Pick's Theorem, some history.

 

*Wik

 Georg Pick was a Jewish Austrian mathematician (murdered during The Holocaust).  In 1880 he received his PhD from University of Vienna under Leo Konigsberger. In 1899 he published a formula for finding the area of a polygon if all the vertices are on lattice points (points whose x and y coordinates on the plane are both integers, although if they are only rational, you can adapt the theorem.)  He presented the formula he created in 1899, in Geometries zur Zahlenlehre in Prague.  He worked at University of Prague and was on the appointments committee and pushed hard for Einstein to get a position in physics there. He was also instrumental in introducing Einstein to Ricci-Curbastro and Levi-Civita which helped him work out the mathematics in the general theory of relativity.  

Hugo Steinhaus caused a surge in the popularity (and knowledge) of the theorem when he used it in his popular, Mathematical Snapshots, in 1969.  (see Google n-gram below)It probably would have been much more popular in 1899 if graph paper had been more in mathematics education, but thirty to fifty years would pass before the paper became popular in geometry and algebra classes. I also think the appearance of the Geoboard created by Egyptian Educator Caleb Gattegno in the fifties was important in applications of Pick's theorem in elementary and middle schools.  I don't know if these still are used much, or any, so I popped an image of one below. My first geoboard was a crude piece of wood with a 6x6 array nails driven by a terrible carpenter, then and now, the writer. 

Pick would never be aware of this late surge of popularity, he died after two weeks in the Theresiestadt prison camp in 1942.  

*Art of Problem Solving

The formula gives the area in two variables, N and B .  N is the number of lattice points inside the polygon (many teachers, and some books, use I, for inside), and B is the number on the boundary.  The area is given by Area = N + B/2 - 1.  The example at the right shows N=3 for the inside points, B= 14 for the inside points, so our Area = 3 + 13/2 - 1, or 8 1/2 square units.  You can, of course verify this to yourself by counting connecting lines inside and dividing into squares, rectangles, and triangles that are easy to compute.

There is not a higher dimensional analogy of this theorem, counting points inside and on the boundary, but the Ehrhart Polynomial, created by French high school teacher Eugene Ehrhart in the 60's, describes an expression for the volume in terms of the number of interior points in the polytope and dilations.  

You can find more about the theorem and links to some of its extensions at Drexel University where I also found that the original theorem was published in "Geometrisches zur Zahlenlehre" Sitzungber. Lotos, Naturwissen Zeitschrift Prague, Volume 19 (1899) pages 311-319.  

Here are some more sources for information om the method:  

W. W. Funkenbusch
“From Euler’s Formula to Pick’s Formula using an Edge Theorem”
The American Mathematical Monthly
Volume 81 (1974) pages 647-648

In this short paper Funkenbusch shows that Pick's theorem is derived from Euler's Gem using the theorem that Edges = 3I +2B -3, with I and B for inside lattice points and B for boundary points.

Dale E. Varberg
“Pick’s Theorem Revisited”
The American Mathematical Monthly
Volume 92 (1985) pages 584-587

Varberg extends the theorem to cases with polygons with holes in them.  An interesting read.

Branko Grünbaum and G. C. Shephard
“Pick’s Theorem”
The American Mathematical Monthly
Volume 100 (1993) pages 150-161

For young students, the best part of this is the introduction in which the authors describe an applied mathematical approach to timber management, which unknown to the speaker, was Pick's Theorem.

My favorite article to date is by an Eighth grade student, Chris Polis, from Papua, New Guinea in 1991.  He found generalizations of Pick's Thm to several different lattices, Triangular, hexagonal, and a tesselation of the plain by isosceles trapezoids.  He found individual formulas for these, then found a general one for all of them including Pick's Thm, \( A = \frac{e + 2 (i - 1)}{P-2}\) where P is the number of edges in the lattice, e is boundary points and i is interior points.  For a square this would give P=4 and the formula becomes \( A = \frac{e + 2 (i - 1)}{4-2}\) which simplifies to Pick's Thm.  (Wonder what he is doing today?)

You can use this Desmos lattice I plucked from google to try the triangular formula with the 9 boundary points on the edge and 3 in the interior, and confirm that the interior has 13 unit equilateral triangles in area.  

I recently saw another use of the theorem on twitter to answer the following question.  Several people found the right answer with trig, but one saw a different easy solution.  

Draw the lattice and use Pick's Theorem.  

Did you notice the triangle is a right triangle?  

Recently I came across an extension of Pick's theorem that would cover shapes with holes and such that make the original theorem not work (although in many cases a little common sense will still solve them.)  

.The new approach was in the article "Counting Parallel Segments: New Variants of Pick’s Area Theorem" by Alexander Belyaev & Pierre-Alain Fayolle.  I found it in the Mathematical Intelligencer ,Volume 41, pages 1–7, (2019).  

The method counts parallel line segments and parts of segments to find the area of such non-traditional figures.  

The line segments in dark blue are called interior segments,  They contribute one unit each to the area.  In the figure (a) there are eight of them.  The red line segments are called exterior segments, and they add 1/2 a unit.  The light blue segments are partial segments and also count 1/2 unit each. [Notice in the one passing through an interior point (blue) there are two partial segments, one above and one below. ]  So the area would be 8 + 2/2 +4/2 = 11. sq units

Try the (b) image for yourself.  I'll put the answer below after a brief side note; do take time to try one diagram that does work with the regular Pick method, such as the one near the beginning of this article.  It should work for them also.

How many red(edge) lines did you use?  How many drk blue(interior) segments ?  How many partials?

I had two reds on the left edge, two drk blue and one partial on the next vertical row, four drk blue on the third vertical row, and one drk blue on the fourth edge.  That should add up to the same 8 1/2 units as we got the "Pick" way.

OK  so did you get 3 + 2/2 + 4/2 for a total of six square units for the figure (b)?

I think this is an interesting method, and the parallels Don't have to be vertical, horizontal or diagonal at any slope.  try those and see if you get the same area each time.

Good luck!

On This Day in Math - February 20

 

A mathematician will recognise Cauchy, Gauss, Jacobi or Helmholtz after reading a few pages, just as musicians recognise, from the first few bars, Mozart, Beethoven or Schubert.
~Ludwig Boltzmann

The 51st day of the year; 51 is the number of different paths from (0,0) to (6,0) made up of segments connecting lattice points that can only have slopes of 1, 0, or -1 but so that they never go below the x-axis. These are called Motzkin Numbers.

\(\pi(51) = 15\), the number of primes less than 51 is given by it's reversal, 15.

Jim Wilder pointed out that 51 is the smallest number that can be written as a sum of primes  with the digits 1 to 5 each used once  2 + 3 + 5 + 41 = 51 (Students might explore similar problems using first n digits 2-9)

A triangle with sides 51, 52 and 53 has an integer area 1170 units².

