Sunday, 9 February 2025

On This Day in Math - February 9

 

da Vinci's Stellated Dodecahedron from divina proportione


To argue with a man who has renounced the use of authority of reason is like administering medicine to the dead.
~Thomas Paine, American Patriot, and a pretty good bridge builder, perhaps pointing out something still very relevant to today's political discussions.


The 40th day of the year; in English forty is the only number whose letters are in alphabetical order.

There are 40 solutions on for the 7 queens problem.  placing seven chess queens on a 7x7 chessboard so that no two queens threaten each other.

-40 is the temperature at which the Fahrenheit and Celsius scales correspond; that is, −40 °F = −40 °C.

Euler first noticed (in 1772) that the quadratic polynomial P(n) = n2 + n + 41 is prime for all non-negative numbers less than 40.
Paul Halcke noted in 1719 that the product of the aliquot parts of 40 is equal to 40 cubed. 1*2*4*5*8*10*20 = 64000 = 403. He found the same is true for 24.

forty (quaranta)  is the root of the word quarantine, for the forty day period of isolation for visitors to Venice to control the plague.

And.... forty is the highest number ever counted to on Sesame Street.

The expanded collection of Number Factoids for Day 1-60  are available here



EVENTS

1498 Luca Pacioli was professor at Milan. He was inspired to start his Divina Proportione on 9 Feb 1498 and completed it on 14 Dec 1498, though it was not published (in an expanded form) until 1509. (J Tennenbaum in 2005 said that it was completed by the Feb 9 date.)  He was a good friend of Leonardo da Vinci. It was Leonardo who drew the pictures for Pacioli's book. Pacioli may have advised Leonardo on the perspective for the painting of The Last Supper. Certainly Pacioli stimulated Leonardo's interest in perspective and it is possible that Leonardo's famous drawing of the proportions of the human body was inspired by Pacioli's comment on classical architecture; "For in the human body they found the two main figures ..., namely the perfect circle and the square." Pacioli seems to have made models of the polyhedra illustrated in his book, though we don't know if Leonardo used these for his drawings. A set was probably given to Pacioli's earlier patron, the Duke of Urbino, in 1494. Another set was paid for by Florence in 1504. *Unknown Internet Source
I was just reminded by a tweet from @IanMegaw that Pacioli also published the first description of double-entry bookkeeping. (thanks Ian)

Another first, or two, for Pacioli is found in a paper on the origins of the game of Nim by Lisa Rougetet.  "The oldest simplified variation of Nim found to date in Europe is in a manuscript of the Renaissance,De Viribus Quantitatis(On the powers of numbers) , a treatise written between 1496 and 1508 by the Italian mathematician Fra Luca Bartolomeo dePacioli , one of the most famous mathematicians of his time. ... TheDe Viribus Quantitatiscan be considered to be one of the first works entirely devoted to mathematical recreations 

I received a comment that included this un-cited quote with two more firsts about the book.

"The world's oldest magic text, De viribus quantitatis (On the Powers of Numbers), was penned by Luca Pacioli, a Franciscan monk who shared lodgings with da Vinci. ...  and contains the first ever reference to card tricks as well as guidance on how to juggle, eat fire and make coins dance. It is also the first work to note that da Vinci was left-handed."





1849 After the death of Lord Kelvin’s Father, James Thomson, his replacement was the subject of a letter from William Hopkins to discuss the merits of G.G. Stokes and Hugh Blackburn for the position, “if you determine … to elect a man who is sure to hereafter to dignify his postion by the highest scientific distinction, Stokes is unquestionably your man.” * The correspondence between Sir George Gabriel Stokes and Sir William Thomson, pg 59

In 1829 the honorary degree of LLD was conferred upon James Thomson by the University of Glasgow, where in 1832 he was appointed professor of mathematics. He held this post till his death on 12 January 1849.



In 1870, the U.S. Weather Bureau (later named the Weather Service) was authorized by Congress, and placed under the direction of the Signal Service. Cleveland Abbe had inaugurated a private weather reporting and warning service at Cincinnati and had been issuing weather reports or bulletins since 1 Sep 1869. Hence, Abbe was the only person in the country who was already experienced in drawing weather maps from telegraphic reports and forecasting from them. Naturally, he was offered an important position in this new service which he accepted, beginning 3 Jan 1871, and was often the official forecaster of the weather. He was the first U.S. meteorologist, and known as the "father of the U.S. Weather Bureau."*TIS  



1883 The very first issue of Science is published. The first item in the “Weekly summary of the progress of science” contains a report by Thomas Craig that “Lindemann gave a proof of the fact that π cannot be a root of an equation of any degree with rational co-eficients. This is a most remarkable paper, as it thus contains the first direct, absolute proof that has ever been given of the impossibility of the quadrature of the circle. ... Lindemann has certainly done a splendid piece of work in thus absolutely proving the impossibility of ‘squaring the circle’; and it is only to be regretted that his work will not carry conviction to the minds of those mistaken individuals, the ‘circle-squarers.’ But it is hardly to be supposed that they will be convinced of the futility of their task, any more than the perpetual-motion inventors were convinced by the discovery and enunciation of the principles of the conservation of energy.” [p. 15] *VFR


1912 The Salt Lake Tribune, reporting on the 1912 theory that Mars was populated by a giant vegetable that possessed a single giant eye, accompanied by a remarkable diagram. *Paul Fairie

