~ Marcus Aurelius

The 192nd day of the year; 192 is the smallest number that together with its double and triple contain every digit from 1-9 exactly once. There are three other values of n so that n, 2n, and 3n contain each non-zero digit exactly once. Can you find them?

192 is the sum of ten consecutive primes (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37)

192 is the number of edges on a 6th dimension hypercube

Diophantus probably knew, and Lagrange proved, that every positive integer can be written as a sum of four perfect squares. Jacobi] proved the stronger result that the number of ways in which a positive integer can be so written equals 8 times the sum of its divisors that are not multiples of 4. Use this theorem to prove that there are 192 ways to express 14 as a sum of four squares.

**EVENTS**

1600 Kepler’s interest in optics arose as a direct result of his observations of the partial solar eclipse of 10 July 1600. Following instructions from Tycho Brahe, he constructed a pinhole camera; his measurements, made in the Graz marketplace, closely duplicated Brahe’ and seemed to show that the moon’s apparent diameter was considerably less than the sun’s. Kepler soon realized that the phenomenon resulted from the finite aperture of the instrument (see Fig. 5); his analysis, assisted by actual threads, led to a clearly defined concept of the light ray, the foundation of modern geometrical optics.

Kepler’s subsequent work applied the idea of the light ray to the optics of the eye, showing for the first time that the image is formed on the retina. He introduced the expression “pencil of light,” with the connotation that the light rays draw the image upon the retina; he was unperturbed by the fact that the image is upside down. *Encyclopedia.com

1610 Galileo receives a letter from Cosimo II agreeing to his salary requests, and confirming him as "First Mathematician of our Stadium in Pisa" but with no requirements that he live or lecture in Pisa, "except when it may please you as an honor." *The Copernican Question: Prognostication, Skepticism, and Celestial Order By Robert S. Westman

1637 First meeting of the Acad´emie Fran¸caise. *VFR

1676 Flamsteed began living at the Observatory with his two servants. On 19 July, his long series of Greenwich observations began? *Rebekah Higgitt, Teleskopos

1794 Star in a crescent moon? Astronomer Royal Investigates. The results are read to the Royal Society..."An Account of an Appearance of Light, like a Star, Seen Lately in the Dark Part of the Moon, by Thomas Stretton, in St. John's Square, Clerkenwell, London; with Remarks upon This Observation, and Mr. Wilkins's. Drawn up, and Communicated by the Rev. Nevil Maskelyne, D. D. F. R. S. and Astronomer Royal" *Phil. Trans. R. Soc. Lond. January 1, 1794 84:435-440;

1796 Date of the entry EγPHKA! num=Δ+Δ+Δ in Gauss’s scientiﬁc diary, recording his discovery that every positive integer is the sum of three triangular numbers. [Thanks to Howard Eves] *VFR

1826 Cauchy presented a proof to the Acad´emie dealing with existence theorems for ﬁrst-order dif-ferential equations. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, pp. 758 and 1401] *VFR

1843 Jacques Philippe Marie Binet, age 57, elected to the Acad´emie des Sciences to succeed Lacroix. He is an example of a mathematician who published much late in life. He worked in mechanics, elasticity, perturbation theory, determinants, and the calculus. [Ivor Grattan-Guiness, Convo¬lutions in French Mathematics, 1800–1840, pp. 191 and 1410] *VFR

1908 at 5:45 in the morning, Kammerlingh Onnes, of Leiden, wins the race to produce liquified helium. 75 liters of liquid air is used to condense 20 liters of liquid oxygen, from which 20 milli-liters of liquid helium under reduced pressure. *Quantum Generations: A History of Physics in the Twentieth Century By Helge Kragh

1925 The “Monkey Trial” of John T. Scopes began in Dayton, Tennessee. Clarence Darrow defended him. The prosecution, conducted by William Jennings Bryan, presented a strong case, and he was convicted of violating a state law prohibiting the teaching of evolution. Although the law was later overturned, this case provided a strong blow to science education. Scopes was not a biologist and never taught evolution. Rather he was a mathematics and physics teacher who volunteered to stand trial to furnish a test case. *VFR

The trial ran for 12 days. A local school teacher, John Scopes, was prosecuted under the state's Butler Act, but was supported by the American Civil Liberties Union. This law, passed a few months earlier (21 Mar 1925) prohibited the teaching of evolution in public schools. The trial was a platform to challenge the legality of the statute. Local town leaders,(wishing for the town to benefit from the publicity of the trial) had recruited Scope to stand trial. He was convicted (25 Jul) and fined $100. On appeal, the state supreme court upheld the constitutionality of the law but acquitted Scopes on the technicality that he had been fined excessively. The law was repealed on 17 May 1967. *TIS

**1950**France honors Lazar Carnot (1753–1823) with a postage stamp. [Scott #B251]. *VFR

**1950**The German Democratic Republic, to celebrate the 250th anniversary of the founding of the Academy of Sciences, Berlin, issued postage stamps picturing Leonhard Euler and Gottfried von Leibniz. [Scott #58, 66]. *VFR

Leibniz was also honored with stamp issues in 1980 and 1996 it seems.