Diophantus (around 250 AD) solved 26y^2 +1 = x^2 for (x,y) = (51, 10)

You may recognize the sequence of Perfect Numbers 6, 28, 496, 8128...(and then they start getting really big).  The fifth is 33,550,336, and if you have this one or the next memorized you are a very unusual person.  But how many are there? As of December of 2018, there are only 51 known.  The largest presently known is   2^82589932 × (2^82589933 − 1).  It has just a little less than 50 million digits.  

And like any odd number, 51 is the sum of two consecutive numbers, 25+26 , and the difference of their squares \(26^2 - 25^2\)

And I just found this unusual reference, "Don’t be baffled if you see the number 51 cropping up in Chinese website names, since 51 sounds like 'without trouble' or 'carefree' in Chinese." at the Archimedes Lab

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EVENTS

1639  Responding to Mersenne's comment that the sum of the divisors of 360 form a ratio of 9/4 with 360, Fermat responds that 2016 has the same property. For 2024 the ratio is 540/253, about 2.134

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1648 A letter from Fermat through Frenicle to Digby reached Wallis saying that Fermat had solved equations of the type x²-Ay² = 1 for all non-square values up to 150. Thus begins the saga of the mis-naming of Pell's equation. *Edward Everett Whitford, The Pell Equation.  

As early as 400 BC in India and Greece, mathematicians studied the numbers arising from the n = 2 case of Pell's equation, probably as approximations for square roots.

Pell's equation for n = 2 and six of its integer solutions *Wik

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1729 A Letter from Gabriel Cramer, Prof. Math. Genev. to James Jurin, M. D. and F. R. S. to be read at the Royal Society, gives an “account of an Aurora Borealis Attended with Unusual Appearances” . The borealis occurred on Feb 15, and the letter was sent on Feb 20. *Transactions of RSI

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1807 Sophie Germain writes to Gauss informing him that she is the person who had written to him using the name M. LeBlanc. In closing she writes her hope that this will not change their correspondence. His Response on April 30 would assure her it had not.

*Sophie Germain: An Essay in the History of the Theory of Elasticity

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In 1835, Charles Darwin, on his H.M.S. Beagle voyage reached Chile, and experienced a very strong earthquake and shortly afterward saw evidence of several feet of uplift in the region. He repeated measurement a few days later, and found the land had risen several feet. He had proved that geological changes occur even in our own time. Lyell's principles were based on the concept of a steady-state, nondirectional earth whereby uplift, subsidence, erosion, and deposition were all balanced. Thereby, Darwin coupled in his mind this dramatic evidence of elevation with accompanying subsidence and deposition. Thus he hypothesized that coral reefs of the Pacific developed on the margins of subsiding land masses, in the three stages of fringing reef, barrier reef, and atoll.*TIS

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1905 Lise Meitner had entered the University of Vienna in October 1901. She was particularly inspired by Boltzmann, and was said to often speak with contagious enthusiasm of his lectures. Her dissertation was supervised by Franz Exner and his assistant Hans Benndorf. Her thesis, titled Prüfung einer Formel Maxwells ("Examination of a Maxwell Formula"), was submitted on 20 November 1905 and approved on 28 November. She was examined orally by Exner and Boltzmann on 19 December, and her doctorate was awarded on 1 February 1906.

 She became the second woman to earn a doctoral degree in physics at the University of Vienna, after Olga Steindler who had received her degree in 1903; the third was Selma Freud, who worked in the same laboratory as Meitner, and received her doctorate later in 1906.Freud is also known as  founder of the first official Salvation Army corps in Vienna.

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1913 It was on, or around, this day that the Three Sisters Radio towers near Arlington, Va went into service. In an area called Radio, near the Columbia Pike and Courthouse Road. Virginia. It was a neighborhood named for the old U.S. Navy Wireless Station. The tallest of the three towers was 45 feet taller than the Washington Monument, and second only to the Eiffel Tower in the world.

The Navy opened Radio Arlington, call sign NAA, in 1913, launching the U.S. military’s global communications system on Fort Myers. A streetcar stop was even named “Radio.’’ Old Radio Arlington marked the first time the term “radio’’ was used in communications, according to Nan and Ross Netherton’s book “Arlington County in Virginia: A Pictorial History,” which was published in 1987. In the days of Marconi and other radio pioneers, the new communications mode was called “wireless telegraphy.’’

*The 625 Sentinal

At Tenwatts Blog I found that there is a marker outside the present Dept. of Defense facility there:

"Three radio towers similar to the Eiffel tower were erected here in 1913. One stood 600 feet, and the other two 450 feet above the 200-foot elevation of the site. The word "radio" was first used instead of "wireless" in the name of this naval communications facility. The first trans-Atlantic voice communication was made between this station and the Eiffel tower in 1915. The nation set its clocks by the Arlington Radio time signal and listened for its broadcast weather reports. The towers were dismantled in 1941, as a menace to aircraft approaching the new Washington National Airport."

I also found the nice postcard showing the three sisters (and some additions) taken from Arlington National Cemetery. His post suggests that the towers were eventually dismantled.

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1947   Before John Glenn, Neil Armstrong, or Sally Ride could step into a spacesuit and rocket into the annals of history, an intrepid bunch of fruit flies had to pave the way.

Of course, the flies didn’t have much of a say when they blasted off on a V-2 rocket launched from New Mexico’s White Sands Proving Grounds on Feb. 20, 1947. But as the first animals intentionally shot into and returned alive from space, the insects’ journey heralded a new age in space research and exploration. The flies’ flight traces to the summer of 1945, when a load of German-made V-2 rocket parts arrived at White Sands, a newly established military testing area that is “almost as isolated as a valley of the Moon, which it resembles,” wrote Lloyd Mallan in Men, Rockets and Space Rats, an on-the-ground chronicle of early spaceflight efforts published in 1955. The U.S. also imported a group of scientists and engineers from postwar Germany to assist with experiments using the V-2s, part of the intelligence operation later known as Operation Paperclip.

Up they went on Feb. 20, 1947. The flies reached an altitude of 109 kilometers in 190 seconds (space starts at 100 kilometers, according to NASA), then parachuted back down to Earth, where they were recovered alive and examined by biologists. “Analysis made by Harvard on recovered seeds and flies has shown that no detectable changes are produced by the radiation,” wrote U.S. Naval Research Laboratory nuclear physicist Ernst H. Krause in a report published that same year.

*APS Org

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1947 Computer pioneer Alan Turing suggests testing artificial intelligence with the game of chess in a lecture to the London Mathematical Society. Computers, he argued, must, like humans, be given training before their IQ is tested. A human mathematician has always undergone an extensive training. This training may be regarded as not unlike putting instruction tables into a machine, he said. One must therefore not expect a machine to do a very great deal of building up of instruction tables on its own.*CHM

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1966 The only verified example of a family producing five single children with coincidental birthdays is that of Catherine (1952), Coral (1953), Charles (1956), Claudia (1961), and Cecelia (1966), born to Ralph and Carolyn Cummins of Clintwood, VA. All on Feb 20th.  What is the probability of this happening? *VFR (RALPH? He should have changed his name.)

