## Tuesday, 29 July 2014

### On This Day in Math - July 29

To call in the statistician after the experiment is done may be
no more than asking hm to perform a postmortem examination:
he may be able to say what the experiment died of.
~Ronald Fisher

The 210th day of the year; (21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210). Students are challenged to find another pair of such PPTs

EVENTS
1654 Pascal wrote a letter to Fermat agreeing to a result of Fermat on a probability problem about repeated rolls of a single die for a wager. "Impatience has seized me as well as it has you, and although I am still abed, I cannot refrain from telling you that I received your letter in regard to the problem of the points  yesterday evening from the hands of M. Carcavi, and that I admire it more than I can tell you. I do not have the leisure to write at length, but, in a word, you have found the two divisions of the points and of the dice with perfect justice. I am thoroughly satisfied as I can no longer doubt that I was wrong, seeing the admirable accord in which I find myself with you."  *York Univ Hist of Stats

1698 In a letter to John Bernoulli, Leibniz introduces the dot for multiplication..(cajori 233; vol 1 pg 267) “The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: “I do not like X as a symbol for multiplication, as it is easily confounded with x; … often I simply relate two quantities by an interposed dot and indicate multiplication by ZC · LM. Hence, in designating ratio I use not one point but two points, which I use at the same time for division.”

1739 D’Alembert, age 21, submitted his ﬁrst mathematical paper to the Academy of Sciences. *VFR As his knowledge of mathematics was mainly due to self-study, he often found that others had already established his mathematical discoveries by more elegant and more direct means. In 1739 d’Alembert submitted his first paper to the French Académie Royale des Sciences, in which he described the errors found in the standard textbook, Analyse démontrée, written by Charles Reyneau. *webpage of Robert Nowland

1773 First schoolhouse West of the Alleghenies.*VFR (built in Schoenbrunn, OH.)

1867 Thomas Hill, president of Harvard College, who was also somewhat of a mathematician, wrote Benjamin Peirce, who was a professor there: “I have the honor of informing you that the University, on Commencement Day, conferred on you the Degree of Doctor of Laws in recognition of the transcendent ability with which you have pursued mathematical physical investigations, and in particular for the luster which she has herself for so many years borrowed from your genius.” [P. 10 of Benjamin Peirce, AMM offprint, 1925] *VFR

1878 This was the height of search for the intra-Mercurial planet Vulcan using eclipses to block the Sun. (Vulcan was a small planet proposed to exist in an orbit between Mercury and the Sun. In an attempt to explain peculiarities of Mercury's orbit, in the 19th-century French mathematician Urbain Jean Joseph Le Verrier hypothesized that they were the result of another planet, which he named Vulcan.) Several observers claim sightings, but they are never confirmed. The problem is finally resolved by Albert Einstein (1879-1955) in his general theory of relativity in 1916. *NSEC

1958 President Eisenhower signed the National Aeronautics and Space Act. NASA opened for business on 1 October 1958, and within a week launched Project Mercury—the start of the U.S. manned space program. *VFR

2005, another candidate for tenth planet was announced by Mike Brown of California Institute of Technology. Its diameter is estimated at 2,100 miles - about 1-1/2 times that of Pluto. Its orbit is eccentric and inclined at about 45 degrees to the main plane of the solar system. It was named 2003 UB313 on a photograph made 31 Oct 2003. Later, its motion was recognized, on 8 Jan 2005. With orbits significantly inclined to the others, the status as a planet of either or even Pluto, is a subject for debate. They are in a region of numerous frozen comet-like objects beyond Neptune - the Kuiper Belt. The object Sedna - somewhat smaller than Pluto - was also found there in 2004. NASA also in an official statement referred to 2003 UB313 as a tenth planet*TIS

BIRTHS

1858 Francesco Gerbaldi (29 July 1858, La Spezia, Italy to 29 June 1934, Pavia, Italy) was an Italian geometer, who proved Gerbaldi's theorem. In geometry, Gerbaldi's theorem, proved by Gerbaldi (1882), states that one can find six pairwise apolar linearly independent nondegenerate ternary quadratic forms. These are permuted by the Valentiner group. (say that three times real fast) *Wik

