Pat'sBlog

Monday, 7 June 2021

On This Day in Math - June 7

Go down deep enough into anything

and you will find mathematics.

~Dean Schlicter

I often quoted a similar idea to my students;
"In any endeavor, at the top of the heap, the bottom line, is math."

The 158th day of the year; 158 is the smallest number such that sum of the number plus its reverse is a non-palindromic prime: 158 + 851 = 1009 and 1009 is a non-palindromic prime. *Number Gossip (What's the next one?)

158 is the sum of the first nine Mersenne prime exponents.

158 is the smallest number such that the sum of the number and its reverse is a prime that is not a palindrom, i.e., 158 + 851 = 1009.*Prime Curios

The decimal expansion of 100! (the product of the first 100 natural numbers) has 158 digits.

EVENTS

1634 In a letter which he wrote at Dover in England to Mersenne on 7 June 1634, Bernard Frenicle describes an experiment to study the trajectory of a body released from the top of the mast of a moving ship. The data which he presents in the letter is quite accurate. Again on a more applied mathematical topic, Frenicle wrote an article which makes comments on Galileo's Dialogue. *SAU

1713 Johan Bernoulli (I) writes to Leibniz from Basil to stir the pot in the great calculus dispute. "...my nephew (Nikolaus) brought from Paris a single copy of the Commercium Epistolicum... sent from London for distribution to the learned.... you are at once accused by a tribunal consisting,..., of the participants and witnesses themselves.. documents against you are produced, sentence is passed; you lose the case, you are condemned." *The Correspondence of Isaac Newton (Thony Christie points out that this was about "At the same time as Newton was making him a member of the Royal Society <= Embarrassing!" )

1742, the German mathematician Christian Goldbach originally of Brandenburg-Prussia wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:
“Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units.”
He then proposed a second conjecture in the margin of his letter:
“Every integer greater than 2 can be written as the sum of three primes.”
He considered 1 to be a prime number, a convention subsequently abandoned. Today the conjecture is usually stated as "Every even integer greater than 2 can be expressed as the sum of two primes." The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is:
Every integer greater than 5 can be written as the sum of three primes.
*Wik

1753 the British Museum was founded, the world's oldest public national museum, when King George II gave his royal assent to an Act of Parliament to accept the collection of Sir Hans Sloane, a London-based physician, following his death. In his will, he had offered the British nation the collection he built over his lifetime: 71,000 objects, mostly plant and animal specimens. In return, he asked a sum of £20,000 for his heirs (which today would be more than £2,000,000). The museum opened to the public 15 Jan 1759 at Bloomsbury. Its current buildings there date from the mid-19th century. The natural history collection moved to its own museum in 1881. The British Museum set up a laboratory in 1920 for scientific study of objects. *TIS A description of how to attain entry and museum protocol from Nicholson's Journal@ Wm_Nicholson

1759 Benjamin Franklin writes to William Heberden, 7 June 1759 On the electical effects of heated Tourmaline crystals. Tourmaline crystals, brought to Europe from the East by the Dutch early in the eighteenth century, began to attract the attention of electrical scientists when they found that, if heated, they had the power of attracting and repelling ashes and other light substances. Franklin’s letter is the earliest known report on such investigations in England. Since it was not published for ten years, however, or read to the Royal Society, his friends John Canton and Benjamin Wilson, accounts of whose experiments became public later in 1759, gained general priority among English investigators of the subject. *Franklin Papers, Natl.Archives

1886 Just two-and-a-half years after Winifred Edgerton entered Columbia, she completed her thesis. At the Board of Trustees meeting that day the following motion was made and passed unanimously: “That in consideration of the extraordinary excellence of the scientific work done by Miss Winifred Edgerton, as attested by the Professors who have had the superintendence of her course in practical Astronomy, and the Pure Mathematics in the Graduate Department, the degree of Doctor of Philosophy can be conferredupon Miss Edgerton cum laude”
Winifred Edgerton thus became the first American woman to receive her Ph.D. in mathematics and the first woman to graduate from Columbia University.*Susan E. Kelly and Sarah A. Rozner *AMS Notices,Volume 59, Number 4

1906 The New York Times reported on an early implementation of what might be considered speed bumps in the U.S. town of Chatham, New Jersey, which planned to raise its crosswalks five inches above the road level: "This scheme of stopping automobile speeding has been discussed by different municipalities, but Chatham is the first place to put it in practice". The average automobile's top speed at the time was around 30 miles per hour (48 km/h) The more conventional speed bumps you are familiar with today seem to have been invented by Arthur Holly Compton, a physicist and winner of the Nobel Prize in physics in 1927. He created his "traffic control bumps," in 1953 after noticing the speed at which motorists passed Brookings Hall at Washington University in St. Louis, Missouri, where he was chancellor. *Wik with HT to arclight ‏@arclight

1914 the Alliance was the first vessel to pass through the Panama Canal.*TIS

1958 France issued a stamp with a portrait of Denis Diderot (1713–1784). [Scott #B 323].

BIRTHS

1761 John Rennie (7 June 1761 – 4 October 1821) Scottish engineer and architect who designed London Bridge. After working as a millwright with Andrew Meikle he studied at Edinburgh University (1780-83). He was employed by Boulton & Watt for five years In 1791, he moved to London and started his own engineering company. Over the next few years he became famous as a bridge-builder, including Leeds Bridge, Southwark Bridge and Waterloo Bridge. He was also designed and built docks at Hull, Liverpool, Greenock and Leith and improving the harbours and dockyards at Portsmouth, Chatham and Plymouth. His last project was London Bridge, though he died in 1821 before it was finished. The bridge was completed by his son, Sir John Rennie.*TIS

