On This Day in Math - February 11

 

Good teachers deserve apples; great teachers deserve chocolate.

(A favorite quotation, written in calligraphy on his office door.)
~Richard Hamming

*Subvillain@subvill41n

The 42nd day of the year; in The Hitchhiker's Guide to the Galaxy, the Answer to the Ultimate Question of Life, The Universe, and Everything is 42. The supercomputer, Deep Thought, specially built for this purpose takes 7½ million years to compute and check the answer. The Ultimate Question itself is unknown.

There is only one scalene triangle in simplest terms with integer sides and integer area of 42, it's perimeter is also 42. (There are only three integer (non-right) triangles possible with area and perimeter equal and all integer sides.)

42 is between a pair of twin primes (41,43) and its concatenation with either of them (4241, 4243) is also a prime, which means that 4242 is also between twin primes.

On September 6, 2019, Andrew Booker, University of Bristol, and Andrew Sutherland, Massachusetts Institute of Technology, found a sum of three cubes for \(42= (–80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 \). This leaves 114 as the lowest unsolved case. At the beginning of 2019, 33 was the lowest unsolved case, but Booker solved that one earlier in 2019.

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EVENTS

1144 "On the 11th February 1144 the Hellenistic science of alchemy entered medieval Europe by way of the Islamic empire. In his translation of Liber de compositione alchemiae (Book about the composition of alchemy) Robert of Chester wrote the following:

"I have translated this Book because, what alchemy is, and what its composition is, almost no one in our Latin [that is: Western] world knows. finished February 11th anno 1144."

From a blog at *RMAT

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1635 Sir Charles Cavendish writes to William Oughtred to thank him for teaching him, "the way of calculating the divisions of your guaging rod." He also passes on praise for Oughtred’s, “Clavis is in great estimation amongst the mathematicians at Paris.“ *Augustus De Morgan, Correspondence of scientific men of the seventeenth century ..., Volume 1

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1800 William Herschel discovered infrared radiation by measuring temperature changes caused by dispersed light beyond the red end of the spectrum. This prism, in Herschel Museum is the prism he used for this incredibly important discovery.

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1801, Giuseppe Piazzi made a 24th observation of the position of Ceres, the asteroid he discovered between the orbits of Mars and Jupiter, on 1 Jan 1801. It was the first and largest of the dwarf planets now known. After this, it moved into the light of the Sun, and was lost to view for most of the rest of the year. To mathematically relocate Ceres, Carl Gauss, age 24, took up the challenge to calculate its orbital path, based on the limited number of observations available. His method was tedious, requiring 100 hours of calculation. He began with a rough approximation for the unknown orbit, and then used it to produce a refinement, which became the subject of another improvement.. And so on. Astronomers using them found his results in close agreement as they located Ceres again 25 Nov-31 Dec 1801.« *TIS

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1825 Charles Bonnycastle arrives in America to take up position as Professor of Natl. Philosophy at newly founded Univ of Virginia. He was found on the arrivals manifest of the ship Competitor from London captained by E.P. Godby which arrived in the District of Norfolk & Portsmouth. Passengers included Robley & Harriet Dungleson (Dr of Medicine), Chas Bonnycastle 27 Profr of Natl Philosophy, Thomas H & Sarah Key (Profr of Mathematics), Robert Lee 16, and F.W. Colquhoun 15. All intended to become inhabitants of the United States

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1826 Founding of the University College of London. UCL was the first university institution to be founded in London and the first in England to be established on an entirely secular basis, to admit students regardless of their religion and to admit women on equal terms with men.

First professor of mathematics at UCL was  Augustus De Morgan.

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#Wik

1889  February 11, 1889, Birthday of #Russian paleontologist Ekaterina Vladimirovna Lermontova, published the first Cambrian stratigraphy of Siberia based on trilobites. Fossil animals & algae, & Cambrian biostratigraphic divisions are named in her honor

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1897 Indiana bill to fix the value of π was introduced to the state senate, and referred to committee. It specified several values for π to simplify computation. On the 13th a bill to postpone consideration of the bill passed, and the bill has never been reintroduced. The official history of the Indiana General Assembly (p. 429) gives the credit to Professor Waldo for his intervention.(see my blog)

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In 1939, the journal Nature published a theoretical paper on nuclear fission. The term was coined by the authors Lise Meitner and Otto Frisch, her nephew. They knew that when a uranium nucleus was struck by neutrons, barium was produced. Seeking an explanation, they used Bohr's "liquid drop" model of the nucleus to envision the neutron inducing oscillations in a uranium nucleus, which would occasionally stretch out into the shape of a dumbbell. Sometimes, the repulsive forces between the protons in the two bulbous ends would cause the narrow waist joining them to pinch off and leave two nuclei where before there had been one. They calculated calculated the huge amounts of energy released. This was the basis for nuclear chain reaction. *TIS (see Szilard below) (I was informed by Daniel Fischer@cosmos4u that this paper is now available here.

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1948 Richard Duffin writes a recommendation for John Nash. Duffin was inducted to the National Academy of Sciences in 1972 and to the American Academy of Arts and Sciences in 1984. He was joint winner of the 1982 John von Neumann Theory Prize, and winner of Sigma Xi's Monie A. Ferst Award for 1984 in recognition of his ability as a teacher and communicator. This recommendation tends to emphasize his ability to communicate with few words. *HT to Sarah Crook@SarahRoseCrook

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1966 The RAND Coporation Takes JOSS Out of Service:
The RAND Corporation takes the Johnniac Open Shop System (JOSS) out of service. JOSS was a conversational time-sharing service that eased the bottleneck experienced by programmers in the batch environment--typical of the time--in which long delays existed between sending information to the computer and getting results back. Timesharing aimed to bring the user back into contact with the machine for online debugging and program development. *CHM

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1986 Soviet Jewish dissident, mathematician and computer expert, Anatoly Shcharansky was released by the Soviet Union in an East-West spy swap. Shcharansky, who helped run a committee monitoring human rights abuse in the Soviet Union, had been jailed since 1978 on charges of spying for the CIA. [UPI press release] *VFR

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1999 Nature Magazine publishes an article on how to dunk a biscuit, by UK/Australian scientist, Len Fisher. The article, : "Physics Takes the Biscuit", would also win Professor Fisher an Ignoble Prize on September 30 at at the annual presentation at Harvard. Fisher followed with a Book on the topic, adjusted perhaps for American Audiences, called How to Dunk a Doughnut: The Science of Everyday Life, which includes other explanations of "everyday" science. You can read the chapter on dunking at Fisher's blog site, along with many other wonderful treats.*lenfisherscience.com, .improbable.com

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2003 NASA's WMAP satellite completes the first detailed cosmic microwave background radiation map of the universe. The image reveals the universe is 13.7 billion years old (within one percent error) and provides evidence that supports the inflationary theory.*Wik

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2010 The Solar Dynamics Observatory (SDO) is a NASA mission which has been observing the Sun since 2010. Launched on 11 February 2010, the observatory is part of the Living With a Star (LWS) program.

The goal of the LWS program is to develop the scientific understanding necessary to effectively address those aspects of the connected Sun–Earth system directly affecting life and society. 

The goal of the SDO is to understand the influence of the Sun on the Earth and near-Earth space by studying the solar atmosphere on small scales of space and time and in many wavelengths simultaneously. SDO has been investigating how the Sun's magnetic field is generated and structured, how this stored magnetic energy is converted and released into the heliosphere and geospace in the form of solar wind, energetic particles, and variations in the solar irradiance. *Wik 

Solar Dynamics Observatory satellite *Wik

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BIRTHS

1657  Bernard Le Bovier, sieur de Fontenelle (11 Feb 1657, 9 Jan 1757) French scientist and author, whose Conversations on the Plurality of Worlds (1686), was one of the first works to present science for the lay reader. He popularized the astronomical theories of Descartes. Many of the characteristic ideas of the Enlightenment are found in embryonic form in his works. From 1697 he became permanent secretary to the Académie des Sciences. He held the office for 42 years, and in this official capacity, he wrote the Histoire du renouvellement del Académie des Sciences (Paris, 3 vols., 1708, 1717, 1722) containing extracts and analyses of the proceedings, written with great simplicity and delicacy. Fontenelle presented many obituary notices to the Académie, including those of Newton and Leibniz. *TIS

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1800 William Henry Fox Talbot (11 Feb 1800; 17 Sep 1877 at age 77)

English inventor, mathematician, chemist, physicist, philologist and Egyptologist who invented the negative-positive photographic process. He improved Thomas Wedgewood's discovery (1802) that brushing silver nitrate solution onto paper produces a light-sensitive medium able to record negative images, but Wedgewood was unable to control the darkening. In February 1835, Fox Talbot found that a strong solution of salt fixed the image. Using a camera obscura to focus an image onto his paper to produce a negative, then - by exposing a second sheet of paper to sunlight transmitted through the negative - he was the first to produce a positive picture of which he was able to make further copies at will. His Pencil of Nature (1844) was the first photographically illustrated book to be commercially published.. *TIS Talbot's family estate, the site of Lacock Abbey, was used for scenes from several of the Harry Potter films.

