## Friday 5 July 2024

### On This Day in Math - July 5

There is no smallest among the small and no largest among the large,
But always something still smaller and something still larger.

Quoted in E Maor, To Infinity and Beyond: a Cultural History of the Infinite

The 186th day of the year;  186 is the product of the first four primes less; the product of the first four positive integers (7 x 5 x 3 x 2 - 4 x 3 x 2 x 1 = 186) . *Prime Curios

There are 186 days between the Spring and Fall Equinox, which is well over 1/2 a year. The reason, we are on the wrong side of the Earth's Elliptic orbit and have to travel a greater distance. From Fall to Spring takes only 179 days. (there is of course, an extra quarter of a day in there somewhere.)

186 is the product of the first four primes less; the product of the first four positive integers, 7# - 4! (7 x 5 x 3 x 2 - 4 x 3 x 2 x 1 = 186) . *Prime Curios Students might not have seen the p# symbol, it represents the Primorial, the product of all the primes from p down to 2.

186 is the sum of consecutive primes, 186 =  89 + 97,

186 is a sphenic (wedge) number, product of 3 distinct primes: 186 = 2*3*31

Another number with a nice palindrome expressions, 3*3*3 + 3+13 + 31*3 + 3*3*3 (easy as 1,2,3, but without the 2)

186 is a palindrome in base 5(1221), and base 8(272).

186 is the median between consecutive primes 181 and 191.

and from Jim Wilder @wilderlab An equation for July 4th: 7⁴ = 2401 (2 + 4 + 0 + 1 )4 And a follow up from World Observer@WKryst2011 points out that there are only two other such year dates. (student's should find both)

186 is a palindrome in base 5(1221), and in base 8(272)

EVENTS

In 1643, an exceptionally strong wind occurred in Essex County, Mass. The description by Governor John Winthrop is the first record suggestive of a tornado in the U.S.: "There arose a sudden gust so violent for one-half hour as it blew down multitudes of trees. It lifted up their meeting house at Newbury, the people being in it. It darkened the air with dust, yet through God’s great mercy it did no hurt, but only killed one Indian with the fall of a tree." (You have to read that several times to realize theindifference of the early settlers to the indigenous culture.) However, no tornado-like funnel shape - so likely to be noted, if seen - was not included in his log. So, it was likely not a tornado, in fact, but a severe straight-line squall with strong downburst winds. Reverend Increase Mather cited a likely tornado in Jul 1680 storm at Cambridge, Mass*TIS

2nd, 6th, 9th, and 12th Governor of the Massachusetts Bay Colony

 *Wik

1687 Halley wrote to Newton that his Principia was ﬁnally published. [Westfall, p. 468] *VFR 1687 – ushering in a tidal wave of changes in thought that would significantly accelerate the already ongoing scientific revolution by giving it tools that produced technologically valuable results, which had theretofore been otherwise unobtainable. (Thony Christie has pointed out that the use of "published" may give a false impression to the modern reader, even though this is the date printed on the title page of the document. The actual words, used by Halley were, "I have at length brought your Book to an end, and hope it will please you. the last errata came just in time to be inserted. I will present from you the books you desire to...." )

1698 Johann Bernoulli, in a letter to Leibniz, deﬁned the notion of a function. The term “function” is due to Leibniz. [Cajori, Historical Introduction to the Mathematical Literature, p. 96]*VFR

The word FUNCTION first appears in a Latin manuscript "Methodus tangentium inversa, seu de fuctionibus" written by Gottfried Wilhelm Leibniz (1646-1716) in 1673. Leibniz used the word in the non-analytical sense, as a magnitude which performs a special duty. He considered a function in terms of "mathematical job"--the "employee" being just a curve. He apparently conceived of a line doing "something" in a given figura ["aliis linearum in figura data functiones facientium generibus assumtis"]. From the beginning of his manuscript, however, Leibniz demonstrated that he already possessed the idea of function, a term he denominates relatio.

A paper "De linea ex lineis numero infinitis ordinatim..." in the Acta Eruditorum of April 1692, pp. 169-170, signed "O. V. E." but probably written by Leibniz, uses functions in a sense to denote the various 'offices' which a straight line may fulfil in relation to a curve, viz. its tangent, normal, etc.

In the Acta Eruditorum of July 1694, "Nova Calculi differentialis..." (page 316), Leibniz used the word function almost in its technical sense, defining function as "a part of a straight line which is cut off by straight lines drawn solely by means of a fixed point, and of a point in the curve which is given together with its degree of curvature." The examples given were the ordinate, abscissa, tangent, normal, etc. [Cf. page 150 of Leibniz' "Mathematische Schriften," vol. III, edited by C. I. Gerhardt, Berlin-Halle (Asher-Schmidt), 1849-63.]

In September 1694, Johann Bernoulli wrote in a letter to Leibniz, "quantitatem quomodocunque formatam ex indeterminatis et constantibus," although there is no explicit reference to the Latin term functio. The letter appears in Mathematische Schriften.

On July 5, 1698, Johann Bernoulli, in another letter to Leibniz, for the first time deliberately assigned a specialized use of the term function in the analytical sense, writing "earum [applicatarum] quaecunque functiones per alias applicatas PZ expressae." (Cajori 1919, page 211) [Cf. page 507 of Leibniz' "Mathematische Schriften," vol. III, edited by C. I. Gerhardt, Berlin-Halle (Asher-Schmidt), 1849-63. Also see pages 506-510 and 525-526] At the end of that month, Leibniz replied (p. 526), showing his approval.

Function is found in English in 1779 in Chambers' Cyclopedia: "The term function is used in algebra, for an analytical expression any way compounded of a variable quantity, and of numbers, or constant quantities" [OED].  *JeffMiller

1766 Ben Franklin writes from England to Rev Ezra Stiles, "I have lately propos’d our ingenious and learned Contriman Mr: Winthorp, as a Member of the Royal Society." On Feb 20, 1766. John Winthrop Esqr. Hollisian Professor of Mathematics and Natural Philosophy was unanimously elected Fellow of the Royal Society in London. *Franklin Papers, Natl Archives

His great-great-grandfather, also named John Winthrop, was founder of the Massachusetts Bay Colony. He graduated in 1732 from Harvard, where, from 1738 until his death, he served as professor of mathematics and natural philosophy. *Wik

 *Wik

1854  the turk consumed
Wolfgang Ritter von Kempelen, an Austrian inventor, was born Jan. 23, 1734. In 1770, von Kempelen unveiled one of the most famous automatons in history, a chess-playing machine known as "The Turk". The automaton, as one can see from a contemporary engraving (first image), consisted of a life-size Turk, wearing a turban, sitting before a large enclosed desk, on top of which was a chessboard. The Turk, wielding a long smoking pipe in one hand and moving pieces with the other, would play against human opponents, and beat them, and it did so for over 80 years, until it met its demise.
The desk had three doors in the front that von Kempelen would open before each performance, behind which one could see a complex array of rods and gears, supposedly the brains of the automaton. In fact, The Turk was an ingenious hoax--a pseudo-automaton. The Turk was controlled by a human "director", seated on a sliding chair down below, mechanically rigged so that no matter which door you opened, the operator was not to be seen. Since the Turk beat a number of good chess players, the operator had to be a master chess player himself, and many chess masters of the day are rumored to have been at the controls at one time or another. So it would seem that the fraudulent nature of the Turk was an open secret among the chess-masters community, who apparently treated it as a society of magicians would treat an illusion – a secret not to be revealed to the public.

The Turk came to the United States in 1826 (von Kempelen had died in 1804) and was seen in Richmond, Va., in 1835 by Edgar Allan Poe, who wrote an essay about it, "Maelzel's Chess Player" (Maelzel had inherited the Turk from von Kempelen), in which Poe claimed to have figured out the hoax. In truth, he had not. But others have, and a full-size facsimile was made some years ago by John Gaughan, a living master illusionist.  We have not seen it operate, but a photograph of the reconstruction is available. The Turk eventually ended up in Peale’s Museum in Philadelphia, where it was destroyed in a fire on July 5, 1854. *Linda Hall Org

1951 – William Shockley invents the junction transistor.

John Bardeen, Walter Brattain and William Shockley invented the first working transistors at Bell Labs, the point-contact transistor in 1947.

After Bardeen and Brattain's December 1947 invention of the point-contact transistor (1947 Milestone), Bell Labs physicist William Shockley began a month of intense theoretical activity. On January 23, 1948 he conceived a distinctly different transistor based on the p-n junction discovered by Russell Ohl in 1940.(1940 Milestone) Partly spurred by professional jealousy, as he resented not being involved with the point-contact discovery, Shockley also recognized that its delicate mechanical configuration would be difficult to manufacture in high volume with sufficient reliability.

Shockley also disagreed with Bardeen's explanation of how their transistor worked. He claimed that positively charged holes could also penetrate through the bulk germanium material - not only trickle along a surface layer. Called "minority carrier injection," this phenomenon was crucial to operation of his junction transistor, a three-layer sandwich of n-type and p-type semiconductors separated by p-n junctions. This is how all "bipolar" junction transistors work today.

