## Sunday, 30 September 2018

### The Harmony of the Harmonic Mean, and Related Problems

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 by which 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 sequence in 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 yeild 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.

### On This Day in Math -September 30

Big whirls have little whirls,
That feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.
~Lewis Richardson

The 273rd day of the year; 273oK(to the nearest integer)is the freezing point of water, or 0oC

OOOOH wait, 273 = 13*7*3, and 1373 is also prime.. and There are only two sphenic numbers consisting of concatenation of distinct prime numbers, this is the smaller of the two.(sphenic or wedge numbers are products of three distinct primes) *Prime curios

EVENTS

1717 Colin Maclaurin (1698–1746), age 19, was appointed to the Mathematics Chair at Marischal College, Aberdeen, Scotland. This is the youngest at which anyone has been elected chair (full professor) at a university. (Guinness) In 1725 he was made Professor at Edinburgh University on the recommendation of Newton. *VFR

1810 The University of Berlin opened. *VFR It is now called The Humboldt University of Berlin and is Berlin's oldest university. It was founded as the University of Berlin (Universität zu Berlin) by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt, whose university model has strongly influenced other European and Western universities.*Wik

1890 In his desk notes Sir George Biddell Airy writes about his disappointment on finding an error in his calculations of the moon’s motion. “ I had made considerable advance ... in calculations on my favourite numerical lunar theory, when I discovered that, under the heavy pressure of unusual matters (two transits of Venus and some eclipses) I had committed a grievous error in the first stage of giving numerical value to my theory. My spirit in the work was broken, and I have never heartily proceeded with it since.” *George Biddell Airy and Wilfrid Airy (ed.), Autobiography of Sir George Biddell Airy (1896), 350.

1893 Felix Klein visits Worlds fair in Chicago, then visits many colleges. On this day the New York Mathematical society had a special meeting to honor him. *VFR

1921 William H Schott patented the "hit-and-miss synchronizer for his clocks. The Shortt-Synchronome free pendulum clock was a complex precision electromechanical pendulum clock invented in 1921 by British railway engineer William Hamilton Shortt in collaboration with horologist Frank Hope-Jones, and manufactured by the Synchronome Co., Ltd. of London, UK. They were the most accurate pendulum clocks ever commercially produced, and became the highest standard for timekeeping between the 1920s and the 1940s, after which mechanical clocks were superseded by quartz time standards. They were used worldwide in astronomical observatories, naval observatories, in scientific research, and as a primary standard for national time dissemination services. The Shortt was the first clock to be a more accurate timekeeper than the Earth itself; it was used in 1926 to detect tiny seasonal changes (nutation) in the Earth's rotation rate. *Wik

1939 an early manned rocket-powered flight was made by German auto maker Fritz von Opel. His Sander RAK 1 was a glider powered by sixteen 50 pound thrust rockets. In it, Opel made a successful flight of 75 seconds, covering almost 2 miles near Frankfurt-am-Main, Germany. This was his final foray as a rocket pioneer, having begun by making several test runs (some in secret) of rocket propelled vehicles. He reached a speed of 238 km/h (148 mph) on the Avus track in Berlin on 23 May, 1928, with the RAK 2. Subsequently, riding the RAK 3 on rails, he pushed the world speed record up to 254 km/h (158 mph). The first glider pilot to fly under rocket power, was another German, Friedrich Staner, who flew about 3/4-mile on 11 Jun 1928.*TIS

2010 The ignoble prizes, presented on this date, included an engineering for collecting Whale Snot, and a MANAGEMENT PRIZE: for demonstrating mathematically that organizations would become more efficient if they promoted people at random. See all the 2010 winners here.

BIRTHS

1550 Michael Maestlin (30 September 1550, Göppingen – 20 October 1631, Tübingen) was a German astronomer who was Kepler's teacher and who publicised the Copernican system. Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound. He found, however, a sun centerd orbit for the comet of 1577 which he claimed supported Copernicus's heliocentric system. He did show that the comet was further away than the moon, which contradicted the accepted teachings of Aristotle. Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system. However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mästlin with introducing him to Copernican ideas while he was a student at Tübingen (1589-94).*SAU The first known calculation of the reciprocal of the golden ratio as a decimal of "about 0.6180340" was written in 1597 by Maestlin in a letter to Kepler. He is also remembered for :
Catalogued the Pleiades cluster on 24 December 1579. Eleven stars in the cluster were recorded by Maestlin, and possibly as many as fourteen were observed.
Occultation of Mars by Venus on 13 October 1590, seen by Maestlin at Heidelberg. *Wik

1715 Étienne Bonnot de Condillac (30 Sep 1715; 3 Aug 1780) French philosopher, psychologist, logician, economist, and the leading advocate in France of the ideas of John Locke (1632-1704). In his works La Logique (1780) and La Langue des calculs (1798), Condillac emphasized the importance of language in logical reasoning, stressing the need for a scientifically designed language and for mathematical calculation as its basis. He combined elements of Locke's theory of knowledge with the scientific methodology of Newton; all knowledge springs from the senses and association of ideas. Condillac devoted careful attention to questions surrounding the origins and nature of language, and enhanced contemporary awareness of the importance of the use of language as a scientific instrument.*TIS

1774 Carl Wilhelm Scheele sent a letter to Antoine Lavoisier announcing the discovery of oxygen (O). Unfortunately the letter from the Swedish chemist was never acknowledged and Joseph Priestly published the discovery first. Scheele was trounced in the announcement of other discoveries as well, he identified molybdenum, tungsten, barium, hydrogen, and chlorine before Humphry Davy, among others. Scheele discovered organic acids tartaric, oxalic, uric, lactic, and citric, as well as hydrofluoric, hydrocyanic, and arsenic acids. (Not bad for a chemist you never heard of.) For this reason, Isaac Asimov nicknamed him “hard-luck Scheele” *rsc.org , *Wik

1775 Robert Adrain (30 September 1775 – 10 August 1843) . Although born in Ireland he was one of the ﬁrst creative mathematicians to work in America. *VFR Adrain was appointed as a master at Princeton Academy and remained there until 1800 when the family moved to York in Pennsylvania. In York Adrain became Principal of York County Academy. When the first mathematics journal, the Mathematical Correspondent, began publishing in 1804 under the editorship of George Baron, Adrain became one of its main contributors. One year later, in 1805, he moved again this time to Reading, also in Pennsylvania, where he was appointed Principal of the Academy.
After arriving in Reading, Adrain continued to publish in the Mathematical Correspondent and, in 1807, he became editor of the journal. One has to understand that publishing a mathematics journal in the United States at this time was not an easy task since there were only two mathematicians capable of work of international standing in the whole country, namely Adrain and Nathaniel Bowditch. Despite these problems, Adrain decided to try publishing his own mathematics journal after he had edited only one volume of the Mathematical Correspondent and, in 1808, he began editing his journal the Analyst or Mathematical Museum.
With so few creative mathematicians in the United States the journal had little chance of success and indeed it ceased publication after only one year. After the journal ceased publication, Adrain was appointed professor of mathematics at Queen's College (now Rutgers University) New Brunswick where he worked from 1809 to 1813. Despite Queen's College trying its best to keep him there, Adrain moved to Columbia College in New York in 1813. He tried to restart his mathematical journal the Analyst in 1814 but only one part appeared. In 1825, while he was still on the staff at Columbia College, Adrain made another attempt at publishing a mathematical journal. Realising that the Analyst had been too high powered for the mathematicians of the United States, he published the Mathematical Diary in 1825. This was a lower level publication which continued under the editorship of James Ryan when Adrain left Columbia College in 1826. *SAU

1870 Jean-Baptiste Perrin (30 Sep 1870; 17 Apr 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids, verified Albert Einstein's explanation of this phenomenon and thereby confirmed the atomic nature of matter. Using a gamboge emulsion, Perrin was able to determine by a new method, one of the most important physical constants, Avogadro's number (the number of molecules of a substance in so many grams as indicated by the molecular weight, for example, the number of molecules in two grams of hydrogen). The value obtained corresponded, within the limits of error, to that given by the kinetic theory of gases. For this achievement he was honoured with the Nobel Prize for Physics in 1926.*TIS

1882 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. The Geiger-Müller counter (developed with Walther Müller) had improved durability, performance and sensitivity to detect not only alpha particles but also beta particles (electrons) and ionizing electromagnetic photons. Geiger returned to Germany in 1912 and continued to investigate cosmic rays, artificial radioactivity, and nuclear fission.*TIS

1883 Ernst David Hellinger (1883 - 1950) introduced a new type of integral: the Hellinger integral . Jointly with Hilbert he produced an important theory of forms. *SAU

