## Friday, 22 November 2019

### On This Day in Math - November 22

Ulam Spiral
PrimeSpiral_1000.gif
mathworld.wolfram.com

I believe there are
15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,
914,527,116,709,366,231,025,076,185,631,031,296
protons in the universe,
and the same number of electrons.
— Sir Arthur Stanley Eddington

The 326th day of the year; 326 is the maximum number of pieces that may be produced in a pizza with 25 straight cuts. These are sometimes called "lazy caterer numbers" and more generally they are centered polygonal numbers.

326 is also the sum of the first 14 consecutive odd primes: 326 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47. *MAA

326 prefixed or followed by any digit still remains composite.  *Derek's Daily Math

EVENTS

1850 J J Sylvester called to the Bar. Rather than practicing law he gave private instruction in mathematics, and counted among his pupils Florence Nightingale. [Osiris, 1(1936), 102] *VFR (This idea of Sylvester tutoring Nightingale, to the best of my knowledge, originates from the Herbert Baker obituary. Karen Hunger Parshall, among others, has questioned the accuracy of this statement.)

1906 An International Radiotelegraphic Convention adopted the S.O.S radio distress signal, ... The Convention met in Berlin in 1906. This body signed an international agreement on November 3, 1906, with an effective date of July 1, 1908. An extensive collection of Service Regulations was included to supplement the Convention, and in particular Article XVI adopted Germany's Notzeichen distress signal as the international standard, stating: "Ships in distress shall use the following signal: · · · — — — · · · repeated at brief intervals".
The first well documented use of the SOS distress call is by the Arapahoe on August 11, 1909, when it suffered a broken shaft in the Atlantic Ocean, near Cape Hatteras, North Carolina. However, an article titled "Notable Achievements of Wireless" in the September, 1910 Modern Electrics suggests that an earlier SOS distress call was transmitted by the Cunard liner Slavonia, on June 10, 1909.
[The wireless operator aboard S.S. Arapahoe, T. D. Haubner, radioed for help. A few months later, Haubner on the S.S. Arapahoe received an SOS from the SS Iroquois, the second use of SOS in America.(*TIS)]
The first radio distress call to be adopted appears to have been "CQD", by the Marconi International Marine Communication Company​, for Marconi-operated shipboard stations. It was announced on January 7, 1904 by the company's "Circular 57" that "...on and after the 1st February, 1904, the call to be given by ships in distress or in any way requiring assistance shall be 'C.Q.D.'." ("CQ" was a general call to all stations; amateur or "ham" radio operators still use it today when soliciting a contact with any station that hears the call.) *Citizens Compendium

BIRTHS

1796 Charles Bonnycastle (22 Nov 1796 - 31 Oct, 1840). The University of Virginia's second Professor of Mathematics, Charles Bonnycastle, was born in Woolwich, England. His father, John, was Professor of Mathematics at the Royal Military Academy there, and so Charles grew up and received his education in an environment that very much influenced his own subsequent career. The contributions that the son made to the thirteenth edition of his father's textbook, Introduction to Algebra (1824), in fact, augmented the credentials he presented to Francis Walker Gilmer, agent for the newly forming University of Virginia.
Bonnycastle actually came to the University at its opening in 1825 as the first professor, not of mathematics, but of natural philosophy (as physics was then called). When Thomas Key, the first Professor of Mathematics, resigned to return to his native England, Bonnycastle shifted over to the mathematical chair and remained in that post until his untimely death on 31 October 1840 at the age of only forty-three. "Old Bonny," as he was fondly called by the students, moved away from what was increasingly becoming the antiquated synthetic approach to mathematical pedagogy that had been so typical of Oxbridge mathematical teaching in the eighteenth and early nineteenth centuries and introduced the more avant-garde analytic approach of late eighteenth-century French authors such as Silvestre Lacroix. In 1834, he published his own textbook, Inductive Geometry, in which he aimed to unite the best of the synthetic and the analytic approaches to geometry for the college- and university-level audience. Bonnycastle also contributed works on mathematical and physical topics to the Transactions of the American Philosophical Society, one of the few venues available in early nineteenth-century America for the publication of original work in the sciences.
Bonnycastle apparently also entrusted a number of mathematical papers to his friend, Princeton physics professor and (after 1846) first Secretary of the Smithsonian Institution, Joseph Henry. Shortly before his death in 1878, Henry deposited these in the library at the University of Virginia. They did not survive the infamous Rotunda fire of 1895. *History of the U V Math Dept. He was buried in University of Virginia Cemetery, Charlottesville, Virginia. His gravestone reads:
Sacred to the memory of
Charles Bonnycastle
late Professor of Mathematics
in the University of Virginia
who was born in London
on the 22nd day of November 1796
was made professor in the University in 1825
and continued in this station until his death
on the 31st of October 1840.
For Michigan residents around Kalamazoo, Charles Bonnycastle's brief stay in the area with his brother Humphrey is still marked by Bonniecastle Lake west of the city.

1803 Giusto Bellavitis (22 Nov 1803 in Bassano, Vicenza, Italy - 6 Nov 1880 in Tezze (near Bassano) Italy ) Bellavitis solved various mechanical problems by original methods, among them Hamilton's quaternions. He developed very personal critical observations about the calculus of probabilities and the theory of errors. He also explored physics, especially optics and electrology, and chemistry. As a young man, Bellavitis weighted the problem of a universal scientific language and published a paper on this subject in 1863. He also devoted time to the history of mathematics and, among other things, he vindicated Cataldi by attributing the invention of continued fractions to him. *SAU