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1839 Josiah Willard Gibbs (11 Feb 1839; 28 Apr 1903 at age 64)
was an American mathematical physicist and chemist known for contributions to vector analysis and as one of the founders of physical chemistry. In 1863, He was awarded Yale University's first engineering doctorate degree. His major work was in developing thermodynamic theory, which brought physical chemistry from an empirical inquiry to a deductive science. In 1873, he published two papers concerning the fundamental nature of entropy of a system, and established the “thermodynamic surface,” a geometrical and graphical method for the analysis of the thermodynamic properties of substances. His famous On the Equilibrium of Homogeneous Substances, published in 1876, established the use of “chemical potential,” now an important concept in physical chemistry.*TIS Gibbs studied at Yale, Paris, Berlin, and Heidelberg before becoming Professor of Mathematical Physics at Yale. He was one of the inventors of vector analysis, and discussed the “Gibbs Phenomenon” in the theory of Fourier Series. *VFR

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1854 Benjamin Osgood Peirce (11 February 1854 Beverly, Massachusetts, USA — 14 January 1914 Cambridge, Massachusetts, USA) was an American mathematician and a holder of the Hollis Chair of Mathematics and Natural Philosophy at Harvard from 1888 until his death in 1914.*Wik

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1862 Francis Sowerby Macaulay FRS (11 February 1862 – 9 February 1937) was an English mathematician who made significant contributions to algebraic geometry. He is most famous for his 1916 book, The Algebraic Theory of Modular Systems, which greatly influenced the later course of algebraic geometry. Both Cohen-Macaulay rings and the Macaulay resultant are named for Macaulay.
Macaulay was educated at Kingswood School and graduated with distinction from St John's College, Cambridge.  He taught top mathematics class in St Paul's School in London from 1885 to 1911. His students included J. E. Littlewood and G. N. Watson.*Wik Littlewood consulted the examinations record and wrote, "In the 25 years from [Macaulay's] appointment to St Paul's in 1885 to his resignation in 1911 there were 41 scholarships (34 at Cambridge) and 11 exhibitions; and in the 20 years available there were 4 senior wranglers, one second, and one fourth among his former pupils." *SAU

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1887 John Jackson (11 Feb 1887 in Paisley, Renfrewshire, Scotland - 9 Dec 1958 in London, England) graduated from Glasgow and Cambridge. He went to the Royal Observatory at Greenwich but his career there was interrupted by World War I. He was then appointed HM Astronomer at the University of Cape Town. *SAU

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1898 Leo Szilard (11 Feb 1898; 30 May 1964 at age 66) Hungarian-American physicist who, with Enrico Fermi, designed the first nuclear reactor that sustained nuclear chain reaction (2 Dec 1942). In 1933, Szilard had left Nazi Germany for England. The same year he conceived the neutron chain reaction. Moving to N.Y. City in 1938, he conducted fission experiments at Columbia University. Aware of the danger of nuclear fission in the hands of the German government, he persuaded Albert Einstein to write to President Roosevelt, urging him to commission American development of atomic weapons. In 1943, Major General Leslie Groves, leader of the Manhattan Project designing the atomic bomb, forced Szilard to sell his atomic energy patent rights to the U.S. government. *TIS 

He became known as one of the “Martians,” Hungarian scientists of seemingly unearthly intelligence working during the war.

Frederik Pohl , talks about Szilard's epiphany about chain reactions in Chasing Science (pg 25),

".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."

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1891 Ivan Ivanovich Privalov (13 Feb 1891 in Nizhny Lomov, Penza guberniya (now oblast), Russia - 13 July 1941 in Moscow, USSR) Privalov, often in collaboration with Luzin, studied analytic functions in the vicinity of singular points by means of measure theory and Lebesgue integrals. He also obtained important results on conformal mappings showing that angles were preserved on the boundary almost everywhere. In 1934 he studied subharmonic functions, building on the work of Riesz. He published the monograph Subharmonic Functions in 1937 which gave the general theory of these functions and contained many results from his papers published between 1934 and 1937. *SAU

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1897 Emil Leon Post (February 11, 1897, Augustów – April 21, 1954, New York City) was a mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. In 1936, Post developed, independently of Alan Turing, a mathematical model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometimes called "Post's machine" or a Post-Turing machine, but is not to be confused with Post's tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but first published in 1943. Post's rewrite technique is now ubiquitous in programming language specification and design, and so with Church's lambda-calculus is a salient influence of classical modern logic on practical computing. Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all.
The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages.
In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful priority method in recursion theory.
Post made a fundamental and still influential contribution to the theory of polyadic, or n-ary, groups in a long paper published in 1940. His major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n − 1. He also demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set. The paper contains many other important results.*Wik

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1909 Claude Chevalley (11 Feb 1909 in Johannesberg, Transvaal, South Africa - 28 June 1984 in Paris, France had a major influence on the development of several areas of mathematics including Ring Theory and Group Theory *SAU

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1915 Richard Wesley Hamming (11 Feb 1915, 7 Jan 1998)  was an American mathematician who devised computer Hamming codes - error-detecting and correcting codes (1947). These add one or more bits to the transmission of blocks of data, used for a parity check, so that errors can be corrected automatically. By making a resend of bad data unnecessary, efficiency improved for modems, compact disks and satellite communications. He also worked on programming languages, numerical analysis and the Hamming spectral window (used to smooth data before Fourier analysis is carried out). He taught at University of Louisville, then during WW II worked (1945) on computers with the Manhattan Project creating the atomic bomb. From 1946, he spent 30 years with Bell Telephone Labs, eventually becoming head of computing science research.*TIS

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1917 Andrzej Alexiewicz (11 February 1917, Lwów, Poland – 11 July 1995) was a Polish mathematician, a disciple of the Lwow School of Mathematics. Alexiewicz was an expert at functional analysis and continued and edited the work of Stefan Banach. *Wik

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DEATHS

1141 Hugh of St. Victor died. For him the word “mathematica” had two meanings: When the ‘t’ is not aspirated it means “the superstition of those who place the destiny of men in the constellations” of the heavens; when the ‘t’ is aspirated it means the science of “abstract quantity.” *VFR

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1555 Giovanni Antonio Magini (in Latin, Maginus) (June 13, 1555; Padua, Italy – February 11, 1617; Bologna, Italy) was an Italian astronomer, astrologer, cartographer, and mathematician.
Dedicating himself to astronomy, in 1582 he wrote Ephemerides coelestium motuum, translated into Italian the following year.
In 1588 he was chosen over Galileo to occupy the chair of mathematics at the University of Bologna after the death of Egnatio Danti.

