Saturday, 30 May 2026

Solving Geometric Problem with Pure Logic

  I recently came across one of those geometry puzzles with multiple choice answers people like to post.

It struck me that a clever student could figure out the answer without calculations.  Before I go on I'll post the puzzle and let you play at it, but no calculating.  



You see these things all over the place if you are on the internet much.  This one at least was more straightforward than the "95% of people can't do this arithmetic problem" and the do something with order of operations using an obelus  (the spit symbol, ÷, used only in English speaking countries for division, strangely first used in a German book, Teutsche Algebra by Johann Rahn.)  

Anyway, that diversion was to take up a little space between the problem and the answer.

Since I knew the answer I read some of the different approaches in the comments. My favorite was a guy whose solution was pure logic.  

Answers a) and c) rule each other out,  since sides b and  a are the legs of the right triangle, if you switched their names, you get a different radius for congruent right triangles.  Answer b)  can't be right either.  The incircle can not be longer than the hypotenuse, because it is inside the triangle, but in any triangle, the two shorter sides are more than the longest side, which means that the b) answer woud be greater than the hypotenuse.  

So if any of them are true, d) must be the one.  

And it is true, but I'm too lazy to write it out, here is a  solution I plucked from  Quora.  



That's the kind of clever thinking that makes teachers smile.

On This Day in Math - May 30



The best review of arithmetic 
consists in the study of algebra.
~Florian Cajori


The 150th day of the year; 150 is the largest gap between consecutive twin prime pairs less than a thousand. It occurs between {659, 661} and {809, 811}. *Prime Curios

150 is a palindrome in base 4(2112), and in base 7(303) .

A Poly divisible number is an n-digit number so that for the first digit is divisible by one, the first two digits are divisible by two, the first three digits are divisible by three, etc up to n. There are 150 three-digit poly divisible numbers. Hat tip to Derek Orr .

In February of 1657 Fermat proposed a new problem to Frenicle: Find a number x which will make (ax2 + 1) a square, where a is a (nonsquare) integer.  Frenicle found solutions of the problem. In the second part of the Solutio (pp. 18–30) he cited his table of solutions for all values of a up to 150 and explained his method of solution. 

150 is the sum of eight consecutive primes starting with 7.

150 is a Harshad(joy-giver) number, divisible by the sum of its digits.

150 year celebration is called sesquicentennial of the event.
And... 150 is the number of degrees in the quincunx astrological aspect explored by Johannes Kepler.

Rubix Cube gotten too easy for you? Try the Professor's Cube, 150 movable facets.


EVENTS

*Wikipedia
1667 After much debate about the presence of a woman at a Royal Society meeting, the Duchess of Newcastle was allowed to observe a demonstration of a "experiments of colours", the "weighing of air in an exhausted receiver", and "the dissolving of flesh with a certain liquor of Mr. Boyle's suggesting." This was probably the first visit by a woman to the Royal Society. The Duchess, Margaret Cavendish, was a competent scientist in her own right. Her prolific writings in the nature of science earned her the nickname “Mad Madge”. I have a note from VFR that she was elected to FRS, but can not confirm, the note says "No other woman was elected FRS until 1945" .


1765 "Ms. Catherine Price, Daughter of the late Dr. Halley " was paid a sum of 100 Pounds for "causing to be delivered to the Commissioners of the Longitude, several of the said Dr. Halley's manuscript papers, which... may lead to discoveries useful to navigation." *Derek Howse, Britain's Board of Longitude, the Finances
Three of the Astronomers Royal are buried a mile from the Observatory in the old churchyard of St Margaret’s Lee. By a curious coincidence, the Greenwich Meridian as defined by WGS84 passes though the remains of the old church at its centre. The three Astronomers Royal are Edmond Halley, Nathaniel Bliss and John Pond. Halley and Pond are buried in the same tomb. The exact spot where Bliss is buried is unknown. His grave is unmarked.

The intriguing story of how they came to be buried at Lee rather than at St Alphege’s in Greenwich in which parish the Observatory is located, leaves many questions unanswered. So too does the available information on the restoration of Halley’s tomb in 1854.
A brief history of the church at Lee
In the time between Halley being buried in 1742 and Pond joining him in the same grave in 1836, the church of St Margaret was largely rebuilt. The original church of St Margaret was of medieval origin. By the start of the nineteenth century, it was in a poor state of repair. Apart from part of the tower, it was pulled down in 1813 to make way for a new building on the same site. This was designed by Joseph Gwilt and incorporated the original tower in modified form. Gwilt’s church was built on the old foundations and like its predecessor suffered from structural problems. The population of Lee grew rapidly after the church was built and it was soon apparent that a much larger church was needed. Most of Gwilt’s church was demolished on 31 May 1841 having been replaced by the present larger church which was built on the opposite side of the road and constructed between 1839–41 to the designs of John Brown. The new church was later altered by the architect James Brooks during the last quarter of the nineteenth century. The current church is grade II* listed whilst the ruined tower of the previous two churches is grade II listed.
The old churchyard as seen from Lee Terrace on 5 October 2017. The arrow (right) shows the location of Halley’s tomb. An information board with a plan of the graveyard can be seen bottom right.






1832 Galois mortally wounded by a gunshot wound to the abdomen in a duel of honor. He was left for dead after the duel but a peasant took him to a hospital. *VFR

The infamous duel with Pescheux d'Herbinville took place near the Glassier pond in the southern suburb of Gentilly. The duel was over Galois's involvement with Stéphanie-Félicie Poterine du Motel, who was d'Herbinville's fiancée, but it has been claimed that the affair was a political frame-up by government agents in order to eliminate Galois He died in the Cochin Hospital – this is now at 27 Rue du Faubourg St. Jacques, 14e, but I don't know how long it has been there. He was buried in a common grave at Montparnasse Cemetery, but no trace of the grave remains.
The Galois memorial in the cemetery of Bourg-la-Reine.
Évariste Galois was buried in a common grave and
 the exact location is still unknown. *Wik




1896   Widely considered to be the real first accident, this occurred on May 30, 1896, during a “horseless wagon race” in New York City. Henry Wells lost control of his vehicle and crashed into a bicyclist named Ebeling Thomas. The bicyclist broke his leg, and the driver was arrested. If only there was a New York Defensive Driving course back then, a lot of chaos could have been avoided. However, as there were several other bicyclists arrested that day for the 1896 equivalent of speeding, perhaps a certain amount of chaos was just par for the course at that time.*Improv




