## Saturday, 24 March 2018

### On This Day in Math - March 25

Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.
~Girolamo Cardano

The 84th day of the year, with nine points equally placed around a circle there are 84 different triangles using three of these points as vertices. Are any of them right triangles?

84 is the only number that is spelled with ten letters that are all different.

Litttle is known of the life of Diophantus, but this problem, supposedly on his tomb, will reveal his age.
**If you get stuck, I posted the answer at the bottom of the blog.

Keith numbers are much rarer than the primes, with only 84 Keith numbers with digits.
Here is a Numberphile video explaining Keith numbers.  Michael Keith (born 1955) is an American mathematician, software engineer, and author of works of constrained writing. Some of his writings use a language he calls "Pilish" where the words are somehow constrained by the digits of Pi.

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.

EVENTS
1539 Tartaglia tells Cardano about his method of solving cubic equations and Cardano signs an oath to keep the method secret, according to Tartaglia. *B L van der Waerden, History of Algebra
"Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccolo Tartaglia of Brescia gave Niccolo

occasion to discover it. He [Tartaglia] gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in [various] forms. This was very difficult." Cardano in Ars Magna (Basel, 1545)

1655 Christiaan Huygens was the first to discover a moon of Saturn, when he viewed Titan (the largest and easiest to see) on 25 Mar 1655. However, the moon wasn't named until almost two centuries later when Sir John Herschel, discoverer of Uranus, assigned names to the seven moons of Saturn that were known at that time. Saturn's largest moon was named simply "Titan," since the word means "one that is great in size, importance, or achievement." *TIS

1792 D’Alembert wrote: “I would like to see our friend Condorcet, who assuredly has great talent and wisdom, express himself in another manner.” Reading Condorcet’s mathematical works is a thankless task, for the notation is inconsistent, the expression of ideas often imprecise and obscure, and the proofs labored. Perhaps this helps explain why he is not a well known mathematician. [DSB 3, 384]*VFR

1822 Gauss reveals plans to contact aliens: Gauss wrote of the heliotrope's potential (an instrument invented by Gauss in 1821 that uses a mirror to reflect sunlight over great distances) as a celestial signaling device in a March 25, 1822 letter to Heinrich Olbers, by which he reveals a belief and interest in finding a method to contact extraterrestrial life: "With 100 separate mirrors, each of 16 square feet, used conjointly, one would be able to send good heliotrope-light to the moon.... This would be a discovery even greater than that of America, if we could get in touch with our neighbors on the moon."
Some have suggested Gauss may have also proposed the constructing an immense right triangle and three squares on the surface of the earth to signal to aliens from the Moon or Mars. See more on that story here.*Wik

In 1857, Frederick Laggenheim took the first photograph of a solar eclipse. This is often reported but seems not to be the first. I found a note on Wikipedia that "The first correctly-exposed photograph of the solar corona was made during the total phase of the solar eclipse of 28 July 1851 by a local daguerreotypist named Berkowski at the Royal Observatory in Königsberg, Prussia (now Kalinigrad in Russia). Berkowski, whose first name was never published, observed at the Royal Observatory. A small 6-cm refracting telescope was attached to the 15.8-cm Fraunhofer heliometer and a 84-second exposure was taken shortly after the beginning of totality. *Wik

In 1903, The Times newspaper reported that the French physicist, Pierre Curie assisted by Marie Curie, communicated to the Academy of Sciences that the recently discovered Radium “possesses the extraordinary property of continuously emitting heat, without combustion, without chemical change of any kind, and without any change to its molecular structure, which remains spectroscopically identical after many months of continuous emission of heat ... such that the pure Radium salt would melt more than its own weight of ice every hour ... A small tube containing Radium, if kept in contact with the skin for some hours ... produces an open sore, by destroying the epidermis and the true skin beneath ... and cause the death of living things whose nerve centres do not lie deep enough to be shielded from their influence.” *TIS

1992 Excel 4.0 Spreadsheet Software Released: Microsoft Corporation releases its Excel 4.0 spreadsheet program. Excel was one in a long line of practical applications that Microsoft and other companies developed for personal computers, making them more appealing to home and office users. The earliest commercial computerized spreadsheet was VisiCalc, written by Ed Frankston and Dan Bricklin and released for the Apple II personal computer in 1979.*CHM

BIRTHS
1538 Christopher Clavius (March 25, 1538 – February 6, 1612) was a German Jesuit mathematician and astronomer who was the main architect of the modern Gregorian calendar. In his last years he was probably the most respected astronomer in Europe and his textbooks were used for astronomical education for over fifty years in Europe and even in more remote lands (on account of being used by missionaries). As an astronomer Clavius held strictly to the geocentric model of the solar system, in which all the heavens rotate about the Earth. Though he opposed the heliocentric model of Copernicus, he recognized problems with the orthodox model. He was treated with great respect by Galileo, who visited him in 1611 and discussed the new observations being made with the telescope; Clavius had by that time accepted the new discoveries as genuine, though he retained doubts about the reality of the mountains on the Moon. Later, a large crater on the Moon was named in his honour.*Wik
Called the Euclid of the sixteenth-century, born in the German town of Bamberg, the see of the prince-bishop of Franconia. He was also the leader of the Gregorian calendar reform. Perhaps his greatest contribution was as an educational reformer.
In his Astrolabium (Rome,1593) he uses a dot to separate whole numbers from decimal fractions, but it would be 20 more years before the decimal point would be widely accepted. Carl Boyer mentions "the Jesuit friend of Kepler" who was the first to use the decimal point with a clear idea of its significance. In the same work, Clavius originated a way of dividing a scale for precise measurements. His idea was adopted by Vernier 42 years later.
In his Algebra (Rome, 1608) Clavius was the first to use parenthesis to express aggregation and the first to use a symbol for an unknown quantity. Other innovations were also seen in the symbols attributed to him by Florian Cajori such as the radical sign, plus and minus signs.

Clavius proposed a proof that there can be no more than three dimensions in geometry, based on the fact that only three concurrent lines can be drawn from a point so that they are mutually perpendicular. He discovered and proved a theorem for a regular polygon with an odd number of sides which two centuries later enabled Carl Friedrich Gauss to construct a 17-sided polygon by ruler and compass.
In hisTriangula sphaerica (Mainz 1611) Clavius summarized all contemporary knowledge of plane and spherical trigonometry. His prostlaphaeresis , the grandparent of logarithms, relied on the sine of the sum and differences of numbers. In this way he was able to substitute addition and subtraction for multiplication, by solving the identity with which we are familiar today: 2 sin x sin y = cos(x-y)-cos(x+y). D. E. Smith gives the details of the proof and emphasizes the impact Clavius' work had on the discovery of logarithms. Smith also underlines the modesty of Clavius in generously giving to one of his contemporaries more credit than is due for his own prostlaphaeresis . *Joseph MacDonnell, S.J., Fairfield Univ webpage
Some really nice detail about Clavius is at Renaissance Mathematicus

1798 Christoph Gudermann (March 25, 1798, September 25, 1852) was born in Vienenburg. He was the son of a school teacher and became a teacher himself after studying at the University of Göttingen, where his advisor was Karl Friedrich Gauss. He began his teaching career in Kleve and then transferred to a school in Münster.
He is most known today for being the teacher of Karl Weierstrass, who took Gudermann's course in elliptic functions, 1839–1840, the first to be taught in any institute. Weierstrass was greatly influenced by this course, which marked the direction of his own research.
Gudermann originated the concept of uniform convergence, in an 1838 paper on elliptic functions, but only observed it informally, neither formalizing it nor using it in his proofs. Instead, Weierstrass elaborated and applied uniform convergence.
His researches into spherical geometry and special functions focused on particular cases, so that he did not receive the credit given to those who published more general works. The Gudermannian function, or hyperbolic amplitude, is named after him.Gudermann died in Münster. *Wik

1833 (Henry Charles) Fleeming Jenkin (25 Mar 1833; 12 Jun 1885 at age 52) British engineer noted for his work in establishing units of electrical measurement. After earning an M.A. (1851), he worked for the next 10 years with engineering firms engaged in the design and manufacture of submarine telegraph cables and equipment for laying them. In 1861 his friend William Thomson (later Lord Kelvin) procured Jenkin's appointment as reporter for the Committee of Electrical Standards of the British Association for the Advancement of Science. He helped compile and publish reports that established the ohm as the absolute unit of electrical resistance and described methods for precise resistance measurements. *TIS

1859 Samuil Shatunovsky (25 March 1859 – 27 March 1929) was a Russian mathematician. focused on several topics in mathematical analysis and algebra, such as group theory, number theory and geometry. Independently from Hilbert, he developed a similar axiomatic theory and applied it in geometry, algebra, Galois theory and analysis. However, most of his activity was devoted to teaching at Odessa University and writing associated books and study materials.*Wik

1865 Pierre-Ernest Weiss (25 Mar 1865, 24 Oct 1940) French physicist who investigated magnetism and determined the Weiss magneton unit of magnetic moment. Weiss's chief work was on ferromagnetism. Hypothesizing a molecular magnetic field acting on individual atomic magnetic moments, he was able to construct mathematical descriptions of ferromagnetic behaviour, including an explanation of such magnetocaloric phenomena as the Curie point. His theory succeeded also in predicting a discontinuity in the specific heat of a ferromagnetic substance at the Curie point and suggested that spontaneous magnetization could occur in such materials; the latter phenomenon was later found to occur in very small regions known as Weiss domains. His major published work was Le magnetisme ( 1926).*TIS

1923 Kenneth Linn Franklin (25 Mar 1923, ) American astronomer who discovered that the giant planet Jupiter emits radio waves (1955). Dr. Bernard F. Burke and Franklin, astronomers at the Carnegie Institution in Washington, were scanning the sky for radio waves from galaxies. By chance, they found a radio signal that resembled short bursts of static, similar to interference by lightning on home radios. After weeks of study, finding the signals were periodic, four minutes earlier each day, they pin-pointed Jupiter as the source. Never before had radio sounds from a planet in our solar system been detected. Later it was discovered that the radio waves were circularly polarized, so a magnetic field was involved.*TIS

1939 Richard Alfred Tapia (March 25, 1939 - ) is a renowned American mathematician and champion of under-represented minorities in the sciences. In recognition of his broad contributions, in 2005, Tapia was named "University Professor" at Rice University in Houston, Texas, the University's highest academic title. The honor has been bestowed on only six professors in Rice's ninety-nine year history. On September 28, 2011, President Barack Obama announced that Tapia was among twelve scientists to be awarded the National Medal of Science, the top award the United States offers its researchers. Tapia is currently the Maxfield and Oshman Professor of Engineering; Associate Director of Graduate Studies, Office of Research and Graduate Studies; and Director of the Center for Excellence and Equity in Education at Rice University.
Tapia's mathematical research is focused on mathematical optimization and iterative methods for nonlinear problems. His current research is in the area of algorithms for constrained optimization and interior point methods for linear and nonlinear programming.*Wik

DEATHS
1818 Caspar Wessel (8 Jun 174525 Mar 1818 at age 72) was a Norwegian mathematician who invented a geometric way of representing complex numbers which pre-dated Argand. *SAU
His fundamental paper, Om directionens analytiske betegning, was published in 1799 by the Royal Danish Academy of Sciences and Letters. Since it was in Danish, it passed almost unnoticed, and the same results were later independently found by Argand and Gauss.
One of the more prominent ideas presented in "On the Analytical Representation of Direction" was that of vectors. Even though this wasn't Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added if we unite them in such a way that the second line begins where the first one ends and then pass a straight line from the first to the last point of the united lines. This line is the sum of the united lines". This is the same idea as used today when summing vectors. Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognized. His paper was re-issued in French translation in 1899, and in English in 1999 as On the analytic representation of direction (ed. J. Lützen et al.).*Wik

1995 James S(amuel) Coleman (12 May 1926, 25 Mar 1995 at age 68) was a U.S. sociologist, a pioneer in mathematical sociology whose studies strongly influenced education policy. In the early 1950s, he was as a chemical engineer with Eastman-Kodak Co. in Rochester, N.Y. He then changed direction, fascinated with sociology and social problems. In 1966, he presented a report to the U.S. Congress which concluded that poor black children did better academically in integrated, middle-class schools. His findings provided the sociological underpinnings for widespread busing of students to achieve racial balance in schools. In 1975, Coleman rescinded his support of busing, concluding that it had encouraged the deterioration of public schools by encouraging white flight to avoid integration. (We can never control the law of unintended consequences)*TIS

* Diophantus died at 84, why else would I have it on the 84th day of the year?

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