Pat'sBlog

Tuesday, 28 July 2026

On This Day in Math - July 28

It appears to me that if one wishes to make progress in mathematics
one should study the masters and not the pupils.

Quoted in O Ore's, Niels Abel, Mathematician Extraordinary

The 209th day of the year; 209=1⁶+2⁵+3⁴+4³+5²+6¹.

Also 209 is a "Self number" A self number, Colombian number or Devlali number (after the town where he lived) is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 209, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. students might want to explore self numbers for patterns
[The earliest use of Colombian number I can find is by B. Recaman (1974). "Problem E2408". Amer. Math. Monthly 81. Would love to know if there are earlier uses.]

209 is the maximum number of pieces that can be made by cutting an annulus with 19 straight cuts.

The curve 42x^2 - y^2 = 209 contains the 'prime points' (3, 13), (5, 29), (7, 43), and (13, 83). *Prime Curios

There is an infinity of pairs x,y where x^2 - y^2 - xy = 209 for x, y integers *Prime Curios

As far as I know, there are only two three digit numbers so that abc^2 = uvwxyz and uvw+xyz = abc. These are called three digit Kaprekar numbers. 209 is involved in each case. The two numbers are 297^2 = 88209, with 88 + 209 = 297 and 703^2 = 494209 where 494 + 209 = 703

See More Math Facts for every Year Day here

EVENTS

1619 Kepler wrote Napier expressing his enthusiasm for Napier’s invention of logarithms. *VFR

Kepler, who used logarithm tables extensively to compile his Ephemerids and therefore dedicated it to Napier, remarked:

... the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ... instead of rearing up his child for the public benefit he deserted it in the birth.

— Johannes Kepler, Rudolphine Tables (1627)

1851 First American eclipse expedition to Europe when George Phillips Bond (1825 - 1865) led a team to Scandinavia. *NSEC In the transcription of his notes he wrote:

1851 A total solar eclipse was photographed for the ﬁrst time. *VFR The first correctly-exposed photograph of the solar corona was made during the total phase of the solar eclipse of 28 July 1851 at Königsberg (now Kaliningrad) by a local daguerreotypist named Berkowski at the Royal Observatory in Königsberg, Prussia (now Kalinigrad in Russia). Berkowski, whose first name was never published, observed at the Royal Observatory. A small 6-cm refracting telescope was attached to the 15.8-cm Fraunhofer heliometer and a 84-second exposure was taken shortly after the beginning of totality.
United Kingdom astronomers, Robert Grant and William Swan, and Austrian astronomer Karl Ludwig von Littrow observed this eclipse and determined that prominences are part of the Sun because the Moon is seen to cover and uncover them as it moves in front of the Sun.*Wik

In 1858, fingerprints were used as a means of identification for the first time.*TIS The English first began using fingerprints in July of 1858, when Sir William James Herschel, Chief Magistrate of the Hooghly district in Jungipoor, India, first used fingerprints on native contracts. On a whim, and without thought toward personal identification, Herschel had Rajyadhar Konai, a local businessman, impress his hand print on a contract.
The idea was merely "... to frighten [him] out of all thought of repudiating his signature." The native was suitably impressed, and Herschel made a habit of requiring palm prints--and later, simply the prints of the right Index and Middle fingers--on every contract made with the locals. Personal contact with the document, they believed, made the contract more binding than if they simply signed it. Thus, the first wide-scale, modern-day use of fingerprints was predicated, not upon scientific evidence, but upon superstitious beliefs.
As his fingerprint collection grew, however, Herschel began to note that the inked impressions could, indeed, prove or disprove identity. While his experience with fingerprinting was admittedly limited, Sir William Herschel's private conviction that all fingerprints were unique to the individual, as well as permanent throughout that individual's life, inspired him to expand their use. *History of Fingerprints, Onin.com

Herschel was the grandson of the astronomer/musician Sir William James Herschel, and son of Sir John Herschel (1792–1871) – a renowned astronomer and polymath.

*Wik

I was reminded by Douglas W Boone that Mark Twain uses fingerprint identification in his book Pudd'nhead Wilson.

"As we see most clearly in the trial scene at the end of the book, fingers turn out to be Tom's deadliest enemy when his fingerprints turn up on the murder weapon, revealing his identity. Referring to fingerprints, Pudd'nhead explains:

Every human being carries with him from his cradle to his grave certain physical marks which do not change their character, and by which he can always be identified—and that without shade of doubt or question. " *SHMOOP

1866 The ﬁrst act (in the USA) legalizing the employment of the metric system was approved (14 Stat. L. 339). The act provided that it “shall be lawful throughout the United States of America to employ the weights and measures of the metric system.” *VFR

1882 The Institute of Accountants and Bookkeepers was organized in New York City. It was the first accounting society in the United States. *FFF

1883 On the 28th of July, about nine o"clock in the morning, Jack Ferry pedaled across the English Channel on a floating tricycle. He started from Dover about nine o’clock in the morning, and arrived at Calais in less than eight hours. The distance as the crow flies was twenty miles, but on account of the currents, the effort required was considerably increased. The construction of his vehicle was illustrated in La Nature. Bulky paddlewheels (probably needing more displacement than shown) replace wheels of a land tricycle. The small wheel behind acted as a rudder. The event was reported in Science, 14 Dec 1883. *TiS

1899 Cantor asks Dedekind whether the set of all cardinal numbers is itself a set, because if it is it would have a cardinal number larger than any other cardinal. *VFR

1948 Allen Turing writes to Jack Good with an estimate of the number of neurons in the human brain. "I have repeatedly looked in books on I looked up an estimate neurology ... and never found any numbers offered. My own estimate is 3x10⁸ to 3x10⁹. " *Turing Archives

I looked up an estimate in 2024 and it gave, "Approximately 86 billion neurons in the human brain. "about 9 x 10^10, pretty close for an estimate 80 years out.

1984 The town of Eighty-four Pennsylvania celebrated it's centennial on this day.

1997 Dell Computer Corp. announced its entry into the workstation market with the Dell Workstation 400. The move to the more powerful desktop computers, most commonly used for engineering, followed Dell's entry into the network server industry as it expanded from personal desktop computers and laptops in order to grab a larger part of the market. Dell offered its workstations for $3,000 to $8,000. (Yikes!!!) *CHM

*CHM

2061 Halley's comet will next reach perihelion. The comet last reached perihelion on 9 February 1986, and will reach it again on 28 July 2061 *Wik

BIRTHS

1721 Pierre Jacquet-Droz, a Swiss clockmaker, was born July 28, 1721, in Neuchâtel, now part of Switzerland, but then part of Prussia. Little is known about Pierre, except that he had a son, Henri-Louis, and an adopted son, Jean-Frederic Leschot, and that together, between 1768 and 1774, they constructed three of the most amazing automata ever devised, anywhere, anytime.