1862 Eduard Brückner (July 29, 1862–May 20, 1927) pioneer climate researcher. He also studied the glaciers of the Alps and particularly the effect of the ice ages on the Earth's surface features. By analyzing direct and indirect observations of climatic fluctuations, he discovered the 35-year Brückner climatic cycle (1887) of swings between damp-cold and warm-dry conditions. He initiated scientific debate on whether climate change should be interpreted as a natural function of the Earth system, or whether it was influenced by man's activities, such as deforestation. He considered the impact of climate change on the balance of power between nations and its economic significance in agricultural productivity, emigration, river transportation and the spreading of diseases.*TIS

1898 Isidor Isaac Rabi (29 July 1898 – 11 January 1988) was an American physicist who was awarded the Nobel Prize for Physics in 1944 for his invention (in 1937) of the atomic and molecular beam magnetic resonance method of measuring magnetic properties of atoms, molecules, and atomic nuclei. He spent most of his life at Columbia University (1929-67), where he performed most of his pioneering research in radar and the magnetic moment associated with electron spin in the 1930s and 1940s. His Nobel-winning work led to the invention of the laser, the atomic clock, and diagnostic uses of nuclear magnetic resonance. He originated the idea for the CERN nuclear research center in Geneva (founded 1954). *TIS
1912 Noel Bryan Slater, often cited NB Slater, (1912 in Blackburn , January 31 1973) was a British mathematician and physicist who worked on including statistical mechanics and physical chemistry, and probability theory.*Wik

DEATHS

1781 Johann Kies (September 14, 1713—July 29, 1781) a German astronomer and mathematician. Born in Tübingen, Kies worked in Berlin in 1751 alongside Jérôme Lalande in order to make observations on the lunar parallax in concert with those of Nicolas Louis de Lacaille at the Cape of Good Hope.
From 1742 to 1754, at the recommendation of the mathematician Leonhard Euler, he was made professor of mathematics at Berlin's Academy of Sciences and astronomer at its observatory.
He subsequently taught also at the Collegium of Tübingen. From 1754 to 1755, Kies served as director of the Astronomisches Rechen-Institut in Heidelberg.
Kies was one of the first to propagate Newton's discoveries in Germany, and dedicated two of his works to the Englishman: De viribus centralibus (Tübingen, 1758) and De lege gravitatis (Tübingen, 1773). Kies is also the author of a work on lunar influences: De influxu lunae in partes terrae mobiles (Tübingen, 1769). He wrote many other works, both in French and in Latin, on astronomy.
Kies corresponded with Euler from 1747 to 1767. Their correspondence consists of 8 letters, all of which were written by Kies.
The crater Kies on the Moon is named in his honor. *TIA
1839 Gaspard de Prony. (July 22, 1755 - July 29, 1839) Cauchy was elected his successor at the Bureau des Longitudes but was not admitted as he refused to take the oath of allegiance. *VFR
In 1793, de Prony began a major task of producing logarithmic and trigonometric tables for the French Cadastre. The effort was begun at the request of the French National Assembly, which, after the French Revolution wanted to bring uniformity to the multiple measurements and standards used throughout the nation. The tables and their production were vast, with values calculated to between fourteen and twenty-nine decimal places. Inspired by Adam Smith's Wealth of Nations, de Prony divided up the labor, bragging that he "could manufacture logarithms as easily as one manufactures pins." At the top of the organizational hierarchy were scientists and mathematicians who devised the formulas. Next were workers who created the instructions for doing the calculations. At the bottom were about ninety "computers" (as they were called) who were not trained in mathematics, but who followed the instructions."
One of de Prony's important scientific inventions was the 'de Prony brake' which he invented in 1821 to measure the performance of machines and engines. He also was first to propose using a reversible pendulum to measure gravity, which was independently invented in 1817 by Henry Kater and became known as the Kater's pendulum. He also created a method of converting sinusoidal and exponential curves into a systems of linear equations. Prony estimation is used extensively in signal processing and finite element modelling of non linear materials. Prony was a member, and eventually president, of the French Academy of Science. He was also elected a foreign member of the Royal Swedish Academy of Sciences in 1810. His name is one of the 72 names inscribed on the Eiffel Tower. *Wik
1898 John Alexander Reina Newlands, (July 22, 1755 - July 29, 1839) was a British chemist who first established an order of elements by the atomic weights, and observed a periodicity in the properties. Every eighth element has similar properties, hence he named the Law of Octaves (7 Feb 1863). It took another quarter century, and the work of others, such as Mendeleev, for the significance of his discovery to be recognized. He died in London.*TIS
1944 David Eugene Smith (January 21, 1860 in Cortland, New York – July 29, 1944 in New York) died in New York City at the age of eighty-four.*VFR Smith attended Syracuse University, graduating in 1881 (Ph. D., 1887; LL.D., 1905). He studied to be a lawyer concentrating in arts and humanities, but accepted an instructorship in mathematics at the Cortland Normal School in 1884. He also knew Latin, Greek, and Hebrew. He became a professor at the Michigan State Normal College in 1891, the principal at the State Normal School in Brockport, New York (1898), and a professor of mathematics at Teachers College, Columbia University (1901).
Smith became president of the Mathematical Association of America in 1920. He also wrote a large number of publications of various types. He was editor of the Bulletin of the American Mathematical Society; contributed to other mathematical journals; published a series of textbooks; translated Klein's Famous Problems of Geometry, Fink's History of Mathematics, and the Treviso Arithmetic. He edited Augustus De Morgan's Budget of Paradoxes (1915) and wrote many books on Mathematics. *Wik
1962 Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation. Anders Hald called him "a genius who almost single-handedly created the foundations for modern statistical science" while Richard Dawkins called him "the greatest of Darwin's successors". In 2010 Dawkins named him "the greatest biologist since Darwin". Fisher was opposed to the conclusions of Richard Doll and A.B. Hill that smoking caused lung cancer. He compared the correlations in their papers to a correlation between the import of apples and the rise of divorce in order to show that correlation does not imply causation.
To quote Yates and Mather, "It has been suggested that the fact that Fisher was employed as consultant by the tobacco firms in this controversy casts doubt on the value of his arguments. This is to misjudge the man. He was not above accepting financial reward for his labours, but the reason for his interest was undoubtedly his dislike and mistrust of puritanical tendencies of all kinds; and perhaps also the personal solace he had always found in tobacco."

After retiring from Cambridge University in 1957 he spent some time as a senior research fellow at the CSIRO in Adelaide, Australia. He died of colon cancer there in 1962.
He was awarded the Linnean Society of London's prestigious Darwin–Wallace Medal in 1958.
Fisher's important contributions to both genetics and statistics are emphasized by the remark of L.J. Savage, "I occasionally meet geneticists who ask me whether it is true that the great geneticist R.A. Fisher was also an important statistician"*Wik The stained glass window is from the Greatroom at Caius College.

1996 Marcel-Paul "Marco" Schützenberger (October 24, 1920 – July 29, 1996) was a French mathematician and Doctor of Medicine. His work had impact across the fields of formal language, combinatorics, and information theory.[1] In addition to his formal results in mathematics, he was "deeply involved in [a] struggle against the votaries of Darwinism,"[2] a stance which has resulted in some mixed reactions from his peers and from critics of his stance on evolution. Several notable theorems and objects in mathematics bear his name (for example Schutzenberger group).*Wik

2004 Walter Feit (October 26, 1930 – July 29, 2004)was an Austrian mathematician who (with John Thompson) proved one of the most important theorems about finite simple groups.*SAU

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

## Monday, 28 July 2014

### On This Day in Math - July 28

It appears to me that if one wishes to make progress in mathematics
one should study the masters and not the pupils.
Quoted in O Ore's, Niels Abel, Mathematician Extraordinary

The 209th day of the year; 209=16+25+34+43+52+61. Also 209 is a "Self number" A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. students might want to explore self numbers for patterns

EVENTS
1619 Kepler wrote Napier expressing his enthusiasm for Napier’s invention of logarithms. *VFR

1851  First American eclipse expedition to Europe when George Phillips Bond (1825 - 1865) led a team to Scandinavia. *NSEC   In the transcription of his notes he wrote:

1851 A total solar eclipse was photographed for the ﬁrst time. *VFR The first correctly-exposed photograph of the solar corona was made during the total phase of the solar eclipse of 28 July 1851 at Königsberg (now Kaliningrad) by a local daguerreotypist named Berkowski at the Royal Observatory in Königsberg, Prussia (now Kalinigrad in Russia). Berkowski, whose first name was never published, observed at the Royal Observatory. A small 6-cm refracting telescope was attached to the 15.8-cm Fraunhofer heliometer and a 84-second exposure was taken shortly after the beginning of totality.
United Kingdom astronomers, Robert Grant and William Swan, and Austrian astronomer Karl Ludwig von Littrow observed this eclipse and determined that prominences are part of the Sun because the Moon is seen to cover and uncover them as it moves in front of the Sun.*Wik

