Saturday, 28 December 2024

Pick Two and Get One of Each?

   Another from my archives: Nice for Alg II level or so, I think.



A colleague from Colorado sent me an interesting probability problem the other day. I like it because it illustrates one of those serendipitous qualities of mathematics. Here is the problem. A Jar has a mixture of Red and White balls so that if you withdraw two, the probability of getting two alike, or two of different color are both equal to one-half. You may want to stop and try it before you read on.
Ok, so we let r = the number of red ones and w be the number of white ones, and the total is r+w. So how could we draw one of each color? Well, red first, and then white, or white first and then red. If we find the probability of each of these conditional events and add them up, that will have to equal 1/2. Ok, the probability of red on the first draw is r/(r+w), and on the second ball the probability is w/(r+w-1) since one of the red balls will be missing. The opposite order is exactly the same with the w and r reversed, so the probability of getting one of each color is 2rw/[(r+w)(r+w-1)]. Setting that equal to 1/2 we get 

If we expand (r+w)2 and subtract the 4rw we get 0=r2-2rw +w2-r-w. NOTICE the symmetry, we could exchange r and w and get the same equation. We know right off that any solution (r,w) will have another solution (w,r).

One of the things that is often hard for students is to think of one variable as a constant and the other as a variable. I like to use the word "pronumeral", like a pronoun only instead of him or her we say "that number". It is like a variable that doesn't vary, we just don't know what it is in a particular case. So think of w as if it were fixed. We have that many white balls in the jar and we are wondering how many red can be put in to make the problem work... see it.. w is a "fixed" unknown, but r is going to "vary". That makes the equation a quadratic in r; Ar2 +Br+c=0 where A=1, B= -2w-1, and C=w2-w.
We can solve this using the quadratic formula, but if this solution is going to be a rational number, and the number of balls in a jar must be rational, then the discriminant, the expression under the squre root radical in the quadratic formula, B2-4AC, must be a perfect square.
 B2= 4w2+4w+1 and 4AC= 4w2-4w; so B2-4AC= 8W+1. If there is a rational solution, it must be when 8W+1 is a perfect square.

 Wait, I know this one! That's a problem from number theory. The numbers that make 8W+1 a perfect square are called triangular numbers; 1, 3, 6, 10, 15. They are the sum of the first n counting numbers. But a neat thing happens if we plug 1 in for W, the solution for r is 3.... and if we use 3 for w, r=6. Each time we substitute one of the triangular numbers into the quadratic, the next comes out as a solution. So the probability of drawing two balls of the same color, (or of two that are not alike) will equal 1/2 whenever the number of balls of each color are consecutrive triangular numbers. A very geometric solution to a very algebraic question. 

On This Day in Math - December 28

   


Shadow Family in Cowtown


Anyone who considers arithmetical methods of producing random digits is,
of course,
in the state of sin.
~John Von Neumann


The 362nd day of the year; 362 and its double and triple all use the same number of digits in Roman numerals.*What's Special About This Number.

3!+6!+2! - 1 =727 and 3!*6!*2! + 1=8641 are both prime *Prime Curios

362 is the sum of 3 nonzero squares in exactly 4 ways.(collect the whole set!)


EVENTS

1579: John Dee reveals alchemical secrets to Roger Cook, a fellow alchemist, they would fall out in 1581. Dee’s magic disc is below (Wellcome/BM)


1612 Galileo observed Neptune, but did not recognize it as a planet. Galileo's drawings show that he first observed Neptune on December 28, 1612, and again on January 27, 1613. On both occasions, Galileo mistook Neptune for a fixed star when it appeared very close—in conjunction—to Jupiter in the night sky; hence, he is not credited with Neptune's discovery. (The official discovery is usually cited as September 23, 1846, Neptune was discovered within 1° of where Le Verrier had predicted it to be.) During the period of his first observation in December 1612, Neptune was stationary in the sky because it had just turned retrograde that very day. This apparent backward motion is created when the orbit of the Earth takes it past an outer planet. Since Neptune was only beginning its yearly retrograde cycle, the motion of the planet was far too slight to be detected with Galileo's small telescope.*Wik



1695  On this day in 1695, after fitting several comets data using Newton's proposal that they followed parabolic paths, Edmund Halley tested his own measurements of the 1682 comet against an elliptical orbit. He wrote to Newton,
I am more and more confirmed that we have seen that Comet now three times since the year 1531.



=======================================================
1828 Henry R. Rowlands becomes the first American to patent a device for walking on water. Since that time there have been at least one-hundred other patents approved in the US for similar devices. All seem to be inspired by the earliest known design (Jesus excepted) by Leonardo da Vinci in the late Fifteenth Century.