The first (in order of discussion – we don't know in what order they were constructed) is a young woman, who sits before a small clavichord, her eyes downcast. When prompted, she places her fingers on the keyboard and plays a melody, just like a human would. Her eyes follow her fingers as they play. She is called the Musician, or the Musicienne. The other two automata look like young boys, perhaps the younger brothers of the Musician. One of them, dressed in a velvet suit, draws pictures with a stick of charcoal, seven different ones in succession, now and then pausing to blow the charcoal dust from the paper before him. He is called the Draughtsman.

The other lad, also in velvet, sits before a small writing pad. He dips a real quill pen into real ink and writes a short sentence of up to 40 characters on the tablet before him. It can be any sentence of 40 characters, depending on how you set up the tabs that control the cams that move the pen. He is the Writer.

The reason so much is known about the Jacquet-Droz automata is that they survive, and they still work! They are the treasured trio of the Museum of Art and History in Neuchâtel; they sit quietly in their own special room under the inquiring eyes of daily tourists, and once a month, at three different times in the afternoon, they play their partitas, draw their charcoal sketches, and pen their French phrases.

The Writer gets the most attention from commentators, for he has a set of input tabs that one can set to get him to write any particular phrase. He is, in a word, programmable, and therefore a good candidate for an early computer, perhaps the first computer. Remove the shirt from his back and gaze upon his brass innards, especially if they are in motion, and you will not doubt that this is a machine of amazing complexity and beauty, and it is hard to believe that it was designed and crafted by a Swiss clockmaker and his two assistants, using only a jeweler’s saw, a drill, a lathe, files, and a few other hand- or foot-powered tools. There are many videos available of the Jacquet-Droz automata in action. I like this one, narrated by an outstanding historian of science, Simon Schaffer.

I have not seen the Neuchâtel automata in action, but Jessica Rifkin has. Professor Rifkin literally wrote the book on Enlightenment automata, and she recalled, in The Restless Clock (2016), that what she found most unnerving about the experience came before the performance, and involved the Musician. The automaton just sat there while the curator explained what was about to happen, but she was breathing, her breast slowly rising and falling as she waited her turn. That would be unsettling, and provides further indication that Jaquet-Droz was probing the limits of the difference between living beings and machines when he designed and built his androids.

If anyone else visits Neuchâtel before I do, please send photos. And if you find the grave of Pierre Jacquet-Droz, please leave a small wreath on my behalf. An artificial wreath with an aroma of lilacs might be most appropriate.

William B. Ashworth, Jr., Consultant for the History of Science, Linda Hall Library and Associate Professor emeritus, Department of History, University of Missouri-Kansas City. Comments or corrections are welcome; please direct to ashworthw@umkc.edu. *Linda Hall Org

1849 Robert Scott studied at Cambridge and was elected to a fellowship. After a short time teaching he studied to be a barrister. He spent most of his career as Bursar and Master of St John's College Cambridge. He published a book on Determinants. *SAU

1867 Charles Dillon Perrine (July 28, 1867; Steubenville, Ohio, – June 21, 1951) U.S. astronomer who discovered the sixth and seventh moons of Jupiter in 1904 and 1905, respectively. In 1904 he published a calculation of the solar parallax (a measure of the Earth-Sun distance) based on observations of the minor planet Eros during one of its close approaches to the Earth. *TIS He was an American astronomer at the Lick Observatory in California (1893-1909) who moved to Cordoba, Argentina to accept the position of Director of the Argentine National Observatory (1909-1936). The Cordoba Observatory under Perrine's direction made the first attempts to prove Einstein's theory of relativity by astronomical observation of the deflection of starlight near the Sun during the solar eclipse of October 10, 1912 in Cristina (Brazil), and the solar eclipse of August 21, 1914 at Feodosia, Crimea, Russian Empire. Rain in 1912 and clouds in 1914 prevented results.*Wik

maser components at amhistorymuseum HT to
‏@bhgross144

1915 Charles Hard Townes (July 28, 1915 – January 27, 2015) was an American Nobel Prize-winning physicist and educator. Townes was known for his work on the theory and application of the maser, on which he got the fundamental patent, and other work in quantum electronics connected with both maser and laser devices. He shared the Nobel Prize in Physics in 1964 with Nikolay Basov and Alexander Prokhorov.
In a career that spanned six decades, Dr. Townes developed radar bombing systems and navigation devices during World War II, advised presidents and government commissions on lunar landings and the MX missile system, verified Einstein’s cosmological theories, discovered ammonia molecules at the center of the Milky Way, and created an atomic clock that measured time to within one second in 300 years. He died at the age of 99 in Berkeley, California*Wik *NY Times

1928 John Bell (28 June 1928 – 1 October 1990) his great achievement was that during the 1960s he was able to breathe new and exciting life into the foundations of quantum theory, a topic seemingly exhausted by the outcome of the Bohr-Einstein debate thirty years earlier, and ignored by virtually all those who used quantum theory in the intervening period. Bell was able to show that discussion of such concepts as 'realism', 'determinism' and 'locality' could be sharpened into a rigorous mathematical statement, 'Bell's inequality', which is capable of experimental test. Such tests, steadily increasing in power and precision, have been carried out over the last thirty years. *SAU

Kai Wenz sent me a link to a post on his LinkedIn site that has more detail about Bell's work....enjoy.

1954 Gerd Faltings (July 28, 1954 - ) was born in Gelsenkirchen-Buer, West Germany. He was awarded the Fields Medal in 1986 for his proofs of the Mordell Conjecture and several related conjectures. He won the Abel prize in 2026 for these achievements. He has also been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n greater than 2 there are at most a finite number of coprime integers x, y, z with xⁿ + yⁿ = zⁿ. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Falting's arguments.
However, Faltings was the natural person that Wiles turned to when he wanted an opinion on the correctness of his repair of his proof of Fermat's Last Theorem in 1994.*TIS

2025 Vicki Powers (July 28, 1958 – February 2, 2025), born Victoria Ann Powers, was an American mathematician specializing in algebraic geometry and known for her work on positive polynomials and on the mathematics of electoral systems. She was a professor in the department of mathematics at Emory University, where she worked starting in 1987.

Powers was the author of the book Certificates of Positivity for Real Polynomials—Theory, Practice, and Applications (Springer, 2021). A review on MathSciNet said that "In the reviewer's opinion this is a very nice and concise presentation of the most important pillars of real algebra up to the present time".

Powers graduated from the University of Chicago in 1980, with a bachelor's degree in mathematics. She completed her Ph.D. in 1985 at Cornell University. Her dissertation, Finite Constructable Spaces of Signatures, was supervised by Alex F. T. W. Rosenberg.

