## Friday 8 March 2024

### On This Day in Math - March 8

 Sheet from Kepler's Harmonices Mundi *Alamy.com

The teaching of Algebra in the early stages ought to consist of a gradual generalisation of Arithmetic; in other words, Algebra ought, in the first instance to be taught as Arithmetica Universalis in the strictest sense.
~George Chrystal

The 67th day of the year; 67 is the largest prime which is not the sum of distinct squares. It is the 19th prime number and the sum of five consecutive primes ending in 19 (7 + 11 + 13 + 17 + 19)

The maximum number of internal pieces possible if a circle is cut with eleven lines. These are sometimes called "lazy caterer's numbers."
$67 = \binom{11}{0} + \binom {11} {1} + \binom {11}{2}$

67 is the largest prime which is not the sum of distinct squares. It is also the smallest prime which contains all ten digits when raised to the tenth power. *Prime Curios

and Jim Wilder ‏@wilderlab sent 67 = 26 + 21+ 20 = 26 + 21 + 20 = 67

And one smoot is equal to 67 inches.  The long and short of it is that a smoot is a unit of measurement that measures exactly 5 feet 7 inches (or 67 inches or 1.7018 meters – sorry, surveyors tend to get carried away with conversions). The smoot was created in 1958 when Lambda Chi Alpha fraternity members at MIT decided to use a pledge, Oliver R. Smoot, Jr., to calculate the length of the Massachusetts Avenue Bridge. Smoot lay down on the bridge, his fraternity brothers marked his head and feet, then he moved down one length and the process was repeated until the entire length of the bridge had been measured. The fraternity painted markings every ten smoots. The length of the bridge was calculated at 364.4 smoots, plus one ear. Succeeding pledge classes repainted the markings; it is a tradition that continues to this day.

EVENTS

1618 Kepler, On how he discovered his Third law:
...and if you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred an eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances ...
* Harmonice mundi (Linz, 1619) Book 5, Chapter 3, trans. Aiton, Duncan and Field, p. 411.

", as Kepler later recalled, on the 8th of March in the year 1618, something marvelous "appeared in my head". He suddenly realized that

III.  The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.

Presumably he used the word “proportion” here to signify the logarithm of the ratio, so he is asserting that log(T1/T2) = (3/2)log(r1/r2), where Tj are the periods and rj are the mean radii of the orbits of any two planets. In the form of a diagram, his insight looks like this:

At first it may seem surprising that it took a mathematically insightful man like Kepler over twelve years of intensive study to notice this simple linear relationship between the logarithms of the orbital periods and radii. In modern data analysis the log-log plot is a standard format for analyzing physical data. However, we should remember that logarithmic scales had not yet been invented in 1605. A more interesting question is why, after twelve years of struggle, this way of viewing the data suddenly "appeared in his head" early in 1618. (Kepler made some errors in the calculations in March, and decided the data didn't fit, but two months later, on May 15 the idea "came into his head" again, and this time he got the computations right, and thought he was dreaming because the fit is so exact.)

Is it just coincidental that John Napier's "Mirifici Logarithmorum Canonis Descripto" (published in 1614) was first seen by Kepler towards the end of the year 1616? We know that Kepler was immediately enthusiastic about logarithms, which is not surprising, considering the masses of computation involved in preparing the Rudolphine Tables. Indeed, he even wrote a book of his own on the subject in 1621. It's also interesting that Kepler initially described his "Third Law" in terms of a 1.5 ratio of proportions, exactly as it would appear in a log-log plot, rather than in the more familiar terms of squared periods and cubed distances. It seems as if a purely mathematical invention, namely logarithms, whose intent was simply to ease the burden of manual arithmetical computations, may have led directly to the discovery/formulation of an important physical law, i.e., Kepler's third law of planetary motion. (Ironically, Kepler's academic mentor, Michael Maestlin, chided him − perhaps in jest? − for even taking an interest in logarithms, remarking that "it is not seemly for a professor of mathematics to be childishly pleased about any shortening of the calculations".) By the 18th of May, 1618, Kepler had fully grasped the logarithmic pattern in the planetary orbits: 'Now, because 18 months ago the first dawn, three months ago the broad daylight, but a very few days ago the full Sun of a most highly remarkable spectacle has risen, nothing holds me back.' "

