## Tuesday 11 June 2024

### On This Day in Math - June 11

All truths are easy to understand once they are discovered;
the point is to discover them.
~Galileo Galilei

The 162nd day of the year; 162 is the smallest number that can be written as the sum of 4 positive squares in 9 ways.*What's Special About This Number? (Can you find all nine ways?... Can you find a smaller number that can be written as the sum of four squares in eight ways?)

the 12th prime (12 = 1*6*2) ; p12 = 37, and the number of primes less than 162, $\pi(162)$ is also 37. There is no smaller number with this property.

162 is the total number of baseball games each team plays during a regular season in Major League Baseball.

Jim Wilder pointed out that 1621= 162 has a digit sum of nine; and 1622= 26244 has a digit sum of 18; and 1623= 4251528 has a digit sum of 27. And 1624 ???

162 has a sum of divisors 1+2+3+6+9+18+27+54+81=201 which is greater than 162. Such numbers have been called abundant since the Ancient Greeks.

A 3x3 magic square with a magic constant for each row and column of 162

53    58   51
52   54    56
57   50    55

Imagine you have seven distinctly colored balls, and three numbered tubs to put them in, but none can be in a tub by itself. There are 162 different ways to distribute the balls. (If students struggle with this large a challenge, they can try to find all eleven ways to put five colored balls in just two tubs, again with no solitary balls.

A T Vandermonde should be remembered for the wonderfully useful approach he had for generalizations on the factorial, and in my mind, created the most useful notation ever (and, he seems to be the first to think of 0!=1) His notation included a method for skipping numbers, so that [p/3]n would indicate p(p-3)(p-6)... (p-3(n-1)); and in his notation 162 = [9/3]3 or 9*6*3. Now that's a notation worth having an exclamation point!

EVENTS

1644 Florentine scientist, Evangelista Torricelli described in a letter (to Michelangelo Ricci) the invention of a barometer, or "Torricellian tube.

"Many have said that a vacuum does not exist, others that it does exist in spite of the repugnance of nature and with difficulty; I know of no one who has said that it exists without difficulty and without a resistance from nature. I argued thus: If there can be found a manifest cause from which the resistance can be derived which is felt if we try to make a vacuum, it seems to me foolish to try to attribute to vacuum those operations which follow evidently from some other cause; and so by making some very easy calculations, I found that the cause assigned by me (that is, the weight of the atmosphere) ought by itself alone to offer a greater resistance than it does when we try to produce a vacuum."

"We live submerged at the bottom of an ocean of air.",

Torricelli correctly reasoned that the space above the mercury contained nothing and therefore was a vacuum. Previous experimenters using water had seen a similar behavior in much longer water-filled tubes, and it had been argued that the column of liquid was held up by the properties of the vacuum above it. Incidentally this is apparently why Torricelli used two tubes, one with a simple blind end and the other with a small sphere on the end. He argued that if a vacuum was responsible for attracting the mercury, the heights of the columns would be different because the differences in shape of the end of the tube would change the properties of the vacuum. However, the heights were the same.

1668  James Gregory (age 29) was elected a fellow of the Royal Society. He presented various papers to the Society on a variety of topics including astronomy, gravitation and mechanics. He was appointed Regius Professor of Mathematics at St Andrews later that year.

He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.

In the Optica Promota, published in 1663, Gregory described his design for a reflecting telescope, the "Gregorian telescope". He also described the method for using the transit of Venus to measure the distance of the Earth from the Sun, which was later advocated by Edmund Halley and adopted as the basis of the first effective measurement of the Astronomical Unit.

In his book Geometriae Pars Universalis (1668) Gregory gave both the first published statement and proof of the fundamental theorem of the calculus (stated from a geometric point of view,but only for a special class of the curves considered by later versions of the theorem), for which he was acknowledged by Isaac Barrow.

About a year after assuming the Chair of Mathematics at Edinburgh, James Gregory suffered a stroke while viewing the moons of Jupiter with his students. He died a few days later at the age of 36.

