Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccolo Tartaglia of Brescia gave Niccolo occasion to discover it. He [Tartaglia] gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in [various] forms. This was very difficult.
~Girolamo Cardano
This is the 267th Day of the Year
267 = 46^2 - 43^2, and also 134^2 - 133^2
267 is the smallest number n such that n+ a googol is prime. (anyone want to find the next one? A quick mental problem for students, How do you know that 269+Googol will not be prime?))
267 can be written as the sum of five cubes in two ways, \( 267 = 1^3 + 2^3 + 2^3 + 5^3 + 5^3 = 2^3 + 2^3 + 2^3 + 3^3 + 6^3 \)
Gauss proved that all numbers are the sum of, at most, three triangular numbers. (see
July 10, events 1796) Can you find three triangular numbers that sum to 267?, can you find a sum with two? How many can you find in two or less?
Many people know that N! has N digits for N= 22, 23, and 24. Surprisingly, to me, there are also three consecutive numbers for which N! has 2N digits, 266, 267, and 268. (this means 266! has 2*266 or 532 digits)
For N! has 3N digits, only two consecutive numbers, 2712 and 2713.For N! having 4N digits, there are again two consecutive occurrences, 27175 and 27176. For 5N we go back to three consecutive digits,
271819, 271820, 271821 Note the increase by a power of ten as a limit, and the higher you go, the closer they approach e * 10^n. It has been conjectured that there are always at two or three consecutive numbers for every digit, but never more. The first 100 such numbers are found at
A058814 - OEIS Thanks to Derek Orr and Frank Kampas for some help and direction on this.
EVENTS
1846 Neptune First observed… “It was on that date, back in 1846, that German Astronomer Johann Galle, assisted by graduate student Heinrich Louis d’Arrest, trained the 24 centimeter (9 inch) Fraunhofer Refractor of the Berlin Observatory on a patch of sky near the Aquarius-Capricorn border (see illustration below) and observed the small, blue disk of Neptune. On July 12th, 2011 Neptune completed exactly one orbit since its discovery. One hundred and sixty five years ago a series of events played out in France, England and Germany that would culminate in a watershed moment in the history science and astronomy, a discovery that would prove to be unique and unrepeatable. These events were rife with centuries-old rivalries, political conspiracy and intrigue, all mixed together with good mathematics, some good science, some bad science, some luck and much mayhem.”
1852 The steam powered airship was made by Baptiste Jules Henri Jacques Giffard His airship, powered with a steam engine, and weighing over 180 kg (400 lb), it was the world's first passenger-carrying airship (then known as a dirigible, which was French ). Both practical and steerable, the hydrogen-filled airship was equipped with a 3 hp steam engine that drove a propeller. The engine was fitted with a downward-pointing funnel. The exhaust steam was mixed in with the combustion gases and it was hoped by these means to stop sparks rising up to the gas bag; he also installed a vertical rudder.
On 24 September 1852 Giffard made the first powered and controlled flight traveling 27 km from Paris to Élancourt. The wind was too strong to allow him to make way against it, so he was unable to return to the start. However, he was able to make turns and circles,[citation needed] proving that a powered airship could be steered and controlled. *Wik
1940 Westinghouse patent application for the Nimatron, a machine to play the game of Nim, is approved. Created by Eduard Condon, Edgewood Tawney, Gerald Tawney, and Willard Dorr, the machine would be featured in the Westinghouse exhibit at the 1940 World's Fair. The machine played 100,000 games at the fair, winning about 90,000. Most of its defeats were apparently administered by attendants to demonstrate that possibility. When the machine did lose it would "present its opponent with a token coin stamped with the words 'Nim Champ'" *historyofinformation.com
Do kids in math classrooms still learn and play Nim today? someone in the classroom from grade 7-12 who would survey a class and tell me if any recognize the name or the game.
On this day in 2017, Maryna Viazovska was given a Clay Research Award in recognition of her groundbreaking work on sphere-packing problems in eight and twenty-four dimensions. In particular, she proved that the \(E_8\) lattice is an optimal solution in eight dimensions.
