The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.

~Tobias Danzig

The 357th day of the year; there are 357 odd numbers in the first 46 rows of Pascal's Arithmetic triangle. (How many evens?)

357 is made up of three consecutive prime digits, and is the product of three distinct primes, 3 x 7 x 17=357

There are 21 year dates for which the sum of the divisors is a square number. 357 is the 20th of them. 1+3+7+17+21+51+119+357=576=24

^{2}

**EVENTS**

**1493**The German version of the

**Nuremberg Chronicle**- in German '

**Schedelsche Weltchronik**' - was published. It is one of the best-documented early printed books - an incunabulum - and one of the first to successfully integrate illustrations and text. Moreover, it was the most extensively illustrated book of the 15th century. See more detail and some of the beautiful graphics at *Yovisto

In

**1672,**astronomer Giovanni Cassini discovered Saturn's moon Rhea, the fifth major satellite of Saturn, which may be one of the most heavily cratered satellites in the solar system. Its surface appears to be saturated with craters, but long, bright linear features can be seen on the trailing hemisphere and linear ridges can be seen in the leading hemisphere. These ancient features may record changes in Rhea's shape due to internal heating or cooling. Rhea is 950 miles (1500 km) in diameter. Its largest crater is 190 miles (300 km) in diameter. Cassini also discovered three more of Saturn's major moons -- Iapetus, Tethys, and Dione. In 1675, he discovered that Saturn's rings are split largely into two parts by a narrow gap - known since as the "Cassini Division." *TS

**1690**Flamsteed observes Uranus, but doesn’t recognize it as undiscovered planet. *VFR

**1750**, Benjamin Franklin was severely shocked while electrocuting a turkey.*TIS (although I heard he described the meat as unusually tender)

**1751**Jacobi called this the birthday of elliptic functions because on this day a work by the Italian mathematician Giulio Carlo Fagnano, which had been sent to the Berlin Academy, was handed to Euler for review. The study of this work led Euler to his important investigations on elliptic integrals and to the discovery of the addition theorem. [Cajori, Historical Introduction to Mathematical Literature, p. 213) *VFR Fagnano had discovered a way to find the length of a lemniscate.

**1763**Price read Bayes’s essay to the Royal Society. *VFR Bayes never published what would eventually become his most famous accomplishment; his notes were edited and published after his death by Richard Price. In his later years he took a deep interest in probability. Stephen Stigler feels that he became interested in the subject while reviewing a work written in 1755 by Thomas Simpson,

^{ }but George Alfred Barnard thinks he learned mathematics and probability from a book by de Moivre *Wik

**1788**First use of "catenary", rather than the longer, more formal Latin "catenaria", may have been in a letter from Thomas Jefferson to Thomas Paine. Paine had written to Jefferson regarding the design of Iron Bridges. Jefferson's response was:

You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abbe Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'The earliest citation for catenary in the OED2 is from the above letter. *Jeff Miller

**1907**William Thompson, Lord Kelvin; died of a severe chill on 17 December 1907.

The Royal Society asked the Dean of Westminster if Kelvin could be buried in the Abbey and he agreed. The funeral was on 23 December and he lies to the south of Sir Isaac Newton's grave in the nave. On the previous night the coffin, covered by a purple pall, had rested in St Faith's chapel. The simple stone reads: WILLIAM THOMSON LORD KELVIN 1824-1907.

In 1913 a stained glass window, designed by J.Ninian Comper, was erected near the grave. This contains large figures of King Henry V and Abbot William Colchester and below is an inscription "In memory of Baron Kelvin of Largs. Engineer, Natural Philosopher. B.1824.D.1907". His coat of arms and those of Glasgow University are shown. The window was the gift of engineers from Great Britain and America. *Wik

**1947**Bardeen and Brattain demonstrate the transistor to the Bell Labs brass

It was the point-contact transistor, made from strips of gold foil on a plastic triangle, pushed down into contact with a slab of germanium. To measure the amplification they hooked up a microphone to one end of the device and a loudspeaker to the other. One by one, the men picked up the microphone and whispered hello; the loudspeaker at the other end of the circuit shouted HELLO!

Later, realizing that another major breakthrough in electronics had occurred in Bell’s lab, Shockley wrote Hearing speech amplified by the transistor was in tradition of Alexander Graham Bell’s famous ‘Mr. Watson, come here, I want you.’*CHM (

*The name transistor came from its electrical property known as trans-resistance.*) The next day Brattain noted the important event into his notebook.