In 1858, fingerprints were used as a means of identification for the first time.*TIS The English first began using fingerprints in July of 1858, when Sir William James Herschel, Chief Magistrate of the Hooghly district in Jungipoor, India, first used fingerprints on native contracts. On a whim, and without thought toward personal identification, Herschel had Rajyadhar Konai, a local businessman, impress his hand print on a contract.
The idea was merely "... to frighten [him] out of all thought of repudiating his signature." The native was suitably impressed, and Herschel made a habit of requiring palm prints--and later, simply the prints of the right Index and Middle fingers--on every contract made with the locals. Personal contact with the document, they believed, made the contract more binding than if they simply signed it. Thus, the first wide-scale, modern-day use of fingerprints was predicated, not upon scientific evidence, but upon superstitious beliefs.
As his fingerprint collection grew, however, Herschel began to note that the inked impressions could, indeed, prove or disprove identity. While his experience with fingerprinting was admittedly limited, Sir William Herschel's private conviction that all fingerprints were unique to the individual, as well as permanent throughout that individual's life, inspired him to expand their use. *History of Fingerprints, Onin.com

1866 The ﬁrst act (in the USA) legalizing the employment of the metric system was approved (14 Stat. L. 339). The act provided that it “shall be lawful throughout the United States of America to employ the weights and measures of the metric system.” *VFR

1882 The Institute of Accountants and Bookkeepers was organized in New York City. It was the first accounting society in the United States. *FFF

1899 Cantor asks Dedekind whether the set of all cardinal numbers is itself a set, because if it is it would have a cardinal number larger than any other cardinal. *VFR

1948 Allen Turing writes to Jack Good with an estimate of the number of neurons in the human brain. "I have repeatedly looked in books on neurology ... and never found any numbers offered. My own estimate is 3x108 to 3x109. " *Turing Archives

2061 Halley's comet will next reach perihelion. The comet last reached perihelion on 9 February 1986, and will reach it again on 28 July 2061 *Wik

BIRTHS
1849 Robert Scott studied at Cambridge and was elected to a fellowship. After a short time teaching he studied to be a barrister. He spent most of his career as Bursar and Master of St John's College Cambridge. He published a book on Determinants. *SAU

1867 Charles Dillon Perrine (July 28, 1867; Steubenville, Ohio, – June 21, 1951) U.S. astronomer who discovered the sixth and seventh moons of Jupiter in 1904 and 1905, respectively. In 1904 he published a calculation of the solar parallax (a measure of the Earth-Sun distance) based on observations of the minor planet Eros during one of its close approaches to the Earth. *TIS

1928 John Bell (28 June 1928 – 1 October 1990)   his great achievement was that during the 1960s he was able to breathe new and exciting life into the foundations of quantum theory, a topic seemingly exhausted by the outcome of the Bohr-Einstein debate thirty years earlier, and ignored by virtually all those who used quantum theory in the intervening period. Bell was able to show that discussion of such concepts as 'realism', 'determinism' and 'locality' could be sharpened into a rigorous mathematical statement, 'Bell's inequality', which is capable of experimental test. Such tests, steadily increasing in power and precision, have been carried out over the last thirty years. *SAU

1954 Gerd Faltings (July 28, 1954 - ) was born in Gelsenkirchen-Buer, West Germany. In 1986 he received a Fields Medal for solving Mordell’s Conjecture using arithmetic algebraic geometry. *VFR He has also been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n greater than 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Falting's arguments.
However, Faltings was the natural person that Wiles turned to when he wanted an opinion on the correctness of his repair of his proof of Fermat's Last Theorem in 1994.*TIS

DEATHS
1818 Gaspard Monge (9 May 1746 – 28 July 1818) died in disgrace in Bourbon Paris, having been stripped of his place in the reorganized Acad´emie of 1816. Although he contributed to diﬀerential equations and the geom¬etry of surfaces, his special interest was descriptive geometry. Employed as a teacher, he made signiﬁcant contributions to educational reform. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 616]
On the fall of Napoleon he was deprived of all his honors, and even excluded from the list of members of the reconstituted Institute. Monge died at Paris on 28 July 1818 and was interred in Le Père Lachaise Cemetery, in Paris, in a mausoleum. He was later transferred to the Panthéon. The mausoleum and Monge's bust remain in Le Père Lachaise Cemetery.
A statue portraying him was erected in his home town of Beaune, Côte-d'Ors in 1849. His name is one of the 72 names inscribed on the Eiffel Tower.