*mathpages.com

1758 Euler's paper on the game of Rencontre,(A type of solitaire card game, although it was sometimes played in a variation with two players... Rencontre takes two players, whom Euler names A and B. (Their descendents still populate mathematics problems worldwide. )The players have identical decks of cards. They both turn over cards, one at a time and at the same time. If they turn over the same card at the same time, there is a coincidence, and A wins. If they go all the way through the deck without a coincidence, then B wins. published in 1753, is E201, "Calcul de la probabilité dans le jeu de rencontre," Mémoires de l'académie de Berlin (1751), 1753, p. 255-270. Regarding this work, the editor says that a memoir entitled "Calcul des probabilités dans les jeux de hasard" was presented to the Academy of Berlin 8 March 1758. He asserts that it is probably memoir 201: "Calcul de la probabilité dans le jeu de rencontre." An analysis of it appeared in the Nova Acta eruditorum, Leipzig 1754, p, 179. Euler's paper can be found here   Euler showed that the probability that A wins (there is a match in first n cards) is 1/n!, which rapidly converges to 1/e, or about 37%.

The function is called a derangement or subfactorial.  A classic form of the problem is how many different can a clerk put n letters in n addressed envelopes so that no letter is in the correct address.

The symbol for subfactorial n is !n, a reversal of the usual factorial notation of today.

The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d'analyse sur les jeux de hazard in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time.

The 9 derangements of the order of numbers one to four (from 24 permutations) are highlighted the probability of randomly picking a derangement is 9/24.

In 1775, Joseph Priestley, having discovered oxygen on 1 Aug 1774, experimented with mice in his home laboratory on whether it is necessary to support life. *TIS

Over the past experiments, Joseph Priestley was looking for different ‘airs’ and trying to observe their properties. In one of the experiments, he noticed that when a burning candle was placed in a jar, it was put out. In such a jar, a mouse would also die because of the lack of air. However, putting a green plant in the same jar and exposing it to sunlight would bring the air back, which would permit the flame to burn and the mouse to breathe.

On August 1, Priestley took a lump of reddish solid substance, which was mercury oxide, and put it inside an inverted container, which was placed in a pool of mercury. Then he took a ‘burning lens’ and focussed the sunlight on the reddish lump hoping the substance to burn and collect the air that was produced.

The produced ‘air,’ he wrote, was “five or six times as good as common air," and it allowed the mouse to breathe and the candle to burn for four times longer than earlier. Priestley had discovered what he called “dephlogisticated air," and which was later named by Antoine Lavoisier as Oxygen.

1838 US mint in New Orleans begins operation (producing dimes).  “Dime” is based on the Latin word “decimus,” meaning “one tenth.” The French used the word “disme” in the 1500s when they came up with the idea of money divided into ten parts. In America, the spelling changed from “disme” to “dime.”

1896 James Dewar responds in answer to questions about his cryogenic experiments and safety precautions from Heike Kamerlingh Onnes, the Dutch Physicist whose laboratory had been shut down in Leiden for being to dangerous. "I may say that I have made all my experiments with high pressure apparatus before the Prince of Wales and the Sister of your Queen Dowager the Duchess of Albany without the slightest hesitation and no suggestions of danger were even suggested." *archive of the Kamerlingh Onnes Laboratory.

1945 A Patent is Filed for the Harvard Mark I: C.D Lake, H.H. Aiken, F.E. Hamilton, and B.M. Durfee file a calculator patent for the Automatic Sequence Control Calculator, commonly known as the Harvard Mark I. The Mark I was a large automatic digital computer that could perform the four basic arithmetic functions and handle 23 decimal places. A multiplication took about five seconds. *CHM

In 1976, the largest recovered single stony meteorite (1,774 kg) fell in Jilin, China, during a meteor shower that dropped more than 4,000 kg of extra-terrestrial rock. *TIS  One piece weighed 1.77 tons, produced an impact pit 6 m deep (only a couple of hundred meters from the nearest house), and is the largest single fragment of stony meteorite ever found.

