived. He was world champion from 1955–1958 and 1975–1991. Tinsley never lost a World Championship match, and lost only seven games (two of them to the Chinook computer program) in his entire 45 year career.[1] He withdrew from championship play during the years 1958–1975, relinquishing the title during that time. (anyone know why?) Tinsley retired from championship play in 1991. In August 1992, he defeated the Chinook computer program 4–2 (with 33 draws) in a match. Chinook had placed second at the U.S. Nationals in 1990, which usually qualifies one to compete for a national title. However, the American Checkers Federation and the English Draughts Association refused to allow a computer to play for the title. Unable to appeal their decision, Tinsley resigned his title as World Champion and immediately indicated his desire to play against Chinook. The unofficial yet highly publicized match was quickly organized, and was won by Tinsley.
In one game, Chinook, playing with white pieces, made a mistake on the tenth move. Tinsley remarked, "You're going to regret that." Chinook resigned after move 36, fully 26 moves later.[2] The ACF and the EDA were placed in the awkward position of naming a new world champion, a title which would be worthless as long as Tinsley was alive. They granted Tinsley the title of World Champion Emeritus as a solution.
In August 1994, a second match with Chinook was organized, but Tinsley withdrew after only six games (all draws) for health reasons. Don Lafferty, rated the number two player in the world at the time, replaced Tinsley and fought Chinook to a draw. Tinsley was diagnosed with pancreatic cancer a week later. Seven months later, he died. *Wik

1683 Jean-Pierre Christin (May 31, 1683 – January 19, 1755) was a French physicist, mathematician, astronomer and musician. His proposal to reverse the Celsius thermometer scale (from water boiling at 0 degrees and ice melting at 100 degrees, to water boiling at 100 degrees and ice melting at 0 degrees) was widely accepted and is still in use today.
Christin was born in Lyon. He was a founding member of the Académie des sciences, belles-lettres et arts de Lyon and served as its Permanent Secretary from 1713 until 1755. His thermometer was known in France before the Revolution as the thermometer of Lyon. *Wik

1872 Charles Greeley Abbot (31 May 1872; 17 Dec 1973 at age 101) was an American astrophysicist who is thought to have been the first scientist to suspect that the radiation of the Sun might vary over time. In 1906, Abbot became director of the Smithsonian Astrophysical Observatory and, in 1928, fifth Secretary of the Smithsonian. To study the Sun, SAO established a network of solar radiation observatories around the world-- usually at remote and desolate spots chosen primarily for their high percentage of sunny days. Beginning in May 1905 and continuing over decades, his studies of solar radiation led him to discover, in 1953, a connection between solar variations and weather on Earth, allowing general weather patterns to be predicted up to 50 years ahead. *TIS

1912 Martin Schwarzschild (31 May 1912; 10 Apr 1997 at age 84) German-born American astronomer who in 1957 introduced the use of high-altitude hot-air balloons to carry scientific instruments and photographic equipment into the stratosphere for solar research.*TIS

1926 John Kemeney  (May 31, 1926 – December 26, 1992) born in Budapest, Hungary. He worked on logic with Alonzo Church at Princeton, was Einstein’s assistant at the IAS, developed the computer language BASIC, and served as President of Dartmouth College. To learn more about him, see the interview in Mathematical People. Profiles and Interviews (1985), edited by Donald J. Albers and G. L. Alexanderson. *VFR
  In his 66-year life, Kemeny had a significant impact on the history of computers, particularly during his years at Dartmouth College, where he worked with Thomas Kurtz to create BASIC, an easy-to-use programming language for his computer students. Kemeny earlier had worked with John von Neumann in Los Alamos, N.M., during the Manhattan Project years of World War II. *CHM

1930 Ronald Valentine Toomer (31 May 1930; 26 Sep 2011 at age 81) was an American engineer who was a legendary creator of steel roller coasters. His early career, was in the aerospace industry, where he helped design the heat shield for Apollo spacecraft and was also involved with NASA's first satellite launches. In 1965, he joined the Arrow Development Company to apply tubular steel technology to the design the Runaway Mine Ride, the world's first all-steel roller coaster. It opened the following year at Six Flags over Texas. By 1975, he designed the Roaring 20's Corkscrew for Knott's Berry Farm, introducing first 360° looping rolls, in fact two of them. Later, his design included seven inversions in the Shockwave roller coaster for Six Flags Great America. He produced over 80 roller coasters before 1998. *TIS

1931 John Robert Schrieffer (31 May 1931; Oak Park, Illinois,USA- )John Robert Schrieffer is an American physicist who shared (with John Bardeen and Leon N. Cooper) the 1972 Nobel Prize for Physics for developing the BCS theory (for their initials), the first successful microscopic theory of superconductivity. Although first described by Kamerlingh Onnes (1911), no theoretical explanation had been accepted. It explains how certain metals and alloys lose all resistance to electrical current at extremely low temperatures. The insight of the BCS theory is that at very low temperatures, under certain conditions, electrons can form bound pairs (Cooper pairs). This pair of electrons acts as a single particle in superconductivity. Schrieffer continued to focus his research on particle physics, metal impurities, spin fluctuations, and chemisorption. *TIS

1832 Evariste Galois (25 October 1811 – 31 May 1832) died of peritonitis from a gunshot wound of the previous day. He died in the Cochin Hospital – this is now at 27 Rue du Faubourg St. Jacques,in the 14th district of Paris. He was buried in a common grave at Montparnasse Cemetery, but no trace of the grave remains.

1841 George Green (14 July 1793 – 31 May 1841) British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Green, 1828).[1] The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. George Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work ran parallel to that of the great mathematician Gauss (potential theory).

Green's life story is remarkable in that he was almost entirely self-taught. He was born and lived for most of his life in the English town of Sneinton, Nottinghamshire, nowadays part of the city of Nottingham. His father (also named George) was a baker who had built and owned a brick windmill used to grind grain. The younger Green only had about one year of formal schooling as a child, between the ages of 8 and 9.
Self taught at a reading library while working full time as the manager of the family mill, He wrote a pivotal paper in applied calculus. George Green is buried in the family grave in the north east corner of St Stephens churchyard, just across the road from Green's Mill and car park. After his death the plaque below was placed in Westminster Abbey near the memorial to Newton. There are also memorials to Faraday, and Lord Kelvin. The Green Family mill has been completely restored and is now a Science center.

1931 Eugène Maurice Pierre Cosserat (4 March 1866 in Amiens, France - 31 May 1931 in Toulouse, France) Cosserat studied the deformation of surfaces which led him to a theory of elasticity. *SAU

1986 (Leo) James Rainwater (9 Dec 1917, 31 May 1986 at age 68)was an American physicist who won a share of the Nobel Prize for Physics in 1975 for his part in determining the asymmetrical shapes of certain atomic nuclei. During WW II, Rainwater worked on the Manhattan Project to develop the atomic bomb. In 1949 he began formulating a theory that not all atomic nuclei are spherical, as was then generally believed. The theory was tested experimentally and confirmed by Danish physicists Aage N. Bohr(4th son of Niels Bohr) and Ben R. Mottelson. For their work the three scientists were awarded jointly the 1975 Nobel Prize for Physics. He also conducted valuable research on X rays and took part in Atomic Energy Commission and naval research projects. *TIS

1998 Michio Suzuki (October 2, 1926 – May 31, 1998) was a Japanese mathematician who studied group theory.
A Professor at the University of Illinois at Urbana-Champaign from 1953 until his death. Suzuki received his Ph.D in 1952 from the University of Tokyo, despite having moved to the United States the previous year. He was the first to attack the Burnside conjecture, that every finite non-abelian simple group has even order.
A notable achievement was his discovery in 1960 of the Suzuki groups, an infinite family of the only non-abelian simple groups whose order is not divisible by 3. The smallest, of order 29120, was the first simple group of order less than 1 million to be discovered since Dickson's list of 1900.
There is also a sporadic simple group called the Suzuki group, which he announced in 1968. The Tits ovoid is also referred to as the Suzuki ovoid. *Wik