Magini supported a geocentric system of the world, in preference to Copernicus's heliocentric system. Magini devised his own planetary theory, in preference to other existing ones. The Maginian System consisted of eleven rotating spheres, which he described in his Novæ cœlestium orbium theoricæ congruentes cum observationibus N. Copernici (Venice, 1589).
In his De Planis Triangulis (1592), he described the use of quadrants in surveying and astronomy. In 1592 Magini published Tabula tetragonica, and in 1606 devised extremely accurate trigonometric tables. He also worked on the geometry of the sphere and applications of trigonometry, for which he invented calculating devices. He also worked on the problem of mirrors and published on the theory of concave spherical mirrors.
He also published a commentary on Ptolemy’s Geographia (Cologne, 1596).
As a cartographer, his life's work was the preparation of Italia or the Atlante geografico d'Italia (Geographic Atlas of Italy), printed posthumously by Magini's son in 1620. This was intended to include maps of every Italian region with exact nomenclature and historical notes. A major project, its production (begun in 1594) proved expensive and Magini assumed various additional posts in order to fund it, including becoming tutor in mathematics to the sons of Vincenzo I of Gonzaga, Duke of Mantua, a major patron of the arts and sciences. He also served as court astrologer. The Duke of Mantua, to whom the atlas is dedicated, assisted him with this project and allowed for maps of the various states of Italy to be brought to Magini. The governments of Messina and Genoa also assisted Magini financially in this project. Magini did not do any of the mapping himself.
He was also interested in pursuits which today would be considered pseudoscientific. A strong supporter of astrology, he defended its use in medicine in his De astrologica ratione (Venice, 1607). Magini collaborated closely with Valentine Naibod, and in this book he published De annui temporis mensura in Directionibus and De Directionibus from Naibod's unfinished manuscript Claudii Ptolemaei Quadripartitae Constructionis Apotelesmata Commentarius novus et Eiusdem Conversio nova. He was also interested in metoposcopy.
He corresponded with Tycho Brahe, Clavius, Abraham Ortelius, and Johann Kepler.
*Wik

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1626 Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to prove Euclid's fifth postulate. Cataldi discovered the sixth and seventh Mersenne primes by 1588. He held the record for the largest known prime for almost two centuries, until Leonhard Euler discovered 2³¹ - 1 was the eighth Mersenne prime.*Wik
(It is known now that "J Scheybl gave the sixth perfect number in 1555 in his commentary to a translation of Euclid's Elements. This was not noticed until 1977 and therefore did not influence progress on perfect numbers." *SAU )

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1650 René Descartes (31 March 1596 in La Haye (now Descartes),Touraine, France
- 11 Feb 1650 in Stockholm, Sweden)was a French philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. His work had a great influence on both mathematicians and philosophers. La Géométrie is by far the most important part of this work. Scott summarises the importance of this work in four points:

He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes arbitrariness.
Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all.
Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method.
Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all.

*SAU  His lifelong habit of laying abed till noon was interrupted by Descartes’ new employer, the athletic, nineteen-year-old Queen Christiana of Sweden, who insisted he tutor her in philosophy in an unheated library early in the morning. This change of lifestyle caused the illness that killed him. [Eves, Circles, 177◦]*VFR

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1868 Jean Bernard Léon Foucault (18 Sep 1819; 11 Feb 1868) French physicist whose Foucault Pendulum experimentally proved that the Earth rotates on its axis (6 Jan 1851). Using a long pendulum with a heavy bob, he showed its plane rotated at a rate related to Earth's angular velocity and the latitude of the site. He studied medicine and physics and became an assistant at the Paris Observatory (1855). He invented an accurate test of a lens for chromatic and spherical aberations. Working with Fizeau, and also independently, he made accurate measurements of the absolute velocity of light. In 1850, Foucault showed that light travels slower in water than in air. He also built a gyroscope (1852), the Foucault's prism (1857) and made improvements for mirrors of reflecting telescopes. *TIS (a brief biography of Foucault is here)

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1914 Alexander Ross Clarke (16 Dec 1828; 11 Feb 1914) English geodesist with the Army Ordnance Survey who made calculations of the size and shape of the Earth (the Clarke ellipsoid) were the first to approximate accepted modern values with respect to both polar flattening and equatorial radius. The figures from his second determination (1866) became a standard reference for U.S. geodesy for most of the twentieth century until satellites could improve accuracy. In 1880, Clarke coined the term "Geodesy" when he published his famous book by that title. He wrote articles on mathematical geography and geodesy and also contributed "The Figure of the Earth" in the Encyclopedia Britannica. His military service with the Ordnance Survey lasted 27 years.*TIS

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1923 Wilhelm Karl Joseph Killing (10 May 1847 in Burbach (near Siegen), Westphalia, Germany - 11 Feb 1923 in Münster, Germany) introduced Lie algebras independently of Lie in his study of non-euclidean geometry. His classification of the simple Lie algebras was one of the finest achievements in the whole of mathematical research.*SAU

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1942 Egbert van Kampen,(28 May 1908 in Berchem, Antwerp, Belgium - 11 Feb 1942 in Baltimore, Maryland, USA) In 1908 he left Europe and traveled to the United States to take up the position which he had been offered at Johns Hopkins University in Baltimore, Maryland. There he met Oscar Zariski who had taught at Johns Hopkins University as a Johnston Scholar from 1927 until 1929 when he had joined the Faculty. Zariski had been working on the fundamental group of the complement of an algebraic curve, and he had found generators and relations for the fundamental group but was unable to show that he had found sufficient relations to give a presentation for the group. Van Kampen solved the problem, showing that Zariski's relations were sufficient, and the result is now known as the Zariski–van Kampen theorem. This led van Kampen to formulate and prove what is nowadays known as the Seifert–van Kampen theorem. *Wik

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1959  Hardy Cross (10 Feb 1885; died 11 Feb 1959 at age 73) U.S. professor of civil and structural engineering whose outstanding contribution was a method of calculating tendencies to produce motion (moments) in the members of a continuous framework, such as the skeleton of a building. By the use of Cross's technique, known as the moment distribution method, or simply the Hardy Cross method, calculation can be carried to any required degree of accuracy by successive approximations, thus avoiding the immense labour of solving simultaneous equations that contain as many variables as there are rigid joints in a frame. He also successfully applied his mathematical methods to the solution of pipe network problems that arise in municipal water supply design; these methods have been extended to gas pipelines. *TIS

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1973  Johannes Hans Daniel Jensen (25 Jun 1907, 11 Feb 1973 at age 65) was a German physicist who proposed the shell theory of nuclear structure of nucleons - protons and neutrons - grouped in onion-like layers of concentric shells. He suggested that the nucleons spun on their own axis while they moved in an orbit within their shell and that certain patterns in the number of nucleons per shell made the nucleus more stable. Scientists already knew that the electrons orbiting the nucleus were arranged in different shells. For his model of the nucleus, Jensen shared the 1963 Nobel Prize in physics (with Maria Goeppert-Mayer, who arrived at the same hypothesis independently in the U.S.; and Eugene P. Wigner for unrelated work.) Through the 1950s, Jensen worked on radioactivity.
*TIS

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1974 Vladimir Ivanovich Smirnov  (10 June 1887 – 11 February 1974) was a Russian mathematician who made significant contributions in both pure and applied mathematics, and also in the history of mathematics.
Smirnov worked on diverse areas of mathematics, such as complex functions and conjugate functions in Euclidean spaces. In the applied field his work includes the propagation of waves in elastic media with plane boundaries (with Sergei Sobolev) and the oscillations of elastic spheres.
Smirnov is also widely known among students for his five volume book A Course in Higher Mathematics (the first volume was written jointly with Jacob Tamarkin).*Wik

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1976 Dorothy Maud Wrinch (12 September 1894 – 11 February 1976) married names Nicholson, Glaser) was a mathematician and biochemical theorist best known for her attempt to deduce protein structure using mathematical principles. *Wik

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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

Trisecting an Angle, an Interesting Historical Comparison and a Mystery

 A correspondent, who chooses to remain nameless recently wrote me with some notes on Jost Burgi's (written in Pitiscus' Trigonometria, 1595)& and Viete's (Supplementum Geometriae, 1593 as described by Robin Hartshorne)  approaches to trisecting a general angle. I was first struck by how Burgi's approach seemed to be very algebraic, while Viete, father of algebra, uses a ancient geometric approach with the neusis, (sort of a marked straight edge approach) .

In addition, even though I own Smith's 'Sourcebook', and  had probably read the Pitiscus/Burgi article several times, I had never worked through his actual math.  Doing so left me somewhat confused, confounded, and perhaps enlightened.


Links to both documents on the internet are below.

First, I will explain Viete's use, somewhat amplified to make it clearer to younger readers, and a bit about why he might have chosen this method.  Then I will do somewhat the same with Burgi's method as presented by Pitiscus.
Viete explains clearly that he will extend the classical compass and straightedge approach by allowing, "from any point to any two straight lines, to draw a straight line cutting off between them any segment fixed in advance."

