Tuesday 6 December 2016

On This Day in Math - December 6

"I have finally found a subject where I do not need to memorize, but can think things out myself – mathematics."
~Herta Taussig Freitag (from her diary, age 12)

The 341st day of the year; 341 is the sum of seven consecutive primes, and 341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)

341 is the smallest of the pseudoprimes base 2,  disproving a Chinese math conjecture from around 500 BC. The conjecture was that p is prime IFF it divides 2p-2.

A pseudoprime n in base b is any composite number n such that \b^{n-1} \equiv 1 Mod n \)  so for this case, \( 2^{341-1} \equiv 1 mod (341) \)
for younger students that really means if you raise two to the 340th  power, and divide by 341, you get a remainder of one. 
Pseudoprimes are also called Poulet numbers, and Sarrus numbers.  "Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" mentioned above. 
"Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers *OEIS

341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)


In 1592, Galileo was appointed Professor of Mathematics at the University of Padua (the University of the Republic of Venice) at a salary of three times that he had received at Pisa. On 7 December 1592 he gave his inaugural lecture and began a period of 18 years at the University, years which he later described as the happiest of his life. *British Journal of Sports Medicine (honest)

In 1631, the transit of Venus occurred as first predicted by Kepler. He correctly predicted that an ascending node transit of Venus would occur in Dec 1631, but no-one observed it - due to the fact that it occurred after sunset for most of Europe. Kepler himself died in 1630. He not only predicted this particular transit but also worked out that transits of Venus involve a cyclical period of approximately 120 years. When such a transit is observed, Venus appears as a small black circle moving across the face of the Sun.*TIS

1710 An advertisement in the Old Bailey Proceedings for a book on mathematics, and more
*** The Marrow of the Mathematicks, made Plain and Easie to the Understanding of any ordinary Capacity. Containing the Doctrines of Arithmetick, Geometry, Astronomy, Gauging, the Use of the Sector, Surveying, Dyaling, and the Art of Navigation, &c. Illustrated with several Cuts, for the better Explanation of the whole Matter. After a New, Compendious, Easy Method By W. Pickering, Merchant-Adventurer.
To which is added,
Measuring Surfaces and Solids, such as Plank, Timber, Stone, &c. Joiners, Carpenters, Bricklayers, Glasiers, Painters and Paviers Work: Each Proposition being wrought Vulgarly, Decimally, Practically and Instrumentally. With a small Tract of Gauging Wine, Ale, or Malt, without Inches, or Division; by which any one may Gauge ten Backs or Floors of Malt, in the same time another shall Guage one, by the Way now used. Altogether New, and submitted to the Censure of the Honourable Commissioners of Excise. By J. L. P. M.
Both Printed for Eben. Tracy, at the Three Bibles on London-Bridge. 1710
Pedro Nunes Nonius original model
(They just don't make titles like they used to) Available on line for free here

1763 From Charles Mason's Journal of the Mason Dixon survey, "Set up a Sector brought by the Commissioners from Maryland and found that the nonius would not touch the middle part of the arch" A nonius is a device, named in honor to its author and inventor Pedro Nunes (Latin: Petrus Nonius), created in 1542 as a system for taking fine measurements on the astrolabe which could largely improve its accuracy. Later on, it was adapted in 1631 by the French mathematician Pierre Vernier, to create the vernier scale. *Wik

1830 First national observatory established at Washington, D.C. Established by the order of the Secretary of the Navy, John Branch, on 6 December 1830 as the Depot of Charts and Instruments, the Observatory rose from humble beginnings. Placed under the command of Lieutenant Louis M. Goldsborough, with an annual budget of $330, its primary function was the restoration, repair, and rating of navigational instruments. It was made into a national observatory in 1842 via a federal law and a Congressional appropriation of $25,000. Lieutenant James Melville Gilliss was put in charge of "obtaining the instruments needed and books." *Wik (Interestingly, Goldsborough was appointed to the Naval Academy at the age of seven, although he did not enter for several years. He rose to the rank of Admiral during the U S Civil War and three naval ships were named for him.)

