## Thursday, 20 April 2017

### On This Day in Math - April 20

Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language.
~Giuseppe Peano

The 110th day of the year; The sum of the first 110 primes has only two prime factors. 2+3+5+7+....+599 + 601 = 29897 = 7 X 4271

110 is the average of first fifty-three primes.

110 is the side of the smallest square that can be tiled with distinct integer-sided squares (see image below). There are 3 distinct Simple Perfect Squared Squares with this property. Two 110's with 22 squares were discovered in 1978, one by Duijvestijn using computer search, the second by Willcocks, who transformed Duijvestijns 110 into a different second 110, and one more 110 with 23 squares was discovered in 1990 by Duijvestijn. It was Gambini who proved 110 is the minimal square. *http://www.archimedes-lab.org

110 = 52 + 62 + 72 (3 consecutive squares)
= 11^2 - 11^1 (difference between powers of the same number)

110 hertz is the standard frequency of the musical note A or La.

110 is also known as "eleventy" according to the number naming system invented by J. R. R. Tolkien.

EVENTS
1543 Copernicus’ De Revolutionibus published, "in his An Annotated Census of Copernicus' De Revolutionibus Owen Gingerich writes, 'The printing was finished on 20 April 1543 when Rheticus autographed a presentation copy of the completed work. (Copernicus himself did not receive the final pages until a month later, the day on which he died.)' *Thony Christie
The book was so technically complex that only true astronomers could read through it so the 400 copies didn't even sale out. In addition Osiander had written a disclaimer (without, it seems, the dying Copernicus' permission) that readers should view it as a useful mathematical fiction with no physical reality, thereby somewhat shielding it from accusations of blasphemy. But eventually it was banned. It was placed on the Index of Forbidden Books by a decree of the Sacred Congregation of March 5, 1616. (while I was researching this note I came across a nice information that I am not sure where else I could use it. De revolutionibus was printed in Hans Petreiuss printing shop in Nuremberg. The building of Petreiuss former printing shop at 9, Öberg Street, (located near Albrecht Durers birthplace) luckily survived the ravages of WWII. Several nice pics of the house, and information about the printer is at the Renaissance Mathematicus, who uses the house in his blog header.

1829 Siméon Denis Poisson reads his Memoir on the Mean Results of Observations before the Academy of Sciences. This paper contains his observations on the function $f(x) = \frac{1}{\pi(1+x^2)}$ which is often credited to Cauchy, whose interest in the function begins some 20 years later. *Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory ... By Andreĭ Nikolaevich Kolmogorov, Adolʹf Pavlovich I͡Ushkevich

1833 The great German geometer Jakob Steiner received an honorary degree from the University of Konigsberg. [DSB 13, 14] *VFR

1861 Charles Darwin writes to Frederick Wollaston Hutton, "I am actually weary of telling people that I do not petend to adduce evidence of one species turning into another, but I believe that this visw is in the main correct." *Mario Livio, Brilliant Blunders, pg 31

In 1902, Marie and Pierre Curie isolated one gram of radium, the first sample of the radioactive element. They had refined it from eight tons of pitchblende ore.*TIS

1951 MIT "Whirlwind" Computer Seen on Television:
MIT demonstrates its Whirlwind machine on Edward R. Murrow's "See It Now" television series. Project director Jay Forrester describes the computer as a "reliable operating system," running 35 hours a week at 90-percent utility using an electrostatic tube memory that stores up to 2,048 16-digit words. The machine used 4,500 vacuum tubes and 14,800 diodes, taking up a total of 3,100 square feet.*TIS

1975 India issued a stamp to celebrate the launching of the Aryabhata satellite the previous day. This has to be a record for a quick celebration with a stamp. [Scott #655] *VFR

BIRTHS

1644 Heinrich Meissner (April 20th 1644 in Hamburg - September 1 1716 Hamburg) was a co-founder of the Hamburg Masters and computing Mathematical Society in Hamburg. This is the oldest existing mathematical society in the world.
From 1688 until shortly before his death he was "writing, arithmetic and upper-master" of the parish school of St. Jacobi .
Meissner founded (Jan 2, 1690) along with Valentin Heins 'art-accounting practicing Society ", which became Hamburg Mathematical Society .
Meissner published a whole series of books and magazines. Worth mentioning are especially the key star and Algebrae, a textbook on algebra in the German language, and the Teutsche Euclid, a translation of the first two books in the "Elements" of Euclid with extensive annotations. *Wik

1839 Francesco Siacci (20 April 1839 – 31 May 1907), an Italian mathematician, ballistician, and officer in the Italian army, was born in Rome, Italy. He was a professor of mechanics in the University of Turin and University of Naples. Siacci is well known for his contributions in the field of ballistics, distinguishing himself with a famous treatise Balistica, published in 1888 and translated to French in 1891. Of great importance is an approximation method he devised to calculate bullet trajectories of small departure angles. Known as Siacci’s method, it was a major innovation in exterior ballistics and was widely used almost exclusively at the beginning of World War I. Several modifications of the method are still in use today, including those of H.P. Hitchcock and R.H. Kent, and James Ingalls. Siacci also studied theoretical mechanics (Siacci’s theorem, rigid body dynamics, canonical transformations, and inverse problems) and mathematics (theory of conic sections, Riccati differential equation, etc.).
Siacci's theorem in dynamics is the resolution of the acceleration vector of a particle into radial and tangential components, which are generally not perpendicular to one another. Siacci formulated this decomposition in two papers which were published in 1879, the first for planar motions, and the second for spatial motions. The theorem is useful in situations where angular momentum is constant (for example, in central forces).*Wik

1928 Gerald Stanley Hawkins (20 Apr 1928; died 26 May 2003 at age 75) was an English astronomer and mathematician who identified Stonehenge to be a prehistoric astronomical observatory. He identified 165 key points in the Stonehenge complex and found that many of them very strongly correlated with the rising and setting positions of the sun and moon. He used a computer to show that there existed at Stonehenge a pattern of alignments with twelve major lunar and solar events. He first published his findings in an article, Stonehenge Decoded, in the journal Nature (1963), and then in a book with the same title (1965). In Beyond Stonehenge he explored the mysteries of Machu Pichu, the Nasca Lines, Easter Island and the Egyptian Temples of Karnak and Amon-Ra. *TIS

DEATHS

1344 Levi ben Gerson He wrote Art of Calculation (or Art of the Computer) in 1321. It deals with arithmetical operations, including extraction of square roots and cube roots. In this work he also looks at the summation of series, permutations and combinations, and basic algebraic identities. He gives formulas for the sum of squares and the sum of cubes of natural numbers as well as studying the binomial coefficients. In proofs, he uses induction making this one of the earliest texts to use this important technique. In fact, it is the Art of Calculation which allows us to give the year of Levi's birth, since he says he finished writing it in 1321, when he was thirty-three years old.
In 1342, at the request of the bishop of Meaux, he wrote The Harmony of Numbers which contains a proof that (1,2), (2,3), (3,4) and (8,9) are the only pairs of consecutive numbers whose only factors are 2 or 3. One year later, he wrote On Sines, Chords and Arcs which examined trigonometry, in particular proving the sine theorem for plane triangles and giving 5 figure sine tables. He calculated his sine tables using Ptolemy's methods and his tables are very accurate. In this work he studied chords, sines, versed sines, cosines but not tangents (which were not in use at this time). Gino Loria suggested that the sine theorem be named after Levi but he was not the first to present the theorem, which was known to Jabir ibn Aflah in the 12th century, but he may have rediscovered it. He also published two geometry books, one being a commentary and introduction to the first five books of Euclid, but not presented axiomatically. The other is the Science of Geometry of which only a fragment has survived. It is interesting to note that Levi was interested in Euclid's parallel postulate and appears to have been part of a lively debate about whether it could be deduced from the other axioms. He proved the parallel postulate with an argument based on an assumption on the convergence or divergence of straight lines that is (as of course it must be) equivalent to the parallel postulate.
He invented the Jacob's staff, an instrument to measure the angular distance between celestial objects. We should note that the term 'Jacob's staff' was not used by Levi but rather by his Christian contemporaries; he used a Hebrew name which translates as 'Revealer of Profundities'. It is described as consisting:
... of a staff of 41/2 feet long and about one inch wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars.
This was far from his only contribution to improvements in astronomical instruments. A striking example is the design of a transversal scale for reading fifteenths of degrees on the graduated outer circle of an astrolabe. We note that, remarkably, it was around 250 years later that Tycho Brahe used a similar transversal scale on his great mural quadrant. Goldstein examines Levi's transversal scale for the Jacob staff. We note that while Levi's method for constructing the scale is theoretically correct, it requires making measurements that seem extremely difficult, so perhaps the theory was never put into practice. *SAU

1786 John Goodricke (17 Sep 1764, 20 Apr 1786 at age 21) English astronomer who was the first to notice that some variable stars were periodic.Born a deaf-mute, after a proper education he was able to read lips and to speak. He was the first to calculate the period of Algol to 68 hours and 50 minutes, where the star was changing its brightness by more than a magnitude as seen from Earth. He was also first to correctly propose that the distant sun is periodically occulted by a dark body. John Goodricke was admitted to the Royal Society on 16 April 1786, when 21 years old. He didn't recognized this honour, because he died four days later, in York, by pneumonia. *TIS
Mike Rendell has written a nice blog with more detail about his short life and discoveries at the Georgian Gentleman.

1794 Jean-Baptiste-Gaspard Bochart de Saron (16 Jan 1730, 20 Apr 1794 at age 64)French lawyer and natural scientist who pursued his interest in astronomy both as a productive amatuer and a patron. He assembled a significant collection of astronomical instruments made by renowned craftsmen. He both utilized then himself and gave access to his academic colleagues. In collaboration with Charles Messier, who provided the data, he calculated orbits of comets, helping his friend find them again after they had disappeared behind the sun. He funded the publication of Laplace's Theory of the Movement and Elliptic Figure of the Planets (1784). Bochart made calculations for what was at first called Herschel's comet, supposing a circular orbit at twelve time the Sun-Saturn distance. This was refined by Laplace, and contributed to the discovery of Uranus. Bochart died as a politician guillotined during the French Revolution.*TIS

1918 Karl Ferdinand Braun (6 Jun 1850, 20 Apr 1918 at age 67) was a German physicist who shared the Nobel Prize for Physics in 1909 with Guglielmo Marconi for the development of wireless telegraphy. He published papers on deviations from Ohm's law and on the calculations of the electromotive force of reversible galvanic elements from thermal sources, and discovered (1874) the electrical rectifier effect. He demonstrated the first cathode-ray oscilloscope (Braun tube) in 1897, after work on high-frequency alternating currents. Cathode-ray tubes had previously been characterized by uncontrolled rays; Braun succeeded in producing a narrow stream of electrons, guided by means of alternating voltage, that could trace patterns on a fluorescent screen. *TIS

1932 Giuseppe Peano, 73, died, after teaching his regular classes the previous day. He axiomatized the natural numbers (1889), elementary geometry (1889), and many other systems. *VFR Peano introduced symbols to represent "belongs to the set of" and "there exists." In Arithmetics principia (1889), a pamphlet he wrote in Latin, Peano published his first version of a system of mathematical logic, giving his Peano axioms defining the natural numbers in terms of sets. *TIS

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

1957 Konrad Hermann Theodor Knopp (22 July 1882 in Berlin, Germany - 20 April 1957 in Annecy, France) Konrad Knopp was a German mathematician who worked on generalised limits and complex functions. He was the co-founder of Mathematische Zeitschrift in 1918. *SAU

2006 Kathleen "Kay" McNulty Mauchly Antonelli (February 12, 1921 – April 20, 2006) was one of the six original programmers of the ENIAC, the first general-purpose electronic digital computer. *Wik

2006 Paul Moritz Cohn FRS (8 January 1924, Hamburg, Germany – 20 April 2006, London, England) was Astor Professor of Mathematics at University College London, 1986-9, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings.*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