The Neusis Approach of Viete (and Archimedes)

Students who may hear about Viete's introduction of letters as variables in mathematical operations may not realize he was an excellent geometer. In fact he chose to follow a homogeneity of dimension restriction that kept him from addressing simple quadratics like x^2 -2x = 3 since it involved adding the area of a square to the length of a line. His ability to adjust such problems to become problems of proportions, so that x^2 + Zx = B^2 where the S and B were considered as known values (or pronumerals) allowed him to effect the same results. But in this case, he was seeking a solution to cubics, the so called "casus irreducibilis", and trying to avoid the use of Cardano's imaginaries (complex numbers) which were still unexplained geometrically. Caspar Wessel's paper on the geometric meaning of the complex numbers would be printed within a few years (1597) but even then, it would make hardly a ripple in the world of math outside his home country. Wessel was as much unknown as Cardano was known.
The relation of trisecting to the cubic equation \(x^3 - 3x = 2a)\) was well known, and Viete knew that with a suitable change of variables, it could be used to solve any cubic with all real roots.The method he would use dates back to the early Greeks, and was known and used by Archimedes. 
I have tried to illustrate the "neusis" approach with static pictures from a very dynamic program, but teachers who are interested in making this clearer to their students can easily construct an interactive model for their students.

We begin with an isosceles triangle with a base angle at the origin and the angle to be trisected and the vertex on a circle centered at the origin.  The marked ruler is represented here by the ray EG, and for our purpose has G fixed on the circle passing through the top vertex of the triangle.  In use, the mathematician would lay his straightedge along one of the two congruent sides of the triangle, and mark EG so that it was the length of a side.  Then holding E on the x-axis, and G on the circle, he would move the points along the respective curves until the straight edge passed through the vertex angle at C .  The resulting at \(\angle \) AEG is the trisector.

For ease in seeing a simple proof, I add one drawing with a second isosceles triangle shown:

There are actually three isosceles triangles in the figure, the two shaded ones, \(\Delta \)ACH , and \(\Delta \)AGE and the one between them \(\Delta \)CAG.   If we label the \( \angle \) CAH to be trisected as y,  then it's supplementary angle, \( \angle \) CAE is 180 - y.  Now if we call the equal angles \( \angle \)AEG and \( \angle \)GAE each x, then the exterior angle \( \angle \)AGC to this isosceles triangle must be 2x, as must it's base angle partner \( \angle \)ACG.  This leaves \( \angle \) CAG as 180 - 4x.  Adding back the value x at \( \angle \)GAE means that \( \angle \)CAE is equal to 180 - 3x, but we already know it is equal to 180-y, so y=3x as we sought.

Burgi   Using Brahmagupta's theorem and double false position. 
Writing at the same time, Pituscus describes Jost Burgi's approach with a much more algebraic or arithmetical approach.  They will create a complex construction involving Brahmagupta's theorem for inscribed quadrilaterals, then make an two educated guess of the trisecting angle's chord, then use the errors and a linear approximation technique called the rule of double false position to find the chord of the trisected angle.  The idea of double false position as used by Piticus/Burgi  is almost certainly worth showing to students for the historical value alone. Fibonacci described double false position in his well know “Liber Abaci” , using it's Arabic name, and says the method is the one “by which the solutions of nearly all problems are found."  I would think a simple illustration to students is essential before exposing them to the approach used by Burgi and is well within the grasp of any student who has worked with linear equations.

 Double False Position
Here is an example using a linear equation for which the method is exact.  For non-linear functions, like the trigonometric functions, it serves as a method of approximation.
Suppose we take a simple expression like 5x-3 = 7 and have no idea how to solve it.  We might begin with a guess of x=1.  Trying that out we get 5(1)-3 = 2, an error that is five too low, or -5.  We decide to guess larger, and try x=4.  This time 5(4)-3  gives us 17, which is 10 above our target.  The method of double false position was to take the cross products of each trial value, and the opposite error, and find their difference, then divide the result by the difference in the errors.  If we label the x values x₁ and x₂ and the errors e₁  and e₂, then the equation can be written as \( \frac{x_1 * e_2 - x_2 * e_1}{e_2 - e_1}\).  If we apply this to the trials above we get \( \frac{1 * 10 - 4 * (-5)}{10 - (-5)}\), and to none of our surprise, it tells us the correct value of x was 2.

As to why Burgi would have used this approach, when he might have used the much simpler neusis method of Archimedes and Viete, I think it was a matter of focus.  Burgi was an astronomer and precision clock maker who had just before this writing, made an astronomical clock which was based on the Copernican system, and along the way created a table of sines (chord lengths) for astronomical use, his Canon Sinuum which seems not to have been published. Burgi had independently invented logarithms, almost certainly prior to Napier, as an aide to his computations.  I believe that Burgi, and Pitiscus use of Burgi in his Trigonometria, was for the purpose of computing chord lengths for Sine Tables.  To obtain more accurate chord lengths, astronomers increased the radius of their calculating circles to seven or more digits.  In the time before decimals were common (although Viete supposedly, "... wrote decimal fractions with the fractional part printed in smaller type than the integral and separated from the latter by a vertical line." )  the use of a large radius allows the use of a chord to be expressed with greater accuracy and in integer values.  When Pitiscus uses 5176381 for the chord length of a 30^o arc, he is giving what we would now write as .5176381 for 2 sin(15^o). 

Burghi's Use of Brahmagupta's Thm
Burgi realized that if you start with the chord of an angle (in this case, the chord of AB in \( \angle \)ADB) and double the angle by adding another chord of the same length, (BC, thus making  \( \angle \) ADB congruent to \( \angle \)BDC, and the chord of  \( \angle \) ADC would be AC). Then with one more congruent chord (CD) we have the chord of triple the arc of AB, and thus an angle AD is the chord for an angle three times \( \angle \) ABD (you can pick the point P for this APD anywhere on the arc AD).

With this groundwork, Burgi recognized that the four points form an inscribed quadrilateral, and so Brahmagupta's theorem would apply. That is, in this case, AC * BD = AB*CD + AD * BC. But since AB = BC = CD, and AC = BD we can rewrite that as \( AC^2 = AB^2 + AB * AD \).  And by a little algebra we arrive at \( \frac{AC^2 - AB^2}{AB} = AD \) 
Now to trisect the angle, we will be able to construct the chord AD from the given angle, and we want to find the length of the chord that would subtend 1/3 of that angle, that is, the length of AB.

The Mystery
As this point, Pitiscus says to, take a third of the chord, add something to it (the chords of the three equal angles are simply longer than the chord of the triple angle, so we must have something a little larger), and use the method of problem 3 (Brahmagupta's thm)  to compute the given chord.  He knows this will have some error, and so he will repeat it again to get a second error and use the method of double false position to find a close approximation.  But......   I can see no way he knew AC, or AC² .  If we are building a table of chords,  and we knew AC or AC² and hence the chord of 2/3 of the angle, we would have bisected this angle to begin with rather than trisecting the original angle since such a method was known at least as early as Hipparchus .  And Burgi was an expert in  prosthaphaeresis, the use of combining addition and subtraction of products to sums and is often credited with inventing at least two of them. How could we... how could HE carry out this operation?

Doubt and Suspicion (or guesses and wild conjectures)
 Immediately after the explanation of this problem, he gives "another method by algebra" and described the relation  which in modern language would be x^3-3x =2a.   Written in terms of chord lengths this would be crd3δ = 3crdδ−crd³ δ which might have been known as early as Ptolemy, and which Burgi and Pitiscus both well knew.

For the example of a 30^o inscribed angle, we know that the chord length is about .5176381 (or 2 sin(15^o) . Burgi's approach is to guess that the chord of 1/3 the arc is a little more than 1/3 this number, and he chooses .173 as his first guess.  Letting this be x in the expression \( f(x) = 3x - x^3 \) which I now suspect is what he did, we arrive at .5138223 (Burgi / Pitiscus use this number without decimal points, as explained before) .
Since he intends to use the method of double false position, he picks a second guess (.174) and gets a second estimate for his triple chord, .5167320
He arrives at an estimate using the double false position method of  .1743114.

You might compare this to the actual value of 2 sin (5^o) to compare. The error depends on how close the original guesses were, since the chord lengths are not linear.