J F Ptak wrote, "The Giant All-Seeing Eyeball was hoisted high in the Tribune, given supposed life by the very highly capable astronomer W.W. Campbell (1862-1930, with his biography here at the National Academy of Science), who is quoted by the paper as being the source of this preposterous theory.  Campbell was not pleased by this--not at all.  And I can well imagine why. "

"I can't consider the Salt Lake paper's story a hoax (as in the case of the great Moon hoax perpetrated in the pages of the New York Sun in 1835), mainly because it is surrounded by other crazy/funny stories--and it didn't try to present the story as a piece of non-fiction, as is classically the case with hoaxes. The Tribune followed up this story on the very next page with one on how the English aristocracy was turning into gorillas. "

J F Ptak



On this day in 1937, Ruth Moufang habilitated, being only the third German woman to habilitate in mathematics. However, the Nazis refused her permission to teach (because she was a woman), so from 1937 she became an industrial mathematician working on elasticity theory. In fact this gives Moufang the unique position of being the first German woman with a doctorate to be employed in industry. She may actually be the first ever such woman anywhere.

 At the end of World War II she was leading the Department of Applied Mathematics at the arms industry of Krupp.

In 1946 she was finally allowed to accept a teaching position at the University of Frankfurt, and in 1957 she became the first woman professor at the university.

In 1933, Moufang showed Desargues's theorem does not hold in the Cayley plane. The Cayley plane uses octonion coordinates which do not satisfy the associative law. Such connections between geometry and algebra had been previously noted by Karl von Staudt and David Hilbert. Ruth Moufang thus initiated a new branch of geometry called Moufang planes.



1946  Lise Meitner and President Harry Truman in Washington where Meitner was being honored as Woman of the Year by the National Woman's Press Club.  



1986 Halley’s comet last reached perihelion. The next return to perihelion will be on 28 July 2061. *Wik (I am still amazed that we can mathematically predict such an event with such precision.)



BIRTHS

1489 Georg Hartmann (sometimes spelled Hartman; February 9, 1489 – April 9, 1564) was a German engineer, instrument maker, author, printer, humanist, churchman, and astronomer. After finishing his studies, he travelled through Italy and finally settled in Nuremberg in 1518. There he constructed astrolabes, globes, sundials, and quadrants. In addition to these traditional scientific instruments Hartmann also made gunner's levels and sights. Hartmann was possibly the first to discover the inclination of Earth's magnetic field. He died in Nuremberg.
His two published works were Perspectiva Communis (Nuremberg, 1542), a reprint of John Peckham's 1292 book on optics and Directorium (Nuremberg, 1554), a book on astrology. He also left Collectanea mathematica praeprimis gnomonicam spectania, 151 f. MS Vienna, Österreichische Nationalbibliothek, Quarto, Saec. 16 (1527–1528), an unpublished work on sundials and astrolabes that was translated by John Lamprey and published under the title of Hartmann's Practika in 2002. *Wik


1775 Farkas Bolyai (9 Feb 1775, 20 Nov 1856) Hungarian mathematician, poet, and dramatist who spent a lifetime trying to prove Euclid's (fifth) postulate that parallel lines do not meet. While studying at the University of Göttingen, he met as a fellow student, the noted German mathematician Carl F. Gauss, with whom he corresponded as a life-long friend. Bolyai taught mathematics, physics and chemistry at Marosvásárhely all his life. He discouraged his son, János Bolyai, from studying the parallel axiom as he had, writing in a letter to him: "For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life." *TIS (In 1804 he believed that he had a proof the Euclid's fifth postulate could be deduced from the other axioms. He sent this proof to C. F. Gauss who found an error. His Son, Janos, would ignore his father's warnings and go on to discover a non-Euclidean Geometry. ).



1880 Lipót Fejér (9 Feb 1880, 15 Oct 1959) Fejér's main work was in harmonic analysis working on Fourier series and their singularities. Fejér collaborated to produce important papers with Carathéodory on entire functions and with Riesz on conformal mappings. *SAU In 1897 he won a prize in one of the first mathematical competitions held in Hungary. *VFR




1907 Harold Scott MacDonald Coxeter (9 Feb 1907 in London, England - 31 March 2003 in Toronto, Canada) graduated from Cambridge and worked most of his life in Canada. His work was mainly in geometry. In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics. Among his most famous geometry books are The real projective plane (1955), Introduction to geometry (1961), Regular polytopes (1963), Non-euclidean geometry (1965) and, written jointly with S L Greitzer, Geometry revisited (1967). He also published a famous work on group presentations, which was written jointly with his first doctoral student W O J Moser, Generators and relations for discrete groups.
His 12 books and 167 published articles cover more than mathematical research. Coxeter met Escher in 1954 and the two became lifelong friends. Another friend, R Buckminister Fuller, used Coxeter's ideas in his architecture. In 1938 Coxeter revised and updated Rouse Ball's Mathematical recreations and essays, a book which Rouse Ball first published in 1892. *SAU
Considered by many as the greatest geometer of the 20th Century.  





1908 Alexander Dinghas (February 9, 1908 – April 19, 1974) was a Turkish mathematician. He is known for his work in different areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry. His most important contribution was his work in function theory, in particular Nevanlinna theory and the growth of subharmonic functions.

