Sunday, 2 October 2022

# 4 from old math term notes

 Argand Diagram Argand Diagrams, the method of drawing complex numbers as vectors on a coordinate plane, are named for Jean R. Argand (1768-1822), an amateur mathematician who described them in a paper in 1806. A similar method, although less complete, had been suggested as early as 120 years before by John Wallis, and developed extensively by Casper Wessel(1745-1818), a Norwegian surveyor. (Actually, at the time Wessel lived, the area where he was born was a part of Denmark. Norway became an independent government in 1905 after years of domination by Denmark and Sweden.) It may be that even then, the method was unknown to Gauss and he had to rediscover it for himself in 1831 although it has been suggested that Gauss may have discovered the idea as early as Wessel. Some parts of his Demonstratio Nova would seem almost miraculously derived without a knowledge of the ideas of the geometry of complex numbers. 

Wessel's paper was published in Danish, and was not circulated in the languages more common to mathematics at that time. It was not until 1895 that his paper came to the attention of the mathematical community, long after the name Argand Diagram had stuck. Incredibly, there were at least three more individuals who may have independently discovered and written on the same idea; Abbe Bruee, C. V. Mourney, and John Warren.

Argand's Book, Essai sur une maniere de representer les quantities imaginaires dans les constructions geometriques, might have suffered the same fate as Wessel except for an unusual chain of events. I give here the version as presented by Michael Crowe in his A History of Vector Analysis

In 1813 J. F. Francais published a short memoir in volume IV of Gergonne's Annales de mathematiques in which Francais presented the geometrical representation of complex numbers. At the conclusion of his paper Francais stated that the fundamental ideas in his paper were not his own, he had found them in a letter written by Legendre to his (Francis') brother who had died. In this letter Legendre discussed the ideas of an unnamed mathematician. Francis added that he hoped this mathematician would make himself known and publish his results.

The unnamed mathematician had in fact already published his ieas, for Legendre's friend was Jean Robert Argand. Hearing of Francais' paper, Argand immediately sent a communication to Gergonne in which he identified himself as the mathematician in Legendre's letter, called attention to his book, summarized its contents, and finally presented an (unsuccessful) attempt to extend his system to three dimensions.

Even with so much interest and attention to the geometry of complex numbers, it was not until Gauss published a short work on the ideas that they became popular.


Translations of both Wallis' and Wessel's papers on the imaginaries can be found in A Sourcebook of Mathematics by David Eugene Smith.

# 3 from old math terms notes

Parallelepiped This word for a solid made by intersecting pairs of parallel planes forming six faces that are each parallelograms is rapidly becoming obsolete, although no good word has emerged to replace it.

A rectangular or orthogonal parallelepiped is the of a room or a shoe box. The word is condensed from the Greek word parallelepipedon for the same shape. The roots are para (beside) + allel (other) + epi (on) and pedon (ground). Parallelepipedon was the word used by Billingsley in his 1570 translation of Euclid, the first known use of the word in English. According to John Conway, this was the common term in use until around 1870 to 1900 when it gave way to parallelepiped; although the OED lists its use by John Playfair as early as 1812. A posting from John Albree of Auburn University cited an earlier use. [In Charles Hutton's *Dictionary* (volume 2, 1795, p.199), the terms "parallelopiped" and "parallelopipedon" are presented equally, and he remarks that such a polyhedron "is only a particular species" of a prism.]


This newer term now seems headed for demise due to changes in school curriculum and the reduced coverage of solid geometry, although one correspondent suggests that "parallelepipedo" would be known by most Spanish students. I was somewhat surprised to find that parallelepiped is present in my computer spell check, which I find often omits technical terms.


The word is pronounced with the accent on the epi syllable. The para root is common in math words and is related to other words like parlor, paragraph, and parable. Allel became our alter, for other, and gives us alternate and alternative. The epi root shows up in epidermis (on the skin), epitaph (over the grave), epicycle (on the circle), and epidemic (on the people). Pedon is from ped for foot, and was also generalized for plane.


On This Day in Math - October 2


Euler calculated without effort, just as men breathe, as eagles sustain themselves in the air.
~François Arago


The 275th day of the year; 275 is the number of partitions of 28 in which no part occurs only once. (Students might try finding the similar number of partitions for 10, or some smaller number to get a sense for how they grow)

275 can be written in 5 ways as a sum of consecutive naturals, for example, 20 + ... + 30

275 is an arithmetic number, because the mean of its divisors, 62, is an integer

275 is the maximum number. of pieces that can be formed from an annulus with 22 straight lines. 



