## 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

(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