**1949**J. R. Arnold and W. F. Libby publish "Age determinations by radiocarbon content: checks with samples of known age", the results of their experiments on using Carbon 14 for dating ancient organic materials in Science Magazine. The experiments were done on samples of wood of known age. For example, two samples taken from the tombs of two Egyptian kings, Zoser and Sneferu, independently dated to 2625 BC plus or minus 75 years, were dated by radiocarbon measurement to an average of 2800 BC plus or minus 250 years. *Wik

**In 1953**, Dr. Robert Oppenheimer was notified that his security clearance had been suspended. (He had directed the Manhattan Project that produced the atomic bombs used during WW II). There were allegations questioning his trustworthiness for association with Communists. By telegram dated 29 Jan 1954, he requested a hearing. On 4 Mar 1954, he submitted his answer to the original notification. Within two weeks, the Commission informed him who would conduct the hearing, to be led by Gordon Gray. The hearing before the Gray Board began 12 Apr 1954. It returned a result on 29 Jun 1954 that by a vote of 4 to1, it had made a decision against reinstating Dr. J. Robert Oppenheimer's access to classified information.*TIS

**1955**The term ALPHAMETIC was coined in 1955 by J. A. H. Hunter (Schwartzman). In the Dec. 23, 1955, Toronto Globe & Mail Hunter wrote, "These alphametics seem set to take the place of crosswords as a new craze... Don't forget that each letter stands for a particular figure" *Earliest Known Uses of Some of the Words of Mathematics

An alphametic is a peculiar type of mathematical puzzle, in which a set of words is written down in the form of an ordinary "long-hand" addition sum, and it is required that the letters of the alphabet be replaced with decimal digits so that the result is a valid arithmetic sum. For an example one can do no better than the first modern alphametic, published by the great puzzlist H.E. Dudeney in the July 1924 issue of Strand Magazine:

SEND

MORE

-----

MONEY

whose (unique) solution is:

9567

1085

-----

10652

*Alphametic Page BTW alph + a = metic is a self referential name since it is, itself, an alphametic.

For more on Some History notes about alphametics puzzles (Send+More = Money) and more

Curtiss remained at NBS until 1953. From 1954 to 1959 he was an executive director of the American Mathematical Society. In 1959 he became a professor of mathematics at the University of Miami, where he remained until his death in 1977. While no one can recall Curtiss at the console of a computer, he always said that "I was involved in the salt mines of computing." *CHM

Varignon gained his first exposure to mathematics by reading Euclid and then Descartes' La Géométrie. He became professor of mathematics at the Collège Mazarin in Paris in 1688 and was elected to the Académie Royale des Sciences in the same year. In 1704 he held the departmental chair at Collège Mazarin and also became professor of mathematics at the Collège Royal. He was elected to the Berlin Academy in 1713 and to the Royal Society in 1718. Many of his works were published in Paris in 1725, three years after his death. His lectures at Mazarin were published in Elemens de mathematique in 1731.

Varignon was a friend of Newton, Leibniz, and the Bernoulli family. Varignon's principal contributions were to graphic statics and mechanics. Except for l'Hôpital, Varignon was the earliest and strongest French advocate of infinitesimal calculus, and exposed the errors in Michel Rolle's critique thereof. He recognized the importance of a test for the convergence of series, but analytical difficulties prevented his success. Nevertheless, he simplified the proofs of many propositions in mechanics, adapted Leibniz's calculus to the inertial mechanics of Newton's Principia, and treated mechanics in terms of the composition of forces in Projet d'une nouvelle mécanique in 1687. Among Varignon's other works was a 1699 publication concerning the application of differential calculus to fluid flow and to water clocks. In 1690 he created a mechanical explanation of gravitation. In 1702 he applied calculus to spring-driven clocks. *Wik

(

The Varignon parallelogram exists even for a skew quadrilateral, and is planar whether or not the quadrilateral is planar. *Wik )

In 1972, he was awarded the Senior Berwick Prize*Wik

1992

Marshak was born in the Bronx, New York City. His parents were immigrants to New York from Minsk. He was educated at Columbia University.

Marshak received his PhD from Cornell University in 1939. Along with his thesis advisor, Hans Bethe, he discovered many of the fusion aspects involved in star formation. This helped him on his work for the Manhattan Project, in Los Alamos, during World War II.

