## Thursday, 23 October 2014

### On This Day in Math - October 23

God exists since mathematics is consistent, and the devil exists since its consistency cannot be proved.
~Hermann Klaus Hugo Weyl

The 296th day of the year; 296 is the number of partitions of 30 with distinct parts. (Even very young students can enjoy exploring the number of partitions of integers, and the difference in the number when the parts must be distinct. The idea can be explored for very young students with number rods, etc)
And just reminded that 1023 is the exponent for a mole, so about 6:02 (am or pm) you can set down to a Mole of molecules of your favorite brew.  Happy Mole Day.

EVENTS

4004 B.C. The date of the creation of the world according to the computation of Archbishop James Usher (1581–1656), who had a curious fascination for the integration of numerology, astronomy, and scripture. It was highly regarded for several centuries primarily because in 1701 it was inserted into the margin of the King James version of the Bible. *Sky and Telescope, vol. 62, Nov. 81, 404–405.

526 Boethius executed.Boethius translated Nicomachus's treatise on arithmetic (De institutione arithmetica libri duo) as well as Euclid and Iamblichus.
There is a long tradition, going back at least to the eighth century, regarding Boethius as having been executed for maintaining the Catholic faith against the Arian Theodoric. While Theodoric was probably paranoid about spies representing the Catholic eastern emperor-in-waiting Justinian (who would, in fact, later “reconquer” the Italian peninsula), and Boethius claims in the Consolation that he was hated for being smarter than everyone else, the truth is probably that he was caught up in the usual machinations of an imperial court.
A member of the Senate was accused of treasonably conspiring with Justinian’s predecessor Justin I against Theodoric. Boethius defended the accused (apparently the only person to do so, although the charges were surely trumped up), and in the Consolation, Boethius says he was only defending the Senate (implying that the accusations were meant to undermine the authority of the Senate by challenging its loyalty to the king).
In any event, the sources we have say that Boethius was condemned by the Senate (who appear to have thrown him under the bus) without being able to speak in his own defense. After an indeterminate time of imprisonment, he was executed.
It was while he awaited death that he wrote his most famous and arguably most influential work, The Consolation of Philosophy.” From “Executed Today” web site

1676 Hooke's diary records that he had “Mercator's Music” copied on 23 October . This is "Mercator's earliest thinking on music, has major sections on the consonances and on the arithmetic of proportions using ratios or using logarithms.  Descartes, Newton, and Nicolaus Mercator all worked on the problem of musical timing ( To divide the octave into tones) using logs in the mid-17th century. *Benjamin Wardhaugh, Historia Mathematica, Volume 35, Issue 1, February 2008, Pages 19–36 A longer, clearer essay on the problem, with lots of interesting notes about Mercator, and historical thought on ratios by Wardhaugh is at the Convergence web site of the MAA

In 1803, John Dalton presented an essay on the absorption of gases by water, at the conclusion of which he gave a series of atomic weights for 21 simple and compound elements. He read his paper at a meeting of the Manchester Literary and Philosophical Society. *TIS

1852 August DeMorgan reported the conjecture of his student, Francis Guthrie: Four colors suffice to color planar maps so that adjacent regions have different color. It was solved by Kenneth Appel and Wolfgang Haken in 1976. In a letter to W.R. Hamilton he recalls, “A student of mine asked me today to give him a reason for a fact which I did not know was a fact – and do not yet. He says that if a figure be anyhow divided, and the compartments differently colored so that figures with any portion of common boundary line are differently colored --- four colors may be wanted, but not more…” *Dave Richeson, Euler’s Gem, pg 132)

1892  The Duck-Rabbit double illusion was first published in Fliegende Blätter, a German humor magazine (Oct. 23, 1892, p. 147). The ambiguous figure in which the brain switches between seeing a rabbit and a duck was "originally noted" by American psychologist Joseph Jastrow (Jastrow 1899, p. 312; 1900; see also Brugger and Brugger 1993). Jastrow used the figure, together with such figures as the Necker cube and Schröder stairs, to point out that perception is not just a product of the stimulus, but also of mental activity (Kihlstrom 2002). Jastrow's cartoon was based on one originally published in Harper's Weekly (Nov. 19, 1892, p. 1114) which, in turn, was based on the earlier illustration in Fliegende Blätter,*Mathworld.Wolfram.com

2014 Today is Mole Day. Celebrated annually on October 23 from 6:02 a.m. to 6:02 p.m., Mole Day commemorates Avogadro's Number (6.02 x 10^23) (Which I recently learned from John D. Cook was approximately 24 factorial or 24!) , which is a basic measuring unit in chemistry. Mole Day was created as a way to foster interest in chemistry. Schools throughout the United States and around the world celebrate Mole Day with various activities related to chemistry and/or moles.
For a given molecule, one mole is a mass (in grams) whose number is equal to the atomic mass of the molecule. For example, the water molecule has an atomic mass of 18, therefore one mole of water weighs 18 grams. An atom of neon has an atomic mass of 20, therefore one mole of neon weighs 20 grams. In general, one mole of any substance contains Avogadro's Number of molecules or atoms of that substance. This relationship was first discovered by Amadeo Avogadro (1776-1858) and he received credit for this after his death. *Mole Day Org web page

