Thursday, 31 May 2012

On This Day in Math - May 31

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

The 152nd day of the year; the eighth prime number is 19, and 8 x 19 = 152.... 152 is also the largest known even number that can be expressed as the sum of two primes in exactly four ways. (Students should find all four ways.)

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

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

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics. Grinstein & Campbell

Wednesday, 30 May 2012

The Cubic Attractiveness of 153

Just in time for June 1st, which is the 153rd day of this leap year.  I recently discovered an interesting quality about the number 153.

Ok, I was amazed. I was lead from a comment (below) that this is all shown in the online integer sequence, but it is still great fun.

Pick any old number you want, and multiply by three (or just pick a number that is a multiple of three).
Now take all the digits and cube them and add the cubes together.
For example, if you picked 231, you would add 23 + 3 3 + 1 3 to get 36. 
Yeah, So what you're probably thinking... but take that new number and do the same thing... cube the digits and add them up... Nothing?  Keep going... eventually you get to 153, and then when you  do it again, you get 153 forever.

In slightly more formal language, it seems that 153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits.

The 231 that I used above goes to 36, which goes to 243, then 99, 1458, 702 and 351, which give 153. 

If you started with something like 72, it gets there pretty quickly,  72 →   351 →  153.  

Other numbers take a little longer.  717, for example, takes 13 iterations of the cubing process to get there...but it does.  
And 153 is special... ONLY the multiples of three go there.  Pick another number that's not a multiple of three and run the process and several things might happen, but what won't happen, is going to 153. 

When I began to explore why this happened I could figure out a few things easy enough.  The first realization was that the iterations couldn't just get bigger and bigger and diverge to infinity.  93 is 729 and so any four digit number has to iterate to a value less than 4 * 729 or 2916, and anything with more digits just keeps getting pushed down until it gets under that limit.
So all the numbers in the world have to end up doing something other than just keep getting bigger.
That only leaves a few options.  They might just keep jumping around in some orbit that cycles through several numbers.  This happens with 46 for example. 
After a few iterations you fall into a three cycle. 

46 → 280 → 520 →  133 → 55  → 250 →133

Other numbers also do this, and there are a few different three cycles and  two cycles, but that is sort of an oddity, at least for numbers less than 1000 (still have work to do here).
Most numbers go to a fixed attractor.  For multiples of three, that seems to always be 153.   For numbers that are not multiples of three, the general fixed points are either 370, or 371.  And they have a modulo three relationship as well.  Numbers that are congruent to 2 Mod3 (they have a remainder of 2 when you divide by three) generally go to 371.  The exceptions are a couple of numbers; 47 , 74, 77,  89 & 98,   which go to 407 (of course 707, 908, 980, etc would also, I counted 30 numbers less than 1000 that go to 407 as a fixed point).  
All the cycles that I have found are numbers that are equivalent to 1 Mod3.  The cycles seem pretty common with less than 1/2 the smaller numbers going to a fixed value of 370 and the occasional few like 1, 10, 100, that have a fixed point at one.  118 is in that group as well, for example.  

The reason for the separation into modulus classes of three is easy enough to explain.  When you cube a number, it's modulus in base three isn't changed.  For example, 4 Mod3 is 1, and 43 = 64 is also equivalent to 1 Mod3.  So the Modulus of a number doesn't change under this process, grouping the results together.  

Now when you add in the fact that there are not really that many of these smaller (less than 3000, say) numbers that can be made with the sum of cubes unless you allow for weird numbers like 11,111 or something to get five.  
I haven't explored much beyond 1000, so I'm not sure if I will come across other cyclic orbits. Or why only the 3n+1 type numbers produce cycles. And I'm not sure what else I may find, but I'm thinking that what I've seen so far makes 153 a very special number. 

On This Day in Math - May 30

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

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 dual 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.

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.

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Tuesday, 29 May 2012

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 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

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 y2 = an2 + bn + c and we can find solutions to the Pell equation, q2 = ap2 +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 dual 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

1919 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.

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.

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Monday, 28 May 2012

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 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.

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

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

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

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Sunday, 27 May 2012

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 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.
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

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 > 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

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Saturday, 26 May 2012

Transit of Venus on a Paper Plate

Ever want to show your students why this the transit of Venus on June 5, only eight years after the last one, will not come back for over 100 years. Even better, would you like an explanation they can actually see and make sense of. Here it is.

Great for Trig and such where two different periodic functions must align.


On This Day in Math - May 26

You cannot feed the hungry on statistics. 
~Heinrich Heine

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.   