1840 Émile Michel Hyacinthe Lemoine (22 Nov 1840 in Quimper, France - 21 Feb 1912 in Paris, France) Lemoine work in mathematics was mainly on geometry. He founded a
new study of properties of a triangle in a paper of 1873 where he studied the point of intersection of the symmedians of a triangle. He had been a founder member of the Association Française pour l'Avancement des Sciences and it was at a meeting of the Association in 1873 in Lyon that he presented his work on the symmedians.
A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A. He proved that the symmedians are concurrent, the point where they meet now being called the Lemoine point. Among other results on symmedians in Lemoine's 1873 paper is the result that the symmedian from the vertex A cuts the side BC of the triangle in the ratio of the squares of the sides AC and AB. He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle. Its centre is at the mid-point of the line joining the Lemoine point to the circumcentre of the triangle. Lemoine gave up active mathematical research in 1895 but continued to support the subject. He had helped to found a mathematical journal, L'intermédiaire des mathématiciens., in 1894 and he became its first editor, a role he held for many years. *sau

1904 Louis-Eugène-Félix Néel (22 Nov 1904; 17 Nov 2000) French physicist, corecipient (with the Swedish astrophysicist Hannes Alfvén) of the Nobel Prize for Physics in 1970 for his pioneering studies of the magnetic properties of solids. His contributions to solid-state physics have found numerous useful applications, particularly in the development of improved computer memory units. About 1930 he suggested that a new form of magnetic behavior might exist - called antiferromagnetism. Above a certain temperature (the Néel temperature) this behaviour stops. Néel pointed out (1947) that materials could also exist showing ferrimagnetism. Néel has also given an explanation of the weak magnetism of certain rocks, making possible the study of the past history of the Earth's magnetic field.*TIS

DEATHS

1784 Paolo Frisi (13 Apr 1728, 22 Nov 1784) Italian mathematician, astronomer, and physicist who is best known for his work in hydraulics (he designed a canal between Milan and Pavia). He was, however, the first to introduce the lightning conductor into Italy. His most significant contributions to science, however, were in the compilation, interpretation, and dissemination of the work of other scientists, such as Galileo Galilei and Sir Isaac Newton. His work on astronomy was based on Newton's theory of gravitation, studying the motion of the earth (De moto diurno terrae). He also studied the physical causes for the shape and the size of the earth using the theory of gravity (Disquisitio mathematica, 1751) and tackled the difficult problem of the motion of the moon. *TIS

1880 James Craig Watson (January 28, 1838 – November 22, 1880) was a Canadian-American astronomer born in the village of Fingal, Ontario Canada. His family relocated to Ann Arbor, Michigan in 1850.
At age 15 he was matriculated at the University of Michigan, where he studied the classical languages. He later was lectured in astronomy by professor Franz Brünnow.
He was the second director of Detroit Observatory (from 1863 to 1879), succeeding Brünnow. He wrote the textbook Theoretical Astronomy in 1868.
He discovered 22 asteroids, beginning with 79 Eurynome in 1863. One of his asteroid discoveries, 139 Juewa was made in Beijing when Watson was there to observe the 1874 transit of Venus. The name Juewa was chosen by Chinese officials (瑞華, or in modern pinyin, ruìhuá). Another was 121 Hermione in 1872, from Ann Arbor, Michigan, and this asteroid was found to have a small asteroid moon in 2002.
He was a strong believer in the existence of the planet Vulcan, a hypothetical planet closer to the Sun than Mercury, which is now known not to exist (however the existence of small Vulcanoid planetoids remains a possibility). He believed he had seen such two such planets during a July 1878 solar eclipse in Wyoming.
He died of peritonitis at the age of only 42. He had amassed a considerable amount of money through non-astronomical business activities. By bequest he established the James Craig Watson Medal, awarded every three years by the National Academy of Sciences for contributions to astronomy.
The asteroid 729 Watsonia is named in his honour, as is the lunar crater Watson. *Wik

1907 Asaph Hall (15 Oct 1829; 22 Nov 1907) American astronomer, discovered and named the two moons of Mars, Phobos and Deimos, and calculated their orbits.Born in Goshen, Conn. and apprenticed as a carpenter at age 16, he had a passion for geometry and algebra. Hall obtained a position at the Harvard Observatory in Cambridge, Mass. in 1857 and became an expert computer of orbits. In August 1862, he joined the staff of the Naval Observatory in Washington, D.C. where he made his discoveries, in mid- Aug 1877, using the Observatory's 26-inch "Great Equatorial" refracting telescope, then the largest of its kind in the world. He stayed there 30 years until 1891. His son, Asaph Hall, Jr., followed him and worked at the Observatory at various times between 1882-1929.*TIS

1944 Sir Arthur Stanley Eddington (28 Dec 1882, 22 Nov 1944) English astrophysicist, and mathematician known for his work on the motion, distribution, evolution and structure of stars. He also interpreted Einstein's general theory of relativity. He was one of the first to suggest (1917) conversion of matter into radiation powered the stars. In 1919, he led a solar eclipse expedition which confirmed the predicted bending of starlight by gravity. He developed an equation for radiation pressure. In 1924, he derived an important mass-luminosity relation. He also studied pulsations in Cepheid variables, and the very high densities of white dwarfs. He sought fundamental relationships between the prinicipal physical constants. Eddington wrote many books for the general reader, including Stars And Atoms  . *TIS  One of my favorite stories about Eddington is this one: Ludwick Silberstein approached Eddington and told him that people believed he was one of only three people in the world who understood general relativity, and that included Einstein. When Eddington didn't respond for a moment he prodded, come on, don't be modest, and Eddington replied, "Oh, no.  It's not that.  I was just trying to figure out who the third was?"  *Mario Livio, Brilliant Blunders

1986 Nikolai Grigor'evich Chudakov (14 Dec 1904 in Lysovsk, Novo-Burassk, Saratov, Russia - 22 Nov 1986 in Saratov, Russia) Chudakov established a number of important results in number theory. He gave an estimate for the bounds of the zeta-function in the critical strip using techniques which had been introduced a few years earlier by Vinogradov. As a consequence of this work he was able to give a substantially improved remainder term in the asymptotic formula for the number of primes less than a fixed number N. Also, by these method, he improved the estimate for the difference between two consecutive primes. In his later work he extended these results to apply to arbitrary arithmetic progressions. In 1947 Chudakov published On Goldbach-Vinogradov's theorem in the Annals of Mathematics. In this paper he proves Vinogradov's theorem that every large odd integer is representable as a sum of three odd primes. *SAU

1996 Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA – November 22, 1996, Water Mill, New York, USA) was an American mathematician. He is best known for his work in lattice theory.During the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane, substantially advanced American teaching and research in abstract algebra. During and after World War II, Birkhoff's interests gravitated towards what he called "engineering" mathematics. Birkhoff's research and consulting work (notably for General Motors) developed computational methods besides numerical linear algebra, notably the representation of smooth curves via cubic splines.
The mathematician George Birkhoff (1884–1944) was his father.*Wik

2007 Andrew Ronald Mitchell (22 June 1921 – 22 November 2007), popularly known as Ron Mitchell, was a British applied mathematician and numerical analyst. He was a professor of mathematics at the University of St Andrews, Dundee, Scotland. He was known for his contribution to the field of numerical analysis of partial differential equations in general and finite difference method and finite element method in particular. Mitchell has authored several influential books on numerical solution of partial differential equations, including "The Finite Element Analysis in Partial Differential Equations" with Richard Wait and "The Finite Difference Method in Partial Differential Equations" with David F. Griffiths.*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, 21 November 2019

### Notes on the History of the Pigeonhole Theorem

This is an update of several posts I wrote as late as 2009, and some additional information acquired since then.

Sometimes problems that seem very hard, can be very easy if they are viewed in the right way, and one of those easy ways to make some hard problems manageable is the Pigeon-Hole Principle. Over the last few weeks seems like lots of problems invovling this idea have shown up, so I thought I would bring it to you.
The basic idea is so easy any sixth grader would agree; if you have two boxes, and you are going to put three balls in the boxes, then at least one box will get more than one ball..... "well, Duh!" they answer... and yet... it seems easier to apply than it might be. Now that you know the secret, try these two problems. I'll post the answer down lower on the page where you must not look until you take a few minutes to ponder the problems.
Here is the first from a recent blog I read: "39 people are attending a large, formal dinner, which must of course occur at a single, circular table. The guests, after milling about for a while, sit down to eat. It is then pointed out to them that there are name cards labeling assigned seats, and not a single one has sat in the seat assigned to them. Prove that there is some way to rotate the table so that at least two people are in the correct seats."
This one seems tougher, but really isn't, it just requires a different way of thinking. "Suppose you pick six unique integers from 1 to 1000. Prove that at least two of them must have a difference that is a multiple of five.

I'll give you the proofs of each of these, and then get to the main topic of the history of this important theorem in discrete mathematics

Ok, The Proofs... for number one... Suppose you handed each person a number that was how many seats they needed to move to the right to find their assigned seat. Since no one is at the right seat, the number can not be zero or thirty-nine. SO each of the people has a number between 1 and 38...wait, there are 39 people...two of them (at least) must be the same distance away from their assigned seats.... admit it…..that’s pretty cool.
For number two it is sort of the same idea, but you have to think about how much each number would have for a remainder if you divided them by five. The only possible choices are 0, 1, 2, 3, or 4... but there are six numbers, so two of them have the same remainder...and two numbers that have the same remainder on division by five, are a multiple of five apart.... think of 1,6, 11, etc for remainders of one. If you want to read more about how remainders can play a part in solving problems, see my blog on "casting out sevens"

The basic idea behind this mathematical principle is what students would call common sense; if there are n objects to be placed in m receptacles (with m less than n), at least two of the items must go into the same container. While the idea is common sense, in the hands of a capable mathematician it can be made to do uncommon things. Here is a link to an article by Alexander Bogomolny in which he uses the principle to argue that there must be at least two persons in New York City with the same number of hairs on their head. This "counting hairs" approach dates back to the earliest version of the principal I have ever seen.

The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. The pigeon seems to be a recent addition, as Jeff Miller's web site on the first use of some math words gives, "Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, A partition calculus in set theory, Bull. Am. Math. Soc. 62 (Sept. 1956)" (although they credit Dedekind for the principle). In a recent discussion on a history group Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet"
that is 'Zasada szufladkowa Dirichleta' ". I received a note that said, "Dirichlet first wrote about it in Recherches sur les formes quadratiques à coefficients et à indéterminées complexes (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580, he uses the principle."

He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

The idea has been around much longer than Dirichlet, however, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story:
The M. Nicole who demonstrated the principal was Pierre Nicole, (1625 -1695), one of the most distinguished of the French Jansenist writers, sometimes compared more favorably than Pascal for his writings on the moral reasoning of the Port Royal Jansenists. It may be that he had picked up the principal from Antoine Arnauld, another Port Royal Jansenist who was an influential mathematician and logician. Here is a segment from his bio at the St. Andrews Math History site.
-------------------------
He published Port-Royal Grammar in 1660 which was strongly influenced by Descartes' Regulae. In Port-Royal Grammar Arnauld argued that mental processes and grammar are virtually the same thing. Since mental processes are carried out by all human beings, he argued for a universal grammar. Modern linguistic theorists consider this work as the beginnings of the modern approach their subject. Arnauld's next work was Port-Royal Logic which was another book of major importance. It was also strongly influenced by Descartes' Regulae and also gave a first hand account of Pascal's Méthode. This work presented a theory of ideas which remained important in philosophy courses until comparatively recent times. In 1667 Arnauld published New Elements of Geometry. This work was based on Euclid's Elements and was intended to give a new approach to teaching geometry rather than new geometrical theorems."
He was a correspondent of Gottfried Wilhelm Leibniz, and of course Pascal, who wrote the Pascal "Provincial Letters" in support of Arnauld. I enjoyed the quote about him from the Wikipedia bio: "His inexhaustible energy is best expressed by his famous reply to Nicole, who complained of feeling tired. 'Tired!' echoed Arnauld, 'when you have all eternity to rest in?"
I have not been able to find any thing in Arnauld's personal writing at this time to confirm that he was aware of or used the Pigeon-hole Principle. I have also seen a comment that there is a book by Henry (or Henrik) van Etten (pseudonym of Jean Leurechon, who coined the term thermometer) , circa 1624, which uses the method for problems involving "if there are more pages than words on any page" and various other illustrations. The writer suggests that the problem is in the French version but not the English translation. Would love to hear from someone who can confirm, and perhaps send a digital image.