1893 Simon Newcomb gives a speech to the New York Mathematical Society with comments on the fourth dimension; "It is a perfectly legitimate exercise .... if we should not stop at three dimensions in geometry, but construct one for space having four... and there is room for an indefinite number of universes". He also called his speculations on the fourth dimension, "the fairyland of geometry."
The speech appears a short time later on February 1, 1894 in Nature. His comments would also be commented on in H. G. Wells, Time Machine. "But some philosophical people have been asking ... - Why not another direction at right angles to the other three? ... Professor Simon Newcomb was expanding on this only a month or so ago." *Alfred M. Bork, The Fourth Dimension in Nineteenth-Century Physics, Isis, Sept 1964 pg 326-338
Newcomb is famously quoted as having believed it impossible to build a "flying machine." He begins an article titled "Is the Airship Possible?" with the remark, "That depends, first of all, on whether we are to make the requisite scientific discoveries." He ends with the remark "the construction of an aerial vehicle ... which could carry even a single man from place-to-place at pleasure requires the discovery of some new metal or some new force."




In 1893, Professor James Dewar gave six well-illustrated lectures on "Air gaseous and liquid," at the Royal Institution, London, 28 Dec 1893 - 9 Jan 1894. Some of the air in the room was liquefied in the presence of the audience and it remained so for some time, when enclosed in a vacuum jacket. Again, 1 Apr 1898.
My favorite stupid joke about Thermos Bottles: "You put hot stuff in a thermos, it stays hot. You put cold stuff in a thermos, it stays cold. BUT How does the Thermos know which is which?"





1895 Wilhelm Conrad Rontgen announces that he has taken an x-ray of his wife’s hand in a paper, "Ein neue Art von Strahlen", to the Würzburg Physical-Medical-Society on 28 Dec and it appeared in their 1895 proceedings. By January he was famous. In the next year some 50 books and 1000 papers appeared on the subject! A journal devoted to the subject was founded in May 1896.





1895 The Lumières held their first public screening of projected motion pictures in 1895. The Lumière brothers, Auguste Marie Louis Nicolas [oɡyst maʁi lwi nikɔla] (19 October 1862, Besançon, France – 10 April 1954, Lyon) and Louis Jean (5 October 1864, Besançon, France – 6 June 1948, Bandol) were the earliest filmmakers in history. (Appropriately, "lumière" translates as "light" in English.)
Their first public screening of films at which admission was charged was held on December 28, 1895, at Salon Indien du Grand Café in Paris. This history-making presentation featured ten short films, including their first film, Sortie des Usines Lumière à Lyon (Workers Leaving the Lumière Factory). Each film is 17 meters long, which, when hand cranked through a projector, runs approximately 50 seconds. *Wik


1923 George David Birkhoff of Harvard received the first Bocher Memorial Prize for his paper “Dynamical systems with two degrees of freedom.” *VFR

1938 Kurt Godel lectures to the annual AMS meeting, Williamsburg, on the consistency of the axiom of choice and the generalized continuum hypothesis. Independence was proved in 1963 by Paul Cohen. *VFR

In 1931, Irene Joliot-Curie reported her study of the unusually penetrating radiation released when beryllium was bombarded by alpha particles seen by the German physicists, Walter Bothe and H. Becker in 1930. Joliot-Curie (daughter of Marie and Pierre Curie) agreed with them that the radiation was energetic gamma rays. She further discovered that if the emitted radiation passed through paraffin (or other hydrogen containing materials), large numbers of protons were released. Since this was, in fact, a previously unknown result for gamma rays, she lacked an explanation. It was to be the experiments of James Chadwick performed during 7-17 Feb that would discover the radiation was in fact new particles - neutrons.*TIS
Irene and Marie Curie




1973 For a really big ellipse, consider the orbit of the comet kohoutek, which reached perihelion on this date. The length of the major and minor axes are 3,600 and 44 Astronomical Units. The comet’s eccentricity is approximately 0.99993. *UMAP Journal, 4(1983), p. 164
Comet Kohoutek is a long-period comet; its previous apparition was about 150,000 years ago, and its next apparition will be in about 75,000 years. The comet was discovered on March 18th on photographic plates taken on March 7th and 9th by Czech astronomer Luboš Kohoutek, for whom the comet is named. *Wik
Comet Kohoutek on 11 January 1974 as seen from Catalina Observatory




In 2005, the first in a network of satellites, named Galileo, was launched by a consortium of European governments and companies. By 2011, Galileo will consist of 30 satellites providing worldwide coverage as an alternative to the U.S. monopoly with its Global Positioning System (GPS). At a cost of $4 billion, it's Europe's biggest-ever space project, with one-third contributed by governments and the balance from eight companies. Since the American GPS is controlled by the military, the European satellite network is designed to ensure independence for civilian use, but also offer more precision for a paid service. Customers are expected to include service for small airports, transportation, and mobile phone manufacturers to build in navigation capabilities.*TIS

2009 Longest flight by a paper-only plane-Takuo Toda sets world record
TOKYO, Japan--Using a specially designed 10cm long paper plane, Japanese origami plane virtuoso Takuo Toda's origami flight in a Japan Airlines hangar near Tokyo's Haneda Airport lasted 26.1s - setting the world record for the Longest flight by a paper-only plane.
This one was made strictly in keeping with traditional rules of the ancient Japanese art; only one sheet of paper was folded by hand, with no scissors or glue. He had previously set a record for time aloft with a plane that included tape. *worldrecordsacademy.org
There is a video here.