EVENTS

479 B.C.: an Annular Solar Eclipse known as "Xerxes' eclipse" as noted by Herodotus occurred. David Dickinson ‏@Astroguyz


1608, the Dutch Estates General examined an application for a patent for "a device to observe things at a distance" presented by a certain Hans Lipperhey (?-1619) an obscure spectacles-maker from Middelburg, in southwestern Holland The patent application was rejected on the grounds that, although the usefulness of the device was recognised, especially for military purposes it was deemed impossible to keep the secret of its construction for very long. And especially considering that, in those same days, another instrument-maker - a certain Sacharias Janssen (1588-1630), he too a spectacles-maker in Middelburg, indicated by Pierre Borel (c. 1620-1671) a few decades later as the true inventor of the telescope - declared that he knew how to build the instrument.*Institute and Museum of the History of Science
I doubt that many modern science/math historians believe that Lipperhey invented the telesccope, and neither did any of the myriad other names suggested over the years.
The first historical construct concerns the ‘invention’ itself, because what happened in
1608 was in fact not an invention at all, but merely a recognition of the great potential of a device, which must have been around for some decades, as a kind of toy or as a device whose purpose was to correct or improve vision. Indications of the awareness of the magnifying power of a combination of two lenses, long before the year 1608, are indeed abundant in the contemporary literature. For instance, in 1538 the Italian scholar Girolamo Fracastoro wrote: ‘If someone looks through two eye-glasses, of which one is placed above the other, he shall see everything larger and more closely.’
After seeing or hearing of Lipperhey’s telescope, many scholars had a kind of déjà vu -feeling. Girolamo Sirtori, who in 1612, only four years after the emergence of the instrument, composed his well-known Telescopium, captured this feeling in the following phrase: "It appeared that this conception was in the minds of many men, so that once they
heard about it, any ingenious person began trying to make one, without [the help of ]
a model."
*Huib J. Zuidervaart, The ‘true inventor’ of the telescope. A survey of 400 years of debate origins of the telescope


1667 Newton became a fellow at Trinity College, Cambridge. *VFR


1759 “Your solution of the isoperimetric problems leaves nothing to be desired and I rejoice that this subject, with which I have been so completely occupied since my first efforsts, has been carried by you to such a high degree of perfection. The importance of the subject has stimulated me to develop, aided by your lights, an analytical solution that I will keep secret as long as your own meditations are not published, lest I take away from you a part of the glory you deserve.” So wrote Euler to the young Lagrange. See Allen Shields, “Lagrange and the M´ecanique Analytique,” The Mathematical Intelligencer, 10:4, Fall 1988, pp. 7– 10. *VFR


1836 Charles Darwin returned from his voyage on the HMS Beagle to the Pacific. It would be 23 years before he published Origin of Species. *TIS


1856 Sylvester was to dine with Charles Wheatstone, the noted physicist and inventor, and had invited Arthur Cayley to attend and meet Wheatstone. Wheatstone had supported Sylvester's successful candidacy for the Royal Society in 1836. James Joseph Sylvester: Life and Work in Letters, *James Joseph Sylvester: Life and Work in Letters
edited by Karen Hunger Parshall


1912 Ernest Rutherford presents his theory of the structure of the Atom to a session of the Manchester Literary and Philosophical Society. He rejected Thompson's "Plum Pudding" model for an atom with most of its mass concentrated into a tiny charged core in its center. *Brody&Brody, The Science Class You Wish You Had


1937 The London Illustrated News had a picture of a wolf bone discovered in Czechoslovakia by Karl Absolom which has 55 notches in groups of 5, the first 25 being separated from the rest by one of double length. Dating from 30,000 BC, this is the earliest record of counting. [Bunt, Jones, Bedient, The Historical Root of Elementary Mathematics, p 2]. *VFR (The head of an ivory Venus figurine was excavated close to the bone.)


1955 The Electronic Numerical Integrator and Computer (ENIAC) retired. After disassembly, parts of this computer were shipped to the Smithsonian for display. *Goldstein, The Computer from Pascal to von Neumann, p. 234–5.
After eleven years of calculating and processing programs, the ENIAC was retired. Designers John Mauchly and J. Presper Eckert had unveiled the machine in February 1946, showing off its 1,000-time improvement in speed over its contemporaries. The ENIAC ran at 5,000 operations a second with a system of plug boards, switches, and punch cards. It occupied 1,000 square feet of floor space. *CHM


1956, the Atomicron, the first atomic clock in the U.S., was unveiled at the Overseas Press Club in New York City. The basis of the timing was the constant frequency of the oscillations of the caesium atom - 9,192,631,830 MHz. It was priced at $50,000. The Atomicron measured 84" high, 22" wide and 18" deep. *TIS


1959 At the New England eclipse of October 2, 1959, Dr. E. H. Land, inventor of the Polaroid Land camera, had accompanied Harvard astronomers on a DC-6 plane that flew above the heavy overcast. On this flight, Dr. Land and his colleagues secured several excellent photographs of the corona, using Polaroid cameras with telephoto lenses. *NSEC


