Thursday, 3 September 2026

Updating the History of the Pigeon Hole Theorem

  


 The Pigeon Hole Principle......The basic idea behind this mathematical principle is what students would call common sense; if there are n objects to be placed in m receptacles (with m less than n), at least two of the items must go into the same container. While the idea is common sense, in the hands of a capable mathematician it can be made to do uncommon things. The late Alexander Bogomolny used the principle to argue that there must be at least two persons in New York City with the same number of hairs on their head. This "counting hairs" approach dates back to the earliest version of the principal I have ever seen.

The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. The pigeon seems to be a recent addition, as Jeff Miller's web site on the first use of some math words gives, "Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, A partition calculus in set theory, Bull. Am. Math. Soc. 62 (Sept. 1956)" (although they credit Dedekind for the principle). In a recent discussion on a history group Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish
we use also:"the principle of the drawers of Dirichlet"
that is 'Zasada szufladkowa Dirichleta' ". I received a note that said, "Dirichlet first wrote about it in Recherches sur les formes quadratiques à coefficients et à indéterminées complexes (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580, he uses the principle."

 Dirichlet


He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

The idea has been around much longer than Dirichlet, however, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story:

Anne-Geneviève de Bourbon


"I asked M. Nicole (See below for description of M. Nicole) one day what was the character of Mme. de Longueville's mind; he told me she had a very keen and very delicate mind in knowledge of the character of individuals, but that it was very small, very weak, very limited on matters of science and reasoning, and on all speculative matters in which there was no question of sentiment ' For example,' added he, ' I told her one day that I could bet and prove that there were in Paris at least two inhabitants who had the same number of hairs upon their head, though I could not point out who were those two persons. She said i could not be certain of it until I had counted the hairs of the two persons. Here is my demonstration/ I said to her: M lay it down as a fact that the best-fiimbhed (not sure what this word was supposed to be, ..Plumbed??) head does not possess more than 200,000 hairs, and the most scantily furnished head b that which has only 1 hair. If, now you suppose that 200,000 heads all have a different number of hairs, they must each have one of the numbers of hairs which are between 1 and 200,000; for if we suppose that there were 2 among these 200,000 who had the same number of hairs, I win my bet But suppose these 200,000 inhabitants all have a different number of hairs, if I bring in a single other inhabitant who has hairs and has no more than 200,000 of them, it necessarily follows that this number of hairs, whatever it b, will be found between 1 and 200,000, and, consequently, b equal in number of hairs to one of the 200,000 heads. Now, as instead of one inhabitant more than 200,000, there are, in all, nearly 800,000 inhabitants in Paris, you see plainly that there must be many heads equal in number of hairs, although I have not counted them.' Mme. de Longuevillle still could not understand that demonstration could be made of the equality in number of hairs, and she always maintained that the only way to prove it was to count them. "

The M. Nicole who demonstrated the principal was Pierre Nicole, (1625 -1695), one of the most distinguished of the French Jansenist writers, sometimes compared more favorably than Pascal for his writings on the moral reasoning of the Port Royal Jansenists. It may be that he had picked up the principal from Antoine Arnauld, another Port Royal Jansenist who was an influential mathematician and logician. Here is a segment from his bio at the St. Andrews Math History site.

P Nicole


-------------------------
He published Port-Royal Grammar in 1660 which was strongly influenced by Descartes' Regulae. In Port-Royal Grammar Arnauld argued that mental processes and grammar are virtually the same thing. Since mental processes are carried out by all human beings, he argued for a universal grammar. Modern linguistic theorists consider this work as the beginnings of the modern approach their subject. Arnauld's next work was Port-Royal Logic which was another book of major importance. It was also strongly influenced by Descartes' Regulae and also gave a first hand account of Pascal's Méthode. This work presented a theory of ideas which remained important in philosophy courses until comparatively recent times. In 1667 Arnauld published New Elements of Geometry. This work was based on Euclid's Elements and was intended to give a new approach to teaching geometry rather than new geometrical theorems."
He was a correspondent of Gottfried Wilhelm Leibniz, and of course Pascal, who wrote the Pascal "Provincial Letters" in support of Arnauld. I enjoyed the quote about him from the Wikipedia bio: "His inexhaustible energy is best expressed by his famous reply to Nicole, who complained of feeling tired. 'Tired!' echoed Arnauld, 'when you have all eternity to rest in?"

I have not been able to find any thing in Arnauld's personal writing at this time to confirm that he was aware of or used the Pigeon-hole Principle. I have also seen a comment that there is a book by Henry (or Henrik) van Etten (pseudonym of Jean Leurechon, who coined the term thermometer) , circa 1624, which uses the method for problems involving "if there are more pages than words on any page" and various other illustrations. The writer suggests that the problem is in the French version but not the English translation. Would love to hear from someone who can confirm, and perhaps send a digital image.






The earliest known hearing aid, called an ear trumpet, was described by Belgian scientist and high school rector Jean Leurechon in his book Récréations-Mathématiues, in 1624. The book described how to make your own ear trumpet as there were no manufacturers of the device at that time.


 

On This Day in Math - September 3


If it's just turning the crank it's algebra,
but if it's got an idea in it, it's topology.

~Solomon Lefschetz

The 246th day of the year; 246 is a sphenic (wedge) composite since it is the product of three distinct prime factors, 246 = 2x3x41. (what would be the next sphenic number?)

246 is also equal to the sum  9C2 + 9C4 + 9C6 (9 choose 2,4,6)

246 = 233 + 13 (13th Fibonacci number plus 13) *Derek Orr@Derektionary

246 is the smallest number whose complete factorization contains the first four digits (and no others) 246 = 2*3*41

Algebra Fact @AlgebraFact points out that: "Tao, et all, have proved that there are infinitely many primes 246 apart." On the way to proving, hopefully, that there are an infinity of twin primes.

246 is an untouchable number, a number that can not be formed from the proper divisors of any number. Paul Erdos proved there are an infinite number of them. The early ones are 2, 5, 62, 88, 96... There are only 29 untouchable year dates, but very unevenly divided by the century groups. 5 below 100, 5 more before 200, then 12 between 200 and 300, and 8 between 300 and 400.