2013 Voyager 1 is a 722-kilogram (1,590 lb) space probe launched by NASA on September 5, 1977 to study the outer Solar System. Operating for 36 years, 3 months, and 23 days as of 28 December 2013, the spacecraft communicates with the Deep Space Network to receive routine commands and return data. At a distance of about 127.21 AU (1.903×1010 km) from the Earth as of 28 December 2013, it is the farthest human made object from Earth. *Wik


 

BIRTHS

1798 Thomas Henderson (28 Dec 1798; 23 Nov 1844) Scottish astronomer, the first Scottish Astronomer Royal (1834), who was first to measure the parallax of a star (Alpha Centauri, observed at the Cape of Good Hope) in 1831-33, but delayed publication of his results until Jan 1839. By then, a few months earlier, both Friedrich Bessel and Friedrich Struve had been recognized as first for their measurements of stellar parallaxes. Alpha Centauri can be observed from the Cape, though not from Britain. It is now known to be the nearest star to the Sun, but is still so distant that its light takes 4.5 years to reach us. As Scottish Astronomer Royal in 1834, he worked diligently at the Edinburgh observatory for ten years, making over 60,000 observations of star positions before his death in 1844. *TIS
Alpha Centauri, on the left forefoot of the Centarur, in Johann Bayer, Uranometria, 1603. For the complete plate, see third image (Linda Hall Library)




1808 Victoire Louis Athanase Dupré (December 28 1808 ; August 10 1869 ) was a French mathematician and physicist.
He worked on number theory and in the 1860s with thermodynamics and from him comes the textbook mécanique Théorie de la Chaleur (1869), which is essentially the distribution of this then-new field of knowledge in France contributed. Together with his son Paul Dupré experimental research, he examined the capillary and the surface tension of liquids. This work also led to a formulation of Young's equation which is known today as the Young-Dupré equation. *Wik



1873 William Draper Harkins (28 Dec 1873; 7 Mar 1951) American nuclear chemist who was one of the first to investigate the structure and fusion reactions of the nucleus. In 1920, Harkins predicted the existence of the neutron, subsequently discovered by Chadwick's experiment. He made pioneering studies of nuclear reactions with Wilson cloud chambers. In the early 1930's, (with M.D. Kamen) he built a cyclotron. Harkins demonstrated that in neutron bombardment reactions the first step in neutron capture is the formation of an "excited nucleus" of measurable lifetime, which subsequently splits into fragments. He also suggested that subatomic energy might provide enough energy to power the Sun over its lifetime.*TIS



1882 Sir Arthur Stanley Eddington (28 Dec 1882; 22 Nov 1944) English astrophysicist, and mathematician known for his work on the motion, distribution, evolution and structure of stars. He also interpreted Einstein's general theory of relativity. He was one of the first to suggest (1917) conversion of matter into radiation powered the stars. In 1919, he led a solar eclipse expedition which confirmed the predicted bending of starlight by gravity. He developed an equation for radiation pressure. In 1924, he derived an important mass-luminosity relation. He also studied pulsations in Cepheid variables, and the very high densities of white dwarfs. He sought fundamental relationships between the principle physical constants. Eddington wrote many books for the general reader, including Stars and Atoms. *TIS  One of my favorite stories about Eddington is this one: Ludwick Silberstein approached Eddington and told him that people believed he was one of only three people in the world who understood general relativity, and that included Einstein. When Eddington didn't respond for a moment he prodded, come on, don't be modest, and Eddington replied, "Oh, no.  It's not that.  I was just trying to figure out who the third was?"  *Mario Livio, Brilliant Blunders
One of Eddington's photographs of the total solar eclipse of 29 May 1919, presented in his 1920 paper announcing its success, confirming Einstein's theory that light "bends"





1898 Carl-Gustaf Arvid Rossby (28 Dec 1898; 19 Aug 1957) Swedish-U.S. meteorologist who first explained the large-scale motions of the atmosphere in terms of fluid mechanics. His work contributed to developing meteorology as a science. Rossby first theorized about the existence of the jet stream in 1939, and that it governs the easterly movement of most weather. U.S. Army Air Corps pilots flying B-29 bombing missions across the Pacific Ocean during World War II proved the jet stream's existence. The pilots found that when they flew from east to west, they experienced slower arrival times and fuel shortage problems. When flying from west to east, however, they found the opposite to be true. Rossby created mathematical models (Rossby equations) for computerized weather prediction (1950). *TIS



1903 John von Neumann is born in Budapest, Hungary.(28 Dec 1903, 8 Feb 1957) His prodigious abilities were recognized in the early childhood. He obtained a degree in chemical engineering attending the University of Berlin (1921-1923) and the Technische Hochschule in Zurich (1923-1926). *CHM
He made important contributions in quantum physics, logic, meteorology, and computer science. He invented game theory, the branch of mathematics that analyses strategy and is now widely employed for military and economic purposes. During WW II, he studied the implosion method for bringing nuclear fuel to explosion and he participated in the development of the hydrogen bomb. He also set quantum theory upon a rigorous mathematical basis. In computer theory, von Neumann did much of the pioneering work in logical design, in the problem of obtaining reliable answers from a machine with unreliable components, the function of "memory," and machine imitation of "randomness." *TIS



1929 Maarten Schmidt (28 December 1929 – 17 September 2022) Dutch-born American astronomer who in 1963 discovered quasars (quasi-stellar objects). The hydrogen spectrum of these starlike objects shows a huge redshift, which indicates they are more distant than normal stars, travelling away at greater speed, and are among the oldest objects observed. In turn, this indicates they existed only when the universe was very young, and provides evidence against the steady state theory of Fred Hoyle. Schmidt is currently seeking to find the redshift above which there are no quasars, and he also studies x-ray and gamma ray sources.*TIS



1940 Peter Michael Neumann OBE (28 December 1940 – 18 December 2020) was a British mathematician. His fields of interest included the history of mathematics and Galois theory.

Neumann was a son of the German-born mathematicians Bernhard Neumann and Hanna Neumann. He gained a BA degree from The Queen's College, Oxford in 1963, and a DPhil degree from the University of Oxford in 1966. On completing his doctrate, Neumann was named a Tutorial Fellow at the Queen's College, Oxford, and in 1967 he became a lecturer at the University of Oxford. His research work was in the field of group theory. In 1987, Neumann won the Lester R. Ford Award of the Mathematical Association of America for his review of Harold Edwards' book Galois Theory.

He was the first Chairman of the United Kingdom Mathematics Trust, from October 1996 to April 2004, succeeded by Bernard Silverman.

Neumann showed in 1997 that Alhazen's problem (reflecting a light ray off a spherical mirror to hit a target) cannot be solved with a straightedge and compass construction. Although the solution is a straightforward application of Galois theory it settles the constructibility of one of the last remaining geometric construction problems posed in antiquity.

In 2003, the London Mathematical Society awarded him the Senior Whitehead Prize. He was appointed Officer of the Order of the British Empire (OBE) in the 2008 New Year Honours.

After retiring in 2008, he became an Emeritus Fellow at the Queen's College.

Neumann's work in the history of mathematics includes his 2011 publication The Mathematical Writings of Évariste Galois, an English language book on the work of French mathematician Évariste Galois (1811–1832).[10] Neumann was a long-standing supporter of the British Society for the History of Mathematics, whose Neumann Prize is named in his honour]

Neumann was the president of the Mathematical Association from 2015 to 2016.*Wik





DEATHS

1663 Francesco Maria Grimaldi (2 Apr 1618, 28 Dec 1663) Italian mathematician and physicist who studied the diffraction of light. He observed the image on a screen in a darkened room of a tiny beam of sunlight after it passed pass through a fine screen (or a slit, edge of a screen, wire, hair, fabric or bird feather). The image had iridescent fringes, and deviated from a normal geometrical shadow. He coined the name diffraction for this change of trajectory of the light passing near opaque objects (though, more specifically, it may have been interferences with two close sources that he observed). This provided evidence for later physicists to support the wave theory of light. With Riccioli, he investigated the object in free fall (1640-50), and found that distance of fall was proportional to the square of the time taken.*TIS



1827 Robert Woodhouse (28 April 1773 – 23 December 1827) was an English mathematician. His earliest work, entitled the Principles of Analytical Calculation, was published at Cambridge in 1803. In this he explained the differential notation and strongly pressed the employment of it; but he severely criticized the methods used by continental writers, and their constant assumption of non-evident principles. This was followed in 1809 by a trigonometry (plane and spherical), and in 1810 by a historical treatise on the calculus of variations and isoperimetrical problems. He next produced an astronomy; of which the first book (usually bound in two volumes), on practical and descriptive astronomy, was issued in 1812, and the second book, containing an account of the treatment of physical astronomy by Pierre-Simon Laplace and other continental writers, was issued in 1818.
He became the Lucasian Professor of Mathematics in 1820, and subsequently the Plumian professor in the university. As Plumian Professor he was responsible for installing and adjusting the transit instruments and clocks at the Cambridge Observatory.[3] He held that position until his death in 1827. *Wik
The frontispiece of Woodhouse's Principles of Analytical Calculation (1803)




