On This Day in Math - September 25

  

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I am undecided whether or not the Milky Way​ is but one of countless others all of which form an entire system. Perhaps the light from these infinitely distant galaxies is so faint that we cannot see them.
~ Johann H Lambert

This is the 268th day of the year, 268 is the smallest number whose product of digits is 6 times the sum of its digits. (A good classroom exploration might be to find numbers in which the product of the digits is n x the sum of the digits for various values of n.. more generally, for what percentage of numbers is the sum a factor of the product at all?)

The two odd numbers adjacent to 6*268 form a pair of twin primes, and the next two odd numbers after 268 are a pair of twin primes. And the 268th prime, is the smaller of a pair of twin primes.

268 is the sum of two consecutive primes, 268 = 131 + 137

Prime Curios offers this little mental conversion, 268 inches of 1/8 inch copper wire weighs 1 pound. There is no AWG standard gauge wire that is 1/8 of an inch diameter, but AWG 8 is really close. For students, what would a similar length of 1/4 inch diameter copper wire weigh?

Many people know that N! has N digits for N= 22, 23, and 24.  Surprisingly, to me, there are also three consecutive numbers for which N! has 2N digits, 266, 267, and 268.  For N! has 3N digits, only two consecutive numbers, 2712 and 2713.For N! having 4N digits, there are again two consecutive  occurrences,  27175 and 27176. For 5N we go back to three consecutive digits,  271819, 271820, 271821  Note the increase by a power of ten as a limit, and the higher you go, the closer they approach  e * 10^n. It has been conjectured that there are always at two or three consecutive numbers for every digit, but never more. The first 100 such numbers are found at A058814 - OEIS Thanks to Derek Orr and Frank Kampas for some help and direction on this. 

EVENTS

1493 Columbus set sail on his second voyage to America.

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303  On Sept. 25, 303 C.E. bishop-martyr St. Fermin (b. 272) is beheaded in Amiens, France; starting in in 1591 the San Fermin Festival in Pamplona, Spain is founded, featuring eight 3/4-ton bulls chasing drunken revelers down a 900-yard cobblestone street corridor; originally held on Sept. 25, it begins running from noon on July 6 to midnight on July 14 starting in 1592; by the 20th cent. there are eight bull runs during the 9-day festival, one each morning at 8 a.m. - advice: stay in front? *HistoryScoper

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1513 On September 25th, 1513, Vasco Nunez de Balboa crossed the Isthmus of Panama and first sighted the Pacific Ocean (it would take four days for his group to work their way down the mountain to the ocean.. Balboa was accompanied by 190 Spaniards and several hundred slaves. His "discovery" spurred those looking for a passage to the Pacific. Balboa spent a number of months exploring the West Coast of Central America. Within six years he would be found guilty of treason and beheaded by rivals for his supposed "river of gold".

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1608 The oldest written mention of the telescope: In a letter of introduction from the Council of Zeeland to Zeeland’s Delegates to the States General (the Netherlands parliament) in Den Haag asking them to organise an audience with Prince Maurice of Nassau for a spectacle maker from Middelburg who had invented a “…certain device by means of which all things at a very great distance can be seen as if they were nearby, by looking through glasses…”; the oldest written mention of the telescope. On an unknown day between 25th and 29th September: Hans Lipperhey (1570 – 1619) the spectacle maker from Middelburg (who was actually a German from Wesel) demonstrates his new invention at the court of Prince Maurice, where a peace conference in the Dutch-Spanish War is taking place along with the first visit to Europe of the Ambassador of Siam. Lipperhey’s demonstration is described in detail in a French flyer describing the Ambassadors visit and the news of the new invention is thus spread rapidly throughout Europe.

26th July 1609 Julian calendar (5th August 1609 Gregorian calendar): Thomas Harriot (1560 – 1621) makes a sketch of the moon using a telescope.

21st August 1609: Galileo demonstrates his telescope to the aristocrats of Venice.

24th August 1609: Galileo presents his telescope to the Doge and Senate of Venice.

25th August 1609: Galileo is granted a lifetime contract as professor for mathematics at the University of Padua with a salary of 1000 Florins but with the subsidiary clause that he would never receive a raise in salary.

When Galileo first used a telescope as an astronomical instrument is not known but it was at least a couple of months later.

It is highly probable that Simon Marius (1573 – 1624) court astronomer in Ansbach Franconia used a telescope as an astronomical instrument before Galileo but it is not possible to determine when.

7th January 1610: Galileo discovers the first three moons of Jupiter.

