A Curious Property of Vulgar Fractions

     

Another from 2008, hope you enjoy:


John Farey was a geologist, not a mathematician, but he is better remembered today for a single short (four paragraphs) paper he wrote in 1816 than for all his good works in geology. In that year he sent a paper to the Philosophical Magazine (or at least it was published in that year) called On a curious property of vulgar fractions and described a pattern that appears in sequences of what we would today call common (vulgar is the Latin term for common) fractions, like 3/8 etc, which are in simplest terms. The observation has almost NO practical use, not even to prove other things mathematically, and yet, it seems to have all kinds of interesting properties that tend to keep us fascinated with it. If you have never been introduced to it, here is a brief description, and some novel relationships that I think are interesting, with some links to places where they are made clearer than I could do in this brief space.

So First... What is it we are talking about? If you take ALL the fractions that could be written in simplest terms with a denominator less than some number n, say n=5 (since that is the one Farey used in his paper), and put them in order from lowest to highest... you get
The "curious" thing that Farey noticed is that if you ignore the way your fifth grade teacher taught you to add fractions, and do it the way YOU would have added them, "add the tops, add the bottoms", then each number in the sequence is the sum of the terms on each side of it... for example ¹/₄ and ²/₅ are on each side of ¹/₃, and if you add by this approach you get ⁽¹⁺²⁾/₍₄₊₅₎ = ³/₉ and that simplifies to ¹/₃ . The number obtained by adding two fractions in this fashion is often called the mediant .

OK, so that is how you make them. Our first question might be, how many of them are there? F(5) obviously has eleven terms (I counted). If we picked a value of N, what would be the number of fractions in the set F(N). A little investigation would show that F(1) = 2 (0 and 1); and F(2) = 3 (0, 1/2, and 1). So how many would be added to the next set... and the next... it turns out that each new set will have all the values of the previous set (of course) and will add one for every value of one through n that is co-prime (has no common divisors) to n. So the set for N=6 will have the eleven terms of F(5) plus 1/6 (one has no common factor with six), and 5/6. (notice that 2/6, 3/6, and 4/6 are already in the sequence in F(5) in simplified form)... thus F(6) has thirteen terms.. and in general we get a recursive formula that say the Order of F(N) = Order of F(n-1) + φ(n). Euler's \(\phi\) function,(sometimes called the totient, like quotient) he number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1.( and no, Euler never used either term.)

 
A second nice curiosity related to Farey Sequences are the Ford Circles. "Ford circles are named after American mathsedematician s the R. Ford, Sr., who described them in an article in American Mathematical Monthly in 1938, volume 45, number 9, pages 586-601" (from Wikipedia). In fact, the wikipedia article is a nice place to see how they work, and I need say no more as it shows the circles for F(5). What is amazing is that each circle is tangent to every other circle for a fraction it will be adjacent to in ANY sequence.... ahhh, go on..say "cool".

I decided to mention this when I came across another curious property of Farey sequences that relates them to lines on the plane and Pick's Theorem. If you treated each fraction a/b as a point (b,a) then none of the lines cross. If you make a triangle with the origin and any two adjacent Farey fractions, since each of the triangles have a determinant of one (meaning the area is 1/2) and therefore, by PIck's theorem, they cannot contain any other lattice points in their interior. A nice explanation of this, including the photo below, was at the Cut-The-Knot web site

On This Day in Math - September 16

  

...[T]o many it is not knowledge but the quest for knowledge that gives greater interest to thought—to travel hopefully is better than to arrive
~Sir James Jeans

The 259th day of the year;259 expressed in base six is a repunit, 1111 (6³+6²+ 6¹+6⁰= 216+36+6+1=259)

Probably 259 is the largest number that can be written in two ways as \(2^x + 3y\). Here \(259 = 2^8 + 3^1 = 2^4 + 3^5\).

259 can be expressed as the sum of four cubes in two different ways, 259 = 1³ + 2³ + 5³ + 5³= 2³ + 2³ + 3³ + 6³

and for my ex-students from Japan, 259  is The number of Pokémon originally available in Pokémon Gold and Silver

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EVENTS

1566 Tycho Brahe departs Wittenberg to avoid the plague. Early In 1566 he left Denmark and arrived at Wittenberg on the 15th of April. The University of Wittenberg had been founded in 1502, and had then for nearly fifty years been one of the most renowned in Europe. He Studied under Caspar Peucer, distinguished as a mathematician, a physician, and a historian. Tycho, however, did not profit very much from Peucer's instruction, as the plague broke out at Wittenberg, so that he was induced to leave it on the 16th September, after a stay of only five months. *TYCHO BRAHE, A PICTURE OF SCIENTIFIC LIFE AND WORK IN THE SIXTEENTH CENTURY BY J. L. E. DREYER

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1636 In a letter to Roberval, Fermat writes, If a and b are rational, and if \( a^2+b^2 = 2(a+b)x= x^2 \), then x and x² are irrational. *Lemmemeyer, EULER, GOLDBACH, AND “FERMAT’S THEOREM”

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In 1662, the first recorded astronomical observation by the (to become) first Astronomer Royal was John Flamsteed's observation of a solar eclipse from his home in Derby at the age of sixteen, about which he corresponded with other astronomers. Flamsteed's interest in astronomy was stirred by the solar eclipse, and besides reading all he could find on the subject he attempted to make his own measuring instruments. *TIS

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1693 In a letter to John Locke, Newton apologized for ill thoughts that he had harbored against Locke. *VFR Locke was strongly denounced by several writers and even called an atheist, notably by John Edwards, but such charges were commonplace against every departure from Orthodoxy. During his period of insanity (following 1693) Isaac Newton made similar charges against Locke; at least he wrote Locke a strange letter apologizing for considering him a Hobbist and having charged him with attacking the root of morality,*Contra Mundum, No. 1 Fall 1991, "At the Origins of English Rationalism", by T.E. Wilder (Locke and Newton were usually friends)

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1787 Jefferson writes his ex law professor, George Wythe in regard to the construction of geometric models in the classroom.  Wythe is considered one of the finest jurists of the period, and had Jefferson, Monroe, and John Marshall as students.

"I have reflected on your idea of wooden or ivory diagrams for the geometrical demonstrations. I should think wood as good as ivory; & that in this case it might add to the improvement of the young gentlemen; that they should make the figures themselves. Being furnished by a workman with a piece of veneer, no other tool than a penknife & a wooden rule would be necessary. Perhaps pasteboards, or common cards might be still more convenient. The difficulty is, how to reconcile figures which must have a very sensible breadth, to our ideas of a mathematical line, which, having neither breadth nor thickness, will revolt more at these than at simple lines drawn on paper or slate. If after reflecting on this proposition you would prefer having them made here, lay your commands on me and they shall be executed."

This is a full 100 years before Kline brought his models to America and influenced their use in American education.  The earliest record of model building dates back to 1873 and deals with a plaster model of Steiner's Roman surface built by German mathematician Ernst Kummer.  The Roman surface, also called the Steiner surface (not to be confused with the class of Steiner surfaces of which the Roman surface is a particular case), is a quartic nonorientable surface.  The Roman surface is one of the three possible surfaces obtained by sewing a Möbius strip to the edge of a disk.  

*Wik

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1804 J L Gay-Lussac sets height record of 22,000+ feet during balloon lift to make measurements of magnetism and electricity.  Earlier on August 24, 1804, Joseph Louis Gay-Lussac and Jean-Baptiste Biot ascended in a hot air balloon to a height of 4,000 meters altitude in order to conduct scientific experiments on gases. These experiments measuring the expansion of gasses led to him publishing the law he dubbed Charles Law, perhaps giving JAC Charles more credit than was due.

Jacques Alexandre César Charles had discover this law around 1787, based on experiments with air and hydrogen gas; however, Charles never published his findings himself. 

Gay-Lussac formally stated the law, but acknowledged that the unpublished experiments were originally conducted by Charles. It is  sometimes called Gay-Lussac’s Law in older French texts—but he himself deferred the credit to Charles. Over time, the name "Charles’s Law" became standard.*PBnotes

*Science History Inst

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1810 Rise Up and Revolt!!!   (And they did.)

The Cry of Dolores[n 1] (Spanish: Grito de Dolores) occurred in Dolores, Mexico, on 16 September 1810, when Roman Catholic priest Miguel Hidalgo y Costilla rang his church bell and gave the call to arms that triggered the Mexican War of Independence. The Cry of Dolores is most commonly known by the locals as El Grito de Independencia (The Independence Cry).

Every year on the eve of Independence Day, the president of Mexico re-enacts the cry from the balcony of the National Palace in Mexico City while ringing the same bell Hidalgo used in 1810. During the patriotic speech, the president calls out the names of the fallen heroes who died during the War of Independence and ends the speech by shouting "¡Viva México!" three times, followed by the Mexican National Anthem.

In the 1810s, what would become Mexico was still New Spain, part of the Spanish crown. Following Napoleon's overthrow of the Spanish branch of the Bourbon monarchy in 1808, Spain's American possessions rose in rebellion, refusing to accept Napoleon's brother, Joseph Bonaparte, as king. In New Spain, the criollo leadership attempted to set a course of autonomy in support of the legitimate heir to the throne, Ferdinand VII, but the peninsular elite, fearing the loss of the colony, carried out a coup, also in the name of Ferdinand. Almost immediately, groups of creoles formed various plots around the viceroyalty, including in Querétaro, of which Father Hidalgo became a part.

When the plot was discovered in early September 1810, some plotters decided to proceed with the uprising.[1] Around 2:30 am on 16 September 1810, Hidalgo ordered the church bells to be rung and gathered his congregation. Flanked by Ignacio Allende and Juan Aldama, he addressed the people in front of his church, urging them to revolt. His speech became known as the "Cry of Dolores".[2]

The liberated country adopted Mexico as its official name. Mexico's independence from Spain took a decade of war. Independence was achieved by the Declaration of Independence of the Mexican Empire 11 years and 12 days later, on 28 September 1821. However, Hidalgo is credited as being the "father of his country"*Wik

Father Miguel Hidalgo y Costilla with the movement's banner; statue in Atotonilco

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In 1835, British naturalist Charles Darwin, aboard the ship HMS Beagle, arrived at the Galapagos archipelago, a cluster of islands on the equator 600 miles west of South America. During his five weeks studying the fauna in the Galapagos, Darwin found the giant tortoises there greatly differed from one another according to which island they came from. Moreover, many islands developed their own races of iguanas. These observations contributed to his theory of “natural selection,” that species evolved over thousands of millions of years. *TIS 

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1848 Weierstrass came to the Catholic Gymnasium in Braunsberg, his third such position. That year he taught mathematics 19 hours per week, took over the geography class after Easter, and received a special note of thanks for helping out in gym! [From the annual report of the Gymnasium in the University of Louisville’s Bullitt Collection of Mathematics. *VFR

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1895 Pearson writes to Yule, "I had a most kindly and encouraging letter from Francis Galton about my Heredity paper. He really is a fine old fellow to take my modification of his views so well." *The History of Statistics: The Measurement of Uncertainty Before 1900
By Stephen M. Stigler

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1986 “Four out of three jocks can’t count,” read a headline in The Harvard Lampoon’s parody of USA Today. *VFR

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BIRTHS

1494 Francisco Maurolico(September 16, 1494-July 21 or July 22, 1575) was an Italian Benedictine who wrote important books on Greek mathematics. He also worked on geometry, the theory of numbers, optics, conics and mechanics.*SAU (His Arithmeticorum libri duo (1575) includes the first known proof by mathematical induction. (First in western mathematics; The 10th Century Persian mathematician Muhammad Al-Karaji was one of the first to use the method of proof by mathematical induction to prove his results, by proving that the first statement in an infinite sequence of statements is true, and then proving that, if any one statement in the sequence is true, then so is the next one. Among other things, Al-Karaji used mathematical induction to prove the binomial theorem.) He proved that the sum of the first n odd numbers is equal to n² .) Maurolico's astronomical observations include a sighting of the supernova that appeared in Cassiopeia in 1572. Tycho Brahe published details of his observations in 1574; the supernova is now known as Tycho's Supernova.*Wik

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1736 Johannes Nikolaus Tetens (16 Sep 1736; 17 Aug 1807) German natural philosopher whose empirical approach strongly influenced the work of Immanuel Kant, and later in his life, Tetens became interested in mathematics, especially in actuarial applications. From 1760, as a teacher of natural philosophy he wrote on diverse topics but later began the development of the field of developmental psychology in Germany. He wrote Philosophische Versuche über die menschliche Natur und ihre Entwickelung (1777) on the origin and structure of knowledge. He changed career after 1789 to the civil service during which time he pursued mathematics. As a statistician he produced an Introduction to the Calculation of Life Annuities (1785) and On the Tetens Mortality Curve (1785)*TIS

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1804 Squire Whipple (16 Sep 1804; 15 Mar 1888) U.S. civil engineer, inventor, and theoretician who provided the first scientifically based rules for bridge construction, was considered one of the top engineers of the 19th Century, and was known as the "father of iron bridges." He began his career as a bridge-builder in 1840 by designing and patenting an iron-bridge truss. During the next ten years he built several bridges on the Erie canal and the New York and Erie railroad. His design of the Whipple truss bridge was the model for hundreds of bridges that crossed the Erie Canal in the late 19-th century. Before developing his design, Whipple worked for several years on surveys, estimates, and reports for the enlargement of the Erie Canal, and in 1840 he patented a scale for weighing canal boats. He later built the first weighing lock scale constructed on the Erie Canal. The invention of the steam engine required bridges which could support heavy live loads and this motivated Squire to turn his attention to bridges. In 1853, he completed a 146-ft span iron railroad bridge near West Troy (now Watervliet), N.Y. His book on the design of bridges using scientific methods (1847) was the first of its kind. The formulas and his methods are still useful. He obtained a patent for his lift draw-bridge in 1872.*TIS

Completed after Whipple's death, the Cairo Rail Bridge was built in two 518 feet (157.9 m) Whipple truss spans, each the largest of that design ever constructed.  In order to comply with regulations meant to allow steam boat travel on the Ohio, the bridge was required to be 53 feet (16 m) above the river's high-water mark. This resulted in the structure extending nearly 250 feet (76 m) from the bottom of the deepest foundation to the top of the highest iron work. The bridge, substructure and superstructure weighed 194.6 million pounds (88,270 t), excluding the approaches.

In 1952, that original superstructure was replaced. A new bridge, completed in 1952, reused many of the original piers but has different spans (Warren-type truss spans etc.).  *PBnotes

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1929 Murray Gell-Mann (September 15/16, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the fundamental building blocks of the strongly interacting particles, and the renormalization group as a foundational element of quantum field theory and statistical mechanics. He played key roles in developing the concept of chirality in the theory of the weak interactions and spontaneous chiral symmetry breaking in the strong interactions, which controls the physics of the light mesons. In the 1970s he was a co-inventor of quantum chromodynamics (QCD) which explains the confinement of quarks in mesons and baryons and forms a large part of the Standard Model of elementary particles and forces.

Murray Gell-Mann received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles.*Wik

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1931 Ennackal Chandy George Sudarshan (16 September 1931 – 13 May 2018)  is a prominent Indian-American physicist, author and professor at the University of Texas at Austin. Sudarshan has made significant contributions to several areas of physics. He was the originator (with Robert Marshak) of the V-A theory of the weak force (also done later by Richard Feynman and Murray Gell-Mann), which eventually paved the way for the electroweak theory. Feynman said in 1963: "The V-A theory that was discovered by Sudarshan and Marshak, publicized by Feynman and Gell-Mann".

He also developed a quantum representation of coherent light (for which Glauber was awarded the 2005 Nobel). *Wik

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1963 Bernard Kippelen (1963, - )serves as Vice Provost for International Initiatives and holds the Steven A. Denning Chair for Global Engagement at the Georgia Institute of Technology. He is also the Joseph M. Pettit professor in Electrical and Computer Engineering.

Kippelen joined Georgia Tech in 2003 as a member of the faculty with the rank of professor. From 2011-2019 he served as Director of the Center for Organic Photonics and Electronics on the Georgia Tech campus in Atlanta. He is a co-founder and co-President of the Institut Lafayette, an innovation platform located on Georgia Tech’s European site Georgia Tech Europe (Metz, France).

As Vice Provost for International Initiatives at Georgia Tech (since 2021), he oversees the international strategy, global campuses, actively participates in philanthropic engagement, corporate development, and partnerships that connect research and education worldwide. This role has provided him with extensive experience in fiscal management ).

He was born in Guebwiller, raised in Soultz, Alsace, France, and is a US and French citizen. He studied at the University Louis Pasteur in Strasbourg (aka University of Strasbourg) where he received a Maitrise in Solid-State Physics in 1985, and a Doctorate in Nonlinear Optics in 1990. From 1990 to 1997 he was Chargé de Recherches at the French CNRS. In 1991, he joined the Optical Sciences Center at the University of Arizona as a postdoctoral fellow, held several research faculty positions, and joined the faculty in 1998 as a tenure-track Assistant Professor. He was promoted to Associate Professor with tenure in 2001.

His most impactful research contributions are in the field of organic optoelectronics, including the development of organic light-emitting diodes (OLEDs), organic field-effect transistors (OFETs), organic photovoltaics, and organic photodiodes. He holds 26 issued patents and has co-authored over 700 scientific communications, including over 300 peer-reviewed journal publications and 14 book chapters. His work has received over 34,000 citations and his h-index is currently 94 (Google Scholar). He served as chair and co-chair of numerous international conferences on organic optoelectronic materials and devices. He has graduated 30 Ph.D. students and advised 23 postdoctoral fellows. He taught passionately many courses both at the undergraduate and graduate levels, both in the classroom and online. He contributed to curriculum development at the frontiers of optics, physics, engineering, materials science, and chemistry. The numerous teaching awards and accolades he received from his past students are a source of pride.

Kippelen is the recipient of an NSF-Career Award (2000), a 3M Corporation Young Faculty Award (2000), a FlexTech Alliance Award (2012), a Printed Electronics USA Award (2012), the Georgia Tech Class of 1934 Outstanding Interdisciplinary Activity Award (2014), and the Georgia Tech Steven A. Denning Faculty Award for Global Engagement (2019). His research was featured in numerous media outlets, including Forbes and Reuters. He was elected a Fellow of Optica (formerly known as the Optical Society of America) (2006), a Fellow of SPIE (2007), and bestowed the title of Knight of the French National Order of Merit (2023). He served as a Deputy Editor for Optics Express (2009-2012) and was the founding Editor of Energy Express (2010-2012).   

Education

Doctorate in Nonlinear Optics, University Louis Pasteur Strasbourg (now University of Strasbourg), 1990

Maitrise in Solid‑State Physics, University Louis Pasteur Strasbourg, 1985

Research Interests

Organic light-emitting devices for displays and lighting

Flexible organic photovoltaic cells for power generation

Organic photodetectors and sensors

Printed electronics

Structure-property relationships in photonic and electronic materials

Teaching Interests

Kippelen's teaching interests include electromagnetics, optoelectronics, and organic optoelectronics, and he has contributed to curriculum development at the frontiers of optics, physics, engineering, materials science, and chemistry.

Distinctions & Awards

Knight of the French National Order of Merit

NSF CAREER Award

3M Young Faculty Award

Senior Member, IEEE

Fellow, the Optical Society of America

Fellow, the International Society for Optical Engineering  *Ga Tech College of Engineering

And from my brief personal experience, a really nice guy.  *PB

Bernard has always had a passion for mountains and climbed Mount Blanc in the French Alps with his wife, Virginie.

***Mont Blanc is the highest mountain in the Alps and Western Europe, rising about 4,809 meters (15,777 feet) above sea level on the border between France and Italy.  (Layman's translation:"  It's high, snowy, dangerous and very cold.... )

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DEATHS

1736 Gabriel Daniel Fahrenheit (24 May 1686, 16 Sep 1736) was a German-Dutch physicist and instrument maker (meteorological). He lived in Holland for most of his life. He invented the alcohol thermometer (1709) and mercury thermometer (1714) and developed the Fahrenheit temperature scale. For the zero of his scale he used the temperature of an equal ice-salt mixture; 30° for the freezing point of water; and 90° for normal body temperature. Later, he adjusted to 32° for the freezing point of water and 212° for the boiling point of water, the interval between the two being divided into 180 parts. He also invented a hygrometer to measure relative humidity and experimented with other liquids discovering that each liquid had a different boiling point that would change with atmospheric pressure.*TIS

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1925 Alexander Alexandrovich Friedmann (16 Jun 1888, 16 Sep 1925) Russian mathematician who was the first to work out a mathematical analysis of an expanding universe consistent with general relativity, yet without Einstein's cosmological constant. In 1922, he developed solutions to the field equations, one of which clearly described a universe that began from a point singularity, and expanded thereafter. In his article On the Curvature of Space received by the journal Zeitschrift für Physik on 29 Jun 1922, he showed that the radius of curvature of the universe can be either an increasing or a periodic function of time. In Jul 1925, he made a record-breaking 7400-m balloon ascent to make meteorological and medical observations. A few weeks later he fell ill and died of typhus. *TIS

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1931 Niels Nielsen (2 Dec 1865 , 16 Sept 1931) was a Danish mathematician who worked on special functions and number theory. *SAU

In 1917 he suffered from an illness from which he never fully recovered. From this date onward he became interested in the number theory, Bernoulli numbers, Stirling numbers, and the history of mathematics, writing two books on Danish mathematicians of the time period 1528-1908, and two other books on French mathematicians. *Wik

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1932 Sir Ronald Ross (born 13 May 1857, 16 Sep 1932) English physician, bacteriologist and mathematician whose discovery of the malarial parasite in the gastrointestinal tract of the Anopheles mosquito led to the realization that malaria was transmitted by Anopheles. For this work, he was awarded the 1902 Nobel Prize for Physiology or Medicine, becoming the first British Nobelist. He began studying malaria in 1892. In 1894 he made an experimental investigation in India of the hypothesis of Alphonse Laveran and Patrick Manson that mosquitoes are connected with the propagation of the disease. After two and a half years' failure, Ross succeeded in demonstrating the life-cycle of the parasites of malaria in mosquitoes, thus establishing the hypothesis of Laveran and Manson. Later, in West Africa he found the species of mosquitoes which convey the deadly African fever.*TIS (He is most remembered for his work on malaria, but his greatest influence may have come from his development and publishing of a mathematical theory of epidemiology.)

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1946 Sir James Hopwood Jeans (11 Sep 1877, 16 Sep 1946) was an English physicist, astronomer, and mathematician who was the first to propose that matter is continuously created throughout the universe. He made other innovations in astronomical theory but is perhaps best known as a writer of popular books about astronomy. *TIS

He served as a secretary of the Royal Society from 1919 to 1929, and was the president of the Royal Astronomical Society from 1925 to 1927, and won its Gold Medal

Jeans was elected Fellow of Trinity College in October 1901, and taught at Cambridge, but went to Princeton University in 1904 as a professor of applied mathematics. He returned to Cambridge in 1910.

From 1923 to 1944 he was associated with Caltech's Mount Wilson Observatory.

He made important contributions in many areas of physics, including quantum theory, the theory of radiation and stellar evolution. His analysis of rotating bodies led him to conclude that Pierre-Simon Laplace's theory that the Solar System formed from a single cloud of gas was incorrect, proposing instead that the planets condensed from material drawn out of the Sun by a hypothetical catastrophic near-collision with a passing star. This theory is not accepted today.

Jeans, along with Arthur Eddington, is a founder of British cosmology. In 1928, Jeans was the first to conjecture a steady state cosmology based on a hypothesized continuous creation of matter in the universe.*Wik

...[T]o many it is not knowledge but the quest for knowledge that gives greater interest to thought—to travel hopefully is better than to arrive

~Sir James Jeans

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1979 Marion Gray (26 March 1902, 16 Sept 1979) graduated from Edinburgh University and then went to Bryn Mawr College in the USA. She completed her doctorate there and returned to posts at Edinburgh and Imperial College London. She returned to the USA and worked for AT&T for the rest of her career. The Gray graph is named after her.*SAU The Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive *Wik

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1989 Allen Shields (May 7, 1927 - September 16, 1989) worked on a wide range of mathematical topics including measure theory, complex functions, functional analysis and operator theory.
An interesting story from George Piranian, of how Shields was appointed to the University of Michigan.

In 1955, on the first day of the American Mathematical Society Summer Meeting in Ann Arbor, George [Piranian] asked Chairman T H Hildebrandt for leave of absence for the Winter Term of 1956. Immediately Hildebrandt declared that he could not grant the request unless George found a replacement. On his way from Hildebrandt's office to one of the lecture sessions, George ran into Allen Shields, whom a year earlier he had met at the Summer Meeting at Laramie. Allen was cooperative, and George dashed back to report that he had found a substitute and that, after fifteen more minutes the substitute would present a ten-minute paper. ... Young as he was, Allen had already mastered the art of beginning his blackboard work in the upper left-hand corner and ending neatly at the lower right, with one minute to spare. Hildebrandt was so impressed that on the spot he offered Allen a one-term appointment. Later, the department persuaded both Shields and Hildebrandt to extend the arrangement.

Soon after George Piranian returned from his leave, he began working with Shields and they published the joint paper The sets of Luzin points of analytic functions (1957). *SAU

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2005 Gordon Gould (17 Jul 1920, 16 Sep 2005) American physicist who coined the word "laser" from the initial letters of "Light Amplification by Stimulated Emission of Radiation." Gould was inspired from his youth to be an inventor, wishing to emulate Marconi, Bell, and Edison. He contributed to the WWII Manhattan Project, working on the separation of uranium isotopes. On 9 Nov 1957, during a sleepless Saturday night, he had the inventor's inspiration and began to write down the principles of what he called a laser in his notebook. Although Charles Townes and Arthur Schawlow, also successfully developed the laser, eventually Gould gained his long-denied patent rights. *TIS

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2011 Mario Wschebor ( 3 December 1939,   16 September 2011)  was a Uruguayan mathematician and statistician who has a remarkable record of outstanding papers, and an international reputation of speaking at conferences around the world. Yet his career was disrupted by having to flee because of political events. He was also passionate about political and social issues and fought throughout his career for an improved university system. *SAU

In the field of mathematics, he made important contributions to probability and statistics . He was the author of numerous scientific publications, including Level Sets and Extrema of Random Processes and Fields (Wiley, UK, 2009), in collaboration with Jean-Marc Azaïs.

He was a prominent university leader, serving as Dean of the Faculty of Sciences at the University of the Republic from 1987 to 1997, after actively participating in its creation. An important contribution from this period is the Document of the Four Deans , in which he advocated the need for the creation of a national tertiary education system. *Wik

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2016   Professor B L Moiseiwitsch BSc PhD MRIA FIoP (died suddenly on 16 September 2016, aged 88)

Obituary supplied by the Moiseiwitsch family

Born in London on 06 December 1927 to Jacob and Chana Moiseiwitsch, Benjamin Lawrence (Benno) was the last surviving of their three children; Daniel (1944) and Victoria (2012).

He had a career interest in theoretical studies in atomic, molecular and optical physics, receiving his PhD in Applied Mathematics from University College London in 1952 before moving almost immediately to Belfast to take up an academic position at Queen’s.

Initially appointed as a lecturer, he was successively promoted to reader, received a personal chair in Applied Mathematics in 1968, was Dean of the Faculty of Science from 1972 to 1975, and became Head of the Department of Applied Mathematics and Theoretical Physics in 1977. He remained in that role until 1989 before retiring in 1993.

During his time at Queen’s he published the first of his three books Variational Principles (1966) and was elected a Member of the Royal Irish Academy (1969).

While he had numerous research collaborations around the world, travelling extensively in North and South America, India and throughout Europe, Professor Moiseiwitsch always returned to Belfast.

After his appointment as Professor Emeritus in Applied Mathematics at Queen’s he continued to be active, publishing his third and most recent book, How to Solve Applied Mathematics Problems, (2011). 

He is survived by his wife of 63 years, Sheelagh, and their four children – Tanya (Coppel), Lisa (Masterman), Julian and Nicholas, by eight grandchildren and four great grandchildren.

*SAU

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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

Narcissisitic Numbers .... The History and Etymology of Math Terms

   Narcissisitic Numbers  The term seems to come from the pen of Joseph S. Madachy in his Mathematics on Vacation, 1966.  His definition is broader than the current use (at least as I know them.) "A Narcissistic number is one which can be represented as some function of its digits ." He includes examples like 145 = 1! + 4! + 5!. It also would include things like n=(sum of digits of n)^(number of digits in n).  

Today the term(s) is used for numbers n (in a particular base) in which the digits are each raised to the power of the number of digits n has and then summed and found to be equal to n.  An early example is 153 = 1^3 + 5^3 + 3^3.  It has been proven that there are only 89 numbers in base ten that have this quality.   

G H Hardy's A Mathematician's Apology mentions the four three-digit solutions, 153, 370, 371, 407, (without using any particular name for them) but dismisses them with, "There is nothing in these odd facts which appeals to the mathematician." 

Another related problem is is there a number A in n digits where the sum of the digits of A raised to the power n = some Other number B and so that the same function of B, gives A again. Are there any that form three-cycles A-> B -> C ->A etc. (Pssst, the answer is yes)

Other names that have been used for these numbers are Armstrong Numbers and PluPerfect digital invariants, which is usually abbreviated PPDI, and is from the Latin past tense for "more than perfect".  

Armstrong numbers is from the name of  Michael F. Armstrong, a computer science teacher who died in 2020.  It seems to have been created shortly after the Narcissistic name, and grew in popularity as one finding these numbers became a popular task for programming classes.  

There is actually a little more to the story of 153.   

Pick any old number you want, and multiply by three (or just pick a number that is a multiple of three).

Now take all the digits and cube them and add the cubes together.

For example, if you picked 231, you would add 2^3 + 3^3 + 1^3 to get 36. 

Yeah, So what you're probably thinking... but take that new number and do the same thing... cube the digits and add them up... Nothing?  Keep going... eventually you get to 153, and then when you  do it again, you get 153 forever.

In slightly more formal language, it seems that 153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits.  And if you are wondering what happens if you use a number that is NOT divisible by three, you can find my exploration of them here, The Cubic Attractiveness of 153,  some fun.

(Find more at OEIS   or in Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer The link takes you to the book on Amazon.

On This Day in Math - September 15

   

Once a sage asked why scholars always flock to the doors of the rich, whilst the rich are not inclined to call at the doors of scholars. "The scholars" he answered , "are well aware of the use of money, but the rich are ignorant of the nobility of science".
~Al-Biruni

The 258th day of the year; 258 is a sphenic (wedge) number (the product of three distinct prime factors..258 = 2·3·43) it is also the sum of four consecutive primes 258 = 59 + 61 + 67 + 71

(Jim Wilder@Wilderlab pointed out that 2,5,&8 are the numbers in the center column of a phone or calculator.)  Jim's comment reminded me of a math type phone joke I saw at  Wolfram Mathworld:
 "I'm sorry, the number you have dialed is an imaginary number. Please rotate by 90^o and try again."

Prime Curios also states that if you take all six permutations of 258, put them in order, and make the 18 digit concatenation of them, it is the composite number between a prime pair. Can you find others like this?  (Sort of an easy way to learn two eight digit primes that happen to be a prime pair.

And 258 is the sum of four consecutive Prime numbers, 59 + 81 + 67 +71 and also the sum of two consecutive primes 127 + 131.

258 = 6^3 + 6^2 + 6, or 1110 in base six. Tomorrow that will be 1111.

The Number Zoo gives a Magic square using 16 consecutive primes, with a constant of 258

see More Math Facts for every Year Day here

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EVENTS

1668  Robert Hooke delivered some of his most perceptive and far sighted views of the geology of what he called this "terraqueous globe"  in lectures beginning on May 28,1664 and ending on 15 September 1668, and published posthumously.  In these lectures he shared ideas novel to his contemporaries, including "the organic origin, and significance of fossils; cyclicity of the processes of sedimentation,  erosion, consolidation, uplift, and denudation; various processes of petrification; subterraneous eruptions  and earthquakes; biologic evolution; the oblate spheroid shape of the Earth; Polar wandering, and universal gravitation. "  *Ellen Tan Drake; Hooke's Ideas of the Terraqueous  Globe and a Theory of Evolution

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1739 Euler, in a letter to Johann Bernoulli, begins the general treatment of the homogeneous linear differential equation with constant coefficients. *VFR  Within a year Euler had completed this treatment by successfully dealing with repeated quadratic factors and turned his attention to the non-homogeneous linear equation. *John E. Sasser, HISTORY OF ORDINARY DIFFERENTIAL EQUATIONS -THE FIRST HUNDRED YEARS

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1749  Euler's interest in lotteries began at the latest in 1749 when he was commissioned by Frederick the Great to render an opinion on a proposed lottery that would be similar to the Lottery in Genoa. The first of two letters began 15 September 1749. A second series began on 17 August 1763. E812. Read before the Academy of Berlin 10 March 1763 but only published posthumously in 1862. "Reflexions sur une espese singulier de loterie nommée loterie genoise." Opera postuma I, 1862, p. 319–335. The paper determined the probability that a particular number be drawn. *Euler’s Correspondence Translated by Richard J. Pulskamp, Department of Mathematics & Computer Science, Xavier University,
Cincinnati, OH

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1782 Lagrange, in a letter to Laplace, told of finishing his M´ecanique analytique. Legendre undertook the editing of the work for the press. *VFR

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1784 Balloon Corner in London earns its name. 'Vincent' Lunardi, "The Daredevil Aeronaut", demonstrated a hydrogen balloon flight at the Artillery Ground of the Honourable Artillery Company in London before over a reported crowd of 200,000 people. With a cat, a dog, and a caged pigeon, he rose into the air with only a partially filled bag and then set down at Welham Green, to release the cat, which seems to have become airsick. He then continued to Standon Green End. A stone marks the event in Welham Green :

"NEAR THIS SPOT AT 3.30 IN THE
AFTERNOON OF SEPTEMBER 15TH
1784 VINCENZO LUNDARDI THE
ITALIAN BALLOONIST MADE HIS
FIRST LANDING WHILST ON HIS
PIONEER FLIGHT IN THE ENGLISH
ATMOSPHERE.
HAVING HANDED OUT A CAT AND DOG
THE PARTNERS OF HIS FLIGHT FROM
LONDON HE RE-ASCENDED AND
CONTINUED NORTH EASTWARD."

The 24 mile flight brought Lunardi fame and began the ballooning fad that inspired fashions of the day—Lunardi skirts were decorated with balloon styles, and in Scotland, the Lunardi Bonnet was named after him (balloon-shaped and standing some 600 mm tall), and is even mentioned by Scotland's national poet, Robert Burns (1759–96), in his poem 'To a Louse', written about a young woman called Jenny, who had a louse scampering in her Lunardi bonnet, *Wik

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1788 Thomas Paine writes to Thomas Jefferson to discuss shapes for Iron Bridges:

Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypotenuse, the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypotenuse and right angle will be always given.,

Jefferson's reply of Dec 23, 1788 is cited by OED as the first use of "catenary".  *Jeff Miller

In 1803 Paine was the first person to receive a patent for an Iron Bridge. His design was poor, as might be expected with no education or training in the craft. and no Payne bridge ever crossed any stream or river, and the first iron bridge would not be built in America for half a century.  

in 1839 cast Iron bridge was completed over the Dublaps Creek, at the town of Brownsville, Pa.  The bridge still stands today.  

A plaque attached to its railing commemorates the bridge.

*Pa Center for the Book

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1846 George Boole, age 30, applied for a professorship at “any of her Majesty’s colleges, now in the course of being established in Ireland.” Although he had “never studied at a college” he had been a teacher for 15 years and was “familiar with the elementary and the practical as well as the higher Mathematics.” Although he was self taught, the testimonies of DeMorgan, Cayley, and William Thomson showed that he was an accomplished mathematician. In August 1849, he was appointed the first professor of mathematics at Queen’s College Cork. The reason for the long delay is unclear. *MacHale, George Boole, His Life and Work, pp. 75-84

*Wik

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1855 J J Sylvester commenced his duties as professor of mathematics and lecturer in natural philosophy at the Royal Military Academy, Woolwich, and one of the richest research periods of his life began. [Osiris, 1(1936), 101] *VFR

The Woolwich academy initially refused to pay Sylvester his full pension, and only relented after a prolonged public controversy, during which Sylvester took his case to the letters page of The Times.

In 1872, he finally received his B.A. and M.A. from Cambridge, having been denied the degrees due to his being a Jew.

In 1876 Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland. His salary was $5,000 (quite generous for the time), which he demanded be paid in gold. After negotiation, agreement was reached on a salary that was not paid in gold.

In 1877, he was elected as a member to the American Philosophical Society.

In 1878 he founded the American Journal of Mathematics. The only other mathematical journal in the US at that time was the Analyst, which eventually became the Annals of Mathematics.

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1947 The world's oldest computing society, the Association for Computing Machinery, is founded. With more than 80,000 members today, ACM organizes conference and educational workshops to exchange information on technology.*CHM

*CHM

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2017 The Cassini space probe, launched in 1997, was named after Giovanni Cassini and became the first probe to orbit Saturn. For over a decade the probe sent back vital information about Saturn and its moons, expanding our knowledge of the planet, its moons and rings. Fittingly, it crashed into the planet on the day following the anniversary of the death of the astronomer for whom it was named. Earth received @CassiniSaturn’s final signal at 7:55am ET. Cassini is now part of the planet it studied *NASA

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BIRTHS

973 Al-Biruni (15 Sept 973, 13 Dec 1048) is one of the major figures of Islamic mathematics. He contributed to astronomy, mathematics, physics, medicine and history. *SAU

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1736 Jean-Sylvain Bailly (15 Sep 1736; 12 Nov 1793) French astronomer who computed an orbit for Halley's Comet (1759) and studied the four satellites of Jupiter then known. He was the first Mayor of Paris (1789-91). He was executed by guillotine in Paris during the French Revolution.*TIS
Bailly published his Essay on the theory of the satellites of Jupiter in 1766, an expansion of a presentation he had made to the Academy in 1763. It was followed up in 1771 by a noteworthy dissertation, On the inequalities of light of the satellites of Jupiter. and in 1778, he was elected a foreign member of the Royal Swedish Academy of Sciences. *Wik

Titlepages of two of Jean-Sylvain Bailly’s history of astronomy publications (Linda Hall Library)

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1852 Edward Bouchet (15 Sept 1852, New Haven, Conn – 28 Oct 1918, New Haven, Conn) was the first African-American to earn a Ph.D. in Physics from an American university and the first African-American to graduate from Yale University in 1874. He completed his dissertation in Yale's Ph.D. program in 1876 becoming the first African-American to receive a Ph.D. (in any subject). His area of study was Physics. Bouchet was also the first African-American to be elected to Phi Beta Kappa.
Bouchet was also among 20 Americans (of any race) to receive a Ph.D. in physics and was the sixth to earn a Ph. D. in physics from Yale.
Edward Bouchet was born in New Haven, Connecticut. At that time there were only three schools in New Haven open to black children. Bouchet was enrolled in the Artisan Street Colored School with only one teacher, who nurtured Bouchet's academic abilities. He attended the New Haven High School from 1866–1868 and then Hopkins School from 1868-1870 where he was named valedictorian (after graduating first in his class).
Bouchet was unable to find a university teaching position after college, most likely due racial discrimination. Bouchet moved to Philadelphia in 1876 and took a position at the Institute for Colored Youth (ICY). He taught physics and chemistry at the ICY for 26 years. The ICY was later renamed Cheyney University. He resigned in 1902 at the height of the W. E. B. Du Bois-Booker T. Washington controversy over the need for an industrial vs. collegiate education for blacks.
Bouchet spent the next 14 years holding a variety of jobs around the country. Between 1905 and 1908, Bouchet was director of academics at St. Paul's Normal and Industrial School in Lawrenceville, Virginia (presently, St. Paul's College). He was then principal and teacher at Lincoln High School in Gallipolis, Ohio from 1908 to 1913. He joined the faculty of Bishop College in Marshall, Texas in 1913. Illness finally forced him to retire in 1916 and he moved back to New Haven. He died there, in his childhood home, in 1918, at age of 66. He had never married and had no children.*Wik

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1857  Anna Winlock (15 Sept 1857– 4 Jan 1904) was an American astronomer and human computer, one of the first members of female computer group known as "the Harvard Computers." She made the most complete catalog of stars near the north and south poles of her era. She is also remembered for her calculations and studies of asteroids. In particular, she did calculations on 433 Eros and 475 Ocllo.

Winlock attended the Cambridge, Ma. Schools as a child and began to develop an interest in both mathematics and the Greek language. By age 10, Anna had watched her father go from Superintendent at the American Nautical Almanac Office in Cambridge, Massachusetts, to the Director of the Harvard College Observatory as well as a professor of Astronomy at the main Harvard College. Upon her graduation she received a letter from her principal expressing his appreciation for her Greek and of her character. Her father influenced her interest in astronomy. When she was twelve, she attended a solar eclipse expedition with her father in his home state of Kentucky. In June 1875, Joseph died shortly after Winlock had graduated from secondary school. Winlock quickly followed in her father's footsteps becoming one of the first female paid staff members of the Harvard College Observatory. *Wik

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1883 Esteban Terrades i Illa (15 September 1883;Barcelona,-  9 May 1950,Madrid,) was a Spanish mathematician, scientist and engineer. He researched and taught widely in the fields of mathematics and the physical sciences, working not only in his native Catalonia, but also in the rest of Spain and in South America. He was also active as a consultant in the Spanish aeronautics, electric power, telephone and railway industries. *Wik

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1886 Paul Pierre Lévy (15 Sep 1886; 15 Dec 1971) was a French mining engineer and mathematician. He contributed to probability, functional analysis, partial differential equations and series. He also studied geometry. In 1926 he extended Laplace transforms to broader function classes. He undertook a large-scale work on generalised differential equations in functional derivatives.*TIS

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1894 Oskar Benjamin Klein (September 15, 1894 (or 1893?) – February 5, 1977) was a Swedish theoretical physicist. Klein retired as professor emeritus in 1962. He was awarded the Max Planck medal in 1959. He is credited for inventing the idea, part of Kaluza–Klein theory, that extra dimensions may be physically real but curled up and very small, an idea essential to string theory / M-theory. *Wik

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1901 Luigi Fantappiè (15 September 1901 – 28 July 1956) was an Italian mathematician, known for work in mathematical analysis and for creating the theory of analytic functionals: he was a student and follower of Vito Volterra. Later in life he proposed scientific theories of sweeping scope.

In 1942 he put forth a unified theory of physics and biology, and the syntropy concept. In 1952 he started to work on a unified physical theory called projective relativity, for which, he asserted, special relativity was a limiting case. Giuseppe Arcidiacono worked with him on this theory.*Wik

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1923 Georg Kreisel FRS (September 15, 1923 – March 1, 2015) is an Austrian-born mathematical logician who has studied and worked in Great Britain and America. Kreisel came from a Jewish background; his family sent him to England before the Anschluss, where he studied mathematics at Trinity College, Cambridge and then, during World War II, worked on military subjects. After the war he returned to Cambridge and received his doctorate. He taught at the University of Reading until 1954 and then worked at the Institute for Advanced Study from 1955 to 1957. Subsequently he taught at Stanford University and the University of Paris. Kreisel was appointed a professor at Stanford University in 1962 and remained on the faculty there until he retired in 1985.
Kreisel worked in various areas of logic, and especially in proof theory, where he is known for his so-called "unwinding" program, whose aim was to extract constructive content from superficially non-constructive proofs.*Wik

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1926 Jean-Pierre Serre (15 September 1926 - ) born in Bages, France. In 1954 he received a Fields Medal for his work on the homotopy groups of spheres. He also received the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. He reformulated some of the main results of complex variable theory in terms of sheaves. See International Mathematical Congresses. An Illustrated History, 1893–1986, edited by Donald J. Albers, G. L. Alexanderson and Constance Reid. Today will be his 100th birthday

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1929 Murray Gell-Mann (September 15, 1929 – May 24, 2019).  American theoretical physicist who predicted the existence of quarks. He was awarded the 1969 Nobel Prize for Physics for his contributions to particle physics. His first major contribution to high-energy physics was made in 1953, when he demonstrated how some puzzling features of hadrons (particles responsive to the strong force) could be explained by a new quantum number, which he called “strangeness”. In 1964, he (and Yuval Ne'eman) proposed the eightfold way to define the structure of particles. This led to Gell-Mann's postulate of the quark, a name he coined (from a word in James Joyce's Finnegan's Wake).*TIS

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DEATHS

1883 physicist J. Plateau (14 October 1801 – 15 September 1883)  Plateau’s problem asks for the minimal surface through a given curve in three dimensions. A minimal surface is the surface through the curve with the least area. Mathematically the problem is still unsolved, but physical solutions are easy: dip a curved wire in a soap solution. The “soap bubble” that results is the minimal surface for that curve. *VFR
In 1829 Joseph Plateau submitted his doctoral thesis to his mentor Adolphe Quetelet for advice. It contained only 27 pages, but formulated a great number of fundamental conclusions. It contained the first results of his research into the effect of colors on the retina (duration, intensity and color), his mathematical research into the intersections of revolving curves (locus), the observation of the distortion of moving images, and the reconstruction of distorted images through counter revolving

discs Prior to going blind was the first person to demonstrate the illusion of a moving image. To do this he used counter rotating disks with repeating drawn images in small increments of motion on one and regularly spaced slits in the other. He called this device of 1832 the phenakistoscope.
Plateau has often been termed a "martyr for science". . In many (popular) publications the blindness of Plateau is ascribed to his experiment of 1829 in which he looked directly into the sun for 25 seconds. Recent research definitely refutes this. The exact date of the blindness is difficult to formulate simply. It was a gradual process during the year 1843 and early 1844. Plateau publishes two papers in which he painstakingly describes the scientific observations of his own blindness. After 40 years of blindness he still has subjective visual sensations. For his experiments, as well as for the related deskwork colleagues and family help him. *Wik

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1898 William Seward Burroughs (born 28 Jan 1855, 15 Sep 1898) American inventor who invented the world's first commercially viable recording adding machine and pioneer of its manufacture. He was inspired by his experience in his beginning career as a bank clerk. On 10 Jan 1885 he submitted his first patent (issued 399,116 on 21 Aug 1888) for his mechanical “calculating machine.” Burroughs co-founded  the American Arithmometer Co in 1886 to develop and market the machine. The manufacture of the first machines was contracted out, and their durability was unsatisfactory. He continued to refine his design for accuracy and reliability, receiving more patents in 1892, and began selling the much-improved model for $475 each. By 1895, 284 machines had been sold, mostly to banks, and 1500 by 1900. The company later became Burroughs Corporation (1905) and eventually Unisys. *TIS

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1947 Annie Scott Dill Maunder (née Russell) FRAS (14 April 1868 – 15 September 1947)

Edward Walter Maunder (12 Apr 1851, 21 Mar 1928 at age 76) English astronomer who was the first to take the British Civil Service Commission examination for the post of photographic and spectroscopic assistant at the Royal Observatory, Greenwich. For the next forty years that he worked there, he made extensive measurements of sunspots. Checking historical records, he found a period from 1645 to 1715 that had a remarkable lack of reports on sunspots. Although he might have questioned the accuracy of the reporting, he instead attributed the shortage of report to an actual dearth of sunspots during that period. Although his suggestion was not generally accepted at first, accumulating research has since indicated there are indeed decades-long times when the sun has notably few sunspots. These periods are now known as Maunder minima.*TIS

On the far side of the Moon lies the Maunder crater, named after two British astronomers - Annie and Walter Maunder.

Annie worked alongside her husband at the end of the 19th Century, recording the dark spots that pepper the Sun.

The name Maunder is still known in scientific circles, yet Annie has somehow slipped from history.

"I think the name Maunder is there and we have all rather forgotten that that's two people," says Dr Sue Bowler, editor of the Royal Astronomical Society magazine, Astronomy and Geophysics.

"She was acknowledged on papers, she published in her own name as well as with her husband, she wrote books, she was clearly doing a lot of work but she also clearly kept to the conventions of the day, I think." *By Helen Briggs BBC News

She is known to have worked closely with her husband on the study of sunspots, and she is often credited with discovering the butterfly pattern. *LH

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1962  William W(eber) Coblentz   (20 Nov 1873, 15 Sep 1962) was an American physicist and astronomer whose work lay primarily in infrared spectroscopy. In 1905 he founded the radiometry section of the National Bureau of Standards, which he headed for 40 years. Coblentz measured the infrared radiation from stars, planets, and nebulae and was the first to determine accurately the constants of blackbody radiation, thus confirming Planck's law.*TIS

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2003 Alexander Ivanovich Skopin (Александр Иванович Скопин) (22 October, 1927–15 September, 2003) was a Russian mathematician known for his contributions to abstract algebra.

Skopin's student work was in abstract algebra, and concerned upper central series of groups and extensions of fields. In the 1970s, Skopin received a second doctorate concerning the application of computer algebra systems to group theory. From that point onward he used computational methods extensively in his research, which focussed on lower central series of Burnside groups. He related this problem to problems in other areas of mathematics including linear algebra and topological sorting of graphs.

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2015 Malcolm John Williamson (Nov 2, 1950 - Sept 15, 2015) discovered in 1974 what is now known as Diffie-Hellman key exchange^.  He was then working at GCHQ.
Williamson studied at Manchester Grammar School, winning first prize in the 1968 British Mathematical Olympiad. He also won a Silver prize at the 1967 International Mathematical Olympiad in Cetinje, Yugoslavia and a Gold prize at the 1968 International Mathematical Olympiad in Moscow. He read mathematics at Trinity College, Cambridge, graduating in 1971. After a year at Liverpool University, he joined GCHQ, and worked there until 1982.
From 1985 to 1989 Williamson worked at Nicolet Instruments in Madison, Wisconsin where he was the primary author on two digital hearing aid patents. *Wik

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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