 1676 Antonie van Leeuwenhoek applied his hobby of making microscopes from his own handmade lenses to observe some water running off a roof during a heavy rainstorm. He finds that it contains, in his words, "very little animalcules." The life he has found in the runoff water is not present in pure rainwater. This was a fundamental discovery, for it showed that the bacteria and one-celled animals did not fall from the sky. When a ball of molten glass is inflated like a balloon, a small droplet of the hot fluid collects at the very bottom the bubble. Leeuwenhoek used these droplets as microscope lenses to view the animalcules. Despite their crude nature, those early lenses enabled Leeuwenhoek to describe an amazing world of microscopic life. *  TIS  Compound microscopes (that is, microscopes using more than one lens) had been invented around 1595, nearly forty years before Leeuwenhoek was born. Several of Leeuwenhoek's predecessors and contemporaries, notably Robert Hooke in England and Jan Swammerdam in the Netherlands, had built compound microscopes and were making important discoveries with them. These were much more similar to the microscopes in use today. Thus, although Leeuwenhoek is sometimes called "the inventor of the microscope," he was no such thing.

1896 The Dow Jones Industrial Average was created by Charles Dow and Edward Jones, New York financial reporters.  Originally consisted of 11 stocks.  They published The Customers Appreciation Letter, which would become the Wall Street Journal.  The first index published was on July 3, 1884 *Kane, Famous First Facts 

1901 Giuseppe Peano terminated his services to the Royal Military Academy in Turin where he had taught for fifteen years. The trouble was with his teaching. Earlier he was a very good teacher and the author of several excellent texts, but as his work in mathematical logic matured he devoted too much time to what the students called “the symbols.” [H. C. Kennedy, Peano,p. 100] *VFR


1667 Abraham De Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) French mathematician who was a pioneer in the development of analytic trigonometry and in the theory of probability. He published The Doctrine of Chance in 1718. The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuities. He died in poverty, and correctly predicted the day of his own death. He found that he was sleeping 15 minutes longer each night and from this the arithmetic progression, calculated that he would die on the day that he slept for 24 hours. *TIS  
  Born in Vitry-le-Fran¸cois, France. Being a Protestant, he emigrated to England following the Edict of Nantes in 1685 where he eked out a living as a tutor of mathe­matics. He became thoroughly Anglicized and pronounced his name “Mowve-re.” *VFR
In Miscellanea Analytica (1730) appears Stirling's formula (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the normal curve as an approximation to the binomial. In the second edition of the book in 1738 de Moivre gives credit to Stirling for an improvement to the formula. De Moivre is also remembered for his formula for (cos x + i sin x)n which took trigonometry into analysis.

1750  William Morgan, FRS (26 May 1750 – 4 May 1833) was a Welsh physician, physicist and statistician, who is considered the father of modern actuarial science. *Wik

1826  Richard Christopher Carrington  (26 May 1826 – 27 November 1875) English astronomer who was the first to map the motions of sunspots and thus discover from them that the Sun rotates faster at the equator than near the poles (equatorial acceleration). He observed that the sunspots were not attached to any solid object, and also discovered the movement of sunspot zones toward the Sun's equator as the solar cycle progresses. On 1 Sep 1859, Carrington was the first to record the observation of a solar flare. *TIS

1837   Washington Augustus Roebling  (May 26, 1837 – July 21, 1926) U.S. civil engineer under whose direction the Brooklyn Bridge, New York City, was completed in 1883. The bridge was designed by Roebling with his father, John Augustus Roebling, from whom he had gained experience building wire-rope suspension bridges. Upon his father's death, he superintended the building of the Brooklyn Bridge (1869-83). He was disabled by decompression sickness after entering a caisson in 1872. He was brought out nearly insensible and his life was saved with difficulty. Because of resulting poor health, he directed operations from his home in Brooklyn overlooking the site. Though he continued to head the family's wire-rope manufacturing business for several years, medical problems forced retirement (1888).*TIS

1899 Otto Neugebauer (May 26, 1899 – February 19, 1990)  historian of ancient and medieval mathematics and astronomy. *VFR
  He was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences in antiquity and into the Middle Ages. By studying clay tablets he discovered that the ancient Babylonians knew much more about mathematics and astronomy than had been previously realized. The National Academy of Sciences has called Neugebauer "the most original and productive scholar of the history of the exact sciences, perhaps of the history of science, of our age." *Wik