1903 Minor planet 511 Davida Discovered 1903 May 30 by R. S. Dugan at Heidelberg. Named by the discoverer in honor of David P. Todd (1855-1939), professor of astronomy and director of the Amherst College Observatory (1881-1920). David Todd was the husband of Mabel Todd, who wrote books about solar eclipses. David has also a drawing of a painting of a solar eclipse in one of his books. *NSEC
511 Davidais is a large C-type asteroid. It is one of the largest asteroids; approximately tied for 7th place, to within measurement uncertainties, and the 5th or 6th most massive.
almost a decagon in this cross section



 In 1971, the U.S. Mars space probe Mariner 9 blasted off from Cape Kennedy, Florida. It carried cameras, infrared spectrometer and radiometer, ultraviolet spectrometer, radio occultation and celestial mechanics instruments. On 13 Nov 1971, it entered orbit as the first artificial satellite of Mars. After waiting for a month-long planet-wide dust storm to clear, it began compiling a global mosaic of high-quality images for 100% of the Martian surface. The photos showed gigantic volcanoes, a grand canyon stretching 4,800 kilometers (3,000 miles) and relics of ancient riverbeds that were carved in the landscape of this seemingly dry and dusty planet. It also sent the first closeup pictures of the two Martian moons, Phobos and Deimos. *TIS




2000 almost 200 years after the (now called) tangrams exploded across Europe, the nation of Finland issued a stamp panel designed as a tangram square.  Only four of the seven shapes were postage stamps.  Each tangram shape featured an idea of education and science.  One triangle showed a Sierpinski triangle.



BIRTHS

1423 Georg von Peurbach (or Peuerbach) (May 30, 1423 in Peuerbach near Linz – April 8, 1461 in Vienna)  He worked on trigonometry astronomy, and was the teacher of Regiomontanus. *VFR
He promoted the use of Arabic numerals (introduced 250 years earlier in place of Roman numerals), especially in a table of sines he calculated with unprecedented accuracy. He died before this project was finished, and his pupil, Regiomontanus continued it until his own death. Peurbach was a follower of Ptolomy's astronomy. He insisted on the solid reality of the crystal spheres of the planets, going somewhat further than in Ptolomy's writings. He calculated tables of eclipses in Tabulae Ecclipsium,observed Halley's comet in Jun 1456 and the lunar eclipse of 3 Sep 1457 from a site near Vienna. Peurbach wrote on astronomy, his observations and devised astronomical instruments. *TIS  The Renaissance Mathematicus has a nice piece about Peurbach and his life... the kind of detail that comes from a passion for his subject.  Check it out. 
Georg von Peuerbach:
Theoricarum novarum
planetarum testus, Paris 1515
*Wik


1743 Robert Patterson (May 30, 1743 – July 22, 1824) was an Irish-American mathematician and director of the United States Mint from 1806 to 1824. He was a professor of mathematics at the University of Pennsylvania from 1779 to 1810, professor of natural history and mathematics and vice provost from 1810 to 1813. At the request of Thomas Jefferson, he advised Meriwether Lewis on the purchase and usage of navigational equipment for the Lewis and Clark Expedition.
He taught at schools in Hinkletown and Northampton, Pennsylvania. He moved to Philadelphia and opened a school to teach navigational mathematics to ship captains. One of his students was Andrew Ellicott who became a notable surveyor. He operated a country store in Bridgeton, New Jersey for two years.

In 1774, he became principal of Wilmington Academy in Wilmington, Delaware. Classes were suspended at the outbreak of the American Revolutionary War and he served in the war for about three years as a military instructor, an assistant surgeon, adjutant to the 1st Delaware Regiment under John Haslet,  and as a brigade major.

He worked as a professor of mathematics at the University of Pennsylvania from 1799 to 1810 as well as professor of natural history and mathematics and vice provost from 1810 to 1813. He was elected president of Philadelphia's Select Council in 1799.

He was elected a member of the American Philosophical Society in 1783, served as secretary in 1784, vice-president in 1799 and president from 1819 to 1823. He published The Newtonian System (1808) and edited various works on mathematics and physics.

In 1803, Thomas Jefferson wrote a letter to Patterson requesting that he meet with Meriwether Lewis and provide further instruction and advice on calculating latitude and longitude during the Lewis and Clark Expedition. Patterson was one of five American Philosophical Society members who were consulted by Lewis prior to the expedition. In anticipation of the visit from Lewis, Patterson began to calculate astronomical formulas for usage on the expedition for the calculation of longitude from lunar observations and for altitude and time.

Patterson advised Lewis on the navigational equipment to purchase for the expedition and trained him on their usage. Jefferson recommended that Lewis use a theodolite for the calculation of latitude and longitude, however Patterson recommended usage of a sextant instead since it would handle better under the rigors of field work. Patterson advised Lewis on the purchase of a chronometer necessary for calculation of latitude and longitude as well as other techniques in case the chronometer malfunctioned. The chronometer was purchased in Philadelphia for $250, the most expensive single item purchased for the expedition.

Patterson was interested in ciphers and regularly exchanged coded correspondence with Thomas Jefferson. One of Patterson's ciphers included in a December 19, 1801, dated letter to Jefferson was decoded in 2007 by Lawren Smithline. The cipher consists of 7 digit pairs and is decoded by decrypting 7 blocks at a time. The cipher was of the Declaration of Independence, of which Jefferson was the primary author. Patterson called it his "perfect cipher" and Jefferson considered adopting it for government use.

Jefferson appointed Patterson as director of the United States Mint in 1805 and he served in this role until his death. Patterson was one of the founders of the Franklin Institute in Philadelphia and served as the first chairman of their board of managers.

He died on July 22, 1824, and was initially interred in a churchyard in Philadelphia and reinterred in 1844 at Laurel Hill Cemetery along with his wife after her death.