1862 Philipp Eduard Anton von Lenard (7 June 1862 – 20 May 1947), was a German physicist and the winner of the Nobel Prize for Physics in 1905 for his research on cathode rays and the discovery of many of their properties. He was a nationalist and anti-Semite; as an active proponent of the Nazi ideology, he had supported Adolf Hitler in the 1920s and was an important role model for the "Deutsche Physik" movement during the Nazi period.
Lenard is remembered today as a strong German nationalist who despised "English physics", which he considered to have stolen its ideas from Germany. He joined the National Socialist Party before it became politically necessary or popular to do so. During the Nazi regime, he was the outspoken proponent of the idea that Germany should rely on "Deutsche Physik" and ignore what he considered the fallacious and deliberately misleading ideas of "Jewish physics", by which he meant chiefly the theories of Albert Einstein, including "the Jewish fraud" of relativity. An advisor to Adolf Hitler, Lenard became Chief of Aryan physics under the Nazis. *Wik

1863 Edward Burr Van Vleck born. (June 7, 1863, Middletown, Connecticut – June 3, 1943, Madison, Wisconsin) The son of astronomer John Monroe Van Vleck, he graduated from Wesleyan University in 1884, attended Johns Hopkins in 1885-87, and studied at Göttingen (Ph.D., 1893). He was assistant professor and professor at Wesleyan (1895-1906), and after 1906 a professor at the University of Wisconsin–Madison, where the mathematics building is named after him. In 1913 he became president of the American Mathematical Society, of whose Transactions he had been first associate editor (1902-05) and then editor (1905-10). He was the author of Theory of Divergent Series and Algebraic Continued Fractions (1903), and of several monographs in mathematical journals. His son, John Hasbrouck van Vleck, was a notable physicist who received the Nobel Prize in 1977.
Edward van Vleck was also an important art collector, particularly in the medium of Japanese woodblock prints (principally Ukiyo-e), known as Van Vleck Collection. He began collecting around 1909, but became a serious collector in the late 1920s, when he acquired approximately 4,000 prints that had been owned by Frank Lloyd Wright. His collection, one of the largest in the world outside the Library of Congress, features more than 2,000 prints by Utagawa Hiroshige as well as many prints by Hokusai, and fine examples of shin hanga made well into the 20th century. His collection now resides at the Chazen Museum of Art in Madison, Wisconsin.*Wik

1868 Sir John Sealy Edward Townsend (7 Jun 1868, 16 Feb 1957 at age 88) British physicist who pioneered in the study of electrical conduction in gases. In 1898 he made the first direct measurement of the unit electrical charge (e). As a postgraduate, he was a research student of J. J. Thomson. In 1897, Townsend developed the falling-drop method for measuring e, using saturated clouds of charged water droplets (extended by Robert Millikan's highly accurate oil-drop method). He was first to explain how electric discharges pass through gases (Electricity in Gases, 1915) whereby motion of electrons in an electric field releases more electrons by collision. These in turn collide releasing even more electrons in a multiplication of charges known as an avalanche. *TIS

1877 Charles Barkla (7 June 1877 to 23 October 1944) was an influential English physicist who became professor of Natural Philosophy in Edinburgh. He was awarded the Nobel Prize for Physics in 1917 for his work on X-ray spectroscopy.

1890 Werner Schmeidler born. He worked in analysis and applied mathematics.*SAU He studied mathematics in Göttingen. From 1939 he was Professor at the Technical University of Berlin , where he held the Chair of Pure and Applied Mathematics until the summer semester 1958. *Wik

1896 Robert Sunderson Mulliken (June 7, 1896 – October 31, 1986), primarily responsible for the early development of molecular orbital theory, i.e. the elaboration of the molecular orbital method of computing the structure of molecules. Dr. Mulliken received the Nobel Prize for chemistry in 1966. He received the Priestley Medal in 1983. *Wik

DEATHS

1843 Alexis Bouvard (27 June 1767 – 7 June 1843) French astronomer and director of the Paris Observatory, who is noted for discovering eight comets and writing Tables astronomiques of Jupiter and Saturn (1808) and of Uranus (1821). Bouvard's tables accurately predicted orbital locations of Jupiter and Saturn, but his tables for Uranus failed, leading him to hypothesize that irregularities were caused by an unknown perturbing body. This spurred observations leading to the discovery of Neptune by Adams and Leverrier.*TIS

1954 Alan Turing (23 June 1912 – 7 June 1954), died by committing suicide because he was persecuted by the British Government for his homosexuality. *VFR Alan Mathison Turing, OBE, FRS (play /ˈtjʊərɪŋ/ tewr-ing; 23 June 1912 – 7 June 1954), was an English mathematician, logician, cryptanalyst and computer scientist. He was highly influential in the development of computer science, providing a formalization of the concepts of "algorithm" and "computation" with the Turing machine, which played a significant role in the creation of the modern computer. Turing is widely considered to be the father of computer science and artificial intelligence.
Turing's homosexuality resulted in a criminal prosecution in 1952, when homosexual acts were still illegal in the United Kingdom. On 10 September 2009, following an Internet campaign, British Prime Minister Gordon Brown made an official public apology on behalf of the British government for the way in which Turing was treated after the war.
My favorite Turing story is this: When he was found, he had a half eaten apple by his bed, which may have been the delivery mechanism for the poison. Some suggest he chose this method because it was a re-enactment of a scene from the 1937 film Snow White, his favorite fairy tale. Years later, to honor Turing, the developers of Apple Computers used the symbol of an apple with a bite out for their logo. The last part, about the Apple logo, seems not to be true, but it is a great story, and a good excuse to tell students about the work of Turing.. so I do.*Wik

2004 Joseph Doob (February 27, 1910–June 7, 2004) was an American mathematician who worked in probability and measure theory. *SAU After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes. So he wrote his famous "Stochastic Processes" book. It was published in 1953 and soon became one of the most influential books in the development of modern probability theory. *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

Sunday, 6 June 2021

A Unique approach for Odd Order Magic Squares

I have been interested in Math History and Recreation Math for a really long time, (yes, I'm that old), so when I came across a new approach on twitter that I had never seen, I was a little surprised. When I read that it was about 400 years old, I was even more surprised (and no, I'm not THAT old).