1871 John Henry Pratt (4 June 1809 - 28 December 1871) was a British clergyman and mathematician who devised a theory of crustal balance which would become the basis for the isostasy principle. *Wik

1896 Horatio (Emmons) Hale (3 May 1817, 28 Dec 1896) was an American anthropologist whose contributions to the science of ethnology, included his theory of the origin of the diversities of human languages and dialectsa theory suggested by his study of child languages (the languages invented by little children). He emphasized the importance of languages as tests of mental capacity and as criteria for the classification of human groups. Hale was the first to discover that the Tutelos of Virginia belonged to the Siouan family, and to identify the Cherokee as a member of the Iroquoian family of speech. He sailed with the scientific corps of the Wilkes Exploring Expedition (1838-42) collecting linguistic materials. He used the drift of the Polynesian tongue as a clue to the migration of this race. *TIS



1919 Johannes Robert Rydberg​, (‘Janne’ to his friends), (November 8, 1854 – December 28, 1919), was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to predict the wavelengths of photons (of light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom.
The physical constant known as the Rydberg constant is named after him, as is the Rydberg unit. Excited atoms with very high values of the principal quantum number, represented by n in the Rydberg formula, are called Rydberg atoms. Rydberg's anticipation that spectral studies could assist in a theoretical understanding of the atom and its chemical properties was justified in 1913 by the work of Niels Bohr (see hydrogen spectrum). An important spectroscopic constant based on a hypothetical atom of infinite mass is called the Rydberg (R) in his honour. *Wik




1923 Gustave Eiffel (15 Dec 1832, 28 Dec 1923) French civil engineer who specialized in metal structures, known especially for the Eiffel Tower in Paris. He built his first of his iron bridges at Bordeaux (1858) and was among the first engineers to build bridge foundations using compressed-air caissons. His work includes designing the rotatable dome for Nice Observatory on the summit of Mont Gros (1886), and the framework for the Statue of Liberty now in New York Harbor. After building the Eiffel Tower (1887-9), which he used for scientific research on meteorology, aerodynamics and radio telegraphy, he also built the first aerodynamic laboratory at Auteuil, outside Paris, where he pursued his research work without interruption during WW I. *TIS




1964 Edwin Bidwell Wilson (25 April 1879 in Hartford, Connecticut, USA - 28 Dec 1964 in Brookline, Massachusetts, USA) Wilson graduated from Yale with a Ph.D. in 1901 and, in the same year, a textbook which he had written on vector analysis was published. Vector analysis (1901) was based on Gibbs' lectures and , "This beautiful work, published when Wilson was only twenty-two years old, had a profound and lasting influence on the notation for and the use of vector analysis." Wilson had been inspired by Gibbs to work on mathematical physics and he began to write papers on mechanics and the theory of relativity. In 1912 Wilson published the first American advanced calculus text. World War I had seen another move in Wilson's research interests for he had undertaken war work which involved aerodynamics and this led him to study the effects of gusts of wind on a plane. In 1920 he published his third major text Aeronautics and gathered round him a group of students working on this topic.
Wilson had already worked in a number of quite distinct areas and his work on aeronautics did not become the major topic for the rest of his career. Not long after the publication of his important text on Aeronautics his interests moved again, this time towards probability and statistics. He did not study statistics for its own, however, but he was interested in applying statistics both to astronomy and to biology. He was the first to study confidence intervals, later rediscovered by Neyman. In 1922 Wilson left the Massachusetts Institute of Technology to become Professor of Vital Statistics at the Harvard School of Public Health. He continued to hold this post until he retired in 1945, when he became professor emeritus. After he retired, Wilson spent a year in Glasgow, Scotland when he was Stevenson lecture on Citizenship. From 1948 he was a consultant to the Office of Naval Research in Boston. *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

Friday, 27 December 2024

On This Day in Math - December 27

  

Jacob Bernoulli's tomb marker

At ubi materia, ibi Geometria.
Where there is matter, there is geometry.
~Johannes Kepler


The 361st day of the year, 2361 is an apocalyptic number, it contains 666. 2361=4697085165547666455778961193578674054751365097816639741414581943064418050229216886927397996769537406063869952 That's 109 digits.

One of Ramanujan's many approximations of pi was  (92+ (192/22))1/4, and 361 = 192

and as 361 is the last year day that is a perfect square, important to point out for students that all perfect squares are also the sum of consecutive triangular numbers, 361= 171 + 190 (The visual of this is a must see for students)


EVENTS

1612 Galileo observed Neptune, but did not recognize it as a planet. Galileo's drawings show that he first observed Neptune on December 28, 1612, and again on January 27, 1613. On both occasions, Galileo mistook Neptune for a fixed star when it appeared very close—in conjunction—to Jupiter in the night sky; hence, he is not credited with Neptune's discovery. (The official discovery is usually cited as September 23, 1846, Neptune was discovered within 1° of where Le Verrier had predicted it to be.) During the period of his first observation in December 1612, Neptune was stationary in the sky because it had just turned retrograde that very day. This apparent backward motion is created when the orbit of the Earth takes it past an outer planet. Since Neptune was only beginning its yearly retrograde cycle, the motion of the planet was far too slight to be detected with Galileo's small telescope.*Wik 




On this day in 1725, Christian Goldbach was the recording secretary at the opening session of the St Petersburg Academy. The form of the Academy was imported ready-made from the Berlin model proposed to Peter the Great by Leibniz several years earlier.
The Academy started as the The Saint Petersburg Academy of Sciences and was based in St Petersburg. The name varied over the years, becoming The Imperial Academy of Sciences and Arts 1747-1803), The Imperial Academy of Sciences (1803- 1836), and finally, The Imperial Saint Petersburg Academy of Sciences (from 1836 and until the end of the empire in 1917). Following the Revolution in 1917 it was renamed the Russian Academy of Sciences. It kept this name only until 1925 when it became the USSR Academy of Sciences. In 1934 it moved from Leningrad (which is what St Petersburg had been renamed) to Moscow. In 1991 its name of the Russian Academy of Sciences was reinstated.*SAU
Some of the initial Academy members were Daniel and Nicolaus Bernoulli, Christian Goldbach, Johann Duvernoy, Christian Gross, and Gerhard Müller. Euler arrived in St. Petersburg in 1727 to take up a post in physiology, a field in which he had little experience. Before long, though, he was transferred to other areas of study; he was made full Professor of Physics in 1731, and Professor of Mathematics in 1733. Euler also took on another role as a member of the Academy's Geography and Cartography department.*Euler Archive




In 1831, Charles Darwin set sail from Plymouth harbour on his voyage of scientific discovery aboard the HMS Beagle, a British Navy ship. The Captain Robert FitzRoy was sailing to the southern coast of South America in order to complete a government survey. Darwin had an unpaid position as the ship's naturalist, at age 22, just out of university. Originally planned to be at sea for two years, the voyage lasted five years, making stops in Brazil, the Galapagos Islands, and New Zealand. From the observations he made and the specimens he collected on that voyage, Darwin developed his theory of biological evolution through natural selection, which he published 28 years after the Beagle left Plymouth. Darwin laid the foundation of modern evolutionary theory. *TIS
HMS Beagle was a Cherokee-class 10-gun brig-sloop of the Royal Navy, one of more than 100 ships of this class. The vessel, constructed at a cost of £7,803, was launched on 11 May 1820 from the Woolwich Dockyard on the River Thames. Wikipedia




In 1956, the formerly believed "law" of conservation of parity was disproved in the first successful results from an experiment conducted by Madame Chien-Shiung Wu at Columbia University on the beta-decay of cobalt-60. It had been suggested in a paper published by Lee and Yang on 1 Oct 1956. There had been problems to overcome working with the cobalt sample and detectors in a vacuum at a working temperature of one-hundredth of a kelvin. Wu's team repeated the experiment, doing maintenance on the apparatus as necessary, until on 9 Jan 1957 further measurements confirmed the initial results. Leon Lederman performed an independent test of parity with Columbia's cyclotron. They held a press conference on 15 Jan 1957.*TIS




1995  France concludes a series of nuclear weapons tests in the South Pacific ( Moruroa and Fangataufa Atoll test site).  In a controversial move, French President Jacques Chirac had lifted a moratorium on testing. Most countries test weapons with computer simulations instead of actual bomb drops, but France claimed that tests that had been suspended several years earlier left the country without sufficient data to conduct future tests on computers.
Tahitians, as well as much of the international community, were outraged. Many expected the tests to harm the underwater geography and sea life of the atoll, as well as pose health risks to Tahitians. A French map from 1980 shows that testing had cracked the atolls in the past, destroyed coral reefs, and altered land plates. Harmful radioactive material had also been shown to spread via wind and rain. Some also saw France’s decision as a dangerous new step in nuclear proliferation in the West.

Both Tahitians and activists around the world responded strongly to France’s testing announcement. Activists in Tahiti were organizing a response while Greenpeace, an international environmental organization, sent a ship to Tahiti to protest the testing.