1790 John Herapath (30 May 1790 – 24 February 1868) was an English physicist who gave a partial account of the kinetic theory of gases in 1820 though it was neglected by the scientific community at the time.  An English physicist and journalist, he was self-educated in mathematics and science. An early interest investigating a theory of lunar motion (1811) led to considering the nature of heat and derived an equation relating the pressure and volume of a gas to the number, mass and speed of its particles. He published a preliminary notice of his theory in Annals of Philosophy in 1816. In his later career, he took an interest in steam-powered transportation, and became the editor (1836) of Railway Magazine and Annals of Science. He published in it his own scientific papers, including one giving a calculation (1932) on the speed of sound in air, which is the first known calculation of the mean molecular speed of a molecule from the kinetic theory of gases, though it is often Joule's later work that is recognized for this accomplishment.*TIS




1800 Karl Wilhelm Feuerbach (30 May 1800 – 12 March 1834) born in Jena, Germany. His mathematical fame rests entirely on three papers. Most important was this contribution to Euclidean geometry: The circle which passes through the feet of the altitudes of a triangle touches all four of the circles which are tangent to the three sides; it is internally tangent to the inscribed circle and externally tangent to each of the circles which touches the sides of the triangle externally. *VFR

The circle is also commonly called the Nine-point circle. It passes through the feet of the altitudes, the midpoints of the three sides, and the point half way between the orthocenter and the vertices.


1814 Eugene Charles Catalan (30 May 1814 – 14 February 1894) was a Belgian mathematician who defined the numbers called after him, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. *SAU The Catalan numbers have a multitude of uses in combinatorics.  There are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. 
One of the ways is Cn is the number of different ways n + 1 factors can be completely parenthesized (or the number of ways of associating n applications of a binary operator, as in the matrix chain multiplication problem). For n = 3, for example, we have the following five different parenthesizations of four factors:
((ab)c)d     (a(bc))d     (ab)(cd)     a((bc)d)     a(b(cd))

The first Catalan numbers for n = 0, 1, 2, 3, ... are

1, 1, 2, 5, 14, 42, 132, 429

A convex polygon with n + 2 sides can be cut into triangles by connecting vertices with non-crossing line segments (a form of polygon triangulation). The number of triangles formed is n and the number of different ways that this can be achieved is Cn. The following hexagons illustrate the case n = 4:

*Wik



*Wik


1874  Beatrice Mabel Cave-Browne-Cave, MBE AFRAeS (30 May 1874 – 9 July 1947) was an English mathematician who undertook pioneering work in the mathematics of aeronautics.  She studied the mathematical tripos at Girton College Cambridge. She taught mathematics at a High School for eleven years before becoming an assistant to Karl Pearson in the Galton Laboratory of University College, London. She later became an assistant to Leonard Bairstow in the Department of Aeronautics at the Imperial College, London. She published two papers with Pearson and two with Bairstow.
Thumbnail of Beatrice Mabel Cave-Browne-Cave*SAU
In 1916, Cave began working for the government on airplane design. She carried out original research for the government on the mathematics of aeronautics which remained classified under the Official Secrets Act for fifty years. She examined the effects of loads on different areas of planes during flight, and her research helped to improve aircraft stability and propeller efficiency. Some of her works are held in UCL archives which include correspondence from her time at the Galton Laboratory for work on bomb trajectories, terminal velocities, timber tests, and detonators, for the Admiralty Air Department and Ministry of Munitions. 

Cave was elected an associate fellow of the Royal Aeronautical Society in 1919 and awarded an MBE in 1920. She later worked as an assistant to Sir Leonard Bairstow, the Zaharoff Professor of Aviation at Imperial College, and she worked on fluid motion. In 1922, Cave's studies on aircraft oscillations were published in an Advisory Committee for Aeronautics technical report. Cave's name was also included alongside Bairstow in his 1922 and 1923 published reports on fluid mechanics.*Wik




1889 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




1908 Hannes Olof Gösta Alfvén (30 May 1908 in Norrköping, Sweden; 2 April 1995 in Djursholm, Sweden) Alfvén developed the theory of magnetohydrodynamics (MHD), the branch of physics that helps astrophysicists understand sunspot formation and the magnetic field-plasma interactions (now called Alfvén waves in his honor) taking place in the outer regions of the Sun and other stars. For this pioneering work and its applications to many areas of plasma physics, he shared the 1970 Nobel Prize in physics. *DEBORAH TODD AND JOSEPH A. ANGELO, JR., A TO Z OF SCIENTISTS IN SPACE AND ASTRONOMY

In his book, Free Radicals, Michael Brooks described Alfven this way:
" He continuously broke new ground after invading fields of research he had no formal qualifications.  He summarily dismissed the views of experts, having no truck with 'received wisdom'.  He never waited to be proved right before moving on to wreak havoc elsewhere. He received little acknowledgement for his contributions; even the physicist who used his work had no idea where it came from.  Alfven's Nobel Prize was for work he had carried out in the 1930's. It is surely no coincidence that this was just a few months after the death of one Sidney Chapman."





1909 Norris Edwin Bradbury (May 30, 1909 – August 20, 1997), was an American physicist who served as director of the Los Alamos National Laboratory for 25 years from 1945 to 1970. He succeeded Robert Oppenheimer, who personally chose Bradbury for the position of director after working closely with him on the Manhattan Project during World War II. Bradbury was in charge of the final assembly of "the Gadget", detonated in July 1945 for the Trinity test. *Wik



1916  Joseph William Kennedy (May 30, 1916 – May 5, 1957) was an American chemist who was a co-discoverer of plutonium, along with Glenn T. Seaborg, Edwin McMillan and Arthur Wahl. During World War II he was head of the CM (Chemistry and Metallurgy) Division at the Manhattan Project's Los Alamos Laboratory, where he oversaw research onto the chemistry and metallurgy of uranium and plutonium. After the war, he was recruited as a professor at Washington University in St. Louis, where he is credited with transforming a university primarily concerned with undergraduate teaching into one that also boasts strong graduate and research programs. He died of cancer of the stomach at the age of 40.





1927 Joan Sylvia Lyttle Birman (born May 30, 1927, in New York City) is an American mathematician, specializing in low-dimensional topology. She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical systems. Birman is research professor emerita at Barnard College, Columbia University, where she has been since 1973.n 1974, Birman was selected as a Sloan Research Fellow by the Alfred P. Sloan Foundation. In 1987, she was selected by the Association for Women in Mathematics to be a Noether Lecturer; this lecture honors women who have made fundamental and sustained contributions to the mathematical sciences. In 1994, she was selected as a Guggenheim Foundation Fellow by the John Simon Guggenheim Memorial Foundation. In 1996, the Mathematical Association of America awarded Birman the Chauvenet Prize, "the highest award for mathematical expository writing" for her 1993 essay New Points of View in Knot Theory.