Maryna Sergiivna Viazovska is a Ukrainian mathematician known for her work in sphere packing. She is full professor and Chair of Number Theory at the Institute of Mathematics of the École Polytechnique Fédérale de Lausanne in Switzerland. She was awarded the Fields Medal in 2022
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| *Wik |
BIRTHS
1501 Girolamo Cardano (24 Sep 1501; 21 Sep 1576) Famous for his Ars magna of 1545, which contained detailed and systematics algebraic solutions to cubic and quartic equations. He was one of the most colorful figures in the whole history of mathematics, as is well illustrated in his autobiography, The Book of My Life. *VFR
Italian physician, mathematician, and astrologer who was the first to give a clinical description of typhus fever. His book, Ars magna ("Great Art," 1545) was one of the great achievements in the history of algebra, in which he published the solutions to the cubic and quartic equations. His mechanical inventions included the combination lock, the compass gimbal consisting of three concentric rings, and the universal joint to transmit rotary motion at various angles (as used in present-day vehicles). He contributed to hydrodynamics and held that perpetual motion is impossible, except in celestial bodies. He published two encyclopedias of natural science and introduced the Cardan grille, a cryptographic tool (1550). *TIS
His gambling led him to formulate elementary rules in probability, making him one of the founders of the field.
One story says that it was by his own hand so as to fulfill his earlier astrological prediction of of his death on this date. *H. Eves, Introduction to the History of Mathematics, Pg 221...
1625 Jan de Witt (September 24, 1625, Dordrecht - August 20, 1672) murdered by a mob from the (William of) Orange faction. For the previous twenty years he served as grand pensionary in Holland, essentially the prime minister of the Netherlands. Consequently this talented mathematician had little time to devote to mathematics. He wrote the first systematic account of the analytic geometry of the straight line and conics. It was published in Van Schooten’s second Latin edition of Descartes’ Geometrie *VFR de Witt and his brother were both killed by a mob which was probably supported by William III of Orange. At the very least, as the Wikipedia articles states, "he protected and rewarded the killers." After a previous attempt on his life, he was lured by a forged letter to the cell where his brother was held, and both were hanged and then their bodies were mutilated. The story of their deaths are a critical element in the plot of Alexander Dumas' "The Black Tulip". *Wik
1766 John Farey, Sr. (1766 – January 6, 1826) was an English geologist and writer. However, he is better known for a mathematical construct, the Farey sequence named after him.
Farey's most famous work is General View of the Agriculture and Minerals of Derbyshire (3 volumes 1811-17) for the Board of Agriculture. In the first of these volumes (1811) he gave an able account of the upper part of the British series of strata, and a masterly exposition of the Carboniferous and other strata of Derbyshire. In this classic work, and in a paper published in the Philosophical Magazine, vol. 51, 1818, p. 173, on 'Mr Smith's Geological Claims stated', he zealously called attention to the importance of the discoveries of William Smith.
As well as being remembered by historians of geology, his name is more widely known by the Farey sequence which he noted as a result of his interest in the mathematics of sound (Philosophical Magazine, vol. 47, 1816, pp 385-6).
Farey died in London. Subsequently his widow offered his geological collection to the British Museum, which rejected it, and it was dispersed.*WikFarey diagram to F9 represented with circular arcs. In the SVG image, hover over a curve to highlight it and its terms.*Wik
1844 Max Noether born (24 September 1844 – 13 December 1921) . One of the leaders of nineteenth century algebraic geometry. Although himself a very distinguished mathematician (He has been called "one of the finest mathematicians of the nineteenth century"). his daughter Emmy Noether was to bring greater innovation to mathematics than did her father. *SAU
Brill and Max Noether developed alternative proofs using algebraic methods for much of Riemann's work on Riemann surfaces. Brill–Noether theory went further by estimating the dimension of the space of maps of given degree d from an algebraic curve to projective space Pn. In birational geometry, Noether introduced the fundamental technique of blowing up in order to prove resolution of singularities for plane curves.