1944 Sir Ralph Fowler (17 January 1889 – 28 July 1944) a brilliant physicist. But it may be for his influence upon others that he is best known. In fact, no less than fifteen Fellows of the Royal Society and three Nobel Laureates were supervised by Fowler between 1922 and 1939. The total number supervised during this time was a staggering sixty-four giving him an average of eleven research students at any given time. One might be led to believe that this did not allow for any depth of relationship to form between him and his students. However, this was far from the truth of the matter. Those who studied under Fowler had a tremendous admiration for him. In particular, E A Milne [1] was especially taken by the man whom he fondly referred to as "the kind of man you can still remain friendly with, even when he has sold you a motor-bike; it is not possible to say more" and whom he called a "prince amongst men".
Aside from Milne, on whom he had a profound impact, he also had the opportunity of influencing the likes of Sir Arthur Eddington, Subramanian Chandrasekhar, Paul Dirac, Sir William McCrea, Lady Jeffreys and others either directly through supervision or indirectly through collaboration. Even in his personal life he was intimately connected with brilliant people having married Eileen, the only daughter of Lord Rutherford whom he met through Rutherford's Cavendish Laboratory at Cambridge. Sometimes his influence was simply the fact that he was known to so many people. It was Fowler who ultimately introduced Paul Dirac to the burgeoning field of quantum theory in 1923 leading Dirac to the forefront of its ultimate discovery in 1925. Fowler also put Dirac and Werner Heisenberg in touch with each other through Niels Bohr. As Sir William McCrea simply put it: "he was the right man in the right place at the right time." *SAU
1968 Otto Hahn (8 Mar 1879; 28 Jul 1968 at age 89) German physical chemist who, with the radiochemist Fritz Strassmann, is credited with the discovery of nuclear fission. He was awarded the Nobel Prize for Chemistry in 1944 and shared the Enrico Fermi Award in 1966 with Strassmann and Lise Meitner. Element 105 carries the name hahnium in recognition of his work.*TIS

2000 Abraham Pais (May 19, 1918 – July 28, 2000) Dutch-American physicist and science historian whose research became the building blocks of the theory of elemental particles. He wrote Subtle Is the Lord: The Science and Life of Albert Einstein, which is considered the definitive Einstein biography. In Holland, his Ph.D. in physics was awarded on 9 Jul 1941, five days before a Nazi deadline banning Jews from receiving degrees. Later, during WW II, while in hiding to evade the Gestapo, he worked out ideas in quantum electrodynamics that he later shared when working with Niels Bohr (Jan - Aug 1946). In Sep 1946, he went to the U.S. to work with Robert Oppenheimer at Princeton, where Pais contributed to the foundations of the modern theory of particle physics.*TIS

2004 Francis Harry Compton Crick (8 June 1916 – 28 July 2004) was a British biophysicist, who, with James Watson and Maurice Wilkins, received the 1962 Nobel Prize for Physiology or Medicine for their determination of the molecular structure of deoxyribonucleic acid (DNA), the chemical substance ultimately responsible for hereditary control of life functions. Crick and Watson began their collaboration in 1951, and published their paper on the double helix structure on 2 Apr 1953 in Nature. This accomplishment became a cornerstone of genetics and was widely regarded as one of the most important discoveries of 20th-century biology. *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

## Sunday, 27 July 2014

### On This Day in Math - July 27

But just as much as it is easy to find the differential of a given quantity,
so it is difficult to find the integral of a given differential.
Moreover, sometimes we cannot say with certainty
whether the integral of a given quantity can be found or not.

~Bernoulli, Johann

This is the 208th day of the year; 208 is the sum of the squares of the first five primes.
and from Today is 7/27 or 27/7 in somecountries. 727 and 277 are both prime.

EVENTS
1630, On July 27 Giovanni Batista Baliani wrote a letter to Galileo Galilei about the explanation of an experiment he had made in which a siphon, led over a hill about twenty-one meters high, failed to work. Galileo responded with an explanation of the phenomena: he proposed that it was the power of a vacuum which held the water up, and at a certain height (in this case, thirty-four feet) the amount of water simply became too much and the force could not hold any more, like a cord that can only withstand so much weight hanging from it.