1742, Benjamin Franklin invented the Franklin stove. The wood fuel burns on an iron surface over a cold air duct which heats air which then passes through baffles in the back wall. The heated air is released through vents on each side of the stove. Rather than patent it, he chose to write about it in a book so that others could freely copy his design. As he wrote, "That as we enjoy great Advantages from the Inventions of others, we should be glad of an Opportunity to serve others by any Invention of ours, and this we should do freely and generously."*TIS

Franklin's original design for the Franklin stove.

 collections.rmg.co.uk

1795 The Board of Longitude awards a 200 pound payment to Ralph Walker for his invention of a compass/sundial combination. "Comparing this reading with the direction in which the compass needle was pointing gave the magnetic variation. This could, in theory, be used to discover the longitude, by finding where supposed ‘magnetic meridians’ intersected with the observed latitude....Nevil Maskelyne didn’t think it was an effective longitude method" *kmcalpine, Royal Museum Greenwich blog
A letter with the directions for the instruments use is here, and the letter from the board to authorize the payment is here. (with HT to Richard Dunn@Lordoflongitude)

In 1895, the first U.S. patent for a gasoline-driven automobile by a U.S. inventor was issued to Charles E. Duryea (No. 540,648). Early in 1896, the Duryea Motor Wagon Co. set up shop in Springfield, Mass. to manufacture multiple units to a gasoline-powered vehicle that he built with his brother, Frank. The company's assembly of 13 identical machines that year is considered to be the first instance of serial production of American cars. The only surviving 1896 Duryea is on exhibit at the Henry Ford Museum. As this is the first U.S. automobile company, and the first to produce any quantity, the Duryea brothers are considered “Fathers of the American Automobile Industry”*TiS

1929
Walt Disney files a trademark application for the image of Mickey Mouse
with the United States Patent Office.  In May of 1928 the company had filed a trademark on the name Mickey Mouse and their trademark Mickey Mouse Logo. *Disney Archives

1955 France issued a postage stamp with a portrait of
Pierre Simon de Laplace (1749–1827)

BIRTHS

1656 Charles René Reyneau (11 June 1656 in Brissac, Maine-et-Loire, France - 24 Feb 1728 in Paris, France) was a French mathematician who published an influential textbook on the newly invented calculus.*SAU (He) "undertook to reduce into one body, for the use of his scholars, the principal theories scattered here and there in Newton, Descartes, Leibnitz, Bernoulli, the Leipsic Acts, the Memoirs of the Paris Academy, and in other works; treasures which by being so widely dispersed, proved much less useful than they otherwise might have been. The fruit of this undertaking, was his “Analyse Demontree,” or Analysis Demonstrated, which he published in 1708, 2 vols. 4to. He gave it the name of “Analysis Demonstrated,” because he demonstrates in it several methods which had not been handled by the authors of them, with sufficient perspicuity and exactness. The book was so well approved, that it soon became a maxim, at least in France, that to follow him was the best, if not the only way, to make any extraordinary progress in the mathematics and he was considered as the first master, as the Euclid of the sublime geometry." (From the 1812 Chalmer's Biography, vol. 26, p. 151)

1687 Maurice Paul Auguste Charles Fabry (11 June 1867, Marseille – 11 December 1945, Paris) was a French physicist who graduated from the École Polytechnique in Paris.
Together with Henri Buisson, he discovered the ozone layer in 1913. In optics, he discovered an explanation for the phenomenon of interference fringes. Together with his colleague Alfred Pérot he invented the Fabry–Pérot interferometer.
In 1921, he was appointed Professor of General Physics at the Sorbonne and the first director of the new Institute of Optics. He was the first general director of the Institut d'optique théorique et appliquée and director of "grande école" École supérieure d'optique (SupOptique).
During his career Fabry published 197 scientific papers, 14 books, and over 100 popular articles. For his important scientific achievements he received the Rumford Medal from the Royal Society of London in 1918. In the United States his work was recognized by the Henry Draper Medal from the National Academy of Sciences (1919) and the Franklin Medal from the Franklin Institute (1921). In 1927 he was elected to the French Academy of Sciences. *Wik

1723  Johann Georg Palitzsch (June 11, 1723 – February 21, 1788) was born.  As a German farmer and amateur astronomer from the village of Prohlis near Dresden, he would observe a comet on Christmas day in 1758 and confirm one of the most significant scientific theories in history.

