## Tuesday 30 April 2024

### On This Day in Math - April 30

 Statue of C. F. Gauss in Braunschweig *Wik

I mean the word proof not in the sense of the lawyers,
who set two half proofs equal to a whole one,
but in the sense of a mathematician,
where half a proof is zero,
and it is demanded for proof that
every doubt becomes impossible.
Karl Friedrich Gauss As quoted in Calculus Gems (1992) by George F. Simmons

The 120th day of the year; All primes (except 2 and 3) are of form 6*n +/- 1. Note that 120 = 6*20 is the smallest multiple of six such that neither 6n+1 or 6n-1 is prime. *Prime Curios Can you find the next

120 = 3¹ + 3² + 3³ + 3⁴

Had to add this one,120 is the smallest number to appear 6 times in Pascal's triangle. *What's Special About This Number
(There are only three days of the year that appear in the arithmetic triangle more than five times. What are the other two?)
120 =2* 3*4*5 = 11^2 - 1.  The product of four consecutive integers is always one less than a square

6 and 28 are prefect numbers because the sum of their proper divisors is equal to the number.  120 is the only year date that is a multi-perfect number.  The sum of its proper divisors is 2 * 120. (known since antiquity, the second smallest was discovered by Fermat in 1636 is 672. Fermat actually showed a method to find an infinite number of  such "sous-doubles".)

120 is the largest number of spheres that can contract a central sphere in eight dimensions. Beyond the fourth dimension, this "kissing number" is only known for the eighth and 24th dimensions.

EVENTS

1006 Chinese and Arabic astronomers noted a supernova. The speed of the still-expanding shock wave was measured nearly a millenium later. This was history's brightest "new star" ever recorded, at first seen to be brighter than the planet Venus. It occurred in our Milky Way galaxy, appearing in the southern constellation Lupus, near the star Beta Lupi. It was also recorded by observers in Switzerland, Italy, Japan, Egypt and Iraq. From the careful descriptions of the Chinese astronomers of how the light varied, that it was of apparently yellow color and visible for over a year, it is possible that the supernova reached a magnitude of up to -9. Modern measurements of the speed of the shock wave have been used to estimate its distance. *TIS The associated supernova remnant from this explosion was not identified until 1965, when Doug Milne and Frank Gardner used the Parkes radio telescope to demonstrate that the previously known radio source PKS 1459-41, near the star Beta Lupi, had the appearance of a 30-arcminute circular shell.

This image is a composite of visible (or optical), radio, and X-ray data of the full shell of the supernova remnant from SN 1006.

1633 Galileo was forced to recant his scientiﬁc ﬁndings(suppositions?) related to the Copernican Theory as “abjured, cursed and detested” by the Inquisition. He was placed under house arrest for the remaining nine years of his life. Legend had it that when Galileo rose from kneeling before his inquisitors, he murmured, “e pur, si mouve”—“even so, it does move.” *VFR [church doctrine held that the Earth, God's chosen place, was the center of the universe and everything revolved around it. Copernicans believed that the sun was the center of the solar system, and "the earth moves" around it.]

In 1683, the Boston Philosophical Society held its first meeting.* Rev. Increase Mather, stimulated by a recent comet sighting, and seeking to discuss how God intervenes in the natural order of things, had met earlier in the month with Samuel Willard and a few others to plan the group. Mather's idea was to model their meetings on the Royal Philosophical Society, established in London about 20 years earlier. Each last Monday of most of the following months, the members met and presented papers to emulate the transactions of the London society. However, the few local intellectuals didn't sustain interest in the society beyond about three years. Mather wrote Kometographia, or, A Discourse concerning Comets (1683).*TIS
(Note the complete title.)

1695 Bernoulli explains to Leibniz his reasons for the use of the term "integral calculus" for Leibniz's new calculus. Leibniz had used, and tried to get others to use "Sums" but Bernoulli's term had become popular. Bernoulli explained that, "I considered the differential as the infinitesimal part of the whole, or Integral." *VFR

1752 A sealed paper delivered by mathematician/instrument maker James Short to the Royal Society on 30 April 1752 was opened after his death and read publicly on 25 Jan. 1770. It described a method of working object-lenses to a truly spherical form. It seems, from the journal of Lelande that this was done by eye. "from there by water to Surrey Street
to see Mr Short who spoke to me about the difficulty in giving his mirrors a parabolic figure. It is done only by guess-work."
*Richard Watkins

Brass telescope made by Short, now in the collection of Thinktank, Birmingham Science Museum.

1807 Gauss writes to Sophie Germanin for the first time since being aware she was a woman, (She had formally written using the name Monsieur LeBlanc). In a letter with much praise, he writes:
"The scientific notes with which your letters are so richly
filled have given me a thousand pleasures. I have studied
them with attention and I admire the ease with which you
penetrate all branches of arithmetic, and the wisdom with
which you generalize and perfect."

1837 Massachusetts became the ﬁrst state to establish a board of education. *VFR  At the time, the public school system was in very bad condition. The informal organization of schools meant that there were no set rules or class studies. Tax support and attendance were irregular.*Wik

1877 Charles Cros, a French poet and amateur scientist, is the first person known to have made the conceptual leap from recording sound as a traced line to the theoretical possibility of reproducing the sound from the tracing and then to devising a definite method for accomplishing the reproduction. On April 30, 1877, he deposited a sealed envelope containing a summary of his ideas with the French Academy of Sciences, a standard procedure used by scientists and inventors to establish priority of conception of unpublished ideas in the event of any later dispute. #Wik
Edison would invent his phonograph in 1887.

1891 Nature magazine publishes Peter Guthrie Tait's "The Role of Quaternions in the Algebra of Vectors."
“By 1893 the battle between the quaternionists and the vector analysts was in full swing. It was really two battles, one of quaternions versus coordinates, and a second one of quaternions versus vectors… In 1890 Tait entered the controversy on both fronts” (Hankins 319). The dialogue that took place through the Nature letters reveal an impassioned debate, that at times, almost degrades to ridicule in a back and forth exchange following from letter to letter.
"Quaternions are an alternate way to describe orientation or rotations in 3D space using an ordered set of four numbers. They have the ability to uniquely describe any three-dimensional rotation about an arbitrary axis and do not suffer from gimbal lock." *All About Circuits  dot com

1895 Georg Cantor, in a letter to Felix Klein, explains the choice of aleph for the cardinality of sets.
In the same letter he comments that, "the usual alphabets seem to me too much used to be fitter for the purpose. On the other hand, I did not want to invent a new symbol, so I chose finally the aleph, which in Hebrew has also the numerical value 1."
*Cantorian Set Theory and Limitation of Size, By Michael Hallett

1897 at the Royal Institution Friday Evening Discourse, Joseph John Thomson (1856-1940) first announced the existence of electrons (as they are now named). Thomson told his audience that earlier in the year, he had made a surprising discovery. He had found a particle of matter a thousand times smaller than the atom. He called it a corpuscle, meaning "small body." Although Thomson was director of the Cavendish Laboratory at the University of Cambridge, and one of the most respected scientists in Great Britain, the scientists present found the news hard to believe. They thought the atom was the smallest and indivisible part of matter that could exist. Nevertheless, the electron was the first elementary particle to be discovered.*TIS  The etymology of the term was from the Greek atomos, combining a (not) and temnein (cut).

1905 Einstein completed his Doctoral thesis, with Alfred Kleiner, Professor of Experimental Physics, serving as pro-forma advisor. Einstein was awarded a PhD by the University of Zurich. His dissertation was entitled "A New Determination of Molecular Dimensions." This paper included Einstein's initial estimates of Avagadro constant as 2.2×10^23 based on diffusion coefficients and viscosities of sugar solutions in water (the error was more in the known estimates of sugar molecules than his method). That same year, which has been called Einstein's annus mirabilis (miracle year), he published four groundbreaking papers, on the photoelectric effect, Brownian motion, special relativity, and the equivalence of mass and energy, which were to bring him to the notice of the academic world. *Wik
Einstein would often say that Kleiner, at first, rejected his thesis for being too short, so he added one more sentence and it was accepted.

1916 Daylight Saving Time has been used in the U.S. and in many European countries since World War I. At that time, in an effort to conserve fuel needed to produce electric power, Germany and Austria took time by the forelock, and began saving daylight at 11:00 p.m. on April 30, 1916, by advancing the hands of the clock one hour until the following October. *WebExhibits.org

1939   Oddly enough, the dream of a self-driving automobile goes as far back as the middle ages, centuries prior to the invention of the car. The evidence for this comes from a sketching by Leonardo De Vinci that was meant to be a rough blueprint for a self-propelled cart. Using wound up springs for propulsion, what he had in mind at the time was fairly simplistic relative to the highly advanced navigation systems being developed today.
Though the Phantom Auto drew large crowds during its tour of various cities throughout the ’20s and ’30s, the pure spectacle of a vehicle seemingly traveling without a driver amounted to little more than a curious form of entertainment for onlookers. Furthermore, the setup didn’t make life any easier since it still required someone to control the vehicle from a distance. What was needed was a bold vision of how cars operating autonomously could better serve cities as part of a more efficient, modernized approach to transportation.
It wasn’t until the World’s Fair in 1939 on April 30, that a renowned industrialist named Norman Bel Geddes would put forth such a vision. His exhibit “Futurama” was remarkable not only for its innovative ideas but also for the realistic depiction of a city of the future. For example, it introduced expressways as a way to link cities and surrounding communities and proposed an automated highway system in which cars moved autonomously, allowing passengers to arrive at their destinations safely and in an expedient manner.
below is a 1941 self-driving car. the thing on the back isn't a tank or a boiler, it's a device to generate gas from solid fuel (eg wood), presumably added in response to gasoline rationing #SciencePunk

1982 Science (pp. 505–506) reported that Stanford magician-statistician Perci Diaconis solved the problem of which arrangements of a deck of cards can occur after repeated perfect rifflee shuffles. The answer involves M12, one of the Mathieu simple groups. Mathematics Magazine 55 (1982),
p. 245].*VFR

Smaller decks of sequential numbered cards show the effect
with four cards
1 2 3 4   ->1 3 2 4 ->  1 2 3 4

1 2 3 4 5 6 -> 1 4 2 5 3 6 -> 1 5 4 3 2 6 -> 1 3 5 2 4 6 -> 1 2 3 4 5 6
(Now you do 52 cards)

1984 30 April-4 May 1984. Teacher Appreciation Week. Celebrated the ﬁrst week of May in Flint, MI. *VFR

1992 The New York Times “in describing the discovery of the new Mersenne prime, felt it necessary to describe the series of primes, which, (according to them) goes: 1, 2, 3, 5, 7, 11, 13, ... . You will notice that they have slipped in what must be another discover (by one of their writers?) of the world’s smallest prime: 1. I’m sure the mathematicians of the world must be tearing their hair out for having missed this one.” [A posting of Ron Rivest to the net.] (In fact it was common prior to the 20th century to consider one as a prime, not that that is an excuse in 1992.)

The Mersenne Primes are numbers that are one less than a power of two,  3, 7, (not 15) , 31  ...  since all composite numbers used as the power will not produce a prime number, they are usually defined with a prime exponent, Mp = 2^p - 1. Not all prime exponents produce primes, but they are an important tool in prime searches.  Most knew "largest known primes" are found using this method at least in part.

They are named after Marin Mersenne, a French friar, who studied them in the early 17th century. He communicated widely among European mathematicians and helped circulate new mathematical discoveries,
 From Time Magazine

1993 CERN announces World Wide Web protocols will be free. *Wik
At the urging of Tim Berners-Lee, the creator of the World Wide Web protocol, the directors of CERN release the source code of World Wide Web into the public domain, making it freely available to anyone, without licensing fees. The decision to make the World Wide Web software and protocols freely available is considered by some as possibly the single most important moment in the history of the Internet. In fact, some historians mark this as the birth of the Web.*This Day in Tech History

2015 Walpurgis night or Witches’ Sabbath is celebrated on the eve of May Day, particularly by university students in northern Europe. *VFR According to the ancient legends, this night was the last chance for witches and their nefarious cohorts to stir up trouble before Spring reawakened the land. They were said to congregate on Brocken, the highest peak in the Harz Mountains - a tradition that comes from Goethe's Faust. *Wik

BIRTHS

1777 Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.
Sometimes referred to as the Princeps mathematicorum (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences".. *Wik
His poorly educated mother couldn’t remember his birthdate, but could relate it to a movable religious feast. To conﬁrm the date of his birth Gauss developed a formula for the date of Easter. *VFR
He transformed nearly all areas of mathematics, for which his talent showed from a very early age. For his contributions to theory in magnetism and electricity, a unit of magnetic field has been named the gauss. He devised the method of least squares in statistics, and his Gaussian error curve remains well-known. He anticipated the SI system in his proposal that physical units should be based on a few absolute units such as length, mass and time. In astronomy, he calculated the orbits of the small planets Ceres and Pallas by a new method. He invented the heliotrope for trigonometric determination of the Earth's shape. With Weber, he developed an electromagnetic telegraph and two magnetometers. *TIS He proved that the heptadecagon (17 gon) was constructable (see April 8) with straight-edge and compass. Dave Renfro has a complete and elementary proof.
Gauss told his close friend Bolyai that the regular 17-gon should adorn his tombstone, but this was not done. There is a 17 pointed star on the base of a monument to him in Brunswick because the stonemason felt everyone would mistake the 17-gon for a circle. Gauss gave the tablet on which he had made the discovery to Bolyai, along with a pipe, as a souvenir.  The pipe has since, apparently, gone missing.

1904 George Robert Stibitz (30 Apr 1904, 31 Jan 1995) U.S. mathematician who was regarded by many as the "father of the modern digital computer." While serving as a research mathematician at Bell Telephone Laboratories in New York City, Stibitz worked on relay switching equipment used in telephone networks. In 1937, Stibitz, a scientist at Bell Laboratories built a digital machine based on relays, flashlight bulbs, and metal strips cut from tin-cans. He called it the "Model K" because most of it was constructed on his kitchen table. It worked on the principle that if two relays were activated they caused a third relay to become active, where this third relay represented the sum of the operation. Also, in 1940, he gave a demonstration of the first remote operation of a computer.*TIS

1916 Claude Shannon (30 April 1916 in Petoskey, Michigan, USA - 24 Feb 2001 in Medford, Massachusetts, USA) founded the subject of information theory and he proposed a linear schematic model of a communications system. His Master's thesis was on A Symbolic Analysis of Relay and Switching Circuits on the use of Boole's algebra to analyse and optimize relay switching circuits. *SAU While working with John von Neumann on early computer designs, (John) Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948. Among several statues to Shannon, one is erected in his hometown of Gaylord, Michigan. The statue is located in Shannon Park in the center of downtown Gaylord. Shannon Park is the former site of the Shannon Building, built and owned by Claude Shannon's father. The lady beside the statue, a true mathematical genius in her own right, is Betty, the wife, and closest collaborator of Claude Shannon.
While working with John von Neumann on early computer designs, (John) Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948. In 2016 as his 100th birth anniversary was approaching, the Petoskey News (Shannon's birthplace) described him as the folks who knew him growing up in Gaylord like to recall him, "Who would have guessed the world would be celebrating the birthday of a unicycling, juggling, mathematic academic and engineer from Gaylord? But that is exactly what is happening next week as local historians, youth and others celebrate a special centennial birthday of a local celebrity.

1944 Lee Cecil Fletcher Sallows (April 30, 1944, ) is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares. Sallows has an Erdős number of 2.
Sallows is an expert on the theory of magic squares and has invented several variations on them, including Alphamagic Squares and geomagic squares. The latter invention caught the attention of mathematician Peter Cameron who has said that he believes that "an even deeper structure may lie hidden beyond geomagic squares"
In 1984 Lee Sallows invented the self-enumerating sentence — a sentence that inventories its own letters. Following failure in his attempt to write a computer program to generate such sentences, he constructed a so-called electronic Pangram Machine, among the results of which was the following sentence that appeared in Douglas Hofstadter's Metamagical Themas column in Scientific American in October 1984:
This Pangram contains four as, one b, two cs, one d, thirty es, six fs, five gs, seven hs, eleven is, one j, one k, two ls, two ms, eighteen ns, fifteen os, two ps, one q, five rs, twenty-seven ss, eighteen ts, two us, seven vs, eight ws, two xs, three ys, & one z.
A golygon is a polygon containing only right angles, such that adjacent sides exhibit consecutive integer lengths. Golygons were invented and named by Sallows and introduced by A.K. Dewdney in the Computer Recreations column of the July 1990 issue of Scientific American.
In 2012 Sallows invented and named 'self-tiling tile sets'—a new generalization of rep-tiles
1944*Wik

DEATHS

1865 Robert Fitzroy British naval officer, hydrographer, and meteorologist who commanded the voyage of HMS Beagle, aboard which Charles Darwin sailed around the world as the ship's naturalist. That voyage provided Darwin with much of the material on which he based his theory of evolution. Fitzroy retired from active duty in 1850 and from 1854 devoted himself to meteorology. He devised a storm warning system that was the prototype of the daily weather forecast, invented a barometer, and published The Weather Book (1863). His death was by suicide, during a bout of depression. *TIS
[FitzRoy is buried in the front church yard of All Saints Church in Upper Norwood, London. His memorial was restored by the Meteorological Office in 1981]

1907 Charles Howard Hinton​ (1853, 30 April 1907) was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualizing the geometry of higher dimensions. He also had a strong interest in theosophy.
Hinton created several new words to describe elements in the fourth dimension. According to OED, he first used the word tesseract in 1888 in his book A New Era of Thought. He also invented the words "kata" (from the Greek "down from") and "ana" (from the Greek "up toward") to describe the two opposing fourth-dimensional directions—the 4-D equivalents of left and right, forwards and backwards, and up and down.
"It MUST be pointed out that there is a very steely concretized example of the Hinton tesseract that is still around, brought into existence by one of Hinton's sons. Sebastian Hinton (1887-1923) was a lawyer in Chicago who hit on an idea of bringing his father's work somewhat into the hands of the public-at-large--specifically, into the hands of children, who would be able to play and climb and swing in Charles' fourth dimensional idea.  He applied for a patent in 1920 which was granted in 1923."

"Evidently the term "monkey bars" didn't take hold until the 1950's, though Hinton referred to "monkey-like play" in his patent application. Also Charles had built a version of these of bamboo for the children to play on and help them understand the concept of moving through three dimensional space.  " *JF Ptak Science Books

Hinton was convicted of bigamy for marrying both Mary Ellen (daughter of Mary Everest Boole and George Boole, the founder of mathematical logic) and Maud Wheldon. He served a single day in prison sentence, then moved with Mary Ellen first to Japan (1886) and later to Princeton University in 1893 as an instructor in mathematics.
In 1897, he designed a gunpowder-powered baseball pitching machine for the Princeton baseball team's batting practice. According to one source it caused several injuries, and may have been in part responsible for Hinton's dismissal from Princeton that year. However, the machine was versatile, capable of variable speeds with an adjustable breech size, and firing curve balls by the use of two rubber-coated steel fingers at the muzzle of the pitcher. He successfully introduced the machine to the University of Minnesota, where Hinton worked as an assistant professor until 1900, when he resigned to move to the U.S. Naval Observatory in Washington, D.C.
At the end of his life, Hinton worked as an examiner of chemical patents for the United States Patent Office. He died unexpectedly of a cerebral hemorrhage on April 30, 1907. One source colorfully suggests that his death came when he died suddenly after being asked to give a toast to "female philosophers" at the Society of Philanthropic Inquiry meeting. *Wik

1892 Bessie Coleman (January 26, 1892 – April 30, 1926) was an early American civil aviator.  She was the first African-American to qualify for a pilot license. She went to France to learn to fly, and on 15 Jun 1921 was issued an international aviation license from the Fédération Aéronautique Internationale. She was sponsored by Robert Abbott, publisher of the Chicago Defender, the nation’s largest African-American weekly, and wealthy real estate dealer, Jessie Binga. She learned aerobatics to make a living at air shows, and became known as “Queen Bell.” In 1923, she was hospitalized her for three months after a crash. She returned to flying and had speaking engagements, and hoped to open a school for flyers. Her life ended at age 34 due to a flying accident.*TIS

1977 Charles Fox (17 March 1897 in London, England - 30 April 1977 in Montreal, Canada) Fox's main contributions were on hypergeometric functions, integral transforms, integral equations, the theory of statistical distributions, and the mathematics of navigation. In the theory of special functions he introduced an H-function with a formal definition. It is a type of generalisation of a hypergeometric function and related ideas can be found in the work of Salvatore Pincherle, Hjalmar Mellin, Bill Ferrar, Salomon Bochner and others. He wrote only one book An introduction to the calculus of variations (1950, 2nd edition 1963, reprinted 1987). *SAU

1989 Gottfried Maria Hugo Köthe (25 December 1905 in Graz; 30 April 1989 in Frankfurt) was an Austrian mathematician working in abstract algebra and functional analysis. Köthe received a fellowship to visit the University of Göttingen, where he attended the lectures of Emmy Noether and Bartel van der Waerden on the emerging subject of abstract algebra. He began working in ring theory and in 1930 published the Köthe conjecture stating that a sum of two left nil ideals in an arbitrary ring is a nil ideal. By a recommendation of Emmy Noether, he was appointed an assistant of Otto Toeplitz in Bonn University in 1929–1930. During this time he began transition to functional analysis. He continued scientific collaboration with Toeplitz for several years afterward. Köthe's best known work has been in the theory of topological vector spaces. In 1960, volume 1 of his seminal monograph Topologische lineare Räume was published (the second edition was translated into English in 1969). It was not until 1979 that volume 2 appeared, this time written in English. He also made contributions to the theory of lattices.*WIK

Gottfried Köthe, Elisabeth Hagemann, Otto Toeplitz, 1930

1940 Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. He is renowned for being the "prime architect" of higher algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.

Quillen was a Putnam Fellow in 1959.
Quillen retired at the end of 2006. He died from complications of Alzheimer's disease on April 30, 2011, aged 70, in Florida. *Wik

2016  Sir Harold Walter Kroto FRS (born Harold Walter Krotoschiner; 7 October 1939 – 30 April 2016) English chemist who shared (with Richard E. Smalley and Robert F. Curl, Jr.) the 1996 Nobel Prize for Chemistry for their joint discovery of the carbon compounds called fullerenes. These new forms of the element carbon contain 60 or more atoms arranged in closed shells. The number of carbon atoms in the shell can vary, and for this reason numerous new carbon structures have become known. Formerly, six crystalline forms of the element carbon were known, namely two kinds of graphite, two kinds of diamond, chaoit (1968) and carbon(VI) (1972). Fullerenes are formed when vaporised carbon condenses in an atmosphere of inert gas. The carbon clusters can then be analysed with mass spectrometry. *TIS

In 1991 After the C60 Fullerenes discovery and Nobel Prize the session of the House of Lords entertained a question from Lord Errol of Hale as to "What steps the government was taking to encourage the use of Fullerines in science and industry?" The question prompted questions on what was a Fullerene, what shape did it have and finally, this brilliant exchange:
Lord Campbell of Alloway: "My Lords, what does it do?"
Lord Reay: "My Lords, it is thought that it may have several possible uses... All that is speculation. It may turn out to have no uses at all."
Earl Russell: "My Lords, can one say that it does nothing in particular, and does it very well." *Siobhan Roberts, King of Infinite Space

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

## Monday 29 April 2024

### An Unusual Prime Series

I found this in an article in The American Mathematical Monthly  Vol. 1, No. 6, Jun., 1894.  The article is taken from a paper by J. W. NICHOLSON., President and Professor of Mathematics at Louisiana State University.

To keep it simple I will present a very small example of the professor's theorem.

Pick a prime number, p (I'll use five because it keeps things short and easy) .

Now take the sum of the squares of every  integer smaller than p and "voila", it is divisible by p

42 + 32 + 22 + 12 = 30, which is divisible by 5.

and it doesn't have to be a square, the same series using cubes  gives:
43 + 33 + 23 + 13 = 100, which is also divisible by 5.

In general, the first baby rule says for prime p, (p-1)n + (p-2) n + ... + 1n will be divisible by p as for any power n smaller than p.

Go ahead, try a few of your own.

Now to kick it up a little... let's add in any constant, c,  to the mix.
It is also true, that for prime p and n smaller than p:

(c+p-1)n+(c+p-2)n ...... + (c+0)n will also be divisible by p.

If I keep p = 5 and use c=2 the series would be :
(2+4)2+(2+3)2++(2+2)2+(2+1)2  + (2+0)2 =90  which is still divisible by 5.

Ok, and one to grow on:  you can use a constant multiplier in front of the p-x terms... for example using a multiplier of three in each case we have
(2+3x4)2+(2+3x3)2+(2+3x2)2+(2+3x1)2  + (2+0)2 =410 which is still divisible by 5.
According to the Professor, this works only if, we use a prime p, As pointed out in the comments, there are some primes for which this is not true (and some non primes for which it is true in at least one of the versions described.    I originally misstated  what he wrote.
No justification was given, and I could provide none, but the brilliant Joshua Zucker sent a few guidelines to assist:
The group of units mod p is cyclic.

So we are summing 1 through p-1, which for p odd, means we have 0 mod p.

Now, any odd power won't cause a problem because we can still pair x with -x to get a sum of 0.

I think we can deal with all the powers by viewing all the numbers as powers of some generator mod p, but I'm too lazy to work out the details.

Anyway, the point of the group being cyclic mod p is that your addition of a constant and multiplication by a constant leaves your numbers the same mod p, so the second half of things is pretty much doing nothing.

### On This Day in Math - April 29

Science is built up of facts, as a house is with stones.
But a collection of facts is no more a science
than a heap of stones is a house.

~Henri Poincare

The 119th day of the year; the largest amount of US money one can have in coins without being able to make change for a dollar is 119 cents. *Tanya Khovanova, Number Gossip

119 is the product of the first two primes ending with 7

119 is the sum of five consecutive primes (17 + 19 + 23 + 29 + 31).

119 is the order of the largest cyclic subgroups of the Monster group.

There are 119 prime numbers which gets displayed on a 12-hour digital clock.

119 is the smallest composite number, and the only year date, that is one less than a factorial.  The next will be 40319 = 8! - 1.  (students might examine the sequence of n! + 1 for patterns)

119 is a Perrin Number, A Fibonacci like sequence that begins with 3, 0, 2 and then each new value is the sum of the two digits before the last known, so it starts 3, 0, 2, 3, 2, 5, 5, 7, ...  The name is for French mathematician Francois Perrin who wrote about it in 1899,

EVENTS

1657 Christiaan Huygens published De Ratiociniis in Ludo Aleae [Reasoning in games of chance] on the calculus of probabilities, the first printed work on the subject.

John Arbuthnot  translated Huygens' "De ratiociniis in ludo aleae " in 1692 and extended it by adding a few further games of chance. This was the first work on probability published in English.

In 1699, the French Academy of Sciences held its first public meeting, in the Louvre. *TIS

1756 Benjamin Franklin was elected a Fellow of the Royal Society on April 29, 1756. Under the rules candidates had to be recommended in writing by three or more Fellows acquainted with him “either in person or by his Works,” the recommendation had to be approved by the Council, and the certificate publicly displayed at “ten several ordinary meetings” before balloting. Nothing more was required of foreign fellows. British (including colonial) fellows, however, had to pay an admission fee (five guineas after 1752) and a sum of £21 “for the use of the Society in lieu of Contributions,” or give bond for that amount. Only then was a British subject deemed to be a fellow and entitled to be registered in the Journal-Book and be included in the printed List of Fellows. To attend meetings and vote in elections British fellows had also to sign the obligation to “endeavor to promote the Good of the Royall Society … and to pursue the Ends for which the same was formed.” *Franklin Papers, Natl. Archives

1831 Wilhelm Eduard Weber is offered the position of full professor of Physics at Gottingen to fill the position of Tobias Mayer, partially on the recommendation of Gauss.

In December 1837, the Hanoverian government dismissed Weber, one of the Göttingen Seven(a group of seven liberal professors at University of Göttingen. In 1837, they protested against the annullment of the constitution of the Kingdom of Hanover by its new ruler, King Ernest Augustus, and refused to swear an oath to the king.),  from his post at the university for political reasons. Weber then travelled for a time, visiting England, among other countries, and became professor of physics in Leipzig from 1843 to 1849, when he was reinstated at Göttingen. One of his most important works, co-authored with Carl Friedrich Gauss and Carl Wolfgang Benjamin Goldschmidt, was Atlas des Erdmagnetismus: nach den Elementen der Theorie entworfen (Atlas of Geomagnetism: Designed according to the elements of the theory), a series of magnetic maps, and it was chiefly through his efforts that magnetic observatories were instituted. He studied magnetism with Gauss, and during 1864 published his Electrodynamic Proportional Measures containing a system of absolute measurements for electric currents, which forms the basis of those in use. Weber died in Göttingen, where he is buried in the same cemetery as Max Planck and Max Born.*Wik

Together with Gauss, he invented the magnetic telegraph in 1833, which connected the observatory with the institute for physics in Göttingen.

1832 Evariste Galois released from prison. On (1831)Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. For this, he was again arrested and this time sentenced to six months in prison for illegally wearing a uniform. He was released on April 29, 1832. During his imprisonment, he continued developing his mathematical ideas.*Wik (He will be shot on the morning of May 30, and die the next day, 1832)

The Galois memorial in the cemetery of Bourg-la-Reine. Évariste Galois was buried in a common grave and the exact location is unknown.

1854 Lincoln University, the ﬁrst university for Blacks, is incorporated. Lincoln University of the Commonwealth of Pennsylvania was chartered in April 1854 as Ashmun Institute. As Horace Mann Bond, '23, the eighth president of Lincoln University, so eloquently cites in the opening chapter of his book, Education for Freedom, this was "the first institution found anywhere in the world to provide a higher education in the arts and sciences for male youth of African descent." The story of Lincoln University goes back to the early years of the 19th century and to the ancestors of its founder, John Miller Dickey, and his wife, Sarah Emlen Cresson. The Institute was re-named Lincoln University in 1866 after President Abraham Lincoln. *Lincoln University web site

Lincoln University has numerous notable alumni, including US Supreme Court Justice Thurgood Marshall; Harlem Renaissance poet Langston Hughes; Medal of Honor recipient and pioneering African-American editor Christian Fleetwood; former US Ambassador to Botswana, Horace Dawson; civil rights activist Frederick D. Alexander; the first president of Nigeria, and Nnamdi Azikiwe; the first president of Ghana

Student Union, Lincoln University.

1901 Math Blunder succeeds, "But a more recent, a veritably shocking, example is at hand. On April 29, 1901, a Mr. Israel Euclid Eabinovitch submitted to the Board of University Studies of the Johns Hopkins University, in conformity with the requirements for the degree of doctor of philosophy, a dissertation in which, after an introduction full of the most palpable blunders, he proceeds to persuade himself that he proves Euclid's parallel postulate by using the worn-out device of attacking it from space of three dimensions, a device already squeezed dry and discarded by the very creator of non-Euclidean geometry, Janos Bolyai. And his dissertation was accepted by the referees. (Science Monthly, Vol 67, page 642)

In 1878, “a monument, in memory of the great physicist, Alessandro Volta, was unveiled at Pavia. Most of the Italian Universities, and several foreign scientific societies had sent deputies to Pavia University for this event. The monument is a masterpiece of the sculptor Tantardini of Milan. The ceremony of unveiling was followed by a dignified celebration at the University, and upon that occasion the following gentlemen were elected honorary doctors of the scientific faculty: Professors Clerk Maxwell (Cambridge) and Sir W. Thomson (Glasgow); M. Dumas (Paris), Dr. W. E. Weber (Leipzig); Professors Bunsen (Heidelberg) and Helmholtz (Berlin), Dr. F. H. Neumann (Koenigsberg), and Dr. P. Riess (Berlin).”*TIS

1925 The ﬁrst woman, F. R. Sabin, is elected to the National Academy of Sciences (Kane, p. 945). *VFR She was a histology professor at Johns Hopkins University.

Julia Robinson was the first woman to serve as president of the American Mathematical Society, and was also the first woman mathematician to be elected to the U.S. National Academy of Sciences, in 1975.

1931 Robert Lee Moore elected to the National Academy of Sciences. *VFR

BIRTHS

1667 John Arbuthnot (baptised April 29, 1667 – February 27, 1735), fellow of the Royal College of Physicians. In 1710, his paper “An argument for divine providence taken form the constant regularity observ’s in the bith of both sexes” gave the ﬁrst example of statistical inference. In his day he was famous for his political satires, from which we still know the character John Bull. *VFR
He inspired both Jonathan Swift's Gulliver's Travels book III and Alexander Pope's Peri Bathous, Or the Art of Sinking in Poetry, Memoirs of Martin Scriblerus,m (Wikipedia) He also translated Huygens' "De ratiociniis in ludo aleae " in 1692 and extended it by adding a few further games of chance. This was the first work on probability published in English.*SAU

Also known for his statistical analysis of the male and female birth rates in England. This is probably the first use of probability in a social statistical analysis and the earliest case of a statistical significance test.  *RMAT

"He also contributed to the development of archaeology and history with his papers on Tables of Grecian, Roman, and Jewish measures, weights and coins; reduced to the English standard.” Title page from my 1705, first edition copy of this publication by John Arbuthnot." *coffeefueled

1850 William Edward Story (April 29, 1850 in Boston, Massachusetts, U.S. - April 10, 1930 in Worcester, Massachusetts, U.S.) He taught at Johns Hopkins with Sylvester and then moved on to Clark University which was, during the early 1890’s, the strongest mathematics department in the country. In the 1890’s he edited the short lived Mathematical Reviews.*VFR

1854 Jules Henri Poincare (29 April 1854 – 17 July 1912) born in Nancy, France. He did important work in function theory, alge­braic geometry, number theory, algebra, celestial mechanics, diﬀerential equations, mathematical physics, algebraic topology, and philosophy of mathematics. There may never be another universal mathematician like Poincar´e. *VFR His Poincaré Conjecture holds that if any loop in a given three-dimensional space can be shrunk to a point, the space is equivalent to a sphere. Its proof remains an unsolved problem in topology. He influenced cosmogony, relativity, and topology. In applied mathematics he also studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, and cosmology. He is often described as the last universalist in mathematics. He studied the three-body-problem in celestial mechanics, and theories of light and electromagnetic waves. He was a co-discoverer (with Albert Einstein and Hendrik Lorentz) of the special theory of relativity. *TIS

1872 Forest Ray Moulton (29 Apr 1872 (in a log cabin near the small town of Leroy, Michigan); 7 Dec 1952 at age 80) American astronomer who collaborated with Thomas Chamberlin in advancing the planetesimal theory of the origin of the solar system (1904). They suggested filaments of matter were ejected when a star passed close to the Sun, which cooled into tiny solid fragments, “planetesimals.” Over a very long period, grains collided and stuck together. Continued accretion created pebbles, boulders, and eventually larger bodies whose gravitational force of attraction accelerated the formation of protoplanets. (This formation by accretion is still accepted, but not the stellar origin of the planetesimals.) Moulton was first to suggest that the smaller satellites of Jupiter discovered by Nicholson and others in the early 20th century were captured asteroids, now widely accepted. *TIS

1894 Marietta Blau (29 April 1894 – 27 January 1970) was an Austrian physicist credited with developing photographic nuclear emulsions that were usefully able to image and accurately measure high-energy nuclear particles and events, significantly advancing the field of particle physics in her time. For this, she was awarded the Lieben Prize by the Austrian Academy of Sciences. As a Jew, she was forced to flee Austria when Nazi Germany annexed it in 1938, eventually making her way to the United States. She was nominated for Nobel Prizes in both physics and chemistry for her work, but did not win. After her return to Austria, she won the Erwin Schrödinger Prize from the Austrian Academy of Sciences. *Wik

Austrian nuclear physicist who began as a strong student in mathematics and physics at school, and studied physics at university, where she wrote her thesis on the absorption of gamma rays (1919). At first, she took a job (1921) with a manufacturer of x-ray tubes in Berlin. By 1923, she progressed to researching radioactivity with the Institut für Radiumforschung back in Vienna. There she developed the photographic emulsion technique for the study of nuclear disintegration caused by cosmic rays, and contributed to development of photomultiplier tubes. Blau was first to use nuclear emulsions to detect neutrons by observing recoil protons. Albert Einstein recognized her as a very capable experimental physicist, and after 1938 when she fled the rise of the Nazis, Einstein helped her career continue in Mexico City and then the U.S. *TIS

1906 Eugène Ehrhart (29 April 1906 Guebwiller – 17 January 2000 Strasbourg) was a French mathematician who introduced Ehrhart polynomials in the 1960s. Ehrhart received his high school diploma at the age of 22. He was a mathematics teacher in several high schools, and did mathematics research on his own time. He started publishing in mathematics in his 40s, and finished his PhD thesis at the age of 60. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem. *Wik

1926 Vera Nikolaevna Maslennikova (29 April 1926, Priluki, Russia - 14 August 2000) Gelfond supervised her diploma work at Moscow and Sobolev directed her Ph.D. at the Steklov Mathematical Institute. She has published more than 80 papers in the theory of partial diﬀerential equations, the mathematical hydrodynamics of rotating ﬂuids, and in function spaces.*VFR She has worked in the field of partial differential equations, the mathematical hydrodynamics of rotating fluids, and in function spaces, having published more than one hundred and forty research papers. *Wik

1928 Laszlo Belady,( April 29, 1928 in Budapest - November 6, 2021) creator of the Belady algorithm (used in optimizing the performance of computers), is born. Belady worked at IBM for 23 years in software engineering before joining the Mitsubishi Electronics Research Laboratory in the mid-1980s. He wins numerous awards, including the J.D. Warnier Prize for Excellence in Information and an IEEE fellowship. *CHM

1930 Yuan Wang (29 April 1930 in Lanhsi, Zhejiang province, China - 14 May 2021) )Most of Wang Yuan's research has been in the area of number theory. He looked at sieve methods and applied them to the Goldbach Conjecture. He also applied circle methods to the Goldbach Conjecture. In 1956 he published (in Chinese) On the representation of large even integer as a sum of a prime and a product of at most 4 primes in which he assumed the truth of the Riemann hypothesis and with that assumption proved that every large even integer is the sum of a prime and of a product of at most 4 primes. He also proved that there are infinitely many primes p such that p + 2 is a product of at most 4 primes. In 1957 Wang Yuan published four papers: On sieve methods and some of their applications; On some properties of integral valued polynomials; On the representation of large even number as a sum of two almost-primes; and On sieve methods and some of the related problems.*SAU

1936 Volker Strassen
(April 29, 1936 - ) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz. Strassen began his researches as a probabilist; his 1964 paper An Invariance Principle for the Law of the Iterated Logarithm defined a functional form of the law of the iterated logarithm, showing a form of scale invariance in random walks. This result, now known as Strassen's invariance principle or as Strassen's law of the iterated logarithm, has been highly cited and led to a 1966 presentation at the International Congress of Mathematicians.
In 1969, Strassen shifted his research efforts towards the analysis of algorithms with a paper on Gaussian elimination, introducing Strassen's algorithm, the first algorithm for performing matrix multiplication faster than the O(n3) time bound that would result from a naive algorithm. In the same paper he also presented an asymptotically-fast algorithm to perform matrix inversion, based on the fast matrix multiplication algorithm. This result was an important theoretical breakthrough, leading to much additional research on fast matrix multiplication, and despite later theoretical improvements it remains a practical method for multiplication of dense matrices of moderate to large sizes. In 1971 Strassen published another paper together with Arnold Schönhage on asymptotically-fast integer multiplication based on the fast Fourier transform; see the Schönhage–Strassen algorithm. Strassen is also known for his 1977 work with Robert M. Solovay on the Solovay–Strassen primality test, the first method to show that testing whether a number is prime can be performed in randomized polynomial time and one of the first results to show the power of randomized algorithms more generally.*Wik

DEATHS

1633 Francis Godwin, an English cleric, was buried Apr. 29, 1633, at about age 71. As Bishop of Hereford, Godwin published a number of mainstream theological tracts, but he left behind at his death a manuscript about a fantasy voyage undertaken by a Spaniard, Domingo Gonsales. While at the island of St. Helena, Gonsales had discovered a species of wild swans, which he called “gansa” and which he discovered could be trained to fly in harness and carry a load. He rigged up an aerial chariot, hooked 25 gansas to it, and off he flew on a brief test flight. He was picked up on St. Helena by a ship from the Indies, and he persuaded the captain to make room for the swans and the chariot. When the ship was attacked by pirates off the Canaries, Gonsales climbed in his gondola, hooked up the gansas, and escaped. But the gansas had minds of their own, and kept flying up and up, until they ultimately escaped the force of the earth’s gravity (an interesting notion, because in 1630, no one believed in a force of gravity). Eventually, they reached the Moon, where Gonsales discovered that the Moon is inhabited by a race of peaceful giants, who swooned when they heard the name “Jesus” and converted instantly to Christianity.

It was a wonderful tale, and it could have been buried with the bishop, but it was salvaged and published in 1638 as The Man in the Moone: or, A Discourse of a Voyage Thither, with a wonderful frontispiece showing Gonsales in his gansa-powered flying machine. This first edition is very scarce (four known copies), but it was republished in 1657, and also translated into French in 1648; we were very fortunate to acquire a 1666 edition of the French translation just last year. It includes the charming illustration of the “little goose coupe”, as well as a depiction of the language of the Lunarians, which is sung without words (the second tune is lunar for “Gonsales”).

If you look closely at the publication date, you have the rare chance to see the first 7 Roman numerals in ascending order, starting from the right. This is one of the reasons why the year 1666 was feared as an annus mirabilis by the English, a fear not disproved by the arrival of the bubonic plaque and the Great Fire of London. *LH

1713 Francis Hauksbee the elder (baptized on 27 May 1660 in Colchester–buried in St Dunstan's-in-the-West, London on 29 April 1713.), also known as Francis Hawksbee, was an 18th-century English scientist best known for his work on electricity and electrostatic repulsion.
Initially apprenticed in 1678 to his elder brother as a draper, Hauksbee became Isaac Newton’s lab assistant. In 1703 he was appointed curator, instrument maker and experimentalist of the Royal Society by Newton, who had recently become president of the society and wished to resurrect the Royal Society’s weekly demonstrations.
Until 1705, most of these experiments were air pump experiments of a mundane nature, but Hauksbee then turned to investigating the luminosity of mercury which was known to emit a glow under barometric vacuum conditions.
By 1705, Hauksbee had discovered that if he placed a small amount of mercury in the glass of his modified version of Otto von Guericke's generator, evacuated the air from it to create a mild vacuum and rubbed the ball in order to build up a charge, a glow was visible if he placed his hand on the outside of the ball. This glow was bright enough to read by. It seemed to be similar to St. Elmo's Fire. This effect later became the basis of the gas-discharge lamp, which led to neon lighting and mercury vapor lamps. In 1706 he produced an 'Influence machine' to generate this effect. He was elected a Fellow of the Royal Society the same year.

Hauksbee continued to experiment with electricity, making numerous observations and developing machines to generate and demonstrate various electrical phenomena. In 1709 he published Physico-Mechanical Experiments on Various Subjects which summarized much of his scientific work.
In 1708, Hauksbee independently discovered Charles' law of gases, which states that, for a given mass of gas at a constant pressure, the volume of the gas is proportional to its temperature.
The Royal Society Hauksbee Awards, awarded in 2010, were given by the Royal Society to the “unsung heroes of science, technology, engineering and mathematics.” *Wik

1862 John Edward Campbell (27 May 1862, Lisburn, Ireland – 1 October 1924, Oxford, Oxfordshire, England) is remembered for the Campbell-Baker-Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras. *SAU His 1903 book, Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups, popularized the ideas of Sophus Lie among British mathematicians.
He was elected a Fellow of the Royal Society in 1905, and served as President of the London Mathematical Society from 1918 to 1920. *Wik  *Renaissance Mathematicus

1864 Charles-Julien Brianchon (19 Dec 1783, 29 Apr 1864 at age 80) French mathematician who published a geometrical theorem (named as Brianchon's theorem) while a student (1806). He showed that in any hexagon formed of six tangents to a conic, the three diagonals meet at a point. (Conics include circles, ellipses, parabolas, and hyperbolas.) In fact, this theorem is simply the dual of Pascal's theorem which was proved in 1639. After graduation, Brianchon became a lieutenant in artillery fighting in Napoleon's army until he left active service in 1813 due to ill health. His last work in mathematics made the first use of the term "nine-point circle." By 1823, Brianchon's interests turned to teaching and to chemistry. *TIS

1872 Jean-Marie-Constant Duhamel (5 Feb 1797, 29 Apr 1872 at age 75) French mathematician and physicist who proposed a theory dealing with the transmission of heat in crystal structures based on the work of the French mathematicians Jean-Baptiste-Joseph Fourier and Siméon-Denis Poisson. *TIS

1894 Giuseppe Battaglini (11 Jan 1826 in Naples, Kingdom of Naples and Sicily (now Italy) - 29 Apr 1894 in Naples, Italy ) Some of Battaglini's results have proved significant. For example, in his doctoral dissertation of 1868, Klein introduced a classification scheme for second-degree line complexes based on Battaglini's earlier work. However, his main importance is his modern approach to mathematics which played a major role in invigorating the Italian university system, particularly in his efforts to bring the non-Euclidean geometry of Lobachevsky and Bolyai to the Italian speaking world. Jules Hoüel played a similar role for non-Euclidean geometry in the French speaking world and the correspondence between the two (see [6]) provides a vivid picture of the reactions of both the French and the Italian mathematical communities against the non-Euclidean geometries. Battaglini and Hoüel also exchanged ideas relating to mathematical education in various European countries. In particular they debated the use of Euclid's Elements as a textbook for teaching elementary geometry in schools. *SAU

1916 – Jørgen Pedersen Gram (June 27, 1850 – April 29, 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark.
Important papers of his include On series expansions determined by the methods of least squares, and Investigations of the number of primes less than a given number. The mathematical method that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. The Gramian matrix is also named after him.
For number theorists his main fame is the series for the Riemann zeta function (the leading function in Riemann's exact prime-counting function). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function.
Gram was the first mathematician to provide a systematic theory of the development of skew frequency curves, showing that the normal symmetric Gaussian error curve was but one special case of a more general class of frequency curves.
He died after being struck by a bicycle.*Wik

1951 Ludwig Josef Johann Wittgenstein (26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947. In his lifetime he published just one book review, one article, a children's dictionary, and the 75-page Tractatus Logico-Philosophicus (1921). In 1999 his posthumously published Philosophical Investigations (1953) was ranked as the most important book of 20th-century philosophy, standing out as "...the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations". Bertrand Russell described him as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating". *Wik He died three days after his birthday. He is buried in a cemetery off Huntington Road in Cambridge, UK.

1966 William Henry Eccles FRS (23 August 1875 – 29 April 1966) British physicist who pioneered in the development of radio communication. He was an early proponent of Oliver Heaviside's theory that an upper layer of the atmosphere reflects radio waves, thus enabling their transmission over long distances. He also suggested in 1912 that solar radiation accounted for the differences in wave propagation during the day and night. He experimented with detectors and amplifiers for radio reception, coined the term "diode," and studied atmospheric disturbances of radio reception. After WW I, he made many contributions to electronic circuit development, including the Eccles-Jordan "flip-flop" patented in 1918 and used in binary counters (working with F.W. Jordan).* *TIS

1970 Paul Finsler (born 11 April 1894, in Heilbronn, Germany,- 29 April 1970 in Zurich, Switzerland)Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory. He joined the faculty of the University of Zurich in 1927, and was promoted to ordinary professor there in 1944.

Finsler's thesis work concerned differential geometry, and Finsler spaces were named after him by Élie Cartan in 1934. The Hadwiger–Finsler inequality, a relation between the side lengths and area of a triangle in the Euclidean plane, is named after Finsler and his co-author Hugo Hadwiger. Finsler is also known for his work on the foundations of mathematics, developing a non-well-founded set theory with which he hoped to resolve the contradictions implied by Russell's paradox.
In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler. It states that if a triangle in the plane has side lengths ab and c and area A, then

$a^{2} + b^{2} + c^{2} \geq (a - b)^{2} + (b - c)^{2} + (c - a)^{2} + 4 \sqrt{3} A \quad \mbox{(HF)}.$

Weitzenböck's inequality is a straightforward corollary of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths ab and c and area A, then

$a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} A \quad \mbox{(W)}.$

Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. a = b = c.
*Wik

2008 Mary Golda Ross (August 9, 1908 – April 29, 2008) was the first known Native American female engineer, and the first female engineer in the history of Lockheed. She was one of the 40 founding engineers of the renowned and highly secretive Skunk Works project at Lockheed Corporation. She worked at Lockheed from 1942 until her retirement in 1973, where she was best remembered for her work on aerospace design – including the Agena Rocket program – as well as numerous "design concepts for interplanetary space travel, crewed and uncrewed Earth-orbiting flights, the earliest studies of orbiting satellites for both defense and civilian purposes." In 2018, she was chosen to be depicted on the 2019 Native American \$1 Coin by the U.S. Mint celebrating American Indians in the space program. *Wik

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