 735 Bede  ( 672/673 – 26 May 735),(often know as the Venerable Bede) Anglo-Saxon theologian, historian and scholar whose writings established the use of BC and AD with dates. He applied a knowledge of astronomy for the purpose of calculating the correct date for Easter. He found that due to an imperfection in Sosigenes' Julian calendar, that the vernal equinox had slipped to a point three days before the traditional date of 21 Mar. However, no action was taken to make the necessary adjustment in the number of leap years per millenium until nine centuries later. Bede held that the earth was a sphere. He preserved Pytheas' suggestion relating tides to the phases of the moon, and followed Seleucus' idea that a high tide is a local effect and does not occur everywhere at the same point in time. *TIS
Bede was first buried at the monastery of St. Paul at Jarrow in 735. However, in about 1022, his bones were brought to Durham where they were placed with those of St. Cuthbert in the Choir. In 1370, Bede's remains were moved to a splendid shrine in the Galilee Chapel. This shrine was destroyed during the Reformation in 1540 and Bede's bones were then buried in a grave where the shrine had stood.
Eventually, in 1831, the present tomb, made from polished Carboniferous limestone, was erected over Bede's grave. It has the following simple inscription cut into its surface:
Translated from the Latin, this means 'In this tomb are the bones of the Venerable Bede' *Religion Facts

1926 Frank Nelson Cole (September 20, 1861 – May 26, 1926) At the time of his death he was a professor of mathematics at Columbia, but was living in a boarding house, under an assumed name, claiming to be a bookkeeper. The AMS Cole prize in algebra is named after him.*VFR 
His main research contributions are to number theory, in particular to prime numbers, and to group theory. In number theory he achieved the distinction of being the first to factor 267 - 1 and he did this using quadratic remainders. In fact
267 - 1 = 147573952589676412927 = 761838257287 × 193707721
which a computer will compute in a few seconds today. His contributions to factoring large numbers was published in 1903. His output of research papers was, however, fairly modest and he published only around 25 papers during his career. These publications include his doctoral dissertation in 1886 and a discussion of the icosahedron in 1887. He published The linear functions of a complex variable in the Annals of Mathematics in 1890 then, between the years 1891 to 1893, he found the complete list of simple groups with orders between 200 and 600. Another publication worth mentioning is The triad systems of thirteen letters which he published in the Transactions of the American Mathematical Society in 1913.*Wik According to a notice in the American Mathematical Monthly, which he had edited for twenty-five  years, he died of a heart attack brought on by an infected tooth.

2003 Gerald Stanley Hawkins ( April 28, 1928 Gt. Yarmouth, Norfolkshire, U.K.  - May 26, 2003) was a British-born American radio-astronomer who used a computer to show that the stones and other archaeological features at Stonehenge formed a pattern of alignments with 12 major lunar and solar events, suggesting that it was used as a sort of neolithic observatory or astronomical calendar. In the 18th century, William Stukely had noticed that the horseshoe of trilithons and 19 bluestones opened up in the direction of the midsummer sunrise. In the 1960s, Hawkins, a British-born radio astronomer, identified 165 key points in the neolithic complex and found that many were strongly correlated with the rising and setting positions of the sun and moon over an 18.6-year cycle. In the 1990s, he studied the geometry of crop circles.  He retired to a Virginia farm in Rappahannock County with his second wife, Julia Dobson, and died there, suddenly, on May 26, 2003.

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Friday, 25 May 2012

On This Day in Math - May 25

Some people are always critical of vague statements. 
I tend rather to be critical of precise statements; 
they are the only ones which can correctly be labeled wrong.

~ Raymond Smullyan on criticism

The 146th day of the year; 146 is 222 in base eight. *What's So Special About This Number

1581 John Dee, mathematician and mystic, first saw spirits in his crystal globe. [Daniel Cohen, Masters of the Occult , Dodd and Mead, 1971, p. 28] *VFR

1694 Isaac Newton to Nathaniel Hawes: “A Vulgar Mechanick can practice what he has been taught or seen done, but if he is in an error he knows not how to find it out and correct it, and if you put him out of his road, he is at a stand; Whereas he that is able to reason nimbly and judiciously about figure, force and motion, is never at rest till he gets over every rub.” Westfall takes this as the title of his fabulous biography of Newton, Never at Rest. Reportedly he received one of the early Golden Fleece Awards of Senator Proxmire for a grant to write this book. [But we have been unable to find a reference for this last statement.] *VFR

1721 John Copson of High Street in Philidelphia posted an offering in the American Weekly Mercury on this date announcing that he would open an office to provide fire insurance for "vessels, goods, and merchandise."  With that, he became the first Insurance agent in continental America.  *Kane, First Famous Facts

1747 Benjamin Franklin describes his electrical experiments in a letter to Peter Collinson. " Hence have arisen some new Terms among us. .... Or rather B is electrised plus and A minus. And we daily in our Experiments electrise Bodies plus or minus as we think proper. These Terms we may use till your Philosophers give us better. To electrise plus or minus, no more needs to be known than this; that the Parts of the Tube or Sphere, that are rub'd, do, in the Instant of the Friction, attract the Electrical Fire, and therefore take it from the Thing rubbing: the same Parts immediately, as the Friction upon them ceases, are disposed to give the Fire they have received, to any Body that has less."  *Collinson, "Experiments and Observations on Electricity Made at Philadelphia in America."