After completing her doctorate, she joined the faculty at the University of Hawaiʻi, but moved to Emory University only two years later, in 1987.

She was on leave from Emory as a Humboldt Fellow and Alexander von Humboldt research professor at the University of Regensburg in 1991–1992, as a visiting professor at the Complutense University of Madrid in 2002–2003, and as a program officer at the National Science Foundation in 2013–2015. From 2012 to 2014, Powers served as a Council Member at Large for the American Mathematical Society.

Powers' work moved from abstract real algebraic geometry to more concrete questions related to positive polynomials in one and several variables and voting theory. Her collaborators included Bruce Reznick, Eberhard Becker, Mari Castle, Claus Scheiderer and Thorsten Wormann.

Powers was married to Colm Mulcahy, an Irish mathematician who had the same doctoral advisor. On February 2, 2025, she died at home from complications of ALS. *Wik

DEATHS

1818 Gaspard Monge (9 May 1746 – 28 July 1818) died in disgrace in Bourbon Paris, having been stripped of his place in the reorganized Acad´emie of 1816. Although he contributed to diﬀerential equations and the geom¬etry of surfaces, his special interest was descriptive geometry. Employed as a teacher, he made signiﬁcant contributions to educational reform. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 616]
On the fall of Napoleon he was deprived of all his honors, and even excluded from the list of members of the reconstituted Institute. Monge died at Paris on 28 July 1818 and was interred in Le Père Lachaise Cemetery, in Paris, in a mausoleum. He was later transferred to the Panthéon. The mausoleum and Monge's bust remain in Le Père Lachaise Cemetery.
A statue portraying him was erected in his home town of Beaune, Côte-d'Ors in 1849. His name is one of the 72 names inscribed on the Eiffel Tower.

1944 Sir Ralph Fowler (17 January 1889 – 28 July 1944) a brilliant physicist. But it may be for his influence upon others that he is best known. In fact, no less than fifteen Fellows of the Royal Society and three Nobel Laureates were supervised by Fowler between 1922 and 1939. The total number supervised during this time was a staggering sixty-four giving him an average of eleven research students at any given time. One might be led to believe that this did not allow for any depth of relationship to form between him and his students. However, this was far from the truth of the matter. Those who studied under Fowler had a tremendous admiration for him. In particular, E A Milne was especially taken by the man whom he fondly referred to as "the kind of man you can still remain friendly with, even when he has sold you a motor-bike; it is not possible to say more" and whom he called a "prince amongst men".
Aside from Milne, on whom he had a profound impact, he also had the opportunity of influencing the likes of Sir Arthur Eddington, Subramanian Chandrasekhar, Paul Dirac, Sir William McCrea, Lady Jeffreys and others either directly through supervision or indirectly through collaboration. Even in his personal life he was intimately connected with brilliant people having married Eileen, the only daughter of Lord Rutherford whom he met through Rutherford's Cavendish Laboratory at Cambridge. Sometimes his influence was simply the fact that he was known to so many people. It was Fowler who ultimately introduced Paul Dirac to the burgeoning field of quantum theory in 1923 leading Dirac to the forefront of its ultimate discovery in 1925. Fowler also put Dirac and Werner Heisenberg in touch with each other through Niels Bohr. As Sir William McCrea simply put it: "he was the right man in the right place at the right time." *SAU
1968 Otto Hahn (8 Mar 1879; 28 Jul 1968 at age 89) German physical chemist who, with the radiochemist Fritz Strassmann, is credited with the discovery of nuclear fission. He was awarded the Nobel Prize for Chemistry in 1944 and shared the Enrico Fermi Award in 1966 with Strassmann and Lise Meitner. Element 105 carries the name hahnium in recognition of his work.*TIS

1968 Otto Hahn (8 Mar 1879; 28 Jul 1968 at age 89) German physical chemist who, with the radiochemist Fritz Strassmann, is credited with the discovery of nuclear fission. He was awarded the Nobel Prize for Chemistry in 1944 and shared the Enrico Fermi Award in 1966 with Strassmann and Lise Meitner. Element 105 carries the name hahnium in recognition of his work.*TIS

"For the rest of his life, Hahn provided a standard explanation: fission was a discovery that relied on chemistry only and took place after Meitner left Berlin; she and physics had nothing to do with it, except to prevent it from happening sooner." *Lise Meitner by Ruth Lewin Sime

The prize-winning science-fiction writer, Frederik Pohl, talking about Szilard's epiphany 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." (Maybe she had a little idea?)

in 1939 during the Fifth Washington Conference on Theoretical Physics at the George Washington University, Nobel Laureate Niels Bohr publicly announced the splitting of the uranium atom. The resulting “fission,” with its release of two hundred million electron volts of energy, heralded the beginning of the atomic age.

The announcement came just weeks after Otto Hahn and Fritz Strassmann, two of Bohr’s colleagues at Copenhagen, reported that they had discovered the element barium after bombarding uranium with neutrons. After receiving the news in a letter, physicist Lise Meitner and her cousin, Otto Frisch, correctly interpreted the results as evidence of nuclear fission. Frisch confirmed this experimentally on January 13, 1939. *atomicheritage.org

Niels Bohr was planning a trip to America to discuss other problems with Einstein who had found a haven at Princeton's Institute for Advanced Studies. Bohr came to America, but the principal item he discussed with Einstein was the report of Meitner and Frisch. Bohr arrived at Princeton on January 16, 1939. He talked to Einstein and J. A. Wheeler who had once been his student. From Princeton the news spread by word of mouth to neighboring physicists, including Enrico Fermi at Columbia. Fermi and his associates immediately began work to find the heavy pulse of ionization which could be expected from the fission and consequent release of energy. *Atomic Archive

1982 Graciela Beatriz Salicrup López (México City, México, April 7, 1935 – July 29, 1982) was a Mexican architect, archaeologist, and mathematician. In the 1970s and 1980s, she was a pioneer in the field of categorical topology. Most of her work was published in Spanish, and her original contributions were not widely recognized until after her premature death.

A professor at Colegio Alemán originally encouraged her, inciting an interest in mathematics that her family did not understand or support, even sending her to see a psychiatrist for "extravagance, disorientation, and a bit of madness," according to her friend Claudia Gomez Wulschner.

When asked how the story ends, Salicrup López states that she married him. She married the psychiatrist Armando Hinojosa Cavazos. They had three children: Ariel who pursued music, David who became an architect like his mother; and Mariana who studied ballet.

Salicrup Lopez had many interests and passions, especially for music and art. She loved the opera and visiting art exhibits. She also enjoyed literature and history.

Salicrup Lopez still wanted to become a mathematician, and finally enrolled in the Faculty of Sciences in 1964 to study mathematics. Between 1966 and 1968 she taught mathematics at the UNAM Faculty of Architecture. Her thesis, accepted in 1969, was on the Jiang Boju subgroup.

After graduating in 1969 Graciela began teaching in the UNAM Faculty of Sciences. In 1970 she was given a position as a researcher in the UNAM Mathematics Institute, where she worked with Dr. Roberto Vázquez, her mentor. That same year she published her first work along with her mentor.

Her work was concerned with the structure of the Top category of topological spaces and with continuous functions. Her work related concepts such as reflexivity or coreflexivity to those of connection and coexistence, both in Top and in certain subcategories of Top (and in some more general concrete categories). The publications she co-authored with Vázquez were always in Spanish, so many mathematicians were not aware of her work.

Shortly before her death, Graciela fell out with her mentor Roberto Vázquez and they stopped collaborating. In the summer of 1982, she was visited by Lamar Bentley and Horst Herrlich, with whom she planned to collaborate. Soon after this Graciela suffered a fall that hurt her badly. She did not recover and died on July 29, 1982.

1988 Caleb Gattegno (1911–1988) was an educator, psychologist, and mathematician. He is considered one of the most influential and prolific mathematics educators of the twentieth century. He is best known for introducing new approaches to teaching and learning mathematics (Visible & Tangible Math), foreign languages (The Silent Way) and reading (Words in Color). Gattegno also developed pedagogical materials for each of these approaches, and was the author of more than 120 books and hundreds of articles largely on the topics of education and human development.

Gattegno's pedagogical approach is characterised by propositions based on the observation of human learning in many and varied situations. This is a description of three of these propositions. He was also influenced by the works of Jean Piaget and worked on introducing the implications of the latter's cognitive theory on education.

In his approach to teaching mathematics, manipulatives, such as Geoboards which he invented and Cuisenaire Rods which he popularized, are part of a way of systematically developing students' mathematical thinking through the exploration of clear and tangible problems. *Wik

2000 Abraham Pais (May 19, 1918 – July 28, 2000) Dutch-American physicist and science historian whose research became the building blocks of the theory of elemental particles. He wrote Subtle Is the Lord: The Science and Life of Albert Einstein, which is considered the definitive Einstein biography. In Holland, his Ph.D. in physics was awarded on 9 Jul 1941, five days before a Nazi deadline banning Jews from receiving degrees. Later, during WW II, while in hiding to evade the Gestapo, he worked out ideas in quantum electrodynamics that he later shared when working with Niels Bohr (Jan - Aug 1946). In Sep 1946, he went to the U.S. to work with Robert Oppenheimer at Princeton, where Pais contributed to the foundations of the modern theory of particle physics.*TIS

2004 Francis Harry Compton Crick (8 June 1916 – 28 July 2004) was a British biophysicist, who, with James Watson and Maurice Wilkins, received the 1962 Nobel Prize for Physiology or Medicine for their determination of the molecular structure of deoxyribonucleic acid (DNA), the chemical substance ultimately responsible for hereditary control of life functions. Crick and Watson began their collaboration in 1951, and published their paper on the double helix structure on 2 Apr 1953 in Nature. This accomplishment became a cornerstone of genetics and was widely regarded as one of the most important discoveries of 20th-century biology. *TIS

2011 Heinrich-Wolfgang Leopoldt (22 August 1927 – 28 July 2011) was a German mathematician who worked on algebraic number theory.

Leopoldt was brought up in Schwerin, the town of his birth, on the south west shore of Schweriner Lake, about 65 km southwest of the city of Rostock. He was studying at the Gymnasium in Schwerin when World War II began in 1939. He continued his secondary education until January 1943 when Germany began the Luftwaffenhelfer programme. This drafted all boys born in 1926 or 1927, which included Leopoldt, into the military where they were supervised by the Hitler youth who began a programme of ideological indoctrination. Members of the Luftwaffe also trained the boys in military duties. After the war in 1945 the future looked very uncertain and Leopoldt decided that his best option was to take up an apprenticeship. However, he was very fortunate that his love for playing music meant that he joined a group one of whom was the mathematics teacher who had taught him in the Gymnasium. The teacher was, of course, already fully aware of Leopoldt's mathematical abilities and he began to teach him the mathematical principles of astronomy. Soon the teacher was encouraging Leopoldt return to the Gymnasium to complete his school education so that he might gain admission to university. Leopoldt took his teacher's advice and went back to the Gymnasium, gaining the qualifications to enter university in 1947. He then enrolled to study mathematics at the Humboldt University in Berlin, matriculating in the autumn of 1947. *SAU

Leopoldt earned his Ph.D. in 1954 at the University of Hamburg under Helmut Hasse with the thesis Über Einheitengruppe und Klassenzahl reeller algebraischer Zahlkörper (On group of unity and class number of real algebraic number fields). As a postdoc, he was from 1956 to 1958 at the Institute for Advanced Study. In 1959, he obtained his habilitation degree at the University of Erlangen and was then at the University of Tübingen. From 1964, he was ordentlicher Professor at the University of Karlsruhe, where he was also Director of the Mathematics Institute.

Leopoldt and Tomio Kubota introduced and investigated p-adic L-functions (now named after them). These functions are a component of Iwasawa theory and are a p-adic version of the Dirichlet L-functions. With Hans Zassenhaus he also worked on computer algebra and its applications in number theory. *SAU

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell

Monday, 27 July 2026

The Tower of Hanoi And two (three?) clever solutions

Back awhile, in a blog about Fibonacci, I mentioned that Edouard Lucas had created the "Tower of Hanoi" game and received comments and mail from people who thought I must be mistaken because the game was "really old". Turns out, it really isn't, but just the creation of a master mathematical story teller. Here are some notes about the man, and the history of the Towers of Hanoi from my Math Words Etymology page.

Also, you can find a java applet to play the game at this site... and if you've never done it (where HAVE you been?) don't start with all 12 discs, that takes 4095 moves to solve (see below).

The Lucas sequence is similar to the Fibonacci sequence. The Lucas sequence is given by {1, 3, 4, 7, 11, 18, ...} . Each term is the sum of the two previous numbers, as in the Fibonacci sequence. Just as in the Fibonacci sequence, the limit of the ratio of consecutive terms is the Golden Ratio. The Lucas numbers can also be constructed from the Fibonacci numbers by the function Ln = Fn-1 + Fn+1, thus the fifth Lucas number, 11, is the sum of the fourth and sixth fibonacci numbers (3+8).

The sequence is named for Edouard Lucas, a French mathematician of the later half of the nineteenth century. He used his sequence and the Fibonacci sequence to develop techniques for testing for prime numbers. Lucas is also remembered for his unusual death, caused by a waiter dropping a plate which shattered sending a piece of plate into his neck. Lucas died several days later from a deadly inflamation of the skin and subcutaneous tissue caused by streptococcus. The disease, officially listed as erysipelas (from the Greek for "red skin") was more commonly known as "Saint Anthony's Fire".

Lucas was also the creator of a popular puzzle called The Tower of Hanoi in 1883. You can see the original box cover above. Note that the author on the box cover is Professor N. Claus de Siam, an anagram of Lucas d' Amiens (his home). The professors college, Li-Sou-Stian, is also an anagram for "Lycee Saint-Louis" where Lucas worked.

France was building an Empire in Indochina (the peninsula stretching from Burma to Viet Nam and Malaysia) and the "mysterious East" was a very fashionable topic. Lucas created a legend (some say he embellished an existing one, but I can find no earlier record of one) of monks working to move 64 gold disks from one of three diamond points to another after which the world would end. The solution for a tower of n disks taks 2n -1 moves, so the game often had less than the 64 disks of the legend. Solving the 64 disks at one move a second would require 18,446,744,073,709,551,615 seconds, which at 31,536,000 seconds a year would take 584 Billion years. (and you thought Monopoly took a long time to finish). The reference in his instructions to Buddhist monks in a temple in Bernares(Varanasi), India seems, even now, to make people believe there was such an activity taking place. Varanasi is considered the holiest of the seven sacred cities (Sapta Puri) in Hinduism, and Jainism, and is important to Buddhism because it was in nearby Sarnath that Buddha gave his first teaching after attaining enlightenment, in which he taught the four noble truths and the teachings associated with it. There is a Buddhist temple there with many relics of the Buddha, but so far as I can find, no monks moving golden disks on needles.

Students/teachers interested in further explorations of the history and math of the famous game should visit the work of Paul K Stockmeyer who maintains the page with the cover illustration mentioned above, and his Papers and bibliography on the Tower of Hanoi problem.

Lucas developed several other mathematical games of his on, including the well known children's pastime of dots and boxes (which he called La Pipopipette), which on large boards is still essentially unsolved, I believe. He also (probably) invented a Mancala type game called Tchuka Ruma.

Lucas is also remembered for suffering an unusual death. At a banquet in 1891 a waiter dropped a dining plate and one of the pieces cut Lucas on the neck and cheek. Within a week he was dead from what was called the "Holy Fire" or St Anthony's Fire, a form of septicemia.

A while after I wrote the above, I learned a little more, and so:

Just browsing through Wikipedia, and they show a solution to the Towers of Hanoi puzzle that I had never seen using a ruler as a solution key.

If you have been off planet for the last 130 years and don't know the Towers problem, you can play online here. You might try that first, and set the number of discs to 6 so that it matches the solution shown below.

And for those who know the game but just want to see how a ruler is used, here is the graphic.

For any move, just move the disc whose size compares to the marks on the ruler. For instance the first five marks on a ruler marked in 32^nds would be 1/32, 1/16/ 3/32, 1/8, 5/32.... The denominators tell you which disk to move. The largest denominator (smallest scale) goes with the smallest disc, etc. If you then apply two fundamentals of any solution, always move the smallest disc From rod A, to B to C and back to A in a cycle, and never put a bigger disc on a smaller one, then you have a solution... That's easier than Gray codes isn't it.

Why have I never encountered this before? The connection was made in 1956 by Donald W. Crow, in relation to traversing the vertices of a cube in n-dimensions[ D. W. Crowe, The n-dimensional cube and the tower of Hanoi, Amer. Math. Monthly, 63 (1956), 29-30.]

POSTSCRIPT:::: For another really insightful solution (maybe the best of them all) See the comment by Jeffo....Thanks guy, why don't I see ideas like that?

Jeffo said...

If the rods are placed in a circular arrangement instead of linear, then a correct solution will involve always moving the smallest disk one rod clockwise every other move. The alternate moves are forced.

On This Day in Math - July 27

But just as much as it is easy to find the differential of a given quantity,
so it is difficult to find the integral of a given differential.
Moreover, sometimes we cannot say with certainty
whether the integral of a given quantity can be found or not.

~Bernoulli, Johann

The 208th Day of the Year
208 is the sum of the squares of the first five primes.

208 is the number of paths from (0,0) to (7,7) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps.

208 is an abundant number, the proper divisors total 226(more than 208)

208 = 6^3 - 2^3

208 is the sum of a cube and a square, as were 204 and 206. 208 = 4^3 + 12^2

208 is an unprimeable number, changing any digit to something else will not make a prime.

208 = 2^4 x 13 and if you play the four-fours game, 208 = 4^4 - 4! - 4!

208 can be written as the sum of two squares in only one way, 12^2 + 8^2

208 = 53^2 - 51^2 = 17^2 - 9^2 = 28^2 - 24^2

(16*10^208-31)/3 is prime, and it has a 5 followed by 206 threes, finished of with 23. It is the largest year date in this sequence. Previous examples include 523, 5323, 53323, and 5333333333333323, for the exponents 1, 2, 3, 4 and 15

$208 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2$, the sum of the first five prime squares, obviously the smallest number to be the sum of five distinct squares of primes.

208 is a junction number since it is the sum of n + SoD(n) for two (or more) numbers. One is 203 since 203 + 2 + 0 + 3 = 208, find the other(s?)

.
See more Math Facts for Every Year Day here

EVENTS

1630, On July 27 Giovanni Batista Baliani wrote a letter to Galileo Galilei about the explanation of an experiment he had made in which a siphon, led over a hill about twenty-one meters high, failed to work. Galileo responded with an explanation of the phenomena: he proposed that it was the power of a vacuum which held the water up, and at a certain height (in this case, thirty-four feet) the amount of water simply became too much and the force could not hold any more, like a cord that can only withstand so much weight hanging from it.

For hundreds of years, It had been known that water pumps could not lift water past a certain point. The distance water could not be pumped beyond was found to be around 34 feet. However, this height varied because it is based of the weight of the air, and was what Europeans like Galileo and Torricelli were trying to discover.

In 1640 Galileo and Torricelli conducted an experiment together with a suction pump at a well. They lowered the tube into the well and began to pump water as high as they could, but found that no matter their efforts the water could not pass more than about 34 feet about the water’s surface. Galileo concluded that, in fact, they were not pumping the water up the tube at all, but rather removing air from the pump creating a vacuum. This new thinking led one of Torricelli’s greatest inventions, the first barometer.