In 1947, at the Shelter Island Conference, Marshak presented his two-meson hypothesis about the pi-meson, which were discovered shortly thereafter.[1]

In 1957, he and George Sudarshan proposed a V-A ("vector" minus "axial vector") Lagrangian for weak interactions, which was later independently discovered by Richard Feynman and Murray Gell-Mann. His biography below, is explicit about it "Perhaps Marshak's most significant scientific contribution was the proposal of the V-A Theory of Weak Interactions (the fourth force in nature) in collaboration with his student George Sudarshan. Unfortunately, the pair published the theory only in a conference proceedings for a meeting in Italy. Six months later, a different derivation of the same concept was published by Feynman and Gell-Mann in a mainstream scientific journal. Marshak had talked with Feynman about the general problem in California some time before. Though the V-A Concept was considered to be one of the most important contributions to theoretical physics, a Nobel Prize was never awarded for it." Sudarshan himself later commented in a TV interview in 2006 that Murray Gell-Mann got the idea from him, in an informal coffee time!

He was Chairman of the Department of Physics at the University of Rochester for fourteen years (1956 to 1970)

He was the President of the City College of New York from 1970-1979.

Marshak died by accidental drowning in Cancún, Mexico in 1992. *Wik

He wrote a Ph.D. in diophantine approximation under J. E. Littlewood at the University of Cambridge, completed in 1939. He had positions at MIT and Stanford before his appointment in 1950 at Princeton University. There he was involved in a major series of collaborative works with Kunihiko Kodaira on the deformation of complex structures, which had a profound influence on the theory of complex manifolds and algebraic geometry, and the conception of moduli spaces.

He later worked on pseudogroups and their deformation theory, based on a fresh approach to overdetermined systems of PDEs (bypassing the Cartan-Kähler ideas based on differential forms by making an intensive use of jets). Formulated at the level of various chain complexes, this gives rise to what is now called Spencer cohomology, a subtle and difficult theory both of formal and of analytical structure. This is a kind of Koszul complex theory, taken up by numerous mathematicians during the 1960s. In particular a theory for Lie equations formulated by Malgrange emerged, giving a very broad formulation of the notion of integrability.

After his death, a mountain peak outside of Silverton, Colorado was named in his honor. *Wik

Credits :

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts

*NSEC= NASA Solar Eclipse Calendar

*RMAT= The Renaissance Mathematicus, Thony Christie

*SAU=St Andrews Univ. Math History

*TIA = Today in Astronomy

*TIS= Today in Science History

*VFR = V Frederick Rickey, USMA

*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell

**In 1968**, American astronauts on Apollo 8 became the first men to orbit the Moon. The three-man crew was Frank Borman (Commander), James A. Lovell, Jr. (Command Module Pilot) and William Anders (Lunar Module Pilot). Not only was this the first manned flight to and from the Moon, but Apollo 8 served to validate many of the technical procedures necessary to support upcoming lunar missions. During ten lunar orbits, the astronauts took star sightings to pinpoint landmarks, surveyed landing sites, took both still and motion pictures and made two television transmissions to Earth. It was also the world's first manned flight to escape the influence of Earth's gravity. Launched on 21 Dec 1968, the mission lasted 6 days 3 hours until recovery on 27 Dec 1968. *TISApollo 8 crew. From left: Borman, Anders, Lovell. NASA. |

**1987**Egypt issued a stamp honoring the mathematician Ali Mustafa Mousharafa (1898–1950). [Scott #1147]. *VFR Dr. Ali Moustafa Mosharafa Pasha was an Egyptian theoretical physicist. He was professor of applied mathematics in the Faculty of Science at Cairo University, and also served as its first dean. *Wik**BIRTHS**

**1872 Georgii Pfeiffer**did important work on partial differential equations.*SAU**1909 John Hamilton Curtiss**is born on December 23, 1909. He obtained an MS degree in statistics from Northwestern University in 1930, and a PhD from Harvard in 1935. He obtained an MS degree in statistics from Northwestern University in 1930, and a PhD from Harvard in 1935. He had taught at Cornell University (1935-1943) and had served in U.S. Navy (1943-1946). In 1946 Curtiss joined National Bureau of Standards, where in 1947 he became a chief of the Applied Mathematical Division (AMD), the first centralized national computing center dedicated to accelerate the progress of USA in the computing industry. AMD's expertise proved the necessity of development of the UNIVAC, the SEAC, and the SWAC computers. At the same time Curtiss played the crucial role in the organization of ACM, then Eastern Association of Computing Machinery, and in 1947 he became the first ACM president.Curtiss remained at NBS until 1953. From 1954 to 1959 he was an executive director of the American Mathematical Society. In 1959 he became a professor of mathematics at the University of Miami, where he remained until his death in 1977. While no one can recall Curtiss at the console of a computer, he always said that "I was involved in the salt mines of computing." *CHM

