Monday, 9 December 2019

On This Day in Math - December 9

Gravestone of Poincar'e

One geometry cannot be more true than another; it can only be more convenient. 
~Henri Poincar´e

The 343rd day of the year, a Friedman number (named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida *Wik), since it can be made up of arithmetical operations of its digits, (3+4)3 = 343.  There will be one more Friedman number this year; can you find it?

Interestingly, the speed of sound in dry air at 20 °C (68 °F) is 343 m/s.

343 is the smallest cube ending in 3.

Benjamin Vitale@BenVitale noticed that \(343 (10^n+1)^2) \) is a palindrome for n greater than one, you get a palindrome... and if you divide or multiply the result by 7, you get a perfect square. Some simple algebra will allow any HS student to confirm.


1675  Newton writes to Leibniz to comment on the response to his theory that light was corpuscular, "I was so persecuted with discussions arising from the publication of my theory of light, that I blamed my own imprudence for parting with so substantial a blessing as my quiet, to run after a shadow.” *A history of physics in its elementary branches By Florian Cajori

1741 Euler sets out on the trail of his most beautiful theorem. Euler to Goldbach: Berlin Dec 9, 1741 ([1], p. 91) “I have lately also found a remarkable paradox. Namely that the value of the expression (2i+2-i)/2 {Euler wrote sqrt of -1 instead of imaginary constant} is approximately equal to 10/13 and that this fraction differs only in parts per million from the truth. The true value of this expression however is the cosine of the arc .6931471805599 (ln(2)) or the arc of 39 degrees 42 min. 51 sec. 52 tenths of sec. and 9 hundredths of sec. in a circle of radius one. “ (from An English translation of portions of seven correspondences between Euler and Goldbach on Euler’s complex exponential paradox and special values of cosine by Elizabeth Volz)

1906 New York Times Headlines, "Life on Mars"!

1911 Henri Poincar´e wrote the editor of a mathematical journal if, contrary to custom, an unfinished piece of work could be published. He explained that at his age he may not be able to finish it, but that his work might provide ideas for another. The paper was published and not long after this “unfinished symphony,” George David Birkhoff (1884–1944) completed the solution. Poincar´e died suddenly on 17 July 1912. [Eves, Squared, 173◦; Bell, Men of Mathematics,
p. 553]. *VFR

1960 Sperry Rand Corporation of St. Paul, Minnesota, announced the first electronic computer to employ thin-film memory, the UNIVAC 1107. Its operational speed was measured in billionths
of a second (nanoseconds), compared to speeds in most other computers of millionths of a second (microseconds). Memory could be accessed more than a million times a second.*VFR

1965 The Kecksburg UFO incident occurred on December 9, 1965, at Kecksburg, Pennsylvania, USA. A large, brilliant fireball was seen by thousands in at least six U.S. states and Ontario, Canada. It streaked over the Detroit, Michigan/Windsor, Ontario area, reportedly dropped hot metal debris over Michigan and northern Ohio, starting some grass fires and caused sonic booms in Western Pennsylvania. It was generally assumed and reported by the press to be a meteor after authorities discounted other proposed explanations such as a plane crash, errant missile test, or reentering satellite debris.
However, eyewitnesses in the small village of Kecksburg, about 30 miles southeast of Pittsburgh, claimed something crashed in the woods. A boy said he saw the object land; his mother saw a wisp of blue smoke arising from the woods and alerted authorities. Another reported feeling a vibration and "a thump" about the time the object reportedly landed. Others from Kecksburg, including local volunteer fire department members, reported finding an object in the shape of an acorn and about as large as a Volkswagen Beetle. Writing resembling Egyptian hieroglyphics was also said to be in a band around the base of the object. Witnesses further reported that intense military presence, most notably the United States Army, secured the area, ordered civilians out, and then removed the object on a flatbed truck. At the time, however, the military claimed they searched the woods and found "absolutely nothing.
A model of the crashed object, originally created for the show Unsolved Mysteries, and put on display near the Kecksburg fire station.*Wik

In 1968, the first demonstration of the use of a computer mouse was given at the American Federation of Information Processing Societies' Fall Joint Computer Conference at Stanford University, California. The mouse's inventor, Doug Engelbart and a small team of researchers from the Stanford Research Institute stunned the computing world with an extraordinary demonstration at a San Francisco computer conference. They debuted the computer mouse, graphical user interface, display editing and integrated text and graphics, hyper-documents, and two-way video-conferencing with shared workspaces. These concepts and technologies were to become the cornerstones of modern interactive computing.*TIS Patent was over a year later


1667 William Whiston (born 9 Dec 1667, 22 Aug 1752)English Anglican priest and mathematician who sought to harmonize religion and science, and who is remembered for reviving in England the heretical views of Arianism. He attended Newton's lectures while at Cambridge and showed great promise in mathematics. Ordained in 1693. While chaplain to the bishop of Norwich (1694-98), he wrote A New Theory of the Earth (1696), in which he claimed that the biblical stories of the creation, flood and final conflagration could be explained scientifically as descriptions of events with historical bases. The Flood, he believed, was caused by a comet passing close to the Earth on 28 Nov 2349 BC. This put stress on the Earth's crust, causing it to crack and allow the water to escape and flood the Earth. After serving as vicar of Lowestoft (1698–1701), he returned to his alma mater, Cambridge University to become assistant to the mathematician Sir Isaac Newton, whom he succeeded in the Lucasian chair in 1703. *TIS ( His translation of the works of Flavius Josephus may have contained a version of the famous Josephus Problem, and in 1702 Whiston's Euclid discusses the classic problem of the Rope Round the Earth, (if one foot of additional length is added, how high will the rope be). I am not sure of the dimensions in Whiston's problem, and would welcome input, I have searched the book and can not find the problem in it, but David Singmaster has said it is there, and he is not a easy source to reject. It is said that Ludwig Wittgenstein was fascinated by the problem and used to pose it to students regularly. )
In 1701, Newton arranged for Whiston to succeed him as Lucasian professor. In 1710 he was deprived of the chair and driven from Cambridge for his unorthodox religious views (it is not acceptable to be a unitarian at the College of the Whole and Undivided Trinity).*VFR
Whiston was expelled from his chair on 30 October 1710; at the appeal of the heads of colleges. Comets were also part of this disaster in his life. He had become famous for his studies that stated that the Biblical flood had been caused by a comet, and gave support for other geological impacts of comets on the Earth. Whiston was removed from his position at Cambridge, and denied membership in the Royal Society for his “heretical” views. He took the “wrong” side in the battle between Arianism and the Trinitarian view, but his brilliance still made the public attend to his proclamations. When he predicted the end of the world by a collision with a comet in October 16th of 1736 the Archbishop of Canterbury had to issue a denial to calm the panic.

1837 Nils Dalén (30 Nov 1869; 9 Dec 1837)Swedish engineer who won the Nobel Prize for Physics in 1912 for his invention of the automatic sun valve, or Solventil, which regulates a gaslight source by the action of sunlight, turning it off at dawn and on at dusk or at other periods of darkness. It rapidly came into worldwide use for buoys and unmanned lighthouses. While recovering from an accident, convalescing at home, he noticed how much time his wife spent caring for their wood-burning stove. He decided to invent a more efficient and cost-effective stove. In 1922, Dalen's Amalgamated Gas Accumulator Co. patented his design and put the first AGA stoves into production. These stoves produced a radiant heat that kept the kitchen warm. The AGA remains popular today.*TIS (My wife's favorite entry. Her first experience with an AGA was to turn materials for a pie into pure carbonized dust.)

1839 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. Sadly, however, he died of consumption in Venice in November at the age of 26 years. *SAU

1883 Nikolai Nikolaevich Luzin, (also spelled Lusin) (9 December 1883, Irkutsk – 28 January 1950, Moscow), was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics.*Wik

1900 (Noël) Joseph (Terence Montgomery) Needham (9 Dec 1900, 24 Mar 1995 at age 94)was an English biochemist, embryologist, and historian of science who wrote and edited the landmark history Science and Civilization in China, a remarkable multivolume study of nearly every branch of Chinese medicine, science, and technology over some 25 centuries. As head of the British Scientific Mission in China (1942-46) he worked to assure adequate liaison between Chinese scientists and technologists and their colleagues in the West. As an historian of science and technology he wanted to break through the parochial, Europe-centred views of most of his colleagues by disclosing the achievements of traditional China and the contributions made by China leading up to the scientific revolution. *TIS

1906 Grace Murray Hopper (9 Dec 1906; 1 Jan 1992), one of the first women to work on the computer, is born in New York City. Hopper, a rear admiral in U.S. Navy, did significant work on the Harvard Mark II, where she discovered the first computer bug -- a moth -- and coined the term to mean a problem with a program. Hopper went on to develop the first compiler, A-0, and the programming language COBOL. Grace Hopper was honored by having the most modern ship in the U.S. Navy named after her, the U.S.S. Hopper, launched in mid-1997. *CHM
Her ideas contributed to the first commercial electronic computer, Univac I, and naval applications for COBOL (co-mmon b-usiness o- riented l-anguage). With a Ph.D. in Mathematics from Yale University (1934), she taught mathematics (Vassar, 1931-43), before she joined the Naval Reserve. In 1944, she was commissioned as a Lieutenant (Junior Grade) 1944, assigned to the Bureau of Ordnance where she became involved in the early development of the electronic computer. For more than four decades, she was a leader in computer applications and programming languages. *TIS (See Sep 9, 1945 for more on "BUG")