1882 Venus crossed the disc of the Sun. The most recent transit of Venus when observed from Earth took place on June 8, 2004. The event received significant attention, since it was the first Venus transit to take place after the invention of broadcast media. No human alive at the time had witnessed a previous Venus transit. The next transit of Venus will occur on June 5–June 6 in 2012,. After 2012, the next transits of Venus will be in December 2117 and December 2125.*Wik
The transit of Venus across the sun was photgraphed on a series of glass plate negatives made by Amherst College astronomer David Peck Todd. He used a solar photographic telescope (made by the renowned optical firm Alvan Clark & Sons) stationed on the summit of Mount Hamilton, California, where the Lick Observatory was under construction. Of the photos, 147 survived, having been archived in the mountain vault. A century later, they were retrieved and an animation made from them premiered at the International Astronomical Union's general assembly in Sydney in Jul 2003. This is perhaps the most complete surviving record of a historical transit of Venus, dating from the time when Chester Arthur was president of the United States.*TIS

1917 Kazimierz Kuratowski gave a talk “On the definitions in mathematics,” which became his first published paper. This work grew out of Jan LLukasiewicz’s crushing criticism of the foundations of StanisLlaw Zaremba’s Theoretical Arithmetic (1912). Kuratowski’s now famous 1921 definition of ordered pair (a nice note for Alg classes) also grew out of LLukasiewicz’s critique. [Kuratowski, A Half Century of Polish Mathematics, p. 24] *VFR

1946 Birthdate of Nicolette Weil, younger daughter of the mathematician Andre Weil. She was born on St. Nicholas’ day, as he planned, or so he jokingly claimed, but she is named after Nicolas Bourbaki. Professor Weil was one of the founders of the Bourbaki group. See Joong Fang, Bourbaki, Paideia Press, 1970, p. 40. His older daughter is named Sylvie and was born 12 September 1942. *VFR

1956 The knapsack problem was first named and discussed by George B. Dantzig, the father of linear programming. *VFR (The part about naming it may be an error; the problem existed long before and *Wik has this note:) "The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. It is not known how the name "knapsack problem" originated,(they should read my blog?) though the problem was referred to as such in the early works of mathematician Tobias Dantzig (1884–1956)(This was George's Father), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined."

You Tube link (sorry, the embed has been disabled, so hit it on your own

In 1957, America's first attempt at putting a satellite into orbit failed when the Vanguard rocket carrying it blew up on the launch pad at Cape Canaveral, Florida. With a series of rumbles audible for miles around, the vehicle, having risen about four feet into the air, suddenly sank. Falling against the firing structure, fuel tanks rupturing as it did so, the rocket toppled to the ground on the northeast or ocean side of the structure in a roaring, rolling, ball-shaped volcano of flame. *TIS

1963 Time magazine published a copy of Salvador Dali’s “Fifty abstract pictures which as seen from two yards change into three Lenines masquerading as Chinese and as seen from six yards appear as the head of a royal tiger.” It is based on the semi-regular tessellation 4–3–4–3–3 made up of squares and triangles.*VFR

1987 Florida rapist Tommy Lee Andrews is the first person to be convicted as a result of DNA fingerprinting. *Wik

2005 At a book signing after a mathematics professor at West Point was asked what he taught, former president Jimmy Carter commented “In retrospect, I possibly received the best insight into human nature by studying differential equations and systems of differential equations. That subject seemed to interrelate rates of change between interconnected entities.” *VFR


1586 Niccolò Zucchi (6 Dec 1586; 21 May 1670) Italian astronomer who, in approximately 1616, designed one of the earliest reflecting telescopes, antedating those of James Gregory and Sir Isaac Newton. A professor at the Jesuit College in Rome, Zucchi developed an interest in astronomy from a meeting with Johannes Kepler. With this telescope Zucchi discovered the belts of the planet Jupiter (1630) and examined the spots on Mars (1640). He also demonstrated (in 1652) that phosphors generate rather than store light. His book Optica philosophia experimentalis et ratione a fundamentis constituta (1652-56) inspired Gregory and Newton to build improved telescopes.*TIS(I belive the observations he made were NOT with his reflecting telescope, which never seemed to work, but with the more common refracting telescope. See more on reflecting telescopes at this blog where Thony Christie takes me to task for giving (too much) credit to one of the  early developers.)