I've written about Magic Squares over the years, from the earliest known 3x3 supposedly found on the back of a turtle in Chinese Mythology, called the Lo Shu Square ( literally: Luo (River) Book/Scroll) and about the magic square on the Passion Facade on the Sagrada Familia Cathedral in Barcelona and then about a magic square relationship to Matrices I just learned this year (2018) from John D. Cook's blog.

I usually not surprised in finding out new relationships in magic squares, but part of what surprised me this time, was that it was a method created by Claude Gaspard Bachet de Méziriac, Who I've read a lot about, and written a little about, and was aware that he worked with recreational math and number theory. He published a Latin translation of the Greek text of Diophantus’s Arithmetica in 1621. This is the translation that Fermat made his famous margin note that became the famous Fermat's Last Theorem. He asked the ﬁrst ferrying problem: Three jealous husbands and their wives wish to cross a river in a boat that will only hold two persons, in such a manner as to never leave a woman in the company of a man unless her husband is present.

So, anyway, if I'm not the only person in the world who never saw it before, here is a really unique method of constructing nXn magic Squares when n is odd. which I found on a animated tweet created in Geogebra by Jason-Automaths@palajsn, and thanks to Vincent Pantaloni for sharing.

You start by constructing a Diamond stack of squares with 1 square in the first row, three squares in the second, etc. until you get to 2n-1 squares for the desired nXn desired, then descending back down to one. Here is the example for the 7x7 square.

Then you start at the 1'st diagonal down the right side and write the numbers in order, 1 to 7. Skip to the third diagonal and do the next 7 digits. Continue in like fashion and you get something like this.

Now here is the slickest little move imaginable, you take the pyramid of six numbers above the the top row of seven squares, and move it down until the number one is just below the center square (25 in this case)...

Make a similar translation of the pyramids on the other three sides across to the similar position on the other side of the center square, and you have a magic square.

If you learned the quick method I did for odd squares, you start at the bottom center with one (apparently the Chinese put North on their calendars at the bottom, and that was an influence on the future evolution of magic squares). Then you just number up and right (or down and right) on the diagonal (as if the edges were connected right and left, top and bottom like a torus). Each time you come to a multiple of n, you drop down one and continue. Notice this works the same way, except that the diagonals go down and right, and at ever multiple of n, you drop down two rows, instead of one, to continue.

On This Day in Math - June 6

Map of 2012 Transit of Venus visibility *Eclipse Maps

No mathematician can be

a complete mathematician unless

he is also something of a poet.

~Karl Weierstrass

The 157th day of the year; 2¹⁵⁷ is the smallest "apocalyptic number," i.e., a number of the form 2ⁿ that contains '666'. *Prime Curios (Can you find an apocalyptic number of the form 3ⁿ)

157 is prime and it's reverse, 751 is also prime. 157 is also the middle value in a sexy triplet (three primes successively differing by six; 151, 157, 163). 751 is also a sexy prime with 757.

157 is also the largest solution I know for a prime, p, such that \( \frac{p^p-p!}{p} \) is prime.

The number 157 in base ten is equal to \(31_{[52]}\), but don't worry if you get that backwards,
\(52_{[31]}\) is also equal to 157 in decimal. Can you find other examples of reversible numeral/base that give the same decimal value?

157 is a prime which is the arithmetic average of the squares of four consecutive primes, (7^2 + 11^2 + 13^2 + 17^2)/4 = 157. There are only two such year days that meet this definition. and this is the smaller.

And from Fermat's Library @fermatslibrary In 1993 Don Zagier found the smallest rational right triangle with area 157. He used sophisticated techniques using elliptic curves paired with a lot of computational power. If he could do that, certainly you ought to be able to find the smallest rational right triangle with area of 1.... (OK trick question, ask your teacher to explain)

EVENTS

1647 Fermat writes to Digby to repeat challenges he had set in January. 1) Find a cube that when added to the sum of its aliquot parts is a square. 2) Find a square that when increased by the sum of its aliquot parts is a cube. He added that \(7^3\) is not the only solution. Can you solve either, or both?

In 1799, the first definitive prototype meter bars (mètre des Archives) and kilograms were constructed in platinum. This followed the legal definition of the metric system by the French National Assembly on 7 Apr 1795, that was itself established during the famous measurements of the Earth's meridian between Dunkerque and Barcelona. The use of a metal bar to define the standard meter continued until replaced in 1960 by a definition based upon a number of wavelengths of light from a certain spectroscopic light source.*TIS

1902 Scottish chemist professor James Dewar exhibits air in the solid state and a jet of liquid air rising six feet above it with beautiful effects, before the Prince and Princess of Wales. *Great Geek Manual

1944, Supreme Allied Commander General Dwight D. Eisenhower gives the go-ahead for largest amphibious military operation in history: Operation Overlord, code named D-Day, the Allied invasion of northern France.

1984 Sweden issued a series of stamps celebrating the centenary of their Patent System. One shows a tetrahedral container pented in 1948. [Scott #1501]. *VFR (I have been unable to find an image of this stamp.. if anyone has one, please advise)

1984 Tetris is a Soviet tile-matching puzzle video game originally designed and programmed by Alexey Pajitnov. It was released on June 6, 1984. A nice post with ten things you did't know about about Tetris is at this blog from Wallifaction, a very good history blog by Adam Richter

2012 Last Chance. The most recent transit of Venus when observed from Earth took place on June 8, 2004. The event received significant attention, since it was the first Venus transit to take place after the invention of broadcast media. No human alive at the time had witnessed a previous Venus transit, since the previous Venus transit took place on December 6, 1882. The next transit of Venus will occur on June 5–June 6 in 2012. After 2012, the next transits of Venus will be in December 2117 and December 2125.

BIRTHS

1436 Johann Mueller (6 June 1436 – 6 July 1476) , AKA Johannes Regiomontanus after the Latinization of his hometown, Konigsburg. He is the founder of trigonometry as an independent science. The spherical law of sines was first presented by Johann Muller, in his De Triangulis Omnimodis in 1464. This was the first book devoted wholly to trigonometry (a word not then invented). David E. Smith suggests that the theorem was Muller's creation.
The ideas behind the law of sines, like those of the law of cosines, predate the word sine by over a thousand years. Theorems in Euclid on lengths of chords are essentially the same ideas we now call the law of sines. What we now call the law of sines for plane triangles was known to Ptolemy. By the tenth century Abu'l Wefa had clearly expounded the spherical law of sines. It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase.

"In Jan 1472 he made observations of a comet which were accurate enough to allow it to be identified with Halley's comet 210 years later (being three returns of the 70 year period comet). He also observed several eclipses of the Moon. His interest in the motion of the Moon led him to make the important observation that the method of lunar distances could be used to determine longitude at sea. However, instruments of the time lacked the necessary accuracy to use the method at sea. " *TIS {There is a nice blog at The Renaissance Mathematicus about the important role Regiomontanus played in scientific publishing.}

1553 Bernardino Baldi (6 June 1553 – 10 October 1617) was an Italian mathematician and writer.
Baldi descended from a noble family from Urbino, Marche, where he was born. He pursued his studies at Padua, and is said to have spoken about sixteen languages during his lifetime, though according to Tiraboschi the inscription on his tomb limits the number to twelve.
The appearance of the plague at Padua forced him to return to his native city. Shortly afterwards he was called to act as tutor to Ferrante Gonzaga, from whom he received the rich abbey of Guastalla. The oldest biography of Nicolaus Copernicus was completed on 7 October 1588 by him. He held office as abbot for 25 years, and then returned once again to Urbino. In 1612 he was employed by the duke as his envoy to Venice. Baldi died at Urbino on 12 October 1617.
He is said to have written upwards of a hundred different works, the chief part of which have remained unpublished. His various works show his abilities as a theologian, mathematician, geographer, antiquary, historian and poet. One of these has been recently found and is now at the Univ. of Oklahoma.
"Baldi is known to have written a treatise on sun dials and timekeeping. However, this treatise was never published and, since 1783, it has been considered lost. Now we are happy to announce that it has been recently acquired by the History of Science Collections, digitized in high resolution, and made available for study in the Collections’ Online Galleries." The Cronica dei Matematici (published at Urbino in 1707) is an abridgment of a larger work on which he had written for twelve years, and was intended to contain the lives of more than two hundred mathematicians. His life has been written of by Affò, Mazzucchelli and others. *Wik

1580 Govaert Wendelen (6 June 1580 – 24 October 1667) was a Flemish astronomer who was born in Herk-de-Stad. He is also known by the Latin name Vendelinus. His name is sometimes given as Godefroy Wendelin; his first name spelt Godefroid or Gottfried.
Around 1630 he measured the distance between the Earth and the Sun using the method of Aristarchus of Samos. The value he calculated was 60% of the true value (243 times the distance to the Moon; the true value is about 384 times; Aristarchus calculated about 20 times).
In 1643 he recognized that Kepler's third law applied to the satellites of Jupiter.
Wendelin corresponded with Mersenne, Gassendi and Constantijn Huygens.
The crater Vendelinus on the Moon is named after him.
Wendelin died in Ghent on 24 October 1667. *Wik

1842 Henry Martyn Taylor (6 June 1842, Bristol – 16 October 1927, Cambridge) born in Bristol, England. He was a fellow at Trinity College, Cambridge, and is most remembered because he devised a Braille notation when he was overtaken by blindness in 1894, when engaged in the preparation of an edition of Euclid for the Cambridge University Press. By means of his ingenious and well thought out Braille notation he was enabled to transcribe many advanced scientific and mathematical works, and in 1917, with the assistance of Mr. Emblen, a blind member of the staff of the National Institute for the Blind, he perfected it. It was recognised as so comprehensive that it was soon adopted as the standard mathematical and chemical notation. It seems that in the US it is more common to use the Nemeth code for mathematics and science symbols, first developed around 1947. I am not sure about usage at the present time in the rest of the world.

1850 Karl Ferdinand Braun (6 June 1850 – 20 April 1918) was a German inventor, physicist and Nobel laureate in physics. Braun contributed significantly to the development of the radio and television technology: he shared with Guglielmo Marconi the 1909 Nobel Prize in Physics.
Braun was born in Fulda, Germany, and educated at the University of Marburg and received a Ph.D. from the University of Berlin in 1872. In 1874 he discovered that a point-contact semiconductor rectifies alternating current. He became director of the Physical Institute and professor of physics at the University of Strassburg in 1895.
In 1897 he built the first cathode-ray tube (CRT) and cathode ray tube oscilloscope. CRT technology has been replaced by flat screen technologies (such as liquid crystal display (LCD), light emitting diode (LED) and plasma displays) on television sets and computer monitors. The CRT is still called the "Braun tube" in German-speaking countries (Braunsche Röhre) and in Japan (ブラウン管: Buraun-kan). *Wik

1857 Aleksandr Mikhailovich Lyapunov (June 6[O.S. May 25] 1857 – November 3, 1918) born in Yaroslavl, Russia. He was the creator of the modern theory of stability of diﬀerential equations especially as applied to mechanical systems. He also proved the Central Limit Theorem under weaker hypotheses than his predecessors. *VFR He was a student of Chebyshev. In 1901, Lyapunov gave the first prominent proof of the Central Limit Theorem, which made the CLT one of the foundations of probability theory today. (Unlike the classical CLT, Lyapunov’s condition only requires the random variables in question to be independent instead of both independent and identically distributed.)