2002 Why no one trusts medical research: On this date Ig Nobel prizes were awarded to two groups of medical researchers, one from the US who proved that Coca-Cola is an effective spermicide (New England Journal of Medicine, 1985), and one from Taiwan who proved that it is not(Human Toxicology, 1987). *Improbable.com




BIRTHS


1568 Marino Ghetaldi (2 Oct 1568, 11 April 1626) was a Croatian mathematician who published work with early applications of algebra to geometry. *SAU His best results are mainly in physics, especially optics, and mathematics. He was one of the few students of François Viète. He took over Viète's work to restore Apollonius' lost works. He followed Pappus's description of the contents of certain lost books and to do this he had to solve the problems which the books were supposed to contain. He published Apollonius redivivus seu restituta Apollonii Pergaei inclinationum geometria and Supplementum Apollonii Galli seu exsuscitata Apollonii Pergaei tactionum geometriae pars reliqua both in Venice in 1607.

* National Maritime Museum

Renowned for the application of algebra in geometry and his research in the field of geometrical optics on which, he wrote 7 works, including the Promotus Archimedus (1603) and the De resolutione et compositione mathematica (1630). He also produced a pamphlet with the solutions of 42 geometrical problems, Variorum problematum colletio, in 1607 and set grounds of algebraization of geometry. His contributions to geometry had been
cited by Dutch physicist Christiaan Huygens and Edmond Halley in England.
Ghetaldić was the constructor of the parabolic mirror (66 cm in diameter), kept today at the National Maritime Museum in London. During his sejourn in Padua he met Galileo Galilei, with whom he corresponded regularly. He was a good friend to the French mathematician François Viète. He was offered the post of professor of mathematics in Leuven in Belgium, at the time one of the most prestigious university centers in Europe.


1791 Aléxis Thérèse Petit (2 Oct 1791, 21 June 1820) was a French mathematician who worked on the theory of heat.*SAU


1825 John James Walker (2 Oct 1825, 15 Feb 1900) The range of Walker's mathematical research was quite impressive. He wrote some articles on theoretical mechanics but his more elaborate papers were on advanced algebra and geometry. Walker was a strong advocate of Hamilton's quaternions and strongly believed that they had not been given as wide a use as they merited. He applied quaternions to a variety of problems, mostly of an elementary nature.
The three most important papers that Walker wrote were on the analysis of plane curves and curved lines. The papers were closely connected and all appeared in the Proceedings of the London Mathematical Society. He wrote further articles on cubic curves and in this area he wrote the memoir On the diameters of cubic curves which was published in the Transactions of the Royal Society in 1889. *SAU


1852 Sir William Ramsay (2 Oct 1852; 23 Jul 1916) Scottish chemist who discovered the "inert gases", neon, krypton and xenon, and co-discovered argon, radon, calcium and barium. Nobel laureate (1904) "in recognition of his services in the discovery of the inert gaseous elements in air, and his determination of their place in the periodic system." Died in High Wycombe, Buckinghamshire.*TIS


1886 Robert Julius Trumpler (2 Oct 1886; 10 Sep 1956) Swiss-born U.S. astronomer who moved to the US in 1915 and worked at the Lick Observatory. In 1922, by observing a solar eclipse, he was able to confirm Einstein's theory of relativity. He made extensive studies of galactic star clusters, and demonstrated (1930) the presence throughout the galactic plane of a tenuous haze of interstellar material that absorbs light generally that dims and reddens the light from of distant clusters. The presence of this obscuring haze revealed how the size of spiral galaxies had been over-estimated. Whereas Harlow Shapley, in 1918, determined the distance to the centre of the Milky Way to be 50,000 light-years away, Trumpler's work reduced this to 30,000 light-years.*TIS


1901 Charles Stark Draper (2 Oct 1901; 25 Jul 1987) American aeronautical engineer, educator, and science administrator who earned degrees from Stanford, Harvard, and MIT then, in 1939, became head of MIT's Instrumentation Laboratory, which was a centre for the design of navigational and guidance systems for ships, airplanes, and missiles from World War II through the Cold War. He developed gyroscope systems that stabilized and balanced gunsights and bombsights and which were later expanded to an inertial guidance system for launching long-range missiles at supersonic jet targets. He was "the father of inertial navigation." The Project Apollo contract for guiding man and spacecraft to the moon was also placed with the Instrumentation Lab. *TIS


1908 Arthur Erdélyi studied in Brno and Prague and came to Scotland before the Second World War to avoid the Nazi invasion of Czechoslovakia. He became a lecturer at Edinburgh and after a period in the USA he returned to Edinburgh as a Professor. He was an expert on Special Functions. He became President of the EMS in 1971. *SAU