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1979 The German Democratic Republic issued a stamp commemorating the centenary of Einstein’s birth. It shows the Einstein tower in Potsdam and his famous formula E = mc². [Scott #1990]*VFR

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In 1996, a bright "new" star was discovered in Sagittarius by Japanese amateur astronomer Yukio Sakurai. It was found not to be a usual nova, but instead was a star going through a dramatic evolutionary state, re-igniting its nuclear furnace for one final blast of energy called the "final helium flash." It was only the second to be identified in the twentieth century. A star like the Sun ends its active life as a white dwarf star gradually cooling down into visual oblivion. Sakurai's Object had a mass a few times that of the Sun. Its collapse after fusing most of its hydrogen fuel to helium raised its temperature so much higher it began nuclear fusion of its helium remains. This was confirmed using its light spectrum to identify the elements present.*TIS

Sakurai's Object is surrounded by a planetary nebula created following the star's red giant phase around 8300 years ago.[26] It has been determined that the nebula has a diameter of 44 arcseconds and expansion velocity of roughly 32 km/s.

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2013 In celebration of the 100th anniversary of the publication of the three-volume version of Bertrand Russell and Alfred North Whitehead's Principia Mathematica, a London theater company staged the world premiere of a musical based on the epic mathematics text.
Performed by the Conway Collective based out of London's historic Conway Hall and written by Tyrone Landau, the play was described as "fascinating and unusual." *MAA DL

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BIRTHS

 1844 Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. He was one of the most important advocates for atomic theory at a time when that scientific model was still highly controversial. *Wik Trivia: Boltzmann's famous equation S = K log W (where S = entropy, K = Boltzmann's constant, and W = probability of a particular state) was inscribed as an epitaph on Boltzmann's tombstone. *Wik After obtaining his doctorate, he became an assistant to his teacher Josef Stefan. Boltzmann's fame is based on his invention of statistical mechanics, independently of Willard Gibbs. Their theories connected the properties and behaviour of atoms and molecules with the large scale properties and behaviour of the substances of which they were the building blocks. He also worked out a kinetic theory of gases, and the Stefan-Boltzmann law concerning a relationship between the temperature of a body and the radiation it emits. His firm belief and defense of atomism (that all matter is made of atoms) against hostile opposition to this new idea, may have contributed to his suicide in 1906. *TIS

He was an incredible physicist, but an equally talented teacher.  "In her (Lise Meitner) second university year (1902), ....A fairly typical curriculum, ... unusual in one respect, it was taught by ... Ludwig Boltzmann."  Fifty years later the former student would recall his lectures as "the most beautiful and stimulating I have ever heard.....He himself was so enthusiastic about everything he taught us that one left every lecture with the feeling that a new and beautiful world had been revealed." Lise Meitner by Ruth Lewin Sime

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1860 Mathias Lerch​ (20 February 1860, Milínov - 3 August 1922, Schüttenhofen) was an eminent Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory. He studied in Prague and Berlin, and held teaching positions at the Czech Technical Institute in Prague, the University of Fribourg in Switzerland, the Czech Technical Institute in Brno, and Masaryk University in Brno; he was the first mathematics professor at Masaryk University when it was founded in 1920. In 1900, he was awarded the Grand Prize of the French Academy of Sciences for his number-theoretic work. The Lerch zeta-function is named after him as is the Appell–Lerch sum.*Wik

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1910 Esther (Klein) Szekeres (20 February 1910 – 28 August 2005) was a Hungarian–Australian mathematician with an Erdős number of 1. She was born to Ignaz Klein in a Jewish family in Budapest, Kingdom of Hungary in 1910. As a young woman in Budapest, Klein was a member of a group of Hungarians including Paul Erdős, George Szekeres and Paul Turán that convened over interesting mathematical problems.
In 1933, Klein proposed to the group a combinatorial problem that Erdős named as the Happy Ending problem as it led to her marriage to George Szekeres in 1937, with whom she had two children.
Following the outbreak of World War II, Esther and George Szekeres emigrated to Australia after spending several years in Hongkew, a community of refugees located in Shanghai, China. In Australia, they originally settled in Adelaide before moving to Sydney in the 1960s.
In Sydney, Esther lectured at Macquarie University and was actively involved in mathematics enrichment for high-school students. In 1984, she jointly founded a weekly mathematics enrichment meeting that has since expanded into a program of about 30 groups that continue to meet weekly and inspire high school students throughout Australia and New Zealand.
In 2004, she and George moved back to Adelaide, where, on 28 August 2005, she and her husband passed away within an hour of each other *Wik

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1926 Kenneth Harry Olsen (February 20, 1926 – February 6, 2011) was an American engineer who co-founded Digital Equipment Corporation (DEC) in 1957 with colleague Harlan Anderson *Wik

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1929 Madan Lal Puri ( Sialkot in Pakistan , 20 February 1929 ) is an India statistician important in the context of nonparametric statistics and also occupied the fuzzy sets .

He built his career in the United States. He was born on February 20, 1929 in Sialkot, and is known for his work in mathematics which has had profound effects on the way statistics is understood and applied. He has won many honors and awards, including the Bicentennial Medal from Indiana University, Bloomington.

*Wik

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1931 John Willard Milnor (20 Feb 1931, )American mathematician who was awarded the Fields Medal in 1962 for his his proof that a 7-dimensional sphere can have 28 different differential structures. This work opened up the new field of differential topology. Milnor's theorem shows that the total curvature of a knot is at least 4. In the 1950's, Milnor did a substantial amount of work on algebraic topology in which he constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. Since the 1970's his interest is in dynamics, especially holomorphic dynamics. Milnor served the American Mathematical Society as vice president (1975-76) and was awarded the Wolf Prize in 1989. *TIS

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1948 Andrew Christopher Fabian, OBE, FRS (20 February 1948 - ) is a British astronomer and astrophysicist. He is a Royal Society Research Professor at the Institute of Astronomy, Cambridge, and Vice-Master of Darwin College, Cambridge. He was the President of the Royal Astronomical Society from May 2008 through to 2010. He is an Emeritus Professor of Astronomy at Gresham College, a position in which he delivered free public lectures within the City of London between 1982 and 1984. He was also editor-in-chief of the astronomy journal Monthly Notices of the Royal Astronomical Society. He was educated at King's College London (BSc, Physics) and University College London (PhD).
His current areas of research include galaxy clusters, active galactic nuclei, strong gravity, black holes and the X-ray background. He has also worked on X-ray binaries, neutron stars and supernova remnants in the past. Much of his research involves X-ray astronomy and high energy astrophysics. His notable achievements include his involvement in the discovery of broad iron lines emitted from active galactic nuclei, for which he was jointly awarded the Bruno Rossi Prize. He is author of over 800 refereed articles and head of the X-ray astronomy group at the Institute of Astronomy. Fabian was awarded the Dannie Heineman Prize for Astrophysics by the American Astronomical Society in 2008 and the Gold Medal of the Royal Astronomical Society in 2012 *Wik

