## Friday 22 September 2023

### On This Day in Math - September 22

Nothing is too wonderful to be true if it be consistent with the laws of nature.

~Michael Faraday in his Laboratory Notebook

The 265th day of the year; 265 is !6 (sub-factorial 6), the number of ways that six ordered objects can be mis-ordered so that each is in the wrong spot. See Subfactorial

The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms. For example, the only derangements of {1,2,3} are {2,3,1} and {3,1,2}, so !3=2  (Pssst, students... You can find !n by dividing n! by e, and rounding to nearest integer)

265 is the sum of two squares in two different ways, including one that is the sum of consecutive squares. $265 = 3^2 + 16^2 = 11^2 + 12^2$

2652 is also the sum of two squares in two different ways, making 265 the hypotenuse of two Pythagorean Triangles. One of them is 23, 264, 265.

265 = 16^2 + 3^2 and 29^2 - 24^2,

265 is a semi-prime, 5 x 53. The sum of the digits of 265 is the same as the sum of the digits of its factors, sometimes called Joke numbers.

To form a 5x5 magic square with a constant of 265, take the standard 5x5 using 1-25, and add forty to each term. You can also multiply all the digits from 1-25 by four, then add one, using the numbers 5, 9, 13,....., 101

EVENTS

1602 In a public address at T¨ubingen University, Michael Mastlin, Kepler’s teacher, on the basis of chronological research put Jesus’ birth more than four years before the conventional date of A.D. 1. This date is now generally accepted. *VFR

1636 From a letter by Fermat to Roberval, it is clear that Fermat conceived the idea of analytic geometry as early as 1629, yet he published nothing on the subject. [Struik, Source Book, p. 397] *VFR  In the same letter he found the first new pair of amicable numbers since the early Greeks found 220 & 284. Fermat's pair was 17296 & 18416. Two numbers are called amicable, if the sum of the aliquot divisors of each, sums to the other. *L E Dickson, History of Theory of Numbers

1792 This date is considered the beginning of the Republican Calendar of France, for on this date the Republic was proclaimed and this was also the date of the autumnal equinox in that year. The new calendar was not oﬃcially approved until 5 October 1793. *Cecil B. Read, “A book printed in the year VII,” The Mathematics Teacher, 59(1966), 138–140.

1822 “Jean-Francoise Champollion the younger wrote his famous Lettre a Monsieur Dacier, secr´etaire perp´etuel de l’Acad´ero¬emie royale des inscriptions et belles-lettres, relative a l’alphabet des hi´glyphes phon´etiques. On that day he opened the great book of Ancient Egypt, sealed for some two thousand years and now at last decipherable.” Quoted from p. 15 of Tutankhamen (1963) by Christiane Desroches-Noblecourt. *VFR

1986 In a decisive victory for the makers of a computer's insides, a federal judge ruled that code used to run computers and other electronic devices could be copyrighted like printed material.*CHM

2006 Britney Crystal Gallivan of Pomona, California was a keynote speaker at the September 22, 2006 National Council of Teachers of Mathematics convention. As a Junior in high school in 2002 she had disproved a commonly held mathematical myth that a piece of paper could not be folded more than eight times. Gallivan demonstrated that a single piece of toilet paper 4000 ft in length can be folded in half twelve times. Not only did she provide the empirical proof, but she also derived an equation that yielded the width of paper or length of paper necessary to fold a piece of paper of thickness t any n number of times. Wik

BIRTHS

1711 Thomas Wright (22 September 1711 – 25 February 1786) was an English astronomer, mathematician, instrument maker, architect and garden designer. He was the first to describe the shape of the Milky Way and speculate that faint nebulae were distant galaxies.
Wright is best known for his publication An original theory or new hypothesis of the Universe (1750), in which he explains the appearance of the Milky Way as "an optical effect due to our immersion in what locally approximates to a flat layer of stars."
Another of Thomas Wright's ideas, which is often attributed to Kant, was that many faint nebulae are actually incredibly distant galaxies.*Wik

1759  William Playfair (22 September 1759 – 11 February 1823), a Scottish engineer and political economist, served as a secret agent on behalf of Great Britain during its war with France. The founder of graphical methods of statistics, Playfair invented several types of diagrams: in 1786 the line, area and bar chart of economic data, and in 1801 the pie chart and circle graph, used to show part-whole relations.[3] As a secret agent, Playfair reported on the French Revolution and organized a clandestine counterfeiting operation in 1793 to collapse the French currency.

 *Linda Hall org

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1765 Paolo Ruffini born. He anticipated Abel by providing an almost correct proof of the insolubility of the quintic. *VFR Italian mathematician and physician who made studies of equations that anticipated the algebraic theory of groups. In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals, General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible. Ruffini used group theory in his work but he had to invent the subject for himself. He also wrote on probability and the application of probability to evidence in court cases.*TIS (He also published the method now often called Horner's method ,,, see Horner below)

1769 Louis Puissant (22 Sept 1769, 10 Jan 1843) He is best remembered for his invention of a new map projection for a new map of France, and he was involved in the production of the map. The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry. Puissant wrote on geodesy, the shape of the earth and spherical trigonometry. *SAU