1926 Michio Suzuki (October 2, 1926 – May 31, 1998) was a Japanese mathematician who studied group theory.
He was a Professor at the University of Illinois at Urbana-Champaign from 1953 to his death. He also had visiting positions at the University of Chicago (1960–61), the Institute for Advanced Study (1962–63, 1968–69, spring 1981), the University of Tokyo (spring 1971), and the University of Padua (1994). Suzuki received his Ph.D in 1952 from the University of Tokyo, despite having moved to the United States the previous year. He was the first to attack the Burnside conjecture, that every finite non-abelian simple group has even order.
A notable achievement was his discovery in 1960 of the Suzuki groups, an infinite family of the only non-abelian simple groups whose order is not divisible by 3. The smallest, of order 29120, was the first simple group of order less than 1 million to be discovered since Dickson's list of 1900.
He classified several classes of simple groups of small rank, including the CIT-groups and C-groups and CA-groups.
There is also a sporadic simple group called the Suzuki group, which he announced in 1968. The Tits ovoid is also referred to as the Suzuki ovoid. *Wik



DEATHS

1853 Dominique François Jean Arago (26 Feb 1786, 2 Oct 1853) was a French physicist and astronomer who discovered the chromosphere of the sun (the lower atmosphere, primarily composed of hydrogen gas), and for his accurate estimates of the diameters of the planets. Arago found that a rotating copper disk deflects a magnetic needle held above it showing the production of magnetism by rotation of a nonmagnetic conductor. He devised an experiment that proved the wave theory of light, showed that light waves move more slowly through a dense medium than through air and contributed to the discovery of the laws of light polarization. Arago entered politics in 1848 as Minister of War and Marine and was responsible for abolishing slavery in the French colonies. *TIS A really great blog about Arago, With the catchy title, "François Arago: the most interesting physicist in the world!" is posted here. Read this introduction, and you will not be able to resist:

When he was seven years old, he tried to stab a Spanish solider with a lance
When he was eighteen, he talked a friend out of assassinating Napoleon
He once angered an archbishop so much that the holy man punched him in the face
He has negotiated with bandits, been chased by a mob, broken out of prison
He is:
François Arago, the most interesting physicist in the world

1929 Andrei Mikhailovich Razmadze (11 Aug 1889, 2 Oct 1929) His work was on the calculus of variations, continuing work by Weierstrass and Hilbert. The fundamental lemma of the calculus of variations is named after him. He also did important work on discontinuous solutions.*SAU


1933 Philipp Forchheimer (7 Aug 1852, 2 Oct 1933) Austrian hydraulic engineer who made significant studies of groundwater hydrology. Early in his academic career, he worked on problems of soil mechanics. Later, he turned to hydraulic problems, establishing the scientific basis of the discipline by applying standard techniques of mathematical physics - in particular Laplace's equation - to problems of groundwater movement. Laplace's equation had already been well developed for heat flow and fluid flow. Forchheimer extended the preexisting mathematical theory to calculations of groundwater flow. He was also the first to both mathematically and experimentally examine the features of dambreak waves in a rectangular channel (with his PhD student Armin Schoklitsch).*TIS


1962 Boris Yakovych Bukreyev (6 September 1859 – 2 October 1962) was a Russian and Soviet mathematician who worked in the areas of complex functions and differential equations.
In 1889, Bukreyev became a professor of mathematics at the University of Kiev, in Ukraine, Russian Empire. He studied Fuchsian functions of rank zero. He was interested in projective and non-Euclidean geometry. He worked on differential invariants and parameters in the theory of surfaces, being interested in the history of mathematics.*Wik


2006 Paul Richard Halmos​ (March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra.
In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics. This was so even though Halmos arrived in the USA at 13 years of age and never lost his Hungarian accent. He chaired the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973. In 1983, he received the AMS's Steele Prize for exposition. Some of his classics were:
How to read mathematics
How to write mathematics
How to speak mathematics.
In the American Scientist 56(4): 375–389, Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways.
Halmos's 1985 "automathography" I Want to Be a Mathematician is an account of what it was like to be an academic mathematician in 20th century America. He called the book “automathography” rather than “autobiography”, because its focus is almost entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means:
“ "Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?”
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if" and to have been the first to use the “tombstone” notation to signify the end of a proof, and this is generally agreed to be the case. The tombstone symbol ∎ (Unicode U+220E) is sometimes called a halmos. *Wik



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

Saturday, 1 October 2022

# 2 from old math terms notes

Abscissa is the formal term for the x-coordinate of a point on a coordinate graph. The abscissa of the point (3,5) is three. The word is a conjunction of ab(from) + scindere (tear). Literally then, abscissa is a line that has been cut or torn from another line. The main root is closely related to the Latin root from which we get the word scissors. I have a note that credits Leibniz with the orinin of the term in 1692, but in 2006 I received a note from Professor Barney Hughes that, "Fibonacci used the word in our meaning several times in his book, De practica geometrie. " .