Dinghas was not a German and his career during the Nazi years was very difficult. However, after the end of World War II, his luck changed. He became professor of mathematics at the Humboldt University of Berlin in 1947. From 1949 until his death he was a professor of mathematics at the Free University of Berlin and director of the Mathematical Institute there. *Wik



1918 Lloyd Noel Ferguson (February 9, 1918 – November 30, 2011) was an American chemist. 

Many chemists came into the field after being given a chemistry set as a kid, but Ferguson's family was too poor for such niceties, so Ferguson built his own chemistry lab in his backyard, financing it by making insect repellants and stain removers and selling them to neighbors.  And he remembers, naturally, that he liked blowing up things.  He skipped several grades in high school and used the years thus gained to work as a porter for the railroads to raise money for college. He was accepted into the University of California at Berkeley as an undergraduate, and did so well that they took him into their PhD program, which most graduate programs will not do except under exceptional circumstances.  Ferguson apparently warranted the exception.  He worked under Melvin Calvin, future Nobelist, and this being the war years (1940-43), they worked on a military project, trying to develop an inorganic compound, like organic hemoglobin, that would release oxygen on demand and then re-oxygenate itself.  This might provide a supply of oxygen for welding onboard ship, where oxygen tanks were too dangerous if damaged by gunfire. 

After he received a PhD in 1943, Ferguson found it difficult to find a teaching position in white America, just as his advisor had predicted, who suggested that he go into industry instead.  But Ferguson found a small North Carolina black college that needed a chemistry professor, and he started his career there.  He then moved on to Howard University, where he founded a PhD program in chemistry, the first such program in any black college anywhere.  After twenty years at Howard, he was invited to become chair of the chemistry department at California State University, Los Angeles, were he would work until retirement in 1986.

His finest achievement was in promoting interest in chemistry in young black Americans.  He founded the National Organization for the Professional Advancement of Black Chemists and Chemical Engineers (NOBCChE) in 1972, mainly for the purpose of youth education, and, for the American Chemical Society, he founded the SEED project in 1968, which provided financial assistance to black high school students who lacked the resources to pursue higher education.   The organization he founded, NOBCChE, later established the Lloyd N. Ferguson Young Scientist Award, given annually to a promising young black chemist. *Linda Hall Org



1919 Irene Anne Stegun (February 9, 1919 – January 27, 2008) was a mathematician at the National Bureau of Standards who, with Milton Abramowitz, edited a classic book of mathematical tables called A Handbook of Mathematical Functions, widely known as Abramowitz and Stegun. When Abramowitz died of a heart attack in 1958, Stegun took over management of the project and finished the work by 1964, working under the direction of the NBS Chief of Numerical Analysis Philip J. Davis, who was also a contributor to the book. *Wik



1927 David John Wheeler FRS (9 February 1927–13 December 2004) the Inventor of the Wheeler Jump, is Born. In 1951, he introduced the concept of the subroutine to computer programming, is born. He concentrated his work on assembly programming language and invoked the subroutine in his Wheeler jump technique. For this work Wheeler received the IEEE Computer Society Pioneer Award. *CHM
He was born in Birmingham and gained a scholarship at Trinity College, Cambridge to read mathematics, graduating in 1948. He completed the world's first PhD in computer science in 1951. His contributions to the field included work on the EDSAC and the Burrows-Wheeler transform. Along with Maurice Wilkes and Stanley Gill he is credited with the invention of the subroutine (which they referred to as the closed subroutine), because of which a jump to subroutine instruction is often called Wheeler Jump. He was responsible for the implementation of the CAP computer, the first to be based on security capabilities. In cryptography, he was the designer of WAKE and the co-designer of the TEA and XTEA encryption algorithms together with Roger Needham. In 2003 he was a Computer History Museum Fellow Award recipient.
Wheeler is often quoted as saying "All problems in computer science can be solved by another level of indirection... Except for the problem of too many layers of indirection." *Wik




DEATHS

1734 Pierre Polinière (8 September 1671, Coulonces, France - 9 February 1734, Coulonces, France) was an early investigator of electricity and electrical phenomena, notably "barometric light", a form of gas-discharge light, which suggested the possibility of electric lighting. He also helped to introduce the scientific method in French universities. *Wik




1811 Nevil Maskelyne (6 Oct 1732, 9 Feb 1811) (SAU gives 5 Oct for birhtdate)
British astronomer noted for his contribution to the science of navigation. In 1761 the Royal Society sent Maskelyne to the island of St Helena to make accurate measurements of a transit of Venus. This in turn gives the distance from the Earth to the Sun, and the scale of the solar system. During the voyage he also experimented with the lunar position method of determining longitude. In 1764 he went on a voyage to Barbados to carry out trials of Harrison's timepiece, followed by appointment as Astronomer Royal (1765). In 1774, he carried out an experiment on a Scottish mountain with the use of a plumb line to determine the Earth's density. He found it was approximately 4.5 times that of water. *TIS (the current scientific value of the Earth's density is about 5.2 times that of water.) He was the fifth English Astronomer Royal. He held the office from 1765 to 1811.*Wik



1865 James Melville Gilliss (6 Sep 1811; 9 Feb 1865) U.S. naval officer and astronomer who founded the Naval Observatory in Washington, D.C., the first U.S. observatory devoted entirely to research. Gilliss joined the Navy as a midshipman at the age of 15. He taught himself astronomy, at a time when there was no fixed astronomical observatory in the U.S., and very little formal instruction. In 1838, when Charles Wilkes left on the famous South Seas Exploring Expedition, Gilliss became officer-in-charge of the Depot of Charts and Instruments, forerunner of the U. S. Naval Observatory. Gilliss' astronomical observations made during this time in connection with determining longitude differences with the Wilkes Expedition, resulted in the first star catalogue published in the United States. *TIS