BIRTHS

1865 Piers Bohl (23 Oct 1865 in Walka, Livonia (now Valka, Latvia) - 25 Dec 1921 in Riga, Latvia) Among Bohl's achievements was, rather remarkably, to prove Brouwer's fixed-point theorem for a continuous mapping of a sphere into itself. Clearly the world was not ready for this result since it provoked little interest.
Bohl also studied questions regarding whether the fractional parts of certain functions give a uniform distribution. His work in this area was carried forward independently by Weyl and Sierpinski. There are many seemingly simple questions in this area which still seem to be open. For example it is still unknown whether the fractional parts of (3/2)n form a uniform distribution on (0,1) or even if there is some finite subinterval of (0,1) which is avoided by the sequence. *SAU

1875 Gilbert Newton Lewis (23 Oct 1875, 23 Mar 1946 at age 70) American chemist who collaborated with Irving Langmuir in developing an atomic theory. He developed a theory of valency, which introduced the covalent bond (c. 1916), whereby a chemical combination is made between two atoms by the sharing of a pair of electrons, one contributed from each atom. This was part of his more general octet theory, published in Valence and the Structure of Atoms and Molecules (1923). Lewis visualized the electrons in an atom as being arranged in concentric cubes. The sharing of these electrons he illustrated in the Lewis dot diagrams familiar to chemistry students. He generalized the concept of acids and bases now known as Lewis acids and Lewis bases. *TIS

1893 Ernest Julius Öpik (23 Oct 1893; 10 Sep 1985) Estonian astronomer best known for his studies of meteors and meteorites, and whose life work was devoted to understanding the structure and evolution of the cosmos. When Soviet occupation of Estonia was imminent, he moved to Hamburg, then to Armagh Observatory, Northern Ireland (1948-81). Among his many pioneering discoveries were: (1) the first computation of the density of a degenerate body, namely the white dwarf 40 Eri B, in 1915; (2) the first accurate determination of the distance of an extragalactic object (Andromeda Nebula) in 1922; (3) the prediction of the existence of a cloud of cometary bodies encircling the Solar System (1932), later known as the Oort Cloud''; (4) the first composite theoretical models of dwarf stars like the Sun which showed how they evolve into giants (1938); (5) a new theory of the origin of the Ice Ages (1952). *TIS

1905 Felix Bloch (23 Oct 1905; 10 Sep 1983) Swiss-born American physicist who shared (with independent discoverer, E.M. Purcell) the Nobel Prize for Physics in 1952 for developing the nuclear magnetic resonance (NMR) method of measuring the magnetic field of atomic nuclei. He obtained his PhD under Werner Heisenberg in 1928, then taught briefly in Germany, but as a Jew, when Hitler came to power, he left Europe for the USA. Bloch's concept of magnetic neutron polarization (1934) enabled him, in conjunction with L. Alvarez, to measure the neutron's magnetic moment. During WW II he worked on the atomic bomb. Thereafter, Bloch and co-workers developed NMR, now widely used technique in chemistry, biochemistry, and medicine. In 1954 he became the first director of CERN.*TIS

1908 Ilya Mikhaylovich Frank (23 Oct 1908; 22 June 1990) Russian physicist who, with Igor Y. Tamm, theoretically explained the mechanism of Cherenkov radiation. In 1934, Cherenkov discovered that a peculiar blue light is emitted by charged particles traveling at very high speeds through water. Frank and Tamm provided the theoretical explanation of this effect, which occurs when the particles travel through an optically transparent medium at speeds greater than the speed of light in that medium. This discovery resulted in the development of new methods for detecting and measuring the velocity of high-speed particles and became of great importance for research in nuclear physics. For this, Frank received the Nobel Prize for Physics in 1958 (jointly with Pavel A. Cherenkov and Igor Y. Tamm). *TIS

1920 Tetsuya Theodore Fujita (23 Oct 1920; 19 Nov 1998) was a Japanese-American meteorologist who increased the knowledge of severe storms. In 1953, he began research in the U.S. Shortly afterwards, he immigrated and established the Severe Local Storms Project. He was known as "Mr. Tornado" as a result of the Fujita scale (F-scale, Feb 1971), which he and his wife, Sumiko, developed for measuring tornadoes on the basis of their damage. Following the crash of Eastern flight 66 on 24 Jun 1975, he reviewed weather-related aircraft disasters and verified the downburst and the microburst (small downburst) phenomena, enabling airplane pilots to be trained on how to react to them. Late in his career, he turned to the study of storm tracks and El Nino. *TIS