He became famous for recovering Comet 1P/Halley (better known as Halley's Comet) on Christmas Day, 1758. The periodic nature of this comet had been deduced by its namesake Edmond Halley in 1705, but Halley had died before seeing if his prediction would come true.

See the full story of his Christmas/birthday observation from Thony Christie.

1862 Lothar Heffter (June 11th 1862 in Koszalin , January 1 1962 in Freiburg )At the age of 99 he published the second edition of his Begr¨undung der Funktionentheorie. *VFR He did research in the theory of linear differential equations , the complex analysis and analytic geometry and worked on the four-color problem. Lazarus Fuchs was his teacher. His main concern was the popularization of mathematics.

1881 Hilda Phoebe Hudson (June 11, 1881 Cambridge – November 26, 1965 London) was an English mathematician who worked on algebraic geometry, in particular on Cremona transformations.
Educated at Newnham College, University of Cambridge, after a year studying at the University of Berlin she returned to Newnham as lecturer in mathematics and later Associate Research Fellow. She was also awarded MA and ScD degrees by Trinity College, Dublin. Most of Hudson's research was in the area of pure mathematics concerned with surfaces and plane curves, her special interest was in cremona transformation. Her monograph Ruler and Compasses was well-received as "a welcome addition to the literature on the boundary between elementary and advanced mathematics". In 1917 she joined an Air Ministry subdivision undertaking aeronautical engineering research, where she applied pioneering work on the application of mathematical modelling to aircraft design for which she was appointed OBE in 1919. As a distinguished mathematician she was one of the few women of her time to serve on the council of the London Mathematical Society. *Wik

1886 David Barnard Steinman (June 11, 1886 - August 21, 1960) Designer of the BIG MAC Bridge between the Upper and Lower Peninsulas of Michigan (top). American engineer whose studies of airflow and wind velocity helped make possible the design of aerodynamically stable bridges. Steinman's thesis for his Ph.D. from Colombia University (1911) was published as "The Design of the Henry Hudson Memorial Bridge as a Steel Arch, and more than 20 years later he built the bridge he had planned over the Harlem River. Steinman designed more than 400 bridges, for instance Sidney Harbor Bridge in Australia, Mackinac Straits Bridge, Carquinez Strait Bridge, San Francisco (1937), Saint Johns Bridge, Portland, Ore, Deer Isle Bridge, Maine, Mount Hope Bridge, Rhode Island. *TIS

1910 Jacques-Yves Cousteau (11 June 1910 – 25 June 1997) French naval officer, oceanographer, marine biologist and ocean explorer, known for his extensive underseas investigations. He was co-inventor of the aqualung which made SCUBA diving possible (1943). Cousteau the developed the Conshelf series of manned habitats, the Diving Saucer, a process of underwater television and numerous other platforms and specialized instruments of ocean science. In 1945 he founded the French Navy's Undersea Research Group. He modified a WWII wooden hull minesweeper into the research vesselCalypso, in 1950. An observation dome added to the foot of Calypso's bow was found to increase the ship's stability, speed and fuel efficiency. *TIS For whom my oldest son is named.

1914 Rufus Philip Isaacs (11 June 1914 in New York City, New York -18 January 1981 in Baltimore) was a game theorist especially prominent in the 1950s and 1960s with his work on differential games.
He worked for the RAND Corporation from 1948 until winter 1954/1955. His investigation stemmed from classic pursuit-evasion type zero-sum dynamic two player games such as the Princess and monster game. In 1942, He married Rose Barcov, and they had two daughters.
His work in pure mathematics included working with monodiffric functions, fractional-order mappings, graph theory, analytic functions, and number theory. In graph theory he constructed the first two infinite families of snarks. In applied mathematics, he worked with aerodynamics, elasticity, optimization, and differential games, which he is most known for. He received his bachelors from MIT in 1936, and received his MA and PhD from Columbia University in 1942 and 1943 respectively. His first post after the war ended was at Notre Dame, but he left in 1947 due to salary issues. While at RAND, much of his work was classified, and thus remained unknown until the publication of his classic text on differential games a decade after leaving RAND. His career after RAND was spent largely in the defense and avionics industries. While at RAND, he worked with researchers including Richard E. Bellman, Leonard D. Berkovitz, David H. Blackwell, John M. Danskin, Melvin Dresher, Wendell H. Fleming, Irving L. Glicksberg, Oliver A. Gross, Samuel Karlin, John W. Milnor, John F. Nash, and Lloyd S. Shapley. His work has significant influence on mathematical optimization including fundamental concepts such as dynamic programming (Richard E. Bellman) and the Pontryagin maximum principle (Breitner 2005) which are widely used in economics and many other fields. *Wik

1921 Rodney Hill FRS (11 June 1921 – 2 February 2011) was an applied mathematician and a former Professor of Mechanics of Solids at Gonville and Caius College, Cambridge.
In 1953 he was appointed Professor of Applied Mathematics at Nottingham University. His 1950 The Mathematical Theory of Plasticity forms the foundation of plasticity theory. Hill is widely regarded as among the foremost contributors to the foundations of solid mechanics over the second half of the 20th century. His early work was central to founding the mathematical theory of plasticity. This deep interest led eventually to general studies of uniqueness and stability in nonlinear continuum mechanics, work which has had a profound influence on the field of solid mechanics—theoretical, computational and experimental alike—over the past decades. Hill was the founding editor of the Journal of the Mechanics and Physics of Solids, still among the principal journals in the field.
His work is recognized worldwide for its concise style of presentation and exemplary standards of scholarship. Publisher Elsevier, in collaboration with IUTAM, established a quadrennial award in the field of solid mechanics, known as the Rodney Hill Prize, first presented at ICTAM in Adelaide in August 2008. The prize consists of a plaque and a cheque for US\$25,000. Its first recipient is Michael Ortiz, for his contribution to nonconvex plasticity and deformation microstructures (California Institute of Technology, USA).
He won the Royal Medal in 1993 for his contribution to the theoretical mechanics of soil and the plasticity of solids and was elected a Fellow of the Royal Society in 1961. He was awarded an Honorary Degree (Doctor of Science) by the University of Bath in 1978. *Wik

1937 David Bryant Mumford (11 June 1937-  ) born in Worth, Sussex, England. In 1974 he won a Fields Medal for his work on “problems of the existence and structure of varieties of moduli, varieties whose points parameterize isomorhphism clases of some type of geometric object.” *VFR In the 1980s he turned to applied mathematics with the question "Is there a mathematical approach to understanding thought and the brain?" This is part of "Pattern Theory," as introduced by Ulf Grenander in the 70's to give a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence. *TIS

DEATHS

1292 Roger Bacon (Ilchester, Somersetshire, about 1214 -  Oxford, perhaps 11 June, 1294) English scholar who was one of the first to propose mathematics and experimentation as appropriate methods of science. He studied mathematics, astronomy, optics, alchemy, and languages. He elucidated the principles of refraction, reflection, and spherical aberration, and described spectacles, which soon thereafter came into use. He developed many mathematical results concerning lenses, proposed mechanically propelled ships, carriages, and flying machines, and used a camera obscura to observe eclipses of the Sun. Bacon was the first European to give a detailed description of the process of making gunpowder.*TIS

1875  Joseph Winlock (February 6, 1826 – June 11, 1875) was an American astronomer and mathematician.

He was born in Shelby County, Kentucky, the grandson of General Joseph Winlock (1758–1831). After graduating from Shelby College in Kentucky in 1845, he was appointed professor of mathematics and astronomy at that institution.

From 1852 until 1857 he worked as a computer for the American Ephemeris and Nautical Almanac, and relocated to Cambridge, Massachusetts. He briefly served as head of the department of mathematics at the United States Naval Academy, but returned as superintendent of the Almanac office. He was elected a Fellow of the American Academy of Arts and Sciences in 1853.

He married Isabella Washington in Shelbyville, Kentucky on December 10, 1856, and they had six children.

In 1863 he was one of the fifty charter members of the National Academy of Sciences. Three year later in 1866 he became director of the Harvard College Observatory, succeeding George Bond, and making many improvements in the facility. He was also appointed professor of astronomy at Harvard. He remained at the university, eventually becoming professor of geodesy until his sudden death in Cambridge on June 11, 1875.

Much of his astronomical work included measurements with the meridian circle, a catalogue of double stars and stellar photometry investigations. He also led solar eclipse expeditions to Kentucky in 1860 and Spain in 1870.

The crater Winlock on the Moon is named after him.

1895 Daniel Kirkwood (September 27, 1814 - June 11, 1895) American mathematician and astronomer who noted in about 1860 that there were several zones of low density in the minor-planet population. These gaps in the distribution of asteroid distances from the Sun are now known as Kirkwood gaps. He explained the gaps as resulting from perturbations by Jupiter. An object that revolved in one of the gaps would be disturbed regularly by the planet's gravitational pull and eventually would be moved to another orbit. Thus gaps appeared in the distribution of asteroids where the orbital period of any small body present would be a simple fraction of that of Jupiter. Kirwood showed that a similar effect accounted for gaps in Saturns rings.*TIS

The asteroid 1951 AT was named 1578 Kirkwood in his honor and so was the lunar impact crater Kirkwood, as well as Indiana University's Kirkwood Observatory. He is buried in the Rose Hill Cemetery in Bloomington, Indiana, where Kirkwood Avenue is named for him. *Wik

1903 Nikolai Vasilievich Bugaev (September 14, 1837 – June 11, 1903) was a prominent Russian mathematician, the father of Andrei Bely.
Bugaev was born in Georgia, Russian Empire into a somewhat unstable family (his father was an army doctor), and at the age of ten young Nikolai was sent to Moscow to find his own means of obtaining an education. He succeeded, graduating in 1859 from Moscow University, where he majored in mathematics and physics. He went on to study engineering, but in 1863 wrote a Master's thesis on the convergence of infinite series. This document was sufficiently impressive to win him a place studying under Karl Weierstrass and Ernst Kummer in Berlin. He also spent some time in Paris studying under Joseph Liouville. He earned his doctoral degree in 1866 and returned to Moscow, where he taught for the remainder of his career. Some of his most influential papers offered proofs of previously unproven assertions of Liouville, but his most original work centered around the development of formal analogies between arithmetic and analytic operations. *Wik

1931 Franklin H(enry) Giddings (March 23, 1855 – June 11, 1931)  American sociologist, one of the first in the United States to turn sociology from a branch of philosophy into a research science dependent on statistics. He was noted for his doctrine of the "consciousness of kind," which he derived from Adam Smith's conception of "sympathy," or shared moral reactions. His explanation of social phenomena was based this doctrine - his theory that each person has an innate sense of belonging to particular social groups. He encouraged statistical studies in sociology. *TIS

1934 Friedrich Wilhelm Franz Meyer (2 Sept 1856 in Magdeburg, Germany - 11 June 1934 in Königsberg, Germany (now Kaliningrad, Russia) studied algebraic geometry, algebraic curves and invariant theory. *SAU

The wide research work of Meyer (more than 130 papers) is centred basically on geometry and, specifically, on invariant theory.

He is mainly known as one of the main editors of the Encyklopädie der mathematischen Wissenschaften published from 1898 to 1933 in 23 separate books. Meyer was directly in charge to edit the geometry volumes

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