**1993**MASH fans will remember that there was always a sign telling how many miles to Toledo and frequently they talked of the hotdogs at Tony Pacos (they are good). On this date the Cake Walk and Jazz Band

*(I believe the band is called "The Cakewalken Jass Band")*celebrated their twenty-ﬁfth anniversary with a live broadcast at Tony Pacos that was broadcast on public radio in Toledo. So what does this have to do with mathematics? Well, Ray Heitger, their clarinetist, leader, and one of the founding members happens to be a math teacher. If you can’t get to Toledo to hear them play, perhaps you can ﬁnd one of their six LPs.*VFR

**Tony Packo's Cafe**is restaurant that started in the Hungarian neighborhood of Birmingham, on the east side of Toledo, Ohio at 1902 Front Street. The restaurant gained notoriety by its mention in several M*A*S*H episodes and is famous for its signature sandwich and large collection of hot dog buns signed by celebrities. In 2011 it listed five restaurants in the Toledo area. *Wik

**BIRTHS**

**1682 Roger Cotes**born (10 July 1682 — 5 June 1716). In January 1706 he was named the ﬁrst Plumian professor of astronomy and natural philosophy at Cambridge. It was Cotes who ﬁrst showed that e was the natural base to choose for the logarithm. *VFR He did not realize his full potential because he died at age 33, leaving anunfinished series of imposing researches on optics and a large number of other unpublished manuscripts. Newton, who seldom spoke well of anyone else, said of Cotes, "If Cotes had lived, we might have known something."

Thony Christie at the Renaissance Mathematicus has a nice post about Cotes.

**1832 Alvan Graham Clark**(July 10, 1832 – June 9, 1897) U.S. astronomer, one of an American family of telescope makers and astronomers who supplied unexcelled lenses to many observatories in the U.S. and Europe during the heyday of the refracting telescope. He began a deep interest in astronomy while still at school, then joined the family firm of Alvan Clark & Sons, makers of astronomical lenses. In 1861, testing a new lens, he looked through it at Sirius and observed faintly beside it, Sirius B, the twin star predicted by Friedrich Bessel in 1844. Carrying on the family business, after the deaths of his father and brother, Clark made the 40" lenses of the Yerkes telescope (still the largest refractor in the world). Their safe delivery was a source of anxiety. He died shortly after their first use. *TIS

**1856 Nikola Tesla**(10 July 1856 – 7 January 1943)Serbian-American inventor and researcher who designed and built the first alternating current induction motor in 1883. [

*This statement seems to be in error,according to Wikipedia which states,*" In 1824, the French physicist François Arago formulated the existence of rotating magnetic fields, termed Arago's rotations, which, by manually turning switches on and off, Walter Baily demonstrated in 1879 as in effect the first primitive induction motor. Practical alternating current induction motors seem to have been independently invented by Galileo Ferraris(1885) and then Tesla (1887).]He emigrated to the United States in 1884. Having discovered the benefits of a rotating magnetic field, the basis of most alternating-current machinery, he expanded its use in dynamos, transformers, and motors. Because alternating current could be transmitted over much greater distances than direct current, George Westinghouse bought patents from Tesla the system when he built the power station at Niagara Falls to provide electricity power the city of Buffalo, NY. [Born in Croatia of Serbian parents. Some sources give birthdate as 9 Jul; he is said to have been born on the stroke of midnight.]