Testing Alternatives

What if we try to follow the approach using Brahmagupta's formula to get the values?  Beginning with .173.   We have to find a way to tease out a guess for AC from this.  Since \( \Delta \) ABC is isosceles, with Arcs AB and BC both being 10 ^o arcs, the inscribed angles \( \angle \) BAC and \( \angle \) BCA are both 5,sup>o
leaving 170 degrees for \( \angle \) ABC.  Using my very modern calculator I arrive at a length for AC² of  .1206146401.  When this and .173 are used in the Brahmagupta formula, we get .5241944515; not miles away from the .5138... result he found, but obviously not the approach he used. 
I tried a few other things as well and all were as far, or farther away from Pitiscus' value than this, and none seem to get me very close to his result other than the cubic.  I would love to hear some other approaches to how he might have done this other than the cubic. 

And my special thanks to my unnamed correspondent for his note,  sending me on several interesting days of exploration and study.  Now on to write about the other ways of trisecting an angle. 

Robin Hartshorne Paper on Viete https://math.berkeley.edu/~robin/Viete/construction.html

D E Smith "Piticus on Bergi's trisection of an Arc \" https://books.google.com/books?id=awAfO7Ff_z0C&q=Pitiscus+trisection#v=snippet&q=Pitiscus%20trisection&f=false

On This Day in Math - February 10

Newton First Day Cover *BFDC.co.uk

It is impossible to be a mathematician without being a poet in soul.

~Sofya Kovalevskaya

The 41st day of the year; Euler (1772) observed that the polynomial f(x)= x² + x + 41 will produce a prime for any integer value of x in the interval 0 to 39. In 1778 Legendre realized that x² - x + 41 will give the same primes for interval (1-40).
 n^2 + n + 41 is prime for n = 0 ... 39 and Is prime for nearly half the values of n up to 10,000,000. *John D. Cook

If you multiply 41 by any three digit number to produce a five digit number, every cyclic representation of that number formed by moving the last digit to the front is also divisible by 41. (for example 41*378 = 15,498. 41 will also divide 81,549; 98154; 49815; and 54,981 *The Moscow Puzzles
41 can be expressed as the sum of consecutive primes in two ways, (2 + 3 + 5 + 7 + 11 + 13), and the  (11 + 13 + 17).

41 is the smallest integer whose reciprocal has a 5-digit repeating decimal.  \(\frac{1}{41} = \overline{.02439}\)

And even more from @Math Year-Round 41=1!+2!+3!+1¹+2²+3³

Found this on Twitter(X) by Abakcus

The Expanded Number Facts collection for year days 1-60 are now available here

EVENTS

1676/7 “The truth of it is, mathematical learning will not go off without a dowry; the booksellers have lost so much by the works of Drs. Wallis and ... Barrow ... that it is no easy task to persuade booksellers to undertake any thing but toys that are mathematical.” [Collins to Baker, The Correspondence of Isaac Newton, Vol. 2, p. 192]

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In 1720, Edmund Halley was appointed second Astronomer Royal of England. Halley succeeded John Flamsteed as Astronomer Royal, a position he would hold until his death in 1742 at the age of 85. *Wik  On the 300th anniversary of his trip to St Helena, the island issued three stamps. The 5p value shows Halley’s comet as it appeared on the Bayeux Tapestry.

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1734 The Marquise de Chatelet writes to ask Maupertuis to explain his claim in a 1732 paper that "God preferred the inverse square law of attraction over all others." Apparently he failed to convince her. *J. B. Shank, The Newton Wars and the Beginning of the French Enlightenment

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1747  "On the acceptance of trigonometry in wasan: Evidence from a text of Aida Yasuaki" [Born 10 February 1747 Yamagata, Japan ]  J. Marshall Unger; Historia Mathematica Volume 52, August 2020, Pages 51-65  .  

"Aida was hard-working and strong-willed and produced as many as fifty to sixty works a year. Nearly 2000 works survived him, including many on non-mathematical subjects. He was a distinguished teacher of traditional mathematics and a successful populariser of that discipline."

*mathshistory.st-andrews

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1860 Joseph Toynbee (English otologist, whose career was dedicated to pathological and anatomical studies of the ear.) writes to ask Faraday for tickets to Thomas Henry Huxley's lecture.

18 Savile Row. Burlington Gardens. W | 10 Feb 1860

Dear Mr. Faraday,

Having been unsuccessful in my attempts to obtain a ticket for Mr. Huxley’s lecture tonight I shall esteem it a favour if you can give me one.

Believe me yours sincerely ; obliged,

Joseph Toynbee

*JAIPREET VIRDI, From the Hands of Quacks

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1860  Published #OTD 1860 by the LinneanSociety, Alfred Russel Wallace's  paper : "On the Zoological Geography of the Malay Archipelago". The paper having been communicated on his behalf by Charles Darwin to the Society on 3/11/1859.  

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1880 On 10 February 1880, an article ran in the Daily Times (of Bethlehem, Pennsylvania) describing a remarkable invention recently demonstrated by a local inventor, Dr. H.E. Licks. The invention allowed images to be transmitted by telegraph. In other words, it resembled what people today would recognize as a television. However, Licks called his invention a "diaphote," from the Greek dia meaning "through" and photos meaning "light".
Despite the excitement it generated, the diaphote turned out to be a hoax. However, it wasn't until 1917 that "Dr. H.E. Licks" revealed the hoax in a book, Recreations in Mathematics. Licks himself is believed to have been the pseudonym of Mansfield Merriman (1848-1925), a professor of engineering at Lehigh University. *Museum of Hoaxes

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1914 The Girl Scout’s patent the trefoil design for their membership badge. The badge was designed by the founder of Girl Guides (renamed in 1913), Juliette "Daisy" Gordon Low who began the program in Savannah, Ga. with a group of 18. This (2012) is the centennial year of the Girl Scouts. The organization has declared 2012 the "Year of the Girl", and the Smithsonian's National Portrait Gallery opened a year-long exhibit "Juliette Gordon Low and 100 Years of Girl Scouts" on January 13, 2012.*Wik

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1946 War Department revealed the development of ENIAC, an electronic numerical integrator and computer. It is 1000 times faster than human computation. *VFR

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In 1958, radar signals were bounced off the planet Venus by MIT engineers at Lincoln Laboratories in experiments conducted during an inferior conjuction with Venus. A maser installed at the Millstone Hill radar site, Westford, Mass., was used. The return echoes were distinguished from the background noise using digital signal processing. The results were confirmation by other researchers, and together led to a more precise determination of the value of the astronomical unit (AU), the Earth's mean distance to the sun. A new value of the AU, 149,600,000 km was adopted at a general meeting of the International Astronomical Union in Hamburg (1964). Radar was also bounced off the Sun (7 Apr 1959). Lincoln Laboratories had been set up to conduct military rather than astronomical research.*TIS

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1996 Deep Blue defeats Kasparov
In the first game of a six game match, IBM's Deep Blue chess computer defeated world champion Garry Kasparov. No computer had ever won a game against a world champion in chess. Kasparov would eventually win the series 4-2, but would lose to Deep Blue in a re-match a year later.*CHM

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2012 2/10/2012 is a palindrome, and even if it's written 2/10/12. Or even in Roman numerals II/X/MMXII. What other years may be expressed as palindromes in two of these three methods (or European date style, dd/mm/yy

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BIRTHS

1747 Aida Yasuaki (February 10, 1747 – October 26, 1817) Japanese mathematician who published about 2000 works. Aida compiled Sampo tensi shinan which appeared in 1788. It is a book of geometry problems, developing formulae for ellipses, spheres, circles etc. Aida explained the use of algebraic expressions and the construction of equations. He also worked on number theory and simplified continued fraction methods due to Seki. *SAU

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1785 Claude Louis Marie Henri Navier (10 Feb 1785 in Dijon, France - 21 Aug 1836 in Paris, France) Claude-Louis Navier was a French mathematician best known for the Navier-Stokes equations describing the behaviour of a incompressible fluid. We should note, however, that Navier derived the Navier-Stokes equations despite not fully understanding the physics of the situation which he was modelling. He did not understand about shear stress in a fluid, but rather he based his work on modifying Euler's equations to take into account forces between the molecules in the fluid. Although his reasoning is unacceptable today. *SAU