1682 Giulio Carlo Fagnano dei Toschi (December 6, 1682 – September 26, 1766) who was born in Sinigaglia, Italy. He was the founder of the geometry of the triangle, studied the lemniscate, and coined the term “elliptic integral.” *VFR
Fagnano is best known for investigations on the length and division of arcs of certain curves, especially the lemniscate; this seems also to have been in his own estimation his most important work, since he had the figure of the lemniscate with the inscription: "Multifariam divisa atque dimensa Deo veritatis gloria", engraved on the title-page of his Produzioni Matematiche, which he published in two volumes (Pesaro, 1750), and dedicated to Pope Benedict XIV. The same figure and words "Deo veritatis gloria" also appear on his tomb.
Failing to rectify the ellipse or hyperbola, Fagnano attempted to determine arcs whose difference should be rectifiable. He also pointed out the remarkable analogy existing between the integrals which represent the arc of a circle and the arc of a lemniscate. Finally he proved the formula π = 2i log((1-i)/(1+i)
One of his son, Giovanni, is the namesake of the optimization problem called Fagnano's Problem in geometry :
For a given acute triangle determine the inscribed triangle of minimal perimeter.
The solution is the orthic triangle.

1848 Johann Palisa (6 Dec 1848; 2 May 1925) Silesian astronomer who was a prolific discoverer of asteroids, 122 in all, beginning with Asteroid 136 Austria (on 18 Mar 1874, using a 6” refractor) to Asteroid 1073 Gellivara in 1923 - all by visual observation, without the aid of photography. In 1883, he joined the expedition of the French academy to observe the total solar eclipse on May 6 of that year. During the eclipse, he searched for the putative planet Vulcan, which was supposed to circle the sun within the orbit of Mercury. In addition to observing the eclipse, Palisa collected insects for the Natural History Museum in Vienna. He also prepared two catalogs containing the positions of almost 4,700 stars. He remains the most successful visual discoverer in the history of minor planet research.*TIS

1856 Walther Franz Anton von Dyck (6 Dec 1856 in Munich, Germany - 5 Nov 1934 in Munich, Germany) Von Dyck made important contributions to function theory, group theory (where a fundamental result on group presentations is named after him) topology and potential theory. *SAU

1880 Pierre Léon Boutroux (6 December 1880 – 15 August 1922) was a French mathematician and historian of science. Boutroux is chiefly known for his work in the history and philosophy of mathematics.
He was born in Paris on 6 December 1880 into a well connected family of the French intelligentsia. His father was the philosopher Émile Boutroux. His mother was Aline Catherine Eugénie Poincaré, sister of the scientist and mathematician Henri Poincaré. A cousin, Raymond Poincaré was to be President of France.
He occupied the mathematics chair at Princeton University from 1913 until 1914. He occupied the History of sciences chair from 1920 to 1922.
Boutroux published his major work Les principes de l'analyse mathématique in two volumes; Volume 1 in 1914 and Volume 2 in 1919. This is a comprehensive view of the whole field of mathematics at the time.*Wik

1900 George Eugene Uhlenbeck (6 Dec 1900; 31 Oct 1988) Dutch-American physicist who, with Samuel A. Goudsmit, proposed the concept of electron spin (Jan 1925) - a fourth quantum number which was a half integer. This provided Wolfgang Pauli's anticipated "fourth quantum number." In their experiment, a horizontal beam of silver atoms travelling through a vertical magnetic field was deflected in two directions according to the interaction of their spin (either "up" or "down") with the magnetic field. This was the first demonstration of this quantum effect, and an early confirmation of quantum theory. As well as fundamental work on quantum mechanics, Uhlenbeck worked on atomic structure, the kinetic theory of matter and extended Boltzmann's equation to dense gases.*TIS

1907 Mathematical Logician Rosser is Born, J. Barkley Rosser is born in Jaksonville, FL. In 1934 Rosser received a Ph.D. in logic from Princeton under the supervision of Alonso Church. Rosser was able to anticipate the potential of early computers in many areas of mathematics as well as the ultimate impact of logic on the future of computing. He contributed to the Church-Rosser theorem that identifies the outer limit of what is achievable in automated theorem proving and, therefore, plays the same role in computing science as the second law of thermodynamics in engineering.
Rosser taught at Cornell and the University of Wisconsin, served as a president of the Association of Symbolic Logic and SIAM. He died on September 5, 1989. *CHM

1908 Herta Taussig Freitag (December 6, 1908 - January 25, 2000) Herta obtained a job at a private high school, the Greer School, in upstate New York. There she met Arthur H. Freitag and they were married in 1950. Herta started teaching at Hollins College (now University) in Roanoke, VA in 1948. She received a Ph.D. degree from Columbia University in 1953 and the title of her dissertation was "The Use of the History of Mathematics in its Teaching and Learning on the Secondary Level."
During Herta's years at Hollins she was a frequent guest speaker at local schools and gave lectures at both Virginia and North Carolina Governor's Schools. She published numerous articles in The Mathematics Teacher, The Arithmetic Teacher, and The Mathematics Magazine. At the request of the National Council of Teachers of Mathematics, Professor Freitag wrote the monograph, The Number Story, with her husband. In 1962 she was the first woman to be President of the Maryland-District of Columbia-Virginia Section of the Mathematical Association of America (MAA). Professor Freitag received the Hollins' Algernon Sydney Sullivan Award, which is awarded for recognition of "extraordinary humane and scholarly achievement." She officially retired from Hollins in 1971 to spend time with her husband, who was ill. After his death in 1978, Hollins welcomed her back to the classroom as a leave replacement in 1979-1980 and as a teacher in the Master of Arts in Liberal Studies (MALS) program for several years. Professor Herta Freitag was the first faculty member to receive the Hollins Medal (1979) and the first recipient of the Virginia College Mathematics Teacher of the Year award (1980).
Professor Freitag was very proud of her perfect attendance at the International Conferences of the Fibonacci Association. Most of her work with Fibonacci numbers occurred after she retired, which demonstrates the fallacy of a commonly held belief that mathematicians complete their best work before the age of 40. Professor Freitag published more than thirty articles in the Fibonacci Quarterly after 1985. The November 1996 issue of the Fibonacci Quarterly was dedicated to "Herta Taussig Freitag as she enters her 89th year, in recognition of her years of outstanding service and achievement in the mathematics community through excellence in teaching, problem solving, lecturing and research." This award was given to celebrate her 89th birthday, since 89 is a Fibonacci number. *Biographies of Women Mathematicians, Agnes Scott College web site

1941 Filep László (6 Dec 1941 in Csaszlo, Szabolcs-Szatmar-Bereg, Hungary - 19 Nov 2004 in Budapest, Hungary) He worked for the degree of dr. univ. submitting his thesis Life and work of Gyula Farkas (1847-1930) to the Kossuth University in Debrecen in 1978. But this was not László's first publication, for he had published a number of articles in the prestigious Hungarian popular scientific magazine, Termeszet Vilaga (The World of Nature). The first of these articles was Farkas Gyula (1847-1930) published in 1976, was followed by A matematika nagy nöalakjai (1977) and Helyunk a tudomany vilagaban (1979). He published many other articles on the history of mathematics such as Lajos David (1881-1962), historian of Hungarian mathematics (1981), Great female figures of Hungarian mathematics in 19th-20th centuries (1983), The development, and the developing of the concept of a , fraction (2001), The genesis of Eudoxus's infinity lemma and proportion theory (2001), From Fejer's disciples to Erdős's epsilons - change over from analysis to combinatorics in Hungarian mathematics (2002), and Irrationality and approximation of √2 and √3 in Greek mathematics (2004). He also published biographies of many mathematicians including Janos Bolyai, John C Harsanyi, John von Neumann, and Paul Erdős. László's research interest was not only in the history of mathematics for he also published a long series of papers on fuzzy groups, some written with his collaborator Iulius Gyula Maurer, beginning in 1987. *SAU