BIRTHS

1571 Johannes Kepler (27 Dec 1571; 15 Nov 1630) German astronomer who formulated three major laws of planetary motion which enabled Isaac Newton to devise the law of gravitation. Working from the carefully measured positions of the planets recorded by Tycho Brahe, Kepler mathematically deduced three relationships from the data: (1) the planets move in elliptical orbits with the Sun at one focus; (2) the radius vector sweeps out equal areas in equal times; and (3) for two planets the squares of their periods are proportional to the cubes of their mean distances from the sun. Kepler suggested that the tides were caused by the attraction of the moon. He believed that the universe was governed by mathematical rules, but recognized the importance of experimental verification.*TIS

Fig. 1: Illustration of Kepler's laws with two planetary orbits.
  1. The orbits are ellipses, with foci F1 and F2 for Planet 1, and F1 and F3 for Planet 2. The Sun is at F1.
  2. The shaded areas A1 and A2 are equal, and are swept out in equal times by Planet 1's orbit.
  3. The ratio of Planet 1's orbit time to Planet 2's is .
*Wik




1654 Jacob Jacques Bernoulli (27 Dec 1654; 16 Aug 1705) was a Swiss mathematician and astronomer who was one of the first to fully utilize differential calculus and introduced the term integral in integral calculus. Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra (1685), work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. He published five treatises on infinite series (1682 - 1704). Jacob was intrigued by the logarithmic spiral and requested it be carved on his tombstone. He was the first of the Bernoulli family of mathematicians. *TIS 
He was an early proponent of Leibnizian calculus, which he made numerous contributions to; along with his brother Johann, he was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.*Wik

(see more about the family of Bernoulli's at the Renaissance Mathematicus )

Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in the narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!

I found an interesting anecdote related to teaching and learning at the MAA website by Paul Bedard (Saint Clair County Community College)

"To learn more about mathematics than was possible at the University, Jakob traveled to Geneva.  However, before he found a tutor, he became one.  He entered the employ of the Waldkirch family in 1676 as tutor to the young, blind Elizabeth Waldkirch.  His task was to help her learn to read and write – not a common accomplishment for the blind at that time.  He continued in this occupation until 1678.  M.B.W. Tent suggested, in her fictionalized account of the lives of Euler and the Bernoullis,  that the elder Waldkirch wanted someone trained in mathematics, since he had learned that the mathematician Girolamo Cardano (1501-1576) had been involved in teaching literacy to the blind.  It is worth noting that it is still true today that many professions seek mathematically trained candidates or use mathematics tests for eligibility, not because the job requires the specific skills involved, but because the assumption is that minds that can grasp mathematics are disciplined and sharp.  This is a fact which the author shares with his students regularly."

"What skills as a teacher might Jakob Bernoulli have gained from this experience?  There is a certain poetry in the idea that the man who would bring light to so much that was dark in mathematics began his teaching career by alleviating the disadvantages of physical blindness.  If Tent was correct that he obtained this opportunity due to being a mathematician, then there is a second level of unexpected appropriateness here. "

"The early tutoring experiences of Jakob Bernoulli suggested to the author an at-home activity to assign our students.  Rather than merely requiring the students to solve problems, ask them to find volunteers and teach the volunteers how to solve the problems.  Each student will write a brief log entry of how the process goes, what explanations worked or failed, and how her “student” responded.  Even if the person “tutored” in this way is unprepared for this level of mathematics, his response to it may be instructive for, or resonate with, our students."





1773 Sir George Cayley (27 Dec 1773; 15 Dec 1857)(6th Baronet ) English aeronautical pioneer who built the first successful man-carrying glider (1853). He made extensive anatomical and functional studies of bird flight. By measuring bird and human muscle masses, he realized it would be impossible for humans to strap on a pair of wings and take to the air. His further studies in the principles of lift, drag and thrust founded the science of aerodynamics from which he discovered stabilizing flying craft required both vertical and horizontal tail rudders, that concave wings produced more lift than flat surfaces and that swept-back wings provided greater stability. Cayley also invented the caterpillar tractor (1825), automatic railroad crossing signals, self-righting lifeboats, and an expansion-air (hot-air) engine.
*TIS (He was a distant cousin of the father of mathematician Arthur Cayley)



1915 Jacob Lionel Bakst Cooper (27 December 1915, Beaufort West, Cape Province, South Africa, 8 August 1979, London, England) was a South African mathematician who worked in operator theory, transform theory, thermodynamics, functional analysis and differential equations.*Wik






DEATHS

1771 Henri Pitot (3 May 1695, 27 Dec 1771) French hydraulic engineer who invented the Pitot tube (1732), an instrument to measure flow velocity either in liquids or gases. With subsequent improvements by Henri Darcy, its modern form is used to determine the airspeed of aircraft. Although originally a trained mathematician and astronomer, he became involved with an investigation of the velocity of flowing water at different depths, for which purpose he first created the Pitot tube. He disproved the prevailing belief that the velocity of flowing water increased with depth. Pitot became an engineer in charge of maintenance and construction of canals, bridges, drainage projects, and is particularly remembered for his kilometer-long Roman-arched Saint-Clément Aqueduct (1772) at Montpellier, France. *TIS