1883 Henry John Stephen Smith (2 Nov 1826 in Dublin, Ireland, 9 Feb 1883 in Oxford, England) was an Irish mathematician whose most important contributions are in number theory where he worked on elementary divisors. Henry John Stephen Smith In addition to solving these cases explicitly, he gave a method which would yield the number of ways that an integer can be expressed as the sum of k squares for any fixed k. He published his results in The orders and genera of quadratic forms containing more than three indeterminates published in the Proceedings of the Royal Society in 1867. Eisenstein had earlier proved the result for 3 squares and Jacobi for 2, 4 and 6 squares. Smith also extended Gauss's theorem on real quadratic forms to complex quadratic forms. *SAU He posthumously received the Grand Prix des Sciences Mathematiques of the Paris Academy of Science for his proof that every positive integer is the sum of five squares. He shared the prize with the eighteen year old Hermann Minkowski.*VFR The prize was awarded on April 2, less than two months after his death.



1937 Francis Sowerby Macaulay FRS (11 February 1862 – 9 February 1937) was an English mathematician who made significant contributions to algebraic geometry. He is most famous for his 1916 book, The Algebraic Theory of Modular Systems, which greatly influenced the later course of algebraic geometry. Both Cohen-Macaulay rings and the Macaulay resultant are named for Macaulay.
Macaulay was educated at Kingswood School and graduated with distinction from St John's College, Cambridge. He taught top mathematics class in St Paul's School in London from 1885 to 1911. His students included J. E. Littlewood and G. N. Watson.*Wik Littlewood consulted the examinations record and wrote, "In the 25 years from [Macaulay's] appointment to St Paul's in 1885 to his resignation in 1911 there were 41 scholarships (34 at Cambridge) and 11 exhibitions; and in the 20 years available there were 4 senior wranglers, one second, and one fourth among his former pupils." *SAU




1970 Leo Moser (April 11, 1921, Vienna—February 9, 1970, Edmonton) was an Austrian-Canadian mathematician, best known for his polygon notation.
A native of Vienna, Leo Moser immigrated with his parents to Canada at the age of three. He received his Bachelor of Science degree from the University of Manitoba in 1943, and a Master of Science from the University of Toronto in 1945. After two years of teaching he went to the University of North Carolina to complete a Ph.D., supervised by Alfred Brauer.[1] There, in 1950, he began suffering recurrent heart problems. He took a position at Texas Technical College for one year, and joined the faculty of the University of Alberta in 1951, where he remained until his death at the age of 48. *Wik In mathematics, Steinhaus–Moser notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus’s polygon notation.

n in a triangle a number n in a triangle means nn.
n in a square a number n in a square is equivalent with "the number n inside n triangles, which are all nested."
n in a pentagon a number n in a pentagon is equivalent with "the number n inside n squares, which are all nested."
*Wik 



1988 Israel Nathan Herstein (March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was a mathematician, appointed as professor at the University of Chicago in 1951. He worked on a variety of areas of algebra, including ring theory, with over 100 research papers and over a dozen books.
He is known for his lucid style of writing, as exemplified by the classic and widely influential Topics in Algebra, an undergraduate introduction to abstract algebra that was published in 1964, which dominated the field for 20 years. A more advanced classic text is his Noncommutative Rings in the Carus Mathematical Monographs series. His primary interest was in noncommutative ring theory, but he also wrote papers on finite groups, linear algebra, and mathematical economics.

 His family emigrated to Canada in 1926, and he grew up in a harsh and underprivileged environment where, according to him, "you either became a gangster or a college professor." During his school years he played football, ice hockey, golf, tennis, and pool. He also worked as a steeplejack and as a barker at a fair. He received his B.S. degree from the University of Manitoba and his M.A. from the University of Toronto. He received his Ph.D from Indiana University in 1948. His advisor was Max Zorn. He held positions at the University of Kansas, Ohio State University, University of Pennsylvania, and Cornell University before permanently settling at the University of Chicago in 1962. He was a Guggenheim Fellow for the academic year 1960–1961.

*Wik




2001 Herbert Alexander Simon (15 Jun 1916, 9 Feb 2001 at age 84) American social scientist who was a pioneer of the development of computer artificial intelligence. In 1956, with his long-time colleague Allen Newell, Simon produced the computer program, The Logic Theorist, a computer program that could discover proofs of geometric theorems. It was the first computer program capable of thinking, and marked the beginning of what would become known as artificial intelligence. It proved many of the theorems of symbolic logic in Whitehead and Russell's Principia Mathematica. He is further known for his contributions in fields including psychology, mathematics, statistics, and operations research, all of which he synthesized in a key theory for which he won the 1978 Nobel Prize for economics. *TIS



2003 Masatoşi Gündüz İkeda (25 February 1926, Tokyo. - 9 February 2003, Ankara), was a Turkish mathematician of Japanese ancestry, known for his contributions to the field of algebraic number theory.