246 is a palindrome in base 5(1441) and base 9(303)

246 can be expressed as the sum of three (not distinct) squares. 14^2 + 5^2 + 5*2

The "aliquot sequence" of a number is the chain of results following from iterating the sum of the aliquot factors. The Chain for 246 is 258, 270, 450, 759, 393, 135, 105, 87, 33, 15, 9, 4, 3, 1, 0. *Wikipedia

See More Math Facts for every Year Day here



EVENTS


1457  Georg Peurbach observed a lunar eclipse from a site near Vienna. He measured the duration of the eclipse and then found the time of the midpoint. It was eight minutes earlier than the time predicted by the Alphonsine Tables.These tables, prepared in Toledo, Spain, for King Alfonso X, were completed in 1252. Based on Ptolemy's theory, they represented the best data available in Peurbach's time. Peurbach produced a new collection of tables of eclipse calculations in Tabulae Ecclipsium which he completed around 1459. When he observed eclipses on 3 July and 27-28 December 1460 he was able to compare the times with the predictions contained in his own tables.

image  Georg von Peuerbach: Theoricarum novarum planetarum testus, Paris 1515



1752 The dates 3 to 13 September did not exist in England in 1752 due to the conversion to the Gregorian calendar. Poor Richard’s Almanac for 1752 carried the catchy heading, “September hath XIX days.” Much of Europe made the change in 1582, and since 1600 was a leap year under the Gregorian but not the Julian calendar, England had to omit eleven days, not ten. *VFR England and the American Colonies dropped the Roman era Julian Calendar, which had become 10 days out of synchrony with the solar cycle, and adopted the Gregorian Calendar. People rioted in the streets thinking the government stole 11 days of their lives. Instituted by Pope Gregory XIII in 1582, the calendar has 365 days with an extra day every four years (the leap year) except in years divisible by 100 but not divisible by 400. Thus, the calendar year has an average length of 365.2422 days. It moved the day's date up from September 3rd to September 14th. Some other countries, including Russia, did not change until the twentieth century.*TIS


1803 John Dalton makes the first entry in his first meteorological notebook. Dalton came to his views on atomism through his interest in meteorology. The volumes contain daily meteorological observations, vol. 1 covering from 1 Apr 1803 to 20 Mar 1816.  By September 3, 1803 he made a logbook entry that day titled, “Observations on the Ultimate Particles of Bodies and their Combinations.” It was the first use of symbols to represent the elements of modern chemistry.




1806 Francois Arago and Jean-Baptiste Biot leave Paris for Spain to finish the measurement of the Paris Meridian.  Arago had been picked to head the completion of the task while a 19yr old student at the Ecole Polytechnique.  He was nominated by his professor, Dennis Poisson and appointed on Feb 2, 1805 to finish the work began by Mechain and Delambre.  *Amir D Aczel, Pendulum, pg 75-78

François Arago was sent to the municipal college of Perpignan, where he began to study mathematics in preparation for the entrance examination of the École Polytechnique. Within two years and a half he had mastered all the subjects prescribed for examination, and a great deal more, and, on going up for examination at Toulouse, he astounded his examiner by his knowledge of J.-L. Lagrange 's work. 

Towards the close of 1803, Arago entered the École Polytechnique, Paris, but apparently found the professors there incapable of imparting knowledge or maintaining discipline. The artillery service was his ambition, and in 1804, through the advice and recommendation of Siméon Poisson, he received the appointment of secretary to the Paris Observatory. He now became acquainted with Pierre-Simon Laplace, and through his influence was commissioned, with Jean-Baptiste Biot, to complete the meridian arc measurements which had been begun by J. B. J. Delambre, and interrupted since the death of P. F. A. Méchain in 1804 *WIK

Arago by Charles de Steuben, *Wik



1806 F. Nichols sends a copy of his printing of Playfair's Geometry to Thomas Jefferson.
"Sir,—
I desire you to accept a copy of Playfair’s Geometry, which I reprinted last winter.
The corrections of grammatical inaccuracies, &. mentioned in the Advertisement, occur after Book I. There are few corrections in Book I.
The book is pretty correctly printed. The principal typographical errors appear at the end of the volume.
Some improvements will be made in another edition, if indeed another edition should ever be wanted.
Playfair’s Geometry is adopted at Cambridge in Massachusetts, Newhaven in Connecticut, & in a few private places of education. I have been informed that it will be introduced into other colleges.
It is a circumstance unfriendly to the reception of a new elementary work of science, designed for the use of students, that some teachers are unwilling to introduce any book into their seminaries, which they have not read at school or college.
I am, with respect, Sir, Your obedient Servant,
F. Nichols."
Jefferson would reply with thanks on 19 Sept.  *Natl. Archives
advertising in the Apr. 7, 1795, issue of the Massachusetts Mercury as "Francis Nichols, late professor of mathematics in the new college at Manchester"; author and bookseller at Philadelphia, 1802-14; member of the American Philosophical Society; bookseller at New York City, 1817-20)*Library of Congress

I have a copy of John Playfairs Geometry printed in 1804 in Edinburgh by Bell & Bradfute.  It was signed on the flyleaf by James Norgate, Caius College Cambridge in 1604.  On the inside of the cover it is signed by Pro James Boyst.  Below his signature is written, "Nov 7, 1946, Lord Plimton (unclear) bought this book off Plummer the Carpenter who bought it at the Boyst sale."  




1821 A Hurricane made landfall on this day in New York City and moved north into Connecticut. It would confuse and inspire William Redfield, and become the first hurricane tracked from beginning to end. While reviewing the storm damage on horseback across Connecticut Redfield noticed that trees in the northern parts of the state fell in the opposite direction from those he had witnessed further south. His interest led him to make a careful study of newspaper reports, letters and ships log an document the storm from beginning to end. He concluded, "This storm was exhibited in the form of a great whirlwind." Redfield was the first to give evidence to support that hurricanes are large circular vortexes. His study of this and other storms led to the classic meteorology paper, "On The Prevailing Storms of the Atlantic Coast." John Farrar, Professor of Mathematics and Natural Philosophy at Harvard University between 1807 and 1836, had made similar observations describing the hurricanes as “a moving vortex and not the rushing forward of a great body of the atmosphere”, after the Great September Gale of 1815..*Wik



1822 A wagon train containing three tons of books arrived at Allegheny College in Meadville, PA. They came from one of the finest private libraries in America, that of James Winthrop. He was a descendant of John Winthrop, first governor of the Massachussettts Bay Colony. His father was Professor of Mathematics and Natural Philosophy at Harvard and so the collection contained a number of important mathematical works. Winthrop was upset that Harvard had not given him an honorary degree and so he gave the books to Allegheny. [Allegheney College Alumni Bulletin, June 1933.] *VFR

It was, in a sense, Allegheny’s first capital campaign, and our first transformative gift. And it put Allegheny on the national map. Allegheny’s first President, Timothy Alden, in 1824 received an unsolicited letter, congratulating the College on its “good fortune of having become the objects of donations so liberal.”

“I had not expected there was such a private collection in the U.S.,” the author wrote. “We are just commencing the establishment of a University in Virginia but cannot flatter ourselves with the hope of such donations as have been bestowed on you.”

The letter was written by former President Thomas Jefferson. Dated Feb. 14, 1824, and penned from Monticello, the letter remains in the Pelletier Library Archives today.



1970  A record hailstone fell on Coffeyville, Kansas, the heaviest authenticated one to fall in the U.S. in the 20th century. Its weight was recorded as 1-lb 11-oz (0.77 kg) with 5.7-in (14.7 cm) diam. It broke the old record from 6 Jul 1928 at Potter, Nebraska, for one weighing about 1-lb 8-oz (0.68 kg), around 7-in diam. A newer U.S. record for size was set on 22 Jun 2003 in Aurora, Nebraska, when a hailstone was found about 7-in diam. (17.8 cm) and 18.75-in (46.6 cm) circumference. The world record was broken on 23 Jul 2010, by a hailstone found in Vivian, South Dakota at 1-lb 15-oz, (0.88 kg), 8.0-in (20 cm) diam., 18.6-in (47.3 cm) circumference. A larger hailstone is said to have fallen on 14 Apr 1986 that weighed 2-lb 4-oz (1.02 kg) during a hailstorm in Bangladesh that killed 92 people. (This might be more correctly referred to as a Hell Stone)


Later the record volume was broken.  According to the National Oceanic and Atmospheric Administration, a hailstone fell in Aurora, Nebraska, on June 22, 2003, which had a diameter of 7 inches and circumference of 18.75 inches. This hailstone, however, only weighed 1.3 pounds.




In 2000, NASA data showed the hole at just under 11 million square miles - the biggest it had ever been. Record low temperatures in the stratosphere are believed to have helped the expansion of the ozone hole during the southern hemisphere’s spring season. Antarctic ozone depletion starts in July, when sunlight triggers chemical reactions in cold air trapped over the South Pole during the Antarctic winter. It intensifies during August and September before tailing off as temperatures rise in late November of early December. Depletion of the ozone layer over Antarctica and the Arctic is being monitored because ozone protects Earth from harmful ultraviolet radiation. By 9 Sep 2000, the hole had grown over Chile, exposing a populated city for the first time. Image, compiled from NASA's Total Ozone Mapping Spectrometer instrument onboard the Earth Probe satellite, reveals how the ozone hole (in deep blue) has extended as far as southern Chile. *TIS


2009 Saturn's rings cross the plane of the Earth's orbit. This was the first such crossing since May 22, 1995, and another will not occur until March 23, 2025. *Wik

This sequence of images from the Hubble telescope documents a rare astronomical alignment: Saturn's magnificent ring system turned edge-on. This event occurs when the Earth passes through Saturn's ring plane, as it does about every 15 years.




BIRTHS


1780 Heinrich Christian Schumacher (September 3, 1780 – December 28, 1850)
Schumacher was a German astronomer. He was director of the Mannheim observatory from 1813 to 1815, and then became professor of astronomy in Copenhagen. From 1817 he directed the triangulation of Holstein, to which a few years later was added a complete geodetic survey of Denmark (finished after his death). For the sake of the survey an observatory was established at Altona, and Schumacher resided there permanently, chiefly occupied with the publication of Ephemerides (11 parts, 1822–1832) and of the journal Astronomische Nachrichten (founded by himself in 1821 and still being published), of which he edited thirty-one volumes. *TIA




1814 James Joseph,(Sylvester) (3 Sep 1814; 15 Mar 1897) youngest child of Abraham Joseph, born in London. The eldest son, an actuary, eventually migrated to the U.S. where, for unknown reasons, he took the surname Sylvester. The rest of the family soon followed suit, so that is how James Joseph Sylvester got his name. *VFR British mathematician who, with Arthur Cayley, founded the theory of algebraic invariants, algebraic-equation coefficients that are unaltered when the coordinate axes are translated or rotated. Beginning in 1833, he studied at St John's College, Cambridge. However, at this time signing a religious oath to the Church of England was required to graduate. Being Jewish, he refused and so he did not graduate. He taught physics at the University of London (1838-41), one of the few places which did not bar him because of his religion. Sylvester did important work on matrix theory, in particular, to study higher dimensional geometry. In 1851 he discovered the discriminant of a cubic equation. Earlier in his life, he tutored Florence Nightingale.*TIS (This idea of Sylvester tutoring Nightingale, to the best of my knowledge, originates from the Herbert Baker obituary. Karen Hunger Parshall, among others, has questioned the accuracy of this statement.)
He was instrumental in shaping graduate study and American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. *Wik
James Joseph Sylvester died, at age 83, after earlier suffering a paralytic stroke while working at his mathematics. *VFR
I came across a nice story about Sylvester on the wonderful "Cut-the-Knot" blog of Alexander Bogomolny. He writes, "Sylvester was one the greatest British mathematicians of the 19th century. He was known for his absentmindedness and poor memory; on one occasion he even denied the truth of one of his own theorems. "

In his youth his family was proud of the brilliant student.  While he was a teenager, his oldest brother, already in America, Aware that a group of Lottery contractors struggling with a difficult problem in combinations he suggested they contact his teenage brother James.  They did, and found his answer so complete that they paid him 500 dollars for his work.  





1869 Austrian chemist Fritz Pregl (3 September 1869 – 13 December 1930) Pegl began research on bile acids in 1904. With only tiny yields to study, he pioneered micro analytical techniques and designed a new balance capable of weighing 20 grams to an accuracy of 0.001 milligrams. Pregl was awarded the Nobel Prize in Chemistry 1923 for his efforts *RSC.org
In 1950, the department of the University of Graz where Fritz Pregl had worked was named the Institute of Medical Chemistry and Pregl Laboratory. Streets in Graz, Innsbruck, Vienna and Klagenfurt were named after him. In Slovenia, Pregl Awards have been bestowed annually since 2007 by the National Institute of Chemistry for the research work and for outstanding doctorates. Slovenian pupils are conferred Pregl Recognition Awards, whereas secondary school students are conferred Pregl Citations for excellent results in national competitions in chemistry. A square in Ljubljana is named after Pregl. The Fritz Pregl Prize has been awarded annually since 1931 in chemistry by the Austrian Academy of Sciences from the funds left at its disposal by Pregl. *Wik



1874 Fredrik (Carl Mülertz) Størmer (3 Sep 1874; died 13 Aug 1957) was a geophysicist and mathematician who developed a mathematical theory of auroral phenomena. An aurora is the light emitted by energetic protons and electrons at the top of Earth's atmosphere when they come in contact with solar wind particles. He also contributed both important photographic observations and mathematical data to the understanding of the polar aurora, of stratospheric and mesospheric clouds, and of the structure of the ionosphere. The discovery of the Van Allen Radiation Belts by James Van Allen confirmed with surprising accuracy Størmer's theoretical analysis of solar charged particle trajectories in Earth's magnetic field.*TIS



1883 Harold DeForest Arnold (September 3, 1883 – July 10, 1933) was an electronics engineer and pioneer of radio communication and telephony. He served as the first director of research at Bell Telephone Laboratories from 1925 to his death.
He initially studied under Albert A. Michelson but when he confided to Robert Andrews Millikan that he would probably have to commit suicide as he could not meet Michelson's requirement, Millikan took Arnold over as his own student. When Frank B. Jewett was looking for someone to work on repeaters for transcontinental telephony, Arnold was suggested by Millikan. Arnold worked at the University of Chicago from 1907 to 1909 and served as a professor at Mount Allison University, from 1909 to 1910 and then at University of Chicago (1910). In 1911 he joined the Western Electric Company under Edwin H. Colpitts. His earliest work was in the development of a vacuum-tube based amplifiers beginning with improvements to Lee De Forest's triode “audion”. He worked on innovations that made it possible to demonstrate the first radio transmission between Arlington, Virginia, and Paris, France, in October 1915. During World War I he served as a captain in the signal corps. He developed and refined manufacturing techniques for vacuum tubes, oxide coatings for filaments, and other innovations for reliability and ease of replacement. Permalloy and Perminvar were developed by his team and this helped improve signal quality in undersea cables. Arnold received the John Scott Medal in 1928 *Wik




1884 Solomon Lefshetz (3 September 1884 – 5 October 1972) born in Moscow. He invented the phrase “algebraic topology.” See A Century of Mathematics in America, Part I, 1988, p. 171. *VFR mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.*Wik



1905 Carl David Anderson (3 Sep 1905; 11 Jan 1991)American physicist who, with Victor Francis Hess of Austria, won the Nobel Prize for Physics in 1936 for his discovery of the positron, or positive electron, the first known particle of antimatter. He examined the photographs of cosmic rays taken as they passed through a Wilson cloud chamber in a strong magnetic field. Besides the curved paths of negative electrons, he found also paths deviating in the opposite direction, corresponding to positively charged particles - yet having the the same mass as an electron! Previously, Dirac had predicted such particles by theoretical solution to electromagnetic field equations. *TIS



*wik



1908 Lev Semenovich Pontryagin (3 September 1908 – 3 May 1988) One of the 23 problems posed by Hilbert in 1900 was to prove his conjecture that any locally Euclidean topological group can be given the structure of an analytic manifold so as to become a Lie group. This became known as Hilbert's Fifth Problem. In 1929 von Neumann, using integration on general compact groups which he had introduced, was able to solve Hilbert's Fifth Problem for compact groups. In 1934 Pontryagin was able to prove Hilbert's Fifth Problem for abelian groups using the theory of characters on locally compact abelian groups which he had introduced. *SAU [He was buried at the Novodevichie Memorial Cemetery in Moscow.]
Alexandre Zagoskin commented, "Lev Pontryagin completely lost his sight at the age of 14 due to an accident and was then helped by his schoolmates and his mother, who read to him the textbooks. His mother learned German to read him research papers when he studied at the university."   




1970 Stanislav Konstantinovich Smirnov (3 September 1970 - ) is a Russian mathematician currently working at the University of Geneva, who was awarded the Fields Medal in 2010. His research focuses on the fields of complex analysis, dynamical systems and probability theory. *Wik





DEATHS


1595 Federico Commandino (1509 – September 5, 1575) died on this day. “In the sixteenth century, Western mathematics emerged swiftly from a millennial decline. This rapid ascent was assisted by Apollonius, Archimedes, Aristarchus, Euclid, Eutocius, Hero, Pappus, Ptolemy, and Serenus—as published by Commandino,” *VFR He translated the works of ancient mathematicians and was responsible for the publication of the works of Archimedes. He also translated the works of Aristarchus of Samos (On the masses and distances of the Sun and the Moon), Pappus of Alexandria (Mathematical collection), Hero of Alexandria (Pneumatics), and Euclid (Elements). Among his pupils was Guidobaldo del Monte. Commandino maintained a correspondence with the astronomer Francesco Maurolico.
The proposition known as Commandino's theorem** first appears in his work on centers of gravity, Liber de centro gravitatis solidorum in 1565. *Wik
**The four medians of a tetrahedron concur in a point which divides each tetrahedron median in the ratio 1:3, the longer segment being on the side of the vertex of the tetrahedron. Student's should compare this to the property of the medians of a triangle which concur at a point that divides the median in a 1:2 ratio.




1894 Josiah Parsons Cooke (October 12, 1827 – September 3, 1894) was an American scientist who worked at Harvard University and was instrumental in the measurement of atomic weights, inspiring America's first Nobel laureate in chemistry, Theodore Richards, to pursue similar research. Cooke's 1854 paper on atomic weights has been said to foreshadow the periodic law developed later by Mendeleev and others. Historian I. Bernard Cohen described Cooke "as the first university chemist to do truly distinguished work in the field of chemistry" in the United States. *Wik




1910 Wilhelm Winkler (29 June 1884 in Prague, Bohemia (Austro-Hungarian Empire, now Czech Republic - 3 Sept 1984 in Vienna, Austria) After attending the trading high school in Gera, Winkler worked as a merchant in Eisenberg, following in the footsteps of his grandfather. In 1875 he gave up this trade and devoted his time entirely to astronomy. Advised by Carl Bruhns, director of the Leipzig University Observatory, he established an observatory on his estate in Gohlis near Leipzig. From 1878 Winkler regularly observed sunspots; other fields of his observational interests were comets, occultations of stars by the Moon, and Jupiter's satellites. *Uni Bonn Web page




1967 William P Milne (22 May 1881 in Longside, Aberdeenshire, Scotland - 3 Sept 1967 in Glack, Aberdeenshire, Scotland) studied at Aberdeen and Cambridge universities. He taught at Clifton College and then became Professor of Mathematics at Leeds. He published papers in Geometry including many in the Proceedings of the EMS. *SAU (Not to be confused with the William J Milne who, as President of the New York State Normal School in Albany, wrote many popular textbooks in arithmetic, algebra and geometry for high school use)



1992 Barbara McClintock (16 Jun 1902, 3 Sep 1992) American scientist regarded as one of the most important figures in the history of genetics. In the 1940s and 1950s McClintock's work on the cytogenetics of maize led her to theorize that genes are transposable - they can move around - on and between chromosomes. McClintock drew this inference by observing changing patterns of coloration in maize kernels over generations of controlled crosses. The idea that genes could move did not seem to fit with what was then known about genes, but improved molecular techniques of the late 1970s and early 1980s allowed other scientists to confirm her discovery. She was awarded the 1983 Nobel Prize in Physiology or Medicine, the first American woman to win an unshared Nobel Prize. *TIS

*Wik



1995 Bent Christiansen (7 May, 1921 - 3 Sept, 1996)  From his Obituary: "Bent was a legend in mathematics education in Denmark and the Nordic countries. His impact on the development of the teaching and learning of mathematics in primary and lower secondary education can hardly be over-estimated. He wrote textbooks and books on mathematics education, especially the very influential 'Goals and means in basic mathematics education' ('Mål og midler I den elementære matematikundervisning', 1967). He gave innumerable in-service courses and invited lectures at meetings and conferences. Naturally, he also served on hosts of national committees, including the Danish National Sub-Commission of ICMI (1961-1972). All this earned him a reputation as a charismatic, enthusiastic and extremely energetic mentor for generations of mathematics teachers, teacher trainers and colleagues."




2016  Jean-Christophe Yoccoz (May 29, 1957, September 3, 2016)  is a French mathematician , born on in Paris where he also died . He was awarded theFields Medalin 1994, and then became a professor at the Collège de Francein 1996. He is particularly known for his work on dynamical systems.


Jean-Christophe Yoccoz is the son of a physicist , Jean Yoccoz, appointed director of the CNRS's National Institute of Nuclear and Particle Physics in 1975 


Upon graduating from the ENS in 1979, he was appointed research fellow and then CNRS research fellow at the École Polytechnique , a position he held until 1988  . During this period, in the early 1980s , he completed his national service in cooperation with the Instituto de Matematica Pura e Aplicada in Rio de Janeiro . He defended his doctoral thesis at Paris-XI University in 1985 under the supervision of Michaël Herman of the Centre de mathématiques de l'École Polytechnique, before being awarded the Salem Prize in 1988. A strong chess player, he notably participated in the French Chess Championship in 1980  .

He was a professor at Paris-Sud University from 1988 to 1996  . He became a member of the Institut Universitaire de France in 1991  . He was awarded the Fields Medal in 1994 at the International Congress of Mathematicians in Zurich  for his work on the theory of dynamical systems . He was elected a member of the French Academy of Sciences on October 24, 1994, in the "  Mathematics  " section. He was also a member of the Brazilian Academy of Sciences . He was a professor at the Collège de France from 1996, holding the chair of differential equations and dynamical systems. From 2012 to 2016, he was a member of the board of directors of the Hugot Foundation at the Collège de France . He was part of the Bourbaki group . He died onSeptember 3, 2016following a long illness. His colleague and friend Pierre-Louis Lions , who had also received the Fields Medal in 1994, said of him upon hearing of his death   : "  He had a great speed of thought and analysis, was capable of flashes of brilliance.  

His work focuses primarily on dynamical systems. His study aims to "  understand the behavior of anything that can move, like the planets around the Sun  " . In this exploration of the boundary between chaos and regularity, Jean-Christophe Yoccoz has, for example, constructed tools, called "  Yoccoz puzzles  ," which allow spaces to be divided into small pieces in order to study them without losing sight of the whole  . He has thus "  significantly advanced the study of Mandelbrot fractals . *Wik 





2021  David Borwein (March 24, 1924 – September 3, 2021) was a Lithuanian-born Canadian mathematician, known for his research in the summability theory of series and integrals. He also did work in measure theory and probability theory, number theory, and approximate subgradients and coderivatives. He latterly collaborated with his son, Jonathan Borwein, and with B.A. Mares Jr. on the properties of single-variable and many-variable sinc integrals.

He formerly resided and worked in St. Andrews, Scotland, before moving to London, Ontario where he eventually became Head of Mathematics at the University of Western Ontario. He was also the president of the Canadian Mathematical Society (CMS). The David Borwein Distinguished Career Award given out by the CMS is named after him. He was an active researcher in summability theory, classical analysis, inequalities, matrix transformations, and was professor emeritus at the University of Western Ontario, department of Mathematics.

His wife of over 60 years, Bessie Borwein, is a prominent anatomist, and is professor emerita of anatomy at the University of Western Ontario.

In 2018 the Canadian Mathematical Society listed him in their inaugural class of fellows.

Borwein died in his sleep on September 3, 2021, at the age of 97.




247 = 50123 - 49876


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

Wednesday, 2 September 2026

A Bird in the Hand is worth... a Pigeon Hole Principle

 


Sometimes problems that seem very hard, can be very easy if they are viewed in the right way, and one of those easy ways to make some hard problems manageable is the Pigeon-Hole Principle. Over the last few weeks seems like lots of problems involving this idea have shown up, so I thought I would bring it to you.
The basic idea is so easy any sixth grader would agree; if you have two boxes, and you are going to put three balls in the boxes, then at least one box will get more than one ball..... "well, Duh!" they answer... and yet... it seems easier to apply than it might be. Now that you know the secret, try these two problems. I'll post the answer down lower on the page where you must not look until you take a few minutes to ponder the problems.
Here is the first from a recent blog I read: "39 people are attending a large, formal dinner, which must of course occur at a single, circular table. The guests, after milling about for a while, sit down to eat. It is then pointed out to them that there are name cards labeling assigned seats, and not a single one has sat in the seat assigned to them. Prove that there is some way to rotate the table so that at least two people are in the correct seats."
This one seems tougher, but really isn't, it just requires a different way of thinking. "Suppose you pick six unique integers from 1 to 1000. Prove that at least two of them must have a difference that is a multiple of five.
Before I give you the answers, I will throw in a little cultural information that may amuse and entertain you. The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. In a discussion on a history group a few years ago Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portuguese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet" that is 'Zasada szufladkowa Dirichleta' ". You just can't have TOO many names for a really useful idea.
Ok, The Proofs... for number one... Suppose you handed each person a number that was how many seats they needed to move to the right to find their assigned seat. Since no one is at the right seat, the number can not be zero or thirty-nine. SO each of the people has a number between 1 and 38...wait, there are 39 people...two of them (at least) must be the same distance away from their assigned seats.... admit it…..that’s pretty cool. (I have a slight question about whether this actually proves the solution of "rotating the table" to put two in their correct seats.  Suppose we know that persons A and B are in each others seats and 5 seats apart.  Rotating the table five seats in either direction would only put one of them in their correct seat.  Maybe all we proved is that there are , at least, two people who are the same distance from their seat.  If we had specified how far away in a clockwise direction they are from their correct seat, we would have a solution to the rotation of the table problem.)
For number two it is sort of the same idea, but you have to think about how much each number would have for a remainder if you divided them by five. The only possible choices are 0, 1, 2, 3, or 4... , five different remainders, but there are six numbers, so two of them have the same remainder...and two numbers that have the same remainder on division by five, are a multiple of five apart.... think of 1,6, 11, etc for remainders of one. If you want to read more about how remainders can play a part in solving problems, see my blog on "casting out sevens" (And other primes) 

And for some history about this beautiful problem solving idea, see this.

On This Day in Math - September 2

   



The importance of the "New Mathematics" lies mainly in the fact that it has taught us the difference between the disc and the circle.

D MacHale, Comic Sections (Dublin 1993)
(I realize that a whole generation has grown up who have no idea what "New Mathematics" means, my apologies to them for a dated quote)


The 245th day of the year; 245 is the fifth StellaOctangula number. The sum of the 5th octahedral number (85) and eight of the fourth tetrahedral numbers (20). 245 =85 + 8 (20)

245 is also the sum of three consecutive squares, 245=82+92+102

There are 245 odd entries in the first 33 rows of the Arithmetic Triangle.

245 is also the 46th prime , 199+46=245 (is there a mathematical significance for these numbers, or just a nice curiosity?  Serious question.)


See Math Facts for every Year Day here.



EVENTS


1666 Five days previously Wren had visited Old St. Paul's Cathedral to determine the reconstruction needs for the decaying old building.    During the night of Sep 2, and for the next five days, the Great Fire of London will burn out about 7/8 of the city of London and greatly alter Wren's work at St Paul's.  [The Great Fire of is supposed to have started in the house of King Charles II's baker on Pudding Lane near London Bridge.*@History Magazine]

 Image from Wikipedia,


In 1752, today was the last day of the Julian calendar in Great Britain and the British colonies; the Gregorian Calendar designed to correct the extra leap year day problem went into effect the next day with tomorrow being September 14, hence 11 days were dropped. Most other countries made the adjustment in 1582. *TIS

Some historians suggested there had been riots, the oft called the Calendar Riots of 1752.  These claims of civil unrest and rioters demanding “Give us our eleven days” may have arisen through a misinterpretation of a contemporary painting by William Hogarth. His 1755 painting entitled: “An Election Entertainment” refers to the elections of 1754 and depicts a tavern dinner organised by Whig candidates. A stolen Tory campaign banner with the slogan, “Give us our Eleven Days” can be seen lower right (on the black  banner on the floor under the seated gentleman’s foot). The Tories can be seen outside the window, demonstrating.Claims of civil unrest and rioters demanding “Give us our eleven days” may have arisen through a misinterpretation of a contemporary painting by William Hogarth. His 1755 painting entitled: “An Election Entertainment” refers to the elections of 1754 and depicts a tavern dinner organised by Whig candidates. A stolen Tory campaign banner with the slogan, “Give us our Eleven Days” can be seen lower right (on the black banner on the floor under the seated gentleman’s foot). The Tories can be seen outside the window, demonstrating.




1808 Gauss writes Wolfgang Bolyai: “It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.” * Mathematical Circles Squared,Howard Eves, pg 113



1885 Gunshots rang out on the afternoon of September 2, 1885, in Rock Springs, Wyoming Territory. Home to hundreds of Chinese coal miners who had come to the United States for work, the settlement’s Chinatown was facing impending bloodshed. After a morning of violence against Chinese workers in one of the nearby mines, more than a hundred white men armed with guns and other weapons had surrounded the neighborhood.

Tensions between Chinese and white coal miners in Rock Springs had been growing for a long time. White miners, organized under the Knights of Labor union, sought to improve workers’ conditions through unionizing and striking against the giant Union Pacific Railroad Company. Fed up with the company’s proposals to cut pay and its requirement that miners buy necessities at its overpriced stores, the Knights of Labor demanded negotiations with the miners’ employers. The union represented the will of oppressed workers, but it also represented a racist sentiment: the Knights of Labor argued that a large part of the miners’ problems was being caused by an influx of Chinese immigrants who were willing to work for less pay than white workers. When the Chinese workers at Rock Springs refused to strike with the white miners, tensions between the groups reached a breaking point. After returning from the mines to their homes to retrieve their weapons, white men, as well as women, stormed Chinatown that September afternoon. Their violent crusade, now known as the Rock Springs Massacre, resulted in the deaths of 28 Chinese people and the injury of 15, making it one of the bloodiest racially motivated massacres against Chinese immigrants in America.

What happened at Rock Springs was symptomatic of much wider racist sentiment in the United States at the time. Anti-Chinese views had existed since the first major waves of Chinese workers had arrived in North America to build the transcontinental railroad. Such workers represented a relatively cheap source of labour willing to work in dangerous conditions, and they soon replaced many of their white counterparts. In fact, the racist expression “not a Chinaman’s chance” is believed to derive from the dangerous working conditions Chinese workers typically found themselves in, such as being lowered along cliff faces to detonate explosives.*Britannica




1905 While a student at Kumbakonam, Ramanujan was so obsessed with his math studies that he failed all his other classes. It seems after a conflict at home, he ran away, causing his mother to send a missing-person letter to the newspaper:


1958 The National Defense Education act was passed in response to Sputnik (4 October 1957). $840 million was appropriated to improve the teaching of mathematics, science, and foreign languages. *VFR

The year 1957 also coincided with an acute shortage of mathematicians in the United States. The electronic computer created a demand for mathematicians as programmers and it also shortened the lead time between the development of a new mathematical theory and its practical application, thereby making their work more valuable. The United States could no longer rely on European refugees for all of its mathematicians, though they remained an important source, so it had to drastically increase the domestic supply. At the time, "mathematics" was interpreted as pure mathematics rather than applied mathematics. The problem in the 1950s and 1960s was that industry, including defense, was absorbing the mathematicians who were also needed at high schools and universities training the next generation. At the university level, even more recently, there have been years when it was difficult to hire applied mathematicians and computer scientists because of the rate that industry was absorbing them.

This chart shows the number of PhDs in the US by year from 1900.  The linearity across the 20th Century of the semi-log graph may make one wonder if the program really had much impact.  In fact the slope of growth rate drops significantly around the 70's (Which could be from age distribution in the country or numerous other causes.)





1997 After its Deep Blue chess-playing computer defeated human world chess champion Gary Kasparov​ in a closely watched match in May, the pioneering computer company decided to make the machine even faster and stronger. On September 2, IBM announced that its RS/6000 SP model, a parallel supercomputer, was now 58 percent faster thanks to a new microprocessor and some software refinements. Kasparov was not available for comment.*CHM

a Deep Blue Processor, *Wik




2023  Dennis Austin, the principal software developer of PowerPoint, passed away from lung cancer on Sept. 1. He was 76. The Washington Post reports:

Released in 1987 by Forethought, a small software firm, PowerPoint was the digital successor to overhead projectors, transforming the labor-intensive process of creating slides -- a task typically assigned to design departments or outsourced -- to one where any employee with a computer could point, click and rearrange information with a mouse. "Our users were familiar with computers, but probably not graphics software," Mr. Austin wrote in an unpublished history of the software's development. "They were highly motivated to look their best in front of others, but they weren't savvy in graphics design."

Working alongside Robert Gaskins, the Forethought executive who conceived the software, it was Mr. Austin's job as the software engineer to make PowerPoint (originally called Presenter) easy to operate. He accomplished this with a "direct-manipulation interface," he wrote, meaning that "what you are editing looks exactly like the final product." Originally targeted for Macintosh computers, which had a graphical interface, Presenter included ways for users to incorporate graphics, clip art and multiple fonts. In addition, the slides could be uniform with graphic borders, corporate logos and slide numbers. The goal, Mr. Austin wrote, was "to create presentations -- not simply slides."

In his book "Sweating Bullets: Notes about Inventing PowerPoint" (2012), Gaskins wrote that "Dennis came up with at least half of the major design ideas," and was "completely responsible for the fluid performance and the polished finish of the implementation." "It's a good bet," Gaskins added, "that if Dennis had not been the person designing PowerPoint, no one would ever have heard of it."



BIRTHS


1841 Paul Matthieu Hermann Laurent born (2 September 1841 Luxembourg City – 19 February 1908 Paris, France). He developed statistical formulas for the calculation of actuarial tables and studied heat conduction. *VFR

In 1883 he became an examiner at the École Polytechnique. Because examiners were forbidden from publishing textbooks on the very subjects they examined, Laurent found a workaround—he published under pseudonyms! In 1895 he released Traité d’arithmétique, attributing it to his friends C. A. Laisant and Émile Lemoine to comply with the rules while still sharing his knowledge
Despite his large body of works, Laurent series expansions for complex functions were not named after him, but after Pierre Alphonse Laurent.





1850 Alfred Pringsheim born (2 September 1850 – 25 June 1941), a German mathematician who worked on real and complex functions. *SAU Pringsheim's theorem concerns the convergence of a power series with non-negative real coefficients. Pringsheim and Ivan Śleszyński, working separately, proved what is now called the Śleszyński–Pringsheim theorem on convergence of certain continued fractions.*Wik




1856 Wilhelm Franz Meyer born (2 September 1856 in Magdeburg; 11 April 1934 in Königsberg . Meyer studied algebraic geometry, algebraic curves and invariant theory.*SAU He was a Founding member of the German Mathematical Society. *Wik

 



1877 Frederick Soddy (2 September 1877 – 22 September 1956) was an English radiochemist and monetary economist who explained, with Ernest Rutherford, that radioactivity is due to the transmutation of elements, now known to involve nuclear reactions. He also proved the existence of isotopes of certain radioactive elements. He received the Nobel Prize for Chemistry in 1921, and named after him is small crater on the far side of the Moon and the radioactive Uranium mineral, Soddyite. He rediscovered the Descartes' theorem in 1936 and published it as a poem. The kissing circles in this problem are sometimes known as Soddy circles.
The Poem begins,

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.

The entire poem is found here, along with a story about how, In a strange "chain reaction" of ideas, Soddy played a part in the US developing an atomic bomb. *(Assorted notes)



1878 (René-)Maurice Fréchet (2 Sep 1878; 4 June 1973) was a French mathematician known chiefly for his contribution to real analysis. He is credited with being the founder of the theory of abstract spaces, which generalized the traditional mathematical definition of space as a locus for the comparison of figures; in Fréchet's terms, space is defined as a set of points and the set of relations. In his dissertation of 1906, he investigated functionals on a metric space and formulated the abstract notion of compactness. In 1907, he discovered an integral representation theorem for functionals on the space of quadratic Lebesgue integrable functions. He also made important contributions to statistics, probability and calculus. *TIS




1891 Ivan Matveyevich Vinogradov (2 Sep 1891[OS? SAU gives 14 Sep], 20 Mar 1983)Soviet mathematician known for his contributions to the analytical theory of numbers, including a partial solution of the Goldbach conjecture proving that every sufficiently large odd integer can be expressed as the sum of three odd primes. He described his methods in his most celebrated piece of work Some Theorems Concerning the Theory of Prime Numbers (1937)*TIS




1892 Frank Wilcoxon, (2 September 1892 - 18 November 1965) whose name should be familiar to anyone who has used classic nonparametric (distribution-free) tests, was born in County Cork, Ireland, to American parents. He spent much of his early life in the Hudson River Valley region of New York.  Sometime around 1908 he ran away to sea, jumped ship after a week of chipping paint when the ship failed to sail, and hid out for years from the imagined consequences of this desertion in the back country of West Virginia, first as an oil well worker, then as a tree surgeon. A trip to Boston to hone the latter skills at a forestry school fizzled when it turned out that the school had closed. Finally returning home, he was sent to the Pennsylvania Military College in 1917, another totally incompatible environment. His twin sister died in childbirth in 1918.

After a WW1 job with the Atlas Powder Company in Michigan, Wilcoxon entered Rutgers in 1920, and completed an MS in chemistry in 1921; he then shifted to Cornell and physical chemistry, and got his PhD in 1924. 

Much of his early work was in research related to chemistry, with his interest in statistics resulting from reading Fisher's well-known book Statistical Methods for Research Workers (which I recall reading somewhere was for some time the most cited book in all of science). During the 1940s Dr. Wilcoxon was perhaps most instrumental in the growth of the fledgling field of nonparametric or distribution-free statistics, introducing his signed-rank test for paired samples and his famous two-sample rank-sum test as an alternative to Student's unpaired two-sample t-test, each of which carry Wilcoxon's name in its appellation. (Wilcoxon's 1945 paper introducing these two tests was titled "Individual Comparisons by Ranking Methods".)

At the time of his death, Wilcoxon was working on a multivariate generalization of his two-sample rank sum test. His proposals were described posthumously (by Bradley, 1967). They have not been taken up by the statistical community.

For a good description of how outliers led the chemist Wilcoxon to develop these tests, read Chapter 16, "Doing Away With Parameters", in David Salsburg's book The Lady Tasting Tea, in which Salsburg writes in a footnote that "the nonparametric approach was not fully understood to be such a drastic revolution until Wilcoxon's work in this field" *David Bee




1909 Deane Montgomery (2 Sept 1909 - 15 March 1992 in Chapel Hill, North Carolina, USA) was a mathematician specializing in topology who was one of the contributors to the final resolution of Hilbert's fifth problem in the 1950s. He served as President of the American Mathematical Society from 1961 to 1962.
Born in the small town of Weaver, Minnesota, he received his B.S. from Hamline University in St. Paul, MN and his Masters and Ph.D. from the University of Iowa in 1933; his dissertation advisor was Edward Chittenden.
In 1941 Montgomery was awarded a Guggenheim Fellowship. In 1988, he was awarded the American Mathematical Society Leroy P. Steele Prize for Lifetime Achievement.*Wik




1913 Israel Moiseevich Gelfand, (2 September [O.S. 20 August] 1913 – 5 October 2009) was a prominent Soviet-American mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow.

His legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, David Kazhdan, as well as his own son, Sergei Gelfand.

Gelfand is known for many developments including:

the book Calculus of Variations (1963), which he co-authored with Sergei Fomin;

Gelfand's formula, which expresses the spectral radius as a limit of matrix norms.

the Gelfand representation in Banach algebra theory;

the Gelfand–Mazur theorem in Banach algebra theory;

the Gelfand–Naimark theorem;

the Gelfand–Naimark–Segal construction;

Gelfand–Shilov spaces;

the Gelfand–Pettis integral;

the representation theory of the complex classical Lie groups;

contributions to the theory of Verma modules in the representation theory of semisimple Lie algebras (with I. N. Bernstein and S. I. Gelfand);

contributions to distribution theory and measures on infinite-dimensional spaces

the first observation of the connection of automorphic forms with representations (with Sergei Fomin);

conjectures about the Atiyah–Singer index theorem;

ordinary differential equations (Gelfand–Levitan theory);

work on calculus of variations and soliton theory (Gelfand–Dikii equations);

contributions to the philosophy of cusp forms;

Gelfand–Fuchs cohomology of Lie algebras;

Gelfand–Kirillov dimension;

integral geometry;

combinatorial definition of the Pontryagin class;

Coxeter functors;

general hypergeometric functions;

Gelfand–Tsetlin patterns;

Gelfand–Lokutsievski method;

and many other results, particularly in the representation theory of classical groups.




1923 René Thom (September 2, 1923 – October 25, 2002) is known for his development of catastrophe theory, a mathematical treatment of continuous action producing a discontinuous result. *SAU
Born in Montbeliard, France. In 1958 he received a Fields Medal for his 1954 creation of cobordism in algebraic topology. His classification of manifolds used homotopy theory in a fundamental way and this work became an important example of general cohomology theory. *VFR Thom is also known for his later work developing the catastrophe theory (1972), a mathematical treatment of continuous action producing a discontinuous result. Thom's theory is an attempt to describe, in a way that is impossible using differential calculus, those situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes. The theory has widespread application in the physical and biological sciences and in the social sciences, but eventually fell from favour.*TIS




1925 Roy Jay Glauber (September 1, 1925 – December 26, 2018)  was an American theoretical physicist. He was the Mallinckrodt Professor of Physics at Harvard University and Adjunct Professor of Optical Sciences at the University of Arizona. Born in New York City, he was awarded one half of the 2005 Nobel Prize in Physics "for his contribution to the quantum theory of optical coherence", with the other half shared by John L. Hall and Theodor W. Hänsch.
In this work, published in 1963, he created a model for photodetection and explained the fundamental characteristics of different types of light, such as laser light (see coherent state) and light from light bulbs (see blackbody). His theories are widely used in the field of quantum optics. *Wik




1948 Christa McAuliffe (2 Sep 1948; died 28 Jan 1986) Astronaut, first teacher in space, who died on

the Challenger Space Shuttle when 73 seconds into its 10th launch, Challenger (STS-51L) exploded in midair, killing its crew of seven. Space shuttle flights were suspended until 1988. An independent U.S. commission blamed the disaster on unusually cold temperatures that morning and the failure of the O-rings, a set of gaskets in the rocket boosters. *TIS





DEATHS


1764 Nathaniel Bliss (28 November 1700 – 2 September 1764) was an English mathematician and astronomer who went on to become Astronomer Royal. He succeeded Edmond Halley as professor of geometry at Oxford University in 1742 and was elected a Fellow of the Royal Society the same year. He succeeded James Bradley to become the fourth Astronomer Royal in 1762, but held the post for too short a period to make a significant impact (1762-1764).*Wik




1768 Antoine Deparcieux (October 28, 1703 – September 2, 1768) was a French mathematician who is best known for an early work on annuities and mortality.*SAU In 1746, he published Essai sur les probabilités de la durée de la vie humaine (An Essay on the Probabilities of the Duration of Human Life). Deparcieux analyzed in detail empirical observations. As a mathematician and physicist, he can be considered, after Halley and Struyck, one of the founders of the estimation of longevity and all the issues surrounding that concept. *Wik





1832 Franz Xaver von Zach (4 June 1754, 2 Sep 1832) German-Hungarian astronomer patronized by Duke Ernst of Saxe-Gotha-Altenburg. Director of observatory near Gotha (1787-1806). There he organized in 1798 the first congress of astronomers with Josef Lalande (1732-1807) as celebrated guest. In last years of the 18th century he formed a group of 24 astronomers chosen from throughout Europe to track down a "missing" planet between the orbits of Mars and Jupiter, where they instead discovered the asteroids. His greatest contribution was in the organizational area, for he maintained an enormous correspondence with all the astronomers of his time, and edited 28 volumes of Monatliche Korrespondenz zur Beforderung der Erd- und Himmelskunde (1800-13).*TIS





1834 Thomas Telford (9 August 1757 Glendinning, Westerkirk, Eskdale, Dumfriesshire, Scotland - 2 September 1834 (aged 77) 24 Abingdon Street, Westminster, London) He is the founder of modern bridge construction, his crowning achievement being the Menai suspension bridge in Wales. Do you know the shape of the cables on a suspension bridge? *VFR


Menai Suspension Bridge *Wik


Telford's reputation in Shropshire led to his appointment in 1793 to manage the detailed design and construction of the Ellesmere Canal, linking the ironworks and collieries of Wrexham via the north-west Shropshire town of Ellesmere, with Chester, utilising the existing Chester Canal, and then the River Mersey.

Among other structures, this involved the spectacular Pontcysyllte Aqueduct over the River Dee in the Vale of Llangollen, where Telford used a new method of construction consisting of troughs made from cast iron plates and fixed in masonry. Extending for over 1,000 feet (300 metres) with an altitude of 126 ft (38 m) above the valley floor, the Pontcysyllte Aqueduct consists of nineteen arches, each with a 45 ft (14 m) span. Being a pioneer in the use of cast-iron for large scaled structures, Telford had to invent new techniques, such as using boiling sugar and lead as a sealant on the iron connections. Eminent canal engineer William Jessop oversaw the project, but he left the detailed execution of the project in Telford's hands. The aqueduct was designated a UNESCO World Heritage Site in 2009. *Wik

A canal boat traverses the Pontcysyllte aqueduct in North Wales



1836 William Henry FRS (12 December 1774 – 2 September 1836) was an English chemist. He was the son of Thomas Henry and was born in Manchester England. He developed what is known today as Henry's Law.

William Henry was apprenticed to Thomas Percival and later worked with John Ferriar & John Huit at the Manchesters Infirmary.  He began to study medicine at University of Edinburgh in 1795, taking his medical in 1807, but ill-health interrupted his practice as a physician, and he devoted his time mainly to chemical research, especially with regard to gases. One of his best-known papers (published in Philosophical Transactions of the Royal Society, 1803) describes experiments on the quantity of gases absorbed by water at different temperatures and under different pressures. His results are known today as Henry's law. His other papers deal with gas-analysis, fire-damp, illuminating gas, the composition of hydrochloric acid and of ammonia, urinary and other morbid concretions, and the disinfecting powers of heat. His Elements of Experimental Chemistry (1799) enjoyed considerable vogue in its day, going through eleven editions in 30 years. He was one of the founders of the Mechanics' Institute, the original precursor of University of Manchester Institute of Science and Technology.

He was elected a Fellow of the Royal Society in February 1809, having been awarded their prestigious Copley Medal in 1808.

He shot himself in his private chapel at Pendlebury, near Manchester, in 1836.




1865 Sir William Rowan Hamilton (4 Aug 1805, 2 Sep 1865) Irish mathematician in the fields of optics, geometrics, and classical mechanics. By age 12, Hamilton had already learned fourteen languages when he met the American, Zerah Colburn, who could perform amazing mental arithmetical feats, and they joined in competitions. It appears that losing to Colburn sparked Hamilton's interest in mathematics. At 15, he began studied the works of LaPlace and Newton so by age 17 had become the greatest living mathematician. He contributed to the development of optics, dynamics, and algebra. His invention of the calculus of quaternions enabled a three-dimensional algebra or geometry which provided a basis for the later development of quantum mechanics. *TIS





2002 Sheila Edmonds   (1 April 1916 – 2 September 2002) was one of the last of the old-style Cambridge dons who devoted their lives to teaching and to their colleges.
Sheila had an excellent undergraduate career ending up a `Wrangler', as students who are placed first class in the examinations for the Mathematical Tripos are called - though this did not result in a Cambridge BA degree because women were ineligible until 1947. The following year, she was awarded a distinction in the notoriously demanding Part III of the Tripos. In a speech she gave at her 80th birthday dinner, she acknowledged that a key to her success was the thorough mathematical training she received from her Director of Studies at Newnham, Margaret Grimshaw, who was 11 years her senior and another of the old-style dons. *Newnham College web page





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