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1885 Hardy Cross (10 Feb 1885; died 11 Feb 1959 at age 73) U.S. professor of civil and structural engineering whose outstanding contribution was a method of calculating tendencies to produce motion (moments) in the members of a continuous framework, such as the skeleton of a building. By the use of Cross's technique, known as the moment distribution method, or simply the Hardy Cross method, calculation can be carried to any required degree of accuracy by successive approximations, thus avoiding the immense labour of solving simultaneous equations that contain as many variables as there are rigid joints in a frame. He also successfully applied his mathematical methods to the solution of pipe network problems that arise in municipal water supply design; these methods have been extended to gas pipelines.
*TIS

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1888 Selig Brodetsky (10 February 1888 - 20 May 1954)was educated at Cambridge and Leipzig. He became a lecturer at Bristol and later lecturer and professor at Leeds. He worked on fluid flow with particular emphasis on aerodynamics. He was President of the Hebrew University of Jerusalem for a short time. *SAU

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1901 Richard Dagobert Brauer (10 Feb 1901; 17 Apr 1977 at age 76) German-American mathematician and educator, a pioneer in the development of algebra theory. He worked with Weyl on several projects including a famous joint paper on spinors (published in 1935 in the American Journal of Mathematics). This work provided a background for Paul Dirac's theory of the spinning electron within the framework of quantum mechanics. With Nesbitt, Brauer introduced the theory of blocks (1937). Brauer used this to obtain results on finite groups, particularly finite simple groups, and the theory of blocks would play a big part in much of Brauer's later work. Starting with his group-theoretical characterisation of the simple groups (1951), he spent the rest of his life formulating a method to classify all finite simple groups. *TIS

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1907 Agnes Mary Clerke (10 Feb 1842, 20 Jan 1907) Irish astronomical writer who was a diligent compiler of facts rather than a practicing scientist. Nevertheless, by 1885, her exhaustive treatise, A Popular History of Astronomy in the Nineteenth Century gained international recognition as an authoritative work. In 1903, with Lady Huggins, she was elected an honorary member of the Royal Astronomical Society, a rank previously held only by two other women, Caroline Herschel and Mary Somerville. Her publications included several books and 55 pieces in the Edinburgh Review. She contributed some astronomer biographies to the Dictionary of National Biography and some astronomical entries in the Encyclopaedia Britannica. *TIS More detail about her here

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DEATHS

1576 Wilhelm Xylander (born Wilhelm Holtzman, graecized to Xylander) (December 26, 1532 – February 10, 1576) was a German classical scholar and humanist.
Xylander was the author of a number of important works. He translated the first six books of Euclid into German with notes, the Arithmetica of Diophantus, and the De quattuor mathematicis scientiis of Michael Psellus into Latin. *Wik

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1865 Heinrich Friedrich Emil Lenz (24 Feb 1804, 10 Feb 1865 at age 61) was the Russian physicist who framed Lenz's Law to describe the direction of flow of electric current generated by a wire moving through a magnetic field. Lenz worked on electrical conduction and electromagnetism. In 1833 he reported investigations into the way electrical resistance changes with temperature, showing that an increase in temperature increases the resistance (for a metal). He is best-known for Lenz's law, which he discovered in 1834 while investigating magnetic induction. It states that the current induced by a change flows so as to oppose the effect producing the change. Lenz's law is a consequence of the, more general, law of conservation of energy. *TIS

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1868 Sir David Brewster (11 Dec 1781; 10 Feb 1868) Scottish physicist noted for his experimental work in optics and polarized light (light in which all waves lie in the same plane.) He is known for Brewster's Law, which relates the refractive index of a material to its polarizing angle (which is the incident angle at which reflected light becomes completely polarized. He patented the kaleidoscope in 1817. Later, he used lenses to improve three-dimensional images viewed with a stereoscope. Brewster also recommended the use of the lightweight, flat Fresnel lens in lighthouses.*TIS A nice blog about Brewster is here.

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1891 Sonya Kovalevsky (Sofya Vasilyevna Kovalevskaya) (15 Jan 1850; 10 Feb 1891) professor of mathematics at Stockholm. In her youth, her bedroom was wallpapered with the pages of a text from her father’s schooldays, namely, Ostrogradsky’s lithographed lecture notes on the calculus. Study of the novel wallpaper introduced her to the calculus at age 11. She became the greatest woman mathematician prior to the twentieth century. *VFR a Russian mathematician and novelist who made valuable contributions to the theory of differential equations.*TIS She died of influenza in 1891 at age forty-one, after returning from a pleasure trip to Genoa. She is buried in Solna(Stockholm suburb), Sweden, at Norra begravningsplatsen (Northern Cemetery). Alfred Nobel is buried in the same cemetery.

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Rontgen Family Grave, Giessen

1923 Wilhelm Conrad Röntgen (27 Mar 1845 - 10 Feb 1923 at age 77) was a German physicist who discovered the highly penetrating form of radiation that became known as X-rays on 8 Nov 1895. He received the first Nobel Prize for Physics (1901), “in recognition of the extraordinary services he has rendered by the discovery of the remarkable rays subsequently named after him.” This high-energy radiation, though first called Röngen rays, became known as X-rays. His discovery initiated revolutionary improvements in making medical diagnoses and enabled many new advances in modern physics. *TIS "In 1901 he became the first physicist to receive a Nobel prize." *VFR

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1927 Alfred George Greenhill (29 Nov 1847 in London, England - 10 Feb 1927 in London, England) graduated from Cambridge and became Professor of Mathematics at the Royal Military Academy at Woolwich. His main work was on Elliptic Functions but he published widely on applications of mathematics to practical problems. He became an honorary member of the EMS in 1908. *SAU

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1936 Eleanor Mildred (Balfour) Sidgwick, (11 March 1845 – 10 February 1936) was an activist for the higher education of women, Principal of Newnham College of the University of Cambridge and a leading figure in the Society for Psychical Research.
She was born in East Lothian, daughter of James Maitland Balfour and Lady Blanche Harriet. She was born into perhaps the most prominent political clan in nineteenth-century Britain, the 'Hotel Cecil': her brother Arthur would eventually himself become prime minister. Another brother, Frank, a biologist, died young in a climbing accident.
One of the first students at Newnham College in Cambridge, in 1876 she married (and became converted to feminism by) the philosopher Henry Sidgwick. In 1880 she became Vice-Principal of Newnham under the founding Principal Anne Clough, succeeding as Principal on Miss Clough's death in 1892. She and her husband resided there until 1900, the year of Henry Sidgwick's death. In 1894 Mrs Sidgwick was one of the first three women to serve on a royal commission, the Bryce commission on Secondary Education.
As a young woman, Eleanor had helped (John William Strutt, who was married to her sister, Evelyn) Lord Rayleigh improve the accuracy of experimental measurement of electrical resistance. She conducted several experiments in electricity and with him published three papers in the Philosophical Transactions of the Royal Society.
She subsequently turned her careful experimental mind to the question of testing the veracity of claims for psychical phenomena. She was elected President of the Society for Psychical Research in 1908 and named 'president of honour' in 1932. Her Husband, Henry, her brother and future Prime Minister, Arthur, and Lord Rayleigh all were also Presidents of the Society.)
She was a member of the Ladies Dining Society in Cambridge, with 11 other members.
In 1916 Mrs Sidgwick left Cambridge to live with one of her brothers near Woking; she remained there until her death in 1936.
She was awarded honorary degrees by the universities of Manchester, Edinburgh, St Andrews and Birmingham.Most of her writings related to Psychical Research, and are contained in the Proceedings of the Society for Psychical Research. However, some related to educational matters, and a couple of essays dealt with the morality of international affairs. *Wik & encyclopedia.com

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1951 George Abram Miller (31 July 1863 – 10 February 1951) was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance.
Miller was born in Lynnville, Lehigh County, Pennsylvania, and died in Urbana, Illinois.*Wik

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1994 Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation.*Wik

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1997 Jerome Namias (19 Mar 1910, 10 Feb 1997 at age 86) American meteorological researcher most noted for having pioneered the development of extended weather forecasts and who also studied the Dust Bowl of the 1930s and the El Niño phenomenon. *TIS In 1971 he joined the Scripps Institution and established the first Experimental Climate Research Center. His prognosis of warm weather during the Arab oil embargo of 1973 greatly aided domestic policy response.*Wik

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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

On This Day in Math - February 9

da Vinci's Stellated Dodecahedron from divina proportione

To argue with a man who has renounced the use of authority of reason is like administering medicine to the dead.
~Thomas Paine, American Patriot, and a pretty good bridge builder, perhaps pointing out something still very relevant to today's political discussions.