1791 Michael Faraday born. (22 Sep 1791; 25 Aug 1867) English physicist and chemist whose many experiments contributed greatly to the understanding of electromagnetism. Although one of the greatest experimentalists, he was largely self-educated. Appointed by Sir Humphry Davy as his assistant at the Royal Institution, Faraday initially concentrated on analytical chemistry, and discovered benzene in 1825. His most important work was in electromagnetism, in which field he demonstrated electromagnetic rotation and discovered electromagnetic induction (the key to the development of the electric dynamo and motor). He also discovered diamagnetism and the laws of electrolysis. He published pioneering papers that led to the practical use of electricity, and he advocated the use of electric light in lighthouses. *TIS

DEATHS

1703 Vincento Viviani died. His problem of cutting four congruent windows in a hemispherical cupolo so that the remainder was quadrable led to Euler’s development of the double integral. *VFR The leading geometer of his time, who founded the Accademia del Cimento. As one of the first important scientific societies, this organization came before England's Royal Society. In 1639, at age 17, he was a pupil of Torricelli and became the student, secretary and assistant of Galileo (now blind) in Arcetri, until Galileo died in 1642. During his long career, Viviani published a number of books on mathematical and scientific subjects. He edited the first edition of Galileo's collected works (1655-1656), and worked tirelessly to have his master's memory rehabilitated. In 1660, together with Borelli, he measured the velocity of sound by timing the difference between the flash and the sound of a cannon. They obtained the value of 350 metres per second. *TIS

1837 William George Horner (1786 – 22 September 1837) was a British mathematician and schoolmaster. The invention of the zoetrope, in 1834 and under a different name (Daedaleum), has been attributed to him. *Wik
Horner is largely remembered only for the method, Horner's method, of solving algebraic equations ascribed to him by Augustus De Morgan and others. He published on the subject in the Philosophical Transactions of the Royal Society of London in 1819, submitting his article on 1 July. But Fuller has pointed out that, contrary to De Morgan's assertion, this article does not contain the method, although one published by Horner in 1830 does. Fuller has found that Theophilus Holdred, a London watchmaker, did publish the method in 1820 and comments"At first sight, Horner's plagiarism seems like direct theft. However, he was apparently of an eccentric and obsessive nature ... Such a man could easily first persuade himself that a rival method was not greatly different from his own, and then, by degrees, come to believe that he himself had invented it. "
This discussion is somewhat moot because the method was anticipated in 19th century Europe by Paolo Ruffini (What a strange coincidence that he dies on Ruffini's birthdate) , but had, in any case, been considered by Zhu Shijie in China in the thirteenth century. In the 19th and early 20th centuries, Horner's method had a prominent place in English and American textbooks on algebra. It is not unreasonable to ask why that should be. The answer lies simply with De Morgan who gave Horner's name and method wide coverage in many articles which he wrote.
Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations. It is also worth noting that he gave a solution to what has come to be known as the "butterfly problem" which appeared in The Gentleman's Diary for 1815. The problem is the following:-
Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn. Suppose AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.

The butterfly problem, whose name becomes clear on looking at the figure, has led to a wide range of interesting solutions. Finally we mention that Horner published Natural magic, a familiar exposition of a forgotten fact in optics (1832). *SAU

1922 Chen Ning Yang (22 Sep 1922, )Chinese-American theoretical physicist who shared the 1957 Nobel Prize for Physics (with Tsung-Dao Lee) for a ground-breaking theory that the weak force between elementary particles did not conserve parity, thus violating a previously accepted law of physics. (Parity holds that the laws of physics are the same in a right-handed system of coordinates as in a left-handed system.) The theory was subsequently confirmed experimentally by Chien-Shiung Wu in observations of beta decay. Yang is also known for his collaboration with Robert L. Mills. They developed the Yang-Mills fields theory - a mathematical idea for describing interactions among elementary particles and fields*TIS

1956 Frederick Soddy (2 Sep 1877, 22 Sep 1956). English chemist and physicist who received the Nobel Prize for Chemistry in 1921 for investigating radioactive substances. He suggested that different elements produced in different radioactive transformations were capable of occupying the same place on the Periodic Table, and on 18 Feb 1913 he named such species "isotopes" from Greek words meaning "same place." He is credited, along with others, with the discovery of the element protactinium in 1917 *TIS
Soddy is also the author of a mathematical poem about the solution to Descarte's four tangent circles theorem from the letter to Princess Elisabeth of Bohemia. The poem is called The Kiss Precise, and begins:

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

The complete poem and more about the history of the problem can be found here.

1970 Vojtěch Jarník (22 Dec 1897 , 22 Sept 1970) a Czech mathematician. His main area of work was in number theory and mathematical analysis; he proved a number of results on lattice point problems. He also developed the graph theory algorithm known as Prim's algorithm. The Vojtěch Jarník International Mathematical Competition, held each year in Ostrava, is named in his honor.*Wik

1979 Charles Ehresmann (19 April 1905, 22 Sept 1979) He was one of the creators of differential topology. Beginning in 1941, Ehresmann made major contributions toward establishing the current view of fibre spaces, manifolds, foliations and jets. His work in the creation and development of fibre spaces followed on from the study of a special case made earlier by Seifert and Whitney.
After 1957 Ehresmann became a leader in category theory and he worked in this area for 20 years. His principal achievements in this area concern local categories and structures defined by atlases, and germs of categories. The article [3] contains a list of 139 articles written by Ehresmann during his productive career as well as listing several volumes which he edited. *SAU

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