**1878 Oliver Dimon Kellogg**(10 July 1878 in Linwood, Pennsylvania, USA - 26 July 1932 in Greenville, Maine, USA) was appointed to the University of Missouri in 1905 where, despite a heavy teaching and administrative load he was able to publish impressive papers on potential theory. In 1908 he published three papers, namely

*Potential functions on the boundary of their regions of definition*and

*Double distributions and the Dirichlet problem,*both in the

*Transactions of the American Mathematical Society,*and

*A necessary condition that all the roots of an algebraic equation be real*in the

*Annals of Mathematics.*In 1912 he published the important work

*Harmonic functions and Green's integral*in the

*Transactions of the American Mathematical Society.*This paper includes what today is called 'Kellogg's theorem' on harmonic and Green's functions. *SAU

1883 Frank Albert Benford, Jr., ((

*see note below about date of birth*)1883 Johnstown, Pennsylvania – December 4, 1948) was an American electrical engineer and physicist best known for rediscovering and generalizing Benford's Law, a statistical statement about the occurrence of digits in lists of data.

Benford is also known for having devised, in 1937, an instrument for measuring the refractive index of glass. An expert in optical measurements, he published 109 papers in the fields of optics and mathematics and was granted 20 patents on optical devices.

His date of birth is given variously as May 29 or July 10, 1883. After graduating from the University of Michigan in 1910, Benford worked for General Electric, first in the Illuminating Engineering Laboratory for 18 years, then the Research Laboratory for 20 years until retiring in July 1948. He died suddenly at his home on December 4, 1948. *Wik

**1928 Errett Albert Bishop**(July 10, 1928 – April 14, 1983) (His) work is so wide ranging that it is difficult to give an overview in a biography such as this. Let us look at the book Selected papers which was published in 1986 and reprints some of Bishop's most significant contributions. The book divided Bishop's papers into five categories:

(1) Polynomial and rational approximation. Examples are extensions of Mergelyan's approximation theorem and the theorem of Frigyes Riesz and Marcel Riesz concerning measures on the unit circle orthogonal to polynomials. Bishop found new methods in dealing with these problems;

(2) The general theory of function algebras. Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures. In 1965 Bishop wrote an excellent survey Uniform algebras examining the interaction between the theory of uniform algebras and that of several complex variables.

(3) Banach spaces and operator theory. An examples of a paper by Bishop on this topic is Spectral theory for operators on a Banach space (1957). He introduced the condition now called the Bishop condition which turned out to be very useful in the theory of decomposable operators.

(4) Several complex variables. Examples of Bishop's papers in this area are Analyticity in certain Banach spaces (1962). He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in Cn, and a new proof of Remmert's proper mapping theorem.

(5) Constructive mathematics. Bishop become interested in foundational issues around 1964, about the time he was at the Miller Institute. He wrote a famous text Foundations of constructive analysis (1967) which aimed to show that a constructive treatment of analysis is feasible.*SAU

**DEATHS**

**1851 Louis-Jacques-Mandé Daguerre**(18 November 1787 – 10 July 1851) French painter and physicist who invented the daguerreotype, the first practical process of photography. Though the first permanent photograph from nature was made in 1826/27 by Joseph-Nicéphore Niepce of France, it was of poor quality and required about eight hours' exposure time. The process that Daguerre developed required only 20 to 30 minutes. The two became partners in the development of Niepce's heliographic process from 1829 until the death of Niepce in 1833. Daguerre continued his experiments, and he discovered that exposing an iodized silver plate in a camera would result in a lasting image after a chemical fixing process.*TIS

1910 Johann Gottfried Galle

__(9 June 1812 – 10 July 1910) German astronomer who on 23 Sep 1846, was the first to observe the planet Neptune, whose existence had been predicted in the calculations of Leverrier. Leverrier had written to Galle asking him to search for the new planet at a predicted location. Galle was then a member of the staff of the Berlin Observatory and had discovered three comets. In 1838, while assistant to Johann Franz Encke, Galle discovered the dark, inner C ring of Saturn at the time of the maxium ring opening. In 1851, he became professor of astronomy at Breslau and director of the observatory there. In 1872, he proposed the use of asteroids rather than regular planets for determinations of the solar parallax, a suggestion which was successful in an international campaign (1888-89). *TIS__

**1916 John Emory McClintock**(19 Sept 1840 in Carlisle, Pennsylvania , USA - 10 July 1916 in Bay Head, New Jersey, USA) was for many years the leading actuary in America. He published 30 papers between 1868 and 1877 on actuarial questions. His publications were not confined to questions relating to

*A simplified solution of the cubic*in 1900 in the

*Annals of Mathematics.*Another work,

*On the nature and use of the functions employed in the recognition of quadratic residues*(1902), published in the

*Transactions of the American Mathematical Society,*is on quadratic residues.*SAU

**1936 Salvatore Pincherle**(March 11, 1853 — July 10, 1936) worked on functional equations and functional analysis. Together with Volterra, he can claim to be one of the founders of functional analysis.*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