The 40th day of the year; in English forty is the only number whose letters are in alphabetical order.

There are 40 solutions on for the 7 queens problem.  placing seven chess queens on a 7x7 chessboard so that no two queens threaten each other.

-40 is the temperature at which the Fahrenheit and Celsius scales correspond; that is, −40 °F = −40 °C.

Euler first noticed (in 1772) that the quadratic polynomial P(n) = n² + n + 41 is prime for all non-negative numbers less than 40.
Paul Halcke noted in 1719 that the product of the aliquot parts of 40 is equal to 40 cubed. 1*2*4*5*8*10*20 = 64000 = 40³. He found the same is true for 24.

forty (quaranta)  is the root of the word quarantine, for the forty day period of isolation for visitors to Venice to control the plague.

And.... forty is the highest number ever counted to on Sesame Street.

The expanded collection of Number Factoids for Day 1-60  are available here

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EVENTS

1498 Luca Pacioli was professor at Milan. He was inspired to start his Divina Proportione on 9 Feb 1498 and completed it on 14 Dec 1498, though it was not published (in an expanded form) until 1509. (J Tennenbaum in 2005 said that it was completed by the Feb 9 date.)  He was a good friend of Leonardo da Vinci. It was Leonardo who drew the pictures for Pacioli's book. Pacioli may have advised Leonardo on the perspective for the painting of The Last Supper. Certainly Pacioli stimulated Leonardo's interest in perspective and it is possible that Leonardo's famous drawing of the proportions of the human body was inspired by Pacioli's comment on classical architecture; "For in the human body they found the two main figures ..., namely the perfect circle and the square." Pacioli seems to have made models of the polyhedra illustrated in his book, though we don't know if Leonardo used these for his drawings. A set was probably given to Pacioli's earlier patron, the Duke of Urbino, in 1494. Another set was paid for by Florence in 1504. *Unknown Internet Source
I was just reminded by a tweet from @IanMegaw that Pacioli also published the first description of double-entry bookkeeping. (thanks Ian)

Another first, or two, for Pacioli is found in a paper on the origins of the game of Nim by Lisa Rougetet.  "The oldest simplified variation of Nim found to date in Europe is in a manuscript of the Renaissance,De Viribus Quantitatis(On the powers of numbers) , a treatise written between 1496 and 1508 by the Italian mathematician Fra Luca Bartolomeo dePacioli , one of the most famous mathematicians of his time. ... TheDe Viribus Quantitatiscan be considered to be one of the first works entirely devoted to mathematical recreations 

I received a comment that included this un-cited quote with two more firsts about the book.

"The world's oldest magic text, De viribus quantitatis (On the Powers of Numbers), was penned by Luca Pacioli, a Franciscan monk who shared lodgings with da Vinci. ...  and contains the first ever reference to card tricks as well as guidance on how to juggle, eat fire and make coins dance. It is also the first work to note that da Vinci was left-handed."

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1849 After the death of Lord Kelvin’s Father, James Thompson, his replacement was the subject of a letter from William Hopkins to discuss the merits of G.G. Stokes and Hugh Blackburn for the position, “if you determine … to elect a man who is sure to hereafter to dignify his postion by the highest scientific distinction, Stokes is unquestionably your man.” * The correspondence between Sir George Gabriel Stokes and Sir William Thomson, pg 59

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In 1870, the U.S. Weather Bureau (later named the Weather Service) was authorized by Congress, and placed under the direction of the Signal Service. Cleveland Abbe had inaugurated a private weather reporting and warning service at Cincinnati and had been issuing weather reports or bulletins since 1 Sep 1869. Hence, Abbe was the only person in the country who was already experienced in drawing weather maps from telegraphic reports and forecasting from them. Naturally, he was offered an important position in this new service which he accepted, beginning 3 Jan 1871, and was often the official forecaster of the weather. He was the first U.S. metereologist, and known as the "father of the U.S. Weather Bureau."*TIS

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1883 The very first issue of Science is published. The first item in the “Weekly summary of the progess of science” contains a report by Thomas Craig that “Lindemann gave a proof of the fact that π cannot be a root of an equation of any degree with rational co-eficients. This is a most remarkable paper, as it thus contains the first direct, absolute proof that has ever been given of the impossability of the quadrature of the circle. ... Lindemann has certainly done a splendid piece of work in thus absolutely proving the impossibility of ‘squaring the circle’; and it is only to be regretted that his work will not carry conviction to the minds of those mistaken individuals, the ‘circle-squarers.’ But it is hardly to be supposed that they will be convinced of the futility of their task, any more than the perpetual-motion inventors were convinced by the discovery and enunciation of the principles of the conservation of energy.” [p. 15] *VFR

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1912 The Salt Lake Tribune, reporting on the 1912 theory that Mars was populated by a giant vegetable that possessed a single giant eye, accompanied by a remarkable diagram. *Paul Fairie

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1986 Halley’s comet last reached perihelion. The next return to perihelion will be on 28 July 2061. *Wik (I am still amazed that we can mathematically predict such an event with such precision.)

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BIRTHS

1489 Georg Hartmann (sometimes spelled Hartman; February 9, 1489 – April 9, 1564) was a German engineer, instrument maker, author, printer, humanist, churchman, and astronomer. After finishing his studies, he travelled through Italy and finally settled in Nuremberg in 1518. There he constructed astrolabes, globes, sundials, and quadrants. In addition to these traditional scientific instruments Hartmann also made gunner's levels and sights. Hartmann was possibly the first to discover the inclination of Earth's magnetic field. He died in Nuremberg.
His two published works were Perspectiva Communis (Nuremberg, 1542), a reprint of John Peckham's 1292 book on optics and Directorium (Nuremberg, 1554), a book on astrology. He also left Collectanea mathematica praeprimis gnomonicam spectania, 151 f. MS Vienna, Österreichische Nationalbibliothek, Quarto, Saec. 16 (1527–1528), an unpublished work on sundials and astrolabes that was translated by John Lamprey and published under the title of Hartmann's Practika in 2002. *Wik

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1775 Farkas Bolyai (9 Feb 1775, 20 Nov 1856) Hungarian mathematician, poet, and dramatist who spent a lifetime trying to prove Euclid's (fifth) postulate that parallel lines do not meet. While studying at the University of Göttingen, he met as a fellow student, the noted German mathematician Carl F. Gauss, with whom he corresponded as a life-long friend. Bolyai taught mathematics, physics and chemistry at Marosvásárhely all his life. He discouraged his son, János Bolyai, from studying the parallel axiom as he had, writing in a letter to him: "For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life." *TIS (In 1804 he believed that he had a proof the Euclid's fifth postulate could be deduced from the other axims. He sent this proof to C. F. Gauss who found an error. His Son, Janos, would ignore his father's warnings and go on to discover a non-Euclidean Geometry. ).