At about 3:00 pm on March 8, 1976 a red fireball moving southwest was sighted by townspeople of Hsinglung, Kirin Province. During flight there were several explosions and in the last stages of flight three distinct fireballs were observed.

2016 Ralph Bohun's, A Discourse Concerning the Origine and Properties of the Wind (1671), was Sold for $£562 (US 734)$ at auction by Bonhams. The Book is mentioned by John Wallis in a letter to Oldenburg of 24 January, 1672(NS) because the book's printing had been temporarily suspended over some wording that appeared "too favourable to the Royal Society" (*Beeley's correspondence of Wallis)

BIRTHS

1804 Alvan Clark (8 Mar 1804, 19 Aug 1887) American astronomer whose family became the first significant manufacturers of astronomical instruments in the U.S. His company manufactured apparatus for most American observatories of the era, including Lick and Pulkovo, and others in Europe. In 1862, while testing a telescope, Clark discovered the companion star to Sirius, which had previously been predicted but until then never sighted. The 18½-in objective telescope he used was subsequently delivered to the Dearborn Observatory, Chicago. His sons, Alvan Graham Clark and George Bassett Clark, continued the business. The unexcelled 40-in refractor telescopes for the 40-in Yerkes observatory was made by Alvan Graham Clark*TIS

Clark's telescopes at Lowell and Yerkes observatories

1851 George Chrystal (8 March 1851 in Old Meldrum (near Aberdeen), Scotland
- 3 Nov 1911 in Edinburgh, Scotland)is best remembered today for Algebra: a two volume work which was completed by 1889. He was also involved in educational reform throughout his career and was a major figure in setting up an educational system in Scotland. He became one of the first honorary members of the EMS in 1883. *SAU Chrystal was (one of?) the first to use the inverted exclamation mark for the subfactorial notation.  Prior, and for sometime after, the Whitworth symbol was used.   The name subfactorial was created by W A Whitworth around 1877. The symbol for the subractorial is !n, a simple reversal of the use of the exclamation for n-factorial n!, although both symbols are relatively newer than the word. Whitworth himself used a symbol something like || n which is still used  in some places.

My "Notes on the History of the Factorial" are here.

1865 Ernest Vessiot (8 March 1865 in Marseilles, France-17 Oct 1952 in La Bauche, Savoie, France) applied continuous groups to the study of differential equations. He extended results of Drach (1902) and Cartan (1907) and also extended Fredholm integrals to partial differential equations. Vessiot was assigned to ballistics during World War I and made important discoveries in this area. He was honored by election to the Académie des Sciences in 1943. *SAU

1866 Pyotr Nikolayevich Lebedev (8 Mar 1866; 1 Apr 1912 at age 46) Russian physicist who, in experiments with William Crookes' radiometer, proved (1910) that light exerts a minute pressure on bodies (as predicted by James Clerk Maxwell's theory of electromagnetism), and furthermore that this effect is twice as great for reflecting surfaces than for absorbent surfaces. He had proposed that light pressure on small particles of cosmic dust could be greater than gravitational attraction, thus explaining why a comet's tail points away from the Sun (though it is now understood the solar wind has a greater influence). He built an extremely small vibrator source capable of generating 4-6 mm waves, which he used to demonstrate the first observation of douible refraction of electromagnetic waves in crystals of rhombic sulphur.*TIS

1879 Otto Hahn (8 Mar 1879; 28 Jul 1968 at age 89) German physical chemist who, with the radiochemist Fritz Strassmann, is credited with the discovery of nuclear fission. He was awarded the Nobel Prize for Chemistry in 1944 and shared the Enrico Fermi Award in 1966 with Strassmann and Lise Meitner. Element 105 carries the name hahnium in recognition of his work.*TIS