During his time in Japan , Ikeda conducted research on the theory of rings and the matrix representation of groups. In the 1970s, he turned to algebraic number theory and carried out important studies on the automorphisms and universality of the pure Galois group of the rational number field. In a study published in the famous mathematics journal Crelle's Journal , he showed that the Galois group has a very special structure. *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

Saturday, 8 February 2025

Antiparallels, an Overlooked HS Beauty

 


I would think it is pretty fundamental in typical HS classrooms that students recognize that a line parallel to one of the sides of a triangle will cut the other two sides in a pair of angles which are congruent to the angles formed at the third side. Eventually they can prove that the triangle formed by the parallel line forms a triangle with the two sides similar to the original triangle.

Almost none of them, and perhaps very few of their teachers, know that there is a second type of line which can be drawn to cut the two sides which will also form a similar triangle, and thus must also form angles congruent to the two base angle of the original triangle. Its called the anti-parallel now, but it used to be called a subcontrary line, at least by Apollonius.

There are several nice ways to produce an antiparallel in a triangle. A nice general way is to use the two vertices of one leg and a point on one of the other two legs to construct a circle. The circle will then cut the other leg in a fourth point which is the other vertex of the antiparallel. These four point are the vertices of a cyclic quadrilateral, for which the opposite angles are supplementary. This makes it easy to see that the antiparallel forms angles on one leg congruent to the original angle on the other leg.
A second way is to draw an altitude from two of the vertices to the opposite sides. The segment connecting the feet of these two altitudes is also antiparallel to the third side.

*image from Wolfram Mathworld

A recent note from Anthony Leonardo gave another nice way to make anti-parallels using Geogebra,  


Now you can construct any non-isoceles triangle, and then by moving point O, construct a moving set of anti-parallels.


If you construct the circumcircle of the triangle, the tangent at the vertex opposite a side will be anti-parallel to that side. (This doesn't strike me as a simple proof, but I may be overlooking something. I often do. If you have a simple proof, high school level for example, I would love to see and share it.)

Just as the median bisects all lines parallel to the base, its reflection in the angle bisector (called the symmedian) will bisect each anti-parallel.

The antiparallel shows up as the solution to an optimization problem that was first proved by Giovanni Fagnano in 1775: For a given acute triangle determine the inscribed triangle of minimal perimeter. Turns out the answer is the triangle formed by the three anti-parallels connecting the feet of the three altitudes, called the orthic triangle.

For slightly more advanced students who have been exposed to cones it is constructive to point out that for an oblique circular cone, (one in which the axis is not perpendicular to the base; and many students graduate from HS without ever having been made aware that such types of cones exist, much less those whose base is non-circular) there is more than one plane which will cut a circle. A cutting plane parallel to the base is one type, and of course by now you suspect that the other type is a plane anti-parallel to the base.

paramanands blogspot


Maybe soon I'll write about the anti-CENTER.
  Oh, heck.  Now might work.  First we need a new term for many students, maltitude.  A maltitude ("midpoint altitude") is a perpendicular drawn to a side of a quadrilateral from the midpoint M of the opposite side.  It some strange (think "fun" ) quadrilaterals a maltitude may have a foot outside the quadrilateral on the opposite side extended.  


*Wolfram Mathworld



In a cyclic quadrilateral the maltitudes all intersect in a single point, the anticenter.  As the image illustrates, the anticenter is the reflection of the circumcenter reflected in the geocenter G.

Addendum: After a comment by 1SAEED9, I realized a proof that the tangent at the vertex opposite a side is antiparallel to that side.

It is easy to see that angles DBA and BCA both subtend the same arc, and thus are the same measure. By using the fact that CBA, DBA and EBD add up to 180 degrees, and the three interior angles of the triangle CBA, CAB, and BCA also add up to 180 degrees. Since BCA and DBA are congruent, when we subtract these from each side, and remove CBA from both sides we are left with the fact that EBD must be congruent to CAB. We can repeat this process on the opposite angle and we are done. Easier than I imagined.


Another interesting occurrence of antiparallels is the Tucker Circles, named for Robert Tucker (1832–1905) was an English mathematician, who was secretary of the London Mathematical Society for more than 30 years.  
One way to construct a Tucker circle (and every triangle has an infinite number of them), pick a point on any side of the triangle.  From this point draw an antiparallel to one of the other two sides.  Then draw a parallel from this new endpoint to the remaining side of the triangle.  Now draw an antiparallel from this point ending on the side of your first point.  Continue parallel, antiparallel and parallel and after six lines your endpoint will be the point you began.  If you connect the six points on the edges of the triangle, they form a cyclic hexagon, a hexagon with all six vertices on a circle.  Pick a different starting point, you get a different cyclic hexagon. 

A special case of the Tucker Circle is the Cosine circle, which happens if you draw all three antiparallels through the Symmedian point, the point which is the reflection of the intersection of the medians over the intersection of the angle bisectors.  These antiparallels are all in the same ratio with the cosine of the opposite angle.





On This Day in Math - February 8

   

Edgeworth box; *daviddfriedman.com



The most important and urgent problems of the technology of today are no longer the satisfactions of the primary needs or of archetypal wishes, but the reparation of the evils and damages by technology of yesterday.
~Dennis Gabor

The 39th day of the year; 39 is the smallest number with multiplicative persistence 3. [Multiplicative persistence is the number of times the digits must be multiplied until they produce a one digit number; 3(9)= 27; 2(7) = 14; 1(4)=4. Students might try to find the smallest number with multiplicative persistence of four, or prove that no number has multiplicative persistence greater than 11]

For the 39th day: 39 = 3¹ + 3² + 3³ *jim wilder ‏@wilderlab

An Armstrong (or Pluperfect digital invariant) number is a number that is the sum of its own digits each raised to the power of the number of digits. For example, 371 is an Armstrong number since \(3^3+7^3+1^3 = 371\). The largest Armstrong number in decimal numbers has 39 digits. (115,132,219,018,763,992,565,095,597,973,522,401 is the largest)
 (Armstrong numbers are named for Michael F. Armstrong who named them for himself as part of an assignment to his class in Fortran Programming at the University of Rochester \)

I find it interesting that 39 = 3*13, and is the sum of all the primes from 3 to 13, 39=3+5+7+11+13  (is there a name for these kinds of numbers?)