1788 Nicole-Reine Lepaute (5 Jan 1723 in Paris, France - 6 Dec 1788 in Saint-Cloud, France) was a French noblewoman who helped Lalande with astronomical calculations. In June 1757 Lalande decided that he would like to attempt to calculate a precise date for the return of Halley's comet. It was known to have been seen in 1305, 1380, 1456, 1531, 1607 and 1682 and Halley, taking into account perturbations to the orbit caused by the gravitational effects of Jupiter, had predicted that the comet would return reaching perihelion in December 1758. However the only way to get a more accurate prediction of its date of return was to calculate the perturbations to the orbit caused by the gravitational effects of both Jupiter and Saturn. Lalande approached Alexis Clairaut for help and Clairaut provided a basic programme of work requiring an extraordinary amount of computation. Lalande then asked Nicole-Reine Lepaute to assist him in the computations. Lalande wrote, "During six months we calculated from morning to night, sometimes even at meals. ... The assistance of Mme Lepaute was such that, without her I should never have been able to undertake the enormous labour, in which it was necessary to calculate the distance of each of the two planets Jupiter and Saturn from the comet, separately for each successive degree for 150 years. *SAU

1893 Rudolf Wolf (7 Jul 1816, 6 Dec 1893) Swiss astronomer and astronomical historian. Wolf's main contribution was the discovery of the 11 year sunspot cycle and he was the codiscoverer of its connection with geomagnetic activity on Earth. In 1849 he devised a system now known as Wolf's sunspot numbers. This system is still in use for studying solar activity by counting sunspots and sunspot groups. In mathematics, Wolf wrote on prime number theory and geometry, then later on probability and statistics - a long paper discussed Buffon's needle experiment. He estimated by Monte Carlo methods.*TIS

1959 Erhard Schmidt (13 Jan 1876 in Dorpat, Estonia (Russian Empire) (now Tartu, Estonia)- 16 Dec 1959 in Berlin, Germany) 1876 Erhard Schmidt (13 January 1876 – 6 December 1959) was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia (now Estonia). His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations.
Together with David Hilbert he made important contributions to functional analysis. He is best known for the Gram-Schmidt orthogonalisation process, which constructs an orthogonal base from any vector space. *Wik

1973 Joseph Leonard Walsh, (September 21, 1895 – December 6, 1973) was an American mathematician. His work was mainly in the field of analysis.
For most of his professional career he studied and worked at Harvard University.*Wik

1990 Lev Arkad'evich Kaluznin (31 Jan 1914 in Moscow, Russia - 6 Dec 1990 in Moscow, Russia) Kaluznin is best known for his work in group theory and in particular permutation groups. He studied the Sylow p-subgroups of symmetric groups and their generalisations. In the case of symmetric groups of degree pn, these subgroups were constructed from cyclic groups of order p by taking their wreath product. His work allowed computations in groups to be replaced by computations in certain polynomial algebras over the field of p elements. Despite the fact that the earliest applications of wreath products of permutation groups was due to C Jordan, W Specht and G Polya, it was Kaluznin who first developed special computational tools for this purpose. Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more. These results have been included in many textbooks on group theory. *SAU

1993 Wolfgang Paul (10 Aug 1913, 6 Dec 1993) German physicist who developed the Paul trap, an electromagnetic device that captures ions and holds them long enough for study and precise measurement of their properties. During the 1950s he developed the so-called Paul trap as a means of confining and studying electrons. The device consists of three electrodes - two end caps and an encircling ring. The ring is connected to an oscillating potential. The direction of the electric field alternates; for half the time the electron is pushed from the caps to the ring and for the other half it is pulled from the ring and pushed towards the caps. For his work he shared the 1989 Nobel Prize for Physics with Hans G. Dehmelt and Norman F. Ramsey.*TIS

1996 Stefan Schwarz (18 May 1914 in Nové Mesto nad Váhom, Austria-Hungarian Empire (now Slovakia) - 6 Dec 1996 in Bratislava, Slovak Republic) In addition to his work on semigroups, number theory and finite fields, Schwarz contributed to the theory of non-negative and Boolean matrices.
Schwarz organized the first International Conference on Semigroups in 1968. At this conference setting up the journal Semigroup Forum was discussed and Schwarz became an editor from Volume 1 which appeared in 1970, continuing as editor until 1982. This was not his first editorial role since he had been an editor of the Czechoslovak Mathematical Journal from 1945 and continued to edit this journal until he was nearly 80 years old. He also founded the Mathematico-Physical Journal of the Slovak Academy of Sciences in 1950 and continued as an editor of the mathematics part of the journal when it split from the physics part to become Mathematica Slovaca until 1990. *SAU

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

No comments: