## Friday, 28 February 2014

### On This Day in Math - February 28

Well David, I have a lot of ideas and throw away the bad ones.
Upon being asked how he had so many good ideas by David Harker, his student.
— Linus Pauling

The 59th day of the year; 59 is the center prime number in a 3x3 prime magic square that has the smallest possible total for each row, column and diagonal, 177. (Can you find the other eight primes and their positions in this magic square?)

EVENTS
1678 In a letter to Robert Boyle, Isaac Newton explained his concept of ether. “I suppose that there is diffused through all places an ethereal substance capable of contraction and dilation, strongly elastic and, in a word, much like air in all respects, but far more subtil.” He thought it was in all bodies of matter, but "rarer in the pores than in free spaces." This he suspects is the cause of light being refracted towards the perpendicular. *Rigaud, Letters of Scientiﬁc men, vol. 2, p. 407

1695 Liebniz writes to Johann Bernoulli encouraging him to use the term calculus summatorus which Liebniz used for integration.

*VFR

1825 Cauchy presented to the Acad´emie a paper on integals of complex-valued functions where the limits of integration were allowed to be complex. Previously, he had done much work on such
integrals when the limits were real. [Grattan-Guinness, 1990, p. 766] *VFR

1953, James Watson, from early on this Saturday, spent his time at the Cavendish Laboratory in Cambridge, shuffling cardboard cutout models of the molecules of the DNA bases: adenine (A), guanine (G), cytosine (C) and thymine(T). After a while, in a spark of ingenuity, he discovered their complementary pairing. He realized that A joined with T had a close resemblance to C joined with G, and that each pair could hold together with hydrogen bonds. Such pairs could also neatly fit like rungs meeting at right-angles between two anti-parallel helical sugar-phosphate backbones of DNA wound around a common axis. Such structure was consistent with the known X-ray diffraction pattern evidence. Each separated helix with its half of the pairs could form a template for reproducing the molecule. The secret of life First announcement by Francis Crick and James Watson that they had reached their conclusion about the double helix structure of the DNA molecule. Their paper, A Structure for Deoxyribose Nucleic Acid, was published in the 25 Apr 1953 issue of journal Nature. *TIS

1956 Jay Forrester at MIT is awarded a patent for his coincident current magnetic core memory. Forrester's invention, given Patent No. 2,736,880 for a "multicoordinate digital information storage device," became the standard memory device for digital computers until supplanted by solid state (semiconductor) RAM in the mid-1970s. *CHM

2001 With a length of 350 feet 6.6 inches and currently the World's Longest documented Slide Rule, The Texas Magnum by Skip Solberg and Jay Francis,was demonstrated on February 28, 2001 in the Lockeed-Martin Aircraft Assembly Facility at Air Force Plant 4 in Fort Worth, Texas. The Texas Magnum holds the world's record for the longest linear slide rule. The Texas Magnum was designed as a traditional Mannheim style slide rule. The A, C, D and L scales are included on the slide rule *International Slide Rule Museum

BIRTHS

1552 Joost Bürgi (28 Feb 1552, 31 Jan 1632) Swiss watchmaker and mathematician who invented logarithms independently of the Scottish mathematician John Napier. He was the most skilful, and the most famous, clockmaker of his day. He also made astronomical and practical geometry instruments (notably the proportional compass and a triangulation instrument useful in surveying). This led to becoming an assistant to the German astronomer Johannes Kepler. Bürgi was a major contributor to the development of decimal fractions and exponential notation, but his most notable contribution was published in 1620 as a table of antilogarithms. Napier published his table of logarithms in 1614, but Bürgi had already compiled his table of logarithms at least 10 years before that, and perhaps as early as 1588. *TIS

1704 Louis Godin (28 February 1704 Paris – 11 September 1760 Cadiz) was a French astronomer and member of the French Academy of Sciences. He worked in Peru, Spain, Portugal and France.
He was graduated at the College of Louis le Grand, and studied astronomy under Joseph-Nicolas Delisle. His astronomical tables (1724) gave him reputation, and the French Academy of Sciences elected him a pensionary member. He was commissioned to write a continuation of the history of the academy, left uncompleted by Bernard le Bovier de Fontenelle, and was also authorized to submit to the minister, Cardinal André-Hercule de Fleury, the best means of discovering the truth in regard to the figure of the earth, and proposed sending expeditions to the equator and the polar sea. The minister approved the plan and appropriated the necessary means, the academy designating Charles Marie de La Condamine, Pierre Bouguer, and Godin to go to Peru in 1734.
When they had finished their task in 1738, at the invitation of the Viceroy of Peru, Godin accepted the professorship in mathematics in Lima, where he also established a course of astronomical lectures. When in 1746 an earthquake destroyed the greater part of Lima, he took valuable seismological observations, assisted the sufferers, and made plans by the use of which the new buildings would be less exposed to danger from renewed shocks.
In 1751 he returned to Europe, but found that he had been nearly forgotten, and superseded as pensioner of the academy; and, as his fortune had been lost in unfortunate speculations, he accepted the presidency of the college for midshipmen in Cadiz in 1752. During the earthquake of Lisbon, 1755, which was distinctly felt at Cadiz, he took observations and did much to allay the apprehensions of the public, for which he was ennobled by the king of Spain. In 1759 he was called to Paris and reinstated as pensionary member of the academy, but he died on his return to Cadiz. *Wik

1735 Alexandre-Théophile Vandermonde (28 Feb 1735 in Paris, France - 1 Jan 1796 in Paris, France). was a French mathematician best known for his work on determinants. *SAU
In 1772 Vandermonde used [P]n to represent the product of the n factors p(p-1)(p-2)... (p-n+1). With such a notation [P]p would represent what we would now write as p!, but I can imagine this becoming, over time, just [p] (De Morgan would do just such a thing in his 1838 essays on probability). Vandermonde seems to have been the first to consider [p]0 (or 0!) and determined it was (as we now do) equal to one. Vandermonde's notation included a method for skipping numbers, so that [p/3]n would indicate p(p-3)(p-6)... (p-3(n-1)). (this method seems better to me than the present method for factorials which skip terms) It even allowed for negative exponents.

1859 Florian Cajori (born 28 Feb 1859)Swiss-born U.S. educator and mathematician whose works on the history of mathematics were among the most eminent of his time.*TIS at times Cajori's work lacked the scholarship which one would expect of such an eminent scientist, we must not give too negative an impression of this important figure. He almost single-handedly created the history of mathematics as an academic subject in the United States and, particularly with his book on the history of mathematical notation, he is still one of the most quoted historians of mathematics today. *SAU

1878 Pierre Joseph Louis Fatou (28 Feb 1878 in Lorient, France - 10 Aug 1929 in Pornichet, France) was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory. Fatou continued his mathematical explorations and studied iterative and recursive processes such as z == z2+C . the Julia set and the Fatou set are two complementary sets defined from a function.
Fatou wrote many papers developing a Fundamental theory of iteration in 1917, which he published in the December 1917 part of Comptes Rendus. His findings were very similar to those of Gaston Maurice Julia, who submitted a paper to the Académie des Sciences in Paris for their 1918 Grand Prix on the subject of iteration from a global point of view. Their work is now commonly referred to as the generalised Fatou–Julia theorem.*Wik  Fatou dust is a term applied to certain iteration sets that have zero area and an infinite number of disconnected components.

1901 Linus Carl Pauling (28 Feb 1901; 19 Aug 1994 at age 93) an American chemist, physicist and author who applied quantum mechanics to the study of molecular structures, particularly in connection with chemical bonding. Pauling was awarded the Nobel Prize for Chemistry in 1954 for charting the chemical underpinnings of life itself. Because of his work for nuclear peace, he received the Nobel Prize for Peace in 1962. He is remembered also for his strong belief in the health benefits of large doses of vitamin C.*TIS

1925 Louis Nirenberg (28 February 1925, Hamilton, Ontario, Canada - ) is a Canadian-born American mathematician, and one of the outstanding analysts of the twentieth century. He has made fundamental contributions to linear and nonlinear partial differential equations and their application to complex analysis and geometry.*Wik

1930 Leon N. Cooper (28 Feb 1930 - ) American physicist who shared (with John Bardeen and John Robert Schrieffer) the 1972 Nobel Prize in Physics, for his role in developing the BCS (for their initials) theory of superconductivity. The concept of Cooper electron pairs was named after him.*Wik

1939 Daniel C. Tsui (28 Feb 1939 - ) Chinese-American physicist who shared (with Horst L. Störmer and Robert B. Laughlin) received the 1998 Nobel Prize for Physics for the discovery and explanation that the electrons in a powerful magnetic field at very low temperatures can form a quantum fluid whose particles have fractional electric charges. This effect is known as the fractional quantum. *TIS

1954 Jean Bourgain(28 Feb 1954 - )Belgian mathematician who was awarded the Fields Medal in 1994 for his work in analysis. His achievements in several fields included the problem of determining how large a section of a Banach space of finite dimension n can be found that resembles a Hilbert subspace; a proof of Luis Antonio Santaló's inequality; a new approach to some problems in ergodic theory; results in harmonic analysis and classical operators; and nonlinear partial differential equations. Bourgain's work was noteworthy for the versatility it displayed in applying ideas from wide-ranging mathematical disciplines to the solution of diverse problems. *TIS

DEATHS
1691 Joseph Moxon (8 August 1627 - February 1691 (Royal Society archives state his death date as 28 February; the Oxford Dictionary of National Biography states that he was buried on 15 February???{I hope one of them was wrong}), hydrographer to Charles II, was an English printer of mathematical books and maps, a maker of globes and mathematical instruments, and mathematical lexicographer. He produced the first English language dictionary devoted to mathematics, "Mathematicks made easie, or a mathematical dictionary, explaining the terms of art and difficult phrases used in arithmetick, geometry, astronomy, astrology, and other mathematical sciences". In November 1678, he became the first tradesman to be elected as a Fellow of the Royal Society. *Wik Thony Christie has written that he was one of the first English Printers to print tables of Logarithms.

1742 Willem 'sGravesande (26 September 1688 – 28 February 1742)was a Dutch mathematician who expounded Newton's philosophy in Europe. In 1717 he became professor in physics and astronomy in Leiden, and introduced the works of his friend Newton in the Netherlands.
His main work is Physices elementa mathematica, experimentis confirmata, sive introductio ad philosophiam Newtonianam or Mathematical Elements of Natural Philosophy, Confirm'd by Experiments (Leiden 1720), in which he laid the foundations for teaching physics. Voltaire and Albrecht von Haller were in his audience, Frederic the Great invited him in 1737 to come to Berlin.
His chief contribution to physics involved an experiment in which brass balls were dropped with varying velocity onto a soft clay surface. His results were that a ball with twice the velocity of another would leave an indentation four times as deep, that three times the velocity yielded nine times the depth, and so on. He shared these results with Émilie du Châtelet, who subsequently corrected Newton's formula E = mv to E = mv2. (Note that though we now add a factor of 1/2 to this formula to make it work with coherent systems of units, the formula as expressed is correct if you choose units to fit it.) *Wik

1863 Jakob Philipp Kulik (1 May 1793 in Lemberg, Austrian Empire (now Lviv, Ukraine) - 28 Feb 1863 in Prague, Czech Republic) Austrian mathematician known for his construction of a massive factor tables.
Kulik was born in Lemberg, which was part of the Austrian empire, and is now Lviv located in Ukraine.In 1825, Kulik mentioned a table of factors up to 30 millions, but this table does no longer seem to exist. It is also not clear if it had really been completed.
From about 1825 until 1863 Kulik produced a factor table of numbers up to 100330200 (except for numbers divisible by 2, 3, or 5). This table basically had the same format that the table to 30 millions and it is therefore most likely that the work on the "Magnus canon divisorum" spanned from the mid 1820s to Kulik's death, at which time the tables were still unfinished. These tables fill eight volumes totaling 4212 pages, and are kept in the archives of the Academy of Sciences in Vienna. Volume II of the 8 volume set has been lost.*Wik

1956 Frigyes Riesz (22 Jan 1880; 28 Feb 1956) Hungarian mathematician and pioneer of functional analysis, which has found important applications to mathematical physics. His theorem, now called the Riesz-Fischer theorem, which he proved in 1907, is fundamental in the Fourier analysis of Hilbert space. It was the mathematical basis for proving that matrix mechanics and wave mechanics were equivalent. This is of fundamental importance in early quantum theory. His book Leçon's d'analyse fonctionnelle (written jointly with his student B Szökefalvi-Nagy) is one of the most readable accounts of functional analysis ever written. Beyond any mere abstraction for the sake of a structure theory, he was always turning back to the applications in some concrete and substantial situation. *TIS

2013 Donald A. Glaser (21 Sep 1926, 28 Feb 2013) American physicist, who was awarded the Nobel Prize for Physics in 1960 for his invention of the bubble chamber in which the behaviour of subatomic particles can be observed by the tracks they leave. A flash photograph records the particle's path. Glaser's chamber contains a superheated liquid maintained in a superheated, unstable state without boiling. A piston causing a rapid decrease in pressure creates a tendency to boil at the slightest disturbance in the liquid. Then any atomic particle passing through the chamber leaves a track of small gas bubbles caused by an instantaneous boiling along its path where the ions it creates act as bubble-development centers.*TIS  With the freedom that accompanies a Nobel Prize, he soon began to explore the new field of molecular biology, and in 1971 joined two friends, Ronald E. Cape and Peter Farley, to found the first biotechnology company, Cetus Corp., to exploit new discoveries for the benefit of medicine and agriculture. The company developed interleukin and interferon as cancer therapies, but was best known for producing a powerful genetic tool, the polymerase chain reaction, to amplify DNA. In 1991, Cetus was sold to Chiron Corp., now part of Novartis. Glaser died in his sleep Thursday morning, Feb. 28, at his home in Berkeley. He was 86. *Philosopy of Science Portal

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, 27 February 2014

### On This Day in Math - February 27

 Andromeda Galaxy which Hubble measured to be 300,000 parsecs away.

Mathematical Knowledge adds a manly Vigour to the Mind, frees it from Prejudice, Credulity, and Superstition.
~John Arbuthnot

The 58th day of the year; 58 is the fourth smallest Smith Number. (Find the first three. A Smith number is a composite number for which the sum of its digits equals the sum of the digits in its prime factorization, including repetition. 58 = 2*29, and 5+8= 2+2+9.)Smith numbers were named by Albert Wilansky of Lehigh University. He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith.