1894 Dirk Jan Struik (30 Sept 1894 , 21 Oct 2000) Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States.
In 1924, funded by a Rockefeller fellowship, Struik traveled to Rome to collaborate with the Italian mathematician Tullio Levi-Civita. It was in Rome that Struik first developed a keen interest in the history of mathematics. In 1925, thanks to an extension of his fellowship, Struik went to Göttingen to work with Richard Courant compiling Felix Klein's lectures on the history of 19th-century mathematics. He also started researching Renaissance mathematics at this time.
Struik was a steadfast Marxist. Having joined the Communist Party of the Netherlands in 1919, he remained a Party member his entire life. When asked, upon the occasion of his 100th birthday, how he managed to pen peer-reviewed journal articles at such an advanced age, Struik replied blithely that he had the "3Ms" a man needs to sustain himself: Marriage (his wife, Saly Ruth Ramler, was not alive when he turned one hundred in 1994), Mathematics, and Marxism.
It is therefore not surprising that Dirk suffered persecution during the McCarthyite era. He was accused of being a Soviet spy, a charge he vehemently denied. Invoking the First and Fifth Amendments of the U.S. Constitution, he refused to answer any of the 200 questions put forward to him during the HUAC hearing. He was suspended from teaching for five years (with full salary) by MIT in the 1950s. Struik was re-instated in 1956. He retired from MIT in 1960 as Professor Emeritus of Mathematics.
Aside from purely academic work, Struik also helped found the Journal of Science and Society, a Marxian journal on the history, sociology and development of science.
In 1950 Stuik published his Lectures on Classical Differential Geometry.
Struik's other major works include such classics as A Concise History of Mathematics, Yankee Science in the Making, The Birth of the Communist Manifesto, and A Source Book in Mathematics, 1200-1800, all of which are considered standard textbooks or references.
Struik died October 21, 2000, 21 days after celebrating his 106th birthday. *Wik

1905 Sir Nevill F. Mott (30 Sep 1905; 8 Aug 1996) English physicist who shared (with P.W. Anderson and J.H. Van Vleck of the U.S.) the 1977 Nobel Prize for Physics for his independent researches on the magnetic and electrical properties of amorphous semiconductors. Whereas the electric properties of crystals are described by the Band Theory - which compares the conductivity of metals, semiconductors, and insulators - a famous exception is provided by nickel oxide. According to band theory, nickel oxide ought to be a metallic conductor but in reality is an insulator. Mott refined the theory to include electron-electron interaction and explained so-called Mott transitions, by which some metals become insulators as the electron density decreases by separating the atoms from each other in some convenient way.*TIS

1913 Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish and American mathematician born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg–Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.
Eilenberg was a member of Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.
Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules. Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974. *Wik

1916 Richard Kenneth Guy (born September 30, 1916, Nuneaton, Warwickshire - ) is a British mathematician, and Professor Emeritus in the Department of Mathematics at the University of Calgary.
He is best known for co-authorship (with John Conway and Elwyn Berlekamp) of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory, but he has also published over 100 papers and books covering combinatorial game theory, number theory and graph theory.
He is said to have developed the partially tongue-in-cheek "Strong Law of Small Numbers," which says there are not enough small integers available for the many tasks assigned to them — thus explaining many coincidences and patterns found among numerous cultures.
Additionally, around 1959, Guy discovered a unistable polyhedron having only 19 faces; no such construct with fewer faces has yet been found. Guy also discovered the glider in Conway's Game of Life.
Guy is also a notable figure in the field of chess endgame studies. He composed around 200 studies, and was co-inventor of the Guy-Blandford-Roycroft code for classifying studies. He also served as the endgame study editor for the British Chess Magazine from 1948 to 1951.
Guy wrote four papers with Paul Erdős, giving him an Erdős number of 1. He also solved one of Erdős problems.
His son, Michael Guy, is also a computer scientist and mathematician. *Wik

1918 Leslie Fox (30 September 1918 – 1 August 1992) was a British mathematician noted for his contribution to numerical analysis. *Wik

DEATHS

1953 Lewis Fry Richardson, FRS (11 October 1881 - 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them. He is also noted for his pioneering work on fractals and a method for solving a system of linear equations known as modified Richardson iteration.*Wik

1985 Dr. Charles Francis Richter (26 Apr 1900, 30 Sep 1985) was an American seismologist and inventor of the Richter Scale that measures earthquake intensity which he developed with his colleague, Beno Gutenberg, in the early 1930's. The scale assigns numerical ratings to the energy released by earthquakes. Richter used a seismograph (an instrument generally consisting of a constantly unwinding roll of paper, anchored to a fixed place, and a pendulum or magnet suspended with a marking device above the roll) to record actual earth motion during an earthquake. The scale takes into account the instrument's distance from the epicenter. Gutenberg suggested that the scale be logarithmic so, for example, a quake of magnitude 7 would be ten times stronger than a 6.*TIS

2014 Martin Lewis Perl (June 24, 1927 – September 30, 2014) was an American physicist who won the Nobel Prize in Physics in 1995 for his discovery of the tau lepton.
He received his Ph.D. from Columbia University in 1955, where his thesis advisor was I.I. Rabi. Perl's thesis described measurements of the nuclear quadrupole moment of sodium, using the atomic beam resonance method that Rabi had won the Nobel Prize in Phyics for in 1944.
Following his Ph.D., Perl spent 8 years at the University of Michigan, where he worked on the physics of strong interactions, using bubble chambers and spark chambers to study the scattering of pions and later neutrons on protons.[1] While at Michigan, Perl and Lawrence W. Jones served as co-advisors to Samuel C. C. Ting, who earned the Nobel Prize in Physics in 1976.
Seeking a simpler interaction mechanism to study, Perl started to consider electron and muon interactions. He had the opportunity to start planning experimental work in this area when he moved in 1963 to the Stanford Linear Accelerator Center (SLAC), then being built in California. He was particularly interested in understanding the muon: why it should interact almost exactly like the electron but be 206.8 times heavier, and why it should decay through the route that it does. Perl chose to look for answers to these questions in experiments on high-energy charged leptons. In addition, he considered the possibility of finding a third generation of lepton through electron-positron collisions. He died after a heart attack at Stanford University Hospital on September 30, 2014 at the age of 87. *Wik

2017 Vladimir Voevodsky (Jun 4, 1966 - Sep 30, 2017) formerly a gifted but restless student who flunked out of college out of boredom before emerging as one of the most brilliant and revolutionary mathematicians of his generation, died on Sept. 30 at his home in Princeton, N.J. He was 51.

Dr. Voevodsky was renowned for founding entirely new fields of mathematics and creating groundbreaking new tools for computers to confirm the accuracy of proofs. In 2002, he was awarded the Fields Medal, which recognizes brilliance and promise in mathematicians under 40.
He was “one of the giants of our time,” Thomas Hales, a mathematician at the University of Pittsburgh, said in an interview. Dr. Voevodsky, he said, transformed every field he touched. In his work using computers, for example, he upended mathematical thinking to such a degree that he changed the meaning of the equals sign.

He added: “His ideas gave a new way for all mathematicians to do what they do, a new foundation. The foundations of math are like a constitutional document that spells out the governing rules all mathematicians agree to play by. He has given us a new constitution.”

Vladimir Voevodsky was born in Moscow. His father, Alexander, directed a laboratory in experimental physics at the Russian Academy of Sciences; his mother, Tatyana Voevodskaya, was a chemistry professor at Moscow University. *obit NYTimes

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, 29 September 2018

### On This Day in Math - September 29

Young man, if I could remember the names of these particles, I would have been a botanist.
~Enrico Fermi

The 272nd day of the year; 272 = 24·17, and is the sum of four consecutive primes (61 + 67 + 71 + 73).

272 is also a Pronic or Heteromecic number, the product of two consecutive factors, 16x17 (which makes it twice a triangular #).

And 272 is a palindrome, and the sum of its digits, 11, is also a palindrome. (can you find the next?)

EVENTS

1609  Almost exactly a year after the first application for a patent of the telescope, Giambaptista della Porta, the Neapolitan polymath, whose Magia Naturalis of 1589, well known all over Europe, because of a tantalizing hint at what might be accomplished by a combination of a convex and concave lens: ‘With a concave you shall see small things afar off, very clearly; witha convex, things neerer to be greater, but more obscurely: if you know how to fit them both together, you shall see both things afar off, and things neer hand, both greater and clearly.’sends a letter to the founder of the Accademia dei Lincei, Prince Federico Cesi in Rome, with a sketch of an instrument that had just reached him, and he wrote:" It is a small tube of soldered silver, one palm in length, and three finger breadths in diameter, which has a convex glass in the end. There is another tube of the same material four finger breadths long, which enters into the first one, and in the end. It has a concave [glass], which is secured like the first one. If observed with that first tube, faraway things are seen as if they were near, but because the vision does not occur along the perpendicular, they appear obscure and indistinct. When the other concave tube, which produces the opposite effect, is inserted, things will be seen clear and erect and it goes in an out, as in a trombone, so that it adjusts to the eyesight of [particular] observers, which all differ. *Albert Van Helden, Galileo and the telescope; Origins of the Telescope, Royal Netherlands Academy of Arts andSciences, 2010
(I assume that we can safely date the invention of the trombone prior to 1609 also)

1801 Gauss’s Disquisitiones Arithmeticae published. It is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own.
The book is divided into seven sections, which are :
Section I. Congruent Numbers in General
Section II. Congruences of the First Degree
Section III. Residues of Powers
Section IV. Congruences of the Second Degree
Section V. Forms and Indeterminate Equations of the Second Degree
Section VI. Various Applications of the Preceding Discussions
Section VII. Equations Defining Sections of a Circle.
Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way. He was also the first mathematician to realize the importance of the property of unique factorization (sometimes called the fundamental theorem of arithmetic), which he states and proves explicitly.
From Section IV onwards, much of the work is original. Section IV itself develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary quadratic forms; and Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible i.e. can be constructed with a compass and unmarked straight edge alone. *Wik

1954 The European Organization for Nuclear Research (CERN) was officially established.The organisation runs the world’s largest particle physics laboratory in Geneva, Switzerland. In September 2011 CERN scientists reported that some particles appeared to be travelling faster than light, although it’s now thought that the experiment was flawed. *rsc.org

In 1988, the space shuttle Discovery blasted off from Cape Canaveral, Fla., marking America's return to manned space flight following the Challenger disaster. *TIS

1994 HotJava ---- Programmers first demonstrated the HotJava prototype to executives at Sun Microsystems Inc. A browser making use of Java technology, HotJava attempted to transfer Sun's new programming platform for use on the World Wide Web. Java is based on the concept of being truly universal, allowing an application written in the language to be used on a computer with any type of operating system or on the web, televisions or telephones.*CHM

BIRTHS

1561  Adriaan van Roomen (29 Sept 1561 , 4 May 1615) is often known by his Latin name Adrianus Romanus. After studying at the Jesuit College in Cologne, Roomen studied medicine at Louvain. He then spent some time in Italy, particularly with Clavius in Rome in 1585.
Roomen was professor of mathematics and medicine at Louvain from 1586 to 1592, he then went to Würzburg where again he was professor of medicine. He was also "Mathematician to the Chapter" in Würzburg. From 1603 to 1610 he lived frequently in both Louvain and Würzburg. He was ordained a priest in 1604. After 1610 he tutored mathematics in Poland.
One of Roomen's most impressive results was finding π to 16 decimal places. He did this in 1593 using 230 sided polygons. Roomen's interest in π was almost certainly as a result of his friendship with Ludolph van Ceulen.
Roomen proposed a problem which involved solving an equation of degree 45. The problem was solved by Viète who realised that there was an underlying trigonometric relation. After this a friendship grew up between the two men. Viète proposed the problem of drawing a circle to touch 3 given circles to Roomen (the Apollonian Problem) and Roomen solved it using hyperbolas, publishing the result in 1596.
Roomen worked on trigonometry and the calculation of chords in a circle. In 1596 Rheticus's trigonometric tables Opus palatinum de triangulis were published, many years after Rheticus died. Roomen was critical of the accuracy of the tables and wrote to Clavius at the Collegio Romano in Rome pointing out that, to calculate tangent and secant tables correctly to ten decimal places, it was necessary to work to 20 decimal places for small values of sine, see [2]. In 1600 Roomen visited Prague where he met Kepler and told him of his worries about the methods employed in Rheticus's trigonometric tables. *SAU

1803   Jacques Charles-François Sturm (29 Sep 1803; 18 Dec 1855) French mathematician whose work resulted in Sturm's theorem, an important contribution to the theory of equations. .While a tutor of the de Broglie family in Paris (1823-24), Sturm met many of the leading French scientists and mathematicians. In 1826, with Swiss engineer Daniel Colladon, he made the first accurate determination of the velocity of sound in water. A year later wrote a prizewinning essay on compressible fluids. Since the time of René Descartes, a problem had existed of finding the number of solutions of a given second-order differential equation within a given range of the variable. Sturm provided a complete solution to the problem with his theorem which first appeared in Mémoire sur la résolution des équations numériques (1829; “Treatise on Numerical Equations”). Those principles have been applied in the development of quantum mechanics, as in the solution of the Schrödinger equation and its boundary values. *TIS  Sturm is also remembered for the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.*SAU

1812  Gustav Adolph Göpel (29 Sept 1812, 7 June 1847) Göpel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms. He wrote on Steiner's synthetic geometry and an important work, Theoriae transcendentium Abelianarum primi ordinis adumbratio levis, published after his death, continued the work of Jacobi on elliptic functions. This work was published in Crelle's Journal in 1847. *SAU

1895 Harold Hotelling​, 29 September 1895 - 26 December 1973   He originally studied journalism at the University of Washington, earning a degree in it in 1919, but eventually turned to mathematics, gaining a PhD in Mathematics from Princeton in 1924 for a dissertation dealing with topology. However, he became interested in statistics that used higher-level math, leading him to go to England in 1929 to study with Fisher.
Although Hotelling first went to Stanford University in 1931, he not many years afterwards became a Professor of Economics at Columbia University, where he helped create Columbia's Stat Dept. In 1946, Hotelling was recruited by Gertrude Cox​ to form a new Stat Dept at the University of North Carolina at Chapel Hill. He became Professor and Chairman of the Dept of Mathematical Statistics, Professor of Economics, and Associate Director of the Institute of Statistics at UNC-CH. (When Hotelling and his wife first arrived in Chapel Hill they instituted the "Hotelling Tea", where they opened their home to students and faculty for tea time once a month.)
Dr. Hotelling's major contributions to statistical theory were in multivariate analysis, with probably his most important paper his famous 1931 paper "The Generalization of Student's Ratio", now known as Hotelling's T^2, which involves a generalization of
Student's t-test for multivariate data. In 1953, Hotelling published a 30-plus-page paper on the distribution of the correlation coefficient, following up on the work of Florence Nightingale David in 1938. *David Bee

1901 Enrico Fermi (29 Sep 1901; 28 Nov 1954) Italian-American physicist who was awarded the Nobel Prize for physics in 1938 as one of the chief architects of the nuclear age. He was the last of the double-threat physicists: a genius at creating both esoteric theories and elegant experiments. In 1933, he developed the theory of beta decay, postulating that the newly-discovered neutron decaying to a proton emits an electron and a particle he called a neutrino. Developing theory to explain this decay later resulted in finding the weak interaction force. He developed the mathematical statistics required to clarify a large class of subatomic phenomena, discovered neutron-induced radioactivity, and directed the first controlled chain reaction involving nuclear fission. *TIS

1925 Paul Beattie MacCready (29 Sep 1925; 28 Aug 2007) was an American engineer who invented not only the first human-powered flying machines, but also the first solar-powered aircraft to make sustained flights. On 23 Aug 1977, the pedal-powered aircraft, the Gossamer Condor successfully flew a 1.15 mile figure-8 course to demonstrate sustained, maneuverable manpowered flight, for which he won the £50,000 (\$95,000) Kremer Prize. MacCready designed the Condor with Dr. Peter Lissamen. Its frame was made of thin aluminum tubes, covered with mylar plastic supported with stainless steel wire. In 1979, the Gossamer Albatross won the second Kremer Prize for making a flight across the English Channel.*TIS

1931   James Watson Cronin (29 Sep 1931, ) American particle physicist, who shared (with Val Logsdon Fitch) the 1980 Nobel Prize for Physics for "the discovery of violations of fundamental symmetry principles in the decay of neutral K-mesons." Their experiment proved that a reaction run in reverse does not follow the path of the original reaction, which implied that time has an effect on subatomic-particle interactions. Thus the experiment demonstrated a break in particle-antiparticle symmetry for certain reactions of subatomic particles.*TIS

1935 Hillel (Harry) Fürstenberg (September 29, 1935, ..)) is an American-Israeli mathematician, a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. He gained attention at an early stage in his career for producing an innovative topological proof of the infinitude of prime numbers. He proved unique ergodicity of horocycle flows on a compact hyperbolic Riemann surfaces in the early 1970s. In 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédi's theorem. The Fürstenberg boundary and Fürstenberg compactification of a locally symmetric space are named after him. *Wik

DEATHS

1939 Samuel Dickstein (May 12, 1851 – September 29, 1939) was a Polish mathematician of Jewish origin. He was one of the founders of the Jewish party "Zjednoczenie" (Unification), which advocated the assimilation of Polish Jews.
He was born in Warsaw and was killed there by a German bomb at the beginning of World War II. All the members of his family were killed during the Holocaust.
Dickstein wrote many mathematical books and founded the journal Wiadomości Mathematyczne (Mathematical News), now published by the Polish Mathematical Society. He was a bridge between the times of Cauchy and Poincaré and those of the Lwów School of Mathematics. He was also thanked by Alexander Macfarlane for contributing to the Bibliography of Quaternions (1904) published by the Quaternion Society.
He was also one of the personalities, who contributed to the foundation of the Warsaw Public Library in 1907.*Wik

1941 Friedrich Engel (26 Dec 1861, 29 Sept 1941)Engel was taught by Klein who recognized that he was the right man to assist Lie. At Klein's suggestion Engel went to work with Lie in Christiania (now Oslo) from 1884 until 1885. In 1885 Engel's Habilitation thesis was accepted by Leipzig and he became a lecturer there. The year after Engel returned to Leipzig from Christiania, Lie was appointed to succeed Klein and the collaboration of Lie and Engel continued.
In 1889 Engel was promoted to assistant professor and, ten years later he was promoted to associate professor. In 1904 he accepted the chair of mathematics at Greifswald when his friend Eduard Study resigned the chair. Engel's final post was the chair of mathematics at Giessen which he accepted in 1913 and he remained there for the rest of his life. In 1931 he retired from the university but continued to work in Giessen.
The collaboration between Engel and Lie led to Theorie der Transformationsgruppen a work on three volumes published between 1888 and 1893. This work was, "... prepared by S Lie with the cooperation of F Engel... "
In many ways it was Engel who put Lie's ideas into a coherent form and made them widely accessible. From 1922 to 1937 Engel published Lie's collected works in six volumes and prepared a seventh (which in fact was not published until 1960). Engel's efforts in producing Lie's collected works are described as, "... an exceptional service to mathematics in particular, and scholarship in general. Lie's peculiar nature made it necessary for his works to be elucidated by one who knew them intimately and thus Engel's 'Annotations' completed in scope with the text itself. "
Engel also edited Hermann Grassmann's complete works and really only after this was published did Grassmann get the fame which his work deserved. Engel collaborated with Stäckel in studying the history of non-euclidean geometry. He also wrote on continuous groups and partial differential equations, translated works of Lobachevsky from Russian to German, wrote on discrete groups, Pfaffian equations and other topics. *SAU

1955 L(ouis) L(eon) Thurstone (29 May 1887, 29 Sep 1955)  was an American psychologist who improved psychometrics, the measurement of mental functions, and developed statistical techniques for multiple-factor analysis of performance on psychological tests. In high school, he published a letter in Scientific American on a problem of diversion of water from Niagara Falls; and invented a method of trisecting an angle. At university, Thurstone studied engineering. He designed a patented motion picture projector, later demonstrated in the laboratory of Thomas Edison, with whom Thurstone worked briefly as an assistant. When he began teaching engineering, Thurstone became interested in the learning process and pursued a doctorate in psychology. *TIS

2003 Ovide Arino (24 April 1947 - 29 September 2003) was a mathematician working on delay differential equations. His field of application was population dynamics. He was a quite prolific writer, publishing over 150 articles in his lifetime. He also was very active in terms of student supervision, having supervised about 60 theses in total in about 20 years. Also, he organized or coorganized many scientific events. But, most of all, he was an extremely kind human being, interested in finding the good in everyone he met. *Euromedbiomath

2010 Georges Charpak (1 August 1924 – 29 September 2010) was a French physicist who was awarded the Nobel Prize in Physics in 1992 "for his invention and development of particle detectors, in particular the multiwire proportional chamber". This was the last time a single person was awarded the physics prize. *Wik

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

## Friday, 28 September 2018

### On This Day in Math - September 28

But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting.

~ Julian Lowell Coolidge

The 271st day of the year; 271 is a prime number and is the sum of eleven consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43).

271 is also the difference of two consecutive cubes, 103 - 93. Such prime numbers are called Cuban Primes, and were named by the British mathematician Allan Joseph Champneys Cunningham in 1923.

Using the English alphabet code, a = 1, b = 2, etc, there are exactly 271 positive numbers that give larger numbers when you write out their English names and add the letters *primecurios

EVENTS

490 B.C. In one of history’s great battles, the Greeks defeated the Persians at Marathon. A Greek soldier was dispatched to notify Athens of the victory, running the entire distance and providing the name and model for the modern “marathon” race. *VFR

1695 After fitting several comets data using Newton's proposal that they followed parabolic paths, Edmund Halley was "inspired" to test his own measurements of the 1682 comet against an elliptical orbit. He writes to Newton, "I am more and more confirmed that we have seen that Comet now three times since Ye Year 1531." *David A Grier, When Computer's Were Human

1791 Captain George Vancouver observed this Wednesday morning a partial solar eclipse. He went on the name the barren rocky cluster of isles, by the name of Eclipse Islands. *NSEC

1858, Donati's comet (discovered by Giovanni Donati, 1826-1873) became the first to be photographed. It was a bright comet that developed a spectacular curved dust tail with two thin gas tails, captured by an English commercial photographer, William Usherwood, using a portrait camera at a low focal ratio. At Harvard, W.C. Bond, attempted an image on a collodion plate the following night, but the comet shows only faintly and no tail can be seen. Bond was subsequently able to evaluate the image on Usherwood's plate. The earliest celestial daguerreotypes were made in 1850-51, though after the Donati comet, no further comet photography took place until 1881, when P.J.C. Janssen and J.W. Draper took the first generally recognized photographs of a comet*TIS “William Usherwood, a commercial photographer from Dorking, Surrey took the first ever photograph of a comet when he photographed Donati’s comet from Walton Common on the 27th September 1858, beating George Bond from Harvard Observatory by a night! Unfortunately, the picture taken by Usherwood has been lost.” *Exposure web site

1889 The first General Conference on Weights and Measures (CGPM) defined the length of a metre. One metre was defined as the distance between two lines on a standard bar of an alloy of platinum (Pt) with 10% iridium (Ir), measured at the melting point of ice. The original international prototype of the metre is still kept at the BIPM, Bureau International des Poids et Mesures, in Sèvres, France. *rsc,org

1917 Richard Courant wrote to Nina Runge, his future wife, that he ﬁnally got the opportunity to talk to Ferdinand Springer about “a publishing project” and that things looked promising. This meeting led to a contract and a series of books now called the "Yellow Series". *VFR

1938 Paul Erdos boards the Queen Mary bound for the USA. Alarmed by Hitler's demands to annex the Sudatenland, Erdos hurriedly left Budapest and made his way through Italy and France to London. He would pass through Ellis Island on his way to a position at Princeton's Institute for Advanced Study on October 4. * Bruce Schechter, My Brain is Open: The Mathematical Journeys of Paul Erdos

1969 Murchison meteorite , a meteorite fell over Murchison, Australia. Only 100-kg of this meteorite have been found. Classified as a carbonaceous chondrite, type II (CM2), this meteorite is suspected to be of cometary origin due to its high water content (12%). An abundance of amino acids found within this meteorite has led to intense study by researchers as to its origins. More than 92 different amino acids have been identified within the Murchison meteorite to date. Nineteen of these are found on Earth. The remaining amino acids have no apparent terrestrial source. *TIS

1980 "We are Star-Stuff." On this day in 1980 the program Cosmos: A Personal Voyage with Carl Sagan premiered. The series was first broadcast by the Public Broadcasting Service in 1980, and was the most widely watched series in the history of American public television for a decade. As of 2009, it was still the most widely watched PBS series in the world. The series is notable for its groundbreaking use of special effects, which allow Sagan to seemingly walk through environments that are actually models rather than full-sized sets. *WIK

2009: mathoverflow.net goes online. *Peter Krautzberger,

2011 President Barack Obama announced that Richard Alfred Tapia was among twelve scientists to be awarded the National Medal of Science, the top award the United States offers its researchers. Tapia is currently the Maxfield and Oshman Professor of Engineering; Associate Director of Graduate Studies, Office of Research and Graduate Studies; and Director of the Center for Excellence and Equity in Education at Rice University. He is a renowned American mathematician and champion of under-represented minorities in the sciences. *Wik

BIRTHS

551 B.C. Birthdate of the Chinese philosopher and educator Confucius. His birthday is observed as “Teacher’s Day” in memory of his great contribution to the Chinese Nation. His most famous aphorism is: “With education there is no distinction between classes or races of men.” *VFR

1605 Ismael Boulliau (28 Sept 1605 , 25 Nov 1694) was a French clergyman and amateur mathematician who proposed an inverse square law for gravitation before Newton. Boulliau was a friend of Pascal, Mersenne and Gassendi and supported Galileo and Copernicus. He claimed that if a planetary moving force existed then it should vary inversely as the square of the distance (Kepler had claimed the first power), "As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances that is, 1/d2. *SAU

1651 Johann Philipp von Wurzelbau (28 September 1651 in Nürnberg; 21 July 1725 Nürnberg )was a German astronomer.
A native of Nuremberg, Wurzelbauer was a merchant who became an astronomer. As a youth, he was keenly interested in mathematics and astronomy but had been forced to earn his living as a merchant. He married twice: his first marriage was to Maria Magdalena Petz (1656–1713), his second to Sabina Dorothea Kress (1658–1733). Petz bore him six children.
He first published a work concerning his observations on the great comet of 1680, and initially began his work at a private castle-observatory on Spitzenberg 4 owned by Georg Christoph Eimmart (completely destroyed during World War II), the director of Nuremberg's painters' academy. Wurzelbauer was 64 when he began this second career, but proved himself to be an able assistant to Eimmart. A large quadrant from his days at Eimmart's observatory still survives.
After 1682, Wurzelbauer owned his own astronomical observatory and instruments, and observed the transit of Mercury, solar eclipses, and worked out the geographical latitude of his native city. After 1683, he had withdrawn himself completely from business life to dedicate himself to astronomy.
By 1700, Wurzelbauer had become the most well-known astronomer in Nuremberg. For his services to the field of astronomy, he was ennobled in 1692 by Leopold I, Holy Roman Emperor and added the von to his name. He was a member of the French and the Prussian academies of the sciences.
The crater Wurzelbauer on the Moon is named after him. *Wik

1698 Pierre-Louis Moreau de Maupertuis (28 Sep 1698; 27 Jul 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

1761 François Budan de Boislaurent (28 Sept 1761, 6 Oct 1840) was a Haitian born amateur mathematician best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers. Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers. Budan's rule was in a memoir sent to the Institute in 1803 but it was not made public until 1807 in Nouvelle méthode pour la résolution des équations numerique d'un degré quelconque. In it Budan wrote, "If an equation in x has n roots between zero and some positive number p, the transformed equation in (x - p) must have at least n fewer variations in sign than the original." *SAU (Sounds like a nice followup extension to Descartes Rule of signs in Pre-calculus classes. Mention the history, how many times do your students hear about a Haitian mathematician?)

1824 George Johnston Allman (28 September 1824 – 9 May 1904) was an Irish professor, mathematician, classical scholar, and historian of ancient Greek mathematics.*Wik

1873  Julian Lowell Coolidge. (28 Sep 1873; 5 Mar 1954) After an education at Harvard (B.A. 1895), Oxford (B.Sc. 1897), Turin (with Corrado Serge) and Bonn (with Eouard Study, Ph.D. 1904), he came back to Harvard to teach until he retired in 1940. He was an enthusiastic teacher with a ﬂair for witty remarks. [DSB 3, 399] *VFR
He published numerous works on theoretical mathematics along the lines of the Study-Segre school. He taught at Groton School, Conn. (1897-9) where one of his pupils was Franklin D Roosevelt, the future U.S. president. From 1899 he taught at Harvard University. Between 1902 and 1904, he went to Turin to study under Corrado Segre and then to Bonn where he studied under Eduard Study. His Mathematics of the Great Amateurs is perhaps his best-known work. *TIS

1881 Edward Ross studied at Edinburgh and Cambridge universities. After working with Karl Pearson in London he was appointed Professor of Mathematics at the Christian College in Madras India. Ill health forced him to retire back to Scotland. *SAU

1901 Kurt Otto Friedrichs (September 28, 1901 – December 31, 1982) was a noted German American mathematician. He was the co-founder of the Courant Institute at New York University and recipient of the National Medal of Science.*Wik

1925 Martin David Kruskal (September 28, 1925 – December 26, 2006) was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His single most celebrated contribution was the discovery and theory of solitons. His Ph.D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic "The Bridge Theorem For Minimal Surfaces." He received his Ph.D. in 1952.
In the 1950s and early 1960s, he worked largely on plasma physics, developing many ideas that are now fundamental in the field. His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include the Kruskal–Shafranov instability and the Bernstein–Greene–Kruskal (BGK) modes. With I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or magnetohydrodynamic) Energy Principle. His interests extended to plasma astrophysics as well as laboratory plasmas. Martin Kruskal's work in plasma physics is considered by some to be his most outstanding.
In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in General Relativity. A spherically symmetric black hole can be described by the Schwarzschild solution, which was discovered in the early days of General Relativity. However, in its original form, this solution only describes the region exterior to the horizon of the black hole. Kruskal (in parallel with George Szekeres) discovered the maximal analytic continuation of the Schwarzschild solution, which he exhibited elegantly using what are now called Kruskal–Szekeres coordinates.
This led Kruskal to the astonishing discovery that the interior of the black hole looks like a "wormhole" connecting two identical, asymptotically flat universes. This was the first real example of a wormhole solution in General Relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other. This is now believed to be the general fate of wormholes in General Relativity.
Martin Kruskal was married to Laura Kruskal, his wife of 56 years. Laura is well known as a lecturer and writer about origami and originator of many new models.[3] Martin, who had a great love of games, puzzles, and word play of all kinds, also invented several quite unusual origami models including an envelope for sending secret messages (anyone who unfolded the envelope to read the message would have great difficulty refolding it to conceal the deed).
His Mother, Lillian Rose Vorhaus Kruskal Oppenheimer was an American origami pioneer. She popularized origami in the West starting in the 1950s, and is credited with popularizing the Japanese term origami in English-speaking circles, which gradually supplanted the literal translation paper folding that had been used earlier. In the 1960s she co-wrote several popular books on origami with Shari Lewis.*wik

1925 Seymour R. Cray (28 Sep 1925; 5 Oct 1996) American electronics engineer who pioneered the use of transistors in computers and later developed massive supercomputers to run business and government information networks. He was the preeminent designer of the large, high-speed computers known as supercomputers. *TIS Cray began his engineering career building cryptographic machinery for the U.S. government and went on to co-found Control Data Corporation​ (CDC) in the late 1950s. For over three decades, first with CDC then with his own companies, Cray consistently built the fastest computers in the world, leading the industry with innovative architectures and packaging and allowing the solution of hundreds of difficult scientific, engineering, and military problems. Many of Cray's supercomputers are on exhibit at The Computer Museum History Center. Cray died in an automobile accident in 1996.*CHM

1961 Enrique Zuazua Iriondo (September 28, 1961 'Eibar, Gipuzkoa, Basque Country, Spain - ) is a Research Professor at Ikerbasque, the Basque Foundation for Science in BCAM - Basque Center for Applied Mathematics that he founded in 2008 as Scientific Director. He is also the Director of the BCAM Chair in Partial Differential Equations, Control and Numerics and Professor in leave of Applied Mathematics at the Universidad Autónoma de Madrid (UAM).
His domains of expertise in Applied Mathematics include Partial Differential Equations, Control Theory and Numerical Analysis. These subjects interrelate and their final aim is to model, analyse, computer simulate, and finally contribute to the control and design of the most diverse natural phenomena and all fields of R + D + i.
Twenty PhD students got the degree under his advice and they now occupy positions in centres throughout the world: Brazil, Chile, China, Mexico, Romania, Spain, etc. He has developed intensive international work having led co-operation programmes with various Latin American countries, as well as with Portugal, the Maghreb, China and Iran, amongst others. *Wik

DEATHS

1694 Gabriel Mouton was a French clergyman who worked on interpolation and on astronomy.*SAU

1869 Count Guglielmo Libri Carucci dalla Sommaja (1 Jan 1803, 28 Sept 1869) Libri's early work was on mathematical physics, particularly the theory of heat. However he made many contributions to number theory and to the theory of equations. His best work during the 1830s and 1840s was undoubtedly his work on the history of mathematics. From 1838 to 1841 he published four volumes of Histoire des sciences mathématiques en Italie, depuis la rénaissanace des lettres jusqu'à la fin du dix-septième siècle. He intended to write a further two volumes, but never finished the task. It is an important work but suffers from over-praise of Italians at the expense of others. *SAU

1953 Edwin Powell Hubble (20 Nov 1889, 28 Sep 1953)American astronomer, born in Marshfield, Mo., who is considered the founder of extragalactic astronomy and who provided the first evidence of the expansion of the universe. In 1923-5 he identified Cepheid variables in "spiral nebulae" M31 and M33 and proved conclusively that they are outside the Galaxy. His investigation of these objects, which he called extragalactic nebulae and which astronomers today call galaxies, led to his now-standard classification system of elliptical, spiral, and irregular galaxies, and to proof that they are distributed uniformly out to great distances. Hubble measured distances to galaxies and their redshifts, and in 1929 he published the velocity-distance relation which is the basis of modern cosmology.*TIS

1992 John Leech is best known for the Leech lattice which is important in the theory of finite simple groups.*SAU

2004 Jacobus Hendricus ("Jack") van Lint (1 September 1932, 28 September 2004) was a Dutch mathematician, professor at the Eindhoven University of Technology, of which he was rector magnificus from 1991 till 1996. His field of research was initially number theory, but he worked mainly in combinatorics and coding theory. Van Lint was honored with a great number of awards. He became a member of Royal Netherlands Academy of Arts and Sciences in 1972, received four honorary doctorates, was an honorary member of the Royal Netherlands Mathematics Society (Koninklijk Wiskundig Genootschap), and received a Knighthood.*Wik

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

## Thursday, 27 September 2018

### 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 . (It seems when I just checked that the Met no longer allows that link, will replace ASAP) 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.

### On This Day in Math -September 27

Algebra exists only for the elucidation of geometry.

~William Edge

The 270th day of the year; the harmonic mean of the factors of 270 is an integer. The first three numbers with this property are 1, 6, and 28 (which are all perfect #s).. what is the next one? Often called harmoni numbers, they are sometimes called Ore numbers for Øystein Ore, who studied them, and showed that all perfect #s are harmonic. . Many of them also have the arithmetic mean of their divisors is an integer, but not all.

270 is the sum of eight consecutive primes, 270 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 ; and the sum of three cubes $270 = 3^3+ 3^3 + 6^3$.

10! = 3628800 has 270 factors. (A good high school student should be able to confirm this quickly.)