In 2003, Birman was elected to the European Academy of Sciences. In 2005, she won the New York City Mayor's Award for Excellence in Science and Technology.

Birman received an honorary doctorate from the Technion Israel Institute of Technology.
In 2012, Birman was elected to the American Academy of Arts and Sciences In 2013, she became a fellow of the American Mathematical Society in the inaugural class.

In 2013 the Association for Women in Mathematics established the Joan & Joseph Birman Research Prize in Topology and Geometry, first awarded in 2015.

In 2015, Birman was named an honorary member of the London Mathematical Society.
The Association for Women in Mathematics included her in the 2020 class of AWM Fellows for "her groundbreaking research connecting diverse fields, and for her award-winning expository writing; for continuously supporting women in mathematics as an active mentor and a research role model; and for sponsoring multiple prize initiatives for women".
In 2021, Birman was elected to the National Academy of Sciences.
She is included in a deck of playing cards featuring notable women mathematicians published by the Association of Women in Mathematics







DEATHS

1778 (François Marie Arouet) Voltaire (
21 November 1694 – 30 May 1778) was a French author who popularized Isaac Newton's work in France by arranging a translation of Principia Mathematica to which he added his own commentary (1737). The work of the translation was done by the marquise de Châtelet who was one of his mistresses, but Voltaire's commentary bridged the gap between non-scientists and Newton's ideas at a time in France when the pre-Newtonian views of Descartes were still prevalent. Although a philosopher, Voltaire advocated rational analysis. He died on the eve of the French Revolution.*TIS




1912 Wilbur Wright  (April 16, 1867 – May 30, 1912), American aviation pioneer, who with his brother Orville, invented the first powered airplane, Flyer, capable of sustained, controlled flight (17 Dec 1903). Orville made the first flight, airborn for 12-sec. Wilbur took the second flight, covering 853-ft (260-m) in 59 seconds. By 1905, they had improved the design, built and and made several long flights in Flyer III, which was the first fully practical airplane (1905), able to fly up to 38-min and travel 24 miles (39-km). Their Model A was produced in 1908, capable of flight for over two hours of flight. They sold considerable numbers, but European designers became strong competitors. After Wilbur died of typhoid in 1912, Orville sold his interest in the Wright Company in 1915 *TIS





1926 Vladimir Andreevich Steklov (9 January 1864 – 30 May 1926)  made many important contributions to applied mathematics. In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface. He wrote General Theory of Fundamental Functions in which he examined expansions of functions as series in an infinite system of orthogonal eigenfunctions. In fact the term "Fundamental Functions", which is due to Poincaré, means eigenfunctions in today's terminology.
Steklov was not the first to examine series expansions in terms of infinite sets of orthogonal eigenfunctions, of course Fourier had examined a special case of this situation many years before. Steklov, however, produced many papers on this topic which led him to a general theory to replace the special cases examined by others. He studied a generalisation of Parseval's equality for Fourier series to his general setting showing this to be a fundamental property. In all his list of publications contains 154 items. *SAU




1943 Anderson McKendrick (September 8, 1876 - May 30, 1943) trained as a medical doctor in Glasgow and came to Edinburgh as Superintendent of the College of Physicians Laboratory. He made some significant mathematical contributions to biology. *SAU
McKendrick's career as a mathematical epidemiologist began in India. In 1911, McKendrick rediscovered the logistic equation and fit it to bacterial growth data. In 1912 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were James Oliver, Diarmid Noel Paton, Ralph Stockman and Cargill Gilston Knott. He served as the Society's Vice President 1933-36. In 1933 he was elected a member of the Aesculapian Club.

In 1914 he published a paper in which he gave equations for the pure birth process and a particular birth–death process. In 1924 he was elected a Fellow of the Royal College of Physicians of Edinburgh. After his return to Scotland he published more. His 1926 paper, 'Applications of mathematics to medical problems' was particularly impressive, including the widely used McKendrick–Von Foerster partial differential equation.

Some of this paper's other results for stochastic models of epidemics and population growth were rediscovered by William Feller in 1939. Feller remarks in his An Introduction to Probability Theory and Its Applications (3rd edition p. 450), "It is unfortunate that this remarkable paper passed practically unnoticed." In 1927 McKendrick began a collaboration with William Ogilvy Kermack (1898–1970) which produced a notable series of papers on the Kermack–McKendrick theory, a general theory of infectious disease transmission.

W. M. Hirsch gives this picture of the man: "McKendrick was a truly Christian gentleman, a tall and handsome man, brilliant in mind, kind and modest in person, a skilful counsellor and administrator who gave of himself and knew how to enable others."*Wik




1964 Leo Szilard (11 Feb 1898; 30 May 1964 at age 66) Hungarian-American physicist who, with Enrico Fermi, designed the first nuclear reactor that sustained nuclear chain reaction (2 Dec 1942). In 1933, Szilard had left Nazi Germany for England. The same year he conceived the neutron chain reaction. Moving to N.Y. City in 1938, he conducted fission experiments at Columbia University. Aware of the danger of nuclear fission in the hands of the German government, he persuaded Albert Einstein to write to President Roosevelt, urging him to commission American development of atomic weapons. In 1943, Major General Leslie Groves, leader of the Manhattan Project designing the atomic bomb, forced Szilard to sell his atomic energy patent rights to the U.S. government. *TIS Frederik Pohl , talks about Szilard's epiphany about chain reactions in Chasing Science (pg 25),
".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."





1992 Antoni Zygmund (December 25, 1900 – May 30, 1992) Polish-born mathematician who created a major analysis research centre at Chicago, and recognized in 1986 for this with the National Medal for Science. In 1940, he escaped with his wife and son from German controlled Poland to the USA. He did much work in harmonic analysis, a statistical method for determining the amplitude and period of certain harmonic or wave components in a set of data with the aid of Fourier series. Such technique can be applied in various fields of science and technology, including natural phenomena such as sea tides. He also did major work in Fourier analysis and its application to partial differential equations. Zygmund's book Trigonometric Series (1935) is a classic, definitive work on the subject.