"For the rest of his life, Hahn provided a standard explanation: fission was a discovery that relied on chemistry only and took place after Meitner left Berlin; she and physics had nothing to do with it, except to prevent it from happening sooner." *Lise Meitner by  Ruth Lewin Sime

The prize-winning science-fiction writer, Frederik Pohl, talking about Szilard's epiphany 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." (Maybe she had a little idea?)

in 1939 during the Fifth Washington Conference on Theoretical Physics at the George Washington University, Nobel Laureate Niels Bohr publicly announced the splitting of the uranium atom. The resulting “fission,” with its release of two hundred million electron volts of energy, heralded the beginning of the atomic age.

The announcement came just weeks after Otto Hahn and Fritz Strassmann, two of Bohr’s colleagues at Copenhagen, reported that they had discovered the element barium after bombarding uranium with neutrons. After receiving the news in a letter, physicist Lise Meitner and her cousin, Otto Frisch, correctly interpreted the results as evidence of nuclear fission. Frisch confirmed this experimentally on January 13, 1939. *atomicheritage.org

Niels Bohr was planning a trip to America to discuss other problems with Einstein who had found a haven at Princeton's Institute for Advanced Studies. Bohr came to America, but the principal item he discussed with Einstein was the report of Meitner and Frisch. Bohr arrived at Princeton on January 16, 1939. He talked to Einstein and J. A. Wheeler who had once been his student. From Princeton the news spread by word of mouth to neighboring physicists, including Enrico Fermi at Columbia. Fermi and his associates immediately began work to find the heavy pulse of ionization which could be expected from the fission and consequent release of energy. *Atomic Archive

1920 George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist. He was for many years the Professor of Applied Mathematics in the University of Cambridge, and was founding head of the Department of Applied Mathematics and Theoretical Physics (DAMTP). In 1956 he founded the influential Journal of Fluid Mechanics which he edited for some forty years. Prior to Cambridge he studied in Melbourne High School.
As an applied mathematician (and for some years at Cambridge a co-worker with Sir Geoffrey Taylor in the field of turbulent flow), he was a keen advocate of the need for physical understanding and sound experimental basis.
His An Introduction to Fluid Dynamics (CUP, 1967) is still considered a classic of the subject, and has been re-issued in the Cambridge Mathematical Library series, following strong current demand. Unusual for an 'elementary' textbook of that era, it presented a treatment in which the properties of a real viscous fluid were fully emphasized. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1959.*Wik

DEATHS

1688 Honoré Fabri (8 April 1608 in Le Grand Abergement, Ain, France - 8 March 1688 in Rome, Italy) was a French Jesuit who worked on astronomy, physics and mathematics. His lectures on natural philosophy were published in 1646 as Tractatus physicus de motu locali. In this work he uses the parallelogram law for forces, correctly applying it to deduce the law of reflection and the motion of a body acted on simultaneously by two forces.*SAU (This seems to be one of the earlier statements of the law)

1974 Olive Clio Hazlett (October 27, 1890 - March 8, 1974) was an American mathematician who spent most of her career working for the University of Illinois. She mainly researched algebra, and wrote seventeen research papers on subjects such as nilpotent algebras, division algebras, modular invariants, and the arithmetic of algebras.*Wik

1927 George Andrew Olah (born Oláh András György; May 22, 1927 – March 8, 2017) was a Hungarian-American chemist. His research involved the generation and reactivity of carbocations via superacids. For this research, Olah was awarded a Nobel Prize in Chemistry in 1994 "for his contribution to carbocation chemistry." He was also awarded the Priestley Medal, the highest honor granted by the American Chemical Society and F.A. Cotton Medal for Excellence in Chemical Research of the American Chemical Society in 1996.

After the Hungarian Revolution of 1956, he emigrated to the United Kingdom, which he left for Canada in 1964, finally resettling in the United States in 1965. According to György Marx, he was one of The Martians.

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 & Campbel