1907 Max Deuring (9 December 1907, Göttingen, Germany – 20 December 1984, Göttingen, Germany) was a mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p. He worked also in analytic number theory.
Deuring graduated from the University of Göttingen in 1930, then began working with Emmy Noether, who noted his mathematical acumen even as an undergraduate. When she was forced to leave Germany in 1933, she urged that the university offer her position to Deuring. In 1935 he published a report entitled Algebren ("Algebras"), which established his notability in the world of mathematics. He went on to serve as Ordinarius at Marburg and Hamburg, then took a position as ordentlicher Lehrstuhl at Göttingen, where he remained until his retirement.*Wik

1916 Irving John Jack Good, Cryptologist and Statistician, is born in London, England. He obtained a PhD in mathematics from Cambridge under the supervision of G. H. Hardy in 1938. During W.W.II he worked on both the Enigma and Teleprinter encrypting machines with Alan Turing at Bletchley *CHM

1917 Leo James Rainwater (9 Dec 1917; 31 May 1986) was an American physicist who won a share of the Nobel Prize for Physics in 1975 for his part in determining the asymmetrical shapes of certain atomic nuclei. During WW II, Rainwater worked on the Manhattan Project to develop the atomic bomb. In 1949 he began formulating a theory that not all atomic nuclei are spherical, as was then generally believed. The theory was tested experimentally and confirmed by Danish physicists Aage N. Bohr and Ben R. Mottelson. For their work the three scientists were awarded jointly the 1975 Nobel Prize for Physics. He also conducted valuable research on X rays and took part in Atomic Energy Commission and naval research projects. *TIS

1917 Sergei Vasilyevich Fomin (9 December 1917 – 17 August 1975) was a Soviet mathematician who was co-author with Kolmogorov of Introductory real analysis, and co-author with I.M. Gelfand of Calculus of Variations (1963), both books that are widely read in Russian and in English.
Fomin entered Moscow State University at the age of 16. His first paper was published at 19 on infinite abelian groups. After his graduation he worked with Kolmogorov. He was drafted during World War II, after which he returned to Moscow. When the war ended Fomin returned to Moscow University and joined Tikhonov's department. In 1951 he was awarded his habilitation for a dissertation on dynamical systems with invariant measure. Two years later he was appointed a professor. Later in life, he became involved with mathematical aspects of biology. *Wik

1926 Henry Way Kendall (9 Dec 1926; 15 Feb 1999)American nuclear physicist who shared the 1990 Nobel Prize for Physics with Jerome Isaac Friedman and Richard E. Taylor for obtaining experimental evidence for the existence of the subatomic particles known as quarks. To study the internal structure of the proton, they worked with the 3-km linear accelerator recently opened at Stanford (SLAC). Electrons were accelerated to an energy of 20,000 million electronvolts and directed against a target of liquid hydrogen. In 1969 Kendall helped found the Union of Concerned Scientists. In 1997, in connection with the Kyoto Climate Summit, he helped produce a statement signed by 2,000 scientists calling for action on global warming.*TIS


1866 James P. Pierpont (June 16, 1866 – December 9, 1938) was a Connecticut-born American mathematician. He did undergraduate studies at Worcester Polytechnic Institute, initially in mechanical engineering, but turned to mathematics. He went to Europe after graduating in 1886. He studied in Berlin, and later in Vienna. He prepared his PhD at the University of Vienna under Leopold Gegenbauer and Gustav Ritter von Escherich. His thesis, defended in 1894, is entitled Zur Geschichte der Gleichung fünften Grades bis zum Jahre 1858. After his defense, he returned to New Haven and was appointed as a lecturer at Yale University, where he spent most of his career. In 1898, he became professor. Initially, his research dealt with Galois theory of equations. After 1900, he worked in real and complex analysis.
In his textbooks of real analysis, he introduced a definition of the integral analogous to Lebesgue integration. His definition was later criticized by Maurice Fréchet. Finally, in the 1920s, his interest turned to non-Euclidean geometry. *Wik

1958 John Jackson (11 Feb 1887 in Paisley, Renfrewshire, Scotland - 9 Dec 1958 in London, England) graduated from Glasgow and Cambridge. He went to the Royal Observatory at Greenwich but his career there was interrupted by World War I. He was then appointed HM Astronomer at the University of Cape Town. *SAU

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, 8 December 2019

All That Glitters is not Golden

Over many years of teaching, I realized that most students, and many teachers had extensive misunderstandings about the "Golden Mean" and it's history.

I want to try to dispel, and expand, on some of these common misunderstandings.  For example, many think that the "Golden Mean" was known to the early Greeks (it was) by that name (it wasn't).   The idea that Euclid labeled the idea, which was found in geometric constructions such as the pentagon (and pentagram), as "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The "extreme and mean ratio" is still frequently used to describe the idea.

Others believe it did not exist until Fibonacci created the Fibonacci numbers from which it was derived as a limit of the ratio of consecutive terms. First, Fibonacci did not create the, so called, Fibonacci sequence. It was known to the Indian Mathematicians as early as Pingalia before 200 BC. Fibonacci's Liber Abaci (1202) included both the means and extreme property, and the famous sequence, but it seems he never realized that the ratio of consecutive terms of the sequence would approach the well known ratio. Luca Pacioli gave the name "Divine Proportion" to his 1509 book about the ratio, illustrated by Leonardo da Vinci. Leonardo first used golden for the ratio by using the latin "secto aurea" (golden section)  The first use in English did not occur until mathematician James Sulley used it in 1875, according to Alfred Posamentior.  And it was "mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;this was rediscovered by Johannes Kepler in 1608." *Wik

Perhaps the most common misconception of students is that the Golden ratio is some how a one-off.  A very special number with nothing else like it.  In truth, it is part of a class of numbers known as Pisot Numbers.  In fact, it is one of an infinite set of  numbers that are solutions to a quadratic equation, sharing many of the "special" qualities of the golden ratio.

The golden ratio is the smallest of these, but others include the "silver ratio" \( 1 + \sqrt{2} \) , and the "bronze" ratio, \( \frac{3+\sqrt(5)}{2} \).  The three numbers are roots  of the quadratic terms \(x^2 - x -1\), \(x^2-2x-1\) and \(x^2 - 3x - 1\). (There is a pattern here, it WILL be back)

I will try to point out how some of those "special" qualities of the golden mean are shared by these other metalic means.

One of the things that impress students is the continued fraction for the golden mean repeats the same number over and over:
\( \phi = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}\)

but very few realize that there is a very similar expansion for the "silver mean",

\( \phi_2 = 2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dots}}}\)

and I bet most of them can figure out how to write the bronze (or third metallic) mean, and all the ones that come after it.

How about looking at a different way to write the positive roots of each of the metallic means I gave above.

The Goilden mean is \( \frac{1+\sqrt{4+(1^2)}}{2}\) ; The Silver mean is \( \frac{2+\sqrt{4+(2^2)}}{2}\); and the third metallic mean is \( \frac{3+\sqrt{4+(3^2)}}{2}\)... the ones following are equally evident.

Students confusion with history is somewhat confused by the fact that they are often introduced to the Golden mean in association with it's relationship with the Fibonacci sequence; 1, 1, 2, 3, 5, 8, ... and the ratio of consecutive terms approaches the Golden mean.
For the Silver mean, there is another well known(but not often to high school students) sequence that is mistakenly attributed to English mathematician John Pell. The sequences is 0, 1, 2, 5, 12, 29,... and bright students can quickly guess the Fibonacci-type recursive formula for this, and will probably anticipate that the sequence for the third metallic ratio would be 0, 1, 3, 10, 33,... and probably all the metallic ratios after that.

In Geometry students are familiar with the fact that the Golden mean can be found in the pentagon, between a diagonal and a side, or between the two sections of the intersection of diagonals.

The Silver mean is found in the ratio between a side and the second shortest diagonal

Unfortunately, that's where the sequence ends.  There are no regular polygons with ratios of sides and diagonals that are in the ratio of any other metallic mean.  As I will point out later, there are non-quadratic numbers that are Pisot numbers (or cubes and higher order) that I have not checked.

Some lesser known facts about the Metallic Means is that there powers approach "almost-integers" as higher (and not so much higher for many) powers.  For example \( \phi^7 \approx 29.03444\) and \( \phi^13 \approx 529.0019\)  as you might expect from experience, odd powers overshoot the mark a smidge, evens undershoot.  The error diminishes logarithmetically.

IF we go to the other metallic ratios, they demonstrate the same behavior more quickly.  For example the silver mean  \( \phi_2^7 \approx 478.00209\) and \( \phi_2^13 \approx 94642.000010\)  .

And the third metallic mean gives ( \( \phi_3^7 \approx 4287.00023) \)

Another interesting, and not well known fact about the Fibonacci sequence is that the digits Mod (n) have a repeat period.  For the Fibonacci period, they repeat their last digit, (mod (10) ) in a 60 digit cycle.
011235831459437 077415617853819 099875279651673 033695493257291
It turns out that this is true of all the metallic sequences, but it may be easier to spot in the shorter binary cycles.  The Fibonacci digits mod(3) cycle 0,1,1 repeatedly, (Even, Odd, Odd).   For the Pell sequence, the cycle is 0,1; and these two sequences alternate between the odd and even metal ratios.

 But base three is not too hard, so let's look at that cycle of remainders on division by three:  
For the Fibonacci sequence the cycle is 0, 1, 1, 2, 0, 2, 2, 1.
For the Pell Sequene the cycle is this cycle sort of the reverse of this, 0,1,2,2,0,2,1,1.  
And the third metallic sequence, cycles 0,1, similar to the binomial cycle.... (can you figure out why there is never a remainder of 2 when a bronze sequence is divided by 3?)  

Other Pisot Numbers, including a Super-Golden Number, 

Wikopedia gives this description of the Pisot Numbers: 
In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938.

There are other sets that are roots of cubic (and higher order) equations, and the smallest possible Pisot Number is called the "Plastic Number" and is the real root of \(x^3-x-1\),
or approximately 1.324717957...

Just as the golden mean has it's value as the limit of the ratio of consecutive terms of the Fibonacci sequence, the plastic number can be derived from the Padovan sequence, first developed around 1994(?), P(0)=P(1)=P(2)=1; and P(n)=P(n-2) + P(n-1)  which begins, 1,1,1,2,2,3,4,5,7,9,12,16...

This sequence has a longer mod(2) cycle.  Like all Pisot numbers, they approach almost-integers, but they do so much slower than the powers of the Golden Mean.

There is even a Super-Golden Ratio which is the real root of \( x^3 -x^2 - 1\ )
or approximately 1.4655712318...

It has its related sequence also, Naryana's cows, which dates back to the 14th Century. Unlike Fibonacci's rabbits, the Cows go through three stages, immature, adolescent, and then mature, so only the matures reproduce. The pattern looks like The first few terms of the sequence are as follows: 1, 1, 1, 2, 3, 4, 6, 9, 13, 19,... . (students could have fun creating four or five stage maturation sequences and look for the limit of their ratios as a limit, and compare the qualities to those from these ratios.  )

As always, comments (and corrections) are welcomed.

The sequence on the far right is a variation of the Padovan Sequence which begins with

On This Day in Math - December 8

The Boole Window, Lincoln Cathedral.

My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.
~Hermann Weyl

The 342nd day of the year; is an Oblong (or promic, pronic, or heteromecic) number, the product of two consecutive integers. There will be no more of them this year. (Promic numbers are related to the infinite nested iteration of roots, as I discovered in this investigation.)

342 is also the sum of three positive cubes.

OH MY! 342 in base 10 is = 666 in base 7

1128 “In the third year of Lothar, emperor of the Romans, in the twenty-eighth year of King Henry of the English…on Saturday, 8 December, there appeared from the morning right up to the evening two black spheres against the sun.” This description of sunspots, and the earliest known drawing of sunspots, appears in John of Worcester’s Chronicle recorded in 1128.
On the night of 13 December 1128, astronomers in Songdo, Korea, witnessed a red vapour that “soared and filled the sky” from the northwest to the southwest. A delay of five days is the average delay between the occurrence of a large sunspot group near the center of the Sun – exactly as witnessed by John of Worcester – and the appearance of the aurora borealis in the night sky at relatively low latitudes *Joe Hanson,
(It seems amazing that this is the same date as Harriott's first recorded observations of sunspots.

1610 Thomas Harriott makes first recorded observations of sunspots using a telescope: ""Decemb. 8 mane ho. That altitude of the sonne being 7 or 8 degrees. It being a frost & a mist. I saw the sonne in this manner. Instrument 10/1 B. I saw it twise of thrise. once with the right ey & other time with the left. In the space of a minutes time. after the sonne was too cleare". Harriot left nearly 200 drawings of sunspots from the period 1610-1612. Interestingly, he does not mention the spots explicitly, even though they are clearly indicated on the drawing. Like the Fabricius father and son team but unlike Galileo and Scheiner, Harriot observed the sun directly through his telescope. His observations were consequently limited to the hour following sunrise, when, as seen from Harriot's residence in Syon, the Sun was greatly dimmed by mist and fog over the river Thames. * paulchar at

1864 the Clifton Suspension Bridge spanning the Avon Gorge and the River Avon, designed by Isambard Kingdom Brunel, was openend for the public. Although Brunel was not able to see the bridge in operation anymore during his lifetime, the Clifton Suspension bridge was the first major commission of the famous engineer of the Great Western Railroad and the then largest steamships in the world. *yovisto

1931 “Prof. Noether’s lectures (she is a woman—a member of a noted mathematical family) are also excellent ... Prof. Noether thinks fast and talks faster. As one listens, one must also think fast—and that is always excellent training. Furthermore, thinking fast is one of the joys of mathematics.” Saunders MacLane, then a student at Gottingen, to his mother. See Emmy Noether, A Tribute to Her Life and Work, ed. by James W. Brewer and Martha K. Smith.

1947 The Eckert-Mauchly Computer Corporation is incorporated. After a dispute with the administration at the University of Pennsylvania over ENIAC patent rights, J. Presper Eckert and John Mauchly started the company that, after several mergers, produced the Binac for Northrop Aircraft and the Univac. Grace Murray Hopper joined Eckert-Mauchly Computer Corp as a senior mathematician in 1949. In 1950, before completing the UNIVAC, the company became a division of Remington Rand, IBM’s main challenger throughout the 1970s. *CHM

1948 Prudential Insurance signs a contract to buy a UNIVAC I. *VFR


1508 Gemma Frisius (Dec 1508 in Dokkum, Friesland, The Netherlands
- 25 May 1555 in Louvain, Brabant (now Belgium)) He applied his mathematical expertise to geography, astronomy and map making. He became the leading theoretical mathematician in the Low Countries. From 1534 Gemma Frisius began to teach his student Gerardus Mercator and over the following years he cooperated with Gaspard Van der Heyden and Gerardus Mercator. They constructed a terrestrial globe in 1536, and they constructed a celestial globe in the following year.*SAU His real name was Jemme Reinerszoon, which means Jemme son of Reiner. As an author he adopted the humanist name of Gemma Frisius. Gemma is a pseudo Latinised form of Jemme and Frisius is a toponym for Friesland where he was born. *Thony Christie

1594 Pierre Petit (8 Dec 1594 in Montluçon, France - 20 Aug 1677 in Lagny-sur-Marne, France) was a French scientist who had a strong influence on the French government. He was one of Mersenne's collaborators. Among many collaborations, Petit worked with Etienne Pascal and his son Blaise Pascal in October 1646 in repeating Torricelli's experiment on the barometric vacuum. *SAU

1795 Peter Andreas Hansen (8 Dec 1795; 28 Mar 1874) Danish astronomer whose most important work was the improvement of the theories and tables of the orbits of the principal bodies in the solar system. At Altona observatory he assisted in measuring the arc of meridian (1821). He became the director (1825) of Seeberg observatory, which was removed to Gotha in a new observatory built for him (1857). He worked on theoretical geodesy, optics, and the theory of probability. The work in celestial mechanics for which he is best known are his theories of motion for comets, minor planets, moon and his lunar tables (1857) which were in use until 1923. He published his lunar theory in Fundamenta ("Foundation") in 1838, and Darlegung ("Explanation") in 1862-64.*TIS

1865 Jacques-Salomon Hadamard (8 Dec 1865; 17 Oct 1963) French mathematician who proved the prime-number theorem (as n approaches infinity, the limit of the ratio of (n) and n/ln n is 1, where (n) is the number of positive prime numbers not greater than n). Conjectured in the 18th century, this theorem was not proved until 1896, when Hadamard and also Charles de la Vallée Poussin, used complex analysis. Hadamard's work includes the theory of integral functions and singularities of functions represented by Taylor series. His work on the partial differential equations of mathematical physics is important. He introduced the concept of a well-posed initial value and boundary value problem. In considering boundary value problems he introduced a generalisation of Green's functions (1932).*TIS (students, I think, find it easier to understand this theorem in the form, as n gets very large
\pi(x)\sim\frac{x}{\ln x}.\!

1883 Ludwig Berwald (8 Dec 1883 in Prague, Bohemia (now Czech Republic) - 20 April 1942 in Łódź, Poland)was a Czech mathematician who made important contributions to differential geometry. He wrote 54 papers up to the time of his deportation. A portion of his work set up the basic theory of Finsler geometry and Spray geometry (i.e., differential geometry of path spaces). Many people working in Finsler geometry consider that Ludwig Berwald is the founder of Finsler geometry. Berwald and E Cartan developed a general theory of two-dimensional Finsler spaces. Berwald wrote a series of major papers On Finsler and Cartan geometries.*SAU

1919 Julia B Robinson (8 Dec 1919 in St Louis, Missouri, USA - 30 July 1985 in USA)worked on computability, decision problems and non-standard models of arithmetic. " What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved. "
A year after Robinson's death, her husband set up the Julia B Robinson Fellowship Fund to provide fellowships for graduate students in mathematics at Berkeley. When Raphael Robinson died in January 1995 almost all his estate went into the Fellowship Fund. *SAU


1632 Albert Girard (1595 in St Mihiel, France - 8 Dec 1632 in Leiden, Netherlands). He introduced the abbreviations sin, tan, and sec for the trigonometric functions. *VFR
With the aid of trigonometric tables Girard solved equations of the third degree having three real roots. For those having only one root he indicated, beside Cardano's rules, an elegant method of numerical solution by means of trigonometric tables and iteration.
He was the first to give a geometric interpretation of negative quantities, writing, "The negative solution is explained in geometry by moving backward, and the minus sign moves back when the + advances." Girard is also famed for being the first to formulate the (now well-known) inductive definition fn+2 = fn+1 + fn for the Fibonacci sequence, and stating that the ratios of terms of the Fibonacci sequence tend to the golden ratio, which appear in this 1634 publication. *SAU
Glen Van Brummelen states in Heavenly Mathematics that Girard was also the first to use the \(\sqrt[3]{x} \)symbol for cube root in his 1629 Invention nouvelle.. It was not adopted by others until Michel Rolle used it in 1690 in Traité d'Algèbre.*Cajori