Around five years after I wrote the above, I was advised of a paper published by A. Heeffer and B. Rittaud that mentioned this Leurechon (They give the date as 1622) contained a single line about the principle, and amazingly, that involved the idea of proving two men had equal hairs on their heads.  " “It is necessary that two men have the same number of hairs, gold, and others.”
Later the authors add, "It is now established that an immensely popular work published at Pont-`a-Mousson in 1624 resulted from these disputationes. Entitled R´ecr´eation mathematicque, this French work is commonly attributed to Jean Leure-chon, but there are good reasons to believe that this attribution is wrong."  This book goes on to explain the solution from the idea posed in the 1622 book.  They say there is an English translation from 1633, which is [Jean Appier Hanzelet],Mathematicall Recreations , T. Cotes (1633).

### After the fact

Shortly after I wrote the original post, I had a classroom encounter with a student who presented me with another teaching moment.

A young man in one of my classes, obviously trying to improve his A+ by sucking up to the teacher, mentioned that he had read my recent blog on the pigeon-hole principle. He went on to suggest that he really doubted the idea that 39 people could randomly seat themselves and ALL be in the wrong seat. "It just seems VERY unlikely." he suggested.

Rather than tell him the answer, I set him the task of simulating the activity with a deck of cards. Pull out any suit, say the spades, and really shuffle the remaining cards well. Now we need to decide on an order for the remaining suits, so let clubs be the numbers one to thirteen in order from Ace, two, up to King for thirteen. Then the ace of diamonds can be 14, up through the King of diamonds for 26. Finally the ace of hearts is 27 up to the king of hearts for 39. Now turn over the cards and as you do count, one, two, etc... and if you get a card that is where it should be, stop.. they didn't all sit in the wrong chairs. You need not go on forever, just ten or so trials should give you an idea of whether the event is really, really uncommon, or not so very uncommon. (I now realize an easier way to do this would be to have two decks of cards, lay 39 out in one row in order, then from the shuffled deck, lay the cards one at a time under where they should appear.)

I didn't tell him that I knew the probability (or a good approximation), and that he should probably get three or four trials in a string of ten shuffles in which none of the cards landed in the right place. Such a mis-ordering of the cards was just the idea behind the first critical study of the idea we now call derangements by Leonhard Euler, the great Swiss mathematician. Euler was studying the probability of winning in the game of rencontre, now called "coincidences" in his paper "Calcul de la Probabilite dans le jeu de Rencontre", published around 1751.

So what did Euler discover? Well for larger values of N, say 39 or so, the probability of having a perfect mis-sorting of the items approaches 1/e, or about 36.8%, more than a third of the time. It is not an unusual event at all. For smaller numbers you can find the probability by using the idea shown here for six items..
. This can be rewritten more easily using the factorial notation as P= 1/2! - 1/3! + 1/4! - 1/5! + 1/6! which is only a tiny bit above 36.8%, already very close to the 1/e value given above for the limiting value. If the number of items is even, the series will be a little more than 1/e, and if it is odd (and the last term is subtracted) then the probability will be a little below 1/e, with the propbability approaching 1/e as a limit as n gets greater and greater.

My student got two completely mismatched sets of 39, and expressed surprise that it was higher than he would have thought, but he didn't sound convinced that what had occurred was not just an unusual anomaly (or else he thought I might have rigged it somehow?)

I decided to simulate a lot more times than would be practical with a deck of cards, so I cranked up Fathom, a wonderful simulation software by the folks at Key Curriculum, and had it repeat the experiment of seating the 39 people at random 1000 times, and then count how many landed in the right place. The results are shown in the graph below.

It happened that no one landed in the right place 371 times.... Hmmmm, I guess Euler got it right.

### On This Day in Math - November 21

The shortest math joke ever: let $\epsilon < 0$

found on Mathematical humor collected by Andrej and Elena Cherkaev

The 325th day of the year; 325 is the smallest number that can be written as the sum of two squares in three different ways. (What is the next such number?)

325 is last year day that is the sum of the first n^2 integers, $325 = \sum\limits_{i=1}^{5^2} i$

On an infinite chessboard, there are 325 different squares that can be reached in 5 knight moves.

EVENTS

1675 Leibniz completes the product rule. In a manuscript only days earlier Leibniz had struggled with the product and quotient rules for differentiation. At ﬁrst he thought d(uv)= du dv. *F Cajori, History of Mathematics, (pg 208)

1751 “The weather was exceedingly tempestuous, and the sky was overcast with clouds..” so begins An Account of the Eclipse of the Moon, Which Happened Nov. 21, 1751; Observed by Mr. James Short, F. R. S. in Surry-Street *Philosophical Transactions 1751-1752 XLIX

1783 The ﬁrst manned free balloon ﬂight, often credited to the brothers Montgolfier was actually the work of J. A. C. Charles, for whom Charles Law is named. This was a hydrogen filled balloon, and not the hot air type promoted by the Montgolfiers. It carried chemist Jean Pilatre de Rozier and the Marquis d’Arlandes on a ﬂight that wafted across Paris for 25 minutes, reached a height of 500 feet and traveled ﬁve and a half miles. The Montgolﬁer brothers had unmanned launches on June 5 and September 19, 1783. Among the onlookers was Benjamin Franklin, American emissary to the court of Louis XVI. When asked of what use is ballooning, Franklin replied with emphatic simplicity, “Of what use is a newborn baby?” [Air & Space, vol. 1, p. 72 and Williams, p. 43]  Charles and the hydrogen promoters were rivals of the Montgolfiers until Charles' partner,  King Louis XVI had offered to send two prisoners on the flight, but Rozier, a professor of physics and chemistry, wanted to deny criminals the glory of being the first men to go into the atmosphere.  *TIS  Pilatre would become the first aviation casualty the following year when he tried to mix the hot air and hydrogen techniques together to cross the English Channel.

1811 Gauss to Bessel: “One should never forget that the functions, like all mathematical constructions, are only our own constructions.” *VFR

1877 Thomas Edison announced the invention of what he called “The Talking Machine”—the phonograph. *VFR  He appears to have envisioned it as a business dictation machine. In Sep 1877, he wrote that its purpose was "to record automatically the speech of a very rapid speaker upon paper; from which he reproduces the same Speech immediately or years afterwards preserving the characteristics of the speakers voice so that persons familiar with it would at once recognize it." The indented tin foil, however, would survive only a few playings. By the first public showing of a phonograph, which took place in New York City in early Feb 1878, its practical applications had not yet been realized.*TIS

1963 Denmark and Greenland issued almost identical stamps to commemorate the 50th anniversary of the atomic theory of Niels Bohr (1885–1962)*VFR

1969 First ARPANET Link Put Into Service ARPANAT was an early computer network developed by J.C.R. Licklider, Robert Taylor, and other researchers for the U.S. Department of Defense’s Advanced Research Projects Agency​ (ARPA). It connected a computer at UCLA with a computer at the Stanford Research Institute​, Menlo Park, CA. In 1973, the government commissioned Vinton Cerf​ and Robert E. Kahn to create a national computer network for military, governmental, and institutional use. The network used packet-switching, flow-control, and fault-tolerance techniques developed by ARPANET. Historians consider this worldwide network to be the origin of the Internet. *CHM

1973 Mexico issued a stamp portraying an Aztec calendar stone and another with the mathematician and astronomer Carlos de Siguenza y Gongora (1645–1700). *VFR

1983 A special purpose computer built by Lee Sallows generated the following self-documenting pangram (it contains each letter of the alphabet and what it asserts about itself is true): This pangram contains four a’s, one b, two c’s, one d, thirty e’s, six f’s, ﬁve g’s, seven h’s, eleven i’s, one j, one k, two l’s, two m’s, eighteen n’s, ﬁfteen o’s, two p’s, one q, ﬁve r’s, twenty-seven s’s, eighteen t’s, two u’s, seven v’s, eight w’s, two x’s, three y’s and one z. See Scientiﬁc American, October 1984, p. 26. *VFR

BIRTHS

1694  (François Marie Arouet) Voltaire (21 Nov 1694; 30 May 1778) was a French author who popularized Isaac Newton's work in France by arranging a translation of Principia Mathematica to which he added his own commentary (1737). The work of the translation was done by the marquise de Châtelet who was one of his mistresses, but Voltaire's commentary bridged the gap between non-scientists and Newton's ideas at a time in France when the pre-Newtonian views of Descartes were still prevalent. Although a philosopher, Voltaire advocated rational analysis. He died on the eve of the French Revolution. *TIS

1867 Dmitrii Matveevich Sintsov (21 November 1867 – 28 January 1946) was a Russian mathematician known for his work in the theory of conic sections and non-holonomic geometry.
He took a leading role in the development of mathematics at Kharkov University, serving as chairman of the Kharkov Mathematical Society for forty years, from 1906 until his death at the age of 78.*Wik

DEATHS

1652 Jan Brożek (Ioannes Broscius, Joannes Broscius or Johannes Broscius;) (1 November 1585 – 21 November 1652) was a Polish polymath: a mathematician, astronomer, physician, poet, writer, musician and rector of the Kraków Academy.
Brożek was born in Kurzelów, Sandomierz Province, and lived in Kraków, Staszów, and Międzyrzec Podlaski. He studied at the Kraków Academy (now Jagiellonian University) and at the University of Padua. He served as rector of Jagiellonian University.
He was the most prominent Polish mathematician of the 17th century, working on the theory of numbers (particularly perfect numbers) and geometry. He also studied medicine, theology and geodesy. Among the problems he addressed was why bees create hexagonal honeycombs; he demonstrated that this is the most efficient way of using wax and storing honey.
He contributed to a greater knowledge of Nicolaus Copernicus' theories and was his ardent supporter and early prospective biographer. Around 1612 he visited the chapter at Warmia and with the knowledge of Prince-Bishop Simon Rudnicki took from there a number of letters and documents in order to publish them, which he never did. He contributed to a better version of a short biography of Copernicus by Simon Starowolski. "Following his death, his entire collection was lost"; thus "Copernicus' unpublished work probably suffered the greatest damage at the hands of Johannes Broscius."
Brożek died at Bronowice, now a district of Kraków. One of the Jagiellonian University's buildings, the Collegium Broscianum, is named for him. *Wik

1782 Jacques de Vaucanson (24 Feb 1709, 21 Nov 1782) French inventor of automata - robot devices of later significance for modern industry. In 1737-38, he produced  a transverse flute player, a pipe and tabor player, and a mechanical duck, which was especially noteworthy, not only imitating the motions of a live duck, but also the motions of drinking, eating, and "digesting." He made improvements in the mechanization of silk weaving, but his most important invention was ignored for several decades - that of automating the loom by means of perforated cards that guided hooks connected to the warp yarns (later reconstructed and improved by J.-M. Jacquard, it became one of the most important inventions of the Industrial Revolution.) He also invented many machine tools of permanent importance.*TIS

1866 Gustav Roch (9 Dec 1839 in Dresden, Germany, 21 Nov 1866 in Venice, Italy) was a German mathematician known for the Riemann-Roch theorem which relates the genus of a topological surface to algebraic properties of the surface. As presented by Roch, the Riemann-Roch theorem related the topological genus of a Riemann surface to purely algebraic properties of the surface. The Riemann-Roch theorem was so named by Max Noether and Alexander von Brill in a paper they jointly wrote 1874 when they refined the information obtained from the theorem. It was extended to algebraic curves in 1929 and then in the 1950s an n-dimensional version, the Hirzebruch-Riemann-Roch theorem, was proved by Hirzebruch and a version for a morphism between two varieties, the Grothendieck-Riemann-Roch theorem, was proved by Grothendieck.
Over the three academic years 1863-64, 1864-65 and 1865-66 Roch gave a number of courses at Halle. These included: Differential and Integral Calculus; Analytic Geometry; and Elliptic and Abelian Functions. Up to this time Roch was still a privatdozent at Halle but in the spring of 1866 the University began to take up referees' reports with a view to appointing him as an extraordinary professor. Heine wrote a strong letter of support and Roch was appointed extraordinary professor at the University of Halle-Wittenberg on 21 August.
However Roch's health was failing and on 13 October he was granted leave for the winter semester of 1866-67 to allow him to regain his health. Roch went to Venice where he hoped the warmer weather would aid his recovery. Sadly, however, it was not to be and he died of consumption in Venice in November at the age of 26 years. Roch's name will live on through the fundamental Riemann-Roch theorem, but it is a tragedy that the young man with so much mathematical promise died when he had only just commenced his career. *SAU

1970 Sir Chandrasekhara Venkata Raman (7 Nov 1888, 21 Nov 1970)Indian physicist whose work was influential in the growth of science in India. He was the recipient of the 1930 Nobel Prize for Physics for the 1928 discovery now called Raman scattering: a change in frequency observed when light is scattered in a transparent material. When monochromatic or laser light is passed through a transparent gas, liquid, or solid and is observed with the spectroscope, the normal spectral line has associated with it lines of longer and of shorter wavelength, called the Raman spectrum. Such lines, caused by photons losing or gaining energy in elastic collisions with the molecules of the substance, vary with the substance. Thus the Raman effect is applied in spectrographic chemical analysis and in the determination of molecular structure. *TIS

1978 Francesco Giacomo Tricomi studied differential equations which became very important in the theory of supersonic flight. *SAU

1980 László Rédei (Rákoskeresztúr, 15 November, 1900—Budapest, 21 November, 1980) was a Hungarian mathematician.
His mathematical work was in algebraic number theory and abstract algebra, especially group theory. He proved that every finite tournament contains an odd number of Hamiltonian paths. He gave several proofs of the theorem on quadratic reciprocity. He proved important results concerning the invariants of the class groups of quadratic number fields. In several cases, he determined if the ring of integers of the real quadratic field Q(√d) is Euclidean or not. He successfully generalized Hajós's theorem. This led him to the investigations of lacunary polynomials over finite fields, which he eventually published in a book. He introduced a very general notion of skew product of groups, both the Schreier-extension and the Zappa-Szép product are special case of. He explicitly determined those finite noncommutative groups whose all proper subgroups were commutative (1947). This is one of the very early results which eventually led to the classification of all finite simple groups.*Wik

1991 Hans Zassenhaus (28 May 1912 in Koblenz-Moselweiss, Germany - 21 Nov 1991 in Columbus, Ohio, USA) did important work on Group Theory and Lie algebras. His work on computational algebraic number theory seems to have started when he visited Caltec in 1959 and collaborated with Taussky-Todd. He put forward a programme to develop methods for computational number theory which, given an algebraic number field, involved calculating its Galois group, an integral basis, the unit group and the class group. He contributed himself in a major way to all four of these tasks.
Zassenhaus worked on a broad range of topics and, in addition to those mentioned above, he worked on nearfields, the theory of orders, representation theory, the geometry of numbers and the history of mathematics. He loved teaching and wrote several articles on the topic such as On the teaching of algebra at the pre-college level. *SA

1993 Bruno Rossi (13 Apr 1905, 21 Nov 1993)Italian pioneer in the study of cosmic radiation. In the 1930s, his experimental investigations of cosmic rays and their interactions with matter laid the foundation for high energy particle physics. Cosmic rays are atomic particles that enter earth's atmosphere from outer space at speeds approaching that of light, bombarding atmospheric atoms to produce mesons as well as secondary particles possessing some of the original energy. He was one of the first to use rockets to study cosmic rays above the Earth's atmosphere. Finding X-rays from space he became the grandfather of high energy astrophysics, being largely responsible for starting X-ray astronomy, as well as the study of interplanetary plasma.  *TIS

1996 Abdus Salam (29 Jan 1926, 21 Nov 1996) Pakistani nuclear physicist who shared the 1979 Nobel Prize for Physics with Steven Weinberg and Sheldon Lee Glashow. Each had independently formulated a theory explaining the underlying unity of the weak nuclear force and the electromagnetic force. His hypothetical equations, which demonstrated an underlying relationship between the electromagnetic force and the weak nuclear force, postulated that the weak force must be transmitted by hitherto-undiscovered particles known as weak vector  bosons, or W and Z bosons. Weinberg and Glashow reached a similar conclusion using a  different line of reasoning. The existence of the W and Z bosons was eventually verified in 1983  by researchers using particle accelerators at CERN. *TIS

Credits
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*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, 20 November 2019

### On This Day in Math - November 20

The history of astronomy is a history of receding horizons.

— Edwin Powell Hubble

The 324th day of the year; 324 is the largest possible product of positive integers with a sum of 16. (Students, Can you find the integers. Try to find the similar maximum product with a sum of 17)).
324 is also the sum of four consecutive primes, 324 = 73 + 79 + 83 + 89

If you have a square array of 324 dots (that's 18x18) you can carefully paint them each in one of four colors so that no four corners of a rectangle (with sides horizontal and vertical) are the same color. you can also do that for any smaller square, but not for any larger. Here is a 17x17 to ponder

EVENTS

1629 In a letter to Marin Mersenne, Descartes … went on to postulate another kind of language in which ideas would be represented so clearly that errors of judgment would be 'almost impossible'. To realize such a language, all of our thoughts would first have to be given a proper order 'like the natural order of the numbers'; and this presupposes the 'true philosophy', by which the analysis and ordering of thoughts would be carried out. Although Descartes pursues the plan no further, he is optimistic that 'such a language is possible and that the knowledge on which it depends can be discovered'. *Donald Rutherford,

1711 Robert Simson submitted to a simple test of his mathematical knowledge and was duly admitted as professor of mathematics at the University of Glasgow. His most inﬂuential work was a definitive edition of Euclid’s Elements in 1749. *VFR  The pedal line of a triangle is sometimes called the "Simson line" after him, although it does not actually appear in any work of Simson.

1843 Sylvester departs US for England and describes his life as "Pretty much a blank." After resigning from Un of Va. after only four months, J. J. Sylvester lived with a brother in New York City while trying to find work in the US. Finally giving up, her returned to England with no job or prospects for one. *James Joseph Sylvester: Life and Work in Letters
edited by Karen Hunger Parshall

1980, Steve Ptacek in Solar Challenger piloted its first solar-powered flight. The aircraft was designed and built by AeroVironment, Inc. (founded in 1971 by ultra-light airplane innovator, Dr. Paul MacCready). An earlier, 71-ft wingspan, solar-powered design, the Gossamer Penguin, after test flights, flew about 1.95 miles at a public demonstration on 7 Aug 1980. Solar Challenger built upon this experience to be a piloted, solar-powered aircraft strong enough to handle both long and high flights when encountering normal turbulence. With only a 46.5-ft wingspan, it had a huge horizontal stabilizer and had enough wing area for 16,128 solar cells. After design modifications, Ptacek flew across the English Channel flight on 7 July 1981.*VFR

2008 Conficker, also known as Downup, Downadup and Kido, is a computer worm targeting the Microsoft Windows operating system that was first detected on this day in November 2008. It uses flaws in Windows software and dictionary attacks on administrator passwords to propagate while forming a botnet, and has been unusually difficult to counter because of its combined use of many advanced malware techniques. The Conficker infected millions of computers including government, business and home computers in over 200 countries, making it the largest known computer worm infection since the 2003 Welchia. *Wik

BIRTHS

1602 Otto von Guericke (20 Nov 1602; 11 May 1686) German physicist who investigated the properties of a vacuum invented (1654) the first piston air pump to produce a vacuum. While mayor of Madgeburg, in 1663, he demonstrated that two 51 cm diameter copper hemispheres with air pumped out of their interior would be so strongly held together by the force of air pressure that teams of horses harnessed to each hemisphere were not able to pull the hemispheres apart. He studied the role of air in combustion and respiration. With his invention of the first electrostatic machine - a rotating ball of sulphur electrified by friction against his hand - he produced sizeable sparks and showed that like charges repel each other.*TIS

1792 Nikolai Ivanovich Lobachevsky born. (November 20, 1792 – February 12, 1856 (O.S.)) was a Russian mathematician and geometer, renowned primarily for his pioneering works on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Russia did not convert to the Gregorian Calendar until after the communist revolution in 1918. The new style dates were (December 1, 1792 – February 24, 1856 *Wik
And if you've never heard Tom Lehrer's fantastic musical creation about Lobachevsky. He admits the topic has no relation to the man, but the name just fit so well.

1873 William W(eber) Coblentz (20 Nov 1873; 15 Sep 1962) an American physicist and astronomer whose work lay primarily in infrared spectroscopy. In 1905 he founded the radiometry section of the National Bureau of Standards, which he headed for 40 years. Coblentz measured the infrared radiation from stars, planets, and nebulae and was the first to determine accurately the constants of blackbody radiation, thus confirming Planck's law. *TIS

1889 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
The late Bill Buegsen was a resident who was proud of the achievements of Marshfield's native son, so he designed a one-fourth replica of the original Hubble Space Telescope. The Hubble Telescope replica was dedicated on July 4, 1994 and is located on Clay Street, on the west side of the Webster County Courthouse in Marshfield, Mo. It took approximately three months to build, is approximately twelve feet long, ten feet wide and weighs twelve hundred pounds. There is also an Elementary school named for Hubble. The city is on the famous Route 66 just 30 minutes east of Springfield, Mo. *Marshfield Tourist Office web site

1893 André Bloch (20 Nov 1893 in Besançon, France - 11 Oct 1948 in Paris, France) attended the École Polytechnique in 1913 then was drafted in 1914. An accident at the front made him unfit for military service. On 17 Nov 1917, at a family meal, he murdered one of his brothers, his uncle and his aunt. He was confined to a psychiatric hospital (Saint-Maurice Hospital) where he worked on a large range of topics, function theory, geometry, number theory, algebraic equations and kinematics.
Bloch wrote many important papers, corresponding with Hadamard, Mittag-Leffler, Pólya and Henri Cartan (Élie Cartan's son). He was a model patient who refused to go out saying Mathematics is enough for me. Bloch explained the murders to his doctor saying It's a matter of mathematical logic. There had been mental illness in my family. He saw it as his eugenic duty! The Académie awarded him the Becquerel Prize just before his death. *SAU

1917 Leonard Jimmie Savage (20 November 1917 – 1 November 1971) was an American mathematician and statistician. Nobel Prize-winning economist Milton Friedman said Savage was "one of the few people I have met whom I would unhesitatingly call a genius." His most noted work was the 1954 book Foundations of Statistics, in which he put forward a theory of subjective and personal probability and statistics which forms one of the strands underlying Bayesian statistics and has applications to game theory.
During World War II, Savage served as chief "statistical" assistant to John von Neumann, the mathematician credited with building the first electronic computer.
One of Savage's indirect contributions was his discovery of the work of Louis Bachelier on stochastic models for asset prices and the mathematical theory of option pricing. Savage brought the work of Bachelier to the attention of Paul Samuelson. It was from Samuelson's subsequent writing that "random walk" (and subsequently Brownian motion) became fundamental to mathematical finance.
In 1951 he introduced the minimax regret criterion used in decision theory.
The Hewitt–Savag *Wik

1924 Benoit Mandelbrot (20 Nov 1924 in Warsaw, Poland - 14 Oct 2010 in Cambridge, Massachusetts, USA) was largely responsible for the present interest in Fractal Geometry. He showed how Fractals can occur in many different places in both Mathematics and elsewhere in Nature.*SAU

1955 Ray Ozzie, who designed the Lotus Notes office management software for Lotus Development Corporation, is born in Chicago, IL. Ozzie graduated from the University of Illinois at Urbana-Champaign (UIUC) in 1979. During this time Ray worked at the Computer-based Education Research Lab (CERL) on the PLATO operating system. He was impressed with PLATO’s real-time communications and has often publicly credited his CERL experience as the inspiration for Lotus Notes. In 1984 Mitch Kapor, founder of Lotus Development Corporation, supported the idea to develop a PLATO-like product for PC by funding Iris Associates, Inc. In August 1986 Lotus Notes was complete becoming the first example of groupware and a commercial success. In 1997 Ozzie left Iris Associates to start a new venture, Rythmix Corp.*CHM

1963 Sir William Timothy Gowers, FRS (20 November 1963, ) is a British mathematician. He is a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, where he also holds the Rouse Ball chair, and is a Fellow of Trinity College, Cambridge. In 1998 he received the Fields Medal for research connecting the fields of functional analysis and combinatorics.*Wik

DEATHS

1713 Thomas Tompion (baptised 25 Jul 1639, 20 Nov 1713) Most famous English clockmaker of his time, especially known for watchmaking improvements. He worked closely with experimental physicist Robert Hooke, and in 1675, following Hooke's design, Tompion made one of the first English watches regulated by a balance spring. In 1695, with Edward Barlow and William Houghton, he patented the cylinder escapement (a controlling device) that allowed use of a horizontal wheel, enabling Tompion to make the first of the flat and more compact watches.*TIS

1764 Christian Goldbach (18 Mar 1690, 20 Nov 1764)Russian mathematician whose contributions to number theory include Goldbach's conjecture, formulated in a letter to Leonhard Euler dated 7 Jul 1742. Stated in modern terms it proposes that: "Every even natural number greater than 2 is equal to the sum of two prime numbers." It has been checked by computer for vast numbers - up to at least 4 x 1014 - but still remains unproved. Goldbach made another conjecture that every odd number is the sum of three primes, on which Vinogradov made progress in 1937. (It has been checked by computer for vast numbers, but remains unproved.) Goldbach also studied infinite sums, the theory of curves and the theory of equations. *TIS

1856 Farkas Bolyai (9 Feb 1775, 20 Nov 1856) Hungarian mathematician, poet, and dramatist who spent a lifetime trying to prove Euclid's (fifth) postulate that parallel lines do not meet. While studying at the University of Göttingen, he met as a fellow student, the noted German mathematician Carl F. Gauss, with whom he corresponded as a life-long friend. Bolyai taught mathematics, physics and chemistry at Marosvásárhely all his life. He discouraged his son, János Bolyai, from studying the parallel axiom as he had, writing in a letter to him: "For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life." *TIS

1882 Henry Draper (7 Mar 1837, 20 Nov 1882) American physician and amateur astronomer who made the first photograph of the spectrum of a star (Vega), in 1872. He was also the first to photograph a nebula, the Orion Nebula, in 1880. For his photography of the transit of Venus in 1874, Congress ordered a gold medal struck in his honour. His father, John William Draper, in 1840 had made the first photograph of the Moon.*TIS

1934 Willem de Sitter (6 May 1872, 20 Nov 1934) Dutch mathematician, astronomer, and cosmologist who developed theoretical models of the universe based on Albert Einstein's general theory of relativity. He worked extensively on the motions of the satellites of Jupiter, determining their masses and orbits from decades of observations. He redetermined the fundamental constants of astronomy and determined the variation of the rotation of the earth. He also performed statistical studies of the distribution and motions of stars, but today he is best known for his contributions to cosmology. His 1917 solution to Albert Einstein's field equations showed that a near-empty universe would expand. Later, he and Einstein found an expanding universe solution without space curvature.*TIS

1960 Hidehiko Yamabe (山辺 英彦 Yamabe Hidehiko?, August 22, 1923 in Ashiya, Hyōgo, Japan – November 20, 1960 in Evanston, Illinois) was a Japanese mathematician. His most notable work includes the final solution of Hilbert's fifth problem.
After graduating from the University of Tokyo in 1947, Yamabe became an assistant at Osaka University. From 1952 until 1954 he was an assistant at Princeton University, receiving his Ph.D. from Osaka University while at Princeton. He left Princeton in 1954 to become assistant professor at the University of Minnesota. Except for one year as a professor at Osaka University, he stayed in Minnesota until 1960. Yamabe died suddenly of a stroke in November 1960, just months after accepting a full professorship at Northwestern University. *Wik

1986 Arne Carl-August Beurling (February 3, 1905 – November 20, 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937–1954) and later at the Institute for Advanced Study in Princeton, New Jersey.
Beurling worked extensively in harmonic analysis, complex analysis and potential theory. The "Beurling factorization" helped mathematical scientists to understand the Wold decomposition, and inspired further work on the invariant subspaces of linear operators and operator algebras.
In the summer of 1940 he single-handedly deciphered and reverse-engineered an early version of the Siemens and Halske T52 also known as the Geheimfernschreiber (secret teletypewriter) used by Nazi Germany in World War II for sending ciphered messages.[1] The T52 was one of the so-called "Fish cyphers", that using, transposition, created nearly one quintillion (893 622 318 929 520 960) different variations. It took Beurling two weeks to solve the problem using pen and paper. Using Beurling's work, a device was created that enabled Sweden to decipher German teleprinter traffic passing through Sweden from Norway on a cable. In this way, Swedish authorities knew about Operation Barbarossa before it occurred. Not wanting to reveal how this knowledge was attained the Swedish warning was not treated as credible by Soviets. *Wik

Credits
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*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