1794 What a difference a day makes! Jean Baptiste Joseph Fourier (1766?-1830) was a student at the École Normale, c1794. He was sentenced to the guillotine by Robespierre on July 28 of 1794, but Robespierre was overthrown the day before his scheduled execution (27 July, 1794) was due. Fourier went on to both political and scientific success. He was unanimously elected the first Secretary of the Institute of Egypt in 1798. He was Governor of Lower Egypt in 1798‑1801 or Commissioner at the Divan of Cairo . He led one of the expeditions of exploration which examined ancient monuments and he suggested the publication of the great report on Egypt. He was was a professor at the École Polytechnique up to 1806. Napoléon made him a baron and during Napoléon's return from Elba in 1815, he made Fourier a count and Prefect of the Rhone, based at Lyons, from 10 Mar to 1 May. In 1815, he was penniless in Paris and giving lessons for his living. The Prefect of Paris found out and made him director of the Bureau de la Statistique of the Préfecture of the Seine. He was elected to the Académie in 1816, but this was vetoed by the government, so he was elected again in 1817 and this was permitted. He was Prefect of the Department of Isère, whose capital is Grenoble, from 1802 to 1817 (1815??) He was Permanent Secretary of the Académie des Sciences in 1822-1830.

*TIS

1829 By a remarkable coincidence, both Cauchy and Sturm sent papers to the Acad´emie des Sciences dealing with differential equations. Both of them used techniques which we recognize as matrix methods. Thus they are early contributors to linear algebra, a ﬁeld which is usually dated to Cayley’s introduction of matrices in 1858. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 1150]

In the 1830s, while teaching at the Collège Rollin in Paris, Sturm was developing his now-famous method for determining the number of real roots of an algebraic equation within a given interval. One evening, during a small mathematics salon hosted by Joseph Liouville, Sturm presented a clever trick involving sign changes in a sequence of polynomials. He claimed that the number of real roots could be precisely counted just by looking at the changes in sign from one term to the next.

Liouville was skeptical and challenged Sturm on the spot, believing that such a rule was too simple to be true for general equations. Sturm calmly worked through several examples, including a few with irrational and multiple roots. As he demonstrated the accuracy of his method, the room grew quiet. Finally, Liouville reportedly leaned back and said, “C’est trop élégant pour être faux” — It’s too elegant to be false.

This moment helped establish Sturm's reputation in Parisian mathematical circles and contributed to the eventual widespread adoption of his theorem — a foundational result in real algebra still taught today.

Charles Sturm

1837 At a meeting of the Berlin Academy of Sciences, Dirichlet presented his ﬁrst paper on analytic number theory. He proved the fundamental theorem that bears his name: Every arithmetical series an + b, n =0, 1, 2,... of integers where a and b are relatively prime, contains inﬁnitely many primes. The result had long been conjectured. Legendre tried hard for a proof but could only establish special cases such as 4n + 1. *VFR

1861 The Athenaeum magazine carried a review of Charles Dodgson's pamphlet entitled The Formula of Plane Trigonometry in which he suggested new symbols for the six basic trig functions. The reviewer was not convinced.

1866 Cyrus W. Field finally succeeded, after two failures, in laying the first underwater telegraph cable 1,686 miles long across the Atlantic Ocean between North America and Europe. Massachusetts merchant and financier Cyrus W. Field first proposed laying a 2,000-mile copper cable along the ocean bottom from Newfoundland to Ireland in 1854, but the first three attempts ended in broken cables and failure. Field's persistence finally paid off in July 1866, when the Great Eastern, the largest ship then afloat, successfully laid the cable along the level, sandy bottom of the North Atlantic. *TIS

*Thought.co

1905 A Karl Pearson letter appears in Nature asking for assistance on a problem"of considerable interest” about random walks (based on a question in a letter he had received from Sir Ronald Ross, who had discovered mosquitoes as the source of malaria spreading, without mentioning him by name), Two days later Lord Rayleigh wrote the periodical to inform them he had solved the problem and posted results in 1880 in Phil. Mag.. Pearson's response launched the common name for random walk used for many years, Drunkards Walk, "the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!” *Jordan Ellenberg , Shape

A favorite quote about dimensional random walks, "A drunk man will find his way home, but a drunk bird may get lost forever." usually attributed to Shizuo Kakutani

*Wik

1936 Einstein writes to John Tate, editor of the Physical Review angrily withdrawing a paper that he had submitted for publication but had been rejected after peer review. Einstein and Rosen's paper claimed that gravitational waves did not exist. It was Einstein who introduced gravitational waves in his theory of general relativity in 1916, within a few months of finding the correct form of the field equations for it. However by 1936 he had changed his mind, and wrote to his friend, Max Born, "Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist,..."
Later he would submit the paper again, but then drastically revise the conclusions before publication. Einstein simply explained why “fundamental” changes in the paper were required because the “consequences” of the equations derived in the paper had previously been incorrectly inferred. The referee of the paper, it is now known, was relativist Howard Percy Robertson. He was on sabbatical at Caltech. When he returned to Princeton he struck up a friendship with Einstein’s then newly arrived assistant Infeld. Robertson then convinced Infield of the problems with the paper he had re-submitted, and after Infield talked to Einstein, the paper was revised. It seems that Einstein had never read the referee's comments.

*physicstoday

1948 Hungary issued a stamp commemorating the centenary of the birth of the physicist Baron Roland E˝otv˝os1 (1848–1919). [Scott #840]. *VFR They issued another in 1991

2007 Ralph Asher Alpher's belated recognition for his work on the "Big Bang" process. In 2005 Alpher was awarded the National Medal of Science. The citation for the award reads "For his unprecedented work in the areas of nucleosynthesis, for the prediction that universe expansion leaves behind background radiation, and for providing the model for the Big Bang theory." The medal was presented to his son Dr. Victor S. Alpher on July 27, 2007 by President George W. Bush, as his father could not travel to receive the award. *Wik

BIRTHS
1667 Johann Bernoulli (27 July 1667 – 1 January 1748; also known as Jean or John) was a Swiss mathematician who studied reflection and refraction of light, orthogonal trajectories of families of curves, quadrature of areas by series and the brachystochrone.*SAU

1733 Jeremiah Fenwicke Dixon (27 July 1733 – 22 January 1779) was an English surveyor and astronomer who is best known for his work with Charles Mason, from 1763 to 1767, in determining what was later called the Mason-Dixon line.
Dixon was born in Cockfield, near Bishop Auckland, County Durham, the fifth of seven children, to Sir George Fenwick Dixon 5th Bt. and Lady Mary Hunter. His father was a wealthy Quaker coal mine owner and aristocrat. His mother came from Newcastle, and was said to have been "the cleverest woman" to ever marry into the Dixon family. Dixon became interested in astronomy and mathematics during his education at Barnard Castle. Early in life he made acquaintances with the eminent intellectuals of Southern Durham: mathematician William Emerson, and astronomers John Bird and Thomas Wright. In all probability it was John Bird, who was an active Fellow of the Royal Society, who recommended Dixon as a suitable companion to accompany Mason.