EVENTS

14 A.D.: A total lunar eclipse marked the death of Augustus: "The Moon in the midst of a clear sky became suddenly eclipsed; the soldiers who were ignorant of the cause took this for an omen referring to their present adventures: to their labors they compared the eclipse of the planet, and prophesied 'that if to the distressed goodness should be restored her wonted brightness and splendor, equally successful would be the issue of their struggle.' Hence they made a loud noise, by ringing upon brazen metal, and by blowing trumpets and cornets; as she appeared brighter or darker they exulted or lamented"
- Tacitus *NASA Lunar Eclipses

1830 American Statesman Charles Sumner (1811-1874) paid little attention as an undergraduate at Harvard, but a year after graduation he became convinced that mathematics was a necessary part of a complete education. To a classmate he wrote: “Just a week ago yesterday, I commenced Walker’s Geometry, and now have got nearly half through. All those problems, theorems, etc., which were such stumbling-blocks to my Freshman-year career, unfold themselves as easily as possible now. You will sooner have thought, I suppose, that ﬁre and water would have embraced than mathematics and myself; but, strange to tell, we are close friends now. I really get geometry with some pleasure. I usually devote four hours in the forenoon to it.” Quoted from Florian Cajori’s Mathematics in Liberal Education (1928), p. 115. *VFR  (Sumner was nearly beaten to death by a South Carolina Congressional Representative after a vindictive speech attacking the Kansas-Nebraska act, and it's authors.  His speech included direct insults, sexual innuendo, and made fun of South Carolina Senator Andrew Butler, one of the authors, by imitating his stroke impaired speech and mannerisms.  Butler's Nephew,  Preston Brooks, having decided that a duel could not take place between a gentleman (himself) and a drunk-lout(Sumner) stopped by Sumner's desk to confront him and nearly beat him to death with his cane.  Sumner lost the fight, but the incident put his star on the rise in the Northern states.)

In 1831, the first annual meeting of the British Association for the Advancement of Science was held in York. The British Association had been established in the same year by Sir David Brewster, R.I. Murchison and others. One of the association's main objectives was to "promote the intercourse of those who cultivate science with each other." The second annual meeting was held at Oxford (1832), and in following years at Cambridge, Edinburgh, Dublin, Bristol, Liverpool, Newcastle, Birmingham, Glasgow, Plymouth, Manchester and Cork respectively, until returning to York in 1844. It is incorporated by Royal Charter dated 21 Apr 1928.*TIS

1905 E=mc2 the day that Einstein's paper outlining the significance of the equation arrived in the offices of the German journal Annalen der Physik.  "Does the inertia of a body depend on its energy content?"

1915 Giacomo Ciamician published his prediction of the use of solar energy in Science. He predicted that humans would one day be able to directly convert sunlight into energy, stored in a ‘fuel’ as an alternative to fossil fuels. Today, photovoltaic solar panels can produce electricity, but scientists are also working towards other methods of harnessing the sun’s power, such as solar fuels that mimic photosynthesis. *rsc.org

1919 Einstein writes to his ailing mother that "H. A. Lorentz has just telegraphed me that the British Expeditions have definitely confirmed the deflection of light by the sun." He adds consolation on her illness and wishes her "good days", and closes with "affectionately, Albert *Einstein Archives

In 1922, scientists at the Naval Aircraft Radio Laboratory near Washington, D.C., demonstrated that if a ship passed through a radio wave being broadcast between two stations, that ship could be detected, the essentials of radar. *TIS

1996 Kevin Mitnick, 33, was indicted on charges resulting from a 2 ½-year hacking spree. Police accused the hacker, who called himself "Condor," of stealing software worth millions of dollars from major computer corporations. The maximum possible sentence for his crimes was 200 years. *CHM    Mitnick served five years in prison — four and a half years pre-trial and eight months in solitary confinement — because, according to Mitnick, law enforcement officials convinced a judge that he had the ability to "start a nuclear war by whistling into a pay phone". He was released on January 21, 2000. During his supervised release, which ended on January 21, 2003, he was initially forbidden to use any communications technology other than a landline telephone. Mitnick fought this decision in court, eventually winning a ruling in his favor, allowing him to access the Internet. Under the plea deal, Mitnick was also prohibited from profiting from films or books based on his criminal activity for seven years. Mitnick now runs Mitnick Security Consulting​ LLC, a computer security consultancy. *Wik

BIRTHS

1677 Johann Doppelmayr was a German mathematician who wrote on astronomy, spherical trigonometry, sundials and mathematical instruments.*SAU

1719 Abraham Kästner was a German mathematician who compiled encyclopaedias and wrote text-books. He taught Gauss. His work on the parallel postulate influenced Bolyai and Lobachevsky*SAU

1814  Daniel Kirkwood (27 Sep 1814; 11 Jun 1895) American mathematician and astronomer who noted in about 1860 that there were several zones of low density in the minor-planet population. These gaps in the distribution of asteroid distances from the Sun are now known as Kirkwood gaps. He explained the gaps as resulting from perturbations by Jupiter. An object that revolved in one of the gaps would be disturbed regularly by the planet's gravitational pull and eventually would be moved to another orbit. Thus gaps appeared in the distribution of asteroids where the orbital period of any small body present would be a simple fraction of that of Jupiter. Kirwood showed that a similar effect accounted for gaps in Saturns rings.*TIS  The asteroid 1951 AT was named 1578 Kirkwood in his honor and so was the lunar impact crater Kirkwood, as well as Indiana University's Kirkwood Observatory. He is buried in the Rose Hill Cemetery in Bloomington, Indiana, where Kirkwood Avenue is named for him. *Wik

1824 Benjamin Apthorp Gould (27 Sep 1824; 26 Nov 1896) American astronomer whose star catalogs helped fix the list of constellations of the Southern Hemisphere Gould's early work was done in Germany, observating the motion of comets and asteroids. In 1861 undertook the enormous task of preparing for publication the records of astronomical observations made at the US Naval Observatory since 1850. But Gould's greatest work was his mapping of the stars of the southern skies, begun in 1870. The four-year endeavor involved the use of the recently developed photometric method, and upon the publication of its results in 1879 it was received as a signicant contribution to science. He was highly active in securing the establishment of the National Academy of Sciences.*TIS

1876 Earle Raymond Hedrick (September 27, 1876 – February 3, 1943), was an American mathematician and a vice-president of the University of California.
Hedrick was born in Union City, Indiana. After undergraduate work at the University of Michigan, he obtained a Master of Arts from Harvard University. With a Parker fellowship, he went to Europe and obtained his PhD from Göttingen University in Germany under the supervision of David Hilbert in 1901. He then spent several months at the École Normale Supérieure in France, where he became acquainted with Édouard Goursat, Jacques Hadamard, Jules Tannery, Émile Picard and Paul Émile Appell, before becoming an instructor at Yale University. In 1903, he became professor at the University of Missouri.
He was involved in the creation of the Mathematical Association of America in 1916 and was its first president.
His work was on partial differential equations and on the theory of nonanalytic functions of complex variables. He also did work in applied mathematics, in particular on a generalization of Hooke's law and on transmission of heat in steam boilers. With Oliver Dimon Kellogg he authored a text on the applications of calculus to mechanics.
He moved in 1920 to UCLA to become head of the department of mathematics. In 1933, he was giving the first graduate lecture on mathematics at UCLA. He became provost and vice-president of the University of California in 1937. He humorously called his appointment The Accident, and told jokingly after this event, "I no longer have any intellectual interests —I just sit and talk to people." He played in fact a very important role in making of the University of California a leading institution. He retired from the UCLA faculty in 1942 and accepted a visiting professorship at Brown University. Soon after the beginning of this new appointment, he suffered a lung infection. He died at the Rhode Island hospital in Providence, Rhode Island. Two UCLA residence halls are named after him: Hedrick Hall in 1963, and Hedrick Summit in 2005.
Earle Raymond Hedrick worked on partial differential equations and on the theory of nonanalytic functions of complex variables. He also did work in applied mathematics, in particular on a generalization of Hooke's law and on transmission of heat in steam boilers. With Oliver Dimon Kellogg he authored a text on the applications of calculus to mechanics. *Wik

1843 Gaston Tarry was a French combinatorialist whose best-known work is a method for solving mazes.  The problem as stated by Euler is as follows:-
How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6 × 6 array such that no row or column duplicates a rank or a regiment?
Although an amateur mathematician, Tarry had an amazing ability to analyse combinatorial problems. One has simply to feel amazement at some of the problems he solved using purely combinatorial and calculating skills. We give some examples below to illustrate both the type of problem which interested him and also to illustrate his undoubted genius. Even more surprising is the fact that his mathematical achievements came after the age of fifty.
He published a solution to the problem of finding the way out of a maze in 1895, a problem which had been of interest from classical times. Tarry was not the first to give a systematic method so solve a maze, for Trémaux had found a method which was reported by Lucas in 1881. However, the method given by Tarry gave a different approach with an algorithm which, in today's terminology, would be described as depth-first search algorithm. It is particularly suitable to computer implementation. Tarry also gave a general method for finding the number of Euler circuits, and found lots of results pertaining to magic squares..

*SAU

1855 Paul Appell (27 September 1855 – 24 October 1930), also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris. The concept of Appell polynomials is named after him, as is rue Paul Appell in the 14th arrondissement of Paris.*Wik

1876 Earle Raymond Hedrick (September 27, 1876 – February 3, 1943), was an American mathematician and a vice-president of the University of California. He worked on partial differential equations and on the theory of nonanalytic functions of complex variables. He also did work in applied mathematics, in particular on a generalization of Hooke's law and on transmission of heat in steam boilers. With Oliver Dimon Kellogg he authored a text on the applications of calculus to mechanics.*Wik

1879 Hans Hahn was an Austrian mathematician who is best remembered for the Hahn-Banach theorem. He also made important contributions to the calculus of variations, developing ideas of Weierstrass. *SAU

1892 Mykhailo Pilipovich Krawtchouk (27 Sept 1892 in Chovnitsy, (now Kivertsi) Ukraine - 9 March 1942 in Kolyma, Siberia, USSR) In 1929 Krawtchouk published his most famous work, Sur une généralisation des polynômes d'Hermite. In this paper he introduced a new system of orthogonal polynomials now known as the Krawtchouk polynomials, which are polynomials associated with the binomial distribution.
However his mathematical work was very wide and, despite his early death, he was the author of around 180 articles on mathematics. He wrote papers on differential and integral equations, studying both their theory and applications. Other areas he wrote on included algebra (where among other topics he studied the theory of permutation matrices), geometry, mathematical and numerical analysis, probability theory and mathematical statistics. He was also interested in the philosophy of mathematics, the history of mathematics and mathematical education. Krawtchouk edited the first three-volume dictionary of Ukrainian mathematical terminology. *SAU

1919 James Hardy Wilkinson (27 September 1919 – 5 October 1986) was a prominent figure in the field of numerical analysis, a field at the boundary of applied mathematics and computer science particularly useful to physics and engineering.
He received the Turing Award in 1970 "for his research in numerical analysis to facilitate the use of the high-speed digital computer, having received special recognition for his work in computations in linear algebra and 'backward' error analysis." In the same year, he also gave the John von Neumann Lecture at the Society for Industrial and Applied Mathematics.   The J. H. Wilkinson Prize for Numerical Software is named in his honour.*Wik

DEATHS

1783 Étienne Bézout was a French mathematician who is best known for his theorem on the number of solutions of polynomial equations.*SAU Bézout's theorem for polynomials states that if P and Q are two polynomials with no roots in common, then there exist two other polynomials A and B such that AP+BQ=1. *Wik

1997 William Edge graduated from Cambridge and lectured at Edinburgh University. He wrote many papers in Geometry. He became President of the EMS in 1944 and an honorary member in 1983. *SAU

2014 Jacqueline Anne ( Barton)Stedall (4 August 1950; Romford, Essex, U.K.–27 September 2014; Painswick, Gloucestershire) was a well-known historian of mathematics. Although her career as a researcher, scholar and university teacher lasted less than 14 years, it was greatly influential. Her nine books, more than 20 articles, input to the online edition of the manuscripts of Thomas Harriot, journal editorships and contributions to Melvyn Bragg’s Radio 4 programme In Our Time showed her exceptional breadth of scholarship.
Jackie Stedall came to Oxford in October 2000 as Clifford-Norton Student in the History of Science at Queen’s College. She held degrees of BA (later MA) in Mathematics from Cambridge University (1972), MSc in Statistics from the University of Kent (1973), and PhD in History of Mathematics from the Open University (2000). She also had a PGCE in Mathematics (Bristol Polytechnic 1991). In due course she became Senior Research Fellow in the Oxford Mathematical Institute and at Queen’s College, posts from which, knowing that she was suffering from incurable cancer, she took early retirement in December 2013.
This was her fifth career. Following her studies at Cambridge and Canterbury she had been three years a statistician, four years Overseas Programmes Administrator for War on Want, seven years a full-time parent, and eight years a schoolteacher before she became an academic. *Obituaries at The Guardian, Oxford Mathemtics, and Wik

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

## Wednesday, 26 September 2018

### On This Day in Math - September 26

"mathematics is not yet ready for such problems"
~Paul Erdos in reference to Collatz's problem

This is the 269th day of the year, (on non-leap years, the 269th day is Sep 26, and the date is written 26/9 in much of Europe. This is the only day of the year which presents itself in this way. (Are there any days that work using month/day?)

269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime, a Higgs prime, a strong prime, and a highly cototient number. So many new terms to look up... Well? Look them up.

269 is the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix

EVENTS
1679 On September 26, 1679, a fierce fire consumed the Stellaburgum — Europe’s finest observatory, built by the pioneering astronomer Johannes Hevelius in the city of Danzig, present-day Poland, decades before the famous Royal Greenwich Observatory and Paris Observatory existed.
And while he rebuilt the observatory, it simply did not compare with the original. He never fully recovered from the loss. His resilience in continuing was in large part fueled by the miraculous salvation of one of his manuscripts — his fixed-star catalog, which contained the results of thousands of calculations of the positions of the stars made over decades of patient observation. The small leather-bound notebook was the sole manuscript to survive the fire, presumably saved by Hevelius’s 13-year-old daughter Katharina Elisabeth, the sole family member in Danzig at the time of the fire, who had a key to her father’s study. Half a millennium later, it was rediscovered. In 1971, it made its way to Utah’s Brigham Young University, becoming the one-millionth acquisition by the institution’s library.
Nearly two centuries before Maria Mitchell, Elisabeth Hevelius essentially became the first Western female astronomer. After his death, Elisabeth, who had assisted him in the catalog all along, took it upon herself to finish Hevelius’s lifelong quest. She completed the book, dedicating it to the generous Polish monarch. The finished catalog included more than 600 new stars that Johannes and Elisabeth had observed, as well as a dozen new constellations, whose names, as given by Hevelius, astronomers still use today.
*History of Astronomy @HistAstro
*Maria Popova at brainpickings.org

1732,  Euler shows that F5, the fifth Fermat "prime" is, in fact, not prime, but divisible by 641. He goes on to show that it is also the sum of two squares, in two different ways.
$2^{32}+1^2 = 65536^2+1= 62264^2+204496^2$

1775 John Adams writes to his wife to entreat her to teach his children geometry and... "I have seen the Utility of Geometry, Geography, and the Art of drawing so much of late, that I must intreat you, my dear, to teach the Elements of those Sciences to my little Girl and Boys. It is as pretty an Amusement, as Dancing or Skaiting, or Fencing, after they have once acquired a Taste for them. No doubt you are well qualified for a school Mistress in these Studies, for Stephen Collins tells me the English Gentleman, in Company with him, when he visited Braintree, pronounced you the most accomplished Lady, he had seen since he left England.—You see a Quaker can flatter, but dont you be proud. *Natl. Archives

1874 James Clerk Maxwell in a letter to Professor Lewis Campbell describes Galton, "Francis Galton, whose mission it seems to be to ride other men's hobbies to death, has invented the felicitous expression 'structureless germs'. " *Lewis Campbell and William Garnett (eds.), The Life of James Clerk Maxwell (1884), 299.

1991 The first two year closed mission of Biosphere 2 began just outside Tucson, Arizona. Four men and four women entered the Biosphere 2 on this day in 1991. For two years, the eight participants lived in this huge glass and steel structure in the Arizona desert completely closed off from the rest of the world. It also contained 4,000 species of plants, animals and microbes. *On This Day in Chemistry

1999 The Kobe meteorite fell on September 26 (local time 20:23), 1999, in Kita-ku in the north of Kobe city, Japan. The meteorite fall was widely observed in Kobe and the surrounding area, and was photographed by an amateur photographer in Imabari city, 200 km southwest of Kobe. The meteorite struck a house with an explosive sound but otherwise caused only minor property damage. The approximately 20 fragments of the meteorite had a total mass of 136 g. *terrapub.co.jp

2011 Astronauts had this view of the aurora on September 26, 2011. Credit: NASA

We’ve had some great views of the aurora submitted by readers this week, but this one taken from the International Space Station especially highlights the red color seen by many Earth-bound skywatchers, too. Karen Fox from the Goddard Space Flight Center says the colors of the aurora depend on which atoms are being excited by the solar storm. In most cases, the light comes when a charged particle sweeps in from the solar wind and collides with an oxygen atom in Earth’s atmosphere. This produces a green photon, so most aurora appear green. However, lower-energy oxygen collisions as well as collisions with nitrogen atoms can produce red photons — so sometimes aurora also show a red band as seen here. *Universe Today

BIRTHS

1688 Willem 'sGravesande (26 September 1688 – 28 February 1742)was a Dutch mathematician who expounded Newton's philosophy in Europe. In 1717 he became professor in physics and astronomy in Leiden, and introduced the works of his friend Newton in the Netherlands.
His main work is Physices elementa mathematica, experimentis confirmata, sive introductio ad philosophiam Newtonianam or Mathematical Elements of Natural Philosophy, Confirm'd by Experiments (Leiden 1720), in which he laid the foundations for teaching physics. Voltaire and Albrecht von Haller were in his audience, Frederic the Great invited him in 1737 to come to Berlin.
His chief contribution to physics involved an experiment in which brass balls were dropped with varying velocity onto a soft clay surface. His results were that a ball with twice the velocity of another would leave an indentation four times as deep, that three times the velocity yielded nine times the depth, and so on. He shared these results with Émilie du Châtelet, who subsequently corrected Newton's formula E = mv to E = mv2. (Note that though we now add a factor of 1/2 to this formula to make it work with coherent systems of units, the formula as expressed is correct if you choose units to fit it.) *Wik

1754 Joseph-Louis Proust (26 Sep 1754; 5 Jul 1826) French chemist who proved (1808) that the relative quantities of any given pure chemical compound's constituent elements remain invariant, regardless of the compound's source, and thus provided crucial evidence in support of John Dalton's “law of definite proportions,” which holds that elements in any compound are present in fixed proportion to each other. *TIS

1784 Christopher Hansteen (26 Sep 1784; 15 Apr 1873) Norwegian astronomer and physicist noted for his research in geomagnetism. In 1701 Halley had already published a map of magnetic declinations, and the subject was studied by Humboldt, de Borda, and Gay-Lussac, among others. Hansteen collected available data and also mounted an expedition to Siberia, where he took many measurements for an atlas of magnetic strength and declination.*TIS

1854 Percy Alexander MacMahon (26 Sept 1854 , 25 Dec 1929) His study of symmetric functions led MacMahon to study partitions and Latin squares, and for many years he was considered the leading worker in this area. His published values of the number of unrestricted partitions of the first 200 integers which proved extremely useful to Hardy and Littlewood in their own work on partitions. He gave a Presidential Address to the London Mathematical Society on combinatorial analysis in 1894. MacMahon wrote a two volume treatise Combinatory analysis (volume one in 1915 and the second volume in the following year) which has become a classic. He wrote An introduction to combinatory analysis in 1920. In 1921 he wrote New Mathematical Pastimes, a book on mathematical recreations. *SAU

1887 Sir Barnes (Neville) Wallis (26 Sep 1887; 30 Oct 1979) was an English aeronautical designer and military engineer whose famous 9000-lb bouncing "dambuster" bombs of WW II destroyed the German Möhne and Eder dams on 16 May 1943. He designed the R100 airship, and the Vickers Wellesley and Wellington bombers. The specially-formed RAF 617 Squadron precisely delivered his innovative cylindrical bombs which were released from low altitude, rotating backwards at high speed that caused them to skip along the surface of the water, right up to the base of the dam. He later designed the 5-ton Tallboy and 10-ton Grand Slam earthquake bombs (which used on many enemy targets in the later years of the war). Postwar, he developed ideas for swing-wing aircraft. *TIS (His courtship with his wife has been written by his daughter, Mary Stopes-Roe from the actual courtship in the entertaining, but perhaps overpriced book, Mathematics With Love: The Courtship Correspondence of Barnes Wallis, Inventor of the Bouncing Bomb.)

1891 Hans Reichenbach (September 26, 1891, April 9, 1953) was a leading philosopher of science, educator and proponent of logical empiricism. Reichenbach is best known for founding the Berlin Circle, and as the author of The Rise of Scientific Philosophy.*Wik

1924 Jean Hoerni, a pioneer of the transistor, is born in Switzerland. A physicist, Hoerni in 1959 invented the planar process, which, combined with Robert Noyce's technique for placing a layer of silicon dioxide on a transistor, led to the creation of the modern integrated circuit. Hoerni's planar process allowed the placement of complex electronic circuits on a single chip. *CHM

1926 Colin Brian Haselgrove (26 September 1926 , 27 May 1964) was an English mathematician who is best known for his disproof of the Pólya conjecture in 1958. the Pólya conjecture stated that 'most' (i.e. more than 50%) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919.. The size of the smallest counter-example is often used to show how a conjecture can be true for many numbers, and still be false. *Wik

1927 Brian Griffiths (26 Sept 1927 , 4 June 2008) He was deeply involved in the 'School Mathematics Project', he served as chairman of the 'Joint Mathematical Council', and chaired the steering group for the 'Low Attainers Mathematics Project' from 1983 to 1986. This project became the 'Raising Achievement in Mathematics Project' in 1986 and he chaired this from its foundation to 1989. *SAU

DEATHS

1766 Giulio Carlo Fagnano dei Toschi died. He is important for the identity
$\pi = 2i\log{1 - i \over 1 +i}$
and for his rectiﬁcation of the lemmiscate. *VFR An Italian mathematician who worked in both complex numbers and on the geometry of triangles.*SAU
The lemniscate is of particular interest because, even if it has little relevance today, it
was
the catalyst for immeasurably important mathematical development in the 18th and 19th centuries. The figure 8-shaped curve first entered the minds of mathematicians in 1680, when Giovanni Cassini presented his work on curves of the form, appropriately known as the ovals of Cassini. Only 14 years later, while deriving the arc length of the lemniscate, Jacob Bernoulli became the first mathematician in history to define arc length in terms of polar coordinates.
The first major result of work on the lemniscate came in 1753, when, after reading Giulio Carlo di Fagnano’s papers on dividing the lemniscate using straightedge and compass, Leonhard Euler proved that:

Jacobi called December 23,1751 "the birthday of elliptic functions", as this was the day that Euler began reviewing the papers of Fagnanao who was being considered for membership in the Berlin Academy. *Raymond Ayoub, The lemniscate and Fagnano's contributions to elliptic integrals

1802 Jurij Vega (23 Mar 1754, 26 Sept 1802) wrote about artillery but he is best remembered for his tables of logarithms and trigonometric functions. Vega calculated π to 140 places, a record which stood for over 50 years. This appears in a paper which he published in 1789.
In September 1802 Jurij Vega was reported missing. A search was unsuccessful until his body was found in the Danube near Vienna. The official cause of death was an accident but many suspect that he was murdered. *SAU

1867 James Ferguson (31 Aug 1797, 26 Sep 1867) Scottish-American astronomer who discovered the first previously unknown asteroid to be detected from North America. He recorded it on 1 Sep 1854 at the U.S. Naval Observatory, where he worked 1848-67. This was the thirty-first of the series and is now known as 31 Euphrosyne, named after one of the Charites in Greek mythology. It is one of the largest of the main belt asteroids, between Mars and Jupiter. He was involved in some of the earliest work in micrometry was done at the old U.S. Naval Observatory at Foggy Bottom in the midst of the Civil War using a 9.6 inch refractor. He also contributed to double star astronomy. Earlier in his life he was a civil engineer, member of the Northwest Boundary Survey, and an assistant in the U.S. Coast Survey *TIS

1868 August Ferdinand Mobius died. He discovered his famous strip in September 1858. Johann Benedict Listing discovered the same surface two months earlier.*VFR (It is somewhat amazing that we call it after Mobius when Listing discovered it first and published, and it seems, Mobius did not. However Mobius did seem to have thought on the four color theorem before Guthrie, or anyone else to my knowledge.)

1877 Hermann Günther Grassmann (15 Apr 1809, 26 Sep 1877) German mathematician chiefly remembered for his development of a general calculus of vectors in Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844; "The Theory of Linear Extension, a New Branch of Mathematics"). *TIS

1910 Thorvald Nicolai Thiele (24 Dec 1838, 26 Sept 1910) He is remembered for having an interpolation formula named after him, the formula being used to obtain a rational function which agrees with a given function at any number of given points. He published this in 1909 in his book which made a major contribution to numerical analysis. He introduced cumulants (under the name of "half-invariants") in 1889, 1897, 1899, about 30 years before their rediscovery and exploitation by R A Fisher. *SAU

1976 Paul (Pál) Turán (18 August 1910, 26 September 1976) was a Hungarian mathematician who worked primarily in number theory. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. *SAU

1978 Karl Manne Georg Siegbahn (3 Dec 1886, 26 Sep 1978) Swedish physicist who was awarded the Nobel Prize for Physics in 1924 for his discoveries and investigations in X-ray spectroscopy. In 1914 he began his studies in the new science of x-ray spectroscopy which had already established from x-ray spectra that there were two distinct 'shells' of electrons within atoms, each giving rise to groups of spectral lines, labeled 'K' and 'L'. In 1916, Siegbahn discovered a third, or 'M', series. (More were to be found later in heavier elements.) Refining his x-ray equipment and technique, he was able to significantly increase the accuracy of his determinations of spectral lines. This allowed him to make corrections to Bragg's equation for x-ray diffraction to allow for the finer details of crystal diffraction. *TIS

1990 Lothar Collatz​ (July 6, 1910, , September 26, 1990) was a German mathematician. In 1937 he posed the famous Collatz conjecture, which remains unsolved. The Collatz-Wielandt formula for positive matrices important in the Perron–Frobenius theorem is named after him. *Wik The Collatz conjeture is an iteration problem that deals with the following algorithm..
If a number n is odd, then f(n)= 3n+1
if n is even, then f(n) = 1/2 (n)
Each answer then becomes the new value to input into the function. The problem, or should I say problems, resolve around what happens to the sequence of outcomes when we keep putting the answer back into the function. For example if we begin with 15 we get the following sequence, also called the orbit of the number:
15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1...
One of the unproven conjectures is that for any number n, the sequence will always end in the number 1. This has been shown to be true for all numbers up to just beyond 1016. A second interesting question is how long it takes for a number to return to the value of 1. For the example above, the number 15 took 17 steps to get back to the unit value. Questions such as which three (or other n) digit number has the longest orbit. There are many vairations of the problem, but if you are interested in a good introduction, check this link from Simon Fraser University"
Collatz's Problem is often also called the Syracuse Algorithm, Hasse's problem, Thwaite's problem, and Ulam's problem after people who have worked and written on the problem. It is unclear where the problem originated, as it seems to have had a long history of being passed by word of mouth before it was ever written down. It is often attributed to Lothar Collatz from the University of Hamburg who wrote about the problem as early as 1932. The name "Syracuse Problem" was applied by after H. Hasse, an associate of Collatz, visited and discussed the problem at Syracuse University in the 1950's. During the 1960's Stan Ulam circulated the problem at Los Alamos laboratory. One famous quote about the problem is from Paul Erdos who stated, "mathematics is not yet ready for such problems". *Personal notes

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