8th January 1610: Marius discovers the first three moons of Jupiter independently of Galileo. *Renaissance Mathematicus,

Lipperhey’s Patent Application

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1654 Fermat writes to Pascal defending his combinatorial method that Pascal had previously regarded as incorrect.*VFR In the same letter he announced the following two results for odd primes expanding his sums of two primes Christmas letter to Fr Mersenne from 14 years earlier:

p = X² + 2y² Iff p is equivalent to 1 or 3 mod 8,  {3, 11, 17, 43,...}

p = X² + 3y² Iff p is equivalent to 1 mod 3 {4, 7, 13, 19, ...}  

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1820 Arago announces electromagnetism ... Francois Arago announced that a copper wire between the poles of a voltaic cell, could laterally attract iron filings to itself (Ann. de Chim. et de Physique., xv. p.93). His discovery came in the same year that Oersted discovered that an electric current flowing in a wire would deflect a neighboring compass needle. Arago in the same publication described how he had successfully succeeded in causing permanent magnetism in steel needles laid at right angles to the copper wire. Arago and André-Marie Ampère, discussed and experimented with forming the copper wire into a helix to intensify the magnetizing action. However, it was not until 1825 that the electromagnet in its familiar form was invented by William Sturgeon. *TIS

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1938  A brief story in The Oregonian of Portland, Oregon:  "Charles Keville walked into the temporary morgue and looked at a body that had been identified as his."

" 'Nope,' he said,  'That ain't me', and walked out again."  *

Dudes Posting Their W’s

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1944 Denmark issued a stamp commemorating the 300th anniversary of the birth of Ole Roemer,*VFR

Danish astronomer who, in 1676, first demonstrated that light travels at a finite speed. Rømer also invented the modern thermometer showing the temperature between two fixed points, namely the points at which water respectively boils and freezes.

Rømer made his discovery regarding the speed of light while working at the Royal Observatory in Paris and studying Jupiter's moon Io. He estimated that light takes about 11 minutes to travel from the Sun to Earth. Using today's knowledge of the Sun-Earth distance, this would amount to a speed of light of approximately 220,000 kilometers per second,[1] compared to today's accepted value of just under 300,000 kilometers per second.

In scientific literature, alternative spellings such as "Roemer", "Römer", or "Romer" are common.

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1960 NEW MATH. New mathematics is found in Time magazine of Feb. 3, 1958, in the heading, "The new mathematics" [OED].
New math is found again in an article which appeared in numerous newspapers on Sept. 25, 1960: “But the ‘new math’ is being promoted energetically by such influential bodies as the U. S. Office of Education, the National Science Foundation, the National Education Association, the Mathematical Association of America, the College Entrance Examination Board and the Carnegie Corporation.”
* Jeff Niller

cartoon about parental confusion during "New Math" introduction.  

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On this day in 1988, Faà di Bruno was beatified by Pope John Paul II in St Peter's Square in Rome.   is best known for his formula for the nth derivative of a composition of functions. Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855, 1857), although he was not the first to state or prove the formula.  In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast had stated the formula in a calculus textbook, which is considered to be the first published reference on the subject.  

#SAU 

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1989 IBM announces plans to develop a new design for transmitting information within a computer, called Micro Channel Architecture, which it said could transfer data at 160 million bytes per second or eight times faster than the fastest speed at the time. Although IBM was hoping to make its system the industry standard, manufacturers of IBM-compatible computers largely chose other methods. *CHM

*CHM

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BIRTHS
1644 Olaus Roemer, Danish astronomer, born. He was the first to measure the speed of light. *VFR (25 Sep 1644;23 Sep 1710) Astronomer who demonstrated conclusively that light travels at a finite speed. He measured the speed by precisely measuring the length of time between eclipses of Jupiter by one of its moons. This observation produces different results depending on the position of the earth in its orbit around the sun. He reasoned that meant light took longer to travel the greater distance when earth was traveling in its orbit away from Jupiter.*TIS "Ole Rømer took part in several other achievements considering measurement. He developed a temperature scale that is now famous as the Fahrenheit scale. Fahrenheit improved and distributed his ideas after visiting Rømer. In his last years, he was even given the position as second Chief of the Copenhagen Police and invented the first street oil lamps in the city of Copenhagen.
Further achievements and inventions may be added to Rømer's biography, like his innovative water supply system and his urban planning concept. " *Yovista.blogspot

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1819 George Salmon (25 September 1819 – 22 January 1904) made many discoveries about ruled surfaces and other surfaces. *SAU His publications in algebraic geometry were widely read in the second half of the 19th century. A Treatise on Conic Sections remained in print for over fifty years, going though five updated editions in English, and was translated into German, French and Italian. *Wik

Salmon statue at Trinity College, Dublin

*MacTutor, SAU

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1825 Carl Harald Cramer,(25 September 1893 ,5 October 1985) was a Swedish mathematical statisticians and is one of the prominent figures in the statistical theory. He was once described by John Kingman as "one of the giants of statistical theory". 
In number theory, Cramér's conjecture,in 1936 states that

where p_n denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. Intuitively, this means the gaps between consecutive primes are always small, and it quantifies asymptotically just how small they can be. This conjecture has not been proven or disproven.
*Wik

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.

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1846 Wladimir (Peter) Köppen (25 Sep 1846; 22 Jun 1940) German meteorologist and climatologist best known for his delineation and mapping of the climatic regions of the world. He played a major role in the advancement of climatology and meteorology for more than 70 years. The climate classification system he developed remains popular because it uses easily obtained data (monthly mean temperatures and precipitation) and straightforward, objective criteria. He recognized five principal climate groups: (A) Humid tropical -winterless climates; (B) Dry - evaporation constantly exceed precipitation; (C) humid mid-latitude, mild winters; (D) humid mid-latitude, severe winters; and (E) Polar - summerless climates. *TIS

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1857 Sergey Petrovich Degayev (also spelled Degaev; Russian: Серге́й Петрович Дегаев; 1857 in Moscow – 1921 in Bryn Mawr, Pennsylvania) was a Russian revolutionary terrorist, Okhrana agent, and the murderer of inspector of secret police Georgy Sudeykin. After emigrating to the United States, Degayev took the name Alexander Pell and became a prominent American mathematician, the founder of school of Engineering at the University of South Dakota. The Dr. Alexander Pell scholarship is named in his honor.

After December 1883, all the posts in the Empire were plastered with posters showing Degayev's photographs and announcing 5000 roubles for information as to his whereabouts and 10,000 roubles for help in catching him. Still the conspirators had a good lead on their hunters and successfully arrived in Paris. At a winter 1884 meeting, Narodnaya Volya, led by V. A. Karaulov, Lev Tikhomirov and German Lopatin, fulfilled its end of the bargain and granted Degayev his life on the condition that he never again appear in the Russian Empire. Lev Tikhomirov personally verified that he boarded a steamship bound for South America.

From South America Degayev moved to the United States; there he joined his wife, Lyubov Degayeva. His brother, Vladimir Degayev, who worked at the time in a Russian consulate in the United States and moonlighted as a foreign correspondent for a few Russian publications printed an article claiming that Sergey Degayev was killed in New Zealand, discouraging searches for him by both Russian police and Russian revolutionaries.

Both Vladimir and Sergey Degayevs were registered in the USA under the name Polevoi after their maternal grandfather Nikolai Polevoy. After his naturalization Alexander (Sergey) took the name Alexander Pell and his wife took the name Emma Pell. At first they were poor; Sergey worked as a stevedore and as an unskilled labourer while his wife worked as a cook and a laundress. In 1895 Alexander was enrolled into a PhD program in Johns Hopkins University with majors in mathematics and astronomy and a minor in English. During his study he was financially supported by his wife who continued to work as a cook. He received his doctorate in 1897 for the dissertation On the Focal surfaces of the Congruences of Tangents to a Given Surface.

The University of South Dakota was established in the frontier town of Vermillion and started its classes in 1882. In 1897 they decided that they needed a professor of mathematics. They asked Professor L. S. Hulburt from Johns Hopkins if he could suggest a suitable candidate. He replied that he "could suggest a first class mathematician who had the disadvantage of having a strong Russian brogue". The reply from South Dakota was "Send your Russian mathematician along, brogue and all".

Alexander Pell (Sergey Degayev) and Emma Pell (Lyubov Degayeva), South Dakota

Alexander Pell was immensely popular among his students who referred to him as the "class father" and "Jolly Little Pell" (who could "crack jokes faster than the freshmen could crack nuts"). He was a good researcher, a member of the American Mathematical Society and the author of many journal publications. He was also an accomplished administrator who organized the School of Engineering of the University of South Dakota and became its first Dean (1905).

Alexander Pell had a habit of providing financial support from his own resources, and providing accommodation in his house to a few of his students. One such student was Anna Johnson, the future accomplished mathematician Anna Johnson Pell Wheeler. Anna Johnson received her A.B. degree under Pell's supervision in 1903 and continued her study at the University of Iowa and then at the University of Göttingen. In 1904 Emma Pell died. 

Three years later Alexander Pell went to Göttingen and married Anna in July 1907. They both returned to Vermillion where Anna taught classes in the theory of functions and differential equations and Alexander was the Dean of Engineering. In 1908 Pell resigned from the University of South Dakota and went with Anna to Chicago. There Anna completed her doctorate under E. H. Moore, while Pell took a position at the Armour Institute of Engineering (currently Illinois Institute of Technology). In 1911 Pell suffered a stroke and was unable to work thereafter. The same year the Pells moved to South Hadley, Massachusetts where Anna taught at Mount Holyoke College. In 1918 they moved again to Bryn Mawr, Pennsylvania where Anna taught at Bryn Mawr College. Alexander Pell died in Bryn Mawr in 1921.

Alexander Pell (Sergey Degayev) and Emma Pell (Lyubov Degayeva), South Dakota

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1888 Stefan Mazurkiewicz (25 Sept 1888 , 19 June 1945) His main work was in topology and the theory of probability. His notion of dimension of a compact set preceded that of Menger and Urysohn by seven years. Mazurkiewicz applied topological methods to the theory of functions, obtaining powerful results. His theory gave particularly strong results when applied to the Euclidean plane, giving deep knowledge of its topological structure. *SAU

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DEATHS
1777 Johann Heinrich Lambert (26 Aug 1728, 25 Sep 1777) Swiss-German mathematician, astronomer, physicist, and philosopher who provided the first rigorous proof that pi ( the ratio of a circle's circumference to its diameter) is irrational, meaning it cannot be expressed as the quotient of two integers. He also devised a method of measuring light intensity. *TIS In 1766 Lambert wrote Theorie der Parallellinien which was a study of the parallel postulate. By assuming that the parallel postulate was false, he managed to deduce a large number of non-euclidean results. He noticed that in this new geometry the sum of the angles of a triangle increases as its area decreases. *SAU

Lambert devised a formula for the relationship between the angles and the area of hyperbolic triangles. These are triangles drawn on a concave surface, as on a saddle, instead of the usual flat Euclidean surface. Lambert showed that the angles added up to less than π (radians), or 180°. The amount of shortfall, called the defect, increases with the area. The larger the triangle's area, the smaller the sum of the angles and hence the larger the defect C△ = π — (α + β + γ). 

Lambert solves a geometrical problem, viz. to reconstruct the course of a ship at sea on the basis of observations with a plane table on the shore. The problem is originally in Marinoni's De Re Ichnographica (1751), which Lambert read early on (1752). Lambert here reduces this problem (the most complicated one in Marinoni) using algebraic methods. Since many observations have to be reduced to solve the problem, these studies constitute one of the starting points for Lambert's work on the theory of errors.

Lambert is a curious historical figure; extremely accomplished, a worthy member of a select company that includes Leonard Euler and Immanuel Kant, except that hardly anyone has ever heard of Lambert, save the optician who measures the luminance of a light source in lamberts, or the cartographer who produces an aviation map using the Lambert conical projection, or the astrophysicist who measures the albedo of a moon of Saturn, using a term for reflectivity that was coined by Lambert. Lambert spent the last 13 years of his working life in Berlin, where he was appointed a member of the Academy of Sciences there. He wrote original works on pyrometry (the measure of heat), photometry (the measure of luminance), hygrometry (measurement of humidity), cartographic projections (he invented 7 new projections, several still in common use), even treatises on non-Euclidean geometry and irrational numbers. *Linda Hall Org

*Wik

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1852 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

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1877 Urbain-Jean-Joseph Le Verrier (11 May 1811, 25 Sep 1877) French astronomer who predicted the position of a previously unknown planet, Neptune, by the disturbance it caused in the orbit of Uranus. In 1856, the German astronomer Johan G. Galle discovered Neptune after only an hour of searching, within one degree of the position that had been computed by Le Verrier, who had asked him to look for it there. In this way Le Verrier gave the most striking confirmation of the theory of gravitation propounded by Newton. Le Verrier also initiated the meteorological service for France, especially the weather warnings for seaports. *TIS (He died the day after the anniversary of the sighting of his most famous prediction. Between that moment of fame in 1846 and his death, he mistakenly attributed the variability of Mercury's orbit to another small planet he named "Vulcan". It took the theory of General Relativity to explain the variations. He was buried in Montparnasse cemetery in Paris. A large globe sits atop grave. Arago described him as, "the man who discovered a planet with the point of his pen."

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1933 Paul Ehrenfest (January 18, 1880, September 25, 1933) was an Austrian and Dutch physicist and mathematician, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem. *Wik

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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