Jeremiah Dixon served as assistant to Charles Mason in 1761 when the Royal Society selected Mason to observe the transit of Venus from Sumatra. However, their passage to Sumatra was delayed, and they landed instead at the Cape of Good Hope where the transit was observed on June 6, 1761. Dixon returned to the Cape once again with Nevil Maskelyne's clock to work on experiments with gravity.
Dixon and Mason signed an agreement in 1763 with the proprietors of Pennsylvania and Maryland, Thomas Penn and Frederick Calvert, sixth Baron Baltimore, to assist with resolving a boundary dispute between the two provinces. They arrived in Philadelphia in November 1763 and began work towards the end of the year. The survey was not complete until late 1766, following which they stayed on to measure a degree of Earth's meridian on the Delmarva Peninsula in Maryland, on behalf of the Royal Society. They also made a number of gravity measurements with the same instrument that Dixon had used with Maskelyne in 1761. Before returning to England in 1768, they were both admitted to the American Society for Promoting Useful Knowledge, in Philadelphia.
Dixon sailed to Norway in 1769 with William Bayly to observe another transit of Venus. The two split up, with Dixon at Hammerfest Island and Bayly at North Cape, in order to minimize the possibility of inclement weather obstructing their measurements. Following their return to England in July, Dixon resumed his work as a surveyor in Durham. He died unmarried in Cockfield on 22 January 1779, and was buried in an unmarked grave in the Quaker cemetery in Staindrop.
Although he was recognized as a Quaker, he was not a very good one, dressing in a long red coat and occasionally drinking to excess. *Wik

Dixon is (supposedly) the one standing

1801 Sir. George Biddell Airy (27 July 1801 – 2 January 1892) born in Alnwick, England. *VFR English astronomer who became the seventh Astronomer Royal (1836-92). In his life he studied interference fringes in optics, made a mathematical study of the rainbow and computed the density of the Earth by swinging a pendulum at the top and bottom of a deep mine, determined the mass of the planet Jupiter and its period rotation, calculated the orbits of comets and cataloged stars. He designed corrective lenses for astigmatism (1825), the first that worked. His motivation was his own astigmatism. Airy had a long-standing battle with Babbage. In 1854, the conflict continued between the two during the battle of the incompatible railway gauges in England. Airy championed the railway narrow gauge and Babbage for the wide gauge. *TIS

In his On the Algebra and Numerical Theory of Errors of Observation (1861), he joined the company of such mathematicians as Gauss, Legendre, and the American, Robert Adrain, in attempting to mathematically understand the behavior of error patterns in the process of taking observations. *MAA

1844 Ágoston Scholtz (27 July 1844 in Kotterbach, Zips district, Austro-Hungary (now Rudnany, Slovakia) - 6 May 1916 in Veszprém,) From 1871 he was a teacher of mathematics and natural philosophy at the Lutheranian Grammar School of Budapest which at that time had been upgraded to become a so called 'chief grammar school', namely one which offered eight years of teaching. This was precisely the school which later was attended by several famous mathematicians such as Johnny von Neumann and Eugene Wigner (or Jenó Pál Wigner as he was called at that time). Scholtz became the school director of the Lutheranian Grammar School in 1875. Unfortunately this excellent school was closed in 1952, and most of its equipment was lost. Due to the initiative and support of its former well-known students, among others Wigner, it was reopened in 1989 after being closed for thirty-seven years. Scholtz's field of research was projective geometry and theory of determinants. His results were recorded by Muir in his famous work The history of determinants *SAU

1848 Roland Baron von Eötvös (27 July 1848 – 8 April 1919) was a Hungarian physicist who studied at Heidelberg where he was taught by Kirchhoff, Helmholtz and Bunsen. Eötvös introduced the concept of molecular surface tension and published on capillarity (1876-86). For the rest of his life he concentrated on study of the Earth's gravitational field. He developed the Eötvös torsion balance, long unsurpassed in precision, which gave experimental proof that inertial mass and gravitational mass, to a high degree of accuracy, are equivalent - which later was a major principle of Albert Einstein.*TIS

1848 Friedrich Ernst Dorn (27 July 1848 – 16 December 1916) was a German physicist who was the first to discover that a radioactive substance, later named radon, is emitted from radium.

Dorn was born in Guttstadt (Dobre Miasto), Province of Prussia (nowadays Warmia in Poland), and died in Halle, Province of Saxony.

He was educated at Königsberg and went on to teach at the university level. In 1885, at Halle University, Dorn took over the position of personal ordinarius professor for theoretical physics from Anton Oberbeck. Since Dorn was already an ordinarius professor, he was allowed to assume the title so as to not appear as having been demoted. In 1895, Dorn succeeded Hermann Knoblauch at Halle as the ordinarius professor for experimental physics and director of the physics institute. Dorn's previous duties were assumed by Carl Schmidt, who had been a Privatdozent and was called as an extraordinarius professor for theoretical physics.

In 1900, Dorn published a paper in which he described experiments that repeated and extended some earlier work on thorium by Ernest Rutherford. Dorn verified Rutherford's observation that a radioactive material was emitted by thorium, and discovered that a similar emission arose from the element radium. Additional work by Rutherford and Soddy showed that the same emission came from both thorium and radium, that it was a gas, and that it was actually a new element.

Dorn called the radioactive gaseous product from radium simply "emanation", but in 1904 Rutherford introduced the name "radium emanation" for the same material. Ramsay later suggested "niton", from the Latin word "nitens" meaning "shining". In 1923 the name was again changed, this time to radon by an international body of scientists.*Wik

1849 John Hopkinson (27 July 1849 – 27 August 1898) British physicist and electrical engineer who worked on the application of electricity and magnetism in devices like the dynamo and electromagnets. Hopkinson's law (the magnetic equivalent of Ohm's law) bears his name. In 1882, he patented his invention of the three-wire system (three phase) for electricity generation and distribution. He presented the principle the synchronous motors (1883), and designed electric generators with better efficiency. He also studied condensers and the phenomena of residual load. In his earlier career, he became (1872) engineering manager of Chance Brothers and Co., a glass manufacturer in Birmingham, where he studied lighthouse illumination, improving efficiency with flashing groups of lights.*TIS

1867 Derrick Norman Lehmer (27 July 1867, Somerset, Indiana, USA — 8 September 1938 in Berkeley, California, USA) was an American mathematician and number theorist.
In 1903, he presented a factorization of Jevons' number (8,616,460,799) at the San Francisco Section of the American Mathematical Society, December 19, 1903.
He published tables of prime numbers and prime factorizations, reaching 10,017,000 by 1909 (In Number Theory and Its History, Ore calls this the "best factor table now (1948) available"). He developed a variety of mechanical and electro-mechanical factoring and computational devices, such as the Lehmer sieve, built with his son Derrick Henry Lehmer.

He is also known for a reversible algorithm that assigns a Lehmer code to every permutation of size n. *SAU

1870 Bertram Borden Boltwood (July 27, 1870 Amherst, Massachusetts - August 15, 1927, Hancock Point, Maine) was an American chemist and physicist whose work on the radioactive decay of uranium and thorium was important in the development of the theory of isotopes. Boltwood studied the "radioactive series" whereby radioactive elements sequentially decay into other isotopes or elements. Since lead was always present in such ores, he concluded (1905) that lead must be the stable end product from their radioactive decay. Each decay proceeds at a characteristic rate. In 1907, he proposed that the ratio of original radioactive material to its decay products measured how long the process had been taking place. Thus the ore in the earth's crust could be dated, and give the age of the earth as 2.2 billion years.*TIS

1871 Ernest Friedrich Ferdinand Zermelo. (27 July 1871; Berlin, German Empire - 21 May 1953 (aged 81) Freiburg im Breisgau, West Germany) In 1904 he formulated the Axiom of Choice in Set Theory. Years later, when he refused to give the Nazi salute, he was threatened with dismissal from his university position. In reply, he resigned. *VFR

1921 Jonas Kubilius (27 July 1921 – 30 October 2011) was a Lithuanian mathematician who worked in probability theory and number theory. He was rector of Vilnius University for 32 years, and served one term in the Lithuanian parliament.

Kubilius's scientific work was in the areas of number theory and probability theory. The Turán–Kubilius inequality and the Kubilius model in probabilistic number theory are named after him. Eugenijus Manstavičius and Fritz Schweiger wrote about Kubilius's work in 1992, "the most impressive work has been done on the statistical theory of arithmetic functions which almost created a new research area called Probabilistic Number Theory. A monograph devoted to this topic was translated into English in 1964 and became very influential." (The monograph is Probabilistic Methods in the Theory of Numbers.)

Kubilius organized the first mathematical olympiad in Lithuania in 1951, and he wrote books of problems for students to use in preparing for the olympiads. He was a past president of the Lithuanian Mathematical Society.

In addition to his scientific and administrative work, Kubilius was a member of the Seimas (Lithuanian parliament) from 1992 to 1996. *Wik

DEATHS
1759 Pierre-Louis Moreau de Maupertuis (17 July 1698 – 27 July 1759) French mathematician, biologist, and astronomer. In 1732 he introduced Newton's theory of gravitation to France. He was a member of an expedition to Lapland in 1736 which set out to measure the length of a degree along the meridian. Maupertuis' measurements both verified Newton's predictions that the Earth would be an oblate speroid, and they corrected earlier results of Cassini. Maupertuis published on many topics including mathematics, geography, astronomy and cosmology. In 1744 he first enunciated the Principle of Least Action and he published it in Essai de cosmologie in 1850. Maupertuis hoped that the principle might unify the laws of the universe and combined it with an attempted proof of the existence of God.*TIS (he died in the home of Johann II Bernoulli. Johan Bernoulli (above) was born on the day Maupertuis died, but Johann II Bernoulli died on the Calendar date on which Maupertuis was born...)

1844 John Dalton, (6 September 1766 – 27 July 1844) English teacher who, from investigating the physical and chemical properties of matter, deduced an Atomic Theory (1803) whereby atoms of the same element are the same, but different from the atoms of any other element. In 1804, he stated his law of multiple proportions by which he related the ratios of the weights of the reactants to the proportions of elements in compounds. He set the atomic weight of hydrogen to be identically equal to one and developed a table of atomic weights for other elements. He was the first to measure the temperature change of air under compression, and in 1801 suggested that all gases could be liquefied by high pressure and low temperature. Dalton recognized that the aurora borealis was an electrical phenomenon.*TIS

*Linda Hall Org

1931 Jacques Herbrand (12 February 1908 – 27 July 1931) was a French mathematician who died young but made contributions to mathematical logic.*SAU Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant. *Wik

1999 Aleksandr Danilovic Aleksandrov (4 Aug 1912 in Volyn, Ryazan, Russia
- 27 July 1999) approached the differential geometry of surfaces [by extending the notion of the objects studied], extending the class of regular convex surfaces to the class of all convex surfaces ... . In order to solve concrete problems Aleksandrov had to replace the Gaussian geometry of regular surfaces by a much more general theory. In the first place the intrinsic properties (i.e. those properties that appear as a result of measurements carried out on the surface) of an arbitrary convex surface had to be studied, and methods found for the proof of theorems on the connection between intrinsic and exterior properties of convex surfaces. Aleksandrov constructed a theory of intrinsic geometry of convex surfaces on that basis. Because of the depth of this theory, the importance of its applications and the breadth of its generality, Aleksandrov comes second only to Gauss in the history of the development of the theory of surfaces. *SAU

2015 John William Scott Cassels (11 July 1922 – 27 July 2015) initially worked on elliptic curves. After a period when he worked on geometry of numbers and diophantine approximation, he returned in the later 1950s to the arithmetic of elliptic curves, writing a series of papers connecting the Selmer group with Galois cohomology and laying some of the foundations of the modern theory of infinite descent. His best-known single result may be the proof that the Tate-Shafarevich group, if it is finite, must have order that is a square; the proof being by construction of an alternating form. Cassels has often studied individual Diophantine equations by algebraic number theory and p-adic methods.
His publications include 200 papers. His advanced textbooks have influenced generations of mathematicians; some of Cassels's books have remained in print for decades. *Wik

2021 Enrique Aurelio Planchart Rotundo (3 April 1937 – 27 July 2021) was a Venezuelan mathematician and academic. He was rector of Simón Bolívar University in Caracas from 2009 until his death in 2021.

Planchart graduated as a Bachelor of Science from the Central University of Venezuela and obtained his Doctorate in Mathematics from the University of California, Berkeley, where he was also a visiting professor in its Department of Mathematics between 1986 and 1987. From 1973 he was part of the Department of Pure and Applied Mathematics of the Simón Bolívar University.

While at Simón Bolívar University, between 1989 and 1999 he directed the National Center for the Improvement of Science Education, and from 1999 he directed the Equal Opportunities Program (PIO). In 1989 he was awarded the National Council for Scientific and Technological Research Award.

Throughout his scientific career, Planchart published nine books and nine journal articles and gave thirty lectures *Wik