2000 Erich Kähler (16 January 1906, Leipzig – 31 May 2000, Wedel) was a German
Kähler was born in Leipzig, and studied there.
As a mathematician he is known for a number of contributions: the Cartan–Kähler theorem on singular solutions of non-linear analytic differential systems; the idea of a Kähler metric on complex manifolds; and the Kähler differentials, which provide a purely algebraic theory and have generally been adopted in algebraic geometry. In all of these the theory of differential forms plays a part, and Kähler counts as a major developer of the theory from its formal genesis with Élie Cartan.
His earlier work was on celestial mechanics; and he was one of the forerunners of scheme theory, though his ideas on that were never widely adopted. *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

Friday, 30 May 2014

On This Day in Math - May 30

Mercury's spheres in Georg von Peuerbach's "Theoricae novae planetarum", 1542       w/hat tip to History of Astronomy@hist_astro

The best review of arithmetic 
consists in the study of algebra.
~Florian Cajori

The 150th day of the year; 150 is the largest gap between consecutive twin prime pairs less than a thousand. It occurs between {659, 661} and {809, 811}. *Prime Curios
A Poly divisible number is an n-digit number so that for the first digit is divisible by one, the first two digits are divisible by two, the first three digits are divisible by three, etc up to n. There are 150 three-digit poly divisible numbers. Hat tip to Derek Orr

1667 After much debate about the presence of a woman at a Royal Society meeting, the Duchess of Newcastle was allowed to observe a demonstration of a "experiments of colours", the "weighing of air in an exhausted receiver", and "the dissolving of flesh with a certain liquor of Mr. Boyle's suggesting." This was probably the first visit by a woman to the Royal Society. The Duchess, Margaret Cavendish, was a competent scientist in her own right. Her prolific writings in the nature of science earned her the nickname “Mad Madge”. I have a note from VFR that she was elected to FRS, but can not confirm, the note says "No other woman was elected FRS until 1945" .

1765 "Ms. Catherine Price, Daughter of the late Dr. Halley " was paid a sum of 100 Pounds for "causing to be delivered to the Commissioners of the Longitude, several of the said Dr. oHalley's manuscript papers, which... may lead to discoveries useful to navigation." *Derek Howse, Britain's Board of Longitude, the Finances

1832 Galois mortally wounded by a gunshot wound to the abdomen in a duel of honor. He was left for dead after the duel but a peasant took him to a hospital. *VFR

The infamous duel with Pescheux d'Herbinville took place near the Glassier pond in the southern suburb of Gentilly. The duel was over Galois's involvement with Stéphanie-Félicie Poterine du Motel, who was d'Herbinville's fiancée, but it has been claimed that the affair was a political frame-up by government agents in order to eliminate Galois He died in the Cochin Hospital – this is now at 27 Rue du Faubourg St. Jacques, 14e, but I don't know how long it has been there. He was buried in a common grave at Montparnasse Cemetery, but no trace of the grave remains.

1903 Minor planet (511) Davida Discovered 1903 May 30 by R. S. Dugan at Heidelberg. Named by the discoverer in honor of David P. Todd (1855-1939), professor of astronomy and director of the Amherst College Observatory (1881-1920). David Todd was the husband of Mabel Todd, who wrote books about solar eclipses. David has also a drawing of a painting of a solar eclipse in one of his books. *NSEC


1423 Georg von Peurbach (or Peuerbach) (May 30, 1423 in Peuerbach near Linz – April 8, 1461 in Vienna)  He worked on trigonometry astronomy, and was the teacher of Regiomontanus. *VFR
He promoted the use of Arabic numerals (introduced 250 years earlier in place of Roman numerals), especially in a table of sines he calculated with unprecedented accuracy. He died before this project was finished, and his pupil, Regiomontanus continued it until his own death. Peurbach was a follower of Ptolomy's astronomy. He insisted on the solid reality of the crystal spheres of the planets, going somewhat further than in Ptolomy's writings. He calculated tables of eclipses in Tabulae Ecclipsium,observed Halley's comet in Jun 1456 and the lunar eclipse of 3 Sep 1457 from a site near Vienna. Peurbach wrote on astronomy, his observations and devised astronomical instruments. *TIS  The Renaissance Mathematicus has a nice piece about Peurbach and his life... the kind of detail that comes from a passion for his subject.  Check it out. 

1800 Karl Wilhelm Feuerbach (30 May 1800 – 12 March 1834) born in Jena, Germany. His mathematical fame rests entirely on three papers. Most important was this contribution to Euclidean geometry: The circle which passes through the feet of the altitudes of a triangle touches all four of the circles which are tangent to the three sides; it is internally tangent to the inscribed circle and externally tangent to each of the circles which touches the sides of the triangle externally. *VFR

The circle is also commonly called the Nine-point circle. It passes through the feet of the altitudes, the midpoints of the three sides, and the point half way between the orthocenter and the vertices.

1814 Eugene Charles Catalan (30 May 1814 – 14 February 1894) was a Belgian mathematician who defined the numbers called after him, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. *SAU The Catalan numbers have a multitude of uses in combinatorics.

1889 Paul Ernest Klopsteg (May 30, 1889 – April 28, 1991) was an American physicist. The asteroid 3520 Klopsteg was named after him and the yearly Klopsteg Memorial Award was founded in his memory.
He performed ballistics research during World War I at the US Army's Aberdeen Proving Grounds in Maryland. He applied his knowledge of ballistics to the study of archery.
He was director of research at Northwestern University Technical Institution. From 1951 through 1958 he was an associate director of the National Science Foundation and was president of the American Association for the Advancement of Science from 1958 through 1959.*Wik

1908 Hannes Olof Gösta Alfvén (30 May 1908 in Norrköping, Sweden; 2 April 1995 in Djursholm, Sweden) Alfvén developed the theory of magnetohydrodynamics (MHD), the branch of physics that helps astrophysicists understand sunspot formation and the magnetic field-plasma interactions (now called Alfvén waves in his honor) taking place in the outer regions of the Sun and other stars. For this pioneering work and its applications to many areas of plasma physics, he shared the 1970 Nobel Prize in physics. *DEBORAH TODD AND JOSEPH A. ANGELO, JR., A TO Z OF SCIENTISTS IN SPACE AND ASTRONOMY

1778 (François Marie Arouet) Voltaire (21 November 1694 – 30 May 1778) was a French author who popularized Isaac Newton's work in France by arranging a translation of Principia Mathematica to which he added his own commentary (1737). The work of the translation was done by the marquise de Châtelet who was one of his mistresses, but Voltaire's commentary bridged the gap between non-scientists and Newton's ideas at a time in France when the pre-Newtonian views of Descartes were still prevalent. Although a philosopher, Voltaire advocated rational analysis. He died on the eve of the French Revolution.*TIS

1912 Wilbur Wright  (April 16, 1867 – May 30, 1912), American aviation pioneer, who with his brother Orville, invented the first powered airplane, Flyer, capable of sustained, controlled flight (17 Dec 1903). Orville made the first flight, airborn for 12-sec. Wilbur took the second flight, covering 853-ft (260-m) in 59 seconds. By 1905, they had improved the design, built and and made several long flights in Flyer III, which was the first fully practical airplane (1905), able to fly up to 38-min and travel 24 miles (39-km). Their Model A was produced in 1908, capable of flight for over two hours of flight. They sold considerable numbers, but European designers became strong competitors. After Wilbur died of typhoid in 1912, Orville sold his interest in the Wright Company in 1915 *TIS