EVENTS

1587 Mary, Queen of Scots, was beheaded after Sir Francis Walsingham did a frequency count on Mary’s cipher, read her message, and uncovered her plot to assassinate Elizabeth I, Queen of England. *VFR  a more complete version of the "Babington Plot" and Walsingham's work in deciphering the code is here




In 1672, Isaac Newton's first paper on optics read before Royal Society in London. He had been elected a member only the previous month, recognizing his original design of the first reflecting telescope. Newton had already spent several years investigating optics, beginning in 1665. His studies of the colors from glass prisms with their dispersion of light were recorded in his essay New Theory about Light and Colors (1672), and expanded later in Opticks (1704).*TIS (Always sensitive to criticism, the controversy over his theories and experiments in light would lead to his not publishing on the topic until 1704.) Thony Christie has a nice post about Newton's research on color and light here.


In 1865, Gregor Mendel, aged 42, who first discovered the laws of genetics, read his first scientific paper to the Brünn Society for the study of Natural Sciences in Moravia (published 1866). He described his investigations with pea plants. Although he sent 40 reprints of his article to prominent biologists throughout Europe, including Darwin, only one was interested enough to reply. Most of the reprints, including Darwin's, were discovered later with the pages uncut, meaning they were never read. Fortunately, 18 years after Mendel's death, three botanists in three different countries researching the laws of inheritance, in spring 1900, came to realize that Mendel had found them first. Mendel was finally acknowledged as a pioneer in the field which became known as genetics*TIS




1913 Hardy wrote a letter to Ramanujan, (actually Littlewood wrote the letter, but surely speaking their joint interest) expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions". *Wik


1945 A Patent is Filed for the Harvard Mark I. C.D Lake, H.H. Aiken, F.E. Hamilton, and B.M. Durfee file a calculator patent for the Automatic Sequence Control Calculator, commonly known as the Harvard Mark I. The Mark I was a large automatic digital computer that could perform the four basic arithmetic functions and handle 23 decimal places. A multiplication took about five seconds. *CHM (We've come a long way, baby.)


In 1969, pieces of a large meteorite were recovered in Chihuahua, Mexico. It fell at 1:05 am as a huge fireball that scattering several tons of material over an area measuring 48 by 7 km. Named after the nearby village of Allende, samples of this carbonaceous chondrite stone contain an aggregated mass of particles several of which can be easily identified as chondrules. This ancient material comes from before our Solar System formed, thus over 4.6 billion years old. Since these remnants represent the most primitive geological material from which planets were formed, and carry information to help explain the evolution of the our galaxy, Allende is one of the most studied meteorites in the world.*TIS


1978 The first issue of the CSHPM (Canadian Society for History and Philosophy of Mathematics) newsletter is issued. The issue announced the establishment of a fund as a memorial to Ken May. The fund will be used to underwrite the Kenneth O. May Lecture series. May had been one of the primary agents in the creation of the CSHPM. *CSHPM newsletter


BIRTHS

411 Proclus Diadochus (8 Feb 411 in Constantinople (now Istanbul), Byzantium (now Turkey) - 17 April 485 in Athens, Greece) was a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians.

A man of great learning, Proclus was regarded with great veneration by his contemporaries. He followed the neoplatonist philosophy which Plotinus founded, and Porphyry and Iamblichus developed around 300 AD. Other developers of these ideas were Plutarch and Syrianus, the teachers of Proclus. Heath writes [4]:-

He was an acute dialectician and pre-eminent among his contemporaries in the range of his learning; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries, and a devout worshipper of divinities both Greek and Oriental. He was much more a philosopher than a mathematician.

Of course, as one might expect, his belief in many religious sayings meant that he was highly biased in his views on many issues of science. For example he mentions the hypothesis that the sun is at the centre of the planets as proposed by Aristarchus but rejects it immediately since it contradicted the views of a Chaldean whom he says that it is unlawful not to believe.

Proclus wrote Commentary on Euclid which is our principal source about the early history of Greek geometry. The book is certainly the product of his teaching at the Academy. This work is not coloured by his religious beliefs and Martin, writing in the middle of the 19th century, says :... for Proclus the "Elements of Euclid" had the good fortune not to be contradicted either by the Chaldean Oracles or by the speculations of Pythagoreans old and new.

Proclus had access to books which are now lost and others, already lost in Proclus's time, were described based on extracts in other books available to Proclus. In particular he certainly used the History of Geometry by Eudemus, which is now lost, as is the works of Geminus which he also used.

 *SAU



1627 Sir Jonas Moore (8 Feb 1627 in Whitelee, Pendle Forest, Lancashire, England - 25 Aug 1679 in Godalming, England) was an English man of science important for his support of mathematics and astronomy.*SAU He seems to have been the first to use "cot" for the cotangent function. He also founded the Royal Mathematical School at Christ's Hospital with Samuel Pepys to train young men in the mathematics of navigation. *Wik He made critical contributions to the draining of the fens in England (making my daily drive from Lakenheath to Stoke Ferry much easier) and was instrumental in convincing Charles II to create the Royal Observatory and appoint Flamsteed as Astronomer Royal. *The day that Jonas died, Renaissance Mathematicus.




1630 Pierre-Daniel Huet (8 Feb 1630, 26 Jan 1721) French scholar, antiquary, scientist, and bishop whose incisive skepticism, particularly as embodied in his cogent attacks on René Descartes, greatly influenced contemporary philosophers. Huet wrote a number of philosophical works that asserted the fallibility of human reason in addition to scientific work in the fields of astronomy, anatomy, and mathematics. *TIS


1677 Jacques Cassini (8 February 1677 – 16 April 1756) was a French astronomer, son of the famous Italian astronomer Giovanni Domenico Cassini.
Cassini was born at the Paris Observatory. Admitted at the age of seventeen to membership of the French Academy of Sciences, he was elected in 1696 a fellow of the Royal Society of London, and became maître des comptes in 1706. Having succeeded to his father's position at the observatory in 1712, he measured in 1713 the arc of the meridian from Dunkirk to Perpignan, and published the results in a volume entitled Traité de la grandeur et de la figure de la terre (1720). He also wrote Eléments d'astronomie (1740), and died at Thury, near Clermont. He published the first tables of the satellites of Saturn in 1716.*Wik

Engraving of Jacques Cassini in his Paris Observatory by L. Coquin




1700 Daniel Bernoulli (29 January 1700 (8 Feb new style), 8 March 1782) was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. Bernoulli's work is still studied at length by many schools of science throughout the world. The son of Johann Bernoulli (one of the "early developers" of calculus), nephew of Jakob Bernoulli (who "was the first to discover the theory of probability"), and older brother of Johann II, He is said to have had a bad relationship with his father. Upon both of them entering and tying for first place in a scientific contest at the University of Paris, Johann, unable to bear the "shame" of being compared as Daniel's equal, banned Daniel from his house. Johann Bernoulli also plagiarized some key ideas from Daniel's book Hydrodynamica in his own book Hydraulica which he backdated to before Hydrodynamica. Despite Daniel's attempts at reconciliation, his father carried the grudge until his death.
He was a contemporary and close friend of Leonhard Euler. He went to St. Petersburg in 1724 as professor of mathematics, but was unhappy there, and a temporary illness in 1733 gave him an excuse for leaving. He returned to the University of Basel, where he successively held the chairs of medicine, metaphysics and natural philosophy until his death.
In May, 1750 he was elected a Fellow of the Royal Society. He was also the author in 1738 of Specimen theoriae novae de mensura sortis (Exposition of a New Theory on the Measurement of Risk), in which the St. Petersburg paradox was the base of the economic theory of risk aversion, risk premium and utility.
One of the earliest attempts to analyze a statistical problem involving censored data was Bernoulli's 1766 analysis of smallpox morbidity and mortality data to demonstrate the efficacy of vaccination. He is the earliest writer who attempted to formulate a kinetic theory of gases, and he applied the idea to explain Boyle's law. He worked with Euler on elasticity and the development of the Euler-Bernoulli beam equation. *Wik





1777 Bernard Courtois,(8 February 1777 – 27 September 1838) was a French chemist. In 1811, the French government was looking for alternate ways to manufacture saltpeter, or potassium nitrate, an ingredient essential for gunpowder. Saltpeter had traditionally been made using wood ash, but France (like England) was running out of wood, and other sources were desperately needed. So Courtois, a professional salpêtrier, was working on extracting potassium nitrate from seaweed, which one can find in great abundance along the coast of Normandy. Courtois began to suspect that there was something else in seaweed ash besides sodium and potassium, something corrosive, because the copper vats in his lab were being attacked by some chemical.  He found that when he added sulfuric acid to the ash residue, a purple vapor was given off, which then formed deposits of shiny purplish-black crystals on the sides of the vats. Courtois had discovered iodine.  He announced the discovery in the journal Annales de Chimie in 1813. *Linda Hall Org



1834 Dmitry Ivanovich Mendeleev (8 Feb 1834; 2 Feb 1907 at age 73) (Also spelled Mendeleyev) Russian chemist who developed the periodic classification of the elements. In his final version of the periodic table (1871) he left gaps, foretelling that they would be filled by elements not then known and predicting the properties of three of those elements.*TIS

Mendeleev's 1871 periodic table

*Wik


1845 Francis Ysidro Edgeworth FBA (8 February 1845, Edgeworthstown – 13 February 1926, Oxford) was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. Edgeworth was a highly influential figure in the development of neo-classical economics. He was the first to apply certain formal mathematical techniques to individual decision making in economics. He developed utility theory, introducing the indifference curve and the famous Edgeworth box, which is now familiar to undergraduate students of microeconomics. He is also known for the Edgeworth conjecture which states that the core of an economy shrinks to the set of competitive equilibria as the number of agents in the economy gets large. In statistics Edgeworth is most prominently remembered by having his name on the Edgeworth series. *Wik In 1881 he published Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. This work, really on economics, looks at the Economical Calculus and the Utilitarian Calculus. In fact most of his work could be said to be applications of mathematical psychics which Edgeworth saw as analogous to mathematical physics. They were applied to the measure of utility, the measure of ethical value, the measure of evidence, the measure of probability, the measure of economic value, and the determination of economic equilibria. He formulated mathematically a capacity for happiness and a capacity for work. His conclusions that women have less capacity for pleasure and for work than do men would not be popular today. *SAU