EVENTS
1477 Founding of the University of Uppsala. A research university in Uppsala, Sweden, and is the oldest university in Sweden and Northern Europe. It ranks among the best universities in Northern Europe and is generally considered one of the most prestigious institutions of higher learning in Europe. Prominent students include Carolus Linnaeus , the father of taxonomy; Anders Celsius, inventor of the centigrade scale, and Niklas Zennström, co-founder of KaZaA and Skype. *Wik

In 1611, Johannes Fabricius, a Dutch astronomer, observed the rising sun through his telescope, and observed several dark spots on it. This was perhaps the first ever observation of sunspots. He called his father to investigate this new phenomenon with him. The brightness of the Sun's center was very painful, and the two quickly switched to a projection method by means of a camera obscura. Johannes was the first to publish information on such observations. He did so in his Narratio de maculis in sole observatis et apparente earum cum sole conversione. ("Narration on Spots Observed on the Sun and their Apparent Rotation with the Sun"), the dedication of which was dated 13 Jun 1611. *TIS

1665 Huygens writes letter to Robert Moray at the Royal Society asking him to pass on his "miraculous" observation of a synchronizing of his pendulum clocks. (See Feb 25). *Steven Strogatz, Synch

1890 Dedekind’s second letter to Keferstein. Hans Keferstein had published a paper on the notion of number with comments and suggestions for change of Dedekind's 1888 book. Dedekind first responded on February 9, and on February 14 and announced that he would push the publication by the "Society". It was in the letter of February 27 that Dedekind gives what is called, "a brilliant presentation of the development of his ideas on the notion of natural number." *Jean Van Heijenoort, From Frege to Gödel: a source book in mathematical logic, 1879-1931, pg 98 The text of the letter is available on-line at Google Books

1924, Harlow Shapley wrote replied to a letter from Edwin Hubble which presented the measurement of 300,000 parsecs as the distance to the Andromeda nebula. That was the first proof that the nebula was far outside the Milky Way, in fact, a separate galaxy. When Shapley had debated Heber Curtis on 26 Apr 1920, he presented his firm, life-long conviction that all the Milky Way represented the known universe (and, for instance, the Andromeda nebula was part of the Milky Way.) On receipt of the letter, Shapley told Payne-Gaposchkin and said “Here is the letter that has destroyed my universe.” In his reply, Shapley said sarcastically that Hubble's letter was “the most entertaining piece of literature I have seen for a long time.” Hubble sent more data in a paper to the AAS meeting, read on 1 Jan 1925. *TIS

1936 France issued a stamp with a portrait (by Louis Boilly) of Andr´e-Marie Amp`ere (1775–1836) to honor the centenary of his death. [Scott #306] *VFR

1942, J.S. Hey discovered radio emissions from the Sun. *TIS Several prior attempts were made to detect radio emission from the Sun by experimenters such as Nikola Tesla and Oliver Lodge, but those attempts were unable to detect any emission due to technical limitations of their instruments. Jansky first thought the radio signals he picked up from space were from the sun. *Wik

1989 In a review of Einstein–Bessso correspondence in the New Yorker, Jeremy Bernstein wrote: “In 1909, Einstein accepted a job as an associate professor at the University of Zurich, ... Einstein makes a familiar academic complaint—that because of his teaching duties he has less free time than when he was examining patents for eight hours a day.” *VFR

BIRTHS
1547 Baha' ad-Din al-Amili (27 Feb 1547 in Baalbek, now in Lebanon - 30 Aug 1621 in Isfahan, Iran) was a Lebanese-born mathematician who wrote influential works on arithmetic, astronomy and grammar. Perhaps his most famous mathematical work was Quintessence of Calculation which was a treatise in ten sections, strongly influenced by The Key to Arithmetic (1427) by Jamshid al-Kashi. *SAU

1881 L(uitzen) E(gbertus) J(an) Brouwer (27 Feb 1881, 2 Dec 1966) was a Dutch mathematician who founded mathematical Intuitionism (a doctrine that views the nature of mathematics as mental constructions governed by self-evident laws). He founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem. (Topology is the study of the most basic properties of geometric surfaces and configurations.) The Brouwer fixed point theorem is named in his honor. He proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, the treatment of general continuous mappings. *TIS He denies the law of the excluded middle. *VFR

1897 Bernard(-Ferdinand) Lyot (27 Feb 1897; 2 Apr 1952 at age 55) French astronomer who invented the coronagraph (1930), an instrument which allows the observation of the solar corona when the Sun is not in eclipse. Earlier, using his expertise in optics, Lyot made a very sensitive polariscope to study polarization of light reflected from planets. Observing from the Pic du Midi Observatory, he determined that the lunar surface behaves like volcanic dust, that Mars has sandstorms, and other results on the atmospheres of the other planets. Modifications to his polarimeter created the coronagraph, with which he photographed the Sun's corona and its analyzed its spectrum. He found new spectral lines in the corona, and he made (1939) the first motion pictures of solar prominences.*TIS

1910 Joseph Doob (27 Feb 1910 in Cincinnati, Ohio, USA - 7 June 2004 in Clark-Lindsey Village, Urbana, Illinois, USA) American mathematician who worked in probability and measure theory. *SAU After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes. So he wrote his famous "Stochastic Processes" book. It was published in 1953 and soon became one of the most influential books in the development of modern probability theory. *Wik

1942 Robert (Bob) Howard Grubbs (b. 27 February 1942 near Possum Trot, Kentucky, ) is an American chemist and Nobel laureate. Grubbs's many awards have included: Alfred P. Sloan Fellow (1974–76), Camille and Henry Dreyfus Teacher-Scholar Award (1975–78), Alexander von Humboldt Fellowship (1975), ACS Benjamin Franklin Medal in Chemistry (2000), ACS Herman F. Mark Polymer Chemistry Award (2000), ACS Herbert C. Brown Award for Creative Research in Synthetic Methods (2001), the Tolman Medal (2002), and the Nobel Prize in Chemistry (2005). He was elected to the National Academy of Sciences in 1989 and a fellowship in the American Academy of Arts and Sciences in 1994. Grubbs received the 2005 Nobel Prize in Chemistry, along with Richard R. Schrock and Yves Chauvin, for his work in the field of olefin metathesis. *Wik

DEATHS
1735 John Arbuthnot (baptized 29 Apr 1667, 27 Feb 1735 at age 67), fellow of the Royal College of Physicians. In 1710, his paper “An argument for divine providence taken form the constant regularity observ’s in the bith of both sexes” gave the ﬁrst example of statistical inference. In his day he was famous for his political satires, from which we still know the character John Bull. *VFR
He inspired both Jonathan Swift's Gulliver's Travels book III and Alexander Pope's Peri Bathous, Or the Art of Sinking in Poetry, Memoirs of Martin Scriblerus. He also translated Huygens' "De ratiociniis in ludo aleae " in 1692 and extended it by adding a few further games of chance. This was the first work on probability published in English.*SAU A nice blog about Arbuthnot and his work is at this post by *RMAT.

1867 James Dunwoody Brownson DeBow (1820 – February 27, 1867) was an American publisher and statistician, best known for his influential magazine DeBow's Review, who also served as head of the U.S. Census from 1853-1857.*Wik

1906 Samuel Pierpont Langley, (22 Aug 1834; 27 Feb 1906)American astronomer, physicist, and aeronautics pioneer who built the first heavier-than-air flying machine to achieve sustained flight. He launched his Aerodrome No.5 on 6 May 1896 using a spring-actuated catapult mounted on top of a houseboat on the Potomac River, near Quantico, Virginia. He also researched the relationship of solar phenomena to meteorology. *TIS

1915 Nikolay Yakovlevich Sonin (February 22, 1849 – February 27, 1915) was a Russian mathematician.
Sonin worked on special functions, in particular cylindrical functions. He also worked on the Euler–Maclaurin summation formula. Other topics Sonin studied include Bernoulli polynomials and approximate computation of definite integrals, continuing Chebyshev's work on numerical integration. Together with Andrey Markov, Sonin prepared a two volume edition of Chebyshev's works in French and Russian. He died in St. Petersburg.*Wik

1975 Hyman Levy (28 Feb 1889 in Edinburgh, Scotland - 27 Feb 1975 in Wimbledon, London, England )graduated from Edinburgh and went on to study in Göttingen. He was forced to leave Germany on the outbreak of World War II and returned to work at Oxford and at the National Physical Laboratory. He held various posts in Imperial College London, finishing as Head of the Mathematics department. His main work was in the numerical solution of differential equations. he published Numerical Studies in Differential Equations (1934), Elements of the Theory of Probability (1936), and Finite Difference Equations (1958). However, Levy was more than a mathematician. He was a philosopher of science and also a political activist. *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

## Wednesday, 26 February 2014

### On This Day in Math - February 26

Euler calculated without effort,
just as men breathe,
as eagles sustain themselves in the air.
~Francois Arago

The 57th day of the year; 57(base ten) is written with all ones in base seven. It is the last day this year that can be written in base seven with all ones.(What is the last day of the year that can be written with all ones in base two,... base three?)

EVENTS
1616 Galileo is warned to abandon Copernican views. On February 19, 1616, the Inquisition had asked a commission of theologians, known as qualifiers, about the propositions of the heliocentric view of the universe after Nicollo Lorin had accused Galileo of Heretical remarks in a letter to his former student, Benedetto Castelli. On February 24 the Qualifiers delivered their unanimous report: the idea that the Sun is stationary is "foolish and absurd in philosophy, and formally heretical since it explicitly contradicts in many places the sense of Holy Scripture..."; while the Earth's movement "receives the same judgement in philosophy and ... in regard to theological truth it is at least erroneous in faith."At a meeting of the cardinals of the Inquisition on the following day, Pope Paul V instructed Bellarmine to deliver this result to Galileo, and to order him to abandon the Copernican opinions; should Galileo resist the decree, stronger action would be taken. On February 26, Galileo was called to Bellarmine's residence, and accepted the orders. *Wik A transcript filed by the 1633 Inquisition indicates he was also enjoined from either speaking or writing about his theory. Yet Galileo remained in conflict with the Church. He was eventually interrogated by the Inquisition in Apr 1633. On 22 Jun 1633, Galileo was sentenced to prison indefinitely, with seven of ten cardinals presiding at his trial affirming the sentencing order. Upon signing a formal recantation, the Pope allowed him to live instead under house-arrest. From Dec 1633 to the end of his life on 8 Jan 1641, he remained in his villa at Florence.*TIS In 1992, the Vatican officially declared that Galileo had been the victim of an error.

1665 A letter from Christiaan Huygens to his father, Constantyn Huygens describes the discovery of synchronization between two pendulum clocks in his room.
While I was forced to stay in bed for a few days and made observations on my two clocks of the new workshop, I noticed a wonderful effect that nobody could have thought of before. The two clocks, while hanging [on the wall] side by side with a distance of one or two feet between, kept in pace relative to each other with a precision so high that the two pendulums always swung together, and never varied. While I admired this for some time, I finally found that this happened due to a sort of sympathy: when I made the pendulums swing at differing paces, I found that half an hour later, they always returned to synchronism and kept it constantly afterwards, as long as I let them go.

1849 Prince Albert visited the Ri for the 1st time to hear a lecture by Faraday. *Royal Institution ‏@ri_science Image
1855 Carl F. Gauss' body lay in state under the dome in the rotunda of the observatory in Gottingen two days after his death.  At nine o'clock a group of 12 students of science and mathematics, including Dedikind, carried the coffin out of the observatory and to his final resting place in St. Alben's Church Cemetary. After the casket was lowered it was covered with  covered with palms and laurel .

1885 “The Burroughs Company brought out their ﬁrst adding machine and announced that it would sell for $27.75$ plus $1.39$ shipping charges, for a total of whatever that came to.” *Tom Koch, 366 Dumb Days in History by Tom Koch

1962 A new teaching method based on “how and why things happen in mathematics rather than on traditional memorization of rules” is announced by the Educational Research Council of Greater Cleveland. This became the Cleveland Program of the New Math.*VFR

In 1896, Henri Becquerel stored a wrapped photographic plate in a closed desk drawer, and a phosphorescent uranium compound laid on top, awaiting a bright day to test his idea that sunlight would make the phosphorescent uranium emit rays. It remained there several days. Thus by sheer accident, he created a new experiment, for when he developed the photographic plate on 1 Mar 1896, he found a fogged image in the shape of the rocks. The material was spontaneously generating and emitting energetic rays totally without the external sunlight source. This was a landmark event. The new form of penetrating radiation was the discovery of the effect of radioactivity. He had in fact reported an earlier, related experiment to the French Academy on 24 Feb 1896, though at that time he thought phosphorescence was the cause.*TIS

1996 Silicon Graphics Inc. buys Cray Research for \$767 million, becoming the leading supplier of high-speed computing machines in the U.S. Over a forty year career, Cray founder Seymour Cray consistently produced most of the fastest computers in the world-- innovative, powerful supercomputers used in defense, meteorological, and scientific investigations. *CHM

2012 New world record distance for paper airplane throw: Joe Ayoob, a former Cal Quaterback, throws a John Collins paper airplane design, (which was named Suzanne), officially breaking the world record by 19 feet, 6 inches. The new world record was 226 feet, 10 inches. The previous record is 207 feet and 4 inches set by Stephen Kreiger in 2003. *ESPN

BIRTHS
1585 Federico Cesi (26 Feb OR 13 Mar 1585 (sources differ, but Thony Christie did some research to suggest the Feb date is the correct one); 1 Aug 1630 at age 45) Italian scientist who founded the Accademia dei Lincei (1603, Academy of Linceans or Lynxes), often cited as the first modern scientific society, and of which Galileo was the sixth member (1611). Cesi first announced the word telescope for Galileo's instrument. At an early age, while being privately educated, Cesi became interested in natural history and that believed it should be studied directly, not philosophically. The name of the Academy, which he founded at age 18, was taken from Lynceus of Greek mythology, the animal Lynx with sharp sight. He devoted the rest of his life to recording, illustrating and an early classification of nature, especially botany. The Academy was dissolved when its funding by Cesi ceased upon his sudden death(at age 45). *TIS It was revived in its currently well known form of the Pontifical Academy of Sciences, by the Vatican, Pope Pius IX in 1847.

1664 Nicolas Fatio de Duillier (alternative names are Facio or Faccio;) (26 February 1664 – 12 May 1753) was a Swiss mathematician known for his work on the zodiacal light problem, for his very close (some have suggested "romantic" ) relationship with Isaac Newton, for his role in the Newton v. Leibniz calculus controversy , and for originating the "push" or "shadow" theory of gravitation.
[Le Sage's theory of gravitation is a kinetic theory of gravity originally proposed by Nicolas Fatio de Duillier in 1690 and later by Georges-Louis Le Sage in 1748. The theory proposed a mechanical explanation for Newton's gravitational force in terms of streams of tiny unseen particles (which Le Sage called ultra-mundane corpuscles) impacting all material objects from all directions. According to this model, any two material bodies partially shield each other from the impinging corpuscles, resulting in a net imbalance in the pressure exerted by the impact of corpuscles on the bodies, tending to drive the bodies together.]

He also developed and patented a method of perforating jewels for use in clocks.
When Leibniz sent a set of problems for solution to England he mentioned Newton and failed to mention Faccio among those probably capable of solving them. Faccio retorted by sneering at Leibniz as the ‘second inventor’ of the calculus in a tract entitled ‘Lineæ brevissimæ descensus investigatio geometrica duplex, cui addita est investigatio geometrica solidi rotundi in quo minima fiat resistentia,’ 4to, London, 1699. Finally he stirred up the whole Royal Society to take a part in the dispute (Brewster, Memoirs of Sir I. Newton, 2nd edit. ii. 1–5).
In 1707, Fatio came under the influence of a fanatical religious sect, the Camisards, which ruined Fatio's reputation. He left England and took part in pilgrim journeys across Europe. After his return only a few scientific documents by him appeared. He died in 1753 in Maddersfield near Worcester, England. After his death his Geneva compatriot Georges-Louis Le Sage tried to purchase the scientific papers of Fatio. These papers together with Le Sage's are now in the Library of the University of Geneva.
Eventually he retired to Worcester, where he formed some congenial friendships, and busied himself with scientific pursuits, alchemy, and the mysteries of the cabbala. In 1732 he endeavoured, but it is thought unsuccessfully, to obtain through the influence of John Conduitt [q. v.], Newton's nephew, some reward for having saved the life of the Prince of Orange. He assisted Conduitt in planning the design, and writing the inscription for Newton's monument in Westminster Abbey. *Wik
1786 Dominique François Jean Arago (26 Feb 1786, 2 Oct 1853) was a French physicist and astronomer who discovered the chromosphere of the sun (the lower atmosphere, primarily composed of hydrogen gas), and for his accurate estimates of the diameters of the planets. Arago found that a rotating copper disk deflects a magnetic needle held above it showing the production of magnetism by rotation of a nonmagnetic conductor. He devised an experiment that proved the wave theory of light, showed that light waves move more slowly through a dense medium than through air and contributed to the discovery of the laws of light polarization. Arago entered politics in 1848 as Minister of War and Marine and was responsible for abolishing slavery in the French colonies. *TIS A really great blog about Arago, With the catchy title, "François Arago: the most interesting physicist in the world!" is posted here. Read this introduction, and you will not be able to resist:

When he was seven years old, he tried to stab a Spanish solider with a lance
When he was eighteen, he talked a friend out of assassinating Napoleon
He once angered an archbishop so much that the holy man punched him in the face
He has negotiated with bandits, been chased by a mob, broken out of prison
He is:
François Arago, the most interesting physicist in the world

1799 Benoit Clapeyron (26 Feb 1799, 28 Jan 1864) French engineer who expressed Sadi Carnot's ideas on heat analytically, with the help of graphical representations. While investigating the operation of steam engines, Clapeyron found there was a relationship (1834) between the heat of vaporization of a fluid, its temperature and the increase in its volume upon vaporization. Made more general by Clausius, it is now known as the Clausius-Clapeyron formula. It provided the basis of the second law of thermodynamics. In engineering, Clayeyron designed and built locomotives and metal bridges. He also served on a committee investigating the construction of the Suez Canal and on a committee which considered how steam engines could be used in the navy.*TIS

1842 Nicolas Camille Flammarion (26 Feb 1842; 3 Jun 1925 at age 83) was a French astronomer who studied double and multiple stars, the moon and Mars. He is best known as the author of popular, lavishly illustrated, books on astronomy, including Popular Astronomy (1880) and The Atmosphere (1871). In 1873, Flammarion (wrongly) attributed the red color of Mars to vegetation when he wrote “May we attribute to the color of the herbage and plants which no doubt clothe the plains of Mars, the characteristic hue of that planet...” He supported the idea of canals on Mars, and intelligent life, perhaps more advanced than earth's. Flammarion reported changes in one of the craters of the moon, which he attributed to growth of vegetation. He also wrote novels, and late in life he turned to psychic research. *TIS

1843 Karl Friedrich Geiser (26 Feb 1843 in Langenthal, Bern, Switzerland, 7 May 1934 in Küsnacht, Zürich, Switzerland) Swiss mathematician who worked in algebraic geometry and minimal sufaces. He organised the first International Mathematical Congress in Zurich.*SAU

1864 John Evershed (26 Feb 1864, 17 Nov 1956) English astronomer who discovered (1909) the Evershed effect - the horizontal motion of gases outward from the centres of sunspots. While photographing solar prominences and sunspot spectra, he noticed that many of the Fraunhofer lines in the sunspot spectra were shifted to the red. By showing that these were Doppler shifts, he proved the motion of the source gases. This discovery came to be known as the Evershed effect. He also gave his name to a spectroheliograph, the Evershed spectroscope.*TIS

DEATHS
1638 Claude-Gaspar Bachet de M´eziriac (9 Oct 1581, 26 Feb 1638), noted for his work in number theory and mathematical recreations. He published the Greek text of Diophantus’s Arithmetica in 1621. He asked the ﬁrst ferrying problem: Three jealous husbands and their wives wish to cross a river in a boat that will only hold two persons, in such a manner as to never leave a woman in the company of a man unless her husband is present. (With four couples this is impossible.)*VFR (I admit that I don't know how this differs from the similar river crossings problems of Alcuin in the 800's, Help someone?)His books on mathematical puzzles formed the basis for almost all later books on mathematical recreations.*SAU

1693 Sir Charles Scarborough MP FRS FRCP (19 December 1615 – 26 February 1693) was an English physician and mathematician.
He was born in St. Martin's-in-the-Fields, London in 1615, the son of Edmund Scarburgh, and was sent to St. Paul's School, whence he proceeded to Caius College, Cambridge, and educated at St Paul's School, Gonville and Caius College, Cambridge (BA, 1637, MA, 1640) and Merton College, Oxford (MD, 1646). While at Oxford he was a student of William Harvey, and the two would become close friends. Scarborough was also tutor to Christopher Wren, who was for a time his assistant.
Following the Restoration in 1660, Scarborough was appointed physician to Charles II, who knighted him in 1669; Scarborough attended the king on his deathbed, and was later physician to James II and William and Mary. During the reign of James II, Scarborough served (from 1685 to 1687) as Member of Parliament for Camelford in Cornwall.
Scarborough was an original fellow of the Royal Society and a fellow of the Royal College of Physicians, author of a treatise on anatomy, Syllabus Musculorum, which was used for many years as a textbook, and a translator and commentator of the first six books of Euclid's Elements (published in 1705). He also was the subject of a poem by Abraham Cowley, An Ode to Dr Scarborough.
Scarborough died in London in 1693. He was buried at Cranford, Middlesex, where there is a monument to him in the parish church erected by his widow. *Wik
1878 Pietro Angelo Secchi (18 Jun 1818, 26 Feb 1878 at age 59) Italian Jesuit priest and astrophysicist, who made the first survey of the spectra of over 4000 stars and suggested that stars be classified according to their spectral type. He studied the planets, especially Jupiter, which he discovered was composed of gasses. Secchi studied the dark lines which join the two hemispheres of Mars; he called them canals as if they where the works of living beings. (These studies were later continued by Schiaparelli.) Beyond astronomy, his interests ranged from archaeology to geodesy, from geophysics to meteorology. He also invented a meteorograph, an automated device for recording barometric pressure, temperature, wind direction and velocity, and rainfall.*TIS

1985 Tjalling Charles Koopmans (August 28, 1910 – February 26, 1985) was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences. Koopmans' early works on the Hartree–Fock theory are associated with the Koopmans' theorem, which is very well known in quantum chemistry. Koopmans was awarded his Nobel prize (jointly with Leonid Kantorovich) for his contributions to the field of resource allocation, specifically the theory of optimal use of resources. The work for which the prize was awarded focused on activity analysis, the study of interactions between the inputs and outputs of production, and their relationship to economic efficiency and prices.*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, 25 February 2014

### On This Day in Math - February 25

 Cathedral Church of St Paul the Apostle *Wik

People must understand that science is inherently neither a potential for good nor for evil.
It is a potential to be harnessed by man to do his bidding.
~Glenn T. Seaborg

The 56th day of the year; There are 56 normalized 5x5 Latin Squares (First row and column have 1,2,3,4,5; and no number appears twice in a row or column. There are a much smaller number of 4x4 squares, try them first)
EVENTS
1598 John Dee demonstrates the solar eclipse by viewing an image through a pinhole. Two versions from Ashmole and Aubrey give different details of who was present. Dee's Diary only contains the notation, "the eclips. A clowdy day, but great darkness about 9 1/2 maine " *Benjamin Wooley, The Queen's Conjuror

1606 Henry Briggs sends a Letter to Mr. Clarke, of Gravesend, dated from Gresham College, with which he sends him the description of a ruler, called Bedwell's ruler, with directions how to use it. (it seems from the letter to be a ruler for measuring the volume of timber. If you have information on where I could see a picture or other image of the device, please advise) *Augustus De Morgan, Correspondence of scientific men of the seventeenth century

1870 Hermann Amandus Schwarz sent his friend Georg Cantor a letter containing the ﬁrst rigorous proof of the theorem that if the derivative of a function vanishes then the function is constant. See H. Meschkowski, Ways of Thought of Great Mathematicians, pp. 87–89 for an English translation of the letter. *VFR

1959 The APT Language is Demonstrated: The Automatically Programmed Tools language is demonstrated. APT is an English-like language that tells tools how to work and is mainly used in computer-assisted manufacturing.
NEW YORKER: Cambridge, Mass. - Feb. 25: The Air Force announced today that it has a machine that can receive instructions in English - figure out how to make whatever is wanted- and teach other machines how to make it. An Air Force general said it will enable the United States to build a war machine that nobody would want to tackle. Today it made an ashtray. *CHM

1976 Romania issued a stamp picturing the mathematician Anton Davidoglu (1876–1958). [Scott #2613] *VFR

BIRTHS
1670 Maria Winckelmann (Maria Margarethe Winckelmann Kirch (25 Feb 1670 in Panitzsch, near Leipzig, Germany - 29 Dec 1720 in Berlin, Germany) was a German astronomer who helped her husband with his observations. She was the first woman to discover a comet.*SAU "German astronomer Maria Kirch (1670 – 1720). Kirch was original educated by her father and her uncle who believed that girls should receive the same education as boys. From them she learnt mathematics and astronomy going on to study with and work together with the amateur astronomer Christoph Arnold. Through Arnold she got to know the astronomer Gottfried Kirch and despite the fact that he was 30 years older than her they married. Kirch was official astronomer of the Berlin Royal Academy of Science and he and Maria ran the Academy’s observatory together for many years. In 1702 she became the first woman to discover a comet but the credit for the discovery was given to her husband. When Gottfried died in 1710 Maria applied for his position arguing correctly that she had done half of the work in the past. Despite her having published independently and having an excellent reputation as well as the active support of Leibniz the Academy refused to award her the post. She worked in various other observatories until 1717 when her son was appointed to his fathers post, Maria once again becoming the assistant. Despite having more than proved her equality to any male astronomer Maria never really received the recognition she deserved." From Thony Christie's Renaissance Mathematicus blog on Daughters of Urania.

1827 Henry William Watson (25 Feb 1827 in Marylebone, London, England - 11 Jan 1903 in Berkswell (near Coventry), England) was an English mathematician who wrote some influential text-books on electricity and magnetism. *SAU

1902 Kenjiro Shoda (February 25, 1902 – March 3, 1977 *SAU gives March 20 for death) was a Japanese mathematician. He was interested in group theory, and went to Berlin to work with Issai Schur. After one year in Berlin, Shoda went to Göttingen to study with Emmy Noether. Noether's school brought a mathematical growth to him. In 1929 he returned to Japan. Soon afterwards, he began to write Abstract Algebra, his mathematical textbook in Japanese for advanced learners. It was published in 1932 and soon recognised as a significant work for mathematics in Japan. It became a standard textbook and was reprinted many times.*Wik

1922 Ernst Gabor Straus (February 25, 1922 – July 12, 1983) was a German-American mathematician who helped found the theories of Euclidean Ramsey theory and of the arithmetic properties of analytic functions. His extensive list of co-authors includes Albert Einstein and Paul Erdős as well as other notable researchers including Richard Bellman, Béla Bollobás, Sarvadaman Chowla, Ronald Graham, László Lovász, Carl Pomerance, and George Szekeres. It is due to his collaboration with Straus that Einstein has Erdős number 2. *Wik

1926 Masatoşi Gündüz İkeda (25 February 1926, Tokyo. - 9 February 2003, Ankara), was a Turkish mathematician of Japanese ancestry, known for his contributions to the field of algebraic number theory. *Wik

DEATHS
1723 Sir Christopher Wren (20 Oct 1632; 25 Feb 1723) Architect, astronomer, and geometrician who was the greatest English architect of his time (Some may suggest Hooke as an equal) whose famous masterpiece is St. Paul's Cathedral, among many other buildings after London's Great Fire of 1666. Wren learned scientific skills as an assistant to an eminent anatomist. Through astronomy, he developed skills in working models, diagrams and charting that proved useful when he entered architecture. He inventing a "weather clock" similar to a modern barometer, new engraving methods, and helped develop a blood transfusion technique. He was president of the Royal Society 1680-82. His scientific work was highly regarded by Sir Isaac Newton as stated in the Principia. *TIS
Thony Christie points out that, "Most people don’t realise that as well as being Britain’s most famous 17th century architect, Wren was also a highly respected mathematician. In fact Isaac Newton named him along with John Wallace and William Oughtred as one of the three best English mathematicians of the 17th century. As a young man he was an active astronomer and was a highly vocal supporter of the then still relatively young elliptical astronomy of Johannes Kepler."

(I love the message on his tomb in the Crypt of St. Pauls: Si monumentum requiris circumspice ...."Reader, if you seek his monument, look about you." Lisa Jardine's book is excellent

1775 William Small (13 October 1734; Carmyllie, Angus, Scotland – 25 February 1775; Birmingham, England). He attended Dundee Grammar School, and Marischal College, Aberdeen where he received an MA in 1755. In 1758, he was appointed Professor of Natural Philosophy at the College of William and Mary in Virginia, then one of Britain’s American colonies.
Small is known for being Thomas Jefferson's professor at William and Mary, and for having an influence on the young Jefferson. Small introduced him to members of Virginia society who were to have an important role in Jefferson's life, including George Wythe a leading jurist in the colonies and Francis Fauquier, the Governor of Virginia.
Recalling his years as a student, Thomas Jefferson described Small as:
"a man profound in most of the useful branches of science, with a happy talent of communication, correct and gentlemanly manners, and a large and liberal mind... from his conversation I got my first views of the expansion of science and of the system of things in which we are placed."
In 1764 Small returned to Britain, with a letter of introduction to Matthew Boulton from Benjamin Franklin. Through this connection Small was elected to the Lunar Society, a prestigious club of scientists and industrialists.
In 1765 he received his MD and established a medical practice in Birmingham, and shared a house with John Ash, a leading physician in the city. Small was Boulton's doctor and became a close friend of Erasmus Darwin, Thomas Day, James Keir, James Watt, Anna Seward and others connected with the Lunar Society. He was one of the best-liked members of the society and an active contributor to their debates.
Small died in Birmingham on 25 February 1775 from malaria contracted during his stay in Virginia. He is buried in St. Philips Church Yard, Birmingham.
The William Small Physical Laboratory, which houses the Physics department at the College of William & Mary, is named in his honor. *Wik
1786 Thomas Wright (22 September 1711 – 25 February 1786) was an English astronomer, mathematician, instrument maker, architect and garden designer. He was the first to describe the shape of the Milky Way and speculate that faint nebulae were distant galaxies.*Wik

1947 Louis Carl Heinrich Friedrich Paschen (22 Jan 1865; 25 Feb 1947) was a German physicist who was an outstanding experimental spectroscopist. In 1895, in a detailed study of the spectral series of helium, an element then newly discovered on earth, he showed the identical match with the spectral lines of helium as originally found in the solar spectrum by Janssen and Lockyer nearly 40 years earlier. He is remembered for the Paschen Series of spectral lines of hydrogen which he elucidated in 1908. *TIS

1950 Nikolai Nikolaevich Luzin, (also spelled Lusin) (9 December 1883, Irkutsk – 28 January 1950, Moscow), was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics.*Wik

1972 Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach-Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.
Author of around 170 scientific articles and books, Steinhaus has left its legacy and contribution on many branches of mathematics, such as functional analysis, geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founders of the game theory and the probability theory preceding in his studies, later, more comprehensive approaches, by other scholars. *Wik
His Mathematical Snapshots is a delight to read, but get the ﬁrst English edition if you can—there are lots of surprises there. *VFR
"When Steinhaus failed to attend an important meeting of the Committee of the Polish Academy of Sciences in 1960, he received a letter chiding him for "not having justified his absence." He immediately wired the President of the Academy that "as long as there are members who have not yet justified their presence, I do not need to justify my absence."
[ Told by Mark Kac in "Hugo Steinhaus -- A Remembrance and a Tribute," Amer. Math. Monthly 81 (June-July 1974) 578. ] * http://komplexify.com

1988 Kurt Mahler (26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a mathematician and Fellow of the Royal Society. Mahler proved that the Prouhet–Thue–Morse constant and the Champernowne constant 0.1234567891011121314151617181920... are transcendental numbers.
He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D. from Johann Wolfgang Goethe University of Frankfurt am Main in 1927. He left Germany with the rise of Hitler and accepted an invitation by Louis Mordell to go to Manchester. He became a British citizen in 1946.
He was elected a member of the Royal Society in 1948 and a member of the Australian Academy of Science in 1965. He was awarded the London Mathematical Society's Senior Berwick Prize in 1950, the De Morgan Medal, 1971, and the Thomas Ranken Lyle Medal, 1977. *Wik

1999 Glenn Theodore Seaborg (April 19, 1912(Ishpeming, Michigan) – February 25, 1999) was an American scientist who won the 1951 Nobel Prize in Chemistry for "discoveries in the chemistry of the transuranium elements", contributed to the discovery and isolation of ten elements, and developed the actinide concept, which led to the current arrangement of the actinoid series in the periodic table of the elements. He spent most of his career as an educator and research scientist at the University of California, Berkeley where he became the second Chancellor in its history and served as a University Professor. Seaborg advised ten presidents from Harry S. Truman to Bill Clinton on nuclear policy and was the chairman of the United States Atomic Energy Commission from 1961 to 1971 where he pushed for commercial nuclear energy and peaceful applications of nuclear science.
The element seaborgium was named after Seaborg by Albert Ghiorso, E. Kenneth Hulet, and others, who also credited Seaborg as a co-discoverer. It was so named while Seaborg was still alive, which proved controversial. He influenced the naming of so many elements that with the announcement of seaborgium, it was noted in Discover magazine's review of the year in science that he could receive a letter addressed in chemical elements: seaborgium, lawrencium (for the Lawrence Berkeley Laboratory where he worked), berkelium, californium, americium
(Once when being aggressively cross-examined during testimony on nuclear energy for a senate committee, the Senator asked, “How much do you really know about Plutonium.” Seaborg quietly answered, “Sir, I discovered it.” , Which he did as part of the team at the Manhattan Project. *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

## Thursday, 20 February 2014

### Solutions to Harmonic Geometry

These are my proposed solutions to the Harmonic geometry problems posed here.
this
1) a square inscribed in a right triangle whose legs are A ft and B feet (with one corner of the square at the right angle of the triangle), what is the length of a side of the square?

For HS students, the easy approach is to construct the line from (0,0) to the point (x,x) on the opposite vertex of the square, which rests on the hypotenuse of the triangle. This line has equation y=x, and if we find the intersection of this with the hypotenuse through (0, A) and (B,0) we will have our x-value for the length of the square. The hypotenuse has equation $y= \frac {-Ax}{B}+ A$ Since at the point of intersection, y= x we can substitute x for y to get $x= \frac {-Ax}{B}+ A$ and eliminating the denominators we get Ax + Bx = AB, which produces $x = \frac{AB}{A+B}$ which is 1/2 the harmonic mean of A and B.
Why this is so can be more easily seen if we extend the line y=x out to the point (B,B) and drop a perpendicular down to (B,0). Now a segment from (0,A) to (B,B) completes the trapezoid we used to explain the solution to the crossed ladders.

2) The three dimensional version of the problem is solved by a similar approach in three-space.  Using the four vertices (0,0,0) ; A=(A,0,0); B=(0,B,0); and C= (0,0,C) for our right rectangular tetrahedron we know that the diagonal of the cube to the face ABC will have coordinates (r,r,r) . For lack of confusion we can let the equation of the plane ABC be mx+ny+pz = d.  Now each point on the plane must satisfy this equation, so we know that (r,r,r) will solve it and mr+nr+zr = d .  Each of the points A, B, and C must also solve the equation, so mA=nB=pC = d as well .  solving each of these for m, n, and p we see that $m = \frac{d}{A}$ ; $n=\frac{d}{B}$ and $p= \frac{d}{C}$ . Now when we substitute these back into mr+nr+zr = d we get  $\frac{dr}{A}+ \frac{dr}{B}+\frac{dr}{C} = d$ .  Now if we factor out the dr in each numerator we can divide both sides by d to get $r ( \frac{1}{A}+ \frac{1}{B}+\frac{1}{C}) = 1$. Now keep the r alone and divide the one by the three fractions we get $\frac{1}{r}= \frac{1}{A}+ \frac{1}{B}+\frac{1}{C}$ which we recognize as 1/3 of the harmonic mean of the lengths A, B, and C.

3) Given a 5-12-13 Right Triangle, Find a relation between the radius of the inscribed circle and the altitudes.   Is this property true of a non-right triangle?

Using the hopefully familiar formula that A= r s (where s is the semi-perimeter) gives 30 = r (15) making the solution of r=2 rather easy. But what of the harmonic mean? If we find the three altitudes we see that they are 5, 12, and 60/13. The harmonic mean of these turns out to be 6, and (as we expect from previous problems) the radius turns out to be 1/3 the Harmonic Mean of the altitudes.
And in ANY triangle:
The radius of the incircle is ⅓ of the Harmonic Mean of the altitudes.
and given by twice the area divided by the perimeter.

4) For the student of the conics, In the ellipse $9x^2+4y^2=36$, find the length of the semi-latus rectum.

Using a=3 and b=2 for the lengths of the semi-major and semi-minor axes lengths we have a focal length of $\sqrt{5}$
and so the coordinate of B must be $( \sqrt{5} , x)\} where x is the length of the semi-latus rectum. Using these in the original equation \( 9x^2+4(5)=36$ gives us x=4/3.... SOOoooo?? you ask.

If you divide the transverse axis into two segments on each side of one foci, their lengths will be $3+\sqrt{5}$ and $3 - \sqrt{b}$
Using the formula $\frac{2ab}{a+b}$ gives the harmonic mean of these two values as 4/3.

The solution may remind students that if a diameter of a circle is divided into two segments, the perpendicular to the diameter will be the geometric mean of the two lengths.  We might remind them that that works for ANY division of the diameter, where this applies only to the division at a focus.

Hopefully I have omitted my usual characteristic mis-calculations and typos.  And I still want to share other approaches to solutions of these.

### Harmonic Geometry, Some Nice Problems about the Harmonic Mean

The problem above is often called the crossed ladders problem, as it is on the Wikipedia site where I found this image. The idea is that the two ladders are each resting against the foot of a building and leaning against the opposite building. Given the heights A and B at which they touch the opposite buildings, the challenge is to find the height, h at which they cross.

I had recently planned a talk on a collection of harmonic mean problems, particularly some geometric ones, but it was cancelled due to weather, so I thought I would point out some of the interesting geometric problems that are related to the harmonic mean in this blog.

For the student/teacher/reader who is not familiar with the Harmonic mean, a couple of notes to lay the foundation seem appropriate.

The harmonic mean is one of the classic old Greek means called Pythagorean means because they were, supposedly, known to the Pythagoras or his followers. The harmonic mean is associated with the musical interval known as the Perfect fourth. An octave change upwards in pitch reflects a doubling of the wavelength frequency (a 1:2 ratio), the harmonic mean of 1 and 2 is 4/3.
The actual definition of the harmonic mean usually sounds convoluted to students. The harmonic mean is the reciprocal of the average of the reciprocals of the values. The algebraic notation is only a little clearer. For a set of numbers xi the harmonic mean, H, is given by $H = \frac{n}{ \frac{1}{x_1} + \frac {1}{x_2} + \dotsb+ \frac {1}{x_n} }$

Fortunately for two values a and b, the whole thing simplifies down to $\frac{2ab}{a+b}$.

The harmonic mean is often shown geometrically in a trapezoid resembling the crossed ladder problem. The harmonic mean is given by the red line, and the solution to the crossed ladders problem is 1/2  the harmonic means of the wall heights.

Another classic geometry illustration  is used to show the relationship between the means of two values in a circle.  When the diameter is divided into two parts, a and b, then the relation between the Arithmetic mean which equals the length of the radius, the Geometric mean and Harmonic mean will always be related by  $A \ge G \ge H$
A slightly different version shows the contra -harmonic mean which is equal to the length of the diameter less the harmonic mean.    The contr-harmonic mean is the ratio of the sum of the squares divided by the sum of the values, or $\frac{x_1^2 + x_2^2 + \dotsb + x_n^2}{x_1 + x_2 + \dotsb + x_n}$

In the trapezoidal view of the harmonic mean of two numbers, the arithmetic mean (green) joins the midpoints of the two non-parallel sides and the contra-harmonic mean (blue) is the same distance away from the arithmetic mean as the harmonic mean.

So with that bit of information let's talk about a few problems.  I will give all the problems here, and provide solutions in a second blog: The key in all of them is not just a solution, but a solution related to the harmonic mean.  (many are more easily solved w/o using harmonic mean)

a square inscribed in a right triangle whose legs are A ft and B feet (with one corner of the square at the right angle of the triangle), what is the length of a side of the square?

A 3-d version of the same problem may be more of a challenge for the more talented students:
A cube inscribed in a right rectangular tetrahedron with legs of a, b, and c has a cube inscribed with one vertex at (0,0,0).

What is the length of an edge of the cube?

Another with a triangle,
Given a 5-12-13 Right Triangle, Find a relation between the radius of the inscribed circle and the altitudes.   Is this property true of a non-right triangle?

For the student of the conics, In the ellipse $9x^2+4y^2=36$, find the length of the semi-latus rectum.

Hopefully I still remember how to solve all these, and will try to post a blog with at least one explanation for each, but would love to hear your solutions, and additional geometric problems involving the harmonic mean.
Enjoy.

The solutions (I hope) are now posted here.

Addendum: Recently came across this neat one from Antonio Gutierrez@gogeometry.  Note the radii of the two Archimedean Twins.

## Tuesday, 18 February 2014

### Using Euler's Gem to find Platonic Solids

I was recently watching a video by Norm Wildberger of New South Wales University on Algebraic Topology and he did a nice explanation with algebra of why there can only be the five platonic solids we know using Euler's beautiful V+F-E=2 for closed polyhedra.  It was simple enough (as was much of this "beginners" course) for a bright high school student.
I was struck by the fact that the only explanations I had ever seen, and given, was the geometric argument that went something like, "Well, six hexagons will lie flat, so they can't form a solid, and n-gons bigger than that won't fit at all."  It always included a little more hand-waving than I was comfortable with, but it was what I knew.

Since many teachers and students might not choose to stumble onto a course called Algebraic Topology, I wanted to replicate his explanation.

He begins by defining the Regular polyhedra with the common statements then proceeds to his proof:
We let F, E, and V represent the number of Faces, Edges and Vertices as usual.

Let n = Number of edges (and vertices) on each regular polygonal face
Let m = Number of edges (or faces) that meet at each vertex
and both of these must be greater than or equal to 3.

The number of edges will be $E = \frac{n*F}{2}$ since every edge is counted twice when counting all the faces.
The number of vertices will be $V= \frac{n*F}{m}$ since each vertex is counted m times when counting all faces.

Replacing V and E in Eulers formula with these expressions on F we have

V +  F - E = 2    becomes  $\frac{n*F}{m} + F - \frac{n*F}{2} =2$

Now if we solve for F we get $F=\frac{4m}{2m + 2n - mn}$

If we require that the number of faces is a positive integer, then the denominator, 2m+ 2n-mn must be greater than zero.  From this we can extract $2n + 2m > mn$  and from this $2n > mn - 2m$ and then to $2n > m (n-2)$

Finally we can divide by n-2 to get $\frac{2n}{n-2} > m$  but calling on the fact that both m & n are greater than or equal to 3, we can add $\frac{2n}{n-2} > m \ge 3$  and applying the transitive law of inequalities, we can get $\frac{2n}{n-2} > 3$.

This becomes$2n > 3n-6$ and the trivial algebra brings us to  $n < 6$  so there can be no Platonic solids with faces with more than five sides.

Returning for a moment's glance at  $2n + 2m > mn$ we see that it is symmetrical in m & n, and it must also be true that $m < 6$ .  m and n are now limited to the possible sets n={3,4,5} and m={3,4,5}  (I find this use of appeal to symmetry to tie m and n together as one of the better reasons to show this to students.)

If we return now to  $F=\frac{4m}{2m + 2n - mn}$ and try different values for n and m from the set nXm  we will see that the only possible solutions are the five we have.
Letting n=3 ( regular triangle faces  ) we get  $\frac{4m}{2m+6-3m} = \frac{4m}{6-m}$   and testing the values m= 3,4,5 we get 12/3 = 4, 16/2 = 8, and 20/1 = 20 faces, the tetrahedron, octahedron, and icosahedron are all that are possible with triangular faces.

If we use n=4 we get  $\frac{4m}{2m+8-4m} = \frac{4m}{8-2m}$ since m can only be in the range 3,4,5, only 3 will produce a positive integer in the denominator, so we must have 12/2 = 6 faces, or a cube as the only possibility.

Returning to check n=5 we have \( \frac{4m}{2m+10-5m} = \frac{4m}{10-3m}\ ) only m=3 produces a positive integer in the denominator, and so there must be 12 pentagonal faces.

I think this would be a wonderful addition to a presentation of the Platonic solids because it utilizes a beautiful theorem, but in a way that provides support for the beautiful way that geometry and algebra compliment each other (and perhaps the mathematical spelling "complement" is even more meaningful here).  It is important to point out to students that the algebra tells us that these are all the possible  Platonic solids.  We must still construct them.

## Wednesday, 12 February 2014

### Good Math Skills overwhelmed by Strong Political View

David Brooks just sent me a note on a research project at Yale a few months ago I had never seen.  It makes it clearer to me why people sometimes accept political ideas that seem to be totally contradictory to their own welfare.
In the research, by Yale law professor Dan Kahan,  he gave identical numerical data supposedly coming from an actual survey to a sample of over 1000 people.  In one of the sets of data it was described as testing skin cream effectiveness.  In the other, it was testing the effectiveness of bans on concealed gun permits.
In each case the exact same numbers were given in two different ways, one that showed improved results, and one that showed lack of improvement.
In the skin cream problems, the results showed that "more numerate" people were much more likely to get the right answer.  There also was no significant difference between  liberals and conservatives of similar numeracy.
But here is how Chris Mooney described the results in a Mother Jones article:
"So how did people fare on the handgun version of the problem? They performed quite differently than on the skin cream version, and strong political patterns emerged in the results—especially among people who are good at mathematical reasoning. Most strikingly, highly numerate liberal Democrats did almost perfectly when the right answer was that the concealed weapons ban does indeed work to decrease crime (version C of the experiment)—an outcome that favors their pro-gun-control predilections. But they did much worse when the correct answer was that crime increases in cities that enact the ban (version D of the experiment).

The opposite was true for highly numerate conservative Republicans: They did just great when the right answer was that the ban didn't work (version D), but poorly when the right answer was that it did (version C).

A graph of the results given in the article may raise questions about the "numeracy" classifications, or perhaps liberals only develop bias above numeracy level 5.5 while conservatives bias runs across a wider range of ability.  But they clearly support the idea that they gave the answer they wanted, rather than the one they surely knew to be true.

Moody concluded with a telling statement, "The Scottish Enlightenment philosopher David Hume famously described reason as a "slave of the passions." Today's political scientists and political psychologists, like Kahan, are now affirming Hume's statement with reams of new data. This new study is just one out of many in this respect, but it provides perhaps the most striking demonstration yet of just how motivated, just how biased, reasoning can be—especially about politics."