Friday, 29 September 2023

The Harmony of the Harmonic Mean

Things happen in threes according to the old myth, and in this case it was true. I was doing some research on the early history of a mathematical problem often called the "cistern" problem. You probably know the type; "If one pipe can fill a cistern in 6 hours and another can fill it in four hours, how long would it take both pipes working together." While I was working on that, I got a nice article sent to me on the first proof that the harmonic sequence diverges... and then, I was reading a blog by Dave Marain Math Notationsin which he posed a problem that asked, in its general form, given a square inscribed in a right triangle (with one corner at the right angle of the triangle), what is the length of a side of the square in terms of the legs of the triangle.

So what do all these have in common with each other. dare I say what makes them in "harmony"?.... the answer is Harmony, or at least the mathematical relationship of the harmonic mean.

To the early Greeks, if Nichomachus can be believed, all the means were descriptive of musical relations. Much is often made of the Harmonic Mean in relation to a musical sense, but this may not represent the Greek view. Euclid used the word enarmozein to describe a segment that just fits in a given circle. The word is a form of the word Harmozein which the more competent Greek Scholars tell me means to join or to fit together. Jeff Miller's Web site on the first use of Mathematical terms contains a reference to the very early origin of the harmonic mean, 'A surviving fragment of the work of Archytas of Tarentum (ca. 350 BC) states, 'There are three means in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call harmonic.' The term harmonic mean was also used by Aristotle. "
My search for the early roots of the cistern problem had taken me back to Heron's Metre'seis around the year fifty of the common era. The problem became a staple in arithmetics and problem books and was used by Alcuin (775) and appears in the Lilavati of Bhaskara (1150). I found the illustration I used on the blog for The First Illustrated Arithmetic a few days ago, from the 1492 arithmetic, Trattato di aritmetica by Filippo Calandri.

The solution to a cistern problem is the harmonic mean of the times taken by each pipe. For example, one problem asks "If one pipe can fill a cistern in three hours, and a second can fill it in five hours, how fast will the two pipes take to fill the cistern if both are opened at once. The solution is given by finding the average rate of fill of the two rates, the harmonic mean of three and five, which is three and three-quarter hours. But as the name "mean" suggest, that's the average rate of the two so working together, they would take one-half the time, one and seven-eighths hours, or about an hour and 53 minutes.

The Harmonic mean is the reciprocal of the mean of the reciprocals of the values, so for values a and b, the harmonic mean is given bywhich for two numbers can be simplified to the more economical
Heron might have been the first recorded example of a cistern problem, but a problem calling on the reader to use the harmonic mean occurs even earlier in the Rhind Mathematical Papyrus, now located in the British Museum, in problem 76. The problem involves making loaves of bread with different qualities, but the solution is still the harmonic mean. (I have learned from David Singmaster's Chronology of Recreational Mathematics that the cistern problem appeared, perhaps 300 years before Heron's use, in China by Chiu Chang Suan Shu (around 150 BC).

The series of terms formed by the reciprocals of the positive integers is a common torment for college students in their first introduction to analysis. The sequencein which each number gets smaller and smaller seems to very slowly approach some upper limit. Even after adding 250,000,000 terms, the sum is still less than twenty, and yet... in the mid 1300's, Nichole d'Oresme showed that it will eventually pass any value you can name. In short, it diverges, slowly, very, very slowly, to infinity. Even when warned, it seems like students want to believe it converges. A well-known anecdote about a teacher trying to get student's to remember that it diverges goes:

"Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”<\blockquote>

By the way, each number in the harmonic series is the harmonic mean of the numbers on each side of it (so 1/2 and 1/4 have a harmonic mean of 1/3), and in fact, of any numbers equally spaced away from it such as 1 and 1/5 also have a harmonic mean of 1/3.

And then, I came across that little problem of a square inscribed in a right triangle. If the two legs are a and b, then the sides of the square will have a length equal to the one-half the harmonic mean of a and b . More generally, a square inscribe in any triangle with one side along a base will have sides equal to one half the harmonic mean of the base and the altitude to that base. There are lots of other interesting problems that yield to the use of the harmonic mean, and I mean to write again on that collection.

So I guess things do come in threes, unless I come across another one, but whether it comes in threes or fours, it all seems to work together, in perfect harmony.

Many students who struggle with a different puzzle type problem might want to investigate how it too, relates to the harmonic mean, the one where they ask, If you drive to grandmother's house at 60 miles per hour and drive home at 40 miles per hour, what was your average speed for the round trip? There are dozens more, so just to get a collection, send your favorite problem related to the harmonic mean, and I'll update as they come along.

Thursday, 28 September 2023

The First Illustrated Arithmetic, and Common Long Division

I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.
Here are some additional notes from my web notes on division that pertain to the long division algorithm and five early methods that were used.

..... is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step.

The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 "The Common School Arithmetic : prepared for the use of academies and common schools in the United States" by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided. In a recent Greasham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school." (I assume he means with the quotient on top.)

I recently found a site called The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk (which surprised me as it was used almost exclusively in English and American textbooks, and even then seldom beyond elementary school), and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.

Tuesday, 26 September 2023

Two verses from a poem by James Clerk Maxwell.

Two verses from a poem by James Clerk Maxwell. Think deeply on them at your peril, student!

A vision of a Wrangle, of a University, of Pedantry, and of Philosophy

Deep St. Mary's bell had sounded,
And the twelve notes gently rounded
Endless chimneys that surrounded
My abode in Trinity.
(Letter G, Old Court, South Attics),
I shut up my mathematics,
That confounded hydrostatics —
Sink it in the deepest sea!

In the grate the flickering embers
Served to show how dull November’s
Fogs had stamped my torpid members,
Like a plucked and skinny goose.
And as I prepared for bed, I
Asked myself with voice unsteady,
If of all the stuff I read, I
Ever made the slightest use.

The rest, I fear you may seek, on the internet, for a peek.

Sunday, 24 September 2023

On This Day in Math - September 24

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.

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

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

*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 born. This statesman for the Netherlands wrote, before 1650, one of the ﬁrst systematic developments of the analytic geometry of the straight line and conics. It was printed in Van Schooten’s second Latin edition of Descartes’ geometry (1659–1661).*VFR A nice short article about his unusual death, and life are at this blog by The Renaissance Mathematicus

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, his daughter Emmy Noether was to bring greater innovation to mathematics than did her father. *SAU

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". There is an interesting biography here .

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. Click to hear the 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; 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)[1] 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

1930 John Watts Young (24 Sep 1930, ) 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

1945 Ian Nicholas Stewart FRS (24 September, 1945 - ) is an Emeritus Professor of 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. *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 Lew Genrichowitsch Schnirelmann (2 January(15January 1905greg.) in Gomel; 24 September 1938 in Moscaw) . 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

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

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

Saturday, 23 September 2023

On This Day in Math - September 23

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work.
~Feynman, Richard Philips Nobel Lecture, 1966.

The 266th day of the year; 266 can be expressed as 222 in base 11.

266 is the sum of four cubes,

266 = 2^{3} + 2^{3} + 5^{3} + 5^{3}

It is also the index of the largest proper subgroups of the sporadic group known as the Janko group J₁

266 is the sum of five consecutive Triangular numbers. 15 + 21 + 28 + 36 + 45 + 55 + 66. On Sept , 1796 Gauss's entry "EγPHKA! num=Δ+Δ+Δ" in his scientiﬁc diary, recording his discovery that every positive integer is the sum of (at most) three triangular numbers. Can you find three for 266? Can you find three or less in more than one way?

266 has a digit sum of 12, a divisor of 266, so it is a Joy-Giver number.

The sum of the divisors of 266 is 18^2 = 324.

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. 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
1574 Tycho Brahe's rising fame while he lived in Copenhagen brings unwanted lecturing demands. In the capital his rising fame had attracted considerable attention, and some young nobles who were studying at the University requested him to deliver a course of lectures on some mathematical subject on which there were no lectures being given at that time. His friends Dancey and Pratensis urged him to consent to this proposal, but Tycho was not inclined to do so, until the King had also requested him to gratify the wishes of the students. He then yielded, and the lectures were commenced on the 23rd of September 1574, with an oration on the antiquity and importance of the mathematical sciences. *TYCHO BRAHE, A PICTURE OF SCIENTIFIC LIFE AND WORK IN THE SIXTEENTH CENTURY BY J. L. E. DREYER