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DEATHS

1762 Tobias Meyer (17 Feb 1723; 20 Feb 1762 at age 38) German astronomer who developed lunar tables that greatly assisted navigators in determining longitude at sea. Mayer also discovered the libration (or apparent wobbling) of the Moon. Mayer began calculating lunar and solar tables in 1753 and in 1755 he sent them to the British government. These tables were good enough to determine longitude at sea with an accuracy of half a degree. Mayer's method of determining longitude by lunar distances and a formula for correcting errors in longitude due to atmospheric refraction were published in 1770 after his death. The Board of Longitude sent Mayer's widow a payment of 3000 pounds as an award for the tables. *TIS Leonhard Euler described him as 'undoubtedly the greatest astronomer in Europe'. 

In 1748, he began a study of the lunar surface. He decided to improve the accuracy of luar maps by measuring the position of each lunar crater with a micrometer built into his telescope. The result was a kind of lunar gazetteer, with longitudes and latitudes for hundreds of lunar features, which he intended to turn into both a lunar globe and a lunar map. The globe was never finished, but a prospectus was issued in 1750 with two sample engravings.

*Wik

More notes on Meyer can be found on this blog at the Board of Longitude Project from the Royal Museums at Greenwich. Another nice blog by Thony Christie, The Renaissance Mathematicus tells of Meyer's measurement of the Moon's distance, and the importance of that measurement.

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1778 Laura Maria Catarina Bassi (31 Oct 1711 in Bologna, Papal States, 20 Feb 1778 in Bologna,

Portrait of Laura Bassi at
 the University of Bologna.*Wik

Papal States) was an Italian physicist and one of the earliest women to gain a position in an Italian university. *SAU She was the first woman in the world to earn a university chair in a scientific field of studies. She received a doctoral degree from the University of Bologna in May 1732, only the third academic qualification ever bestowed on a woman by a European university, and the first woman to earn a professorship in physics at a university in Europe. She was the first woman to be offered an official teaching position at a university in Europe.
In 1738, she married Giuseppe Veratti, a fellow academic with whom she had twelve children. After this, she was able to lecture from home on a regular basis and successfully petitioned the University for more responsibility and a higher salary to allow her to purchase her own equipment.
One of her principal patrons was Pope Benedict XIV. He supported less censorship of scholarly work, such as happened with Galileo, and he supported women figures in learning, including Agnesi.
She was mainly interested in Newtonian physics and taught courses on the subject for 28 years. She was one of the key figures in introducing Newton's ideas of physics and natural philosophy to Italy. She also carried out experiments of her own in all aspects of physics. In order to teach Newtonian physics and Franklinian electricity, topics that were not focused in the university curriculum, Bassi gave private lessons.[6] In her lifetime, she authored 28 papers, the vast majority of these on physics and hydraulics, though she did not write any books. She published only four of her papers.[2] Although only a limited number of her scientific works were left behind, much of her scientific impact is evident through her many correspondents including Voltaire, Francesco Algarotti, Roger Boscovich, Charles Bonnet, Jean Antoine Nollet, Giambattista Beccaria, Paolo Frisi, Alessandro Volta. Voltaire once wrote to her saying "There is no Bassi in London, and I would be much happier to be added to your Academy of Bologna than that of the English, even though it has produced a Newton". *Wik

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1846 Antonio Abetti (19 Jun 1846, 20 Feb 1928 at age 81) Italian astronomer who was an authority on minor planets. At first a civil engineer, he became an astronomer at the University of Padua (1868-93), with an interest in positional astronomy and made many observations of small planets, comets and star occultations. In 1874, Abetti went to Muddapur, Bengal, to observe the transit of Venus across the sun's disk where his use of a spectroscope was the first use of this kind. Later, he became director at the Arcetri Observatory and Professor of astronomy at the University of Florence (1894-1921). The observatory had been founded by G. B. Donati in 1872, and Abetti equipped it with a new telescope that he had built in the workshops at Padua. He was active after retirement, until his death, and was followed by his son Giorgio.*TIS

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1955 Arthur Lee Dixon FRS (27 November 1867 — 20 February 1955) was a British mathematician and holder of the Waynflete Professorship of Pure Mathematics at the University of Oxford. The younger brother of Alfred Cardew Dixon, he was educated at Kingswood School and Worcester College, Oxford, becoming a Tutorial Fellow at Merton College in 1898 and the Waynflete Professor in 1922. Dixon was the last mathematical professor at Oxford to hold a life tenure, and although he was not particularly noted for his mathematical innovations he did publish many papers on analytic number theory and the application of algebra to geometry, elliptic functions and hyperelliptic functions. Elected a Fellow of the Royal Society in 1912 and serving as President of the London Mathematical Society from 1924 to 1926, *Wik

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1972 Maria Goeppert-Mayer (28 Jun 1906, 20 Feb 1972 at age 65) German physicist who shared one-half of the 1963 Nobel Prize for Physics with J. Hans D. Jensen of West Germany for their proposal of the shell nuclear model. (The other half of the prize was awarded to Eugene P. Wigner of the United States for unrelated work.) In 1939 she worked at Columbia University on the separation of uranium isotopes for the atomic bomb project. In 1949, she devised the shell nuclear model, which explained the detailed properties of atomic nuclei in terms of a structure of shells occupied by the protons and neutrons. This explained the great stability and abundance of nuclei that have a particular number of neutrons (such as 50, 82, or 126) and the same special number of protons. *TIS

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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

The Kiss Precise, Soddy's Circle Theorem

  Soddy's formula is another example of Stigler's law of eponymy, "No scientific discovery is named after its original discoverer." Stigler named the sociologist Robert K. Merton as the discoverer of "Stigler's law", so as to avoid this law about laws disobeying its very own decree.

Soddy's formula is about the relationship of the radii of four mutually tangent circles. The formula is sometimes called the "Kissing Circles Theorem". If four circles are all tangent to each other, then they must intersect at six distinct points. The first demonstration of this relationship between four mutually tangent circles (actually, one can be a line) was in 1643. Rene Descartes sent a letter to Princess Elisabeth of Bohemia in which he showed that the four radii, r₁, r₂, r₃, r₄, must be such that \( \frac{1}{r^2_{1}}+ \frac{1}{r^2_{2}}+\frac{1}{r^2_3}+\frac{1}{r^2_4}= \frac{1}{2}(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4} )^2\)

For this reason the theorem is often called Descarte's circle theorem. The figure shows four circles all externally tangent to each other, but could also be drawn with three tangent circles all inside, and tangent to, a fourth circle. The bend of this externally tangent circle is given a negative value, and thus the same equation provides its radius also.