On February 16, 1948, physicist John Shive achieved transistor action in a sliver of germanium with point contacts on opposite sides, not next to each other, demonstrating that holes were indeed flowing through the germanium. Shockley applied for a patent on the junction transistor that June and published his detailed theory of its operation in 1949. Still, it was two more years before Bell Labs scientists and engineers developed processes that allowed his junction transistor to be manufactured in production quantities (1951 Milestone). CHM

2012  Gresham College  announces the appointment of Raymond Flood, Fellow of Kellogg College, Oxford, as Professor of Geometry and Other Mathematical Sciences. The Geometry chair at Gresham College is the oldest in England, dating back to the College’s founding in 1597. Gresham College was the first higher education institution in England besides the Universities of Oxford and Cambridge, and it was created with the guiding principle of providing free education to the traders and people of London so that England could maintain a position at the forefront of a global economy. Gresham College’s central position in science and mathematics has seen the Royal Society formed within the College, and past luminaries including Henry Briggs, Sir Christopher Wren, Robert Hooke, Sir Christopher Zeeman and Sir Roger Penrose. *Gresham Press Release

2022  Maryna Viazovska became the second woman to be awarded the Fields Medal.  The Ukrainian mathematician accepted her Fields Medal at the International Congress of Mathematicians in Helsinki, Finland.   The IMU cited Viazovska’s many mathematical accomplishments, in particular her proof that an arrangement called the E8 lattice is the densest packing of spheres in eight dimensions.

 Viazovska in 2013 at Oberwolfach *Wik

BIRTHS

1750 François-Pierre-Amédée Argand, known as Ami Argand (5 July 1750 – 14[1] or 24 October 1803) Argand must be the most influential inventor whom no one knows. In 1784, he invented a new kind of oil lamp. Oil lamps before Argand were smoky and produced less light than a candle. Candles, however, were expensive, affordable only by the wealthy. The Argand lamp used a cylindrical cloth wick inside a close-fitting glass chimney, so that air flowed up both inside and outside the wick. It required a thick oil for fuel, such as whale oil or rapeseed oil, which had to be gravity-fed from a container higher than the wick. But the bright side was apparent to all: the Argand lamp produced a brilliant and smokeless light, as bright as 7 or 8 candles. Argand was troubled by patent theft in France, but he arranged to have his Argand lamp produced in England by no less than James Watt and Matthew Boulton, who had been successfully manufacturing steam engines.

The Argand lamp was an immediate world-wide success; Thomas Jefferson, for example, ordered a number for Monticello, as did many other Americans, and soon everyone had Argand lamps (or the French knock-offs) in their parlors, dens, and libraries. The Argand lamp ruled the light waves for seventy years (and provided much of the impetus for the whale-fishing industry of Melville’s day).  Around 1850, the kerosene lamp was introduced – with its much cheaper fuel – and the Argand lamp faded from use. But the kerosene lamp, right up to this day, utilizes many of the same combustion innovations that made the Argand lamp so successful.

Argand himself, as too often happens, was not as successful as his lamps. In addition to his patent problems back home, he lost everything in the French revolution, and he died in 1803 without realizing the lighting revolution he had wrought. There is a portrait by Charles Willson Peale in the Detroit Institute of Arts that shows his younger brother, James Peale, admiring a miniature portrait by the light of an Argand lamp *Linda Hall Library

1820 William John Macquorn Rankine, (5 July 1820 – 24 December 1872) Scottish engineer and physicist and one of the founders of the science of thermodynamics, particularly in reference to steam-engine theory. As the chair (1855) of civil engineering and mechanics at Glasgow, he developed methods to solve the force distribution in frame structures. Rankine also wrote on fatigue in the metal of railway axles, on Earth pressures in soil mechanics and the stability of walls. He was elected a Fellow of the Royal Society in 1853. Among his most important works are Manual of Applied Mechanics (1858), Manual of the Steam Engine and Other Prime Movers (1859) and On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance. *TIS While many students never encounter it, there is a temperature scale named after Rankine. It is the Fahrenheit scale equivalent of the Kelvin scale for Celsius.

1865  Benjamin Franklin Finkel (July 5, 1865 – February 5, 1947) was a mathematician and educator most remembered today as the founder of the American Mathematical Monthly magazine. Born in Fairfield County, Ohio and educated in small country schools, Finkel received both bachelor's and master's degrees from Ohio Northern University, then known as Ohio Normal University (1888 and 1891, respectively).

In 1888 he copyrighted A Mathematical Solution Book. The purpose of the book was to provide mathematics teachers a text utilizing a systematic method of problem solving, "The Step Method", representing a chain of reasoning, in logical order, to arrive at the correct result. The first edition was postponed until 1893, due to financial problems of the original publisher. The book's preface stated that the work was based upon eight years of teaching in the public schools. This was followed by following editions in 1897, 1899 and 1902.  [The book treats very basic Arithmetic. *PB]

In 1895 he became professor of mathematics and physics at Drury University, then known as Drury College. He was a University Scholar in Mathematics at the University of Chicago from 1895–1896. In 1906 he was awarded a doctorate from the University of Pennsylvania, where he had earlier earned an additional master's degree in 1904 and a Harrison fellow appointment in 1905.

He was a member of the American Mathematical Society, 1891; the London Mathematical Society, 1898; and Circolo Matematico di Palermo, 1902. He retained his professorship at Drury College until his death in 1947. *Wik

1867 Andrew Ellicott Douglass (July 5, 1867, Windsor, Vermont – March 20, 1962, Tucson, Arizona) American astronomer and archaeologist who coined the name dendrochronology for tree-ring dating, a field he originated while working at the Lowell Observatory, Flagstaff, Ariz. (1894-1901). He showed how tree rings could be used to date and interpret past events. Douglass also sought a connection between sunspot activity and the terrestrial climate and vegetation. The width of tree rings is a record of the rainfall, with implications on the local food supply in dry years. Archaeologist Clark Wissler collaborated in this work by furnishing sections of wooden beams from Aztec Ruin and Pueblo Bonito so Douglass could cross-date the famous sites. Thus the study of tree rings enables archaeologists to date prehistoric remains. *TIS

1888 Louise Freeland Jenkins (July 5, 1888 – May 9, 1970) was an American astronomer.
She was born in Fitchburg, Massachusetts. In 1911 she graduated from Mount Holyoke College, then she received a Master's degree in astronomy in 1917 from the same institution. From 1913 to 1915 she worked at the Allegheny Observatory in Pittsburgh.
About 1921 she moved to Japan, becoming a teacher at the Women's Christian College, a missionary school. She returned to the United States in 1925 after her father died. A year later she returned to teach at a school in Himeji. (Hinomoto Gakuen girl's high school.)
In 1932 she returned to the US and became a staff member at Yale University Observatory. She was co-editor of the Astronomical Journal starting in 1942, and continued in this post until 1958. She would return to visit Japan later in her life.
She was noted for her research into the trigonometric parallax of nearby stars. She also studied variable stars.
The crater Jenkins on the Moon is named in her honor. *Today in Astronomy

1933 Jean-Paul Pier (July 5, 1933 – December 14, 2016) was a Luxembourgish mathematician, specializing in harmonic analysis and the history of mathematics, particularly mathematical analysis in the 20th century.

Pier was a graduate student in Luxembourg and at the universities of Paris and Nancy. He earned a University of Luxembourg doctorate in mathematical sciences and a French doctorate in pure mathematics. He also spent six months at the Grenoble Nuclear Research Center (1961) and a year at the University of Oregon (1966-1967).

He taught mathematics at the Lycée de Garçons in Esch-sur-Alzette from 1956 to 1980. In 1971 he created the Séminaire de mathématiques[3] at the Centre universitaire de Luxembourg (now the University of Luxembourg). He was a professor at the Centre from its creation in 1974 until 1998, when he retired as professor emeritus.

Pier was primarily responsible for the creation in January 1989 of the Luxembourg Mathematical Society,[5] of which he was president from 1989 to 1993 and again from 1995 to 1998. He was during the academic year 1994–1995 a visiting professor at the Université catholique de Louvain.

1946 Gerardus (Gerard) 't Hooft (July 5, 1946 - ) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating the quantum structure of electroweak interactions".
His work concentrates on gauge theory, black holes, quantum gravity and fundamental aspects of quantum mechanics. His contributions to physics include a proof that gauge theories are renormalizable, dimensional regularization, and the holographic principle. *Wik

DEATHS

1865 Oskar Bolza died (12 May 1857–5 July 1942). Bolza was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Bolza published The elliptic s-functions considered as a special case of the hyperelliptic s-functions in 1900 which related to work he had been studying for his doctorate under Klein. However, he worked on the calculus of variations from 1901. Papers which appeared in the Transactions of the American Mathematical Society over the next few years were: New proof of a theorem of Osgood's in the calculus of variations (1901); Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so-called isoperimetric problems (1902); Weierstrass' theorem and Kneser's theorem on transversals for the most general case of an extremum of a simple definite integral (1906); and Existence proof for a field of extremals tangent to a given curve (1907). His text Lectures on the Calculus of Variations published by the University of Chicago Press in 1904,[3] became a classic in its field and was republished in 1961 and 2005.[4] After the death of his friend Maschke in 1908, Bolza became unhappy in the United States and, in 1910, he and his wife returned to Freiburg in Germany where he was appointed as an honorary professor. Chicago gave him the title of 'non-resident professor of mathematics' which he retained for the rest of his life.*Wik He returned to Germany in 1910, where he researched function theory, integral equations and the calculus of variations. In 1913, he published a paper presenting a new type of variational problem now called "the problem of Bolza." The next year, he wrote about variations for an integral problem involving inequalities, which later become important in control theory. Bolza ceased his mathematical research work at the outbreak of WW I in 1914.*TIS

1911 George Johnstone Stoney (15 February 1826 – 5 July 1911) Irish physicist who introduced the term electron for the fundamental unit of electricity. At the Belfast meeting of the British Association in Aug 1874, in a paper: On the Physical Units of Nature, Stoney called attention to a minimum quantity of electricity. He wrote, "I shall express 'Faraday's Law' in the following terms ... For each chemical bond which is ruptured within an electrolyte a certain quantity of electricity traverses the electrolyte which is the same in all cases." Stoney offered the name electron for this minimum electric charge. When J.J. Thomson identified cathode rays as streams of negative particles, each carrying probably Stoney's minimum quantity of charge, the name was applied to the particle rather than the quantity of charge.*TIS

Stoney proposed the first system of natural units in 1881.[5][11] He realized that a fixed amount of charge was transferred per chemical bond affected during electrolysis, the elementary charge e, which could serve as a unit of charge, and that combined with other known universal constants, namely the speed of light c and the Newtonian constant of gravitation G, a complete system of units could be derived. He showed how to derive units of mass, length, time and electric charge as base units. Due to the form in which Coulomb's law was expressed, the constant 4πε0 was implicitly included, ε0 being the vacuum permittivity.

Like Stoney, Planck independently derived a system of natural units (of similar scale) some decades after him, using different constants of nature.

Hermann Weyl made a notable attempt to construct a unified theory by associating a gravitational unit of charge with the Stoney length. Weyl's theory led to significant mathematical innovations but his theory is generally thought to lack physical significance *Wik

 *Wik

1926 Peter Scott Lang graduated from Edinburgh University and after a period as assistant in Edinburgh he became Regius Professor of Mathematics at St Andrews. He held this position for 42 years. *SAU

1932 Rene Louis Baire died (21 January 1874 – 5 July 1932) a French mathematician most famous for his Baire category theorem, which helped to generalize and prove future theorems. His theory was published originally in his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables") in 1899.*Wik French mathematician whose study of irrational numbers and whose concept to divide the notion of continuity into upper and lower semi-continuity greatly influenced the French School of Mathematics. His doctoral thesis led to the solution of the problem of the characteristic property of limited functions of continuous functions and helped establish the theory of functions of real variables.*TIS

In Dijon, at the Université de Bourgogne, a lecture hall is named after Baire. *HT to Henk Broer

1977 Henry Scheffé (New York City, USA, 11 April 1907 – Berkeley, California, USA, 5 July 1977) worked in several different areas of Statistics, including linear models, analysis of variance and nonparametrics.*SAU He is known for the Lehmann–Scheffé theorem and Scheffé's method.

After teaching mathematics at Wisconsin, Oregon State University, and Reed College, Scheffé moved to Princeton University in 1941. At Princeton, he began working in statistics instead of in pure mathematics, and assisted the U.S. war effort as a consultant with the Office of Scientific Research and Development. Scheffé moved several more times, to Syracuse University in 1944, the University of California, Los Angeles in 1946, and Columbia University in 1948, where he chaired the statistics department. He settled at the University of California, Berkeley from 1953 until he retired in 1974; he took a turn as department chair there as well, from 1965 to 1968. After retiring from Berkeley, he spent more years on the faculty of Indiana University.

In 1951 he was elected as a Fellow of the American Statistical Association. Scheffé was president of the Institute of Mathematical Statistics in 1954, and also served as vice president of the American Statistical Association from 1954 to 1956.

Scheffé died in Berkeley.

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