1794 Jean Baptiste Joseph Fourier (1766?-1830) was a student at the École Normale, c1794. He was sentenced to the guillotine by Robespierre on July 28 of 1794, but Robespierre was overthrown the day before his scheduled execution (27 July, 1794) was due. Fourier went on to both political and scientific success. He was unanimously elected the first Secretary of the Institute of Egypt in 1798. He was Governor of Lower Egypt in 1798‑1801  or Commissioner at the Divan of Cairo .  He led one of the expeditions of exploration which examined ancient monuments and he suggested the publication of the great report on Egypt.  He was was a professor at the École Polytechnique up to 1806.  Napoléon made him a baron and during Napoléon's return from Elba in 1815, he made Fourier a count and Prefect of the Rhone, based at Lyons, from 10 Mar to 1 May.  In 1815, he was penniless in Paris and giving lessons for his living.  The Prefect of Paris found out and made him director of the Bureau de la Statistique of the Préfecture of the Seine.  He was elected to the Académie in 1816, but this was vetoed by the government, so he was elected again in 1817 and this was permitted.    He was Prefect of the Department of Isère, whose capital is Grenoble, from 1802 to 1817 (1815??)  He was Permanent Secretary of the Académie des Sciences in 1822-1830.

1829 By a remarkable coincidence, both Cauchy and Sturm sent papers to the Acad´emie des Sciences dealing with differential equations. Both of them used techniques which we recognize as matrix methods. Thus they are early contributors to linear algebra, a ﬁeld which is usually dated to Cayley’s introduction of matrices in 1858. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 1150]

1837 At a meeting of the Berlin Academy of Sciences, Dirichlet presented his ﬁrst paper on analytic number theory. He proved the fundamental theorem that bears his name: Every arithmetical series an + b, n =0, 1, 2,... of integers where a and b are relatively prime, contains inﬁnitely many primes. The result had long been conjectured. Legendre tried hard for a proof but could only establish special cases such as 4n + 1. *VFR

1866 Cyrus W. Field finally succeeded, after two failures, in laying the first underwater telegraph cable 1,686 miles long across the Atlantic Ocean between North America and Europe. Massachusetts merchant and financier Cyrus W. Field first proposed laying a 2,000-mile copper cable along the ocean bottom from Newfoundland to Ireland in 1854, but the first three attempts ended in broken cables and failure. Field's persistence finally paid off in July 1866, when the Great Eastern, the largest ship then afloat, successfully laid the cable along the level, sandy bottom of the North Atlantic. *TIS

1948 Hungary issued a stamp commemorating the centenary of the birth of the physicist Baron Roland E˝otv˝os1 (1848–1919). [Scott #840]. *VFR They issued another in 1991

BIRTHS

1667 Johann Bernoulli (27 July 1667 – 1 January 1748; also known as Jean or John) was a Swiss mathematician who studied reflection and refraction of light, orthogonal trajectories of families of curves, quadrature of areas by series and the brachystochrone.*SAU

1733 Jeremiah Fenwicke Dixon (27 July 1733 – 22 January 1779) was an English surveyor and astronomer who is best known for his work with Charles Mason, from 1763 to 1767, in determining what was later called the Mason-Dixon line.

Dixon was born in Cockfield, near Bishop Auckland, County Durham, the fifth of seven children, to Sir George Fenwick Dixon 5th Bt. and Lady Mary Hunter. His father was a wealthy Quaker coal mine owner and aristocrat. His mother came from Newcastle, and was said to have been "the cleverest woman" to ever marry into the Dixon family. Dixon became interested in astronomy and mathematics during his education at Barnard Castle. Early in life he made acquaintances with the eminent intellectuals of Southern Durham: mathematician William Emerson, and astronomers John Bird and Thomas Wright. In all probability it was John Bird, who was an active Fellow of the Royal Society, who recommended Dixon as a suitable companion to accompany Mason.

Jeremiah Dixon served as assistant to Charles Mason in 1761 when the Royal Society selected Mason to observe the transit of Venus from Sumatra. However, their passage to Sumatra was delayed, and they landed instead at the Cape of Good Hope where the transit was observed on June 6, 1761. Dixon returned to the Cape once again with Nevil Maskelyne's clock to work on experiments with gravity.

Dixon and Mason signed an agreement in 1763 with the proprietors of Pennsylvania and Maryland, Thomas Penn and Frederick Calvert, sixth Baron Baltimore, to assist with resolving a boundary dispute between the two provinces. They arrived in Philadelphia in November 1763 and began work towards the end of the year. The survey was not complete until late 1766, following which they stayed on to measure a degree of Earth's meridian on the Delmarva Peninsula in Maryland, on behalf of the Royal Society. They also made a number of gravity measurements with the same instrument that Dixon had used with Maskelyne in 1761. