Saturday, 25 October 2014

On This Day in Math - October 25

Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
E. Torricelli

The 298th day of the year; If you multiply 298 by (298 + 3) you get a palindromic number, 89,698. Can every number be similarly adjusted to make a palindrome?


EVENTS

1666 William Lilly, astrologer, was called before the House of Commons to explain the embarrassing success of his 1651 prediction of the plague (of 1665) and “exorbitant fire” of 1666. The House ultimately attributed the fire to the papists. *W W Rouse Ball, Mathematical Recreations and Essays,6th edition, p. 390 Lilly caused much controversy in 1652 for allegedly predicting the Great Fire of London some 14 years before it happened. For this reason many people believed that he might have started the fire, but there is no evidence to support these claims. He was tried for the offense in Parliament but was found to be innocent.*Wik

In 1671, Giovanni Cassini discovered Iapetus, one of Saturn's moons. Iapetus is the third largest and one of the stranger of the 18 moons of Saturn. Its leading side is dark with a slight reddish color while its trailing side is bright. The dark surface might be composed of matter that was either swept up from space or oozed from the moon's interior. This difference is so striking that Cassini noted that he could see Iapetus only on one side of Saturn and not on the other. In Greek mythology Iapetus was a Titan, the son of Uranus, the father of Prometheus and Atlas and an ancestor of the human race. Cassini (1625-1712), first director of the Paris Royal Observatory, also discovered other moons of Saturn (Tethys, Dione, Rhea) and the major gap in its rings. *TIS

1713 Leibniz, in a letter to Johann Bernoulli, observed that an alternating series whose terms monotonically decrease to zero in absolute value is convergent. In a letter of January 10, 1714, he gave an incorrect proof (Big Kline, p. 461). Examination of the proof reveals that it is the one we give today, except he fails to say anything about the completeness of the reals. *VFR

1846 William Thompson (Lord Kelvin) writes to Sir George Stokes regarding the "recent proceedings about Oceanus, or Neptune, or Le Verrier. " commenting that "Cambridge is behind the rest of the world on scientific subjects.". John C. Adams, later became a fellow at Pembroke College, and he and Stokes became close friends. *The correspondence between Sir George Gabriel Stokes and Sir William Thompson, pg 2

1881 Clerk Seaton writes to the chairman of the committee on the census that he has discovered a paradox with the apportionment. Seaton had discovered the Alabama Paradox.
It seemed so easy. The 1787 US Constitution laid out simple rules for deciding how many representatives each state shall receive:
"Representatives and direct taxes shall be apportioned among the several States which may be included within this Union, according to their respective numbers, ... The number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative ..."
It may have seemed easy, but for the 200+ years of US government, the question of "Who gets how many?" continues to perplex and promote controversy.
When congress discussed mathematical methods of applying this constitutional directive there were two methods of prime consideration, Jefferson's method, and Hamilton's method. Congress selected Hamilton's method and in the first use of the Presidential veto (make a note of this for extra points in History or Government class) President Washington rejected the bill. Congress submitted and passed another bill using Jefferson's method. The method used has changed frequently over the years with a method by Daniel Webster adopted in 1842, (the original 65 Representatives had grown to 223) and then replaced with Hamilton's method in 1852 (234 Representatives). In a strange "Only in America" moment in 1872, the congress reapportioned without actually adopting an official method and some analysis suggest that the difference caused Rutherford Hayes to Win instead of Samuel Tilden who would have won had Hamilton's method been used. Since 1931 the US House has had 435 Representatives with the brief exception of when Alaska and Hawaii became states. Then there was a temporary addition of one seat for each until the new apportionment after the 1960 Census. In 1941 the Huntington-Hill Method was adopted and has remained in continuous (and contentious) use ever since.
In 1880 the first of what are called the apportionment paradoxes was discovered. Here is how they state it at the Wikipedia web site:
After the 1880 census, C. W. Seaton, chief clerk of the U. S. Census Office, computed apportionments for all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. They also show a nice example (with small numbers) so you might check their site.
*pballew.net

1944 Max Planck writes to Hitler to plead for the life of his son, Erwin. In the note, the discoverer of the energy quantum pleads for the life of his son, who was involved in the attempted to kill Hitler three months before. Max Planck had already lost his eldest son, who was killed in the Battle of Verdun, during World War I.
Planck writes in his letter that he is ‘confident’ that the Führer will lend his ear to ‘an imploring 87-year-old’. This plea, apparently written from the Planck family’s bombed-out home in a suburb of Berlin, was ignored by the authorities. Erwin was executed on 23 January 1945, and his death certificate recorded: ‘parents unknown’. *Graham Farmelo

2001 Microsoft Releases Windows XP​, the family of 32-bit and 64-bit operating systems produced by Microsoft for use on personal computers. The name "XP" stands for “Experience.” The successor to both Windows 2000 Professional​ and Windows ME, Windows XP was the first consumer-oriented operating system Microsoft built on the Windows NT​ kernel and architecture. Over 400 million copies were in use by January 2006, according to an International Data Corporation​ analyst. It was succeeded by Windows Vista, which was released to the general public in January 2007*CHM

2011   Scientists in California and Sweden have solved a 250-year-old mystery — a coded manuscript written by a secret society.  The University of Southern California announced Tuesday, Oct 25th, that researchers had broken the Copiale Cipher — the writing used in a 105-page 18th century document from Germany.
Kevin Knight, of USC, and Beata Megyesi and Christiane Schaefer, of Uppsala University, did the work.
They used a statistical computer program to decipher part of the manuscript, which was found in East Berlin after the Cold War and is now in a private collection.
The book, written in symbols and Roman letters, details complicated initiation ceremonies of a society fascinated by ophthalmology. They include making mystical signs and plucking a hair from a candidate's eyebrow. The convoluted text swears candidates to loyalty and secrecy. *Associated Press,



BIRTHS

1789 Samuel Heinrich Schwabe (25 Oct 1789; 11 Apr 1875) Amateur German astronomer who discovered the 10-year sunspot activity cycle. Schwabe had been looking for possible intramercurial planets. From 11 Oct 1825, for 42 years, he observed the Sun virtually every day that the weather allowed. In doing so he accumulated volumes of sunspot drawings, the idea being to detect his hypothetical planet as it passed across the solar disk, without confusion with small sunspots. Schwabe did not discover any new planet. Instead, he published his results in 1842 that his 17 years of nearly continuous sunspot observations revealed a 10-year periodicity in the number of sunspots visible on the solar disk. Schwabe also made (1831) the first known detailed drawing of the Great Red Spot on Jupiter.*TIS

1796 James Curley (Irish: Séamus MacThoirealaigh (26 October 1796 – 24 July 1889) was an Irish-American astronomer. He was born at Athleague, County Roscommon, Ireland. His early education was limited, though his talent for mathematics was discovered, and to some extent developed, by a teacher in his native town. He left Ireland in his youth, arriving in Philadelphia on 10 October 1817. Here he worked for two years as a bookkeeper and then taught mathematics at Frederick, Maryland. In 1826 he became a student at the old seminary in Washington, DC, intending to prepare himself for the Catholic priesthood, and at the same time taught one of its classes. The seminary, however, which had been established in 1820, was closed in the following year and he joined the Society of Jesus on 29 September 1827. After completing his novitiate he again taught in Frederick and was sent in 1831 to teach natural philosophy at Georgetown University. He also studied theology and was ordained priest on 1 June 1833. His first Mass was said at the Visitation Convent, Georgetown, where he afterwards acted as chaplain for fifty years.He spent the remainder of his life at Georgetown, where he taught natural philosophy and mathematics for forty-eight years. He planned and superintended the building of the Georgetown Observatory in 1844 and was its first director, filling this position for many years. One of his earliest achievements was the determination of the latitude and longitude of Washington, D.C. in 1846. His results did not agree with those obtained at the Naval Observatory, and it was not until after the laying of the first transatlantic cable in 1858 that his determination was found to be nearer the truth. *Wik

1811 Evariste Galois born in the little village of Bourg-la-Reine, near Paris, France. *VFR (25 Oct 1811; 31 May 1832) famous for his contributions to the part of higher algebra known as group theory. His theory solved many long-standing unanswered questions, including the impossibility of trisecting the angle and squaring the circle. Galois fought a duel with Perscheux d'Herbinville on 30 May 1832, the reason for the duel not being clear but certainly linked with a love affair. Galois was wounded in the duel, and died in hospital the following day, at age 20. His funeral was held on 2 June. It was the focus for a Republican rally and riots followed which lasted for several days. He was commemorated as a revolutionary and geometrician on a French postal stamp issued on 10 Nov 1984.*TIS

1877 Henry Norris Russell (25 Oct 1877; 18 Feb 1957) American astronomer and astrophysicist who showed the relationship between a star's brightness and its spectral type, in what is usually called the Hertzsprung-Russell diagram, and who also devised a means of computing the distances of binary stars. As student, professor, observatory director, and active professor emeritus, Russell spent six decades at Princeton University. From 1921, he visited Mt. Wilson Observatory annually. He analyzed light from eclipsing binary stars to determine stellar masses. Russell measured parallaxes and popularized the distinction between giant stars and "dwarfs" while developing an early theory of stellar evolution. Russell was a dominant force in American astronomy as a teacher, writer, and advisor. *TIS

1886 Lester Randolph Ford (25 Oct 1886 in Missouri, USA - 11 Nov 1967 in Charlottesville, Virginia, USA) was an American mathematician who lectured for several years in Edinburgh before moving back to the USA. He wrote some important text-books and is best known for his contributions to the Mathematical Association of America and the American Mathematical Monthly. *SAU (Ford circles are named after him. If you have never explored this idea, and the related idea of mediants, do it today)

1910 William Higinbotham (Oct 1910; 10 Nov 1994) American physicist who invented the first video game, Tennis for Two, as entertainment for the 1958 visitor day at Brookhaven National Laboratory, where he worked (1947-84) then as head of the Instrumentation Division. It used a small analogue computer with ten direct-connected operational amplifiers and output a side view of the curved flight of the tennis ball on an oscilloscope only five inches in diameter. Each player had a control knob and a button. Late in WW II he became electronics group leader at Los Alamos, New Mexico, where the nuclear bomb was developed. After the war, he became active with other nuclear scientists in establishing the Federation of American Scientists to promote nuclear n)on-proliferation.*TIS (raise your hand if you are old enough to remember "Pong")

1945 David N. Schramm (25 Oct 1945; 19 Dec 1997) American theoretical astrophysicist who was an authority on the particle-physics aspects of the Big Bang theory of the origin of the universe. He considered the nuclear physics involved in the synthesis of the light elements created during the Big Bang comprising mainly hydrogen, with lesser quantities of deuterium, helium, lithium, beryllium and boron. He predicted, from cosmological considerations, that a third family of neutrinos existed - which was later proven in particle accelerator experiments (1989). Schramm worked to evaluate undetected dark matter that contributed to the mass of the universe, and which would determine whether the universe would ultimately continue to expand. He died in the crash of the small airplane he was piloting. *TIS



DEATHS

1400 Geoffrey Chaucer died. Although rightly famous for his Canterbury Tales, he also wrote two astronomical works. [DSB 3, 217] *VFR In his lifetime he was far more known for his “Treatise on the Astrolabe”

1647 Evangelista Torricelli (15 Oct 1608- 25 Oct 1647) an Italian physicist and mathematician who invented the barometer and whose work in geometry aided in the eventual development of integral calculus. Inspired by Galileo's writings, he wrote a treatise on mechanics, De Motu ("Concerning Movement"), which impressed Galileo. He also developed techniques for producing telescope lenses. The barometer experiment using "quicksilver" filling a tube then inverted into a dish of mercury, carried out in Spring 1644, made Torricelli's name famous. The Italian scientists merit was, above all, to admit that the effective cause of the resistance presented by nature to the creation of a vacuum (in the inverted tube above the mercury) was probably due to the weight of air*TIS

1733 Girolamo Saccheri (5 Sep 1667, 25 Oct 1733) Italian mathematician who worked to prove the fifth postulate of Euclid, which can be stated as, "Through any point not on a given line, one and only one line can be drawn that is parallel to the given line." Euclid saw the proof was not self-evident, yet neither did he provide one; instead he accepted it as an assumption. Subsequently many mathematicians tried to prove this fifth postulate from the remained axioms - and failed. Saccheri took the novel approach of first assuming that the postulate was wrong, then followed the all consequences seeking any one contradiction that then leaves the only original postulate as the only possible solution. In the process, he came close to discovering non-Euclidian geometry, but gave up too early.*TIS

1884 Carlo Alberto Castigliano (9 November 1847, Asti – 25 October 1884, Milan) was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial derivatives of strain energy.*Wik

1905 Otto Stolz (3 May 1842 in Hall (now Solbad Hall in Tirol), Austria - 25 Oct 1905 in Innsbruck, Austria) Stolz's earliest papers were concerned with analytic or algebraic geometry, including spherical trigonometry. He later dedicated an increasing part of his research to real analysis, in particular to convergence problems in the theory of series, including double series; to the discussion of the limits of indeterminate ratios; and to integration.*SAU

1914 Wilhelm Lexis studied data presented as a series over time thus initiating the study of time series.*SAU

1933 Albert Wangerin worked on potential theory, spherical functions and differential geometry. *SAU

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

2002 René Frédéric Thom (September 2, 1923 – October 25, 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). He received the Fields Medal in 1958.*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

Friday, 24 October 2014

On This Day in Math - October 24

Monument to Gauss and Weber in Gottingen

Now it is quite clear to me that there are no solid spheres in the heavens, and those that have been devised by authors to save the appearances, exist only in their imagination, for the purpose of permitting the mind to conceive the motion which the heavenly bodies trace in their courses.
~Tycho Brahe

The 297th day of the year; 2972 = 88209 and 88+209 = 297. (Numbers that have this property are a type of Kaprekar number; there are only three such numbers of three digits, now you know one of them.)

And this one from * Jim Wilder @wilderlab 297³=26,198,073 and 26+198+073=297

EVENTS

1676 Newton summarized the stage of development of his method in the “Epistola posterior,” which he sent to Oldenburg to transmit to Leibniz. *VFR (see Oct 26, 1676) This may be the first time Newton used irrational exponents in communication to others. It is one of the earlier uses by anyone. In the letter to Oldenburg, Newton remarks that Leibniz had developed a number of methods, one of which was new to him.

1729 Euler mentioned the gamma function in a letter to Goldbach. In 1826 Legendre gave the function its symbol and name. * F. Cajori, History of Mathematical Notations, vol. 2, p. 271 (the Oct 13 date is for the Julian Calendar still used in Russia when Euler wrote from there. It was the 24th in most of the rest of the world using the Gregorian Calendar.)

1826 Abel wrote Holmboe his impressions of continental mathematics and mathematicians.
Upon reaching Paris from Berlin, he worked on what would be called the Paris Treatise that he submitted to the Academy in October 1826. In this memoir, Abel obtained among other things, an important addition theorem for algebraic integrals. It is also in this treatise that we see the first occurrence of the concept of the genus of an algebraic function. Cauchy and Legendre were appointed referees of this memoir. In Paris, Abel was disappointed to find little interest in his work, which he had saved for the Academy. He wrote to Holmboe, “I showed the treatise to Mr. Cauchy, but he scarcely deigned to glance at it."
*Krishnaswami Alladi, NEILS HENRIK ABEL, Norwegian mathematical genius (paper on UFL website)

In 1851, William Lassell discovered Ariel and Umbriel, satellites of Uranus. All of Uranus's moons are named after characters from the works of William Shakespeare or Alexander Pope's The Rape of the Lock. The names of all four satellites of Uranus then known were suggested by John Herschel in 1852 at the request of Lassell. Ariel has an approx. diameter of 1160-km, an orbital period of 2.52 days, and orbital radius of 191,240-km from Uranus. The name Umbriel comes from Alexander Pope's The Rape of the Lock. Umbriel has a diameter of 1170-km, an orbital period of about 4 days and orbit radius of 266,000-km. Lassell, a British astronomer, had previously also discovered Neptune's largest satellite, Triton and (with Bond) discovered Saturn's moon Hyperion. He was a successful brewer before turning to astronomy.*TIS *Wik

1902 In Science, George Bruce Halsted wrote that his student R. L. Moore, who had proved that one of Hilbert’s betweenness axioms was redundant, “was displaced in favor of a local schoolmarm,” Miss Mary E. Decherd. *VFR Halstead was contentious in many ways, and Moore's rejection may have been a response to the fact that Halstead had suggested him. Halstead would be fired himself on December 11 of the same year. *D. Reginald Traylor , Creative Teaching: The Heritage of R. L. Moore, pg 35-37

1904 Emmy Noether matriculated at the University of Erlangen. *VFR The University was only yards from her house. Images of both are at this site from The Renaissance Mathematicus.

1989 “Welcome to the White House on this glorious fall day. I’m sorry if I’m just a little bit late. I was sitting in there trying to solve a few quadratic equations. [Laughter] Somewhat more difficult than balancing the budget, I might say. And then I thought it might be appropriate to have a moment of silence in memory of those substitute teachers back home. [Laughter].” Remarks by President George Bush (the elder) at the Presentation Ceremony for the Presidential Awards for Excellence in Science and Math Teaching.

1994 Lynchburg College Professor Thomas Nicely, Reports a flaw in the Pentium chip by Intel that he discovered while he was trying to calculate Brun's constant,(The sum of the reciprocals of all the twin primes, 1/3+1/5+1/5+1/7+1/11+1/13.... which converges to about 1.902).
The Pentium chip occasionally gave wrong answers to a floating-point (decimal) division calculations due to errors in five entries in a lookup table on the chip. Intel spent millions of dollars replacing the faulty chips.
Nicely first noticed some inconsistencies in the calculations on June 13, 1994 shortly after adding a Pentium system to his group of computers, but was unable to eliminate other factors until October 19, 1994. On October 24, 1994 he reported the issue to Intel. According to Nicely, his contact person at Intel later admitted that Intel had been aware of the problem since May 1994, when the flaw was discovered during testing of the FPU for its new P6 core, first used in the Pentium Pro. *Wik


BIRTHS

1632 Antonie van Leeuwenhoek (24 Oct 1632; 26 Aug 1723.) Dutch microscopist who was the first to observe bacteria and protozoa. His researches on lower animals refuted the doctrine of spontaneous generation, and his observations helped lay the foundations for the sciences of bacteriology and protozoology.*TIS "The 31th of May, I perceived in the same water more of those Animals, as also some that were somewhat bigger. And I imagine, that [ten hundred thousand] of these little Creatures do not equal an ordinary grain of Sand in bigness: And comparing them with a Cheese-mite (which may be seen to move with the naked eye) I make the proportion of one of these small Water-creatures to a Cheese-mite, to be like that of a Bee to a Horse: For, the circumference of one of these little Animals in water, is not so big as the thickness of a hair in a Cheese-mite. "

1804 Wilhelm Eduard Weber (24 Oct 1804; 23 Jun 1891) German physicist who investigated terrestrial magnetism. For six years, from 1831, Weber worked in close collaboration with Gauss. Weber developed sensitive magnetometers, an electromagnetic telegraph (1833) and other magnetic instruments during this time. His later work (1855) on the ratio between the electrodynamic and electrostatic units of charge proved extremely important and was crucial to Maxwell in his electromagnetic theory of light. (Weber found the ratio was 3.1074 x 108 m/sec but failed to take any notice of the fact that this was close to the speed of light.) Weber's later years were devoted to work in electrodynamics and the electrical structure of matter. The magnetic unit, weber, is named after him.*TIS

1821 Philipp Ludwig von Seidel (23 October 1821, Zweibrücken, Germany – 13 August 1896, Munich)  formulated the notion of uniform convergence.*VFR 
 Lakatos credits von Seidel with discovering, in 1847, the crucial analytic concept of uniform convergence, while analyzing an incorrect proof of Cauchy's. In 1857, von Seidel decomposed the first order monochromatic aberrations into five constituent aberrations. They are now commonly referred to as the five Seidel Aberrations.   The Gauss–Seidel method is a useful numerical iterative method for solving linear systems. *Wik

1853 Heinrich Maschke (24 October 1853 in Breslau, Germany (now Wrocław, Poland) – 1 March 1908 Chicago, Illinois, USA) was a German mathematician who proved Maschke's theorem.*Wik

1873 Sir Edmund Taylor Whittaker (24 Oct 1873; 24 Mar 1956) English mathematician who made pioneering contributions to the area of the special functions, which is of particular interest in mathematical physics. Whittaker is best known work is in analysis, in particular numerical analysis, but he also worked on celestial mechanics and the history of applied mathematics and physics. He wrote papers on algebraic functions and automorphic functions. His results in partial differential equations (described as most sensational by Watson) included a general solution of the Laplace equation in three dimensions in a particular form and the solution of the wave equation. On the applied side of mathematics he was interested in relativity theory and he also worked on electromagnetic theory. *TIS

1898 Lillian Rose Vorhaus Kruskal Oppenheimer (October 24, 1898 in New York City – July 24, 1992) was an American origami pioneer. She popularized origami in the West starting in the 1950s, and is credited with popularizing the Japanese term origami in English-speaking circles, which gradually supplanted the literal translation paper folding that had been used earlier. In the 1960s she co-wrote several popular books on origami with Shari Lewis.  Lillian taught origami to Persi Diaconis when he was working as a magician;
She was the mother of three sons William Kruskal(developed the Kruskal-Wallis one-way analysis of variance), Martin David Kruskal(co-inventor of solitons and of surreal numbers), and Joseph Kruskal ( Kruskal's algorithm for computing the minimal spanning tree (MST) of a weighted graph) who all went on to be prominent mathematicians. Her grandson Clyde P. Kruskal (son of Martin) is an American computer scientist,working on parallel computing architectures, models, and algorithms. *Wik

1906 Aleksandr Osipovich Gelfond (24 Oct 1906; 7 Nov 1968) Russian mathematician who originated basic techniques in the study of transcendental numbers (numbers that cannot be expressed as the root or solution of an algebraic equation with rational coefficients). He profoundly advanced transcendental-number theory, and the theory of interpolation and approximation of complex-variable functions. He established the transcendental character of any number of the form ab, where a is an algebraic number different from 0 or 1 and b is any irrational algebraic number, which is now known as Gelfond's theorem. This statement solved the seventh of 23 famous problems that had been posed by the German mathematician David Hilbert in 1900. *TIS

1922 Werner Buchholz​  (October 24, 1922 in Detmold, Germany - ). He was a member of the teams that designed the IBM 701​ and Stretch models. Buchholz used term byte to describe eight bits—although in the 1950s, when the term first was used, equipment used six-bit chunks of information, and a byte equaled six bits. Buchholz described a byte as a group of bits to encode a character, or the numbers of bits transmitted in parallel to and from input-output. *CHM

1932 Pierre-Gilles de Gennes (24 Oct 1932; 18 May 2007) French physicist who was awarded the 1991 Nobel Prize for Physics for "discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers." He described mathematically how, for example, magnetic dipoles, long molecules or molecule chains can under certain conditions form ordered states, and what happens when they pass from an ordered to a disordered state. Such changes of order occur when, for example, a heated magnet changes from a state in which all the small atomic magnets are lined up in parallel to a disordered state in which the magnets are randomly oriented. Recently, he has been concerned with the physical chemistry of adhesion. *TIS



DEATHS

1601 Tycho Brahe (14 December 1546 – 24 October 1601) Kepler inherited his vast accurate collection of astronomical data. He used this to derive his laws of planetary motion. *VFR In 1901, on the three hundredth anniversary of his death, the bodies of Tycho Brahe and his wife Kirstine were exhumed in Prague. They had been embalmed and were in remarkably good condition, but the astronomer’s artificial nose was missing, apparently filched by someone after his death. It had been made for him in gold and silver when his original nose was sliced off in a duel he fought in his youth at Rostock University after a quarrel over some obscure mathematical point. He always carried a small box of glue in his pocket for use when the new nose became wobbly. Tycho Brahe was famous for the most accurate and precise observations achieved by any astronomer before the invention of the telescope. Born to an aristocratic family in Denmark in 1546, he was one of twin boys – the other twin was still-born – and while still a baby Tycho was stolen from his parents by a rich, childless uncle, who paid for his education and sent him to Leipzig University to study law. His imagination had been fired, however, by a total eclipse of the sun in 1560 and he was determined to be an astronomer. He found that the existing tables recording the positions of planets and stars were wildly inaccurate and dedicated himself to correcting them. *History Today Was Tycho Murdered? Read an excellent blog on "The crazy life and crazier death of Tycho Brahe, history’s strangest astronomer".

1635 Wilhelm Shickard  (22 April 1592 – 24 October 1635) He invented and built a working model of the first modern mechanical calculator. *VFR 
Schickard's machine could perform basic arithmetic operations on integer inputs. His letters to Kepler explain the application of his "calculating clock" to the computation of astronomical tables.
In 1935 while researching a book on Kepler, a scholar found a letter from Schickard and a sketch of his calculator, but did not immediately recognize the designs or their great importance. Another twenty years passed before the book's editor, Franz Hammer, found additional drawings and instructions for Schickard's second machine and released them to the scientific community in 1955.A professor at Schickard's old university, Tübingen, reconstructed the calculator based upon Schickard's original plans; it is still on display there today. 
He was a friend of Kepler and did copperplate engravings for Kepler's Harmonice Mundi. He built the first calculating machine in 1623, but it was destroyed in a fire in the workshop in 1624.


1655 Pierre Gassendi (22 Jan 1592, 24 Oct 1655) French scientist, mathematician, and philosopher who revived Epicureanism as a substitute for Aristotelianism, attempting in the process to reconcile Atomism's mechanistic explanation of nature with Christian belief in immortality, free will, an infinite God, and creation. Johannes Kepler had predicted a transit of Mercury would occur in 1631. Gassendi used a Galilean telescope to observed the transit, by projecting the sun's image on a screen of paper. He wrote on astronomy, his own astronomical observations and on falling bodies. *TIS

1870 Charles Joseph Minard (27 Mar 1781; 24 Oct 1870 at age 89) French civil engineer who made significant contributions to the graphical representations of data. His best-known work, Carte figurative des pertes successives en hommes de l'Armee Français dans la campagne de Russe 1812-1813, dramatically displays the number of Napoleon's soldiers by the width of an ever-reducing band drawn across a map from France to Moscow. At its origin, a wide band shows 442,000 soldiers left France, narrowing across several hundred miles to 100,000 men reaching Moscow. With a parallel temperature graph displaying deadly frigid Russian winter temperatures along the way, the band shrinks during the retreat to a pathetic thin trickle of 10,000 survivors returning to their homeland. *TIS Minard advocated the graphing idea that the ratio of information to ink should be as high as possible.


1930 Paul Emile Appell (27 Sept 1855 in Strasbourg, France - 24 Oct 1930) Appell's first paper in 1876 was based on projective geometry continuing work of Chasles. He then wrote on algebraic functions, differential equations and complex analysis. In 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum which had been though to be purely a mathematical curiosity. He showed that the double periodicity follows from physical considerations. *SAU

1940 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


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

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
Might find the task laborious,
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.