1926 Vladimir Andreevich Steklov (9 January 1864 – 30 May 1926)  made many important contributions to applied mathematics. In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface. He wrote General Theory of Fundamental Functions in which he examined expansions of functions as series in an infinite system of orthogonal eigenfunctions. In fact the term "Fundamental Functions", which is due to Poincaré, means eigenfunctions in today's terminology.
Steklov was not the first to examine series expansions in terms of infinite sets of orthogonal eigenfunctions, of course Fourier had examined a special case of this situation many years before. Steklov, however, produced many papers on this topic which led him to a general theory to replace the special cases examined by others. He studied a generalisation of Parseval's equality for Fourier series to his general setting showing this to be a fundamental property. In all his list of publications contains 154 items. *SAU

1943 Anderson McKendrick (September 8, 1876 - May 30, 1943) trained as a medical doctor in Glasgow and came to Edinburgh as Superintendent of the College of Physicians Laboratory. He made some significant mathematical contributions to biology. *SAU

1964 Leo Szilard (11 Feb 1898; 30 May 1964 at age 66) Hungarian-American physicist who, with Enrico Fermi, designed the first nuclear reactor that sustained nuclear chain reaction (2 Dec 1942). In 1933, Szilard had left Nazi Germany for England. The same year he conceived the neutron chain reaction. Moving to N.Y. City in 1938, he conducted fission experiments at Columbia University. Aware of the danger of nuclear fission in the hands of the German government, he persuaded Albert Einstein to write to President Roosevelt, urging him to commission American development of atomic weapons. In 1943, Major General Leslie Groves, leader of the Manhattan Project designing the atomic bomb, forced Szilard to sell his atomic energy patent rights to the U.S. government. *TIS Frederik Pohl , talks about Szilard's epiphany about chain reactions in Chasing Science (pg 25),
".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."

1992 Antoni Zygmund (December 25, 1900 – May 30, 1992) Polish-born mathematician who created a major analysis research centre at Chicago, and recognized in 1986 for this with the National Medal for Science. In 1940, he escaped with his wife and son from German controlled Poland to the USA. He did much work in harmonic analysis, a statistical method for determining the amplitude and period of certain harmonic or wave components in a set of data with the aid of Fourier series. Such technique can be applied in various fields of science and technology, including natural phenomena such as sea tides. He also did major work in Fourier analysis and its application to partial differential equations. Zygmund's book Trigonometric Series (1935) is a classic, definitive work on the subject.

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

Thursday, 29 May 2014

On This Day in Math - May 29

No matter how correct a mathematical theorem
may appear to be, one ought never to be satisfied
that there was not something imperfect
about it until it also gives
the impression of being beautiful.
~ George Boole

The 149th day of the year; There are 149 ways to put 8 queens on a 7-by-7 chessboard so that each queen attacks exactly one other queen. *Prime Curios
also 149 = 62 + 72 + 82.

And Derek Orr noted that the sum of the digits of 149, \(1 + 4 + 9 = 14 = 1^2 + 2^2 + 3^2 \)

1733  Euler names the "Pell Equations" and gives a method of multiple solutions. "Euler’s first excursion into Pell’s equation was his 1732 paper E-29, bearing a title that translates
as “On the solution of problems of Diophantus about integer numbers.” The main result of this paper is to show how certain quadratic Diophantine equations can be reduced to the Pell equation. In particular, he shows that if we can find a solution to the Diophantine equation \(y^2 = an^2 + bn + c \) and we can find solutions to the Pell equation, \(q^2 = ap^2 +1\), then we can use the solutions to the Pell equation to construct more solutions to the original Diophantine equation. He also shows how to use two solutions to a Pell equation to construct more solutions, and notes that solutions to a Pell equation give good rational approximations for the square root of a.  (Ed Sandifer, Euler and Pell, How Euler Did It. MAA) .

1832 Almost certain that he would die in a duel the next day, Evariste Galois first wrote “Letter to all Republicans,” and then wrote to a friend (Auguste Chevalier) describing his mathematics. It ended: “Eventually there will be, I hope, some people who will find it profitable to decipher this mess.” [Burton, History of Mathematics, p. 322]. See Smith, Source Book, pp. 278–285 for the letter. *VFR

The appropriate pages from Smith's book is below:

1898 the heirs of Alfred Nobel sign a "reconciliation agreement" so that lawyers and accountants can execute his will. The will's major bequest was to create the Nobel Prizes, but first, there were disputes to be settled.*TIS

1919Proof of the general theory of relativity was observed during a total solar eclipse. São Tomé and Príncipe, officially the Democratic Republic of São Tomé and Príncipe, is a Portuguese-speaking island nation in the Gulf of Guinea, off the western equatorial coast of Central Africa. Príncipe was the site where astronomical observations of the total solar eclipse of 29 May 1919 confirmed Einstein's prediction of the curvature of light. The expedition was sponsored by the Royal Society and led by Sir Arthur Stanley Eddington. A solar eclipse permitted observation of the bending of starlight passing through the sun's gravitational field, as predicted by Einstein's theory of relativity. Separate expeditions of the Royal Astronomical Society travelled to Brazil and off the west coast of Africa. Both made measurements of the position of stars visible close to the sun during a solar eclipse. These observations showed that, indeed, the light of stars was bent as it passed through the gravitational field of the sun. The verification of predictions of Einstein's theory, proved during the solar eclipse was a dramatic landmark scientific event. *Wik

1957 Romania issued two stamps picturing a slide rule to publicize the 2nd Congress of the Society of Engineers and Technicians, which began in Bucharest on this day. [Scott #1159-60].
For the younger set... If you never used (saw) a slide rule, there is actually an online java app that you can simulate the use of one at this page.

1675 Humphry Ditton (May 29, 1675 – October 15, 1715) was born at Salisbury and died in London in 1715 at Christ's Hospital, where he was mathematical master. He does not seem to have paid much attention to mathematics until he came to London about 1705. W. W. Rouse Ball states that Ditton's 1706 book on fluxions occupied a place in English education equivalent to L'Hospital's book in France.

1794 Johann Heinrich von Mädler (May 29, 1794 – March 14, 1874)  German astronomer who (with Wilhelm Beer) published the most complete map of the Moon of the time, Mappa Selenographica, 4 vol. (1834-36). It was the first lunar map to be divided into quadrants, and it remained unsurpassed in its detail until J.F. Julius Schmidt's map of 1878. Mädler and Beer also published the first systematic chart of the surface features of the planet Mars (1830).*TIS

1882 Harry Bateman (29 May 1882 – 21 January 1946) He spent much of his life collecting special functions and integrals that solved partial differential equations. He kept the references on index cards stored in shoe boxes—eventaully these began to crowd him out of his office. [DSB 1, 500] *SAU

1885 Finlay Freundlich (May 29, 1885 – July 24, 1964) was a distinguished German astronomer who worked with Einstein on measurements of the orbit of Mercury to confirm the general theory of relativity. He left Germany to avoid Nazi rule and became the Napier Professor of Astronomy at St Andrews.