2008 Gilbert Agnew Hunt, Jr. (March 4, 1916 – May 30, 2008) was an American mathematician and amateur tennis player active in the 1930s and 1940s.  He was born in Washington, D.C. and attended Eastern High School.

Hunt reached the quarterfinals of the U.S. National Championships in 1938 and 1939.

Hunt received his bachelor's degree from George Washington University in 1938 and his Ph.D. from Princeton University in 1948 under Salomon Bochner. Hunt became a mathematics professor at Princeton University specializing in probability theory, Markov processes, and potential theory.

The Hunt process is named after him. He was an Invited Speaker at the ICM in 1962 in Stockholm. His doctoral students include Robert McCallum Blumenthal and Richard M. Dudley.

Hunt's theorem

Hunt's theorem states that for a large class of positive kernels 

\(𝑉 f =  \int_{0}^{\inf} Pt f \,dt \   \)   

V is called the "potential kernel" of the semigroup.




2011 Rosalyn Sussman Yalow (July 19, 1921 – May 30, 2011) was an American biophysicist who shared (with Andrew V. Schally and Roger Guillemin) the 1977 Nobel Prize for Physiology or Medicine, making her the second woman to win the Nobel Prize in medicine, “for the development of radioimmuno assays (RIA) of peptide hormone.” RIA brought about a revolution in biological and medical research. With her coworkers, she applied RIA to study of the physiology of the peptide hormones insulin, ACTH, growth hormone, and also to throw light upon the pathogenesis of diseases caused by abnormal secretion of these hormones. This was pioneering work that opened diabetes research in new directions. She has been called the “Madame Curie of the Bronx..” *TiS





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


On This Day in Math - May 30

Friday, 29 May 2026

The Distracted Goalie

   



The great Physicist, Niels Bohr, was brother to an outstanding mathematician, Harald who founded the field of almost periodic functions. In their youth both were very good athletes, with Harald clearly the more dedicated sportsman. Harald had been a member of the Danish National Football team while still a student and had earned a silver medal as such in the 1908 Olympic games; the first time the Olympic games had football. Harald scored two goals in the opening game defeating the French nine-zero. Denmark lost to the UK in the final game. He was such an accomplished football player that it is said when he defended his PhD thesis there were more football fans in the audience than mathematicians.
Brother Niels was also a good athlete, but often seemed to have his focus somewhere other than sports. Both brothers played several games for the Copenhagen-based Akademisk Boldklub, with Niels in goal. The story is told that during one game when almost all the action was happening in the attacking half for his club, a long clearing kick from the other end of the field began to roll toward his goal. Niels stood near the goalpost and seemed unaware of the ball rolling toward his goal with players rushing in from many yards behind it. Alerted by the screaming crown behind him, Niels made the save and cleared away the threat.
After the game his explanation was that he had been distracted by a math problem and was carrying out calculations on the edge of the goal post.
Apparently he kept his love for the game.  The photo at the top shows Niels Bohr with  a group that is unidentified, but he looks to me to be the one handling the ball.  A goal-keeper to the end, it seems.

An Anon. comment suggested, "I think there's also Gamow (upright), Pauli (back to camera) and possibly Heisenberg opposite Bohr. Must have been during a conference."  If anyone can spot brother Harald, and if someone recognizes young version of later great science/math wizards, share, please

On This Day in Math - May 29

  ​



No matter how correct a mathematical theorem
may appear to be, one ought never to be satisfied
that there was not something imperfect
about it until it also gives
the impression of being beautiful.
~ George Boole


The 149th day of the year; There are 149 ways to put 8 queens on a 7-by-7 chessboard so that each queen attacks exactly one other queen. *Prime Curios

also 149 = 62 + 72 + 82.(note that the digits 1, 4, 9 are squares also)

And Derek Orr noted that the sum of the digits of 149, \(1 + 4 + 9 = 14 = 1^2 + 2^2 + 3^2 \)

149 is the smallest 3-digit prime with distinct digits in each position such that inserting a zero between any two digits creates a new prime (that is, 1049 & 1409 are both prime).

149 is the 35th prime number, and a twin prime with 151.

149 is an Emirp since 941, its reversal, is also a prime.

149 in binary is 10010101. The zeros are in prime positions 2, 3, 5, and 7, when read left-to-right. These are the four single digit prime numbers.*Prime Curios

149 is a strictly non-palindromic number, it is not a palindrome in any base from 2 to 147.

149 is a full reptend prime, its reciprocal is 148 digits long, 1/149 repeats 0067114093959731543624161073825503355704697986577181208053691275167785234899328859060402684563758389261744966442953020134228187919463087248322147651 indefinitely.





EVENTS

1733  Euler names (or mis-names) the "Pell Equations" and gives a method of multiple solutions. "Euler’s first excursion into Pell’s equation was his 1732 paper E-29, bearing a title that translates
as “On the solution of problems of Diophantus about integer numbers.” The main result of this paper is to show how certain quadratic Diophantine equations can be reduced to the Pell equation. In particular, he shows that if we can find a solution to the Diophantine equation \(y^2 = an^2 + bn + c \) and we can find solutions to the Pell equation, \(q^2 = ap^2 +1\), then we can use the solutions to the Pell equation to construct more solutions to the original Diophantine equation. He also shows how to use two solutions to a Pell equation to construct more solutions, and notes that solutions to a Pell equation give good rational approximations for the square root of a.  (Ed Sandifer, Euler and Pell, How Euler Did It. MAA) .
As early as 400 BC in India and Greece, mathematicians studied the numbers arising from the n = 2 case of Pell's equation, The first general method for solving the Pell's equation (for all N) was given by Bhāskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method,  
Several European mathematicians rediscovered how to solve Pell's equation in the 17th century. Pierre de Fermat found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to Kenelm Digby, Bernard Frénicle de Bessy said that Fermat found the smallest solution for N up to 150 and challenged John Wallis to solve the cases N = 151 or 313. Both Wallis and William Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker.

John Pell's connection with the equation is that he revised Thomas Branker's translation[ of Johann Rahn's 1659 book Teutsche Algebra into English, with a discussion of Brouncker's solution of the equation. Leonhard Euler mistakenly thought that this solution was due to Pell, as a result of which he named the equation after Pell.