1632 Philippe van Lansberge (25 Aug 1561 in Ghent, Netherlands (now Belgium)
Died: 8 Dec 1632 in Middelburg, Netherlands) was a Flemish clergyman who wrote on mathematics and astronomy. He calculated π to 28 places by a new method.*SAU

1864 George Boole (2 Nov 1815, 8 Dec 1864)English mathematician who helped establish modern symbolic logic and an algebra of logic, now called Boolean algebra. By replacing logical operations by symbols, Boole showed that the operations could be manipulated to give logically consistent results. Boole's logical algebra is essentially an algebra of classes, being based on such concepts as complement and union of classes. The study of mathematical or symbolic logic developed mainly from his ideas, and is basic to the design of digital computer circuits. Boolean also algebras find important applications in such diverse fields as topology, measure theory, probability and statistics. Boole also wrote important works on differential equations and other branches of mathematics. *TIS George Boole died from a feverish cold, or perhaps pneumonia, he got after walking two miles from his home in Queen’s College at Cork, Ireland, in a drenching rain to teach his class. [Eves, Circles, 289◦] *VFR

1894 Pafnuty Lvovich Chebyshev (4 May 1821, 8 Dec 1894) Russian mathematician who founded the St. Petersburg mathematical school (sometimes called the Chebyshev school), who is remembered primarily for his work on the theory of prime numbers, including the determination of the number of primes not exceeding a given number. He wrote about many subjects, including the theory of congruences in 1849, probability theory, quadratic forms, orthogonal functions, the theory of integrals, the construction of maps, and the calculation of geometric volumes. Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling. The Chebyshev parallel motion is three linked bars approximating rectilinear motion. *TIS
(I always loved the little jingle by Nathan Fine, "Chebyshev said, and I say it again. There is always a prime between n and 2n.")

1896 Ernst Engel (26 Mar 1821, 8 Dec 1896) German statistician, the head of the Prussian Statistical Bureau (1860-82), known for the "Engel curve," or Engel's law, which states that the proportion of expenditure on food will fall as income rises, i.e. food is a necessary good. Engel's law applies to goods as a whole. Demand for food, clothing and shelter - and for most manufactured products - doesn't keep pace with increases in incomes. Engel curves are useful for separating the effect of income on demand from the effects of changes in relative prices. Engel also examined the relationship between the size of the Prussian rye harvest and the average price of rye over a number of years prior to 1860, probably the first empirical study of the relationship between price and supply. *TIS

1933 John Joly (1 Nov 1857, 8 Dec 1933) Irish geologist, physicist and inventor whose interests spanned several fields. Using Edmond Halley's method of measuring the degree of salinity of the oceans, and then by examining radioactive decay in rocks, he estimated Earth's age at 80-90 million years (1898). Later, he revised this figure to 100 million years. He published Radioactivity and Geology (1909) in which he demonstrated that the rate of radioactive decay has been more or less constant through time. He also developed a method for extracting radium (1914) and pioneered its use for cancer treatment, and invented a constant- volume gas thermometer, a photometer, and a differential steam calorimeter for measuring the specific heat capacity of gases at constant volume. *TIS

1955 Hermann Weyl (9 Nov 1885, 8 Dec 1955)German-American mathematician whose widely varied contributions in mathematics linked pure mathematics and theoretical physics. He made significant contributions to quantum mechanics and the theory of relativity. He attempted to incorporate electromagnetism into the geometric formalism of general relativity. Weyl published Die Idee der Riemannschen Fläche (1913) which united analysis, geometry and topology. He produced the first guage theory in which the Maxwell electromagnetic field and the gravitational field appear as geometrical properties of space-time. He evolved (1923-38) the concept of continuous groups using matrix representations. Applying group theory to quantum mechanics he set up the modern subject.
(My favorite Weyl quote, "God exists since mathematics is consistent, and the devil exists since its consistency cannot be proved." ) *TIS

1961 Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician.
Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry. Together with Federigo Enriques, he won the Bordin prize from the French Academy of Sciences.
He contributed in a major way to birational geometry, the theory of algebraic surfaces, in particular of the curves lying on them, the theory of moduli spaces and the theory of functions of several complex variables. He wrote prolifically, and some of his work has subsequently been shown to be not rigorous according to the then new standards set in particular by Oscar Zariski and David Mumford. At the personal level, according to Roth (1963) he was easily offended, and he was involved in a number of controversies. He died in Rome of cancer.*Wik

1966 Arthur Byron Coble (November 3, 1878, Williamstown, Pennsylvania – December 8, 1966) was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface. He was President of the American Mathematical Society from 1933 to 1934.*Wik

1973 Griffith Conrad Evans (May 11, 1887 – December 8, 1973) was a mathematician working for much of his career at the University of California, Berkeley. He is largely credited with elevating Berkeley's mathematics department to a top-tier research department,[1] having recruited many notable mathematicians in the 1930s and 1940s.*Wik

1986 Harrison (Scott) Brown (26 Sep 1917, Sheridan, Wyoming, 8 Dec 1986) was an American geochemist known for his role in isolating plutonium for its use in the first atomic bombs and for his studies regarding meteorites and the Earth's origin. He was one of 67 concerned Manhattan Project scientists at Oak Ridge to sign a July 1945 petition to the President, which said, in part, "...Therefore we recommend that before this weapon be used without restriction in the present conflict, its powers should be adequately described and demonstrated, and the Japanese nation should be given the opportunity to consider the consequences of further refusal to surrender." *TIS (I like that he graduated from Galileo HS in San Francisco)

2012 Barry S. Altman (founder of Commodore USA and designer of the C64x), died on December 8, 2012. *Commodore USA

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

Saturday, 7 December 2019

On This Day in Math - December 7

God made the integers, all else is Man’s work
~Leopold Kronecker

The 341st day of the year; 341 is the sum of seven consecutive primes,

and 341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)

341 is the smallest of the pseudoprimes base 2,  disproving a Chinese math conjecture from around 500 BC. The conjecture was that p is prime IFF it divides 2p-2.

A pseudoprime n in base b is any composite number n such that \(b^{n-1} \equiv 1 Mod n \)  so for this case, \( 2^{341-1} \equiv 1 mod (341) \)
for younger students that really means if you raise two to the 340th  power, and divide by 341, you get a remainder of one.
Pseudoprimes are also called Poulet numbers, and Sarrus numbers.  "Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" mentioned above. 
"Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers *OEIS

341 is also the smallest number with seven representations as a sum of three positive squares (collect the whole set!)

1676 The first public release of Ole Rømer's conjecture that the speed of light was finite is published in the Journal des sçavans.. He may have included a conjecture about the speed of light being finite in his presentation to the Royal Academy of Sciences in Paris on August 22 of that year, "This second inequality appears to be due to light taking some time to reach us from the satellite; light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit." His final calculations of 220,000 Km/sec were presented to the Academy on 22 November, but the record of that meeting has been lost. *Wik

1725 The first meeting of the Petersburg Academy of Science was held in a meeting room of the palace of Baron Peter Pavlovich Shafirov. The meeting featured discussion of the physics theories of Wolff and Leibniz.

In 1869, the Thames Tunnel between Rotherhithe and Wapping in London, the world's first tunnel under a navigable river, was re-opened with the East London Railway line. Work had started on 2 Mar 1825. Excavation was engineered by Marc Brunel, for which he invented the tunneling shield to reduce the danger of collapse while digging through soft sediments. Beginning his own engineering career, his son Isambad Brunel assisted. They persevered through 18 years, including floods, human disasters, and delays caused by financing difficulties. Planned ramps for use by carts and freight traffic were never added due to cost, but it was opened for pedestrian use on 25 Mar 1843. It remains in use as the oldest part of the London Underground.*TIS

In 1872, the H.M.S. Challenger embarked from Portsmouth, England on the world's first scientific voyage around the world. Physicists, chemists, and biologists collaborated with expert navigators to map the sea. The Challenger was a corvette class ship, a military vessel that traveled under sail but had auxiliary steam power. The ship was fitted with a natural history laboratory where specimens were examined, identified, dissected and drawn; a chemistry laboratory; and scientific equipment. During the 4 year journey, ending on 24 May 1876, the voyage zig-zagged around the globe to visit every continent, sounded the ocean bottom to a depth of 26,850-ft, found many new species, and provided collections for scores of biologists.*TIS [First is always subject to quibbles about definitions. Thony Christie points out that Cooks 1768 voyage could claim equal status. ]

1873 Cantor wrote Dedekind that the “aggregate” of real numbers is uncountable. Five days earlier he wrote that he “had never seriously concerned himself with the problem, since it seemed to have no practical value.” *VFR
 According to Dedekind's notes, Dedekind sent a new version of Cantor's proof, making its core simpler and more precise the following day . He said "this presentation was transcribed, almost word for word, in Cantor's article". When Cantor posed the problem of the denumerability of R, on November 29, Dedekind answered that he was unable to solve it, but at the same time he stated and proved the theorem on the denumerability of the set of algebraic numbers [Cantor & Dedekind 1937, 18]. Although Dedekind's letter is no longer extant, the point is confirmed by Cantor's next letter, acknowledging receipt of the proof on December 2 [Cantor & Dedekind 1937, 13]. Now, as Dedekind wrote, "after a short time, this theorem and its proof were reproduced almost literally, including the use of the technical term 'height' [HOhe], in Cantor's article" *HISTORIA MATHEMATICA 20 (1993), 343-363 On the Relations between Georg Cantor and Richard Dedekind Jos~ FERREIROS

In 1934, Wiley Post is credited with discovering the jet stream when he flew into the stratosphere over Bartlesville, Oklahoma. With the financial backing of Oklahoma oil pioneer Frank Phillips, Post planned flights to test the "thin air" in the stratosphere above 50,000 feet. The Winnie Mae, made of plywood, could not be pressurized so Post developed the pressurized flying suit, forerunner of the modern space suit. Made by B.F. Goodrich, it was of double ply rubberized parachute fabric, with pigskin gloves, rubber boots, and aluminium helmet, pressurized to 0.5 bar. In Mar 1935, Post flew from Burbank California to Cleveland Ohio in the stratosphere using the jet stream. At times, his ground speed exceeded 550 kph in a 290 kph aircraft.*TIS

1948 The first transistor is developed at Bell Labs. See 10 July 1973. *VFR

1962 The Atlas computer was developed at Manchester, and the first production version of the machine ran on 7 December 1962. At the time of that switch-on, the Atlas was believed to be the most powerful machine in the world. *BBC NEWS

1972 Apollo 17, the last manned moon flight was launched. *VFR Flight Commander Eugene Cernan was the last man on the moon. With him on the voyage of the command module America and the lunar module Challenger were Ronald Evans (command module pilot) and Harrison H. "Jack" Schmitt (lunar module pilot). In maneuvering Challenger to a landing at Taurus-Littrow, located on the southeast edge of Mare Serenitatis, Cernan and Schmitt activated a base of operations from which they completed three highly successful excursions to the nearby craters and the Taurus mountains, making the Moon their home for over three days. The mission returned on 19 Dec. *TIS (In 2004 President George Bush had made a pledge to return to the moon, and beyond, by 2020. But in September of 2009 the Augustine Commission, also known as the Human Space Flight committee, predicted a cost of an additional three-billion dollars a year, effectively killing the idea of manned flights beyond Earth orbit.)


903 'Abd al-Rahman al-Sufi (December 7, 903 – May 25, 986) was a Persian astronomer also known as 'Abd ar-Rahman as-Sufi, or 'Abd al-Rahman Abu al-Husayn, 'Abdul Rahman Sufi, 'Abdurrahman Sufi and known in the west as Azophi; the lunar crater Azophi and the minor planet 12621 Alsufi are named after him. Al-Sufi published his famous Book of Fixed Stars in 964, describing much of his work, both in textual descriptions and pictures. He identified the Large Magellanic Cloud, which is visible from Yemen, though not from Isfahan; it was not seen by Europeans until Magellan's voyage in the 16th century. He also made the earliest recorded observation of the Andromeda Galaxy in 964 AD; describing it as a "small cloud".[3] These were the first galaxies other than the Milky Way to be observed from Earth.
He observed that the ecliptic plane is inclined with respect to the celestial equator and more accurately calculated the length of the tropical year. He observed and described the stars, their positions, their magnitudes and their colour, setting out his results constellation by constellation. For each constellation, he provided two drawings, one from the outside of a celestial globe, and the other from the inside (as seen from the earth).
Al-Sufi also wrote about the astrolabe, finding numerous additional uses for it : he described over 1000 different uses, in areas as diverse as astronomy, astrology, horoscopes, navigation, surveying, timekeeping, Qibla, Salah prayer, etc *Wik

1637 William Neile (7 Dec 1637 in Bishopsthorpe (near York), England - 24 Aug 1670 in White Waltham, Berkshire, England) Neile entered Wadham College, Oxford, in 1652 (but did not matriculate until 1655) where he was taught mathematics by John Wilkins and Seth Ward. He was a gentleman-commoner, meaning that he paid the highest fees and was ranked near the top of the social order just below the nobles. Gentleman-commoners had many privileges enjoying fine suites of rooms in College, and sat with the College Fellows at meals and in the common rooms. Certainly Neile was fortunate in being part of a family that was in the forefront of scientific work for certainly while Neile was a student, his father was observing with Christopher Wren in the observatory he had constructed on the roof of his house, the 'Hill House', at White Waltham. Paul Neile was also building a telescope for Gresham College at this time. In 1657 William Neile became a pupil of law at the Middle Temple in London. He went on to become a member of the privy council of King Charles II.
In 1657, while still a student at Oxford, he became the first person to find the arc length of an algebraic curve when he rectified the semicubical parabola. He communicated his results to William Brouncker and Christopher Wren at the Gresham College Society, the Society based at Gresham College, London, which a few years later became the Royal Society. Neile's work on this appeared in John Wallis's De Cycloide in 1659. As well as his mathematical work Neile made astronomical observations using instruments on the roof of his father's house, the 'Hill House' at White Waltham in Berkshire. He died in this house at the age of 32 and was buried in the local parish church. *SAU (The evolute of the parabola is a particular case of the semicubical parabola also called Neile's parabola or the cuspidal cubic. The "semi" is because it is a three-halves power, hence semi-cubic)(The wording of the plaque honoring Neile and his grave stone below are contained in The antiquities of Berkshire, By Elias Ashmole, which is available at Google Books

1823 Leopold Kronecker (7 Dec 1823; 29 Dec 1891) German mathematician who worked to unify arithmetic, algebra and analysis, with a particular interest in elliptic functions, algebraic equations, theory of numbers, theory of determinants and theory of simple and multiple integrals. However the topics he studied were restricted by the fact that he believed in the reduction of all mathematics to arguments involving only the integers and a finite number of steps. He believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs, and believed that transcendental numbers did not exist. The Kronecker delta function is named in his honour.*TIS

1647 Giovanni Ceva (7 Dec 1647 in Milan, Italy - 15 June 1734 in Mantua, Italy) For most of his life Giovanni Ceva worked on geometry. He discovered one of the most important results on the synthetic geometry of the triangle between Greek times and the 19th Century. The theorem states that lines from the vertices of a triangle to the opposite sides are concurrent precisely when the product of the ratio the sides are divided is 1. He published this in De lineis rectis (1678).
Ceva also rediscovered and published Menelaus's theorem. He also studied applications of mechanics and statics to geometric systems. Although he wrongly concluded that the periods of oscillation of two pendulums were in the same ratio as their lengths, he later corrected the error.
Ceva published Opuscula mathematica in 1682. In Geometria Motus (1692) he, to some extent, anticipated the infinitesimal calculus. De Re Nummeraria in 1711 is one of the first works in mathematical economics; it attempts to solve the conditions of equilibrium for the monetary system of a state like Mantua.
Ceva also did important work on hydraulics. On this topic he published Opus hydrostaticum (1728). He held official positions in Mantua and used his knowledge of hydraulics to argue successfully against a project which proposed to divert the river Reno into the river Po. *SAU (Ceva's theorem is understandable to most high school math students and (imho) should be more commonly taught.)

1830 (Antonio) Luigi (Gaudenzio Giuseppe) Cremona (7 Dec 1830; 10 Jun 1903)
was an Italian mathematician who was an originator of graphical statics (the use of graphical methods to study forces in equilibrium) and work in projective geometry. Cremona's work in statics is of great importance and he gave, in a clearer form, some theorems due to Maxwell. In a paper of 1872 Cremona took an idea of Maxwell's on forces in frame structures that had appeared in an engineering journal in 1867 and interpreted Maxwell's notion of reciprocal figures as duality in projective 3-space. These reciprocal figures, for example, have three forces in equilibrium in one figure represented by a triangle while in the reciprocal figure they are represented by three concurrent lines.*TIS

1905 Gerard Peter Kuiper (7 Dec 1905; 23 Dec 1973) Dutch-born American astronomer, who discovered Miranda, a moon of Uranus, and Nereid, a moon of Neptune. The Kuiper Belt is so-named after his original suggestion of its existence outside the orbit of Neptune before it was confirmed as a belt of small bodies. He measured the diameter of Pluto. In the Martian atmosphere Kuiper detected carbon dioxide, but the absence of oxygen (1947). In the 1960s, Kuiper pioneered airborne infrared observing using a Convair 990 aircraft and served as chief scientist for the Ranger spacecraft crash-landing probes of the moon. By analyzing Ranger photographs, he identified landing sites on the lunar surface most suitable for safe manned landings. *TIS

1910 Richard Brooke Roberts (7 Dec 1910; 4 Apr 1980) American biophysicist who contributed most to the discovery of "delayed neutrons" - that uranium fission does not release all the neutrons it produces at one time, but some come off at measurably later times. Some are emitted seconds to minutes later. This is crucial in the operation of a fission reactor. In uranium-235 fission in a thermal reactor, the proportion of delayed neutrons is about 0.65 percent. If the reactivity stays below the proportion of delayed neutrons, the reactor can be controlled. The delayed neutrons modify the rate of fission sufficiently to give time for the insertion of control rods. Without the margin of safety provided by the delayed neutrons, nuclear reactors might not be practical at all.*TIS

1924 Mary Ellen Rudin (born December 7, 1924, Hillsboro, Texas) is an American mathematician.
Born Mary Ellen Estill, she attended the University of Texas, completing her B.A. in 1944 and her Ph.D. in 1949, under Robert Lee Moore. In 1953, she married the mathematician Walter Rudin. Following her mentor Moore, her research centers on point-set topology. She was appointed as Professor of Mathematics at the University of Wisconsin in 1971, and is currently a Professor Emerita there. She served as vice-president of the American Mathematical Society, 1980–1981. In 1984 she was selected to be a Noether Lecturer. She is an honorary member of the Hungarian Academy of Sciences (1995).
Rudin is best known in topology for her constructions of counterexamples to well-known conjectures. Most famously, she was the first to construct a Dowker space, thus disproving a conjecture of Dowker's that had stood, and helped drive topological research, for more than twenty years. She also proved the first Morita conjecture and a restricted version of the second. Her latest major result is a proof of Nikiel's conjecture. Rudin's Erdős number is 1.
"Reading the articles of Mary Ellen Rudin, studying them until there is no mystery takes hours and hours; but those hours are rewarded, the student obtains power to which few have access. They are not hard to read, they are just hard mathematics, that's all." (Steve Watson)
She resides in Madison, Wisconsin, in the Rudin House, a home designed by architect Frank Lloyd Wright.*SAU

1928 Noam Chomsky is born in Philadelphia, Pennsylvania. He received his Ph.D. from the University of Pennsylvania in 1955. Since then he has taught at MIT, where he now holds the Ferrari P. Ward Chair of Modern Languages and Linguistics. Chomsky's work on the syntax of natural languages influenced the early development of programming languages. He is most famous for his work on the hierarchy of grammar that bears his name. Chomsky has been awarded an Honorary Doctorate by the University of London and the University of Chicago. In 1988 he received the Kyoto Prize in Basic Science
Chomsky has always been interested in politics. Since 1965 he has become one of the leading critics of U.S. foreign policy and divided his efforts between linguistic studies and his social concerns.*CHM

1936 Oleksandr Mikolaiovich Sharkovsky (7 Dec 1936 in Kiev, Ukraine, )attended his local university of Kiev, graduating in 1958. In 1961 he was appointed to the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev. He also taught at the University of Kiev from 1967.
Sharkovsky's main areas of interest are the theory of dynamical systems, the theory of stability and the theory of oscillations. He also works in the theory of functional and functional differential equations, and the study of difference equations and their application.
He is perhaps best known for an important theorem on continuous functions which he proved in 1964. Although the result did not attract a great deal of interest at the time of its publication, during the 1970s other surprising results were proved which turned out to be special cases of Sharkovsky's theorem. *SAU


1912 Sir George Howard Darwin (9 Jul 1845, 7 Dec 1912) the second son of the famous biologist Charles Darwin, was an English astronomer who championed a theory (no longer accepted) that the Moon was once part of the Earth, in what is now the Pacific Ocean. His was the first mathematical analysis of the evolution of Earth's Moon. He suggested that since the effect of the tides has been to slow the Earth's rotation and to cause the Moon to recede from the Earth, then by extrapolating back 4.5 billion years ago the Moon and the Earth would have been very close, with a day being less than five hours. Before this time the two bodies would actually have been one, until the Moon was torn away from the Earth by powerful solar tides that would have deformed the Earth every 2.5 hours*TIS

1928 James Whitbread Lee Glaisher (5 November 1848 – 7 December 1928) son of James Glaisher, the meteorologist, was a prolific English mathematician.
He was educated at St Paul's School and Trinity College, Cambridge, where he was second wrangler in 1871.[1] Influential in his time on teaching at the University of Cambridge, he is now remembered mostly for work in number theory that anticipated later interest in the detailed properties of modular forms. He published widely over other fields of mathematics.
He was the editor-in-chief of Messenger of Mathematics. *Wik

1943 Elizabeth Ruth Naomi Belville (5 March 1854 – 7 December 1943), also known as the Greenwich Time Lady, was a businesswoman from London. She, her mother Maria Elizabeth, and her father John Henry, sold people the time. This was done by setting a watch to Greenwich Mean Time, as shown by the Greenwich clock, and then selling people the time by letting them look at the watch. *Wik A nice blog about time, and the time lady by Greg Ross at Futility Closet. and a book by David Rooney.

1952 Forest Ray Moulton (29 Apr 1872, 7 Dec 1952) American astronomer (born in the tiny town of Leroy, Michigan, population 267 in the 2000 census) who collaborated with Thomas Chamberlin in advancing the planetesimal theory of the origin of the solar system (1904). They suggested filaments of matter were ejected when a star passed close to the Sun, which cooled into tiny solid fragments, "planetesimals". Over a very long period, grains collided and stuck together. Continued accretion created pebbles, boulders, and eventually larger bodies whose gravitational force of attraction accelerated the formation of protoplanets. (This formation by accretion is still accepted, but not the stellar origin of the planetesimals.) Moulton was first to suggest that the smaller satellites of Jupiter discovered by Nicholson and others in the early 20th century were captured asteroids - now widely accepted. *TIS The crater Moulton on the Moon, the Adams-Moulton methods for solving differential equations and the Moulton plane in geometry are named after him. In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues' theorem does not hold. *Wik

1970 Rube Goldberg (4 Jul 1883, 7 Dec 1970) American cartoonist who satirized the American preoccupation with technology. His name became synonymous with any simple process made outlandishly complicated because of his series of "Invention" cartoons which use a string of outlandish tools, people, plants and steps to accomplish everyday simple tasks in the most complicated way. Goldberg applied his training as a graduate engineer and used his engineering, story-telling, and drawing skills to make sure that the "Inventions" could work, even though dozens of arms, wheels, gears, handles, cups, and rods were put in motion by balls, canary cages, pails, boots, bathtubs, paddles, and even live animals for simple tasks like squeezing an orange for juice or closing a window in case it should start to rain. *TIS

1979 Cecilia Helena Payne-Gaposchkin (10 May 1900, 7 Dec 1979) was an English-born American astronomer who was the first to apply laws of atomic physics to the study of the temperature and density of stellar bodies, and the first to conclude that hydrogen and helium are the two most common elements in the universe. During the 1920s, the accepted explanation of the Sun's composition was a calculation of around 65% iron and 35% hydrogen. At Harvard University, in her doctoral thesis (1925), Payne claimed that the sun's spectrum was consistent with another solution: 99% hydrogen with helium, and just 1% iron. She had difficulty persuading her superiors to take her work seriously. It was another 20 years before Payne's original claim was confirmed, by Fred Hoyle. *TIS

1982 George Bogdanovich Kistiakowsky (November 18, 1900 – December 7, 1982) was a Ukrainian-American physical chemistry professor at Harvard who participated in the Manhattan Project and later served as President Dwight D. Eisenhower's Science Advisor.
Born in Kiev in the old Russian Empire, Kistiakowsky fled Russia during the Russian Civil War. He made his way to Germany, where he earned his PhD in physical chemistry under the supervision of Max Bodenstein at the University of Berlin. He emigrated to the United States in 1926, where he joined the faculty of Harvard University in 1930, and became a citizen in 1933.
During World War II, he was the head of the National Defense Research Committee (NDRC) section responsible for the development of explosives, and the technical director of the Explosives Research Laboratory (ERL), where he oversaw the development of new explosives, including RDX and HMX. He was involved in research into the hydrodynamic theory of explosions, and the development of shaped charges. In October 1943, he was brought into the Manhattan Project as a consultant. He was soon placed in charge of X Division, which was responsible for the development of the explosive lenses necessary for an implosion-type nuclear weapon. He watched an implosion weapon that was detonated in the Trinity test in July 1945. A few weeks later a Fat Man implosion weapon was dropped on Nagasaki.
From 1962 to 1965, he chaired the National Academy of Sciences's Committee on Science, Engineering, and Public Policy (COSEPUP), and was its vice president from 1965 to 1973.
In later years he was active in an antiwar organization, the Council for a Livable World. Kistiakowsky severed his connections with the government in protest against the US involvement in the war in Vietnam. In 1977, he assumed the chairmanship of the Council for Livable World, campaigning against nuclear proliferation. He died of cancer in Cambridge, Massachusetts, on December 17, 1982. His body was cremated, and his ashes were scattered near his summer home on Cape Cod, Massachusetts. His papers are in the Harvard University archives.*Wik

2011 Tonny Albert Springer (February 13, 1926, The Hague – December 7, 2011, Zeist) was a mathematician at Utrecht university who worked on linear algebraic groups, Hecke algebras, complex reflection groups, and who introduced Springer representations and the Springer resolution.*Wik

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

Friday, 6 December 2019

On This Day in Math - December 6

"I have finally found a subject where I do not need to memorize, but can think things out myself – mathematics."
~Herta Taussig Freitag (from her diary, age 12)

The 340th day of the year; 340 is the sum of the first four powers of four.

340 can also be written as the sum of consecutive primes in three different ways.

340! +1 is prime. There are only thirteen day numbers of the year for which n! +1 is prime, and 340 is the last of these.

Jim Wilder@wilderlab pointed out that 340 = 41 + 42 + 43 + 44. Just think, tomorrow will be even a longer string of consecutive powers of four!


In 1592, Galileo was appointed Professor of Mathematics at the University of Padua (the University of the Republic of Venice) at a salary of three times that he had received at Pisa. On 7 December 1592 he gave his inaugural lecture and began a period of 18 years at the University, years which he later described as the happiest of his life. *British Journal of Sports Medicine (honest)

In 1631, the transit of Venus occurred as first predicted by Kepler. He correctly predicted that an ascending node transit of Venus would occur in Dec 1631, but no-one observed it - due to the fact that it occurred after sunset for most of Europe. Kepler himself died in 1630. He not only predicted this particular transit but also worked out that transits of Venus involve a cyclical period of approximately 120 years. When such a transit is observed, Venus appears as a small black circle moving across the face of the Sun.*TIS

1710 An advertisement in the Old Bailey Proceedings for a book on mathematics, and more
*** The Marrow of the Mathematicks, made Plain and Easie to the Understanding of any ordinary Capacity. Containing the Doctrines of Arithmetick, Geometry, Astronomy, Gauging, the Use of the Sector, Surveying, Dyaling, and the Art of Navigation, &c. Illustrated with several Cuts, for the better Explanation of the whole Matter. After a New, Compendious, Easy Method By W. Pickering, Merchant-Adventurer.
To which is added,
Measuring Surfaces and Solids, such as Plank, Timber, Stone, &c. Joiners, Carpenters, Bricklayers, Glasiers, Painters and Paviers Work: Each Proposition being wrought Vulgarly, Decimally, Practically and Instrumentally. With a small Tract of Gauging Wine, Ale, or Malt, without Inches, or Division; by which any one may Gauge ten Backs or Floors of Malt, in the same time another shall Guage one, by the Way now used. Altogether New, and submitted to the Censure of the Honourable Commissioners of Excise. By J. L. P. M.
Both Printed for Eben. Tracy, at the Three Bibles on London-Bridge. 1710
Pedro Nunes Nonius original model
(They just don't make titles like they used to) Available on line for free here

1763 From Charles Mason's Journal of the Mason Dixon survey, "Set up a Sector brought by the Commissioners from Maryland and found that the nonius would not touch the middle part of the arch" A nonius is a device, named in honor to its author and inventor Pedro Nunes (Latin: Petrus Nonius), created in 1542 as a system for taking fine measurements on the astrolabe which could largely improve its accuracy. Later on, it was adapted in 1631 by the French mathematician Pierre Vernier, to create the vernier scale. *Wik

1830 First national observatory established at Washington, D.C. Established by the order of the Secretary of the Navy, John Branch, on 6 December 1830 as the Depot of Charts and Instruments, the Observatory rose from humble beginnings. Placed under the command of Lieutenant Louis M. Goldsborough, with an annual budget of 330 US Dollars, its primary function was the restoration, repair, and rating of navigational instruments. It was made into a national observatory in 1842 via a federal law and a Congressional appropriation of 25,000 dollars. Lieutenant James Melville Gilliss was put in charge of "obtaining the instruments needed and books." *Wik (Interestingly, Goldsborough was appointed to the Naval Academy at the age of seven, although he did not enter for several years. He rose to the rank of Admiral during the U S Civil War and three naval ships were named for him.)

1882 Venus crossed the disc of the Sun. 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. 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.*Wik
The transit of Venus across the sun was photgraphed on a series of glass plate negatives made by Amherst College astronomer David Peck Todd. He used a solar photographic telescope (made by the renowned optical firm Alvan Clark & Sons) stationed on the summit of Mount Hamilton, California, where the Lick Observatory was under construction. Of the photos, 147 survived, having been archived in the mountain vault. A century later, they were retrieved and an animation made from them premiered at the International Astronomical Union's general assembly in Sydney in Jul 2003. This is perhaps the most complete surviving record of a historical transit of Venus, dating from the time when Chester Arthur was president of the United States.*TIS

1917 Kazimierz Kuratowski gave a talk “On the definitions in mathematics,” which became his first published paper. This work grew out of Jan LLukasiewicz’s crushing criticism of the foundations of StanisLlaw Zaremba’s Theoretical Arithmetic (1912). Kuratowski’s now famous 1921 definition of ordered pair (a nice note for Alg classes) also grew out of LLukasiewicz’s critique. [Kuratowski, A Half Century of Polish Mathematics, p. 24] *VFR

1946 Birthdate of Nicolette Weil, younger daughter of the mathematician Andre Weil. She was born on St. Nicholas’ day, as he planned, or so he jokingly claimed, but she is named after Nicolas Bourbaki. Professor Weil was one of the founders of the Bourbaki group. See Joong Fang, Bourbaki, Paideia Press, 1970, p. 40. His older daughter is named Sylvie and was born 12 September 1942. *VFR

1956 The knapsack problem was first named and discussed by George B. Dantzig, the father of linear programming. *VFR (The part about naming it may be an error; the problem existed long before and *Wik has this note:) "The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. It is not known how the name "knapsack problem" originated,(they should read my blog?) though the problem was referred to as such in the early works of mathematician Tobias Dantzig (1884–1956)(This was George's Father), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined."

You Tube link (sorry, the embed has been disabled, so hit it on your own

In 1957, America's first attempt at putting a satellite into orbit failed when the Vanguard rocket carrying it blew up on the launch pad at Cape Canaveral, Florida. With a series of rumbles audible for miles around, the vehicle, having risen about four feet into the air, suddenly sank. Falling against the firing structure, fuel tanks rupturing as it did so, the rocket toppled to the ground on the northeast or ocean side of the structure in a roaring, rolling, ball-shaped volcano of flame. *TIS

1963 Time magazine published a copy of Salvador Dali’s “Fifty abstract pictures which as seen from two yards change into three Lenines masquerading as Chinese and as seen from six yards appear as the head of a royal tiger.” It is based on the semi-regular tessellation 4–3–4–3–3 made up of squares and triangles.*VFR

1987 Florida rapist Tommy Lee Andrews is the first person to be convicted as a result of DNA fingerprinting. *Wik

2005 At a book signing after a mathematics professor at West Point was asked what he taught, former president Jimmy Carter commented “In retrospect, I possibly received the best insight into human nature by studying differential equations and systems of differential equations. That subject seemed to interrelate rates of change between interconnected entities.” *VFR


1586 Niccolò Zucchi (6 Dec 1586; 21 May 1670) Italian astronomer who, in approximately 1616, designed one of the earliest reflecting telescopes, antedating those of James Gregory and Sir Isaac Newton. A professor at the Jesuit College in Rome, Zucchi developed an interest in astronomy from a meeting with Johannes Kepler. With this telescope Zucchi discovered the belts of the planet Jupiter (1630) and examined the spots on Mars (1640). He also demonstrated (in 1652) that phosphors generate rather than store light. His book Optica philosophia experimentalis et ratione a fundamentis constituta (1652-56) inspired Gregory and Newton to build improved telescopes.*TIS(I belive the observations he made were NOT with his reflecting telescope, which never seemed to work, but with the more common refracting telescope. See more on reflecting telescopes at this blog where Thony Christie takes me to task for giving (too much) credit to one of the  early developers.)

1682 Giulio Carlo Fagnano dei Toschi (December 6, 1682 – September 26, 1766) who was born in Sinigaglia, Italy. He was the founder of the geometry of the triangle, studied the lemniscate, and coined the term “elliptic integral.” *VFR
Fagnano is best known for investigations on the length and division of arcs of certain curves, especially the lemniscate; this seems also to have been in his own estimation his most important work, since he had the figure of the lemniscate with the inscription: "Multifariam divisa atque dimensa Deo veritatis gloria", engraved on the title-page of his Produzioni Matematiche, which he published in two volumes (Pesaro, 1750), and dedicated to Pope Benedict XIV. The same figure and words "Deo veritatis gloria" also appear on his tomb.
Failing to rectify the ellipse or hyperbola, Fagnano attempted to determine arcs whose difference should be rectifiable. He also pointed out the remarkable analogy existing between the integrals which represent the arc of a circle and the arc of a lemniscate. Finally he proved the formula π = 2i log((1-i)/(1+i)
One of his son, Giovanni, is the namesake of the optimization problem called Fagnano's Problem in geometry :
For a given acute triangle determine the inscribed triangle of minimal perimeter.
The solution is the orthic triangle.

1848 Johann Palisa (6 Dec 1848; 2 May 1925) Silesian astronomer who was a prolific discoverer of asteroids, 122 in all, beginning with Asteroid 136 Austria (on 18 Mar 1874, using a 6” refractor) to Asteroid 1073 Gellivara in 1923 - all by visual observation, without the aid of photography. In 1883, he joined the expedition of the French academy to observe the total solar eclipse on May 6 of that year. During the eclipse, he searched for the putative planet Vulcan, which was supposed to circle the sun within the orbit of Mercury. In addition to observing the eclipse, Palisa collected insects for the Natural History Museum in Vienna. He also prepared two catalogs containing the positions of almost 4,700 stars. He remains the most successful visual discoverer in the history of minor planet research.*TIS

1856 Walther Franz Anton von Dyck (6 Dec 1856 in Munich, Germany - 5 Nov 1934 in Munich, Germany) Von Dyck made important contributions to function theory, group theory (where a fundamental result on group presentations is named after him) topology and potential theory. *SAU

1880 Pierre Léon Boutroux (6 December 1880 – 15 August 1922) was a French mathematician and historian of science. Boutroux is chiefly known for his work in the history and philosophy of mathematics.
He was born in Paris on 6 December 1880 into a well connected family of the French intelligentsia. His father was the philosopher Émile Boutroux. His mother was Aline Catherine Eugénie Poincaré, sister of the scientist and mathematician Henri Poincaré. A cousin, Raymond Poincaré was to be President of France.
He occupied the mathematics chair at Princeton University from 1913 until 1914. He occupied the History of sciences chair from 1920 to 1922.
Boutroux published his major work Les principes de l'analyse mathématique in two volumes; Volume 1 in 1914 and Volume 2 in 1919. This is a comprehensive view of the whole field of mathematics at the time.*Wik

1900 George Eugene Uhlenbeck (6 Dec 1900; 31 Oct 1988) Dutch-American physicist who, with Samuel A. Goudsmit, proposed the concept of electron spin (Jan 1925) - a fourth quantum number which was a half integer. This provided Wolfgang Pauli's anticipated "fourth quantum number." In their experiment, a horizontal beam of silver atoms travelling through a vertical magnetic field was deflected in two directions according to the interaction of their spin (either "up" or "down") with the magnetic field. This was the first demonstration of this quantum effect, and an early confirmation of quantum theory. As well as fundamental work on quantum mechanics, Uhlenbeck worked on atomic structure, the kinetic theory of matter and extended Boltzmann's equation to dense gases.*TIS

1907 Mathematical Logician Rosser is Born, J. Barkley Rosser is born in Jaksonville, FL. In 1934 Rosser received a Ph.D. in logic from Princeton under the supervision of Alonso Church. Rosser was able to anticipate the potential of early computers in many areas of mathematics as well as the ultimate impact of logic on the future of computing. He contributed to the Church-Rosser theorem that identifies the outer limit of what is achievable in automated theorem proving and, therefore, plays the same role in computing science as the second law of thermodynamics in engineering.
Rosser taught at Cornell and the University of Wisconsin, served as a president of the Association of Symbolic Logic and SIAM. He died on September 5, 1989. *CHM

1908 Herta Taussig Freitag (December 6, 1908 - January 25, 2000) Herta obtained a job at a private high school, the Greer School, in upstate New York. There she met Arthur H. Freitag and they were married in 1950. Herta started teaching at Hollins College (now University) in Roanoke, VA in 1948. She received a Ph.D. degree from Columbia University in 1953 and the title of her dissertation was "The Use of the History of Mathematics in its Teaching and Learning on the Secondary Level."
During Herta's years at Hollins she was a frequent guest speaker at local schools and gave lectures at both Virginia and North Carolina Governor's Schools. She published numerous articles in The Mathematics Teacher, The Arithmetic Teacher, and The Mathematics Magazine. At the request of the National Council of Teachers of Mathematics, Professor Freitag wrote the monograph, The Number Story, with her husband. In 1962 she was the first woman to be President of the Maryland-District of Columbia-Virginia Section of the Mathematical Association of America (MAA). Professor Freitag received the Hollins' Algernon Sydney Sullivan Award, which is awarded for recognition of "extraordinary humane and scholarly achievement." She officially retired from Hollins in 1971 to spend time with her husband, who was ill. After his death in 1978, Hollins welcomed her back to the classroom as a leave replacement in 1979-1980 and as a teacher in the Master of Arts in Liberal Studies (MALS) program for several years. Professor Herta Freitag was the first faculty member to receive the Hollins Medal (1979) and the first recipient of the Virginia College Mathematics Teacher of the Year award (1980).
Professor Freitag was very proud of her perfect attendance at the International Conferences of the Fibonacci Association. Most of her work with Fibonacci numbers occurred after she retired, which demonstrates the fallacy of a commonly held belief that mathematicians complete their best work before the age of 40. Professor Freitag published more than thirty articles in the Fibonacci Quarterly after 1985. The November 1996 issue of the Fibonacci Quarterly was dedicated to "Herta Taussig Freitag as she enters her 89th year, in recognition of her years of outstanding service and achievement in the mathematics community through excellence in teaching, problem solving, lecturing and research." This award was given to celebrate her 89th birthday, since 89 is a Fibonacci number. *Biographies of Women Mathematicians, Agnes Scott College web site

1941 Filep László (6 Dec 1941 in Csaszlo, Szabolcs-Szatmar-Bereg, Hungary - 19 Nov 2004 in Budapest, Hungary) He worked for the degree of dr. univ. submitting his thesis Life and work of Gyula Farkas (1847-1930) to the Kossuth University in Debrecen in 1978. But this was not László's first publication, for he had published a number of articles in the prestigious Hungarian popular scientific magazine, Termeszet Vilaga (The World of Nature). The first of these articles was Farkas Gyula (1847-1930) published in 1976, was followed by A matematika nagy nöalakjai (1977) and Helyunk a tudomany vilagaban (1979). He published many other articles on the history of mathematics such as Lajos David (1881-1962), historian of Hungarian mathematics (1981), Great female figures of Hungarian mathematics in 19th-20th centuries (1983), The development, and the developing of the concept of a , fraction (2001), The genesis of Eudoxus's infinity lemma and proportion theory (2001), From Fejer's disciples to Erdős's epsilons - change over from analysis to combinatorics in Hungarian mathematics (2002), and Irrationality and approximation of √2 and √3 in Greek mathematics (2004). He also published biographies of many mathematicians including Janos Bolyai, John C Harsanyi, John von Neumann, and Paul Erdős. László's research interest was not only in the history of mathematics for he also published a long series of papers on fuzzy groups, some written with his collaborator Iulius Gyula Maurer, beginning in 1987. *SAU


1788 Nicole-Reine Lepaute (5 Jan 1723 in Paris, France - 6 Dec 1788 in Saint-Cloud, France) was a French noblewoman who helped Lalande with astronomical calculations. In June 1757 Lalande decided that he would like to attempt to calculate a precise date for the return of Halley's comet. It was known to have been seen in 1305, 1380, 1456, 1531, 1607 and 1682 and Halley, taking into account perturbations to the orbit caused by the gravitational effects of Jupiter, had predicted that the comet would return reaching perihelion in December 1758. However the only way to get a more accurate prediction of its date of return was to calculate the perturbations to the orbit caused by the gravitational effects of both Jupiter and Saturn. Lalande approached Alexis Clairaut for help and Clairaut provided a basic programme of work requiring an extraordinary amount of computation. Lalande then asked Nicole-Reine Lepaute to assist him in the computations. Lalande wrote, "During six months we calculated from morning to night, sometimes even at meals. ... The assistance of Mme Lepaute was such that, without her I should never have been able to undertake the enormous labour, in which it was necessary to calculate the distance of each of the two planets Jupiter and Saturn from the comet, separately for each successive degree for 150 years. *SAU

1893 Rudolf Wolf (7 Jul 1816, 6 Dec 1893) Swiss astronomer and astronomical historian. Wolf's main contribution was the discovery of the 11 year sunspot cycle and he was the codiscoverer of its connection with geomagnetic activity on Earth. In 1849 he devised a system now known as Wolf's sunspot numbers. This system is still in use for studying solar activity by counting sunspots and sunspot groups. In mathematics, Wolf wrote on prime number theory and geometry, then later on probability and statistics - a long paper discussed Buffon's needle experiment. He estimated by Monte Carlo methods.*TIS

1959 Erhard Schmidt (13 Jan 1876 in Dorpat, Estonia (Russian Empire) (now Tartu, Estonia)- 16 Dec 1959 in Berlin, Germany) 1876 Erhard Schmidt (13 January 1876 – 6 December 1959) was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia (now Estonia). His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations.
Together with David Hilbert he made important contributions to functional analysis. He is best known for the Gram-Schmidt orthogonalisation process, which constructs an orthogonal base from any vector space. *Wik

1973 Joseph Leonard Walsh, (September 21, 1895 – December 6, 1973) was an American mathematician. His work was mainly in the field of analysis.
For most of his professional career he studied and worked at Harvard University.*Wik

1990 Lev Arkad'evich Kaluznin (31 Jan 1914 in Moscow, Russia - 6 Dec 1990 in Moscow, Russia) Kaluznin is best known for his work in group theory and in particular permutation groups. He studied the Sylow p-subgroups of symmetric groups and their generalisations. In the case of symmetric groups of degree pn, these subgroups were constructed from cyclic groups of order p by taking their wreath product. His work allowed computations in groups to be replaced by computations in certain polynomial algebras over the field of p elements. Despite the fact that the earliest applications of wreath products of permutation groups was due to C Jordan, W Specht and G Polya, it was Kaluznin who first developed special computational tools for this purpose. Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more. These results have been included in many textbooks on group theory. *SAU

1993 Wolfgang Paul (10 Aug 1913, 6 Dec 1993) German physicist who developed the Paul trap, an electromagnetic device that captures ions and holds them long enough for study and precise measurement of their properties. During the 1950s he developed the so-called Paul trap as a means of confining and studying electrons. The device consists of three electrodes - two end caps and an encircling ring. The ring is connected to an oscillating potential. The direction of the electric field alternates; for half the time the electron is pushed from the caps to the ring and for the other half it is pulled from the ring and pushed towards the caps. For his work he shared the 1989 Nobel Prize for Physics with Hans G. Dehmelt and Norman F. Ramsey.*TIS

1996 Stefan Schwarz (18 May 1914 in Nové Mesto nad Váhom, Austria-Hungarian Empire (now Slovakia) - 6 Dec 1996 in Bratislava, Slovak Republic) In addition to his work on semigroups, number theory and finite fields, Schwarz contributed to the theory of non-negative and Boolean matrices.
Schwarz organized the first International Conference on Semigroups in 1968. At this conference setting up the journal Semigroup Forum was discussed and Schwarz became an editor from Volume 1 which appeared in 1970, continuing as editor until 1982. This was not his first editorial role since he had been an editor of the Czechoslovak Mathematical Journal from 1945 and continued to edit this journal until he was nearly 80 years old. He also founded the Mathematico-Physical Journal of the Slovak Academy of Sciences in 1950 and continued as an editor of the mathematics part of the journal when it split from the physics part to become Mathematica Slovaca until 1990. *SAU

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