1930 Gyula Farkas (28 March 1847 in Sárosd, Fejér County, Hungary - 27 Dec 1930 in Pestszentlorinc, Hungary) He is remembered for Farkas theorem which is used in linear programming and also for his work on linear inequalities. In 1881 Gyula Farkas published a paper on Farkas Bolyai's iterative solution to the trinomial equation, making a careful study of the convergence of the algorithm. In a paper published three years later, Farkas examined the convergence of more general iterative methods. He also made major contributions to applied mathematics and physics, particularly in the areas of mechanical equilibrium, thermodynamics, and electrodynamics.*SAU




1952 Mary Engle Pennington (October 8, 1872 – December 27, 1952) was an American bacteriological chemist, food scientist and refrigeration engineer. She was a pioneer in the preservation, handling, storage and transportation of perishable foods and the first female lab chief at the U.S. Food and Drug Administration. She was awarded 5 patents, received the Notable Service Medal from President Herbert Hoover and the Garvin-Olin Medal from the American Chemical Society. She is an inductee of the National Inventor's Hall of Fame, the National Women's Hall of Fame and the ASHRAE Hall of Fame.




1973 Raymond Woodard Brink (4 Jan 1890 in Newark, New Jersey, USA - 27 Dec 1973 in La Jolla, California, USA) was an American mathematician who studied at Kansas State University, Harvard and Paris. He taught at the University of Minnesota though he spent a year in Edinburgh in 1919. He worked on the convergence of series. *SAU
He also authored numerous math textbooks. He served as president of the Mathematical Association of America from 1941–42.*Wik





1992 Alfred Hoblitzelle Clifford (July 11, 1908 – December 27, 1992) was an American mathematician who is known for Clifford theory and for his work on semigroups. The Alfred H. Clifford Mathematics Research Library at Tulane University is named after him.*Wik



1995 Boris Vladimirovich Gnedenko (January 1, 1912 - December 27, 1995) was a Soviet mathematician and a student of Andrey Nikolaevich Kolmogorov. He was born in Simbirsk (now Ulyanovsk), Russia, and died in Moscow. He is perhaps best known for his work with Kolmogorov, and his contributions to the study of probability theory. Gnedenko was appointed as Head of the Physics, Mathematics and Chemistry Section of the Ukrainian Academy of Sciences in 1949, and also became Director of the Kiev Institute of Mathematics in the same year.*Wik



1996 Sister Mary Celine Fasenmyer, R.S.M., (October 4, 1906, Crown, Pennsylvania – December 27, 1996, Erie, Pennsylvania) was a mathematician. She is most noted for her work on hypergeometric functions and linear algebra.*Wik

For ten years after her graduation she taught and studied at Mercyhurst College in Erie, where she joined the Sisters of Mercy. She pursued her mathematical studies in Pittsburgh and the University of Michigan, obtaining her doctorate in 1946 under the direction of Earl Rainville, with a dissertation entitled Some Generalized Hypergeometric Polynomials.
After earning her Ph.D., Fasenmyer published two papers which expanded on her doctorate work. These would be further elaborated by Doron Zeilberger and Herbert Wilf into "WZ theory", which allowed computerized proof of many combinatorial identities. After this, she returned to Mercyhurst to teach and did not engage in further research.
Fasenmyer is most remembered for the method that bears her name, first described in her Ph.D. thesis concerning recurrence relations in hypergeometric series.The thesis demonstrated a purely algorithmic method to find recurrence relations satisfied by sums of terms of a hypergeometric polynomial, and requires only the series expansions of the polynomial. The beauty of her method is that it lends itself readily to computer automation. The work of Wilf and Zeilberger generalized the algorithm and established its correctness.
The hypergeometric polynomials she studied are called Sister Celine's polynomials.
*Wik





2006 Peter Ladislaw Hammer (December 23, 1936, Timișoara – December 27, 2006, Princeton, New Jersey) was an American mathematician native to Romania. He contributed to the fields of operations research and applied discrete mathematics through the study of pseudo-Boolean functions and their connections to graph theory and data mining.
 He did both his undergraduate and graduate studies at the University of Bucharest, earning a diploma in 1958 and a doctorate in 1965 under the supervision of Grigore Moisil. For a while in the 1960s he published under the name of Petru L. Ivănescu. In 1967, he and his wife (Anca Ivănescu) escaped Romania and defected to Israel. Hammer taught at the Technion from 1967 to 1969, then moved to Canada at McGill University in Montreal from 1969 to 1972, at the University of Waterloo from 1972 to 1983, and finally at Rutgers University in New Brunswick, New Jersey for the remainder of his career. He was killed in a car accident on December 27, 2006.

Hammer founded the Rutgers University Center for Operations Research, and created and edited the journals Discrete Mathematics, Discrete Applied Mathematics, Discrete Optimization, Annals of Discrete Mathematics, Annals of Operations Research, and SIAM Monographs on Discrete Mathematics and Applications





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