1906 Gerrit Bol (May 29, 1906 in Amsterdam, Nov 1, 1989) was a Dutch mathematician, who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points.
Bol earned his PhD in 1928 at Leiden University under Willem van der Woude. In the 1930s, he worked at the University of Hamburg on the geometry of webs under Wilhelm Blaschke and later projective differential geometry. In 1931 he earned a habilitation.
In 1942–1945 during World War II, Bol fought on the Dutch side, and was taken prisoner. On the authority of Blaschke, he was released. After the war, Bol became professor at the Albert-Ludwigs-University of Freiburg, until retirement there in 1971. *Wik
1929 Peter Ware Higgs (29 May 1929 -  ) is an English theoretical physicist, the namesake of the Higgs boson. In the late 1960s, Higgs and others proposed a mechanism that would endow particles with mass, even though they appeared originally in a theory - and possibly in the Universe! - with no mass at all. The basic idea is that all particles acquire their mass through interactions with an all-pervading field, called the Higgs field. which is carried by the Higgs bosons. This mechanism is an important part of the Standard Model of particles and forces, for it explains the masses of the carriers of the weak force, responsible for beta-decay and for nuclear reactions that fuel the Sun. No Higgs boson has yet been detected; its mass (over 1 TeV) exceeds the capacity of any current accelerator. *TIS

1911 George Szekeres (29 May 1911 – 28 August 2005) was a Hungarian-born mathematician who worked for most of his life in Australia on geometry and combinatorics*SAU

1957 Jean-Christophe Yoccoz ( May 29, 1957 -   )  French mathematician who was awarded the Fields Medal in 1994 for his work in dynamical systems. Such studies began with Poincaré about the turn of the 20th century, who considered the stability of the solar system. It evolves according to Newton's laws but will it remain stable or, might a planet be ejected from the system? The techniques apply also in biology, chemistry, mechanics, and ecology where stability is an issue. This work also produces aesthetically appealing objects, such as the Julia and Mandelbrot fractal sets. Yoccoz was primarily concerned with establishing criteria that gave precise bounds on the validity of stability theorems. A combinatorial method for studying the Julia and Mandelbrot sets was named "Yoccoz puzzles." *TIS

1660 Frans van Schooten (1615 in Leiden – 29 May 1660 in Leiden) was a Dutch mathematician who was one of the main people to promote the spread of Cartesian geometry. Van Schooten's father was a professor of mathematics at Leiden, having Christiaan Huygens, Johann van Waveren Hudde, and René de Sluze as students.
Van Schooten read Descartes' Géométrie (an appendix to his Discours de la méthode) while it was still unpublished. Finding it hard to understand, he went to France to study the works of other important mathematicians of his time, such as François Viète and Pierre de Fermat. When Frans van Schooten returned to his home in Leiden in 1646, he inherited his father's position and one of his most important pupils, Huygens.
Van Schooten's 1649 Latin translation of and commentary on Descartes' Géométrie was valuable in that it made the work comprensible to the broader mathematical community, and thus was responsible for the spread of analytic geometry to the world. Over the next decade he enlisted the aid of other mathematicians of the time, de Beaune, Hudde, Heuraet, de Witt and expanded the commentaries to two volumes, published in 1659 and 1661. This edition and its extensive commentaries was far more influential than the 1649 edition. It was this edition that Gottfried Leibniz and Isaac Newton knew.
Van Schooten was one of the first to suggest, in exercises published in 1657, that these ideas be extended to three-dimensional space. Van Schooten's efforts also made Leiden the centre of the mathematical community for a short period in the middle of the seventeenth century. *Wik    Thony Christie (aka The Renaissance Mathematicus) sent me a comment to tell me that it was van Schooten who first used rectangular coordinates in his translations and extensions of Descartes Geometry.  The MAA Digital Library has seven images from van Schooten's "Exercitationes mathematicae". The copy was once the property of his student, Johann Hudde, and include problems from the book of another of his famous students, Christian Huygen's Ludo aleae.

Thony Christie added a note about van Schooten's contributions:
If you read La Géométrie you will search for rectangular co-ordinates in vain, Descartes did not use them. (Neither did Fermat who developed/invented algebraic geometry independently from Descarte). The first person to use them was van Schooten in his extended translation of Descartes work. (Thanks Thony)

1829 Sir Humphrey Davy (Baronet) (17 December 1778 – 29 May 1829) English chemist who discovered several chemical elements and compounds, invented the miner's safety lamp, and epitomized the scientific method. With appointment to the Pneumatic Institution to study the physiological effects of new gases, Davy inhaled gases (1800), such as nitrous oxide (laughing gas) and a nearly fatal inhalation of water gas, (a mixture of hydrogen and carbon monoxide). Davy discovered alkali metals, potassium and sodium, an isolation made with electric current for the first time (1807); as well as alkaline earth metals: calcium, strontium, barium, and magnesium (1808). He discovered boron at the same time as Gay-Lussac. He recognized chlorine as an element, which prior workers confused as a compound. *TIS Davy died in Switzerland in 1829 of heart disease inherited from his father's side of the family. He spent the last months of his life writing "Consolations In Travel", an immensely popular, somewhat freeform compendium of poetry, thoughts on science and philosophy (and even speculation concerning alien life) which became a staple of both scientific and family libraries for several decades afterward. He is buried in the Plainpalais Cemetery in Geneva.

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

Wednesday, 28 May 2014

How Long, How Far and Stuff Like That, a brief History of some measures of length

A few days ago John D Cook posted a link in his Units @UnitFact·Tweet (if you have any interest in measures and their history, a must follow) that said, "Units of length in LOTR" which more clever people (not me) realized was Lord of the Rings. It contained a link to a web site called tolkiengateway
In essence, it discussed a section from one of the "Rings" books in which a magic rope was used to descend a cliff. As part of the blog he computes the length rope by using ells, a very old measure of distance, to conclude the elven rope must have been 112.5 feet long.
I have been instructed by those who know these things that the LOTR took place in Middle Earth, and the period is "before the age of man."

It did get me thinking again about the history of units of distance, and I thought I might begin with the ell. Now the word ell, comes from the Latin Ulna for a bone in the forearm. It is the red one in the picture. It is also called the "elbow bone" and it gives a clear image of the original length it indicated, a measure from the elbow to the end of the middle finger. The term actually came from the Latin through the Old Germanic alinâ of the same meaning. The same measure was common in the Roman standards by a different name,cubitum, from which the English "cubit" is derived, and which was defined to be 1 1/2 Pedes (feet).
(If Cubitum sonds suspiciously like a good source for "cube", your close, but that came from the Greek, κύβος to be Latinized into cubus)

Of course many more people know cubit than ell, most probably because of the famous Bible story of Noah and the flood in which he was instructed to build an ark 300 cubits, by 50 cubits, by 30 cubits.  Noah didn't speak Latin, but the same measure in Hebrew is אמה which I am told is pronounced something like ammah. This is essentially the primary unit of measure in all the Old Testament. So how long was that? Well, it seems there are two versions of the Hebrew cubit, the standard cubit or short cubit (about 17.5 inches) and a long cubit, which is given in Ezekiel 43:13 “These are the measurements of the altar in cubits (the cubit is one cubit and a hand breadth): the base one cubit high and one cubit wide, with a rim all around its edge of one span. " A hand breadth is the distance across the four fingers when they are pressed together, about four inches today, but the Roman standard was 1/4 of a Pes (foot), so the long cubit would be in the neighborhood of 20.5 inches.
The rim of one span is a different unit of hand measure, the length between the tip of the thumb and little finger when the hand is spread apart. If you try this on your own arm, you should find that it takes about two spans to reach from the tip of the middle finger to the bend of the elbow, and hence a span was 1/2 cubit.

After the collapse of the Roman Empire the measures they had established so carefully began to be altered and twisted in every little Dukedom and Middle Earth Kingdom. By the 16th century or so, many of them had adopted a standard measure of cloth which is the double ell, which was very nearly a yard. It was so common a measure that the "double" sort of slipped away and it just became the Scottish ell (not to be confused with Scottish Ale) It was common in my youth for clerks to approximate a yard of material by
taking the cloth and holding it between thumb and forefinger of each hand at the center of their chest (or often at their nose), then letting the cloth slip between one hand as the other drew it out to the tip of the arm. If you fold the arm at the elbow and bring it to touch the chest (or nose) you can see this was about two ells, or a double ell. In Scotland the statutes set it at 37 inches, with a special longer Barony ell for land measure. Cloth was such an important item of trade in markets in England and Scotland, that the cross in the Market square would often have an official (double) ell  inlaid in Metal, or etched in the stone to measure off cloth and other items sold by length.

Sometimes units in one culture are merged with units from another.  After 1066 the Norman Conquerors merged their  residue of the Latin inch (from a word for the 12th part of something, ounce is from the same source,) and merged it with the Anglo Saxon unit of a Barley corn.  Three barley corns was established as one inch.  In 1324 Edward II declared that the inch would be, "three grains of barley, dry and round, placed end to end".  Barley corns are still the standard measure for UK shoe sizes, with each increase in length of 1 size relating to one barley corn.  The adult size zero is a shoe 25 barley corns long.  (In shoe size, like in building floors, the UK starts at 0 while the US begins at 1).

Greater distances extended the common smaller standards.  A Roman Legion used a standard pace (one left, one right) of 5 feet.  Our current unit for a mile comes from the Roman  mille passes which means one thousand paces.  1500 of these passes would cover a Gallic league, which Mr. Cooks Units tweet recently reminded us all that "A league is an hour's walk. (There are many slightly different ways to make this more precise, but an hour's walk is the idea behind them.)"  So those Roman centurians (yep, 100 in each unit) marched along at a clip of about 25 paces per minute, or one stride per calculations make this about 1.5 miles per hour. I remember learning way back in my youth in the military that the US standard march is 100 (2.5 ft)strides per minute, and that these were essentially the standards for the Roman Legions, so I expect that the League was an hours walk for the normal folk, not for trained soldiers. (I'm sure my drill sgt in boot camp would have loved to hear my view on how we should march at a pace to cover 1.5 miles per hour)

On the extreme end, a light year, the distance a beam of light can travel in a vacuum in a Julian year (how many things are still defined for the Julian yr.?) of 365.25 days.  The term was introduced before it was a well defined distance. In 1838 Fredrich Bessell made the first successful measurement of the distance to a star other than the sun, the star 61 Cygni (actually he didn't see below).  At that time the longest standardized unit of measure was the Astronomical Unit, the distance from the Earth to the sun.  Bessell used the parallax of 61 Cygni to determine that it was 660000 astronomical units away from the Earth, then he interjected that this meant that light took 10.3 years to cover that distance (this was an estimate, as the precise speed of light was not known at that time).  He never used the term "light year", but others soon did.  Otto Ule used it in an article in 1851, for example.  The true astronomers seem to dislike the term; Arthur Eddington called the  term an inconvenient and irrelevant unit.  To overcome this issue they created the unit Parsec.  I love this unit of measure because it is based on two imaginary similar triangles with a common vertex.  When the Earth moves through it's orbit around the sun, we imagine a right triangle with the right angle at the sun and legs of 1 AU from the sun to the earth, and another leg perpendicular to the earth's elliptic orbit reaching out to (wait for it) an imaginary star 1 Parsec away.  The parallax of this "imaginary near star" against the distant stars would be 1 second of arc. Of course, there are no stars as close as 1 parsec.   So when we measure a star that it farther out, the parallax angle is smaller, and so a star that is 2 parsecs away, will have a 1/2 arc sec of parallax.  The Earth's nearest star neighbor outside our solar system is Proxima Centauri, which is 1.3 parsecs away (or about 4 light years).  That means, if I understand all this imaginary astronomy, that it should have a parallax of about .75 arc sec.  (if that is a gross error, can one of my astrophysics readers give me a quick correction)
To give you an idea how accurately they can now measure cosmological angles, 1 arc second would be about the thickness of a US dime (the side view) seen from about 4 kilometers (2 1/2 miles). 

The parsec unit was likely first suggested in 1913 by British astronomer Herbert Hall Turner. Named from an abbreviation of the parallax of one arcsecond.

The Centauri star system is actually three stars, cleverly labeled alpha, beta, and Proxima is about .05 parsecs closer than the other two, hence its name.

The Centari alpha/beta binary was actually probably the first star to have it's parallax accurately measured. Scottish astronomer Thomas Henderson studied the trigonometric parallaxes of the AB system from April of 1832 until May of the following year. Trouble was, the parallax was so much greater than stars normally have that he was afraid he had made a mistake. So in 1839, after seeing Bessel's results, he published.

There are, of course, new measurements for the very short units, but it is late, and this old teacher needs a break. Good night, all.

On This Day in math - May 28

Twice two makes four seems to me
simply a piece of insolence. 
Twice two makes four is a pert coxcomb who stands
with arms akimbo barring your path and spitting.
I admit that twice two makes four is an excellent thing,
but if we are to give everything its due,
twice two makes five is sometimes
a very charming thing too.  
~Fyodor Mikhailovich Dostoevsky

The 148th day of the year; 148 "Primelicious", 21 + 1 is prime,24 + 1 is prime, and 28 + 1, and the three results add to a prime, 3+17+257 = 277. Looking for more Primelicious numbers.

A Vampire number is a number whose digits can be regrouped into two smaller numbers that multiply to make the original (1260 = 21*60).  There are 148 vampire numbers with six digits.   

585 BC Thales predicted the total eclipse of the sun that took place on this date. See Herschel, Outline of Astronomy (1902), pp. 833 and 839. [Eves, Circles, 33◦] *VFR  WW Rouse Ball says it is uncertain whether the date is the 585 date, or Sep 30, 609 BC.  Heath, and most others, seem to settle on the 585 BC date.

1765 The Longitude Board at Greenwich awards Leonhard Euler an amount of 300 Pounds, "Reward for Theorems furnished by him to assist Professor Mayer in the Construction of Lunar Tables upon the Principles of Gravitation laid down by Sir Isaac Newton."
Tobias Mayer had died in 1762, but his widow received an amount of 3000 Pounds for his work in the same meeting for his construction of the tables, which she signed over to the Committee. *Derek Howse, Britain's Board of Longitude:The Finances, 1714-1828

In 1937, the Golden Gate Bridge, San Francisco was ceremonially opened to vehicles by President Franklin Delano Roosevelt who pressed a telegraph key in the White House. Within the first hour after the toll gates opened, 1,800 cars crossed the bridge. By day's end, 32,300 vehicles and 19,350 pedestrians had paid to pass over the bridge. A firework display that night celebrated the opening of the bridge. The previous day, a Pedestrian Day had been held which first opened the bridge for public use. The building and design of the bridge had been supervised by chief engineer Joseph B. Strauss. Construction had started on 5 Jan 1933. It was the first bridge to span the mouth of a major U.S. ocean harbour.*TIS

1959 Committee formed which developed COBOL. COBOL is one of the oldest programming languages. Its name is an acronym for COmmon Business-Oriented Language, defining its primary domain in business, finance, and administrative systems for companies and governments.
The COBOL specification was created by a committee of researchers from private industry, universities, and government during the second half of 1959. The specifications were to a great extent inspired by the FLOW-MATIC language invented by Grace Hopper - commonly referred to as "the mother of the COBOL language." The IBM COMTRAN language invented by Bob Bemer was also drawn upon, but the FACT language specification from Honeywell was not distributed to committee members until late in the process and had relatively little impact. FLOW-MATIC's status as the only language of the bunch to have actually been implemented made it particularly attractive to the committee.*Wik

1981 The New Scientist (pp 506-507) describes a mathematical theory of how coloration develops in animals. Zebras have stripes rather that spots because coloring is determined at an early stage of the development of the fetus. [Mathematics Magazine 54 (1981), p 215.] *VFR

In 1998, NASA released a picture of what California astronomer Susan Terebey said may be the first extrasolar planet ever seen, dubbed TMR-1C. Digitized pictures taken by the Hubbell Space Telescope seemed to show an image of a planet apparently flung from a pair of young stars in the constellation Taurus, 450 light years from Earth. Located at one end of a bright trail that led from the newborn stars, the faint object appeared as if it was their offspring, a planet a few times as massive as Jupiter that had been expelled from its birthplace. However, by the following year, scrutiny of its spectrum suggested to other astronomers that it could be merely a background star. Telescopic tracking for several years should resolve the answer.*TIS

2013 David L. Donoho has been awarded the 2013 Shaw Prize in Mathematical Sciences for his profound contributions to modern mathematical statistics and in particular the development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets.
The Anne T and Robert M Bass Professor of the Humanities and Sciences, and Professor of Statistics at Stanford University, Dr. Donoho is well known for his role in developing new mathematical and statistical tools to deal with problems ranging from large data-sets in high dimensions to contamination with noise. *SIAM

1676 Jacopo Riccati (28 May 1676 – 15 April 1754) was an Italian mathematician who wrote on philosophy, physics and differential equations. He is chiefly known for the Riccati differential equation. *SAU   The general Riccati diferential equation is of the form dy/dx = A+ By + Cy2 where A, B, and C represent functions of x..(there are actually several types of diff equations known by this term..)  He had two sons who also contributed to mathematics.  Vincenzo was a professor in Bologna, and Giordano published works in Geometry and on Newton's works.  Jacopo (and both sons) died in Treviso.

1710 Johann(II) Bernoulli (28 May 1710 in Basel, Switzerland - 17 July 1790 in Basel, Switzerland)
was a member of the Swiss mathematical family. He worked mainly on heat and light. He was one of three sons of Johann Bernoulli. In fact he was the most successful of the three. He originally studied law and in 1727 he obtained the degree of doctor of jurisprudence. He worked on mathematics both with his father and as an independent worker. He had the remarkable distinction of winning the Prize of the Paris Academy on no less than four separate occasions. On the strength of this he was appointed to his father's chair in Basel when Johann Bernoulli died. *Wik

1850 Wooster Woodruff Beman (May 28, 1850 - January 1, 1922). He attended school in Valparaiso, Ind., and entered the University of Michigan in 1866, receiving his B.A. degree in 1870. After teaching for a year at Kalamazoo College as instructor in Greek and mathematics, he returned to the University of Michigan as an instructor while also working for his master's degree, which he received in 1873. In 1874, he became assistant professor, in 1882 associate professor, and in 1887 full professor.
In addition to his teaching, Beman wrote books and articles on the history and teaching of elementary mathematics. Among his works are "Nature and Meaning of Numbers" (from the German), and "Continuity and Irrational Numbers." He was the joint author, with D. E. Smith, of "Plane and Solid Geometry," "Higher Arithmetic," "New Plane and Solid Geometry," "Elements of Algebra," "Academic Algebra," translations of "Famous Problems of Elementary Geometry," and "A Brief History of Mathematics." *Michigan Historical Collections. They also were editors of T. Sundara Row's Geometric Exercises in Paper Folding:

1888 Jim Thorpe (May 28, 1888 – March 28, 1953) World-class athlete He was born in a one-room cabin near Prague in Indian Territory, now Oklahoma. Thorpe's versatile talents earned him the distinction of being chosen, in 1950, the greatest football player and the greatest American athlete of the first half of the twentieth century by American sports writers and broadcasters. Thorpe won the gold medal in both the decathlon and pentathlon events at the Stockholm Olympics, but was stripped of his medals when a reporter revealed he had played semi-professional baseball. It was not until after his death that Thorpe's amateur status was restored, and his name reentered in the Olympic record book. (Library of Congress web page)
So why is this on a math page…Well it seems that Jim Thorpe may have indirectly influenced the naming of the # key on the telephone. One of several stories for how it is named is this one: In the 1960's when Bell Telephone added two new buttons for push button telephones, they used the * symbol and the # symbol. Although most people call the * an asterisk, the telephone folks decided to use "star". The other symbol, #, has been called lots of different names such as crosshatch, and now the common term on twitter seems to be "hashtag".  Others have  referred to it as tic-tac-toe, the pound sign, and the number sign (leave it to the telephone company to put the number sign on one of the two keys without a number); but the term now "officially" used by the American telephone industry for the symbol is octothorpe although it is more often called the pound key in conversations with the public.
It seems that the name was made up more or less spontaneously by Bell Engineer Don MacPherson while meeting with their first potential customer. The octo part was chosen because of the eight points at the ends of the line segments, and the thorpe was in honor of Jim Thorpe, the great Native American athlete. Why honor Thorpe? At the time MacPherson was working with a group that was trying to restore Thorpe's Olympic medals, which had been taken from him when it was found he had played semi-professional baseball prior to his track victories in the Olympics in Sweden. [It's not math, but I love the story that when the King of Sweden gave him the gold medal, the king said, "You are surely the greatest athlete on the earth". The modest Thorpe smiled and replied, "Thanks, King."]
There are a host of other names for the # symbol, and many of them can be found at this page from Wikipedia which includes several different stories about the creation of "octothorpe" or "octothorn" and also has this rather interesting clip:
"The pronunciation of # as `pound' is common in the US but a bad idea. The British Commonwealth has its own, rather more apposite, use of `pound sign. On British keyboards the UK pound currency symbol once frequenlty replaced #, with # being elsewhere on the keyboard. The US usage derives from an old-fashioned commercial practice of using a # suffix to tag pound weights on bills of lading. The character is usually pronounced `hash' outside the US. There are more culture wars over the correct name of this character than any other, which has led to the “ha-ha” only serious suggestion that it be pronounced `shibboleth' (see Judges 12:6 in the Old Testament)." (pballew Etymology page)

1908 Egbert van Kampen, In 1908 he left Europe and traveled to the United States to take up the position which he had been offered at Johns Hopkins University in Baltimore, Maryland. There he met Oscar Zariski who had taught at Johns Hopkins University as a Johnston Scholar from 1927 until 1929 when he had joined the Faculty. Zariski had been working on the fundamental group of the complement of an algebraic curve, and he had found generators and relations for the fundamental group but was unable to show that he had found sufficient relations to give a presentation for the group. Van Kampen solved the problem, showing that Zariski's relations were sufficient, and the result is now known as the Zariski–van Kampen theorem. This led van Kampen to formulate and prove what is nowadays known as the Seifert–van Kampen theorem. *Wik

1912 Ruby Violet Payne-Scott, (28 May 1912 – 25 May 1981) was an Australian pioneer in radiophysics and radio astronomy, and was the first female radio astronomer.
One of the more outstanding physicists that Australia has ever produced and one of the first people in the world to consider the possibility of radio astronomy, and thereby responsible for what is now a fundamental part of the modern lexicon of science, she was often the only woman in her classes at the University of Sydney.
Her career arguably reached its zenith while working for the Australian government's Commonwealth Scientific and Industrial Research Organisation (then called CSIR, now known as CSIRO) at Dover Heights, Hornsby and especially Potts Hill in Sydney. Some of her fundamental contributions to solar radio astronomy came at the end of this period. She is the discoverer of Type I and Type III bursts and participated in the recognition of Type II and IV bursts.
She played a major role in the first-ever radio astronomical interferometer observation from 26 January 1946, when the sea-cliff interferometer was used to determine the position and angular size of a solar burst. This observation occurred at either Dover Heights (ex Army shore defence radar) or at Beacon Hill, near Collaroy on Sydney's north shore (ex Royal Australian Air Force surveillance radar establishment - however this radar did not become active until early 1950).[4]
During World War II, she was engaged in top secret work investigating radar. She was the expert on the detection of aircraft using PPI (Plan Position Indicator) displays. She was also at the time a member of the Communist Party and an early advocate for women's rights. The Australian Security Intelligence Organisation (ASIO) was interested in Payne-Scott and had a substantial file on her activities, with some distortions.

1912 Hans Zassenhaus, algebraist. (28 May 1912–21 November 1991) was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra.
He was born in Koblenz–Moselweiss, and became a student and then assistant of Emil Artin. He was subsequently a professor at McGill University, the University of Notre Dame, and Ohio State University, and was one of the founding editors of the Journal of Number Theory. He died in Columbus, Ohio. *Wik

1930 Frank Donald Drake ( May 28, 1930 -  ) is an American astronomer who formulated the Drake Equation (1961) to estimate the number of technological civilizations that may exist in our galaxy. In 1960, Drake led the first search, the two-month Project Ozma to listen for patterns in radio waves with a complex, ordered pattern that might be assumed to represent messages from some extraterrestrial intelligence. Carl Sagan and Drake designed the plaques on Pioneer 10 and Pioneer 11 for the purpose of greeting and informing any extraterrestrial life that might find the vessels after they left the solar system. *TIS


1997 Ronald Vernon Book (April 1937 – May 28, 1997 in Santa Barbara, California) worked in theoretical computer science. He published more than 150 papers in scientific journals.