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1880 Lipót Fejér (9 Feb 1880, 15 Oct 1959) Fejér's main work was in harmonic analysis working on Fourier series and their singularities. Fejér collaborated to produce important papers with Carathéodory on entire functions and with Riesz on conformal mappings. *SAU In 1897 he won a prize in one of the first mathematical competitions held in Hungary. *VFR

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1907 Harold Scott MacDonald Coxeter (9 Feb 1907 in London, England - 31 March 2003 in Toronto, Canada) graduated from Cambridge and worked most of his life in Canada. His work was mainly in geometry. In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics. Among his most famous geometry books are The real projective plane (1955), Introduction to geometry (1961), Regular polytopes (1963), Non-euclidean geometry (1965) and, written jointly with S L Greitzer, Geometry revisited (1967). He also published a famous work on group presentations, which was written jointly with his first doctoral student W O J Moser, Generators and relations for discrete groups.
His 12 books and 167 published articles cover more than mathematical research. Coxeter met Escher in 1954 and the two became lifelong friends. Another friend, R Buckminister Fuller, used Coxeter's ideas in his architecture. In 1938 Coxeter revised and updated Rouse Ball's Mathematical recreations and essays, a book which Rouse Ball first published in 1892. *SAU

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1908 Alexander Dinghas (February 9, 1908 – April 19, 1974) was a Turkish mathematician. He is known for his work in different areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry. His most important contribution was his work in function theory, in particular Nevanlinna theory and the growth of subharmonic functions.*Wik

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1919 Irene Anne Stegun (February 9, 1919 – January 27, 2008) was a mathematician at the National Bureau of Standards who, with Milton Abramowitz, edited a classic book of mathematical tables called A Handbook of Mathematical Functions, widely known as Abramowitz and Stegun. When Abramowitz died of a heart attack in 1958, Stegun took over management of the project and finished the work by 1964, working under the direction of the NBS Chief of Numerical Analysis Philip J. Davis, who was also a contributor to the book. *Wik

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1927 David John Wheeler FRS (9 February 1927–13 December 2004) the Inventor of the Wheeler Jump, is Born. In 1951, he introduced the concept of the subroutine to computer programming, is born. He concentrated his work on assembly programming language and invoked the subroutine in his Wheeler jump technique. For this work Wheeler received the IEEE Computer Society Pioneer Award. *CHM
He was born in Birmingham and gained a scholarship at Trinity College, Cambridge to read mathematics, graduating in 1948. He completed the world's first PhD in computer science in 1951. His contributions to the field included work on the EDSAC and the Burrows-Wheeler transform. Along with Maurice Wilkes and Stanley Gill he is credited with the invention of the subroutine (which they referred to as the closed subroutine), because of which a jump to subroutine instruction is often called Wheeler Jump. He was responsible for the implementation of the CAP computer, the first to be based on security capabilities. In cryptography, he was the designer of WAKE and the co-designer of the TEA and XTEA encryption algorithms together with Roger Needham. In 2003 he was a Computer History Museum Fellow Award recipient.
Wheeler is often quoted as saying "All problems in computer science can be solved by another level of indirection... Except for the problem of too many layers of indirection." *Wik

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DEATHS

1734 Pierre Polinière (8 September 1671, Coulonces, France - 9 February 1734, Coulonces, France) was an early investigator of electricity and electrical phenomena, notably "barometric light", a form of gas-discharge light, which suggested the possibility of electric lighting. He also helped to introduce the scientific method in French universities. *Wik

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1811 Nevil Maskelyne (6 Oct 1732, 9 Feb 1811) (SAU gives 5 Oct for birhtdate)
British astronomer noted for his contribution to the science of navigation. In 1761 the Royal Society sent Maskelyne to the island of St Helena to make accurate measurements of a transit of Venus. This in turn gives the distance from the Earth to the Sun, and the scale of the solar system. During the voyage he also experimented with the lunar position method of determining longitude. In 1764 he went on a voyage to Barbados to carry out trials of Harrison's timepiece, followed by appointment as Astronomer Royal (1765). In 1774, he carried out an experiment on a Scottish mountain with the use of a plumb line to determine the Earth's density. He found it was approximately 4.5 times that of water. *TIS (the current scientific value of the Earht's density is about 5.2 times that of water.) He was the fifth English Astronomer Royal. He held the office from 1765 to 1811.*Wik

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1865 James Melville Gilliss (6 Sep 1811; 9 Feb 1865) U.S. naval officer and astronomer who founded the Naval Observatory in Washington, D.C., the first U.S. observatory devoted entirely to research. Gilliss joined the Navy as a midshipman at the age of 15. He taught himself astronomy, at a time when there was no fixed astronomical observatory in the U.S., and very little formal instruction. In 1838, when Charles Wilkes left on the famous South Seas Exploring Expedition, Gilliss became officer-in-charge of the Depot of Charts and Instruments, forerunner of the U. S. Naval Observatory. Gilliss's astronomical observations made during this time in connection with determining longitude differences with the Wilkes Expedition, resulted in the first star catalogue published in the United States. *TIS

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1883 Henry John Stephen Smith (2 Nov 1826 in Dublin, Ireland, 9 Feb 1883 in Oxford, England) was an Irish mathematician whose most important contributions are in number theory where he worked on elementary divisors. He proved that any integer can be expressed as the sum of 5 squares and as the sum of 7 squares, showing in how many ways this could occur. In addition to solving these cases explicitly, he gave a method which would yield the number of ways that an integer can be expressed as the sum of k squares for any fixed k. He published his results in The orders and genera of quadratic forms containing more than three indeterminates published in the Proceedings of the Royal Society in 1867. Eisenstein had earlier proved the result for 3 squares and Jacobi for 2, 4 and 6 squares. Smith also extended Gauss's theorem on real quadratic forms to complex quadratic forms. *SAU He posthumously received the Grand Prix des Sciences Mathematiques of the Paris Academy of Science for his proof that every positive integer is the sum of five squares. He shared the prize with the eighteen year old Hermann Minkowski.*VFR The prize was awarded on April 2, less than two months after his death.

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1937 Francis Sowerby Macaulay FRS (11 February 1862 – 9 February 1937) was an English mathematician who made significant contributions to algebraic geometry. He is most famous for his 1916 book, The Algebraic Theory of Modular Systems, which greatly influenced the later course of algebraic geometry. Both Cohen-Macaulay rings and the Macaulay resultant are named for Macaulay.
Macaulay was educated at Kingswood School and graduated with distinction from St John's College, Cambridge. He taught top mathematics class in St Paul's School in London from 1885 to 1911. His students included J. E. Littlewood and G. N. Watson.*Wik Littlewood consulted the examinations record and wrote, "In the 25 years from [Macaulay's] appointment to St Paul's in 1885 to his resignation in 1911 there were 41 scholarships (34 at Cambridge) and 11 exhibitions; and in the 20 years available there were 4 senior wranglers, one second, and one fourth among his former pupils." *SAU

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1970 Leo Moser (April 11, 1921, Vienna—February 9, 1970, Edmonton) was an Austrian-Canadian mathematician, best known for his polygon notation.
A native of Vienna, Leo Moser immigrated with his parents to Canada at the age of three. He received his Bachelor of Science degree from the University of Manitoba in 1943, and a Master of Science from the University of Toronto in 1945. After two years of teaching he went to the University of North Carolina to complete a Ph.D., supervised by Alfred Brauer.[1] There, in 1950, he began suffering recurrent heart problems. He took a position at Texas Technical College for one year, and joined the faculty of the University of Alberta in 1951, where he remained until his death at the age of 48. *Wik In mathematics, Steinhaus–Moser notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus’s polygon notation.

 a number n in a triangle means nⁿ.

 a number n in a square is equivalent with "the number n inside n triangles, which are all nested."

 a number n in a pentagon is equivalent with "the number n inside n squares, which are all nested."

*Wik 

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1988 Israel Nathan Herstein (March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was a mathematician, appointed as professor at the University of Chicago in 1951. He worked on a variety of areas of algebra, including ring theory, with over 100 research papers and over a dozen books.
He is known for his lucid style of writing, as exemplified by the classic and widely influential Topics in Algebra, an undergraduate introduction to abstract algebra that was published in 1964, which dominated the field for 20 years. A more advanced classic text is his Noncommutative Rings in the Carus Mathematical Monographs series. His primary interest was in noncommutative ring theory, but he also wrote papers on finite groups, linear algebra, and mathematical economics.*Wik

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2001 Herbert Alexander Simon (15 Jun 1916, 9 Feb 2001 at age 84) American social scientist who was a pioneer of the development of computer artificial intelligence. In 1956, with his long-time colleague Allen Newell, Simon produced the computer program, The Logic Theorist, a computer program that could discover proofs of geometric theorems. It was the first computer program capable of thinking, and marked the beginning of what would become known as artificial intelligence. It proved many of the theorems of symbolic logic in Whitehead and Russell's Principia Mathematica. He is further known for his contributions in fields including psychology, mathematics, statistics, and operations research, all of which he synthesized in a key theory for which he won the 1978 Nobel Prize for economics. *TIS

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2003 Masatoşi Gündüz İkeda (25 February 1926, Tokyo. - 9 February 2003, Ankara), was a Turkish mathematician of Japanese ancestry, known for his contributions to the field of algebraic number theory. *Wik

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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