1853 Alexander Ziwet (February 8, 1853 - November 18, 1928) born in Breslau. He became professor at the University of Michigan, an editor of the Bulletin of the AMS, and a collector of mathematics text who enriched the Michigan library. *VFR His early education was obtained in a German gymnasium. He afterwards studies in the universities of Warsaw and Moscow, one year at each, and then entered the Polytechnic School at Karlsruhe, where he received the degree of Civil Engineer in 1880.
He came immediately to the United States and received employment on the United States Lake Survey. Two years later he was transferred to the United States Coast and Geodetic Survey, computing division, where he remained five years.
In 1888 he was appointed Instructor in Mathematics in the University of Michigan. From this position he was advanced to Acting Assistant Professor in 1890, to Assistant Professor in 1891, to Junior Professor in 1896, and to Professor of Mathematics in 1904.
He was a member of the Council of the American Mathematical Society and an editor of the "Bulletin" of the society. In 1893-1894 he published an "Elementary Treatise on Theoretical Mechanics" in three parts, of which a revised edition appeared in 1904. He also translated from the Russian of I. Somoff "Theoretische Mechanik" (two volumes, 1878, 1879).
*Burke A. Hinsdale and Isaac Newton Demmon, History of the University of Michigan (Ann Arbor: University of Michigan Press, 1906), pp. 320-321.

While an Assistant Professor at UM, Ziwet attended and took notes, (published in 1894, and recently reprinted by the AMS), for a famous series of "colloquium" lectures of Felix Klein, featuring some of the important mathematical developments of the late 19th century, including Lie theory, function theory, algebraic geometry (of curves and surfaces), number theory, and non euclidean geometry. These lectures, held under the hospitality of Northwestern University, followed a Congress of Mathematics sponsored by the World's Fair Auxillary, 21-26 August, 1893. This occasion launched the greatly influential role that Klein played in the development of American mathematics.




1875 Thomas John l'Anson Bromwich (8 Feb 1875 in Wolverhampton, England - 26 Aug 1929 in Northampton, England) He worked on infinite series, particularly during his time in Galway. In 1908 he published his only large treatise An introduction to the theory of infinite series which was based on lectures on analysis he had given at Galway. He also made useful contributions to quadratic and bilinear forms and many consider his algebraic work to be his finest. In a series of papers he put Heaviside's calculus on a rigorous basis treating the operators as contour integrals*SAU G. H. Hardy described him as the “best pure mathematician among the applied mathematicians at Cambridge, and the best applied mathematician among the pure mathematicians.” *VFR





1928 Ennio de Giorgi (Lecce, February 8, 1928 – Pisa, October 25, 1996) was an Italian mathematician who worked on partial differential equations and the foundations of mathematics.*SAU


DEATHS

1909 Giacinto Morera (born Novara, 18 July 1856 – died Turin, 8 February 1909), was an Italian engineer and mathematician. He is remembered for Morera's theorem in the theory of functions of a complex variables and for his work in the theory of linear elasticity. *Wik

In complex analysis, a branch of mathematics, *WikMorera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.The assumption of Morera's theorem is equivalent to f locally having an antiderivative on D.The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions.


A curve C in a domain D, as required by the
statement of Morera's theorem*Wik



1957 John von Neumann (28 Dec 1903, 8 Feb 1957 at age 53)Hungarian-American mathematician who made important contributions in quantum physics, logic, meteorology, and computer science. He invented game theory, the branch of mathematics that analyses strategy and is now widely employed for military and economic purposes. During WW II, he studied the implosion method for bringing nuclear fuel to explosion and he participated in the development of the hydrogen bomb. He also set quantum theory upon a rigorous mathematical basis. In computer theory, von Neumann did much of the pioneering work in logical design, in the problem of obtaining reliable answers from a machine with unreliable components, the function of “memory,” and machine imitation of “randomness.”*TIS

In his classic, How to Solve It,  Polya tells this story about von Neumann as his student







1974 Fritz Zwicky (14 Feb 1898, 8 Feb 1974 at age 76) Swiss-American astronomer and physicist who proposed dark matter exists in the universe, and made valuable contributions to the theory and understanding of supernovas (stars that for a short time are far brighter than normal).*TIS

He worked most of his life at the California Institute of Technology in the United States of America, where he made many important contributions in theoretical and observational astronomy. In 1933, Zwicky was the first to use the virial theorem to postulate the existence of unseen dark matter, describing it as "dunkle Materie"

Zwicky married Dorothy Vernon Gates (1904-1991), a member of a prominent local family and a daughter of California State Senator Egbert James Gates. Her money was instrumental in the funding of the Palomar Observatory during the Great Depression. Nicholas Roosevelt, cousin of President Theodore Roosevelt, was his brother-in-law by marriage to Tirzah Gates. 

He is remembered as both a genius and a curmudgeon. One of his favorite insults was to refer to people whom he did not like as "spherical bastards", because, as he explained, they were bastards no matter which way one looked at them.




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



2005 Germund Dahlquist (January 16, 1925 – February 8, 2005) was a Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations.*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