Monday 10 June 2024

On This Day in Math - June 10


Mathematics is one of the essential emanations of the human spirit,
a thing to be valued in and for itself, like art or poetry.
~Oswald Veblen

The 161st day of the year, Every number greater than 161 is the sum of distinct primes of the form 6n - 1.  *Prime Curios (which numbers less than 161 are also the sum of distinct primes of the form 6n-1? or which are not?)

and for the gamblers out there, There are 161 ways to bet on a roulette wheel.
161 is not only a palindrome, when is rotated 180o it gives a palindromic prime, (191) (Such  reversible numbers, or words, which form a different number, or word, are called "ambigrams".)

161 is the sum of five consecutive prime numbers: 23 + 29 + 31 + 37 + 41 = 161


 1742 You will probably remember from your grade school education that George Washington spent several of his youthful years as a professional surveyor. But how much mathematics did he know and how did he use it as a surveyor? Thanks to two “cyphering books” (sometimes referred to as copybooks) he compiled as a teenager, we are able to show what he learned of trigonometry and surveying. The combined use of these subjects is perplexing to the modern reader, so we shall illustrate and explain the methods he used. Finally, in contrast to what one would expect, we will argue that he did not use trigonometry in surveying.

At age ten, Washington owned a copy of the thirteenth edition (1727) of  "The Young Man’s Companion; or, Arithmetic Made Easy", by William Mather. It is signed at the top of the title page: “George Washington” and the date “1742” is written to the right of the subtitle.

It is possible that Austin (his brother), hearing from his father of Washington’s growing interest in mathematics, brought the Mather book to him when he returned to Virginia on 10 June 1742.  We regard 1742 as the true beginning of Washington’s mathematical education.

The authors bring up the word decuple, which they say they had not heard, nor had I before reading their article.  It is clear from the suspect source of it appearing in young Washington's cyphering book; William Hawney, The Compleat Measurer: or, the Whole Art of Measuring  (1721).  

Left Washington's cypherbook
Right Hawney's Arithmetic

Checking on Google Ngram viewer, I found that after a great reduction in use around the time of Washington's early math instruction, it still seems to be used in education of four to eight year-olds: "Numbers close to a decuple , for instance , can be identified by using that decuple as a referent , e.g. 64 = 60 + 4 ; 37 40-3 . "

In a footnote, the authors expressed that it was very uncommon at this time for a cypherbook to deal with decimal fractions.  They said that Washington's was the earliest they had ever heard. A leader of his nation in one more way.

*George Washington's Use of Trigonometry and Logarithms, Joel S Silverberg, Fred Rickey,Thomas E. Dexter

A survey by Washington

Benjamin West *Wik

1752 This is the most common date given, where one is supplied, for the supposed Electrical kite experiment by Benjamin Franklin. The event is poorly documented. Franklin seems never to have written about it, and the only record seems to come from the pen of Joseph Priestly some fifteen years later who was told about it by Ben. Many now think the entire event never took place.

The standard account of Franklin's experiment was disputed following an investigation and experiments based on contemporaneous records by science historian Tom Tucker, the results of which were published in 2003. According to Tucker, Franklin never performed the experiment, and the kite as described is incapable of performing its alleged role.

Further doubt about the standard account has been cast by an investigation by the television series MythBusters. The team found evidence that Franklin would have received a fatal current through his heart had the event actually occurred. Nevertheless, they confirmed that certain aspects of the experiment were feasible - specifically, the ability of a kite with sufficiently damp string to receive and send to the ground the electrical energy delivered by a lightning strike.

Despite this, mainstream historians still support the view that the experiment was performed by Franklin *Wik

1827 William Rowan Hamilton, age 21, appointed astronomer royal at Dunsink Observatory and Andrews professor of astronomy at Trinity College, Ireland. This was a unique event in that he was still an undergraduate. *VFR

1854 The first known published mention of the Four Color Problem was printed in the Athenaeum on this date, appearing in the Miscellanea portion. The letter was signed with the initials F. G., which many supposed might have been one of the two Guthrie brothers involved in discovering the story and revealing it to DeMorgan, but others suspect it may have been Francis Galton, who had requested admission to the esteemed Athenaeum Club during this period. Certainly many of the members would have heard the story of the four colors problem from DeMorgan who had first circulated it to William R. Hamilton. (see October 23, 1852) *PB Notes (unknown)

1854, G.F. Bernhard Riemann proposed that space is curved in a lecture titled Über die Hypothesen welche der Geometrie zu Grunde liegen. He described the old-fashioned Euclidean plane geometry and solid geometry, respectively, as two-, and three-dimensional examples of what we now call Riemann spaces with zero curvature. Saying that the space is curved, rather than flat or Euclidean, is another way saying that the familiar properties of Euclidean geometry - such as the Pythagorean theorem - do not hold. He went on to suggest that all physical laws become simpler when expressed in higher dimensions. Einstein in 1915 used Rieman’s work in his theory of General Relativity which incorporated time as the fourth dimension.*TIS Weber recounted how with unusual emotion Gauss praised Riemann’s profundity on their way home. John Derbyshire in his Prime Obsession calls it "one of the top ten mathematical papers ever delivered anywhere."

1919 In a letter to Irving Langmuir, Ernest Rutherford writes, "I am a great believer in the simplicity of things and as you probably know I am inclined to hang on to broad & simple ideas like grim death until evidence is too strong for my tenacity." Nelson Ernest Rutherford *Quoted in Nathan Reingold and Ida H. Reingold, Science in America: A Documentary History 1900-1939 (1981), 354.

1924 Oswald Veblen describes his ideas for the Institute for Mathematical Research in a letter to Vernon Kellogg. The senior men would devote themselves “entirely to research, and to the guidance of the research of the younger men.” (History and philosophy of modern mathematics By William Aspray)

1977 The first Apple II computer was delivered. This was the first computer I ever used in a classroom.

Image : Apple II in a common 1977 configuration, with a 9" monochrome monitor, game paddles, and a Red Book-recommended Panasonic RQ-309DS cassette deck  *Wik


In 2000, the Millennium Bridge - a footbridge across the River Thames - was opened by Queen Elizabeth. The radical new design was the work of architect Sir Norman Foster with sculptor Sir Anthony Caro and engineering support from Arup. It was the first new crossing of the River Thames in over 100. As the first few thousand people crossed the bridge, it developed an unexpected and potentially dangerous lateral "wobble". This caused people to unwittingly walk "in step", which increased the oscillation. The design had been adapted from a computer model typical for a car bridge, but which did not take into account the lateral forces associated with human walking. After structural damping was added to stop the oscillation, the bridge re-opened in 2002*TIS 


940 Abu’l-Wafa (June 10, 940 AD, Buzhgan - July 1, 998 AD, Baghdad) He worked with a rusty compass.*VFR The professors cryptic remark about a "rusty compass" refers to Abu'l wafa's preference, when possible, to do his geometric constructions with a compass with a fixed opening.
Abu'l-Wafa is best known for the first use of the tan function and compiling tables of sines and tangents at 15' intervals. This work was done as part of an investigation into the orbit of the Moon, written down in Theories of the Moon. He also introduced the sec and cosec and studied the interrelations between the six trigonometric lines associated with an arc. He is also often credited as one of the likely originators of the spherical law of sines and established several trigonometric identities such as sin(a ± b) in their modern form, where the Ancient Greek mathematicians had expressed the equivalent identities in terms of chords.

\(\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \)
"A text written by Abu'l-Wafa for practical use was A book on those geometric constructions which are necessary for a craftsman. This was written much later than his arithmetic text, certainly after 990. The book is in thirteen chapters and it considered the design and testing of drafting instruments, the construction of right angles, approximate angle trisections, constructions of parabolas, regular polygons and methods of inscribing them in and circumscribing them about given circles, inscribing of various polygons in given polygons, the division of figures such as plane polygons, and the division of spherical surfaces into regular spherical polygons.
Another interesting aspect of this particular work of Abu'l-Wafa's is that he tries where possible to solve his problems with ruler and compass constructions. When this is not possible he uses approximate methods. However, there are a whole collection of problems which he solves using a ruler and fixed compass, that is one where the angle between the legs of the compass is fixed. It is suggested in Dictionary of Scientific Biography that:-
Interest in these constructions was probably aroused by the fact that in practice they give more exact results than can be obtained by changing the compass opening.

 His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy's were only accurate to 3 places." *SAU

In 2015, Google celebrated his 1075th birthday with a Google Doodle, and included, "contributions to science include one of the first known introductions to negative numbers, and the development of the first wall quadrant, a tool used by astronomers to examine the sky."

It is interesting that during this period there were two types of arithmetic books written, those using Indian symbols and those of finger-reckoning type. Abu'l-Wafa's text is of this second type with no numerals; all the numbers are written in words and all calculations are performed mentally. It is now believed that mathematicians wrote for two differing types of readers. Abu'l-Wafa himself was an expert in the use of Indian numerals but these :-"... did not find application in business circles and among the population of the Eastern Caliphate for a long time."

Hence he wrote his text using finger-reckoning arithmetic since this was the system used for by the business community. *SAU *Wik *PB

1710 James Short (June 10, 1710, Edinburgh, Scot. -  June 14, 1768, London, Eng) British optician and astronomer who produced the first truly parabolic and elliptic (hence nearly distortionless) mirrors for reflecting telescopes.

 During his working life of over 35 years, Short made about 1,360 instruments - not only for customers in Britain but also for export: one is still preserved in Leningrad, another at Uppsala and several in America. Short was principal British collator and computer of the Transit of Venus observations made throughout the world on 6th June 1761. His instruments traveled on Endeavour with Captain Cook to observe the next Transit of Venus on 3rd June 1769, but Short died before this event took place.

1803 Henri-Philibert-Gaspard Darcy (June 10, 1803 – January 3, 1858) French hydraulic engineer who first derived the equation (now known as Darcy's law) that governs the laminar (nonturbulent) flow of fluids in homogeneous, porous media. In 1856, modern studies of groundwater began when Darcy was commissioned to develop a water-purification system for the city of Dijon, France. He constructed the first experimental apparatus to study the flow characteristics of water through the earth. From his experiments, he derived the Darcy's Law equation, describing the flow of water in nature, which is fundamental to understanding groundwater systems.

1861 Pierre(-Maurice-Marie) Duhem (10 June 1861 – 14 September 1916)French physicist, mathematician, and philosopher of science who emphasized a history of modern science based on evolutionary metaphysical concepts. He had a wide variety of mathematical interests from mechanics and physics to philosophy and the history of mathematics. Duhem studied magnetism following the work of Gibbs and Helmholtz and also worked on thermodynamics and hydrodynamics producing over 400 papers. He maintained that the role of theory in science is to systematize relationships rather than to interpret new phenomena.  *TIS

1887 Vladimir Ivanovich Smirnov (10 June 1887 – 11 February 1974) was a Russian mathematician who made significant contributions in both pure and applied mathematics, and also in the history of mathematics.
Smirnov worked on diverse areas of mathematics, such as complex functions and conjugate functions in Euclidean spaces. In the applied field his work includes the propagation of waves in elastic media with plane boundaries (with Sergei Sobolev) and the oscillations of elastic spheres.
Smirnov is also widely known among students for his five volume book A Course in Higher Mathematics (the first volume was written jointly with Jacob Tamarkin).*Wik

1904 John Semple (10 June 1904 in Belfast, Ireland - 23 October 1985 in London, England) studied at Queen's University Belfast and Cambridge. He held a post in Edinburgh for a year before becoming Professor of Pure Mathematics at Queen's College Belfast. He moved to King's College London where he spent the rest of his career. His most important work was in Algebraic geometry. *SAU

1932 Pierre Emile Jean Cartier (10 June 1932 in Sedan , Ardennes - ) is a French mathematician . His main interest is the algebraic geometry , presentation and category theory . 1957-1959 he worked at the Institute for Advanced Study . From 1961 he was a professor at the University of Strasbourg (then Faculté de Science). In 1971 he was appointed professor at the Institut des Hautes Études Scientifiques in Paris. He was also from 1974 Director of Research CNRS. In 1982 he became a professor at the Ecole Polytechnique and 1988 at the ENS. Pierre Cartier led the Cartier operator and is the namesake of the Cartier divisor . *Wik

1987 James Alexander Maynard FRS (born 10 June 1987) is an English mathematician working in analytic number theory and in particular the theory of prime numbers. In 2017, he was appointed Research Professor at Oxford. Maynard is a fellow of St John's College, Oxford. He was awarded the Fields Medal in 2022 and the New Horizons in Mathematics Prize in 2023.
In August 2014, Maynard (independently of Ford, Green, Konyagin and Tao) resolved a longstanding conjecture of Erdős on large gaps between primes, and received the largest Erdős prize ($10,000) ever offered.

In 2014, he was awarded the SASTRA Ramanujan Prize. In 2015, he was awarded a Whitehead Prize and in 2016 an EMS Prize.

In 2016, he showed that, for any given decimal digit, there are infinitely many prime numbers that do not have that digit in their decimal expansion.

In 2019, together with Dimitris Koukoulopoulos, he proved the Duffin–Schaeffer conjecture.
Maynard was awarded the Fields Medal 2022 for "contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation".
Maynard was elected a Fellow of the Royal Society (FRS) in 2023.


1836 Andre-Marie Ampere(20 January 1775 – 10 June 1836)French mathematician and physicist who founded and named the science of electrodynamics, now known as electromagnetism. His interests included mathematics, metaphysics, physics and chemistry. In mathematics he worked on partial differential equations. Ampère made significant contributions to chemistry. In 1811 he suggested that an anhydrous acid prepared two years earlier was a compound of hydrogen with an unknown element, analogous to chlorine, for which he suggested the name fluorine. He produced a classification of elements in 1816. Ampère also worked on the wave theory of light. By the early 1820's, Ampère was working on a combined theory of electricity and magnetism, after hearing about Oersted's experiments.TIS It is said that Ampere was capable of intense concentration leading to absent-mindedness. Once walking in Paris he had an insight and pulled a piece of chalk out of his pocket and finding the back of a cab he began to cover the back of the cab with equations, and was then shocked to see his solution begin to pull away and disappear down the street.

1903 (Antonio) Luigi (Gaudenzio Giuseppe) Cremona ( Pavia , 7 December 1830 - Rome , 10 June 1903 ) was an Italian mathematician who was an originator of graphical statics (the use of graphical methods to study forces in equilibrium) and work in projective geometry. Cremona's work in statics is of great importance and he gave, in a clearer form, some theorems due to Maxwell. In a paper of 1872 Cremona took an idea of Maxwell's on forces in frame structures that had appeared in an engineering journal in 1867 and interpreted Maxwell's notion of reciprocal figures as duality in projective 3-space. These reciprocal figures, for example, have three forces in equilibrium in one figure represented by a triangle while in the reciprocal figure they are represented by three concurrent lines.*TIS

1948 Philippa Garrett Fawcett (4 April 1868 - 10 June 1948)
Fawcett's performance in the Trinity Intercollege Examination which she sat after two years at Cambridge was outstanding and it was clear that she would excel in the Tripos Examinations of 1890. At this time only the men were ranked in the Tripos Examination but women who took the examination were made aware of their place by being told they were placed between the nth and (n+1)st man or equal to the nth man. Expectations were high that Fawcett would perform well and her mother wrote in a letter to a friend):-

I am going to Cambridge tomorrow week and shall have my last sight of [Philippa] till after the exam. I have made up my mind not to be too anxious about it. There are a great many better things in the world than beating other people in examinations.

However, beat other people is exactly what Fawcett did in the twelve three hour examination papers. The Senior Moderator of the Mathematical Tripos Examinations of 1890 was Walter Rouse Ball and it was his duty to read the women's list after the men's ranked list had been read. When Rouse Ball came to read the women's list he read out first:-
Miss Philippa Garrett Fawcett - above the Senior Wrangler.
Fawcett had become the first woman at Cambridge to come top in the Mathematical Tripos Examinations. A description of the event is recorded in the North Hall Diary of Newnham College:-
The great event of the year was Philippa Garrett Fawcet's achievement in the Mathematical Tripos. For the first time a woman has been placed above the Senior Wrangler. The excitement in the Senate House when the lists were read was unparalleled. The deafening cheers of the throng of undergraduates redoubled as Miss Fawcett left the Senate House by the side of the Principal. On her arrival at the College she was enthusiastically greeted by a crowd of fellow-students, and carried in triumph into Clough Hall. Flowers, letters, and telegrams poured in upon her throughout the day. The College was profusely decorated with flags. In the evening the whole College dined in Clough Hall. After dinner toasts were proposed: the healths drunk were those of the Principal, Miss Fawcett, her Coach (Mr Hobson) and Senior and Junior Optimes. At 9.30 p.m. the College gardens were illuminated, and a bonfire was lighted on the hockey-ground, round which Miss Fawcett was three times carried amid shouts of triumph and strains of "For she's a jolly good fellow." *SAU

Following Fawcett's great achievement in the Mathematical Tripos, she won a scholarship at Cambridge through which she conducted research in Fluid Dynamics. Her published papers include "Note on the Motion of Solids in a Liquid".

She went on to be a College Lecturer in Mathematics at Newnham College, Cambridge a position she held for 10 years. In this capacity, her teaching abilities received considerable praise. One student wrote:
“ What I remember most vividly of Miss Fawcett's coaching was her concentration, speed, and infectious delight in what she was teaching. She was ruthless towards mistakes and carelessness... My deepest debt to her is a sense of the unity of all truth, from the smallest detail to the highest that we know. ”
Fawcett left Cambridge in 1902, when she was appointed as a lecturer to train mathematics teachers at the Normal School, Johannesburg, South Africa. Here, she remained until 1905, setting up schools in South Africa. She then returned to England to take a position in the administration of education for London County Council. Here, she attained the highest LCC rank ever for a woman, in her work developing secondary schools.
Philippa Fawcett maintained strong links with Newnham College throughout her life. The Fawcett building (1938) was named in recognition of her contribution to Newnham, and that of her family. She died on 10 June 1948, two months after her 80th birthday, just one month after the Grace that allowed women to be awarded the Cambridge BA degree received royal assent, and fifty eight years after coming above the `Senior Wrangler'. *Wik

Philippa on Newham Field Hockey team

1974 Jaroslav Hájek (4 Feb 1926 in Podebrady, Bohemia (now Czech Republic) - 10 June 1974 in Prague, Czechoslovakia) He was among the pioneers of unequal probability sampling. The name "Hájek predictor" now labels his contributions to the use of auxiliary data in estimating population means. In 1967 Hájek published (jointly with Z Sidak) Theory of rank tests but it was a work which had in fact been written four years before in 1963. Their methods use three lemmas of Le Cam in order to treat rank statistics under local alternatives and they established the efficiency of rank tests. *SAU

1992 Morris Kline (May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.
Kline grew up in Brooklyn and in Jamaica, Queens. After graduating from Boys High School in Brooklyn, he studied mathematics at New York University, earning a bachelor's degree in 1930, a master's degree in 1932, and a doctorate in 1936. He continued at NYU as an instructor until 1942.
During World War II, Kline was posted to the Signal Corps (United States Army) stationed at Belmar, New Jersey. Designated a physicist, he worked in the engineering lab where radar was developed. After the war he continued investigating electromagnetism, and from 1946 to 1966 was director of the division for electromagnetic research at the Courant Institute of Mathematical Sciences.
Kline resumed his mathematical teaching at NYU, becoming a full professor in 1952. He taught at New York University until 1975, and wrote many papers and more than a dozen books on various aspects of mathematics and particularly mathematics teaching. He repeatedly stressed the need to teach the applications and usefulness of mathematics rather than expecting students to enjoy it for its own sake. Similarly, he urged that mathematical research concentrate on solving problems posed in other fields rather than building structures of interest only to other mathematicians. *Wik

*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

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