2003 Ilya Prigogine (25 Jan 1917; 28 May 2003) Russian-born Belgian physical chemist who received the Nobel Prize for Chemistry in 1977 for contributions to nonequilibrium thermodynamics, or how life could continue indefinitely in apparent defiance of the classical laws of physics. The main theme of Prigogine's work was the search for a better understanding of the role of time in the physical sciences and in biology. He attempted to reconcile a tendency in nature for disorder to increase (for statues to crumble or ice cubes to melt, as described in the second law of thermodynamics) with so-called "self-organisation", a countervailing tendency to create order from disorder (as seen in, for example, the formation of the complex proteins in a living creature from a mixture of simple molecules). *TIS

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

Tuesday, 27 May 2014

On This Day in Math - May 27

 The mathematics are distinguished by

a particular privilege, that is,

in the course of ages, they may always advance

and can never recede. 

 ~Edward Gibbon, Decline and Fall of the Roman Empire

The 147th day of the year; if you iterate the process of summing the cubes of the digits of a number starting with 147, you eventually start repeating 153. This seems to be true for all multiples of three.

If there are no fouls, the maximum score on a snooker break is 147.

And Derek Orr@Derektionary pointed out that "147 is the smallest number formed by a column of numbers on a phone button pad"

669 BC "If the Sun at its rising is like a crescent and wears a crown like the Moon: the king wll capture his enemy's land; evil will leave the land, and (the land) will experience good . . . " Refers to a solar eclipse of 27 May 669 BC. BY Rasil the older, Babylonian scribe to the king. *NSEC
1638  In a letter to Fr Marin Mersenne, Descartes claimed to have a general rule to find  number n with a sum of its factors S(n) given only the ratio of n:S(n) = p/q.  He showed that n:S(n) = 4/9 is solved for n= 360 .  Fermat responded to Mersenne that 2016 has the same property.. (for students, S(6) would = 1+2+3+6 = 12)  (History of the theory of numbers  By Leonard Eugene Dickson)

 1832 In a letter to Legendre, Jacobi stated that the solutions to x2-ay2=1 can be expressed in terms of the sine and cosine of

1849 Chebyshev defends his doctoral dissertation on the theory of numbers at Petersburg University.*VFR

1919 Astronomical party arrives at  São Tomé and Príncipe, officially the Democratic Republic of São Tomé and Príncipe, is a Portuguese-speaking island nation in the Gulf of Guinea, off the western equatorial coast of Central Africa.  Príncipe was the site where astronomical observations of the total solar eclipse of 29 May 1919 confirmed Einstein's prediction of the curvature of light.  The expedition was sponsored by the Royal Society and led by Sir Arthur Stanley Eddington.

1937 Golden Gate bridge opened.*VFR In 1937, the Golden Gate Bridge, San Francisco was first opened to the public as a Pedestrian Day. By 6 am, 18,000 people were waiting for the toll gates to open. Many crossed in unique ways, hoping to be prize-winners as the first to establish a record, whether by walking backwards or on stilts, tap-dancing, roller-skating or playing instruments. It was a sprinter, Donald Bryan, from San Francisco Junior College, who became the first person to cross the entire span. At 10 am, Chief engineer Joseph Strauss gave no speech, but instead read a poem he had written for the event. By the end of the day, about 200,000 people had joined the celebration. The bridge was ceremonially opened to traffic the next day.*TIS

1332 Ibn Khaldūn or Ibn Khaldoun (full name, Arabic: أبو زيد عبد الرحمن بن محمد بن خلدون الحضرمي‎, Abū Zayd ‘Abdu r-Raḥmān bin Muḥammad bin Khaldūn Al-Ḥaḍrami, May 27, 1332 AD/732 AH – March 19, 1406 AD/808 AH) was a Muslim historiographer and historian who is often viewed as one of the fathers of modern historiography,sociology and economics.
He is best known for his Muqaddimah (known as Prolegomenon in English), which was discovered, evaluated and fully appreciated first by 19th century European scholarship, although it has also had considerable influence on 17th-century Ottoman historians like Ḥajjī Khalīfa and Mustafa Naima who relied on his theories to analyze the growth and decline of the Ottoman Empire. Later in the 19th century, Western scholars recognized him as one of the greatest philosophers to come out of the Muslim world. *Wik

1660 Francis Hauksbee the elder (baptized on 27 May 1660 in Colchester–buried in St Dunstan's-in-the-West, London on 29 April 1713.), also known as Francis Hawksbee, was an 18th-century English scientist best known for his work on electricity and electrostatic repulsion.
Initially apprenticed in 1678 to his elder brother as a draper, Hauksbee became Isaac Newton’s lab assistant. In 1703 he was appointed curator, instrument maker and experimentalist of the Royal Society by Newton, who had recently become president of the society and wished to resurrect the Royal Society’s weekly demonstrations.
Until 1705, most of these experiments were air pump experiments of a mundane nature, but Hauksbee then turned to investigating the luminosity of mercury which was known to emit a glow under barometric vacuum conditions.
By 1705, Hauksbee had discovered that if he placed a small amount of mercury in the glass of his modified version of Otto von Guericke's generator, evacuated the air from it to create a mild vacuum and rubbed the ball in order to build up a charge, a glow was visible if he placed his hand on the outside of the ball. This glow was bright enough to read by. It seemed to be similar to St. Elmo's Fire. This effect later became the basis of the gas-discharge lamp, which led to neon lighting and mercury vapor lamps. In 1706 he produced an 'Influence machine' to generate this effect. He was elected a Fellow of the Royal Society the same year.

Hauksbee continued to experiment with electricity, making numerous observations and developing machines to generate and demonstrate various electrical phenomena. In 1709 he published Physico-Mechanical Experiments on Various Subjects which summarized much of his scientific work.
In 1708, Hauksbee independently discovered Charles' law of gases, which states that, for a given mass of gas at a constant pressure, the volume of the gas is proportional to its temperature.
The Royal Society Hauksbee Awards, awarded in 2010, were given by the Royal Society to the “unsung heroes of science, technology, engineering and mathematics.” *Wik

1862 John Edward Campbell (27 May 1862, Lisburn, Ireland – 1 October 1924, Oxford, Oxfordshire, England) is remembered for the Campbell-Baker-Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras. *SAU His 1903 book, Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups, popularized the ideas of Sophus Lie among British mathematicians.
He was elected a Fellow of the Royal Society in 1905, and served as President of the London Mathematical Society from 1918 to 1920. *Wik & *Renaissance Mathematicus

1967 Sir John Douglas Cockcroft (27 May 1897, 18 Sep 1967) British physicist, who shared (with Ernest T.S. Walton of Ireland) the 1951 Nobel Prize for Physics for pioneering the use of particle accelerators to study the atomic nucleus. Together, in 1929, they built an accelerator, the Cockcroft-Walton generator, that generated large numbers of particles at lower energies - the first atom-smasher. In 1932, they used it to disintegrate lithium atoms by bombarding them with protons, the first artificial nuclear reaction not utilizing radioactive substances. They conducted further research on the splitting of other atoms and established the importance of accelerators as a tool for nuclear research. Their accelerator design became one of the most useful in the world's laboratories. *TIS He was the first Master of Churchill College and is buried at the Parish of the Ascension Burial Ground in Cambridge, together with his wife Elizabeth and son John, known as Timothy, who had died at the age of two in 1929.*Wik

1907 Herbert Karl Johannes Seifert (May 27, 1907, Bernstadt – October 1, 1996, Heidelberg) was a German mathematician known for his work in topology. Seifert did other important work related to knot invariants. In 1934 he published results, using surfaces today called Seifert surfaces, which he used to calculate homological knot invariants. Another topic which Seifert worked on was the homeomorphism problem for 3-dimensional closed manifolds. *SAU