The equation can be written much more easily, and usually is, using a notation of "bend". For each value let the "bend" equal the reciprocal of the radius, then \(\frac{1}{r_1} =b_1\)With this notation the formula can be written as \(b^2_1 + b^2_2+b^2_3+b^2_4=\frac{1}{2}(b_1+b_2+b_3+b_4)^2\).

It seems that it may also have been discovered about the same time in Japan. In the book, Sacred Mathematics: Japanese Temple Geometry, by Fukagawa Hidetoshi and Tony Rothman, there is an illustration of a complicated pattern of nested congruent circles, for which knowledge of the theorem would seem to be required, on a wooden tablet. It was a practice during the Edo period in Japan that people from every segment of society would inscribed geometry solutions on wooden tablets called sangaku and hang them as offerings in temples and shrines.
The Theorem was rediscovered and published in the 1841 The Lady's and Gentleman's diary by an amateur English Mathematician named Phillip Beecroft. Beecroft also observed that there exist four other circles that would each be mutually tangent at the same four points. These circles would have tangents perpendicular to the original circles tangents at each point of intersection. Both sets of Beecroft's circles are shown in this illustration from Mad Math.
Beecroft's circles are related to the use of a geometrical inversion in a circle which will invert the inner tangent circle to become an outer tangent circle. The circle of inversion between the two is the circle Beecroft uses that passes through the three points of tangent in the other three circles. (A nice explanation and illustration of this is at this AMS site.

In 1936 Sir Fredrick Soddy rediscovered the theorem again. Soddy may also be known to students of Science for receiving the Nobel Prize for Chemistry in 1921 for the discovery of the decay sequences of radioactive isotopes.  According to Oliver Sacks' wonderful book, Uncle Tungsten, Soddy also created the term "isotope" and was the first to use the term  "chain reaction".  In a strange "chain reaction" of ideas, Soddy played a part in the US developing an atomic bomb.  Soddy's book, The  Interpretation of Radium, inspired  H G Wells to write The World Set Free in 1914, and he dedicated the novel to Soddy's book. Twenty years later, Wells' book set Leo Szilard to thinking about the possibility of Chain reactions, and how they might be used to create a bomb, leading to his getting a British patent on the idea in 1936.  A few years later Szilard encouraged his friend, Albert Einstein, to write a letter to President Roosevelt about the potential for an atomic bomb.  The prize-winning science-fiction writer, Frederik Pohl, talks about Szilard's epiphany in Chasing Science (pg 25),

".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb.  There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row.  Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."

Perhaps Soddy's name is appropriate for the formula if only for the unique way he presented his discovery. He presented it in the form of a poem which is presented below.

The Kiss Precise
by
Frederick Soddy

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.

To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.

If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.

Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four,
The square of the sum of all five bends
Is thrice the sum of their squares.

In _Nature_, June 20, 1936

One may notice in the last verse that Soddy generalizes the theorem to five spheres. The extended theorem becomes: \[ b_1^2+b_2^2+b_3^2+b_4^2+b_5^2 = \frac13(b_1+b_2+b_3+b_4+b_5)^2. \]

Later   another verse was written by Thorold Gosset to describe the even more general   case in N dimensions for N+2 hyperspheres of the Nth dimension.

On August 15, 1936, only a few months after Soddy's poem had been published in Nature,  Gosset sent a copy of the poem to Donal Coxeter on the occasion of his wedding in the Round Church in Cambridge.  Gossett enclosed in his wedding congratulations, and his extension of the poem to the higher dimensions which were Coxeter's special area of study. It would be published in Nature the following year,
 

The Kiss Precise (Generalized) by Thorold Gosset

And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears,
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares -
As n + 2 such osculate
Each with an n + 1 fold mate
The square of the sum of all the bends
Is n times the sum of their squares.

In _Nature_
    January 9, 1937.

Fred Lunnon sent me a kind note correcting a typing oversight, and adding that

"The original result generalizes nicely to curved n-space with
curvature v [e.g. v^2 = +1 for elliptic space, -1 for hyperbolic]
in the form



        \((\sum_i x_i)^2 - n \sum_i x_i^2 = 2n v^2\)




where \(x_i\) denote the curvatures of n+2 mutually tangent spheres.
Example: n = 2, v = 0, x = [-1,2,2,3] is one solution, corresponding to
a unit circle in the plane enclosing circles of radii 1/2,1/2,1/3.



See Ivars Petersen "Circle Game" in Science News (2001) \bf 159 (16) p.254"

 

Fred admits he wasn't the first to prove this, but did manage to replicate it on his own (which impresses the heck out of me)... but THEN....... wait for it.... He wrote another poem verse to accompany this extension to higher dimensions...

  The Kiss Precise (Further Generalized) by Fred Lunnon



    How frightfully pedestrian

    My predecessors were

    To pose in space Euclidean

    Each fraternising sphere!

    Let Gauss' k squared be positive

    When space becomes elliptic,

    And conversely turn negative

    For spaces hyperbolic:

    Squared sum of bends is sum times n

    Of twice k squared plus squares of bends.

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On This Day in Math - February 19

  

Copernicus statue at Olsztyn Castle

It is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician.
~Karl Weierstrass

The 50th day of the year; 50 is the smallest number that can be written as the sum of two squares in two distinct ways 50 = 49 + 1 = 25 + 25. *Tanya Khovanova, Number Gossip (What is the next, or what is the smallest number that can be written as the sum of two squares in three distinct ways? For solution from Ben Vitale, see bottom of post)

50 is also expressible as the sum of distinct primes in two ways so that all consecutive primes 2-23 are used :50 = 2 + 5 + 7 + 17 + 19 = 3 + 11 + 13 + 23.

Heron of Alexandria used \( \sqrt{50} = 7 + \frac{1}{14} = \frac {99}{14}\)  A solution to the Pell equation \( x^2 - 50 y^2 =1 \)  in the First century AD
 
The number 50 is somewhat responsible for the area of number theory about partitions. In 1740 Philip Naudé the younger (1684-1747) wrote Euler from Berlin to ask “how many ways can the number 50 be written as a sum of seven different positive integers?” Euler would give the answer, 522, within a few days but would return to the problem of various types of partitions throughout the rest of his life.

EVENTS

1003 al-Biruni observed two lunar eclipses from Gurgān,(Azerbaijan)  one on 19 February and the other on 14 August. On 4 June of the following year, 1004, he observed a third lunar eclipse.  *Encylopedia . com

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Tartaglia's Masterpiece

1512 The French invaded Brescia, in Northern Italy, during the War of the League of Cambrai. The militia of Brescia defended their city for seven days. When the French finally broke through, they took their revenge by massacring the inhabitants of Brescia. By the end of battle, over 45,000 residents were killed. During the massacre, a French soldier sliced Niccolò's jaw and palate with a saber. This made it impossible for Niccolò to speak normally, prompting the nickname "Tartaglia" Tartaglia is perhaps best known today for his conflicts with Gerolamo Cardano over the solution of cubics. (see this blog for the unfortunate common mistake about Tartaglia's family name.)

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1549 Osiander wrote of Michael Stifel: “He has devised new numbers for the alphabet, namely the triangular numbers, and his fantasies are more absurd than before.” *VFR  In 1553 Stifel published the Cours (algebra) Christoph Rudolph's. He was also steadily engaged on the computation of the numbers in Daniel and the Apocalypse, and became the zealous opponent of Andreas Osiander. Earlier, in 1532, Stiefel published a treatise on the numbers in Daniel, entitled Ein Rechenbuchlein vom End Christi, in which he fixed the last day and hour to be Oct. 19, 1533, at 8 o'clock in the morning with the result that the peasants neglected their labors and lost their harvests, but sued for damages when the prediction was not fulfilled.

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1600 The Inquisition brought Giordano Bruno to the Campo dei Fiori in Rome’s center where they chained him to an iron stake and burned him alive for his beliefs that the earth rotated on its axis. *Amir Aczel, Pendulum, pg 9 (This date seems wrong. Thony Christie noted that " Bruno was executed on 17th Feb and not for his cosmology but for his heretical theology." Thanks... several other sources agree with Feb 17th date))

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1616 On February 19, 1616, the Inquisition asked a commission of theologians, known as qualifiers, about the propositions of the heliocentric view of the universe after Nicollo Lorin had accused Galileo of Heretical remarks in a letter to his former student, Benedetto Castelli. On February 24 the Qualifiers delivered their unanimous report: the idea that the Sun is stationary is "foolish and absurd in philosophy, and formally heretical since it explicitly contradicts in many places the sense of Holy Scripture..."; while the Earth's movement "receives the same judgement in philosophy and ... in regard to theological truth it is at least erroneous in faith."At a meeting of the cardinals of the Inquisition on the following day, Pope Paul V instructed Bellarmine to deliver this result to Galileo, and to order him to abandon the Copernican opinions; should Galileo resist the decree, stronger action would be taken. On February 26, Galileo was called to Bellarmine's residence, and accepted the orders.*Wik

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1671/72 Newton’s first publication appears as a letter in the Philosophical Transactions. It deals with his new theory of light, showing that a prism separates white light into its component colors. Huygens, Hooke and others objected so strongly that he vowed not to publish again. Fortunately that vow was not kept. *VFR The full text of that publication is here.

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1787 William Herschel's 40-foot telescope, also known as the Great Forty-Foot telescope, was a reflecting telescope constructed between 1785 and 1789 at Observatory House in Slough, England. It used a 48-inch (120 cm) diameter primary mirror with a 40-foot-long (12 m) focal length (hence its name "Forty-Foot"). It was the largest telescope in the world for 50 years. It may have been used to discover Enceladus and Mimas, the 6th and 7th moons of Saturn. It was dismantled in 1840 by Herschel's son John Herschel due to safety concerns; today the original mirror and a 10-foot (3.0 m) section of the tube remain.

 The first observation with the telescope was on 19 February 1787, when Herschel pointed the then-incomplete telescope towards the Orion nebula, which he observed by crawling into the telescope and using a hand-held eyepiece: "The apparatus for the 40-foot telescope was by this time so far completed that I could put the mirror into the tube and direct it to a celestial object; but having no eye-glass fixed, not being acquainted with the focal length which was to be tried, I went into the tube, and laying down near the mouth of it I held the eye-glass in my hand, and soon found the place of the focus. The object I viewed was the nebula in the belt of Orion, and I found the figure of the mirror, though far from perfect, better than I had expected. It showed four small stars in the nebula and many more. The nebula was extremely bright."

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In 1855, M. Le Verrier presented the first weather map at the French Academy of Sciences.*TIS A storm on November 14, 1854 destroyed the French warship Henri IV and damaged other British and French vessels on the Black Sea involved in the Crimean War. A report from the state-supported Paris Observatory indicated that barometric readings showed that the storm had passed across Europe in about four days. Urban Leverrier, director of the Paris Observatory, concluded that had there been a telegraph line between Vienna and the Crimea, the British and French fleets could have received warnings. An earlier map is mentioned, but not shown in a letter dated Dec 1, 1816 in Gilbert's Annalen der Physik from Heinrich Wilhelm Brandes *Report of The International Meterological Congress, 1893

Meteorological map of the 16th of January 1865 — credits : Observatoire de Paris

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1876 Sylvester began his duties at the newly founded Johns Hopkins, *TIS

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1880, the photophone was demonstrated by Alexander Graham Bell and Charles Sumner Tainter. In their device, a mirrored silver disc was made to vibrate by speech from a speaking tube. Light reflected off the disc was focused by a parabolic dish onto a selenium photocell. The variations in the reflected light were converted into electrical signals carried to headphones.

 It was invented jointly by Alexander Graham Bell and his assistant Charles Sumner Tainter on February 19, 1880, at Bell's laboratory at 1325 L Street in Washington, D.C. Both were later to become full associates in the Volta Laboratory Association, created and financed by Bell.

While honeymooning in Europe with his bride Mabel Hubbard, Bell likely read of the newly discovered property of selenium having a variable resistance when acted upon by light, in a paper by Robert Sabine as published in Nature on 25 April 1878. In his experiments, Sabine used a meter to see the effects of light acting on selenium connected in a circuit to a battery. However Bell reasoned that by adding a telephone receiver to the same circuit he would be able to hear what Sabine could only see.

A photophone receiver and headset, one half of Bell and Tainter's optical telecommunication system of 1880

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1901 Messages from Mars reported in Collier's Magazine. While conducting experiments on high-frequency electrical transmission in 1899 in his Colorado Springs, Colorado laboratory, Nikola Tesla picked up cosmic radio waves on his instruments. Announcing this development, he publicly opined that the messages came from outer space, possibly from inhabitants of Mars. In a Collier’s Weekly article dated February 19, 1901, Tesla wrote, “At the present stage of progress, there would be no insurmountable obstacle in constructing a machine capable of conveying a message to Mars … What a tremendous stir this would make in the world! How soon will it come?” Later discoveries revealed that Tesla had actually picked up common radio waves emitted by interstellar gas clouds. *History. Com

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1940 Edwin Hubble wrote in a letter to Harlow Shapley that he had determined the distance to the "Andromeda nebula". He included this graph. HT Massimo

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1946 Alan Turing Presents the “Proposal for the Development in the Mathematics Division of an Automatic Computing Engine (ACE).”
This research proposal was presented to a meeting of the Executive Committee of the National Physical Laboratory (NPL) in Teddington, England, and approved at a second meeting held a month later.
Turing based this research on von Neumann’s First Draft of a Report on the EDVAC. He had studied it in summer 1945 when he was recruited by J.R. Womersley to join the staff of the NPL. *CHM

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1957 William H. Spencer obtained a patent on February 19, 1957 for a new "Method for Making Elastic Bands" which produced rubber bands in an Open Ring design. 

 The rubber band was patented in England on March 17, 1845 by Stephen Perry

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1971 The first warrant is issued to search a computer storage. Although the requirements for obtaining such a warrant were similar to those for searching a home, they ushered in a new era that would lead to increasingly sophisticated methods of encryption to hide computer files from law enforcement agents.*CHM

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1972 The New Yorker published an article by A. Adler on “Mathematics and Creativity” that was not well received by the mathematical community. See The [old] Mathematical Intelligencer, no. 2. *VFR An abstract is here

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BIRTHS

1473 Nicolaus Copernicus Polish astronomer who proposed that the planets have the Sun as the fixed point to which their motions are to be referred; that the Earth is a planet which, besides orbiting the Sun annually, also turns once daily on its own axis; and that very slow, long-term changes in the direction of this axis account for the precession of the equinoxes *TIS
An advance copy of his work De revolutionibus orbium coelestium was presented to Copernicus. On the same day he died. *VFR
Over 450 years after his death, Copernicus was reburied in the cathedral at Frombork on Poland’s Baltic coast. The astronomer whose ideas were once declared heresy by the Vatican—was reburied with full religious honors.

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1837 Aleksandr Nikolayevich Korkin (3 March [O.S. 19 February] 1837–September 1, 1908, all New Style) was a Russian mathematician. He made contribution to the development of partial differential equations. After Chebyshev, Korkin was the most important initiator of the formation of the Saint Petersburg Mathematical School*Wik

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1863 Axel Thue(19 Feb 1863 in Tönsberg, Norway - 7 March 1922 in Oslo, Norway) Thue studied Diophantine equations, showing that, for example, y³ - 2x² = 1 cannot be satisfied by infinitely many pairs of integers. Edmund Landau, in 1922, described Thue's work as, ".. the most important discovery in elementary number theory that I know. "
Thue's Theorem states, " If f (x, y) is a homogeneous polynomial with integer coefficients, irreducible in the rationals and of degree > 2 and c is a non-zero integer then  f (x, y) = c has only a finite number of integer solutions." *SAU

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1866 Thomas Jefferson Jackson See (19 Feb 1866 in Montgomery City, Missouri - 4 July 1962 in Oakland, California, USA) was an U S astronomer who studied in the University of Missouri and in Berlin. He fell out with his astronomical colleagues and was eventually banned from publishing. He spend the last part of his life arguing against Einstein's Theory of Relativity. *SAU

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1889 Sir Ernest Marsden (19 Feb 1889, 15 Dec 1970) British-born New Zealand nuclear physicist who worked under Ernest Rutherford investigating atomic structure with Hans Geiger. Marsden visually counted scintillations from alpha particles after passing through gold foil and striking a phosphorescent screen. That some of these were observed scattered at surprisingly large angles led to Rutherford's theory of the nucleus as the massive, tiny centre of the atom. Later, Marsden's own experiments, working in New Zealand, hinted suggested transmutation of elements was possible when alpha particles bombarding nitrogen nuclei produced scattered particles of greater speed than the original radiation. *TIS

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1924 Arnljot Høyland (19 February 1924 – 21 December 2002) was a Norwegian mathematical statistician.

Høyland was born in Bærum. He studied at the University of Oslo and later at the University of California, Berkeley in the USA. While a student he worked for the intelligence department at the Norwegian High Command, a military officer with the rank of Major. He lectured at the University of Oslo from 1959 to 1965, and then at the Norwegian Institute of Technology, eventually as a Professor of mathematical statistics. He published the textbooks Sannsynlighetsregning og statistisk metodelære (two volumes) in 1972 and 1973.

In 1944 Høyland composed the melody for Alf Prøysen's song "Julekveldsvise". 

He was decorated Knight, First Class of the Order of St. Olav in 1995. *Wik

Alf Prøysen (left) and Arnljot Høyland (right),

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1964  Jennifer Anne Doudna ForMemRS (born February 19, 1964  in Washington, D.C) is an American biochemist who has done pioneering work in CRISPR gene editing, and made other fundamental contributions in biochemistry and genetics. Doudna was one of the first women to share a Nobel in the sciences. She received the 2020 Nobel Prize in Chemistry, with Emmanuelle Charpentier, "for the development of a method for genome editing.

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DEATHS

1553 Erasmus Reinhold (October 22, 1511 – February 19, 1553) was a German astronomer and mathematician, considered to be the most influential astronomical pedagogue of his generation. He was born and died in Saalfeld, Saxony.
He was educated, under Jacob Milich, at the University of Wittenberg, where he was first elected dean and later became rector. In 1536 he was appointed professor of higher mathematics by Philipp Melanchthon. In contrast to the limited modern definition, "mathematics" at the time also included applied mathematics, especially astronomy. His colleague, Georg Joachim Rheticus, also studied at Wittenberg and was appointed professor of lower mathematics in 1536.
Reinhold catalogued a large number of stars. His publications on astronomy include a commentary (1542, 1553) on Georg Purbach's Theoricae novae planetarum. Reinhold knew about Copernicus and his heliocentric ideas prior to the publication of De revolutionibis and made a favourable reference to him in his commentary on Purbach. However, Reinhold (like other astronomers before Kepler and Galileo) translated Copernicus' mathematical methods back into a geocentric system, rejecting heliocentric cosmology on physical and theological grounds.
It was Reinhold's heavily annotated copy of De revolutionibus in the Royal Observatory, Edinburgh that started Owen Gingerich on his search for copies of the first and second editions which he describes in The Book Nobody Read.[5] In Reinhold's unpublished commentary on De revolutionibus, he calculated the distance from the Earth to the sun. He "massaged" his calculation method in order to arrive at an answer close to that of Ptolemy.*Wik

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1622 Sir Henry Savile (30 Nov 1549 in Bradley (near Halifax), Yorkshire, England - 19 Feb 1622 in Eton, Berkshire, England) Savile was an English mathematician who founded professorships of geometry and astronomy at Oxford. It is interesting to read Savile's comments in these lectures on why he felt that mathematics at that time was not flourishing. Students did not understand the importance of the subject, Savile wrote, there were no teachers to explain the difficult points, the texts written by the leading mathematicians of the day were not studied, and no overall approach to the teaching of mathematics had been formulated. Of course, as we shall see below, fifty years later Savile tried to rectify these shortcomings by setting up two chairs at the University of Oxford. *SAU

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1799 Jean-Charles Borda, (4 May 1733 in Dax, France - 19 Feb 1799 in Paris, France) a major figure in the French navy who participated in sev­eral scientific voyages and the American revolution. Besides his contributions to navigational instruments he did important work on fluid mechanics, even showing that Newton’s theory of fluid resistance was untenable. He is best known for the voting system he created in 1770.*VFR (The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all votes have been counted the candidate with the most points is the winner. Because it sometimes elects broadly acceptable candidates, rather than those preferred by the majority, the Borda count is often described as a consensus-based electoral system, rather than a majoritarian one.The Borda count is a popular method for granting awards for sports in the United States, and is used in determining the Most Valuable Player in Major League Baseball, and by the Associated Press and United Press International to rank teams in NCAA sports, to determine the winner of the Heisman Trophy.) [He was one of the main driving forces in the introduction of the decimal system. Borda made good use of calculus and experiment to unify areas of physics. For his surveying, he also developed a series of trigonometric tables. In 1782, while in command of a flotilla of six French ships, he was captured by the British. Borda's health declined after his release. He is one of 72 scientists commemorated by plaques on the Eiffel tower.]*TIS

With the advent of the metric system, after the French Revolution it was decided that the quarter circle should be divided into 100 angular units, currently known as the gradian, instead of 90 degrees, and the gradian into 100 centesimal minutes of arc (centigrades) instead of 60 arc-minutes. This required the calculation of trigonometric tables and logarithms corresponding to the new unit and instruments for measuring angles in the new system.

Borda constructed instruments for measuring angles in the new units (the instrument could no longer be called a "sextant") which was later used in the arc measurement of the meridian between Dunkirk and Barcelona by Delambre to determine the radius of the Earth and thus define the length of the metre.  *Wik 

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1897 Karl (Theodor Wilhelm) Weierstrass (31 Oct 1815, 19 Feb 1897 at age 81) was a German mathematician who is known as the "father of modern analysis" for his rigour in analysis led to the modern theory of functions, and considered one of the greatest mathematics teachers of all-time. He was doing mathematical research while a secondary school teacher, when in 1854, he published a paper on Abelian functions in the famous Crelle Journal. The paper so impressed the mathematical community that he shortly received an honorary doctorate and by 1856, he had a University appointment in Berlin. In 1871, he demonstrated that there exist continuous functions in an interval which have no derivatives nowhere in the interval. He also did outstanding work on complex variables.*TIS Weierstrass died peacefully at the age of 82 at his home in Berlin after a long illness culminating in influenz. It is reported that his last wish was that the priest say nothing in his praise at the funeral, but to restrict the services to the customary prayers. *VFR

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1908 Paul Matthieu Hermann Laurent (2 Sept 1841 in Echternach, Luxembourg - 19 Feb 1908 in Paris, France) He developed statistical formulas for the calculation of actuarial tables and studied heat conduction. *VFR

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1916 Ernst Mach (18 Feb 1838; 19 Feb 1916 at age 77) Austrian physicist and philosopher who established important principles of optics, mechanics, and wave dynamics. His early physical works were devoted to electric discharge and induction. Between 1860 and 1862 he studied in depth the Doppler Effect by optical and acoustic experiments. He introduced the "Mach number" for the ratio of speed of object to speed of sound is named for him. When supersonic planes travel today, their speed is measured in terms that keep Mach's name alive. His lifetime interest, however, was in psychology and human perception. He supported the view that all knowledge is a conceptual organization of the data of sensory experience (or observation). *TIS

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1842 Joseph Valentin Boussinesq (13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.
In 1897 he published Théorie de l' écoulement tourbillonnant et tumultueux des liquides, a work that greatly contributed to the study of turbulence and hydrodynamics.

John Scott Russell experimentally observed solitary waves in 1834 and reported it during the 1844 Meeting of the British Association for the advancement of science. Subsequently, this was developed into the modern physics of solitons. In 1871, Boussinesq published the first mathematical theory to support Russell's experimental observation, and in 1877 introduced the Korteweg–De Vries equation. In 1876, Lord Rayleigh published his mathematical theory to support Russell's experimental observation. At the end of his paper, Rayleigh admitted that Boussinesq's theory came before his.

In 1897, he published Théorie de l'écoulement tourbillonnant et tumultueux des liquides ("Theory of the swirling and agitated flow of liquids"), a work that greatly contributed to the study of turbulence and hydrodynamics.

The word "turbulence" was never used by Boussinesq. He used sentences such as "écoulement tourbillonnant et tumultueux" (vortex or tumultuous flow). The first mention of the word "turbulence" in French or English scientific fluid mechanics literature (the word "turbulence" existed in other context) can be found in a paper by Lord Kelvin in 1887 *Wik

(While Kelvin is credited with introducing the term, Leonardo da Vinci is recognized for early observations and descriptions of turbulent flow in water, even using the Italian word "turbolenza" to describe it.)

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1938 Edmund Georg Hermann Landau (14 Feb 1877 in Berlin, Germany - 19 Feb 1938 in Berlin, Germany) Although famous as a number theorist, he is best known for his textbooks which are written in an austere definition-theorem-proof style. His Grundlagen der Analysis is an excellent treatment of the development of our number systems from the Peano postualates. Reading this book is a good way to learn mathematical German. But if you are lazy, it has been translated into English. *VFR Landau gave the first systematic presentation of analytic number theory and wrote important works on the theory of analytic functions of a single variable.*SAU Legend has it that at the age of three, when is mother forgot her umbrella in a carriage, he replied, "It was number 354," and the umbrella was quickly re-acquired.

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1940 Otto Toeplitz died in Jerusalem, after having left Germany in the Spring of 1939. He made lasting contributions to the theory of integral equations and the theory of functions of infinitely many variables. Today he is best remembered for two popular works which have been translated into English: The Enjoyment of Mathematics (original 1930, 1957), and The Calculus: A Genetic Approach (first published 1949; English 1963). These are some of the most successful attempts to bring higher mathematics to the general public. The later shows his deep interest in the history of mathematics; every calculus teacher could profit from reading it. *VFR

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1990 Otto Neugebauer, historian of ancient and medieval mathematics and astronomy. *VFR
(May 26, 1899 – February 19, 1990) He was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences in antiquity and into the Middle Ages. By studying clay tablets he discovered that the ancient Babylonians knew much more about mathematics and astronomy than had been previously realized. The National Academy of Sciences has called Neugebauer "the most original and productive scholar of the history of the exact sciences, perhaps of the history of science, of our age." *Wik

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  1937 Robert Coleman Richardson (June 26, 1937 – February 19, 2013) was an American physicist who (with Douglas Osheroff and David Lee) was awarded the 1996 Nobel Prize for Physics for their discovery of superfluidity in the isotope helium-3. As helium is reduced in temperature toward almost absolute zero, a strange phase transition occurs, and the helium takes on the form of a superfluid. The atoms had until that point had moved with random speeds and directions. But as a superfluid, the atoms then move in a co-ordinated manner! 

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@BenVitale: smallest number w/ 3 representations: \( 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2+ 15^2\)


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