DEATHS

1581 Michael Neander (April 3, 1529 – October 23, 1581) German mathematician and astronomer was born in Joachimsthal, Bohemia, and was educated at the University of Wittenberg, receiving his B.A. in 1549 and M.A. in 1550.
From 1551 until 1561 he taught mathematics and astronomy in Jena, Germany. He became a professor in 1558 when the school where he taught became a university. From 1560 until his death he was a professor of medicine at the University of Jena. He died in Jena, Germany. The crater Neander on the Moon is named after him. *Wik

1944 Charles Glover Barkla (7 Jun 1877, 23 Oct 1944) was a British physicist who was awarded the Nobel Prize for Physics in 1917 for his work on X-ray scattering. This technique is applied to the investigation of atomic structures, by studying how X-rays passing through a material and are deflected by the atomic electrons. In 1903, he showed that the scattering of x-rays by gases depends on the molecular weight of the gas. His experiments on the polarization of x-rays (1904) and the direction of scattering of a beam of x-rays (1907) showed X-rays to be electromagnetic radiation like light (whereas, at the time, William Henry Bragg who held that X-rays were particles.) Barkla further discovered that each element has its own characteristic x-ray spectrum. *TIS

1985 John Semple studied at Queen's University Belfast and Cambridge. He held a post in Edinburgh for a year before becoming Professor of Pure Mathematics at Queen's College Belfast. He moved to King's College London where he spent the rest of his career. His most important work was in Algebraic geometry, in particular work on Cremona transformations and work extending results of Severi . He wrote two famous texts Algebraic projective geometry (1952) and Algebraic curves (1959) jointly with G T Kneebone. *SAU

2007 David George Kendall FRS (15 January 1918 – 23 October 2007)[2] was an English statistician, who spent much of his academic life in the University of Oxford and the University of Cambridge. He worked with M. S. Bartlett during the war, and visited Princeton University after the war. He was appointed the first Professor of Mathematical Statistics in the Statistical Laboratory, University of Cambridge in 1962, in which post he remained until his retirement in 1985. He was elected to a professorial fellowship at Churchill College, and he was a founding trustee of the Rollo Davidson Trust.
Kendall was a world expert in probability and data analysis, and pioneered statistical shape analysis including the study of ley lines. He defined Kendall's notation for queueing theory.
The Royal Statistical Society awarded him the Guy Medal in Silver in 1955, followed in 1981 by the Guy Medal in Gold. In 1980 the London Mathematical Society awarded Kendall their Senior Whitehead Prize, and in 1989 their De Morgan Medal. He was elected a fellow of the Royal Society in 1964. *Wik

2011 John McCarthy (September 4, 1927 – October 23, 2011) was an American computer scientist and cognitive scientist who received the Turing Award in 1971 for his major contributions to the field of Artificial Intelligence (AI). He was responsible for the coining of the term "Artificial Intelligence" in his 1955 proposal for the 1956 Dartmouth Conference and was the inventor of the LISP programming language.*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

## Wednesday, 22 October 2014

### The Kiss Precise, Soddy's Circle Theorem

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

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

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

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

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

In 1936 Sir Fredrick Soddy rediscovered the theorem again. Soddy may also be known to students of Science for receiving the Nobel Prize for Chemistry in 1921 for the discovery of the decay sequences of radioactive isotopes.  According to Oliver Sacks' wonderful book, Uncle Tungsten, Soddy also created the term "isotope" and was the first to use the term  "chain reaction".  In a strange "chain reaction" of ideas, Soddy played a part in the US developing an atomic bomb.  Soddy's book, The  Interpretation of Radium, inspired  H G Wells to write The World Set Free in 1914, and he dedicated the novel to Soddy's book. Twenty years later, Wells' book set Leo Szilard to thinking about the possibility of Chain reactions, and how they might be used to create a bomb, leading to his getting a British patent on the idea in 1936.  A few years later Szilard encouraged his friend, Albert Einstein, to write a letter to President Roosevelt about the potential for an atomic bomb.  The prize-winning science-fiction writer, Frederik Pohl, talks about Szilard's epiphany in Chasing Science (pg 25),
".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb.  There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row.  Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."

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

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

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

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

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

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

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

In _Nature_, June 20, 1936

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

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

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

The Kiss Precise (Generalized) by Thorold Gosset

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

In _Nature_
January 9, 1937.

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

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

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

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

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

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

How frightfully pedestrian

My predecessors were

To pose in space Euclidean

Each fraternising sphere!

Let Gauss' k squared be positive

When space becomes elliptic,

And conversely turn negative

For spaces hyperbolic:

Squared sum of bends is sum times n

Of twice k squared plus squares of bends.