1832 Galois writes to his friend, Auguste Chevalier, about his broken romance for Stephanie. The fatal duel looms only five days away.

1842 Johann Christian Doppler (1803–1853) presented a lecture on
the Doppler effect. It was first experimentally verified in 1845 using a locomotive drawing an open car with several trumpeters. *VFR

1844 the first news communicated by telegraph in the U.S. was sent 80 miles to the Baltimore Patriot, Maryland, from Washington, D.C. giving the information that "One o'clock. There has just been made a motion in the House to go into committee of the whole on the Oregon question. Rejected. Ayes 79 - Nays 86." This was just one day after Samuel Morse transmitted his famous "What hath God wrought!" message from the U.S. Supreme Court room and opened America's first telegraph line linking Washington and Baltimore.*TIS

1946 The Soviet Union issued two stamps to celebrate the 125th anniversary of Pafnuti Lvovich Chebyshev (1821–1894). [Scott #1050-1].*VFR (see May 16th Births)

1961 the formal announcement of an American lunar landing was made by President John F. Kennedy speaking to the Congress: "I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the Moon and returning him safely to the Earth. No single space program in this period will be more impressive to mankind, or more important in the long-range exploration of space; and none will be so difficult or expensive to accomplish." *TIS

2142 The next total solar eclipse in Ostend, Belgium. The last total solar eclipse took place more than 11 centuries ago, 29 September 878. But only 9 years later, on 14 June 2151, there will be another one. *NASA Solar Eclipse Catalog (I know I can hardly wait!)

1828 Karl Peterson (25 May 1828 in Riga, Russia (now Latvia) - 19 April 1881 in Moscow, Russia)
was a Latvian mathematician worked in differential geometry and partial differential equations.  .. by means of a uniform general method, he deduced nearly all the devices known at that time for finding general solutions of different classes of equations.
Largely because he was not at a university his results were not well known but they did influence Egorov in Moscow, but Peterson gained an international reputation only when Darboux and Bianchi used his results.

1919 Raymond Merrill Smullyan ( May 25, 1919 -  ) is an American mathematician, concert pianist, logician, Taoist philosopher, and magician. His first career (like Persi Diaconis a generation later) was stage magic. He then earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959. He is one of many logicians to have studied under Alonzo Church. Smullyan is the author of many books on recreational mathematics, recreational logic, etc. Most notably, one is titled "What Is the Name of This Book?". *Wik For example the book is described on the cover as follows:"Beginning with fun-filled monkey tricks and classic brain-teasers with devilish new twists, Professor Smullyan spins a logical labyrinth of even more complex and challenging problems as he delves into some of the deepest paradoxes of logic and set theory, including Gödel's revolutionary theorem of undecidability."
Martin Gardner described this book in Scientific American as:"The most original, most profound and most humorous collection of recreational logic and mathematics problems ever written."

1555 Regnier Gemma Frisius (December 9, 1508–May 25, 1555) was a Dutch mathematician who applied his mathematical expertise to geography, astronomy and map making. He became the leading theoretical mathematician in the Low Countries.*SAU Frisius created or improved many instruments, including the cross-staff, the astrolabe and the astronomical rings. His students included Gerardus Mercator (who became his collaborator), Johannes Stadius, John Dee, Andreas Vesalius and Rembert Dodoens.*Wik

1676 Johann Rahn (10 March 1622 in Zürich, Switzerland - 25 May 1676 in Zürich, Switzerland)
was a Swiss mathematician who was the first to use the symbol "÷",called an obelus, for a division symbol in Teutsche Algebra (1659). The invention is also sometimes credited to British Mathematician John Pell who was Rahn's tutor for a while. The book, written in German, contains an example of Pell's equation also. Here is more on the various symbols used for division

1956 Johann Radon (16 December 1887 – 25 May 1956) worked on the calculus of variations, differential geometry and measure theory. 
Radon is known for a number of lasting contributions, including:
  • his part in the Radon–Nikodym theorem;
  • the Radon measure concept of measure as linear functional;
  • the Radon transform, in integral geometry, based on integration over hyperplanes — with application to tomography for scanners (see tomographic reconstruction);
  • Radon's theorem, that d + 2 points in d dimensions may always be partitioned into two subsets with intersecting convex hulls
  • the Radon-Hurwitz numbers.
  • He is possibly the first to make use of the so called Radon-Riesz property. *Wik

1989 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.

*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*SAU=St Andrews Univ. Math History
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell