## Wednesday, 28 April 2021

### On This Day in Math - April 28

One of the principal objects of theoretical research
in my department of knowledge
is to find the point of view from which
the subject appears in its greatest simplicity.
Willard Gibbs (1839 - 1903)

The 118th day of the year. 118 is the smallest n such that the range n, n + 1, ... 4n/3 contains at least one prime from each of these forms: 4x + 1, 4x - 1, 6x + 1 and 6x - 1.

There are four unique partitions of 118 into three integers that all have the same product.  No smaller example exists.  14 × 50 × 54 = 15 × 40 × 63 = 18 × 30 × 70 = 21 × 25 × 72 = 37800.

And there are 118 partitions of the number 16.

59, one factor of 118, has 118 divisors itself.

EVENTS

1664 Trinity College, Cambridge awards a scholarship to Isaac Newton to study for his Master's Degree, thus ending his period as a lowly sizar earning his tuition by cleaning up after wealthier students. Within months his formal education would be put on hold as the college closed under the assault of the plague.

1673 Leeuwenhoeck writes his first letter to the Royal Society, which would be published the next month, May 19, in Philosophical Transactions number 94, "A Specimen of Some Observations Made by a Microscope, Contrived by M. Leewenhoeck in Holland, Lately Communicated by Dr. Regnerus de Graaf." Constantijn Huygens, who lived not far from Delft, visited Leeuwenhoek and read the letter. A week before Leeuwenhoek sent it, Huygens sent his own letter to Robert Hooke that acted as a cover letter and recommendation similar to de Graaf's letter in April. Over the rest of Leeuwenhoeck's life, the Society would publish 116 articles containing excerpts from 113 letters. *lensonleeuwenhoek

1686 Newton shows the handwritten copy of his Principia to the Royal Society. *VFR
28 April 1686 "Dr. Vincent presented a manuscript treatise entitled Philosophiae Naturalis principia mathematica, and dedicated to the Society by Mr. Isaac Newton,..." Minutes of the RS written by Halley clerk to the Society. (It was actually only the manuscript of Book I) *Thony Christie

1693 Leibniz, in a letter to L’Hopital, explains his discovery of determinants. This work was ﬁfty years before that of Cramer who was the real driving force in the development of determinants. Leibniz’s work had no inﬂuence because it was not published until 1850 in his Mathematische Schriften. [Smith, Source Book, p. 267] *VFR
Leibniz was convinced that good mathematical notation was the key to progress so he experimented with different notation for coefficient systems. His unpublished manuscripts contain more than 50 different ways of writing coefficient systems which he worked on during a period of 50 years beginning in 1678. Only two publications (1700 and 1710) contain results on coefficient systems and these use the same notation as in his letter to de l'Hôpital mentioned above.
Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant. He proved various results on resultants including what is essentially Cramer's rule. He also knew that a determinant could be expanded using any column - what is now called the Laplace expansion. As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory. In the 1730's Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death. It contains the first published results on determinants proving Cramer's rule for 2 X 2 and 3X 3 systems and indicating how the 4 X 4 case would work. Cramer gave the general rule for n X  n systems in his book Introduction to the analysis of algebraic curves (1750). It arose out of a desire to find the equation of a plane curve passing through a number of given points. The rule appears in an Appendix to the book but no proof is given] *SAU (edited and corrected with suggestions by Dave Renfro)
Dave adds:   a 715 page book (xxiii + 680 + xii pages), which is freely available on the internet. Cramer's rule itself appears in Appendix 2 (pp. 657-676). Cramer's book itself was motivated by Newton's work in classifying cubic curves, and I believe he was one of three mathematicians that devoted an extensive study to Newton's classification in the 1700s. (I don't remember who the other two were, but I believe one of them was Euler.) There is an excellent annotated and translation of Newton's work published in 1860 and freely available on the internet:

"Sir Isaac Newton's Enumeration of Lines of the Third Order, Generation of
Curves by Shadows, Organic Description of Curves, and Construction of
Equations by Curves", Translated from the Latin, with notes and examples,
by C.R.M. Talbot, 1860.

http://name.umdl.umich.edu/ABQ9451.0001.001
Thanks again to Dave for the corrections.

1817 Gauss wrote the astronomer H. W. M. Olbers, “I am becoming more and more convinced that the necessity of our [Euclidean] geometry cannot be proved, at least not by human intellect nor for the human intellect.” [G. E. Martin, Foundations of Geometry and the Non-Euclidean Plane, p. 306] *VFR

1983 Greece issued a stamp portraying Archimedes and his Hydrostatic Principle

1897 In a letter to Fuchs, Dedekind expressed skepticism of a tale about Gauss attempting to light his pipe with a copy of his DA
Schering in Gottingen in response to a note from Fuchs that he had found materials related to Guass' Disquisitiones Arithmetica in the papers of Dirichlet had described a story that he had shared with Kronecker a decade before,
"The piece of Guass's Disquisitiones Arithmeticiae, which is found among Dirichlet's papers, is probably that portion which, as Dirichlet told me himself, he saved from the hand of Gauss when the latter lit his pipe with his manuscript of the Disquisitiones Arithmeticae on the day of his doctoral jubilee."
Dedekind reasoned, if Guass had saved the paper for fifty years he obviously valued it, and that if the anecdote were true, Dirichlet surely would have shared it with him as well.
*Uta Merzbach, An Early Version of Gauss's Disquisitiones Arithmeticae, Mathematical Perspectives, 1981

2004 At 11:50 AM a paper was submitted electronically to the American Mathematical Monthly which purports to be the shortest journal entry ever, essentially two words," n2 + 2 can". After some correspondence back and forth, (the journal suggested, "a line or two of explanatin might help") the paper was accepted as a "filler" in the January 2005 issue. *wfnmc.org

2012 Mountain View, Ca—January 19, 2012—
The Computer History Museum (CHM), the world’s leading institution exploring the history of computing and its ongoing impact on society, today announced its 2012 Fellow Award honorees: Edward A. Feigenbaum, pioneer of artificial intelligence and expert systems; Steve Furber and Sophie Wilson, chief architects of the ARM processor architecture; and Fernando J. Corbató, pioneer of timesharing and the Multics operating system. The four Fellows will be inducted into the Museum’s Hall of Fellows on Saturday, April 28, 2012, at a formal ceremony where Silicon Valley insiders, technology leaders, and Museum supporters will gather to celebrate the accomplishments of the Fellows and their impact on society. This year’s celebration commemorates the 25th Anniversary of the Fellow Awards and will reunite pioneers from more than two decades. *CHM

BIRTHS
1765 Sylvestre François Lacroix (April 28, 1765, Paris – May 24, 1843, Paris) was a French mathematician. He displayed a particular talent for mathematics, calculating the motions of the planets by the age of 14. In 1782 at the age of 17 he became an instructor in mathematics at the École Gardes de Marine in Rochefort, France. He returned to Paris and taught courses in astronomy and mathematics at the Lycée. In 1787 he was the co-winner of that year's Grand Prix of the French Académie des Sciences, but was never awarded the prize. The same year the Lycée was abolished and he again moved to the provinces.
In Besançon he taught course in mathematics, physics, and chemistry at the École d'Artillerie. In 1793 he became examiner of the Artillery Corps, replacing Pierre-Simon Laplace in the post. By 1794 he was aiding his old instructor, Gaspard Monge, in creating material for a course on descriptive geometry. In 1799 he was appointed professor at the École Polytechnique. Lacroix produced most of his texts for the sake of improving his courses. The same year he was voted into the newly formed Institut National des Sciences et des Arts. In 1812 he began teaching at the Collège de France, and was appointed chair of mathematics in 1815.
During his career he produced a number of important textbooks in mathematics. Translations of these books into the English language were used in British universities, and the books remained in circulation for nearly 50 years. In 1812 Babbage set up The Analytical Society for the translation of Differential and Integral Calculus and the book was translated into English in 1816 by George Peacock. *Wik He coined the term “analytic geometry.” *VFR

1773 Robert Woodhouse (28 April 1773 – 23 December 1827) was an English mathematician. He was born at Norwich and educated at Caius College, Cambridge, (BA 1795) of which society he was subsequently a fellow. He was elected a Fellow of the Royal Society in December 1802.
His earliest work, entitled the Principles of Analytical Calculation, was published at Cambridge in 1803. In this he explained the differential notation and strongly pressed the employment of it; but he severely criticized the methods used by continental writers, and their constant assumption of non-evident principles. This was followed in 1809 by a trigonometry (plane and spherical), and in 1810 by a historical treatise on the calculus of variations and isoperimetrical problems. He next produced an astronomy; of which the first book (usually bound in two volumes), on practical and descriptive astronomy, was issued in 1812, and the second book, containing an account of the treatment of physical astronomy by Pierre-Simon Laplace and other continental writers, was issued in 1818.
He became the Lucasian Professor of Mathematics in 1820, and subsequently the Plumian professor in the university. As Plumian Professor he was responsible for installing and adjusting the transit instruments and clocks at the Cambridge Observatory. He held that position until his death in 1827.
On his death in Cambridge he was buried in Caius College Chapel.*Wik He was interested in the “metaphysics of the calculus,” i.e., questions such as the proper theoretical foundations of the calculus, the role of geometric and analytic methods, and the importance of notation. *VFR

1774 Francis Baily (28 April 1774 – 30 August 1844) was an English astronomer. He is most famous for his observations of 'Baily's beads' during an eclipse of the Sun. Bailey was also a major figure in the early history of the Royal Astronomical Society, as one of the founders and president four times.
Baily was born at Newbury in Berkshire in 1774 to Richard Baily. After a tour in the unsettled parts of North America in 1796–1797, his journal of which was edited by Augustus de Morgan in 1856, Baily entered the London Stock Exchange in 1799. The successive publication of Tables for the Purchasing and Renewing of Leases (1802), of The Doctrine of Interest and Annuities (1808), and The Doctrine of Life-Annuities and Assurances (1810), earned him a high reputation as a writer on life-contingencies; he amassed a fortune through diligence and integrity and retired from business in 1825, to devote himself wholly to astronomy.
His observations of "Baily's Beads", during an annular eclipse of the sun on 15 May 1836, at Inch Bonney in Roxburghshire, started the modern series of eclipse expeditions. The phenomenon, which depends upon the irregular shape of the moon's limb, was so vividly described by him as to attract an unprecedented amount of attention to the total eclipse of 8 July 1842, observed by Baily himself at Pavia. *Wik

1831 Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical discipline. His name is known in Graph theory mainly for Tait's conjecture.*Wik (His conjecture was proved wrong by counterexample in 1946 by W. T. Tutte. The problem is related to the four color theorem.) He helped develop quaternions, an advanced algebra that gave rise to vector analysis and was instrumental in the development of modern mathematical physics. *TIS
Below is The First Seven Orders of Knottiness"-table compiled by P.G. Tait in 1884 with a big hat-tip to Ben Gross@bhgross144 .

1854 Phoebe Sarah Hertha Ayrton (28 April 1854 – 23 August 1923), was a British engineer, mathematician, physicist, and inventor. Known in adult life as Hertha Ayrton, born Phoebe Sarah Marks, she was awarded the Hughes Medal by the Royal Society for her work on electric arcs and ripples in sand and water.
In 1880, Ayrton passed the Mathematical Tripos, but Cambridge did not grant her an academic degree because, at the time, Cambridge gave only certificates and not full degrees to women. Ayrton passed an external examination at the University of London, which awarded her a Bachelor of Science degree in 1881.
In 1899, she was the first woman ever to read her own paper before the Institution of Electrical Engineers (IEE). Her paper was entitled "The Hissing of the Electric Arc". Shortly thereafter, Ayrton was elected the first female member of the IEE; the next woman to be admitted to the IEE was in 1958. She petitioned to present a paper before the Royal Society but was not allowed because of her sex and "The Mechanism of the Electric Arc" was read by John Perry in her stead in 1901. Ayrton was also the first woman to win a prize from the Society, the Hughes Medal, awarded to her in 1906 in honour of her research on the motion of ripples in sand and water and her work on the electric arc. By the late nineteenth century, Ayrton's work in the field of electrical engineering was recognised more widely, domestically and internationally. At the International Congress of Women held in London in 1899, she presided over the physical science section. Ayrton also spoke at the International Electrical Congress in Paris in 1900. Her success there led the British Association for the Advancement of Science to allow women to serve on general and sectional committees. *Wik

1868 Georgy Fedoseevich Voronoy (also voronoi)(28 April 1868 – 20 November 1908) introduced what are today called Voronoi diagrams or Voronoi tessellations. Today they have wide applications to the analysis of spatially distributed data, so have become important in topics such as geophysics and meteorology. Although known under different names, the notion occurs in condensed matter physics, and in the study of Lie groups. (Two dimensional diagrams of Voronoi type were considered as early at 1644 by René Descartes and were used by Dirichlet (1850) in the investigation of positive quadratic forms. They were also studied by Voronoi (1907), who extended the investigation of Voronoi diagrams to higher dimensions. They find widespread applications in areas such as computer graphics, epidemiology, geophysics, and meteorology. A particularly notable use of a Voronoi diagram was the analysis of the 1854 cholera epidemic in London, in which physician John Snow determined a strong correlation of deaths with proximity to a particular (and infected) water pump on Broad Street. *Mathworld)

1900 Jan Hendrik Oort (28 April 1900 – 5 November 1992) was a Dutch astronomer who made significant contributions to the understanding of the Milky Way and who was a pioneer in the field of radio astronomy. His New York Times obituary called him “one of the century's foremost explorers of the universe;” the European Space Agency website describes him as, “one of the greatest astronomers of the 20th century,” and states that he “revolutionised astronomy through his ground-breaking discoveries.” In 1955, Oort’s name appeared in Life Magazine’s list of the 100 most famous living people. He has been described as “putting the Netherlands in the forefront of postwar astronomy.”

Oort determined that the Milky Way rotates and overturned the idea that the Sun was at its center. He also postulated the existence of the mysterious invisible dark matter in 1932, which is believed to make up roughly 84.5% of the total matter in the Universe and whose gravitational pull causes “the clustering of stars into galaxies and galaxies into connecting strings of galaxies.” He discovered the galactic halo, a group of stars orbiting the Milky Way but outside the main disk. Additionally Oort is responsible for a number of important insights about comets, including the realization that their orbits “implied there was a lot more solar system than the region occupied by the planets.”

The Oort cloud, the Oort constants, and the Asteroid, 1691 Oort, were all named after him. *Wik

1906 Kurt Godel (April 28, 1906 – January 14, 1978) Austrian-born US mathematician, logician, and author of Gödel's proof. He is best known for his proof of Gödel's Incompleteness Theorems (1931) He proved fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. This ended a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis.*TIS

1906 Richard Rado FRS(28 April 1906 – 23 December 1989) was a Jewish German mathematician. He earned two Ph.D.s: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He made contributions in combinatorics and graph theory. He wrote 18 papers with Paul Erdős. In 1964, he discovered the Rado graph (The Rado graph contains all finite and countably infinite graphs as induced subgraphs..)
In 1972, he was awarded the Senior Berwick Prize*Wik

1923 Fritz Joseph Ursell FRS (28 April 1923 – 11 May 2012) was a British mathematician noted for his contributions to fluid mechanics, especially in the area of wave-structure interactions.[5] He held the Beyer Chair of Applied Mathematics at the University of Manchester from 1961–1990, was elected Fellow of the Royal Society in 1972 and retired in 1990.
Ursell came to England as a refugee in 1937 from Germany. From 1941 to 1943 he studied at Trinity College, Cambridge, graduating with a bachelor degree in mathematics. *Wik

DEATHS
1843 William Wallace (23 September 1768, Dysart—28 April 1843, Edinburgh) worked on geometry and discovered the (so-called)
Simson line of a triangle.*SAU In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace.*Wik
Mary Sommerville was one of his students.  He succeeded John Playfair as Math Chair in Edinburgh. He also invented a complicated type of pantograph called the eidograph.

1903 Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist and chemist known for contributions to vector analysis and as one of the founders of physical chemistry. In 1863, He was awarded Yale University's first engineering doctorate degree. His major work was in developing thermodynamic theory, which brought physical chemistry from an empirical inquiry to a deductive science. In 1873, he published two papers concerning the fundamental nature of entropy of a system, and established the "thermodynamic surface," a geometrical and graphical method for the analysis of the thermodynamic properties of substances. His famous On the Equilibrium of Homogeneous Substances, published in 1876, established the use of "chemical potential," now an important concept in physical chemistry. *TIS
He is buried at the  Grove Street Cemetery in New Haven Connecticut, USA.

1946 Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 – April 28, 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, which was part of his PhD thesis The Theory of Speculation, (published 1900).
His thesis, which discussed the use of Brownian motion to evaluate stock options, is historically the first paper to use advanced mathematics in the study of finance. Thus, Bachelier is considered a pioneer in the study of financial mathematics and stochastic processes. *Wik Bachelier is now recognised internationally as the father of financial mathematics, but this fame, which he so justly deserved, was a long time coming. The Bachelier Society, named in his honour, is the world-wide financial mathematics society and mathematical finance is now a scientific discipline of its own. The Society held its first World Congress on 2000 in Paris on the hundredth anniversary of Bachelier's celebrated PhD Thesis, Théorie de la Spéculation *SAU

1986 R H Bing (October 20, 1914, Oakwood, Texas – April 28, 1986, Austin, Texas) He wrote papers on general topology, particularly on metrization; planar sets where he examined in particular planar webs, cuttings and planar embeddings. He worked on topological classification of the 2-sphere, the 3-sphere, pseudo arcs, simple closed curves and Hilbert space. He studied partitions and decompositions of locally connected continua. He considered several different aspects of 3-manifolds including decompositions, maps, approximating surfaces, recognizing tameness, triangulation and the Poincaré conjecture. *SAU Oakwood had a population of 471 at the 2000 census.

1991 Paul Ernest Klopsteg (May 30, 1889 – April 28, 1991) was an American physicist. The asteroid 3520 Klopsteg was named after him and the yearly Klopsteg Memorial Award was founded in his memory.
He performed ballistics research during World War I at the US Army's Aberdeen Proving Grounds in Maryland. He applied his knowledge of ballistics to the study of archery.
He was director of research at Northwestern University Technical Institution. From 1951 through 1958 he was an associate director of the National Science Foundation and was president of the American Association for the Advancement of Science from 1958 through 1959.*Wik

1999 Arthur Leonard Schawlow (May 5, 1921 – April 28, 1999) was an American physicist. He is best remembered for his work on lasers, for which he shared the 1981 Nobel Prize in Physics with Nicolaas Bloembergen and Kai Siegbahn.
In 1991 the NEC Corporation and the American Physical Society established a prize: the Arthur L. Schawlow Prize in Laser Science. The prize is awarded annually to "candidates who have made outstanding contributions to basic research using lasers."
In 1951, he married Aurelia Townes, younger sister to physicist Charles Hard Townes, and together they had three children; Arthur Jr., Helen, and Edith. Arthur Jr. was autistic, with very little speech ability.
Schawlow and Professor Robert Hofstadter at Stanford, who also had an autistic child, teamed up to help each other find solutions to the condition. Arthur Jr. was put in a special center for autistic individuals, and later Schawlow put together an institution to care for people with autism in Paradise, California. It was later named the Arthur Schawlow Center in 1999, shortly before his death on the 29th of April 1999.
Schawlow died of leukemia in Palo Alto, California.*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