### On This Day in Math - October 22

One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges, have been afflicted with is, that mathematics can be mastered by the favored few, but lies beyond the grasp and power of the ordinary mind.
~Florian Cajori, The Teaching and History of Mathematics in the United States

The 295th day of the year; 295 may be interesting only because it seems to be the least interesting day number of the year. (Willing to be contradicted, send your comments)
[Here are several of the best I received from David Brooks:
295 can be partitioned in 6486674127079088 ways.
295 is a 31-gonal number.
295 is a semiprime number (the product of two distinct prime numbers).
]

And Derek Orr pointed out that "295 is the second proposed Lychrel number." A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proved to exist, but many, including 196, are suspected on heuristic and statistical grounds. The name "Lychrel" was coined by Wade VanLandingham as a rough anagram of Cheryl, his girlfriend's first name.

EVENTS
1668 Leibniz writes to the German emperor to request permission to publish a "Nucleus Libareaus". This was the beginnings of the foundation of Acta Eruditorium, the first German scientific journal.

1685 Abraham De Moivre was a student of physics at the University, Collège d'Harcourt, in the 1680s. After the Revocation of the Edict of Nantes, (October 22, 1685 ) he went into seclusion in the priory of St. Martin (possibly that which became the Conservatoire National des Arts et Métiers ??) and then emigrated to England, having no contact with France until he was elected a Foreign Associate of the Academy of Sciences just before his death.*VFR

1922 M. C. ESCHER visited here(Alhambra) on 18 - 24 Oct 1922 and was impressed by the patterns, but he didn't really use them in his art until after his second visit on 22-26 May 1936 *VFR

1746 Princeton chartered as the College of New Jersey -- the name by which it was known for 150 years -- Princeton University was British North America's fourth college. Located in Elizabeth for one year and then in Newark for nine, the College of New Jersey moved to Princeton in 1756. It was housed in Nassau Hall, which was newly built on land donated by Nathaniel FitzRandolph. Nassau Hall contained the entire College for nearly half a century. *Princeton Univ web page

In 1797, the first parachute jump was made by André-Jacques Garnerin, released from a balloon 2,230-ft above the Parc Monceau, Paris. He rode in a gondola fixed to the lines of a 23-ft diameter parachute, which was supported by a wooden pole and had its 32 white canvas gores folded like a closed umbrella. Lacking any vent in the top of the parachute, Garnerin descended with violent oscillations, and suffered the first case of airsickness. For his next jump, he added a hole in the top of the parachute. He made his fifth jump on 21 Sep 1802 over London, from a height of 3,000-ft. This was the first parachute descent made in England. He landed near St. Pancras Church. Having eliminated the center vent for this jump, he again suffered a fit of vomiting. *TIS See A larger TIS article here.

1850 Fechner’s law introduced. [Springer’s 1985 Statistics Calendar] A pioneering though in many situations incorrect formulation of the relationship between the physical strength of a stimulus and its strength as perceived by humans, proposed by G. T. Fechner in 1860. Fechner postulated that sensation increases as the log of the stimulus. For example, by Fechner's law, if light A was twice as bright as light B (measured by an instrument), it would appear to the human eye to be log 2 (times a constant to allow for such factors as the units used) brighter than light B. Later experiments have shown conclusively that the Fechner's law doesn't generally apply.

1908 First meeting of the Spanish Association for the Advancement of Science was held October 22–29. Sixteen papers were read in the section of mathematics.*VFR

1938 In the back of a beauty shop in the Astoria section of Queens New York, Chester A. Carlson and his assistant Otto Kornei, conducted the ﬁrst successful experiment in electrophotography. The message, “10.-22.-38 ASTORIA,” was even less inspiring than Alexander Graham Bell’s ﬁrst phone conversation, but the effect was just as great. In 1949 Haloid Corporation marketed the Xerox Model A, a crude machine that required fourteen manual operations. Today ﬁve million copiers churn out 2,000 copies each year for every American citizen. *VFR

BIRTHS

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

1587 Joachim Jungius (22 Oct 1587 in Lübeck, Germany - 23 Sept 1657 in Hamburg) a German mathematician who was one of the first to use exponents to represent powers and who used mathematics as a model for the natural sciences. Jungius proved that the catenary is not a parabola (Galileo assumed it was). *SAU (I can not find the first use by Jungius anywhere, but Cajori gives Descartes 1637 use in Geometrie as the first example of the common form today. A year earlier, James Hume produced a copy of Viete's Algebra in which he used exponents as powers of numbers, but his exponents were Roman Numerals.)

1792 Guillaume-Joseph-Hyacinthe-Jean-Baptiste Le Gentil de la Galaziere  (12 Sep 1725; 22 Oct 1792) was a French astronomer who attempted to observe the transit of Venus across the sun by travelling to India in 1761. He failed to arrive in time due to an outbreak of war. He stayed in India to see the next transit which came eight years later. This time, he was denied a view because of cloudy weather, and so returned to France. There, he found his heirs had assumed he was dead and taken his property.*TIS A more detailed blog about his life is at Renaissance Mathematicus

1843 John S Mackay graduated from St Andrews University and taught at Perth Academy and Edinburgh Academy. He was a founder member of the EMS and became the first President in 1883 and an honorary member in 1894. He published numerous papers on Geometry in the EMS Proceedings.*SAU

1881 Clinton Joseph Davisson (22 Oct 1881; 1 Feb 1958) American experimental physicist who shared the Nobel Prize for Physics in 1937 with George P. Thomson of England for discovering that electrons can be diffracted like light waves. Davisson studied the effect of electron bombardment on surfaces, and observed (1925) the angle of reflection could depend on crystal orientation. Following Louis de Broglie's theory of the wave nature of particles, he realized that his results could be due to diffraction of electrons by the pattern of atoms on the crystal surface. Davisson worked with Lester Germer in an experiment in which electrons bouncing off a nickel surface produced wave patterns similar to those formed by light reflected from a diffraction grating, and supporting de Broglie's electron wavelength = (h/p). *TIS

1895 Rolf Herman Nevanlinna​ (22 October 1895 – 28 May 1980) was one of the most famous Finnish mathematicians. He was particularly appreciated for his work in complex analysis.Rolf Nevanlinna's most important mathematical achievement is the value distribution theory of meromorphic functions. The roots of the theory go back to the result of Émile Picard in 1879, showing that a complex-valued function which is analytic in the entire complex plane assumes all complex values save at most one.*Wik

1905 Karl Guthe Jansky (22 Oct 1905; 14 Feb 1950) was an American electrical engineer who discovered cosmic radio emissions in 1932. At Bell Laboratories in NJ, Jansky was tracking down the crackling static noises that plagued overseas telephone reception. He found certain radio waves came from a specific region on the sky every 23 hours and 56 minutes, from the direction of Sagittarius toward the center of the Milky Way. In the publication of his results, he suggested that the radio emission was somehow connected to the Milky Way and that it originated not from stars but from ionized interstellar gas. At the age of 26, Jansky had made a historic discovery - that celestial bodies could emit radio waves as well as light waves. *TIS

1907 Sarvadaman D. S. Chowla (22 October 1907, London–10 December 1995, Laramie, Wyoming) was a prominent Indian mathematician, specializing in number theory. Among his contributions are a number of results which bear his name. These include the Bruck–Chowla–Ryser theorem, the Ankeny–Artin–Chowla congruence, the Chowla–Mordell theorem, and the Chowla–Selberg formula, and the Mian–Chowla sequence.*Wik

1916 Nathan Jacob Fine (22 October 1916 in Philadelphia, USA - 18 Nov 1994 in Deerfield Beach, Florida, USA) He published on many different topics including number theory, logic, combinatorics, group theory, linear algebra, partitions and functional and classical analysis. He is perhaps best known for his book Basic hypergeometric series and applications published in the Mathematical Surveys and Monographs Series of the American Mathematical Society. The material which he presented in the Earle Raymond Hedrick Lectures twenty years earlier form the basis for the material in this text.*SAU

1927 Alexander Ivanovich Skopin (22 Oct 1927 in Leningrad (now St Petersburg), Russia - 15 Sept 2003 in St Petersburg, Russia) He was a Russian mathematician known for his contributions to abstract algebra. Skopin's student work was in abstract algebra, and concerned upper central series of groups and extensions of fields. In the 1970s, Skopin received a second doctorate concerning the application of computer algebra systems to group theory. From that point onward he used computational methods extensively in his research, which focused on lower central series of Burnside groups. He related this problem to problems in other areas of mathematics including linear algebra and topological sorting of graphs. *Wik

1941 Stanley Mazor was born in Chicago on October 22, 1941. He studied mathematics and programming at San Francisco State University. He joined Fairchild Semiconductor in 1964 as a programmer and then a computer designer in the Digital Research Department where he shares patents on the Symbol computer. In 1969, he joined Intel. In 1977, he began his teaching career in Intel's Technical Training group, and later taught classes at Stanford, University of Santa Clara, KTH in Stockholm and Stellenbosch, S.A. In 1984 he was at Silicon Compiler Systems. He co-authored a book on chip design language while at Synopsys 1988-1994. He was invited to present The History of the Microcomputer at the 1995 IEEE Proceedings. He is currently the Training Director at BEA Systems. *CHM

DEATHS

1950 Ada Isabel Maddison (13 April 1869 in Cumberland, England - 22 Oct 1950 in Martin's Dam, Wayne, Pennsylvania, USA) A British mathematician best known for her work on differential equations. Although Maddison passed an honors exam for the University of Cambridge, she was not given a degree there. Instead, she went to Bryn Mawr in Pennsylvania. In 1893, the University of London awarded her a bachelor's degree in mathematics with honors. After further study at the University of Göttingen, Maddison went back to Bryn Mawr, where she taught as well as doing time consuming administrative work. Her will endowed a pension fund for Bryn Mawr's administrative staff.*Wik

1977 Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of combinatorial geometry. Among his main contributions to algebraic geometry are studies of birational invariants of algebraic varieties, singularities and algebraic surfaces. His work was in the style of the old Italian School, although he also appreciated the greater rigor of modern algebraic geometry. Another contribution of his was the introduction of finite and non-continuous structures into geometry. In his best known paper he proved the following theorem: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conics. Some critics felt that his work was no longer geometry, but today it is recognized as a separate sub-discipline: combinatorial geometry.
In 1938 he lost his professorship as a result of the anti-Jewish laws enacted under Benito Mussolini's government; he spent the next 8 years in Great Britain (mostly at the University of Manchester), then returned to Italy to resume his academic career *Wik

1979 Reinhold Baer (22 July 1902 in Berlin, Germany - 22 Oct 1979 in Zurich, Switzerland) Baer's mathematical work was wide ranging; topology, abelian groups and geometry. His most important work, however, was in group theory, on the extension problem for groups, finiteness conditions, soluble and nilpotent groups. *SAU

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

## Tuesday, 21 October 2014

### On This Day in Math - October 21

“Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.”~Ron Graham on Martin Gardner

The 294th day of the year; 294 is a practical number because all numbers strictly less than 294 can be formed with sums of distinct divisors of 294.

EVENTS
1621 Kepler's Mother, Katherine, during her trial for witchcraft was shown the "instruments of torture."
"The whole case was now passed on the law faculty of the University of Tübingen, Kepler’s Alma Mater, who decided that Katharine should be taken to the hangman and shown the instruments of torture and ordered to confess. On 21st October 1621 this was duly carried out but the stubborn old lady refused to bend she said,"
“Do with me what you want. Even if you were to pull one vein after another out of my body, I would have nothing to admit.” Then she fell to her knees and said a Pater Noster. God would she said, bring the truth to light and after her death disclose that wrong and violence had been done to her. He would not take the Holy Ghost from her and would stand by her.

1743 In the United States, on October 21, 1743, Benjamin Franklin tracked a hurricane for the first time. It was the first recorded instance in which the progressive movement of a storm system was recognized.

1796 The date of a still uninterpreted cryptic entry "Vicimus GEGAN"" in Gauss’s scientiﬁc diary. There is a another insertion that also remains uninterpreted. He wrote "REV. GALEN" in the diary on April 8, 1799 *VFR
 *Genial Gauss Gottingen

1803 John Dalton's Atomic Theory was first presented on 21st October 1803 to the Manchester Literary and Philosophical Society of which he was President 1816-1844. *Open Plaques

1805 British Admiral Nelson defeated the combined French and Spanish ﬂeets in the Battle of Trafalgar by adopting the tactic of breaking the enemy line in two and concentrating his ﬁrepower on a few ships (orthodox tactics had the opponents facing each other in roughly parallel lines—the “line-ahead” formation). For an analysis of why this works see David H. Nash, “Differential equations and the Battle of Trafalgar”, The College Mathematics Journal, 16(1985), 98–102. *VFR

1845 After two unsuccessful attempts to present his work in person to the Royal Astronomer Sir George Biddell Airy, John Couch Adams left a copy of his calculation regarding a hypothetical planet at the Royal Observatory. Airy criticized the work and didn’t search for the planet until later. Consequently he didn’t discover Neptune. See 23 September 1846.

1854 Florence Nightingale embarked for the Crimea on 21 October with thirty-eight nurses: ten Roman Catholic Sisters, eight Anglican Sisters of Mercy, six nurses from St. John's Institute, and fourteen from various hospitals. *Victorian Web Org

1965 Greece issued a postage stamp picturing Hipparchus and an astrolabe to commemorate the opening of the Evghenides Planetarium in Athens. [Scott #835]. *VFR

1976, the United States made a clean sweep of the Nobel Prizes, winning or sharing awards in chemistry, physics, medicine, economics, and literature. (No peace prize was awarded.)

1988 Science (pp. 374-375) reported that the 100-digit number 11104 + 1 was factored by using computers working in parallel using a quadratic sieve method. [Mathematics Magazine 62 (1989), p 70].*VFR

2015 Marty McFly and Doctor Emmet Brown "return" to this date in the future in the 1989 Sci-fi-sequel, Back to the Future II. The "future" included rocket powered skateboards... Do Razors count?

BIRTHS

1687 Nicolaus(I) Bernoulli (21 Oct 1687 in Basel, Switzerland - 29 Nov 1759 in Basel) Nicolaus Bernoulli was one of the famous Swiss family of mathematicians. He is most important for his correspondence with other mathematicians including Euler and Leibniz. *SAU (Can't tell your Bernoulli's without a scorecard? Check out "A Confusion of Bernoulli's" by the Renaissance Mathematicus.)

1823 Birthdate of Enrico Betti. In algebra, he penetrated the ideas of Galois by relating them to the work of Ruffini and Abel. In analysis, his work on elliptic functions was further developed by Weierstrass. In “Analysis situs”, his research inspired Poincar´e, who coined the term “Betti numbers” to characterize the connectivity of surfaces. *VFR He was the first to give a proof that the Galois group is closed under multiplication. Betti also wrote a pioneering memoir on topology, the study of surfaces and space. Betti did important work in theoretical physics, in particular in potential theory and elasticity.*TIS

1833 Alfred Bernhard Nobel (21 Oct 1833; 10 Dec 1896) a Swedish chemist and inventor of dynamite and other, more powerful explosives, was born in Stockholm. An explosives expert like his father, in 1866 he invented a safe and manageable form of nitroglycerin he called dynamite, and later, smokeless gunpowder and (1875) gelignite. He helped to create an industrial empire manufacturing many of his other inventions. Nobel amassed a huge fortune, much of which he left in a fund to endow the annual prizes that bear his name. First awarded in 1901, these prizes were for achievements in the areas of physics, chemistry, physiology or medicine, literature, and peace. The sixth prize, for economics, was instituted in his honour in 1969. *TIS (The well-known anecdote that there is no Nobel prize in mathematics as he thought Mittag-Leffler might win it seems to have no basis in fact

1855 Giovanni Battista Guccia (21 Oct 1855 in Palermo, Italy - 29 Oct 1914 in Palermo, Italy) Guccia's work was on geometry, in particular Cremona transformations, classification of curves and projective properties of curves. His results published in volume one of the Rendiconti del Circolo Matematico di Palermo were extended by Corrado Segre in 1888 and Castelnuovo in 1897. *SAU

1882 Harry Schultz Vandiver (21 Oct 1882 in Philadelphia, Pennsylvania, USA - 9 Jan 1973 in Austin, Texas, USA) Harry developed an antagonism towards public education and left Central High School at an early age to work as a customshouse broker for his father's firm. D H Lehmer writes:
He was self-taught in his youth and must have had little patience with secondary education since he never graduated from high school. This impatience, especially with mathematical education, was to last the rest of his life.
When he was eighteen years old he began to solve many of the number theory problems which were posed in the American Mathematical Monthly, regularly submitting solutions. In addition to solving problems, he began to pose problems himself. By 1902 he was contributing papers to the Monthly. For example he published two short papers in 1902 A Problem Connected with Mersenne's Numbers and Applications of a Theorem Regarding Circulants.
In 1904 he collaborated with Birkhoff on a paper on the prime factors of a^n - b^n published in the Annals of Mathematics. In fact the result they proved was not new, although they were not aware of the earlier work which had been published by A S Bang in 1886. Also in the year 1904, Vandiver published On Some Special Arithmetic Congruences in the American Mathematical Monthly and, although still working as an agent for his father's firm, he did attend some graduate lectures at the University of Pennsylvania. He also began reading papers on algebraic number theory and embarked on a study of the work of Kummer, in particular his contributions to solving Fermat's Last Theorem. Over the next few years he published papers such as Theory of finite algebras (1912), Note on Fermat's last theorem (1914), and Symmetric functions formed by systems of elements of a finite algebra and their connection with Fermat's quotient and Bernoulli's numbers (1917).
The outbreak of World War I in 1914 did not directly affect the United States since the Democratic president Woodrow Wilson made a declaration of neutrality. This policy was controversial but popular enough to see him re-elected in 1916. However US shipping was being disrupted (and sunk) by German submarines and, under pressure from Republicans, Wilson declared war on Germany on 6 April 1917. Vandiver joined the United States Naval Reserve and continued to serve until 1919 when the war had ended. After leaving the Naval Reserve, Birkhoff persuaded Vandiver to become a professional mathematician and to accept a post at Cornell University in 1919. Despite having no formal qualifications, his excellent publication record clearly showed his high quality and he was appointed as an instructor. He also worked during the summer with Dickson at Chicago on his classic treatise History of the Theory of Numbers. In 1924 he moved to the University of Texas where he was appointed as an Associate Professor. He spent the rest of his career at the University of Texas, being promoted to full professor in 1925, then named as distinguished professor of applied mathematics and astronomy in 1947. He continued in this role until he retired in 1966 at the age of 84. *SAU

1893 Bill Ferrar graduated from Oxford after an undergraduate career interrupted by World War I. He lectured at Bangor and Edinburgh before moving back to Oxford. He worked in college administration and eventually became Principal of Hertford College. He worked on the convergence of series. *SAU

1914 Martin Gardner born in Tulsa, Oklahoma. From 1957 to 1980 he wrote the “Mathematical Games” column in Scientiﬁc American. Many of these columns have been collected together into the numerous books that he has written. If you want to know more about the person who has done more to popularize mathematics than any other, see the interview with Gardner in Mathematical People. Proiles and Interviews (1985), edited by Donald J. Albers and G. L. Alexanderson, pp. 94–107. *VFR (My favorite tribute to Martin was this one from Ron Graham, “Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.”)

DEATHS
1872 Jacques Babinet (5 March 1794 – 21 October 1872) was a French physicist, mathematician, and astronomer who is best known for his contributions to optics. A graduate of the École Polytechnique, which he left in 1812 for the Military School at Metz, he was later a professor at the Sorbonne and at the Collège de France. In 1840, he was elected as a member of the Académie Royale des Sciences. He was also an astronomer of the Bureau des Longitudes.
Among Babinet's accomplishments are the 1827 standardization of the Ångström unit for measuring light using the red Cadmium line's wavelength, and the principle (Babinet's principle) that similar diffraction patterns are produced by two complementary screens. He was the first to suggest using wavelengths of light to standardize measurements. His idea was first used between 1960 and 1983, when a meter was defined as a wavelength of light from krypton gas.
In addition to his brilliant lectures on meteorology and optics research, Babinet was also a great promoter of science, an amusing and clever lecturer, and a brilliant, entertaining and prolific author of popular scientific articles. Unlike the majority of his contemporaries, Babinet was beloved by many for his kindly and charitable nature. He is known for the invention of polariscope and an optical goniometer. *Wik

1881 Heinrich Eduard Heine (16 March 1821 in Berlin, Germany - 21 Oct 1881 in Halle, Germany) Heine is best remembered for the Heine-Borel theorem. He was responsible for the introduction of the idea of uniform continuity.*SAU

1967 Ejnar Hertzsprung (8 Oct 1873, 21 Oct 1967) Danish astronomer who classified types of stars by relating their surface temperature (or color) to their absolute brightness. A few years later Russell illustrated this relationship graphically in what is now known as the Hertzsprung-Russell diagram, which has become fundamental to the study of stellar evolution. In 1913 he established the luminosity scale of Cepheid variable stars.*TIS

1969 WacLlaw Sierpinski (14 March 1882 in Warsaw, - 21 Oct 1969 in Warsaw) His grave carries—according to his wish—the inscription: Investigator of inﬁnity. [Kuratowski, A Half Century of Polish Mathematics, p. 173; Works, p. 14] *VFR Sierpinski's most important work is in the area of set theory, point set topology and number theory. In set theory he made important contributions to the axiom of choice and to the continuum hypothesis. *SAU

2000 Dirk Jan Struik (30 Sept 1894 , 21 Oct 2000) Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States.
In 1924, funded by a Rockefeller fellowship, Struik traveled to Rome to collaborate with the Italian mathematician Tullio Levi-Civita. It was in Rome that Struik first developed a keen interest in the history of mathematics. In 1925, thanks to an extension of his fellowship, Struik went to Göttingen to work with Richard Courant compiling Felix Klein's lectures on the history of 19th-century mathematics. He also started researching Renaissance mathematics at this time.
Struik was a steadfast Marxist. Having joined the Communist Party of the Netherlands in 1919, he remained a Party member his entire life. When asked, upon the occasion of his 100th birthday, how he managed to pen peer-reviewed journal articles at such an advanced age, Struik replied blithely that he had the "3Ms" a man needs to sustain himself: Marriage (his wife, Saly Ruth Ramler, was not alive when he turned one hundred in 1994), Mathematics, and Marxism.
It is therefore not surprising that Dirk suffered persecution during the McCarthyite era. He was accused of being a Soviet spy, a charge he vehemently denied. Invoking the First and Fifth Amendments of the U.S. Constitution, he refused to answer any of the 200 questions put forward to him during the HUAC hearing. He was suspended from teaching for five years (with full salary) by MIT in the 1950s. Struik was re-instated in 1956. He retired from MIT in 1960 as Professor Emeritus of Mathematics.
Aside from purely academic work, Struik also helped found the Journal of Science and Society, a Marxian journal on the history, sociology and development of science.
In 1950 Stuik published his Lectures on Classical Differential Geometry.
Struik's other major works include such classics as A Concise History of Mathematics, Yankee Science in the Making, The Birth of the Communist Manifesto, and A Source Book in Mathematics, 1200-1800, all of which are considered standard textbooks or references.
Struik died October 21, 2000, 21 days after celebrating his 106th birthday. *Wik

2002 Bernhard Hermann Neumann (15 Oct 1909 in Berlin, Germany - 21 Oct 2002 in Canberra, Australia) Neumann is one of the leading figures in group theory who has influenced the direction of the subject in many different ways. While still in Berlin he published his first group theory paper on the automorphism group of a free group. However his doctoral thesis at Cambridge introduced a new major area into group theory research. In his thesis he initiated the study of varieties of groups, that is classes of groups defined which are by a collection of laws which must hold when any group elements are substituted into them. *SAU

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