1882 Clement Vavasor Durell (6 June 1882 in Fulbourn, near Cambridge, England, -10 December 1968 in South Africa) Durell was educated at Felsted School and, while still at school, he published his first note in the Mathematical Gazette, the journal of the Mathematical Association. The note was A geometrical method of trisecting any angle with the aid of a rectangular hyperbola written jointly with W F Beard.
Durell joined the Mathematical Association in 1900, the year in which he entered Clare College, Cambridge, to study mathematics. He was a First Class student in the Mathematical Tripos examinations, graduating in 1904. He was appointed as a mathematics teacher at Gresham's School immediately after graduating, and in the following year of 1905 he moved to take up the post of mathematics master at Winchester College.
Soon after taking up this post Durell's first textbook Elementary Problem Papers (1906) was published. He was promoted to senior mathematics master at Winchester College in 1910 and began publishing a series of articles in the Mathematical Gazette. Before the outbreak of World War I, Durell published The arithmetic syllabus in secondary schools (1911) and Analysis and projective geometry (1911) in the Mathematical Gazette. During World War I, Durell served in the Royal Garrison Artillery as a lieutenant. After the end of the war he returned to Winchester College and began publishing a series of articles in the Mathematical Gazette and a remarkable series of textbooks which would make him the best known writer of English school mathematics texts.

As well as writing articles for the Mathematical Gazette such as The use of limits in elementary geometry (1925) and The teaching of loci in the elementary geometry course to school certificate stage (1936), he was also actively involved with the committee work of the Mathematical Association and its report production. He wrote reports The teaching of geometry in schools (1925), Memo from the Girls' Schools' Committee: Mathematics for girls (1926), and Questionnaire on the teaching of mathematics in evening continuation schools (1926). Among the books he wrote around this time were: Readable relativity (1926), A Concise Geometry (1928), Matriculation Algebra (1929), Arithmetic (1929), Advanced Trigonometry (1930), A shorter geometry (1931), The Teaching of Elementary Algebra (1931), Elementary Calculus (1934), A School Mechanics (1935), and General Arithmetic (1936). In a catalogue produced by the Mathematical Association's publishers G Bell & Sons in 1934, they listed 20 textbooks by Durell and write

There can indeed be few secondary schools in the English-speaking world in which some at least of Mr Durell's books are not now employed in the teaching of mathematics.

*SAU

1906 Max Zorn (June 6, 1906 in Krefeld, Germany – March 9, 1993 in Bloomington, Indiana, United States) To his chagrin, he is most famous for discovering something yellow and equivalent to the Axiom of Choice. *VFR (with a smile, I'm sure) He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a powerful tool in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, ordered sets, etc. Zorn's lemma was first discovered by K. Kuratowski (see June 18) in 1922, and then independently by Zorn in 1935.*Wik Interesting that he was born on 6/6/6.

DEATHS

1834 Erastus Lyman De Forest (27 June 1834 in Watertown, Connecticut, USA - 6 June 1888 in Watertown, Connecticut, USA) His parents were Lucy Starr Lyman and Dr John De Forest. He was named after his mother's father, Erastus Lyman, who was from Litchfield, Connecticut. Both sides of the family were well off and Erastus was born into a privileged place in society. John De Forest graduated from Yale College and wished his son to follow in his footsteps as indeed he did, entering Yale at the age of sixteen to study mathematics. He was awarded his B.A. in 1854 and his father celebrated the occasion by endowing the De Forest Mathematical Prize at Yale. Erastus's maternal grandfather celebrated the occasion by making him a large bequest.

De Forest remained at Yale to study engineering and at this time was a fellow student with J Willard Gibbs who entered Yale in the year that De Forest was awarded his B.A. In 1856 De Forest was awarded a Ph.B. by Yale and then in February of the following year he set off with his aunt for New York to begin a journey with her to Havana. However, before the ship was due to depart De Forest vanished leaving his luggage. When his family could find no trace of him they put an advertisement in the New York Times asking for information. They received a reply which told them his body was in East River but a search revealed nothing.

For two years De Forest's family continued to make desperate efforts to locate him but receiving not a shred of information they came to believe that he must have been murdered. It was more than two years after he vanished that John De Forest received a letter from his son, posted in Australia. De Forest, depressed with his privileged life, had travelled to California where he had got a job at a mine. After a while he was appointed as a teacher in a private school where he taught for about a year before going to Australia where again he taught, this time at Melbourne Church of England Grammar School in South Yarra. After more than four years away, he returned to the United States in 1861 visiting India and England on his way. He returned to Europe in 1863 for a lengthy trip which lasted until 1865.

From his return to Connecticut in 1865 he devoted himself to the study of mathematics. After publishing papers interpolation and its applications, he was asked by his uncle, who was President of the Knickerbocker Life Insurance Company of New York, to examine the liabilities that the company's life policies involved. De Forest became deeply involved in improving mortality tables, publishing over 20 papers on the topic between 1870 and 1885.

The remarkable contributions of De Forest to statistics had little or no influence on the subject since those who later developed similar ideas were totally unaware of his contributions. This was for a number of reasons. De Forest was not associated with any institution so lacked the visibility that such a position would have meant. He worked in the United States at a time when little of mathematical significance was happening in that country. Also he published his work in somewhat obscure American journals. His contributions were recognized, however, by Pearson whose attention was drawn to De Forest's papers. Pearson acknowledged De Forest's priority in deriving the chi square distribution. The book contains reprints of four of De Forest's papers as well as a biographical article written by J Anderson. His life and work are both discussed by Stigler . Stigler uses information on De Forest available to him from a well researched but unpublished work on De Forest by H H Wolfenden.

De Forest never married and cared for his father for many years until his death in 1885, from which time his own health began to deteriorate. Shortly before he died he founded the Erastus L De Forest Professorship of Mathematics at Yale. *SAU

1898 Henry Perigal, Jr. FRAS MRI (1 April 1801 – 6 June 1898) was a British stockbroker and amateur mathematician, known for his dissection-based proof of the Pythagorean theorem and for his unorthodox belief that the moon does not rotate.
In his booklet Geometric Dissections and Transpositions (London: Bell & Sons, 1891) Perigal provided a proof of the Pythagorean theorem based on the idea of dissecting two smaller squares into a larger square. The five-piece dissection that he found may be generated by overlaying a regular square tiling whose prototile is the larger square with a Pythagorean tiling generated by the

two smaller squares. Perigal had the same dissection printed on his business cards, and it also appears on his tombstone.

As well as being interested in mathematics, Perigal was an accomplished lathe worker, and made models of mathematical curves for Augustus De Morgan. He believed (falsely) that the moon does not rotate with respect to the fixed stars, and used his knowledge of curvilinear motion in an attempt to demonstrate this belief to others. *Wik

1928 Luigi Bianchi (January 18, 1856 – June 6, 1928) He did fundamental work on Lie groups. *VFR He was a leading member of the vigorous geometric school which flourished in Italy during the later years of the 19th century and the early years of the twentieth century.
In 1898, Bianchi worked out the Bianchi classification of nine possible isometry classes of three-dimensional Lie groups of isometries of a (sufficiently symmetric) Riemannian manifold. As Bianchi knew, this is essentially the same thing as classifying, up to isomorphism, the three-dimensional real Lie algebras. This complements the earlier work of Lie himself, who had earlier classified the complex Lie algebras.
Through the influence of Luther P. Eisenhart and Abraham Haskel Taub, Bianchi's classification later came to play an important role in the development of the theory of general relativity. Bianchi's list of nine isometry classes, which can be regarded as Lie algebras, Lie groups, or as three dimensional homogeneous (possibly nonisotropic) Riemannian manifolds, are now often called collectively the Bianchi groups.
In 1902, Bianchi rediscovered what are now called the Bianchi identities for the Riemann tensor, which play an even more important role in general relativity. (They are essential for understanding the Einstein field equation.) According to Tullio Levi-Civita, these identities had first been discovered by Ricci in about 1880, but Ricci apparently forgot all about the matter, which led to Bianchi's rediscovery! *Wik

1943 Guido Fubini (19 January 1879 – 6 June 1943) He is best known for a theorem on the exchange of order of integration. his research focused primarily on topics in mathematical analysis, especially differential equations, functional analysis, and complex analysis; but he also studied the calculus of variations, group theory, non-Euclidean geometry, and projective geometry, among other topics. With the outbreak of World War I, he shifted his work towards more applied topics, studying the accuracy of artillery fire; after the war, he continued in an applied direction, applying results from this work to problems in electrical circuits and acoustics. *Wik

1972 Abraham Adrian Albert (9 November 1905 Chicago, Illinois, USA - 6 June 1972 Chicago, Illinois, USA) A Adrian Albert's parents were Russian. His father, Elias Albert, came to the United States from England and had set up a retail business. His mother, Fannie Fradkin, had come to the United States from Russia. Adrian was the second of Elias and Fannie's three children, but he also had both a half-brother and half-sister from his mother's side.
Albert completed his B.S. degree in 1926 and was awarded his Master's degree in the following year. He remained at the University of Chicago undertaking research under L E Dickson's supervision.
By the time that he received his doctorate Albert was a married man, having married Freda Davis on 18 December 1927.
In his doctoral thesis Albert had made considerable progress in classifying division algebras. It was an impressive piece of work and it led to him being awarded a National Research Council Fellowship to enable him to undertake postdoctoral study at Princeton. He spent nine months at Princeton in 1928-29 and this was an important period for Albert since during his time there Lefschetz suggested that he look at open problems in the theory of Riemann matrices. These matrices arise in the theory of complex manifolds and Albert went on to write an important series of papers on these questions over the following years.

Albert was then offered a post as an instructor at Columbia University and he worked there for two years from 1929 to 1931. His first paper A determination of all normal division algebras in sixteen units was published in 1929. It was based on the second half of his doctoral thesis but Albert had, by this time, pushed the ideas further classifying division algebras of dimension 16 over their centres. The case of dimension 9, the next smaller case, had been solved by Wedderburn.

Albert returned to the University of Chicago in 1931 where he was appointed as assistant professor. He remained on the staff there for the rest of his life being promoted to associate professor in 1937 and full professor in 1941. During the years 1958 to 1962 he was chairman of the Chicago Department.

Shortly after beginning his second three year term as Chairman of the Department Albert was asked to take on the post of Dean of Physical Sciences. He served Chicago for 9 year in the role until 1971.
His main work was on associative algebras, non-associative algebras, and Riemann matrices. He worked on classifying division algebras building on the work of Wedderburn but Brauer, Hasse and Emmy Noether got the main result first. Albert's major contribution is, however, detailed in a joint paper with Hasse. Albert's book Structure of Algebras, published in 1939, remains a classic. The content of this treatise was the basis of the Colloquium Lectures which he gave to the American Mathematical Society in 1939.
Albert's work on Riemann matrices was, as we mentioned above, a consequence of suggestions made by Lefschetz.
During the Second World War Albert contributed to the war effort as associate director of the Applied Mathematics Group at Northwestern University which tackled military problems. Another interest of Albert's, which appears to have been prompted by the War, was that of cryptography. He lectured to the American Mathematical Society on Some mathematical aspects of cryptography at the Society's meeting in November 1941.
Albert investigated just about every aspect of non-associative algebras.
Albert received many honours for his outstanding achievements. He was elected to the National Academy of Sciences in 1943, the Brazilian Academy of Sciences in 1952, and the Argentine Academy of Sciences in 1963. He served as chairman of the Mathematics Section of the National Research Council from 1958 to 1961, and President of the American Mathematical Society in 1965-66. *SAU

1977 Stefan Bergman (5 May 1895 in Częstochowa, Russian Empire (now Poland)- 6 June 1977 in Palo Alto, California, USA) Stefan Bergman (5 May 1895 – 6 June 1977) was a Polish-born American mathematician whose primary work was in complex analysis. He is best known for the kernel function he discovered while at Berlin University in 1922. This function is known today as the Bergman kernel. Bergman taught for many years at Stanford University, and served as an advisor to several students.
Bergman received his Ph.D. at Berlin University in 1921 for a dissertation on Fourier analysis. His adviser, Richard von Mises, had a strong influence on him, lasting for the rest of his career. In 1933, Bergman was forced to leave his post at the Berlin University because he was a Jew. He fled first to Russia, where he stayed until 1939, and then to Paris. In 1939, he emigrated to the United States, where he would remain for the rest of life. He was elected a Fellow of the American Academy of Arts and Sciences in 1951. In 1962 he was an invited speaker at the International Congress of Mathematicians in Stockholm (On meromorphic functions of several complex variables). He died in Palo Alto, California, aged 82.
The Stefan Bergman Prize in mathematics was initiated by Bergman's wife in her will, in memory of her husband's work. The American Mathematical Society supports the prize and selects the committee of judges. The prize is awarded for, "the theory of the kernel function and its applications in real and complex analysis; or function-theoretic methods in the theory of partial differential equations of elliptic type with a special attention to Bergman's and related operator methods." *Wik

1985 András P Huhn (Szeged, 26 January 1947 – Szeged, 6 June 1985) was a Hungarian mathematician. Huhn's theorem on the representation of distributive semilattices is named after him. At the height of his creative powers at the age of 38, Huhn was killed in a tragic accident. *Wik

1946 Jean-Louis Loday (12 January 1946 in Le Pouliguen, Pays de la Loire, France
- 6 June 2012 in Les Sables-d'Olonne, France) was a French mathematician who worked on cyclic homology and who introduced Leibniz algebras (sometimes called Loday algebras) and Zinbiel algebras. He occasionally used the pseudonym Guillaume William Zinbiel, formed by reversing the last name of Gottfried Wilhelm Leibniz.
Loday died in a tragic boating accident, falling from his boat off Les Sables-d'Olonne. *Wik *SAU

Saturday, 5 June 2021

On This Day in Math - June 5

Don't worry about people stealing your ideas.

If your ideas are any good,

you'll have to ram them down people's throats.

~Howard Aiken

The 156th day of the year; 156 is the number of graphs with six vertices. *What's So Special About This Number.

\( ( \pi(1)+\pi(5)+\pi(6)) * (p_1 + p_5 + p_6) = 156 \). 156 is the smallest number for which this is true, and the only even number for which it is true. (The symbols \( \pi(n)\) and \(p_n \) represent the number of primes less than or equal to n, and the nth prime respectively)

156 is evenly divisible by 12, the sum of its digits. Numbers which are divisible by the sum of their digits are sometimes called Niven Numbers and often called Harshad (Joy-giver) numbers..

According to an article in the Journal of Recreational Mathematics the origin of the name is as follows. In 1977, Ivan Niven, a famous number theorist presented a talk at a conference in which he mentioned integers which are twice the sum of their digits. Then in an article by Kennedy appearing in 1982, and in honor of Niven, he christened numbers which are divisible by their digital sum “Niven numbers.” One might try to find the smallest strings of consecutive Niven Numbers with more than a single digit. *http://trottermath.net/niven-numbers/

I wonder about the relative order of the classes of numbers which are n times their digit sum for various n.

EVENTS

1661 Newton admitted to Trinity College. He was admitted as a "sizar", which meant he earned part of the cost of his education by doing menial chores. His mother was quite wealthy enough to pay his tuition, but was unsure about his prospects at college since he seemed to be such a poor farmer. Mama and Junior seemed to have an unsteady relationship. He once admitted to his diary in a list of sins, "Threatening my father and mother Smith to burn them and the house over them."

1828 The final meeting of the Board of Longitude in Greenwich. This was the 243rd meeting of the Board since it's creation in 1714. John Barrow, Second Secretary of the Admiralty chaired the meeting. On July 15th the Board was dissolved by Parliament.

1833 Ada Lovelace ﬁrst meets Charles Babbage at the home of Mary Sommerville. She is known to have assisted Charles Babbage in the design of an "analytical engine", an early mechanical computing device. She is often credited with writing the first computer program.
Ada's mother, Lady Byron, had intentionally schooled Ada in the Sciences and Mathematics to counteract the "poetic tendencies" she might have inherited fom her father. Ada knew Mary Somerville and Augustus de Morgan socially and received some math instruction from both. She died of cancer in the womb in November of 1852, only 36 years of age, and was buried beside Lord Byron, the father she never knew, in the parish church of St. Mary Magdalene, Hucknall in the UK.
In 1980, 165th years after Ada's birth, the US Defense Department announced a powerful new computer language. They named it Ada in honour of the Countess of Lovelace's important role in the history of computing. It may be of interest to students of mathematics and computer science that Ada Lovelace husband,also named William, was the Baron of Ockham (ancestor of 14th century William of Occam, for whom Occam’s Razor is named) in the 19th century.

1873 The term “radian” ﬁrst appeared in print. Some suggest it may have been intended as an abbreviation for "RADIus ANgle".
Here is a quote from Cajori's History of Mathematical Notations, vol 2 (1929) as provided by Julio Cabellion to the Historia-Matematica Newsgroup:
"An isolated matter of interest is the origin of the term 'radian', used with trigonometric functions. It first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between 'rad', 'radial' and 'radian'. In 1874, T. Muir adopted 'radian' after a consultation with James Thomson.+" (+) _Nature_, Vol. 83, pp. 156, 217, 459, 460.
The concept of a radian measure, as opposed to the degree of an angle, but not the term, should probably be credited to Roger Cotes, although it appeared as early as around 1400 by the Persian mathematician al-Khashi. According to a recent post to a math history newsgroup by Bob Stein; "He (Cotes) then calculated this as approximately 57.295 degrees. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure."

1929 The US Post Office issued a 2 cent stamp commemorating the Golden Jubilee of Edison's electric Lamp. On Dec 31, 1879 Edison gave the first public demonstration of his new incandescent lamp when he lit up a street in Menlo Park, New Jersey. The Pennsylvania Railroad Company ran special trains to Menlo Park on the day of the demonstration in response to public enthusiasm over the event.
Although the first incandescent lamp had been produced 40 years earlier, no inventor had been able to come up with a practical design until Edison embraced the challenge in the late 1870s. His patent would be approved on January 27, 1880. *.history.com

1943 Contract signed to develop ENIAC with the Moore School at the University of Pennsylvania.

1977, first personal computer, the Apple II, went on sale. They were the invention of Steve Wozniak and Steve Jobs. They have the 6502 microprocessor, ability to do Hi-res and Lo-res color graphics, sound, joystick input, and casette tape I/O. They have a total of eight expansion Slots for adding peripherials. Clock speed is 1MHz and, with Apple's Language Card installed, standard memory size is 64kB. (The Apple I designation referred to an earlier computer that was not much more than a board. You had to supply your own keyboard, monitor and case.) The Apple II was one of three prominent personal computers that came out in 1977. Despite its higher price, it quickly pulled ahead of the TRS-80 and the Commodore Pet. *TIS Model pictured must be after 1979 when the floppy disk drive (1978) and spreadsheet program VisiCalc (1979) made it a blockbuster.

1995 The first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST–JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) [6] (1.7×10−7 K). For their achievements Cornell, Wieman, and Wolfgang Ketterle at MIT received the 2001 Nobel Prize in Physics. This Bose–Einstein condensate was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Interestingly, Bose first letter to Einstein was written on June 4,1924 so the discovery was one day over exactly 71 years later. *Wik

BIRTHS

1819 John Couch Adams (5 June 1819 – 21 January 1892); In 1878 he published his calculation of Euler’s constant (Euler-Mascheronie constant) to 263 decimal places. (he also calculated the Bernoulli numbers up to the 62 nd) *VFR The Euler-Mascheronie constant is the limiting value of the difference between the sum of the first n values in the harmonic series and the natural log of n. (not 263 places, but the approximate value is 0.5772156649015328606065...)
He also predicted the location of the then unkown planet of Neptune, but it seems he failed to convince Airy to search for the planet. Independently, Urbanne LeVerrier predicted its locatin in Germany, and then assisted Galle in the Berlin Observatory in locating the planet on 23 September 1846. As a side note, when he was appointed to a Regius position at St. Andrews in Scotland, he was the last professor ever to have to swear and oath of “abjuration and allegience”, swearing fealty to Queen Victoria, and abjuring the Jacobite succession. The need for the oath was removed by the 1858 Universities Scotland Act. Adams made many other contributions to astronomy, notably his studies of the Leonid meteor shower (1866) where he showed that the orbit of the meteor shower was very similar to that of a comet. He was able to correctly conclude that the meteor shower was associated with the comet.

1883 John Maynard Keynes born. (5 June, 1883–21 April, 1946) a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments. He greatly refined earlier work on the causes of business cycles, and advocated the use of fiscal and monetary measures to mitigate the adverse effects of economic recessions and depressions. His ideas are the basis for the school of thought known as Keynesian economics, as well as its various offshoots. *Wik In one logic class of Whitehead he was the only student. Keynes worked on the foundations of probability.

1888 Gregor Michailowitch Fichtenholz, ( 5 June 1888 in Odessa; 25 June 1959 in Leningrad)who was the founder of the Leningrad school of function theory. *VFR
1900 Dennis Gabor (5 Jun 1900, 8 Feb 1979 at age 78) Hungarian-born British electrical engineer who won the Nobel Prize for Physics in 1971 for his invention of holography, a system of lensless, three-dimensional photography that has many applications. He first conceived the idea of holography in 1947 using conventional filtered-light sources. Because such sources had limitations of either too little light or too diffuse, holography was not commercially feasible until the invention of the laser (1960), which amplifies the intensity of light waves. He also did research on high-speed oscilloscopes, communication theory, physical optics, and television. Gabor held more than 100 patents. *TIS

1904 George McVittie studied at Edinburgh and Cambridge. He then held posts at Leeds, Edinburgh and London and became Professor of Astronomy at the University of Illinois. His main work was in Relativity and Cosmology. *SAU More detail of his life can be found in this obituary.

DEATHS

1716 Roger Cotes (10 July 1682 — 5 June 1716) died at age 33 of a violent fever. Sir Isaac Newton, speaking of Mr. Cotes, said, “If he had lived we might have known something.” See Ronald Gowing’s Roger Cotes, Natural Philosopher, pp. 136 and 142. *VFR
A really nice bio about Cotes is at the Renaissance Mathematicus blog by Thony Christie.

1940 Augustus Edward Hough Love (17 April 1863, Weston-super-Mare – 5 June 1940, Oxford), British geophysicist and mathematician who discovered a major type of earthquake wave that was subsequently named for him. Love assumed that the Earth consists of concentric layers that differ in density and postulated the occurrence of a seismic wave confined to the surface layer (crust) of the Earth which propagated between the crust and underlying mantle. His prediction was confirmed by recordings of the behaviour of waves in the surface layer of the Earth. He proposed a method, based on measurements of Love waves, to measure the thickness of the Earth's crust. In addition to his work on geophysical theory, Love studied elasticity and wrote A Treatise on the Mathematical Theory of Elasticity, 2 vol. (1892-93). *TIS

1965 Tadashi Nakayama or Tadasi Nakayama (July 26, 1912 – June 5, 1964) was a mathematician who made important contributions to representation theory. He received his degrees from Tokyo University and Osaka University and held permanent positions at Osaka University and Nagoya University. He had visiting positions at Princeton University, Illinois University, and Hamburg University. Nakayama's lemma and Nakayama algebras and Nakayama's conjecture are named after him.