**1936 Peter L. Hammer**(December 23, 1936 - December 27, 2006) 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.*Wk**1938 Robert E(lliot) Kahn**(23 Dec 1938, ) American computer scientist who co-created the packet-switching protocols that enable computers to exchange information on the Internet. In the late 1960s Kahn realized that a packet-switching network could effectively transmit large amounts of data between computers. Along with fellow computer scientists Vinton Cerf, Lawrence Roberts, Paul Baran, and Leonard Kleinrock, Kahn built the ARPANET, the first network to successfully link computers around the country. Kahn and Cerf also developed the Transmission Control Protocol (TCP) and the Internet Protocol (IP), which together enable communication between different types of computers and networks; TCP/IP is the standard still in use today. *TIS**1943 Mikhail Leonidovich Gromov**(23 December 1943 - ), is a French–Russian mathematician known for important contributions in many different areas of mathematics. He is considered a geometer in a very broad sense of the word. In 2009 he was awarded the Abel Prize "for his revolutionary contributions to geometry." *Wik**DEATHS**

**1722 Pierre Varignon**(Caen 1654 – December 23, 1722 Paris) was a French mathematician. He was educated at the Jesuit College and the University in Caen, where he received his M.A. in 1682. He took Holy Orders the following year.Varignon gained his first exposure to mathematics by reading Euclid and then Descartes' La Géométrie. He became professor of mathematics at the Collège Mazarin in Paris in 1688 and was elected to the Académie Royale des Sciences in the same year. In 1704 he held the departmental chair at Collège Mazarin and also became professor of mathematics at the Collège Royal. He was elected to the Berlin Academy in 1713 and to the Royal Society in 1718. Many of his works were published in Paris in 1725, three years after his death. His lectures at Mazarin were published in Elemens de mathematique in 1731.

Varignon was a friend of Newton, Leibniz, and the Bernoulli family. Varignon's principal contributions were to graphic statics and mechanics. Except for l'Hôpital, Varignon was the earliest and strongest French advocate of infinitesimal calculus, and exposed the errors in Michel Rolle's critique thereof. He recognized the importance of a test for the convergence of series, but analytical difficulties prevented his success. Nevertheless, he simplified the proofs of many propositions in mechanics, adapted Leibniz's calculus to the inertial mechanics of Newton's Principia, and treated mechanics in terms of the composition of forces in Projet d'une nouvelle mécanique in 1687. Among Varignon's other works was a 1699 publication concerning the application of differential calculus to fluid flow and to water clocks. In 1690 he created a mechanical explanation of gravitation. In 1702 he applied calculus to spring-driven clocks. *Wik

(

**Varignon's theorem**is a statement in Euclidean geometry by Pierre Varignon that was first published in 1731. It deals with the construction of a particular parallelogram from an arbitrary quadrangle.*The midpoints of the sides of an arbitrary quadrangle form a parallelogram. If the quadrangle is convex or reentrant, i.e. not a crossing quadrangle, then the area of the parallelogram is half as big as the area of the quadrangle*.

^{}The Varignon parallelogram exists even for a skew quadrilateral, and is planar whether or not the quadrilateral is planar. *Wik )

**1834 Thomas Robert Malthus**(13 Feb 1766; 23 Dec 1834 at age 68) English economist and demographer who can be regarded as a pioneer sociologist. He was one of the first to systematically analyze human society when he published his theories in An Essay on the Principle of Population. Malthus predicted population would always outrun the food supply and that would result in famine, disease or war to reduce the number of people. As Malthus observed the Industrial Revolution was causing a rapid increase in population, he indicated keeping improved social conditions would require imposing strict limits on reproduction. Reading the book inspired Charles Darwin to reflect upon the survival of the fittest individuals in the process of natural selection in evolving populations of any organism. Alfred Russell Wallace likewise acknowledged his theory was stimulated by the book by Malthus. *TIS**1890 Edward Sang,**(30 Jan 1805 in Kirkcaldy, Fife, Scotland - 23 Dec 1890) A native of Fife, Sang wrote extensively on mathematical, mechanical, optical and actuarial topics. *SAU**1907 Pierre (-Jules-César) Janssen**(22 Feb 1824, 23 Dec 1907) was a French astronomer who in 1868 devised a method for observing solar prominences without an eclipse (an idea reached independently by Englishman Joseph Norman Lockyer). Janssen observed the total Sun eclipse in India (1868). Using a spectroscope, he proved that the solar prominences are gaseous, and identified the chromosphere as a gaseous envelope of the Sun. He noted an unknown yellow spectral line in the Sun in 1868, and told Lockyer (who subsequently recognized it as a new element he named helium, from Greek "helios" for sun). Janssen was the first to note the granular appearance of the Sun, regularly photographed it, and published a substantial solar atlas with 6000 photographs (1904). *TIS**1973 Gerard Peter Kuiper**(7 Dec 1905, 23 Dec 1973) Dutch-born American astronomer, who discovered Miranda, a moon of Uranus, and Nereid, a moon of Neptune. The Kuiper Belt is so-named after his original suggestion of its existence outside the orbit of Neptune before it was confirmed as a belt of small bodies. He measured the diameter of Pluto. In the Martian atmosphere Kuiper detected carbon dioxide, but the absence of oxygen (1947). In the 1960s, Kuiper pioneered airborne infrared observing using a Convair 990 aircraft and served as chief scientist for the Ranger spacecraft crash-landing probes of the moon. By analyzing Ranger photographs, he identified landing sites on the lunar surface most suitable for safe manned landings. *TIS**1989 Richard Rado**FRS(28 April 1906 – 23 December 1989) was a Jewish German mathematician. He earned two Ph.D.s: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He made contributions in combinatorics and graph theory. He wrote 18 papers with Paul Erdős. In 1964, he discovered the Rado graph (The Rado graph contains all finite and countably infinite graphs as induced subgraphs..)In 1972, he was awarded the Senior Berwick Prize*Wik