*MAA

1647 Descartes, on a visit on September 23-24 to France from Holland, met with Pascal. On this occasion Descartes may have recommended the experiment of noting the variation in the height of the barometer with altitude. [J. F. Scott, The Scientiﬁc Work of Ren´e Descartes, p. 6] *VFR
His visit only lasted two days and the two argued about the vacuum which Descartes did not believe in. Pascal had done a series of experiments on atmospheric pressure and proved to his satisfaction that a vacuum existed.Descartes wrote, rather cruelly, in a letter to Huygens after this visit that Pascal, " ...has too much vacuum in his head. " *SAU

Also present were Professor Roberval, of the College de France, a voluble anti-Cartesian, and Pascal's younger sister Jacqueline. Pascal brought out a calculating machine, his recent invention, and demonstrated its ability to add and subtract. Descartes was impressed. The talk turned to the vacuum. Pascal described his experiment; Descartes expressed doubt - a polite skirmish that might have ended there. But Roberval injected his opinion, and a heated argument ensued. Descartes took his leave.

The next morning, however, he returned - not Descartes the philosopher this time, but Descartes the physician. He sat for three hours by his patient's side, listened to his complaints, examined him, prescribed soups and rest. When Pascal was sick of staying in bed, Descartes said, he would be nearly well. Their views would remain opposed, but it was the supreme rationalist in his role as kindly doctor whom Pascal would later remember, and who may have been in his mind when he observed, "The heart has its reasons which reason knows nothing of"

*The Independent UK, Saturday 15 June 1996

The Pascaline, also called Arithmetic Machine, the first calculator or adding machine to be produced in any quantity and actually used. It was built by Blaise Pascal between 1642 and 1644. It could only do addition and subtraction, with numbers being entered by manipulating its dials. Pascal invented the machine for his father, a tax collector, so it was the first business machine too (if one does not count the abacus). He built 50 of them over the next 10 years.

1673 Hooke in his diary, "bought Pappus in Cornhill for 11sh. at ye crown." *Robert Hooke ‏@HookesLondon
Suspect but am not sure that this was Commandino's translation of Pappus's Mathematicae Collectiones

1763 The Princess Louise sailed for Barbados on 23 September. During the voyage Maskelyne and Charles Green took many lunar-distance observations (with Maskelyne later claiming that his final observation was within half of degree of the truth) and struggled a couple of times with the marine chair. Maskelyne’s conclusion was that the Jupiter’s satellites method of finding longitude would simply never work at sea because the telescope magnification required was far too high for use in a moving ship.
On 29 December 1763 he wrote his brother Edmund, reporting his safe arrival on 7 November after “an agreeable passage of 6 weeks”. He noted that he had been “very sufficiently employed in making the observations recommended to me by the Commissioners of Longitude” and that it was at times “rather too fatiguing”. *Board of Longitude project, Greenwich

A marine chair made by Christopher Irwin that was intended to steady an observer to allow him to measure the positions of Jupiter's satellites at sea. (Eclipses of Jupiter's moons were already used as a celestial timekeeper to determine longitude on land: these were the observations Maskelyne made at Barbados.)

1740 In a letter to Euler dated August 29th, 1740, Philippe Naudé (the Younger) asked Euler in how many ways a number n can be written as a sum of positive integers. In his answer written on September 12th (23rd), Euler explained that if we denote
this “partition number” by p(n), then

*Correspondence of Leonhard Euler with Christian Goldbach, Springer

1793 The new decimalized calendar was presented to the Jacobin-controlled National Convention on 23 September 1793, which adopted it on 24 October 1793 and also extended it proleptically to its epoch of 22 September 1792. The French Republican Calendar was a calendar created and implemented during the French Revolution, and used by the French government for about 12 years from late 1793 to 1805, and for 18 days by the Paris Commune in 1871. There were twelve months, each divided into three ten-day weeks called décades. The tenth day, décadi, replaced Sunday as the day of rest and festivity. The five or six extra days needed to approximate the solar or tropical year were placed after the months at the end of each year. The new system was designed in part to remove all religious and royalist influences from the calendar, and was part of a larger attempt at decimalisation in France. *Wik