Euler


Pell




1832 Almost certain that he would die in a duel the next day, Evariste Galois first wrote “Letter to all Republicans,” and then wrote to a friend (Auguste Chevalier) describing his mathematics. It ended: “Eventually there will be, I hope, some people who will find it profitable to decipher this mess.” [Burton, History of Mathematics, p. 322]. See Smith, Source Book, pp. 278–285 for the letter. *VFR

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




1898 the heirs of Alfred Nobel sign a "reconciliation agreement" so that lawyers and accountants can execute his will. The will's major bequest was to create the Nobel Prizes, but first, there were disputes to be settled.*TIS




1919 Proof of the general theory of relativity was observed during a total solar eclipse. São Tomé and Príncipe, officially the Democratic Republic of São Tomé and Príncipe, is a Portuguese-speaking island nation in the Gulf of Guinea, off the western equatorial coast of Central Africa. Príncipe was the site where astronomical observations of the total solar eclipse of 29 May 1919 confirmed Einstein's prediction of the curvature of light. The expedition was sponsored by the Royal Society and led by Sir Arthur Stanley Eddington. A solar eclipse permitted observation of the bending of starlight passing through the sun's gravitational field, as predicted by Einstein's theory of relativity. Separate expeditions of the Royal Astronomical Society travelled to Brazil and off the west coast of Africa. Both made measurements of the position of stars visible close to the sun during a solar eclipse. These observations showed that, indeed, the light of stars was bent as it passed through the gravitational field of the sun. The verification of predictions of Einstein's theory, proved during the solar eclipse was a dramatic landmark scientific event. *Wik





1957 Romania issued two stamps picturing a slide rule to publicize the 2nd Congress of the Society of Engineers and Technicians, which began in Bucharest on this day. [Scott #1159-60].
For the younger set... If you never used (saw) a slide rule, there is actually an online java app that you can simulate the use of one at this page.  The other instrument is a vernier caliper, used for measuring outside dimensions, inside diameter, and often a small depth measure.








2017 The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. (Time Hack, that is before 1700)
In 1998, Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. So now, 300 years have passed and we have a proof........Maybe.

In 2002  Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem for publication . 

In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Mathematics, Pi.  (time hack, 317+ years after Kepler, we have a proof.






BIRTHS

1675 Humphry Ditton (May 29, 1675 – October 15, 1715) was born at Salisbury and died in London in 1715 at Christ's Hospital, where he was mathematical master. He does not seem to have paid much attention to mathematics until he came to London about 1705. W. W. Rouse Ball states that Ditton's 1706 book on fluxions occupied a place in English education equivalent to L'Hospital's book in France.





1859 John Walker (29 May 1781 – 1 May 1859) was an English inventor who invented the friction match.
He made them from small wooden sticks which he coated with sulphur, then tipped with a mixture of potassium chlorate, antimony sulphide and a binder of gum arabic. After searching for a suitable mixture with the intent of making a useful way to start a fire, he was successful on 27 Nov 1826. Beginning on 7 Apr 1827, he sold them in boxes of 50 for a shilling, with a folded slip of sandpaper as a striking surface. He called them Congreves, to honour Sir William Congreve, known for his invention of military rockets. He declined to patent the matches, yet was still able to make a comfortable income from them.  *TIS

He did not name the matches "Congreves" in honour of the inventor and rocket pioneer, Sir William Congreve as it is sometimes stated. The congreves were the invention of Charles Sauria, a French chemistry student at the time. He did not divulge the exact composition of his matches.

Two and a half years after Walker's invention was made public Isaac Holden arrived, independently, at the same idea of coating wooden splinters with sulphur. The exact date of his discovery, according to his own statement, was October 1829. Before that date Walker's sales-book contains an account of no fewer than 250 sales of friction matches, the first entry dated 7 April 1827.
 The credit for his invention was attributed only after his death.







1794 Johann Heinrich von Mädler  (29 May 1794, 14 Mar 1874 at age 79) German astronomer who (with Wilhelm Beer) published the most complete map of the Moon of the time, Mappa Selenographica, 4 vol. (1834-36). It was the first lunar map to be divided into quadrants, and it remained unsurpassed in its detail until J.F. Julius Schmidt's map of 1878. Mädler and Beer also published the first systematic chart of the surface features of the planet Mars (1830). *TIS






1882 Harry Bateman (29 May 1882 – 21 January 1946) He spent much of his life collecting special functions and integrals that solved partial differential equations. He kept the references on index cards stored in shoe boxes—eventually these began to crowd him out of his office. [DSB 1, 500] *SAU

With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poincare to a more expansive conformal group of spacetime leaving Maxwell's equations invariant. Moving to the US, he obtained a Ph.D. in geometry with Frank Morley and became a professor of mathematics at California Institute of Technology. There he taught fluid dynamics to students going into aerodynamics with Theodore von Karman. Bateman made a broad survey of applied differential equations in his Gibbs Lecture in 1943 titled, "The control of an elastic fluid".*Wik




1885 Finlay Freundlich (May 29, 1885 – July 24, 1964) was a distinguished German astronomer who worked with Einstein on measurements of the orbit of Mercury to confirm the general theory of relativity. He left Germany to avoid Nazi rule and became the Napier Professor of Astronomy at St Andrews.




1906 Gerrit Bol (May 29, 1906 in Amsterdam, Nov 1, 1989) was a Dutch mathematician, who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points.
Bol earned his PhD in 1928 at Leiden University under Willem van der Woude. In the 1930s, he worked at the University of Hamburg on the geometry of webs under Wilhelm Blaschke and later projective differential geometry. In 1931 he earned a habilitation.
In 1942–1945 during World War II, Bol fought on the Dutch side, and was taken prisoner. On the authority of Blaschke, he was released. After the war, Bol became professor at the Albert-Ludwigs-University of Freiburg, until retirement there in 1971. *Wik




1911 George Szekeres (29 May 1911 – 28 August 2005) was a Hungarian-born mathematician who worked for most of his life in Australia on geometry and combinatorics. *SAU

Szekeres worked closely with many prominent mathematicians throughout his life, including Paul Erdős, Esther Szekeres (née Esther Klein), Paul Turán, Béla Bollobás, Ronald Graham, Alf van der Poorten, Miklós Laczkovich, and John Coates.

The so-called Happy Ending problem is an example of how mathematics pervaded George's life. During 1933, George and several other students met frequently in Budapest to discuss mathematics. At one of these meetings, Esther Klein proposed the following problem:

Given five points in the plane in general position, prove that four of them form a convex quadrilateral.

After allowing George, Paul Erdős, and the other students to scratch their heads for some time, Esther explained her proof. Subsequently, George and Paul wrote a paper (1935) that generalizes this result; it is regarded as one of the foundational works in the field of combinatorial geometry. Erdős dubbed the original problem the "Happy Ending" problem because it resulted in George and Esther's marriage in 1937.

George and Esther died within an hour of each other, on the same day, 28 August 2005, in Adelaide, Australia.*Wik

My story of the "Happy Ending" story is here.



1918 David Rees FRS (29 May 1918 – 16 August 2013) was a British professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter from 1958 to 1983. During the Second World War, Rees was active on Enigma research in Hut 6 at Bletchley Park.

Rees won a scholarship to Sidney Sussex College, Cambridge, supervised by Gordon Welchman and graduating in summer 1939. On completion of his education, he initially worked on semigroup theory; the Rees factor semigroup is named after him. He also characterised completely simple and completely 0-simple semigroups, in what is nowadays known as Rees's theorem. The matrix-based semigroups used in this characterisation are called Rees matrix semigroups.

Later in 1939, Welchman drafted Rees into Hut 6, Bletchley Park, for the war effort. He was credited with the first decode using the Herivel tip. He was subsequently seconded to the Enigma Research Section, where the Abwehr Enigma was broken, and later to the Newmanry, where the Colossus computer was built.

After the war, Rees was appointed an assistant lecturer at Manchester University in 1945 and a full lecturer at University of Cambridge in 1948. In 1949, he was a Fellow of Downing College.

At the behest of Douglas Northcott he switched his research focus to commutative algebra. In 1954, in a joint paper with Northcott, Rees introduced the Northcott–Rees theory of reductions and integral closures, which has subsequently been influential in commutative algebra. In 1956 he introduced the Rees decomposition of a commutative algebra.

In 1958, Rees and his family moved to Exeter, where he had been appointed to the Chair of Pure Mathematics. In 1959, he was awarded a DSc by the University of Cambridge.

According to Craig Steven Wright, Rees was the third part of the Satoshi team that created Bitcoin.*Wik




1929 Günter Lumer (1929–2005) was a mathematician known for his work in functional analysis. He is the namesake of the Lumer–Phillips theorem on semigroups of operators on Banach spaces, and was the first to study L-semi-inner products. Born in Germany and raised in France and Uruguay, he spent his professional career in the United States and Belgium.

Following short-term positions at the University of California, Los Angeles and Stanford University, he joined the faculty at the University of Washington in 1961. He moved to the University of Mons-Hainaut in 1973, and then to the International Solvay Institutes for Physics and Chemistry in Brussels in 1999, where he remained until his death in 2005




1929 Peter Ware Higgs (29 May 1929 -  8 April 2024) is an English theoretical physicist, the namesake of the Higgs boson. In the late 1960s, Higgs and others proposed a mechanism that would endow particles with mass, even though they appeared originally in a theory - and possibly in the Universe! - with no mass at all. The basic idea is that all particles acquire their mass through interactions with an all-pervading field, called the Higgs field. which is carried by the Higgs bosons. This mechanism is an important part of the Standard Model of particles and forces, for it explains the masses of the carriers of the weak force, responsible for beta-decay and for nuclear reactions that fuel the Sun. The particle was discovered on 4 July 2012 at the Large Hadron Accelerator.





1957 Jean-Christophe Yoccoz ( May 29, 1957 -   )  French mathematician who was awarded the Fields Medal in 1994 for his work in dynamical systems. Such studies began with Poincaré about the turn of the 20th century, who considered the stability of the solar system. It evolves according to Newton's laws but will it remain stable or, might a planet be ejected from the system? The techniques apply also in biology, chemistry, mechanics, and ecology where stability is an issue. This work also produces aesthetically appealing objects, such as the Julia and Mandelbrot fractal sets. Yoccoz was primarily concerned with establishing criteria that gave precise bounds on the validity of stability theorems. A combinatorial method for studying the Julia and Mandelbrot sets was named "Yoccoz puzzles." *TIS




DEATHS

1660 Frans van Schooten (1615 in Leiden – 29 May 1660 in Leiden) was a Dutch mathematician who was one of the main people to promote the spread of Cartesian geometry. Van Schooten's father was a professor of mathematics at Leiden, having Christiaan Huygens, Johann van Waveren Hudde, and René de Sluze as students.
Van Schooten read Descartes' Géométrie (an appendix to his Discours de la méthode) while it was still unpublished. Finding it hard to understand, he went to France to study the works of other important mathematicians of his time, such as François Viète and Pierre de Fermat. When Frans van Schooten returned to his home in Leiden in 1646, he inherited his father's position and one of his most important pupils, Huygens.
Van Schooten's 1649 Latin translation of and commentary on Descartes' Géométrie was valuable in that it made the work comprehensible to the broader mathematical community, and thus was responsible for the spread of analytic geometry to the world. Over the next decade he enlisted the aid of other mathematicians of the time, de Beaune, Hudde, Heuraet, de Witt and expanded the commentaries to two volumes, published in 1659 and 1661. This edition and its extensive commentaries was far more influential than the 1649 edition. It was this edition that Gottfried Leibniz and Isaac Newton knew.
Van Schooten was one of the first to suggest, in exercises published in 1657, that these ideas be extended to three-dimensional space. Van Schooten's efforts also made Leiden the centre of the mathematical community for a short period in the middle of the seventeenth century. *Wik    Thony Christie (aka The Renaissance Mathematicus) sent me a comment to tell me that it was van Schooten who first used rectangular coordinates in his translations and extensions of Descartes Geometry.  The MAA Digital Library has seven images from van Schooten's "Exercitationes mathematicae". The copy was once the property of his student, Johann Hudde, and include problems from the book of another of his famous students, Christian Huygen's Ludo aleae.

Thony Christie added a note about van Schooten's contributions:
If you read La Géométrie you will search for rectangular co-ordinates in vain, Descartes did not use them. (Neither did Fermat who developed/invented algebraic geometry independently from Descarte). The first person to use them was van Schooten in his extended translation of Descartes work. (Thanks Thony)





1829 Sir Humphrey Davy (Baronet) (17 December 1778 – 29 May 1829) English chemist who discovered several chemical elements and compounds, invented the miner's safety lamp, and epitomized the scientific method. With appointment to the Pneumatic Institution to study the physiological effects of new gases, Davy inhaled gases (1800), such as nitrous oxide (laughing gas) and a nearly fatal inhalation of water gas, (a mixture of hydrogen and carbon monoxide). Davy discovered alkali metals, potassium and sodium, an isolation made with electric current for the first time (1807); as well as alkaline earth metals: calcium, strontium, barium, and magnesium (1808). He discovered boron at the same time as Gay-Lussac. He recognized chlorine as an element, which prior workers confused as a compound. *TIS Davy died in Switzerland in 1829 of heart disease inherited from his father's side of the family. He spent the last months of his life writing "Consolations In Travel", an immensely popular, somewhat freeform compendium of poetry, thoughts on science and philosophy (and even speculation concerning alien life) which became a staple of both scientific and family libraries for several decades afterward. He is buried in the Plainpalais Cemetery in Geneva.



1908 William Arnold Anthony (November 17, 1835 – May 29, 1908) was an American physicist  and electrical engineer who initiated and developed one of the first courses in electrical engineering in the U.S. (1883), while teaching in the Physics Department at Cornell University, Ithaca, N.Y. During 1872-75, Anthony, with the aid of student George Moler, built the first American Gramme dynamo for direct current, used to power arc lamps that lighted the Cornell campus, the first American electrical outdoor-lighting system. Anthony also built a mammoth tangent galvanometer, a device which utilized the earth's magnetic field for the measurement of current. He designed the dynamo for first underground electricity distributing system. Anthony contributed to development of gas-filled electric lamps.*TIS



1917 Sir William Davidson Niven KCB FRS (24 March 1842 – 29 May 1917) was a Scottish mathematician and electrical engineer.

Niven studied mathematics at Trinity College, where he graduated as third Wrangler in the mathematical tripos of 1866 (the means that he was ranked third among the First Class students). The following year he was elected to a fellowship at Trinity College but after holding the fellowship for some time he left Cambridge to take up an appointment as Professor of Mathematics at the Royal Indian Engineering College, Cooper's Hill, Surrey. This College, explicitly set up to train engineers for work in India, was opened on 5 August 1872 and Niven became its first mathematics professor. 

In 1882 Niven became Director of Studies at the Royal Naval College, Greenwich, succeeding Thomas Archer Hirst. He was appointed a Companion of the Order of the Bath (Civil division) in Queen Victoria's Diamond Jubilee Honours of 1897. He retired in 1903, when he was knighted by being appointed a Knight Commander of the Order of the Bath.

Niven was a colleague of James Clerk Maxwell (1831–1879), whose scientific papers he edited after his death. Among Niven's students was Alfred North Whitehead, to whom he taught mathematics, by instructing him in the physics of Maxwell.

In retirement Niven lived at Eastburn, Sidcup, Kent, where he died in 1917.



1999 John Peter Louis Knopfmacher ( 20 January 1937 in Johannesburg – 29 May 1999 in Graz ) was a South African mathematician.*Wik

 John Knopfmacher studied accounting and then mathematics at the University of the Witwatersrand . While still a student, his first publication on perfect numbers appeared in the Mathematical Gazette in 1960. His son, Arthur, related:

"He related to me that as only a first year student, he became inspired by the famous problem of odd perfect numbers and derived what he believed to be a proof that none existed. One of his lecturers realised that while not a proof of this, it was in fact a new proof of the formula that describes all even perfect numbers, and this result became his first publication in a mathematics journal." *SAU

He received his bachelor's degree in 1958 and his master's degree in 1961. He then went to the University of Manchester , where he received his doctorate in 1965 under John Frank Adams ( Extensions in Varieties of Groups and Algebras ).  After returning (1965) to the University of Witwatersrand, he became a lecturer , in 1966 a senior lecturer, in 1971 a reader and associate professor, and in 1979 a professor. From 1984 to 1994 he was head of the mathematics department at his university and became City of Johannesburg professor. In 1992 he founded the Centre for Applicable Analysis and Number Theory at the university, which was named after him in 1999. He retired in 1997 and moved to Melbourne . He was most recently a visiting professor at the University of Graz .

He worked on algebra ( non-associative algebras, finite groups , Lie algebras ) and topology before turning to analytic number theory in 1970. In particular, he established an abstract (algebraic) approach to analytic number theory, which was also the title of his 1975 monograph. For this purpose, he developed the theory of arithmetic semigroups (arithmetic of free commutative semigroups with unity and real-valued multiplicative norm with some additional properties). The theory also allows, for example, consideration over finite fields.

He was married to Rose Hendler from 1959 until their divorce in 1991 and had three children. His son Arnold Knopfmacher is also a mathematician, with whom John Knopfmacher published around 35 joint works. 

In 1995 he received the Lifetime Achievement Medal of the South African Mathematical Society and was for many years the editor of its journal Quaestiones Mathematicae. In 1991 he was made a Fellow of the Royal Society of South Africa.*Wik




2005 Kazimierz Urbanik (February 5, 1930 – May 29, 2005) was a prominent member of the Polish School of Mathematics. He founded the journal Probability and Mathematical Statistics and served as rector of the University of Wrocław.

Urbanik began teaching at the University of Wrocław in 1956. By 1960, he was promoted to professor, and in 1965 he became a member of the Polish Academy of Sciences, becoming its youngest member. He was an invited speaker at the International Congress of Mathematicians in 1966. He directed the university's Institute of Mathematics for most of the years from 1967 to 1996, and was rector of the university from 1975 to 1981. In 1980, he founded the journal Probability and Mathematical Statistics, and became its first editor-in-chief.

His research contributions include over 180 papers. His work in probability theory included work on random variables in compact groups, connections between measurability and connectivity, generalized convolutions, and decomposability semigroups. He also studied stochastic processes, information theory, universal algebra, and functional analysis. He was the doctoral advisor of 17 students.





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