1992

**Robert Eugene Marshak**(October 11, 1916 – December 23, 1992) was an American physicist dedicated to learning, research, and education.Marshak was born in the Bronx, New York City. His parents were immigrants to New York from Minsk. He was educated at Columbia University.

Marshak received his PhD from Cornell University in 1939. Along with his thesis advisor, Hans Bethe, he discovered many of the fusion aspects involved in star formation. This helped him on his work for the Manhattan Project, in Los Alamos, during World War II.

In 1947, at the Shelter Island Conference, Marshak presented his two-meson hypothesis about the pi-meson, which were discovered shortly thereafter.[1]

In 1957, he and George Sudarshan proposed a V-A ("vector" minus "axial vector") Lagrangian for weak interactions, which was later independently discovered by Richard Feynman and Murray Gell-Mann. His biography below, is explicit about it "Perhaps Marshak's most significant scientific contribution was the proposal of the V-A Theory of Weak Interactions (the fourth force in nature) in collaboration with his student George Sudarshan. Unfortunately, the pair published the theory only in a conference proceedings for a meeting in Italy. Six months later, a different derivation of the same concept was published by Feynman and Gell-Mann in a mainstream scientific journal. Marshak had talked with Feynman about the general problem in California some time before. Though the V-A Concept was considered to be one of the most important contributions to theoretical physics, a Nobel Prize was never awarded for it." Sudarshan himself later commented in a TV interview in 2006 that Murray Gell-Mann got the idea from him, in an informal coffee time!

He was Chairman of the Department of Physics at the University of Rochester for fourteen years (1956 to 1970)

He was the President of the City College of New York from 1970-1979.

Marshak died by accidental drowning in Cancún, Mexico in 1992. *Wik

**2001 Donald Clayton Spencer**(April 25, 1912 – December 23, 2001) was an American mathematician, known for major work on deformation theory of structures arising in differential geometry, and on several complex variables from the point of view of partial differential equations. He was born in Boulder, Colorado, and educated at the University of Colorado and MIT.He wrote a Ph.D. in diophantine approximation under J. E. Littlewood at the University of Cambridge, completed in 1939. He had positions at MIT and Stanford before his appointment in 1950 at Princeton University. There he was involved in a major series of collaborative works with Kunihiko Kodaira on the deformation of complex structures, which had a profound influence on the theory of complex manifolds and algebraic geometry, and the conception of moduli spaces.

He later worked on pseudogroups and their deformation theory, based on a fresh approach to overdetermined systems of PDEs (bypassing the Cartan-Kähler ideas based on differential forms by making an intensive use of jets). Formulated at the level of various chain complexes, this gives rise to what is now called Spencer cohomology, a subtle and difficult theory both of formal and of analytical structure. This is a kind of Koszul complex theory, taken up by numerous mathematicians during the 1960s. In particular a theory for Lie equations formulated by Malgrange emerged, giving a very broad formulation of the notion of integrability.

After his death, a mountain peak outside of Silverton, Colorado was named in his honor. *Wik

Credits :

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts

*NSEC= NASA Solar Eclipse Calendar

*RMAT= The Renaissance Mathematicus, Thony Christie

*SAU=St Andrews Univ. Math History

*TIA = Today in Astronomy

*TIS= Today in Science History

*VFR = V Frederick Rickey, USMA

*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell

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