The Months of the French Decimal Calendar

Autumn:

Vendémiaire (from French vendange, derived from Latin vindemia, "vintage"), starting 22, 23, or 24 September

Brumaire (from French brume, "mist", from Latin brūma, "winter solstice; winter; winter cold"), starting 22, 23, or 24 October

Frimaire (from French frimas, "frost"), starting 21, 22, or 23 November

Winter:

Nivôse (from Latin nivosus, "snowy"), starting 21, 22, or 23 December

Pluviôse (from French pluvieux, derived from Latin pluvius, "rainy"), starting 20, 21, or 22 January

Ventôse (from French venteux, derived from Latin ventosus, "windy"), starting 19, 20, or 21 February

Spring:

Germinal (from French germination), starting 21 or 22 March

Floréal (from French fleur, derived from Latin flos, "flower"), starting 20 or 21 April

Prairial (from French prairie, "meadow"), starting 20 or 21 May

Summer:

Messidor (from Latin messis, "harvest"), starting 19 or 20 June

Thermidor (or Fervidor*) (from Greek thermon, "summer heat"), starting 19 or 20 July

Fructidor (from Latin fructus, "fruit"), starting 18 or 19 August

1815 The Great September Gale of 1815 came ashore in New England on this date. This was the first hurricane, although the word had not been created yet, to hit New England in 180 yrs. In the aftermath of the Great Gale, the concept of a hurricane as a "moving vortex" was presented by John Farrar, Hollis Professor of Mathematics and Natural Philosophy at Harvard University. In an 1819 paper he concluded that the storm "appears to have been a moving vortex and not the rushing forward of a great body of the atmosphere". The word "hurricane" comes from Spanish huracán, from the Taino hurakán, “god of the storm.” While the Taino have been essentially wiped out by disease brought by the Spanish, there are still several words from the language remaining in English. Two of my favorites, Barbecue and Hammock. *Assorted sources (The Merriem Webster gives the first use of Hurricane in 1555, the same year as another Taino word, Yuca, was first used in English.)

Engraving: The Great Storm of 1815 strikes Providence, Rhode Island. From an old painting in possession of the Rhode Island Historical Society.

1831 Faraday writes to Richard Phillips, “ I am busy just now again on Electro-Magnetism and think I have got hold of a good thing but can't say; it may be a weed instead of a fish that after all my labour I may at last pull up.” (It was a fish Michael!) * Michael Faraday, Bence Jones (ed.), The Life and Letters of Faraday (1870), Vol. 2, 3

One of Faraday's 1831 experiments demonstrating induction. The liquid battery (right) sends an electric current through the small coil (A). When it is moved in or out of the large coil (B), its magnetic field induces a momentary voltage in the coil, which is detected by the galvanometer (G).

*Wik

1846 Neptune first seen. Le Verrier's most famous achievement is his prediction of the existence of the then unknown planet Neptune, using only mathematics and astronomical observations of the known planet Uranus. Encouraged by physicist Arago, Director of the Paris Observatory, Le Verrier was intensely engaged for months in complex calculations to explain small but systematic discrepancies between Uranus's observed orbit and the one predicted from the laws of gravity of Newton. At the same time, but unknown to Le Verrier, similar calculations were made by John Couch Adams in England. Le Verrier announced his final predicted position for Uranus's unseen perturbing planet publicly to the French Academy on 31 August 1846, two days before Adams's final solution, which turned out to be 12° off the mark, was privately mailed to the Royal Greenwich Observatory. Le Verrier transmitted his own prediction by 18 September letter to Johann Galle of the Berlin Observatory. The letter arrived five days later, and the planet was found with the Berlin Fraunhofer refractor that same evening, 23 September 1846, by Galle and Heinrich d'Arrest within 1° of the predicted location near the boundary between Capricorn and Aquarius. Le Verrier will be known by the phrase attributed to Arago: "the man who discovered a planet with the point of his pen." [Le Verrier also noted that the perihelion of Mercury was advancing more rapidly than Newtonian physics could account for, but he proposed in 1845 that this was due to a planet between Mercury and the sun which he called Vulcan…..oops] *Wik (It is a strange twist of fate that he died on the date on which his most famous prediction was verified, See below under deaths)(Another coincidence is that the Director of the Berlin Observatory where Galle observed the new planet, was Johann Encke, whose birth was on this date. One story says he wasn't interested in the proposed planet, but yielded to Galle's request to seek it out because Encke was hurrying home for a birthday celebration.)

Within 17 days of the discovery of Neptune, William Lassell of Liverpoole would discover the planet's largest moon, to be named Triton, on October 10.

Berlin Fraunhofer refractor

1884 Patent ﬁled for Hollerith tabulating machine. It was used in the 1890 census and became the model for computer cards. *VFR

The tabulating machine was an electromechanical machine designed to assist in summarizing information stored on punched cards. Invented by Herman Hollerith.

*Wik *CHM

1983 The Los Angeles Times reported that David Slowinski of Cray research has found the 29th Mersenne prime, 2^132,049-1. It turned out that this was actually the 30th, as the 29th would turn out to be 2^110,503 -1 found by Walter Colquitt ; Luke Welsh almost five years later on Jan 28, 1988 *VFR & Wik

BIRTHS
1623 Stefano degli Angeli (Venice, September 23, 1623 – Padova, October 11, 1697) was an Italian mathematician, philosopher, and Jesuate.

He was member of the Catholic Order of the Jesuats (Jesuati). In 1668 the order was suppressed by Pope Clement IX. Angeli was a student of Bonaventura Cavalieri. From 1662 until his death he taught at the University of Padua.

From 1654 to 1667 he devoted himself to the study of geometry, continuing the research of Cavalieri and Evangelista Torricelli based on the method of Indivisibles. He then moved on to mechanics, where he often found himself in conflict with Giovanni Alfonso Borelli and Giovanni Riccioli.

Showing an early interest in mathematics and the concept of infinitesimals, Angeli studied and wrote on the behavior of various curves and physical applications of mathematics. In his Accessionis ad Stereometriam et Mecanicam (1662), he examined various solids and determined their centers of gravity.

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1768 William Wallace (23 September 1768, Dysart in Fife – 28 April 1843, Edinburgh) was a Scottish mathematician and astronomer who invented the eidograph. (A form of pantograph for reproducing images on a different scale) He mainly worked in the field of geometry and in 1799 became the first to publish the concept of the Simson line, which erroneously was attributed to Robert Simson by Poncelet. In 1807 he proved a result about polygons with an equal area, that later became known as the Bolyai–Gerwien theorem. His most important contribution to British mathematics however was, that he was one of the first mathematicians introducing and promoting the advancement of the continental European version of calculus in Britain.

Wallace's grave in Greyfriars Kirkyard, Edinburgh, 2012

He was assisted in his studies by John Robison (1739–1805) and John Playfair, to whom his abilities had become known. After various changes of situation, dictated mainly by a desire to gain time for study, he became assistant teacher of mathematics in the academy of Perth in 1794, and this post he exchanged in 1803 for a mathematical mastership in the Royal Military College at Great Marlow (afterwards at Sandhurst with a recommendation by Playfair). In 1819 he was chosen to succeed John Leslie (or John Playfair?) in the chair of mathematics at Edinburgh.
He developed a reputation for being an excellent teacher. Among his students was Mary Somerville. In 1838 he retired from the university due to ill health. He died in Edinburgh and is buried in Greyfriars Churchyard. *Wik

"There was an especially active period of invention in Scotland in the 1820s when heated controversy surrounded instruments such as Andrew Smith’s apograph and his “new” pantograph, as well as John Dunn’s pantograph. The most successful and long-lived of these new designs was the eidograph devised by the Edinburgh professor of mathematics William Wallace in 1821. Like the pantograph, the eidograph incorporated tracing, drawing, and fixed points, all three of which remained in a single line during operation. However Wallace’s arrangement of these components was novel. The fixed weight was placed centrally and supported a graduated bar at each end of which was a pivoted, adjustable rod, one bearing the tracer and the other the drawing point. A fine chain (later a steel band) was used to link the two rods and ensure that they moved in parallel. Wallace was able to dispense with the pantograph’s castors because his instrument was balanced around the central weight. At the time Wallace was working on the eidograph, Edinburgh was a center for publishing and engraving, and among its characteristic products were multivolume encyclopedias. These were expected to be heavily illustrated with engraved plates whose images would usually be copied from existing publications. Al-though it was never developed commercially, Wallace devised a special form of eidograph to produce reversed images that were engraved directly onto copper plates for printing. The simpler form of eidograph was manufactured by the London maker Robert Bate and then, in a reengineered version, by Alexander Adie. It was further improved by W.F. Stanley in the second half of the nineteenth century and, in parallel with the pantograph, continued to figure in instrument makers’ catalogs into the twentieth century."

(Ref: Bud J. & Warner D.J. (Ed). Instruments of Science – An Historical Encyclopedia. The Science Museum, London and The National Museum of American History, 1998.)

The Eidograph (sometimes ideograph) is from the same Greek root as Idol. The reproduced image is called an eidolan.

1785 Georg Scheutz (1785-1873), who with his son built a commercially available calculator based on Charles Babbage's Difference Engine, is born in Stockholm. After reading about the Difference Engine in 1833, Scheutz and son Edvard worked on a version that could process 15-digit numbers and calculate using fourth-order differences. The result won the gold medal at the Paris Exhibition in 1855 and was used by the Dudley Observatory in New York to calculate a few tables. A second copy was used by the British Registrar General to calculate tables for the developing life insurance industry. *CHM

1791 Johann Franz Encke (23 Sep 1791; 26 Aug 1865) German astronomer who in 1819 established the period of the comet now known by as Encke's Comet. At at 3.3 years it has the shortest period of any known. *TIS It was first recorded by Pierre Méchain in 1786, but it was not recognized as a periodic comet until 1819 when its orbit was computed by Encke. Comet Encke is believed to be the originator of several related meteor showers known as the Taurids (which are encountered as the Northern and Southern Taurids across November, and the Beta Taurids in late June and early July). Near-Earth object 2004 TG10 may be a fragment of Encke. Some also think it may have already had a part of it break off and hit the earth. "In 1908 Comet Encke was making a close pass near the Earth. It is believed that a 100 meter (m) diameter chunk of ice from Encke broke off and plowed into the atmosphere over the Stony Tunguska River in Siberia. The result was an air-burst explosion liberating the equivalent of 600 Hiroshima-size nuclear bombs, so much energy that sensitive instruments around the world recorded the resulting shock waves. Trees in the Siberian forests were leveled for dozens of miles around, and horses 400 miles away were knocked from their feet. There was no known loss of human life, but this is only because the impact site was so isolated. If the same ice chunk had, by chance, struck over a major population center, Tokyo, or New York, or Bombay, mega-deaths would have resulted. " *greatdreams.com

1819 Armand-Hippolyte-Louis Fizeau (23 Sep 1819; 18 Sep 1896) French physicist who was the first to measure the speed of light successfully without using astronomical calculations (1849). Fizeau sent a narrow beam of light between gear teeth on the edge of a rotating wheel. The beam then traveled to a mirror 8 km/5 mi away and returned to the wheel where, if the spin were fast enough, a tooth would block the light. Knowing this time from the rotational speed of the wheel, and the mirror's distance, Fizeau directly measured the speed of light. He also found that light travels faster in air than in water, which confirmed the wave theory of light, and that the motion of a star affects the position of the lines in its spectrum. With Jean Foucault, he proved the wave nature of the Sun's heat rays by showing their interference (1847).*TIS

1851 Ellen Amanda Hayes (September 23, 1851 – October 27, 1930) was an American mathematician and astronomer. Born in Granville, Ohio (pop 1,127 in the 1880 census) she graduated from Oberlin College in 1878 and began teaching at Adrian College. From 1879 to her 1916 retirement, she taught at Wellesley College, where she became head of the mathematics department in 1888 and head of the new department in applied mathematics in 1897.Hayes was also active in astronomy, determining the orbit of newly discovered 267 Tirza while studying at the Leander McCormick Observatory at the University of Virginia.
She wrote a number of mathematics textbooks. She also wrote Wild Turkeys and Tallow Candles (1920), an account of life in Granville, and The Sycamore Trail (1929), a historical novel.
Hayes was a controversial figure not just for being a rare female mathematics professor in 19th century America, but for her embrace of radical causes like questioning the Bible and gender clothing conventions, suffrage, temperance, socialism, the 1912 Lawrence Textile Strike, and Sacco and Vanzetti. She was the Socialist Party candidate for Massachusetts Secretary of State in 1912, the first woman in state history to run for statewide office. She did not win the race, but did receive more votes than any Socialist candidate on the ballot, including 2500 more than their gubernatorial candidate.
Hayes was concerned about under-representation of women in mathematics and science and argued that this was due to social pressure and the emphasis on female appearance, the lack of employment opportunities in those fields for women, and schools which allowed female students to opt out of math and science courses.
Her will left her brain to the Wilder Brain Collection at Cornell University. Her ashes were buried in Granville, Ohio. *Wik

1869 Typhoid Mary Mallon (23 Sep 1869; 11 Nov 1938) famous typhoid carrier in the New York City area in the early 20th century. Fifty-one original cases of typhoid and three deaths were directly attributed to her (countless more were indirectly attributed), although she herself was immune to the typhoid bacillus (Salmonella typhi). The outbreak of Typhus in Oyster Bay, Long Island, in 1904 puzzled the scientists of the time because they thought they had wiped out the deadly disease. Mallon's case showed that a person could be a carrier without showing any outward signs of being sick, and it led to most of the Health Code laws on the books today. She died not from typhoid but from the effects of a paralytic stroke dating back to 25 Dec 1932.*TIS

1921 Albert Messiah (23 September 1921, Nice -) is a French physicist.
He spent the Second World War in the French Resistance: he embarked June 22, 1940 in Saint-Jean-de-Luz to England and participated in the Battle of Dakar with Charles de Gaulle in September 1940. He joined the Free French Forces in Chad, and the 2nd Armored Division in September 1944, and participated in the assault of Hitler's Eagle's nest at Berchtesgaden in 1945.
After the war, he went to Princeton to attend the seminar of Niels Bohr on quantum mechanics. He returned to France and introduced the first general courses of quantum mechanics in France, at the University of Orsay. His textbook on quantum mechanics (Dunod 1959) has trained generations of French physicists.
He was the director of the Physics Division at the CEA and professor at the Pierre and Marie Curie University. *Wik

1968 Wendelin Werner (September 23, 1968 - ) is a German-born French mathematician working in the area of self-avoiding random walks, Schramm-Loewner evolution, and related theories in probability theory and mathematical physics. In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain he received the Fields Medal. He is currently professor at ETH Zürich. *Wik

DEATHS
1657 Joachim Jungius was a German mathematician who was one of the first to use exponents to represent powers and who used mathematics as a model for the natural sciences. *SAU

In 1669, Jungius demonstrated that the form adopted by the chain wasn’t a parabola and one year later, Jakob Bernoulli (1654-1705) proposed a contest looking for the first mathematician who could find out the real forma of a hanging chain. The problem was solved by Johann Bernoulli (1667-1748), Christiann Huygens (1629-1695) and Gottfried W. Leibnitz (1646-1717) each independently.

1877 Urbain-Jean-Joseph Le Verrier (11 Mar 1811; 23 Sep 1877 at age 66) French astronomer who predicted by mathematical means the existence of the planet Neptune. He switched from his first subject of chemistry to to teach astronomy at the Ecole Polytechnique in 1837 and worked at the Paris Observatory for most of his life. His main activity was in celestial mechanics. Independently of Adams, Le Verrier calculated the position of Neptune from irregularities in Uranus's orbit. As one of his colleagues said, " ... he discovered a star with the tip of his pen, without any instruments other than the strength of his calculations alone. In 1856, the German astronomer Johan G. Galle discovered Neptune after only an hour of searching, within one degree of the position that had been computed by Le Verrier, who had asked him to look for it there. In this way Le Verrier gave the most striking confirmation of the theory of gravitation propounded by Newton. Le Verrier also initiated the meteorological service for France, especially the weather warnings for seaports. Incorrectly, he predicted a planet, Vulcan, or asteroid belt, within the orbit of Mercury to account for an observed discrepancy (1855) in the motion in the perihelion of Mercury. *TIS

1822 Joseph-Louis-François Bertrand (11 Mar 1822; 5 Apr 1900 at age 78) was a French mathematician and educator and educator remembered for his elegant applications of differential equations to analytical mechanics, particularly in thermodynamics, and for his work on statistical probability and the theory of curves and surfaces. In 1845 Bertrand conjectured that there is at least one prime between n and (2n-2) for every n>3, as proved five years later by Chebyshev. In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French. He wrote a number of notes on the reduction of data from observations. *TIS At age 11 he started to attend classes at the Ecole Polytechnique, where his Uncle Duhamel was a well-known professor of mathematics. At 17 he received his doctor of science degree. *VFR

1897 “Bourbaki is a pen name of a group of younger French mathematicians who set out to publish an encyclopedic work covering most of modern mathematics.” So wrote Samuel Eilenberg in Mathematical Reviews, 3(1942), 55–56. He was the ﬁrst to reveal in print that Bourbaki was a pseudonym—but the name was appropiated from a real general, Charles Denis Sauter Bourbaki, who died on this date at the age of 81. See Joong Fang, Bourbaki, Paideia Press, 1970, pp. 24, *VFR

1919 Heinrich Bruns was interested in astronomy, mathematics and geodesy and worked on the three body problem.*SAU

1971 James Waddell Alexander (19 Sept 1888, 23 Sept 1971) In a collaboration with Veblen, he showed that the topology of manifolds could be extended to polyhedra. Before 1920 he had shown that the homology of a simplicial complex is a topological invariant. Alexander's work around this time went a long way to put the intuitive ideas of Poincaré on a more rigorous foundation. Also before 1920 Alexander had made fundamental contributions to the theory of algebraic surfaces and to the study of Cremona transformations.
Soon after arriving in Princeton, Alexander generalised the Jordan curve theorem and continued his work, now exclusively on topology, with an important paper on the Jordan-Brouwer separation theorem. This latter paper contains the Alexander Duality Theorem and Alexander's lemma on the n-sphere. In 1924 he introduced the now famous Alexander horned sphere.
In 1928 he discovered the Alexander polynomial which is much used in knot theory. In the same year the American Mathematical Society awarded Alexander the Bôcher Prize for his memoir, Combinatorial analysis situs published in the Transactions of the American Mathematical Society two years earlier. Knot theory and the combinatorial theory of complexes were the main topics on which he worked over the following few years.
The theory which is now called the Alexander-Spanier cohomology theory, was introduced in 1935 by Alexander but was generalised by Spanier in 1948 to the form seen today. Also around 1935 Alexander discovered cohomology theory, at essentially the same time as Kolmogorov, and the theory was announced in the 1936 Moscow Conference. *SAU

2004 Bryce Seligman DeWitt (January 8, 1923 – September 23, 2004) was a theoretical physicist who studied gravity and field theories.
He approached the quantization of general relativity, in particular, developed canonical quantum gravity and manifestly covariant methods that use the heat kernel. B. DeWitt formulated the Wheeler–DeWitt equation for the wavefunction of the Universe with John Archibald Wheeler and advanced the formulation of the Hugh Everett's many-worlds interpretation of quantum mechanics. With his student Larry Smarr he originated the field of numerical relativity.
He received his bachelor's, master's and doctoral degrees from Harvard University. His Ph.D. (1950) supervisor was Julian S. Schwinger. Afterwards he worked at the Institute for Advanced Study, the University of North Carolina at Chapel Hill and the University of Texas at Austin. He was awarded the Dirac Prize in 1987, the American Physical Society's Einstein Prize in 2005, and was a member of the National Academy of Sciences and the American Academy of Arts and Letters.
He was born Carl Bryce Seligman but he and his three brothers added "DeWitt" from their mother's side of the family, at the urging of their father, in 1950. This is similar to Spanish naming customs, where a person bears two surnames, one being from their father and the other from their mother. Twenty years later this change of name so angered Felix Bloch that he blocked DeWitt's appointment to Stanford University and DeWitt instead moved to Austin, Texas. He served in World War II as a naval aviator. He was married to mathematical physicist Cécile DeWitt-Morette. He died September 23, 2004 from pancreatic cancer at the age of 81. He is buried in France, and was survived by his wife and four daughters. *Wik

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