1896 Aleksandr Grigorievich Stoletov (August 10, 1839 – May 27, 1896) was a Russian physicist, founder of electrical engineering, and professor in Moscow University. He was the brother of general Nikolai Stoletov. By the end of the 20th century his disciples had headed the chairs of Physics in five out of seven major universities in Russia.
His major contributions include pioneer work in the field of ferromagnetism and discovery of the laws and principles of the outer photoelectric effect.*Wik

1928 Arthur Moritz Schönflies (April 17, 1853 – May 27, 1928) worked first on geometry and kinematics but became best known for his work on set theory and crystallography. He classified the 230 space groups in 1891 He studied under Kummer and Weierstrass, and was influenced by Felix Klein.
The Schoenflies problem is to prove that an (n − 1)-sphere in Euclidean n-space bounds a topological ball, however embedded. This question is much more subtle than initially appears. *Wik *SAU

1960  Milton B. Porter  Professor at Univ of Texas, he was the dissertation adviser for Goldie Horton, the first woman to get a PhD in Mathematics at Univ of Texas.  Eighteen years later he married her.  He died in Austin Texas.

1962 FELIX ADALBERT BEHREND (23 April 1911 in Charlottenburg, Berlin, Germany -27 May 1962 in Richmond, Victoria, Australia) Felix Behrend's sympathies within pure mathematics were wide, and his creativeness ranged over theory of numbers, algebraic equations, topology, and foundations of analysis. A problem that caught his fancy early and that still occupied him shortly before his death was that of finite models in Euclidean 3-space of the real projective plane. He remained productive for much of the two years of his final illness, and left many unfinished notes in which his work on foundations of analysis is continued. (From his obituary by B H Neumann)

1964 Colin Brian Haselgrove  (26 September 1926 – 27 May 1964) In 1958 Haselgrove published his most famous number theory result in A disproof of a conjecture of Pólya. The conjecture of Pólya claims that for every x greater than 1 there are at least as many numbers less than or equal to x having an odd number of prime factors as there are numbers with an even number of prime factors. R S Lehman and W G Spohn had verified the conjecture for all numbers x up to 800,000 but Haselgrove found a counterexample using methods based on those developed by Ingham with the help of computations carried out on the EDSAC 1 computer at Cambridge. He also verified the calculations using Manchester University's Mark I computer before publishing the results. In the same paper Haselgrove announced that he had also disproved a number theory conjecture of Turán. *SAU

2012 Friedrich Ernst Peter Hirzebruch (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in the Germany of the postwar period.
Amongst many other honours, Hirzebruch was awarded a Wolf Prize in Mathematics in 1988 and a Lobachevsky Medal in 1989. The government of Japan awarded him the Order of the Sacred Treasure in 1996. He also won an Einstein Medal in 1999, and received the Cantor medal in 2004.*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

Monday, 26 May 2014

More Geometric Solutions to Quadratics

A few years back I wrote a paper with the tongue-in-cheek title, Twenty Ways to Solve a Quadratic with reference to the song, "Fifty Ways to Leave Your Lover". It included a variety (actually about 20) of approaches, including some graphic approaches, and some history notes on each method.
Recently while looking into the work of Karl George Christian von Staudt (the 147th anniversary of his death is coming up June 1) I found two more graphic methods I had not previously known. I also found an earlier use of one that I had written about in the article above, plus a very early use that I omitted.

I will begin with two very early examples and conclude with the von Staudt example.

One of the earliest graphic examples of a solution has to be from Euclid's Elements, in book 2 proposition 11. Euclid's description of the task is to cut a given straight line into two parts so that the rectangle formed by the whole and one of the parts is equal to the square on the second part. If we call the parts of the line b and x, then what we seek is x^2 =(x+b)b or x^2 = bx+b^2.

The construction in the Elements is pretty brief, finding a midpoint and a couple of compass constructions.
To see the image, and Euclid's solution I leave you to the always wonderful web page on the Elements by Professor David Joyce.

When I wrote the paper on solving quadratics, I credited a method (#13 in the list) this way:
13. Real roots by Lill circle. One of the most unusual graphic methods I have ever seen comes from a more general
method of solving algebraic equations first proposed, to my knowledge, by M.E. Lill, in Resolution graphique des équations numériques de tous les degrés..., Nouv. Ann. Math. Ser. 2 6 (1867) 359--362. Lill was supposedly an Artillery Captain, but his method was included in Calcul graphique et nomographie by a more famous French engineer, Maurice d’Ocagne, who called it the “Lill Circle”.
It turns out that before Lill, a young Scottish mathematician named Thomas Carlyle was credited with using the same circle to solve quadratics (To be fair, Lill extended his method to all polynomials, and even complex roots).
In Sir John Leslie's (who died in 1832) "Elements of Geometry" he writes this method "...was suggested to me by Thomas Carlyle, an ingenious young mathematician, and formally my pupil."

Carlyle skipped right to the diameter of the circle in question. Given a quadratic, x2+bx + c=0, plot points at (0,1) and (-b,c) and construct a diameter. The circle with this diameter will intersect the x-axis at the real solutions (if any exist) of the quadratic.
The circle for y=x2 +5x - 6 is shown with the actual parabolic function in red.
Students might be challenged to explain why one end of the diameter will always lie on the function.

In 1908 a little book of less than 100 pages titled Graphic Algebra, written by Arthur Schultze, was published by Macmillan & Co. Schultze was a high school math dept. head at the New York High School of Commerce, and an associate Professor at NYU.
The book is free online in several formats.

On page 47 I found a graphic method of solving quadratic equations I had never seen before. The process uses a standard graph of xy=1. [One of the common approaches when calculators and computers did not exist was to alter an equation to the solution of two equations such as a familiar conic and a straight line.]

I will illustrate his approach with the example he uses in his book. To solve the quadratic equation x2 + 2x - 8 =0 He first makes the simple step of dividing all terms by x to get \( x+2 - \frac{8}{x} = 0 ( x\neq 0)\)

Now by substitution of y= 1/x (or xy=1) we get x+2-8y=0. So where both of these equations are true, must be a solution to the original equation. Simply picking a couple of convenient points to plot the line x+2-8y = 0 he determines that when y= 0, x = -2; and when y= 1, x = 6. So we graph the equation xy=1 and then plot the points (-2,0) and ((6,1) and hope for an intersection.

The x-coordinates of the two intersections (red) give us the solutions x={-4; 2}.

Schultze's little book also uses the method from my paper (15. Using the graph of y = x^2 and y = -bx – c to find real roots.) as a graphic solution using a simple conic (y=x2)

And for me, the treat of the day was Karl George Christian von Staudt's graphic method of solving quadratics because it is so different from all the others I've ever learned. Staudt was a student at Gottingen and worked with Gauss, who was at the time the director of the observatory, becoming a very good mathematical astronomer in his own right. He began as a high school teacher as well. His book Geometrie der Lage (1847) was a landmark in projective geometry. It was the first work to completely free projective geometry from any metrical basis. In 1857 von Staudt contributed a route to number through geometry called the Algebra of throws, and he made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently.
So here is the solution method used by von Staudt. The challenge to teacher (and students) is to see if you can figure out WHY it works (I found this very difficult):
Using the equation x2 - 5x + 6 = 0 we begin by constructing a one unit circle centered at (0,1). Then we construct the points \( \frac {c}{-b} , 0)\) and \( \frac{4}{-b},2)\) For b= -5 and c= 6 we get the points \( \frac {6}{5} , 0)\) and \( \frac{4}{5},2)\)
Construct the straight line through these two points (in red in the image).
The line intersects the circle in the points I, and J, and it is not essential to know their coordinates, but J is (1,1)
Now use (2,0) to project each of these points onto the x-axis. The result is the solutions to the equation. In this case x= {2, 3}