Noether made major contributions to the theory of algebraic surfaces. Noether's formula is the first case of the Riemann-Roch theorem for surfaces. The Noether inequality is one of the main restrictions on the possible discrete invariants of a surface. *Wik
1862 Winifred Edgerton Merrill (September 24, 1862 – September 6, 1951) made a vast impact on the male orientated world of mathematics. She left behind the Victorian ideal that a wellborn woman should stay at home, and went about continuing her education in mathematics to Ph.D. level. This was a fantastic achievement and Merrill became the first American woman to obtain a Ph.D. in mathematics. *SAU
She earned her B.A. degree from Wellesley College in 1883, and taught for a time at Sylvanus Reed's School. She continued her interest in astronomy by independently using data from the Harvard observatory to calculate the orbit of the Pons-Brooks comet of 1883. She then appealed to Columbia University for permission to use their telescope. On February 4, 1884 the members of the board of trustees agreed, considering her an "exceptional case" and cautioning her "not to disturb the male students." She was required to work as a laboratory assistant to the director of the observatory.
She studied math and astronomy at Columbia which at the time was an all-male institution. Her teachers included Professor John Krom Rees, Professor J. Howard Van Amringe and Professor William Guy Peck. After her first appeal to receive a degree was rejected by the trustees, she was advised by President Frederick A. P. Barnard to speak to each of the trustees individually. At the next meeting, she was awarded the PhD with high honors from Columbia University in 1886, by a unanimous vote.
1870 Georges Claude (24 Sep 1870; 23 May 1960) The French engineer, chemist, and inventor of the neon light, Georges Claude, was born in Paris. He invented the neon light, which was the forerunner of the fluorescent light. Claude was the first to apply an electrical discharge to a sealed tube of neon gas, around 1902 and make a neon lamp ("Neon" from Greek "neos," meaning "new gas.") He first publicly displayed the neon lamp on 11 Dec 1910 in Paris. His French company Claude Neon, introduced neon signs to the U.S. with two "Packard" signs for a Packard car dealership in Los Angeles, purchased by Earle C. Anthony for $24,000. *TIS
1891 William F. Friedman (24 Sep 1891; 12 Nov 1969) one of the world's greatest cryptologists, who helped decipher enemy codes from World War I to World War II. He was born as Wolfe Friedman.in Kishinev, Russia. He emigrated to the U.S. in 1893. Originally trained as an agricultural geneticist, he had become interested in cryptology. During World War I, with his wife Elizebeth, he set up a cryptology school for military personnel, which led to appointment by the U.S. as head of the Signal Intelligence Service (1930). He broke the Japanese "Purple" code (1937-40), thus allowing Americans to read much of Japan's secret messages during World War II. *TIS There is a bust of him at the National Cryptologic Museum in Fort Meade Maryland on which he is identified as the "Dean of American Cryptology".
William and Elizabeth around 1917
1896 Tadeusz Ważewski (24 September 1896 – 5 September 1972) was a Polish mathematician.
Ważewski made important contributions to the theory of ordinary differential equations, partial differential equations, control theory and the theory of analytic spaces. He is most famous for applying the topological concept of retract, introduced by Karol Borsuk to the study of the solutions of differential equations. *Wik
Ważewski studied at the Jagiellonian University in 1914–1920. He started from physics but very quickly turned to mathematics. Ważewski was a pupil of Zaremba.
He spent three years in Paris and got a doctoral diploma from Sorbona.
Ważewski’s research started from topology. In his doctoral dissertation he obtained interesting results on dendrites (locally connected continua not containing simple closed curves). *Ciesielski & Pogoda, EMS Newsletter December 2012
1898 Charlotte Moore Sitterly (24 Sep 1898; 3 Mar 1990) astrophysicist who organized, analyzed, and published definitive books on the solar spectrum and spectral line multiplets. From 1945 to age 90, she conducted this work at the U.S. National Bureau of Standards and the Naval Research Laboratory. She detected that technetium, an unstable element (previously known only as a result of laboratory experiments with nuclear reactions) exists in nature. She made major contributions to the compilation of tables for atomic-energy levels associated with optical spectra, which are now standard reference material. As instruments carried in space rockets provided new data in the ultraviolet, she extended these tables beyond the optical range. She was awarded the Bruce Medal in 1990.*TIS
Inscription. Prominent authority on astronomy and author of more than one hundred books and articles. Sitterly was a career physicist with the Bureau of Standards, U.S. Department of Commerce. She received the American Astronomical Society award in 1937 and was the first woman elected to the Royal Astronomical Society of Great Britain, 1949. Born here in Ercildoun, Dr. Sitterly was a lifelong Quaker and attended Fallowfield Friends Meeting nearby. Erected 2005 by Pennsylvania Historical and Museum Commission.
1904 Evan T Davies graduated from the University of Wales at Aberystwyth and then studied in Rome and Paris. After lecturing at King's College London he was appointed to a professorship in Southampton. He worked in Differential Geometry and the Calculus of Variations.*SAU
1906 Pol(idore) Swings, (24 Sep 1906; 28 Oct, 1983) Belgian astrophysicist, made spectroscopic studies to identify elements and structure of stars and comets. He discovered the first interstellar molecule, the CH radical (1937). In comet atmospheres he studied the "Swings bands" - certain carbon emission lines. In 1941, with a slit spectrograph he identified a "Swings effect" in the violet CN bands (3875 A) - a fluorescence partly due to solar radiation that shows emmission line excitation differences dependant on the Doppler shift caused by a comet's motion relative to the Sun. He co-authored an Atlas of Cometary Spectra with Leo Haser in 1956. *TIS
1923 Raoul Bott,(September 24, 1923 – December 20, 2005) was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. *Wik
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group.
1925 Geoffrey Ronald Burbidge (24 September 1925 – 26 January 2010), an English/American astronomer and astrophysicist, was born in Chipping Norton, Oxfordshire . Burbidge earned his PhD from University College London. He met there Margaret Burbidge, an observational astronomer, and they were married. Eventually both moved to the United States and taught and worked at Mount Wilson Observatories and Caltech, before they both accepted permanent positions at the University of California, San Diego, where Geoffrey taught from 1962 until his retirement. He was director of Kitt Peak Observatory in Arizona from 1978 to 1984.
Before they moved to the United States, Geoffrey and Margaret worked at Cambridge University with Fred Hoyle and an American physicist, Willy Fowler. This was not long after the cosmogonical wars had begun, with George Gamow championing the Big Bang hypothesis, while Hoyle was in favor of a steady-state theory, which did not require a Big Bang to explain the expansion of the universe. Gamow’s Big Bang proposal also sought to explain the origin of the elements as byproducts of the Big Bang, so that heavier elements were cooked up in rapid succession by the addition of protons and neutrons in a process now known as nucleosynthesis. Since steady-state advocates had no Big Bang to serve as a cosmic cooker, they needed to explain nucleosynthesis some other way.
The two Burbidges, Fowler, and Hoyle worked on the problem in Cambridge in 1954-55, and published a now-famous paper in 1957 which put forward the idea of stellar nucleosynthesis. Elements, they argued, are slowly built up by fusion in the cores of stars, so that hydrogen fuses to helium, then to carbon, oxygen, silicon, all of the elements up to iron. The remaining elements are byproducts of the energy released in supernova explosions, which not only produce the rest of the elements up to uranium, but explode them out into the interstellar medium, where they become the building blocks for future stars.
The paper was written at Caltech in 1956, primarily by the Burbidges and Fowler, and published in Reviews of Modern Physics in 1957. It was over 100 pages long, with lots of supporting data to show that chemical abundances in the universe have changed over time (which should not happen according to Gamow). The paper was called “Synthesis of the elements in stars,” and the authors were listed, in order, as E. Margaret Burbidge, G. R. Burbidge, William A. Fowler, and F. Hoyle (second image). This was almost immediately abbreviated to B2FH by readers (pronounced B-squared-F-H), and the paper is usually referred to as the B2FH paper. I don’t know if Geoffrey ever did so, but if asked which author he was, he could have responded: “I was the square” (or, less dramatically, “I was the 2”).
The authors of B2FH, left to right: Margaret Burbidge, Geoffrey Burbidge, Willy Fowler, and Fred Hoyle, photograph, 1971, exhibition at St. John’s College, Cambridge, 1971, photo by Don Clayton (joh.cam.ac.uk)
The paper has become a milestone paper because B2FH were right – only hydrogen and helium were byproducts of the Big Bang; all of the 90 other elements were made in stars – normal stars, red giants, white dwarfs, and supernovas. Nearly all nucleosynthesis in the cosmos is stellar nucleosynthesis. Gamow turned out to be right about the Big Bang, but not about the origin of the elements. Credit for that goes to the Burbidges, Hoyle, and Fowler. We include a group photo of the four, 14 years after B2FH, on the occasion of Fowler’s birthday. *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.
1930 John Watts Young (September 24, 1930 – January 5, 2018) astronaut who was the commander of the first ever Space Shuttle mission (STS-1, 12 Apr 1981), walked on the Moon during the Apollo 16 mission (21 Apr 1972), made the first manned flight of the Gemini spacecraft with Virgil Grissom. *TIS
He is the only astronaut to fly on four different classes of spacecraft: Gemini, the Apollo command and service module, the Apollo Lunar Module and the Space Shuttle.
1945 Ian Nicholas Stewart FRS (24 September, 1945 - ) is an Emeritus Professor Mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer.
While in the sixth form at school, Stewart came to the attention of the mathematics teacher. The teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students; Stewart placed first in the examination. This teacher arranged for Stewart to be admitted to Cambridge on a scholarship to Churchill College, where he obtained a BA in mathematics. Stewart then went to the University of Warwick for his doctorate, on completion of which in 1969 he was offered an academic position at Warwick, where he presently professes mathematics. He is well known for his popular expositions of mathematics and his contributions to catastrophe theory.
While at Warwick he edited the mathematical magazine Manifold. He also wrote a column called "Mathematical Recreations" for Scientific American magazine for several years.
Stewart has held visiting academic positions in Germany (1974), New Zealand (1976), and the U.S. (University of Connecticut 1977–78, University of Houston 1983–84). *Wik
DEATHS
1054 Hermann of Reichenau (1013 July 18 – 1054 September 24), was a German mathematician who important for the transmission of Arabic mathematics, astronomy and scientific instruments into central Europe. Hermann introduced three important instruments into central Europe, knowledge of which came from Arabic Spain. He introduced the astrolabe, a portable sundial and a quadrant with a cursor.
His works include De Mensura Astrolabii and De Utilitatibus Astrolabii (some parts of these works may not have been written by Hermann).
Hermann's contributions to mathematics include a treatise dealing with multiplication and division, although this book is written entirely with Roman numerals. He also wrote on a complicated game based on Pythagorean number theory which was derived from Boethius.
The game was played with counters on a board; capture of the opponent's pieces was dependent on the determination of arithmetical ratios and arithmetic, geometrical, and harmonic progressions. This game, which enjoyed a considerable vogue during the Middle Ages, has been attributed to Pythagoras, Boethius, and Gerbert.*SAU
1651 Etienne Pascal died (Clermont, May 2, 1588 - Paris, September 24, 1651). The Pascal limacon is named after him, and not after his famous son who later came blazing on the scene. *VFR Étienne is famed as the discoverer of the curve the Limaçon of Pascal. The curve, so named by Roberval, can be used to trisect an angle. He discovered the curve in around 1637. (Limacon is from the Latin word for a snail the curve is a roulette formed when a circle rolls around the outside of another circle.) In a letter (see Lettre d'Étienne Pascal et Roberval à Fermat, samedi 16 août 1636) he actively argued in favour of Fermat's De maximis et minimis in opposition to Descartes who viewed the work in a very negative light. *SAU
1938 Lev Genrikhovich Schnirelman ( 2 January 1905 in Gomel ; 24 September 1938 in Moscow ) He was a Belarussian mathematician who made important contributions to the Goldbach conjecture. Using these ideas of compactness of a sequence of natural numbers he was able to prove a weak form of the Goldbach conjecture showing that every number is the sum of ≤ 20 primes.*SAU
On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until all terms are units.
Goldbach was following the now-abandoned (mostly) convention of considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Every integer greater than 2 can be written as the sum of three primes.
1945 Hans (Wilhelm) Geiger (30 Sep 1882, 24 Sep 1945) was a German physicist who introduced the Geiger counter, the first successful detector of individual alpha particles and other ionizing radiations. After earning his Ph.D. at the University of Erlangen in 1906, he collaborated at the University of Manchester with Ernest Rutherford. He used the first version of his particle counter, and other detectors, in experiments that led to the identification of the alpha particle as the nucleus of the helium atom and to Rutherford's statement (1912) that the nucleus occupies a very small volume in the atom. Geiger returned to Germany in 1912 and continued to investigate cosmic rays, artificial radioactivity, and nuclear fission. *TIS
1885 Pauline Sperry (March 5, 1885 – September 24, 1967) born in Peabody, Massachusetts. After graduating Phi Beta Kappa from Smith College in 1906 she taught several years before doing graduate work at the University of Chicago under the projective differential geometer Ernest Julius Wilczynski (1876–1932). Her doctoral thesis, "Properties of a certain projectively defined two-parameter family of curves on a general surface", drew on his work as the founder of the American school of projective differential geometry. After receiving her Ph.D. in 1916 she taught at the University of California at Berkeley, becoming the first woman to be promoted to assistant professor in mathematics (in 1923). In 1950 she was fired for refusing to sign a loyalty oath.
At the height of McCarthyism, the Board of Regents required university employees to sign a loyalty oath. Sperry, Hans Lewy, and others who refused were barred from teaching without pay in 1950. In the case Tolman v. Underhill, the California Supreme Court ruled in 1952 the loyalty oath unconstitutional and reinstated those who refused to sign. Sperry was reinstated with the title emeritus associate professor and later awarded back pay. *Wik
1999 Anneli Cahn Lax (23 Feb 1922 in Katowice, Poland - 24 Sept 1999 in New York City, New York, USA) Anneli Cahn was born in Katowice, then a German city, but now part of Poland, on February 23, 1922. Her family fled Hitler’s regime in 1935 and settled in New York. She married Peter Lax, a fellow mathematician, in 1948. Their lives together included a shared love for mathematics. Perhaps her most important contribution to mathematics was as editor of the New Mathematics Library. The launch of the Soviet satellite Sputnik in 1957 was a shock to the American scientific community, a shock felt on every level. Much thought was devoted to the education of a new generation who would accelerate the pace of American scientific productivity. Out of this endeavor grew the New Mathematical Library. The notion was to make accessible to interested high school students, and to a more general public, deep results in mathematics described by research mathematicians. (This sort of work had long been going on in Eastern Europe.) Lax was asked to take over as general editor for this series, and under her guidance it grew to be the foremost mathematical expository series in the language. Upon her death it was renamed in her honor. *Mark Saul, Obituary for the AMS VOl 47,#7
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
he was able to express the Nth term of the series as 
. 
the word bighellone. Julio Gonzalez Cabillon has written, "The name 'Fibonacci' most probably originated with the historian of mathematics Guillaume Libri (1803-1869)."
sequence and the Golden Ratio at this page from England. The page is also the source of the photo at right of the statue to Fibonacci in the cemetary at the Duomo (Cathedral) in Pisa. 
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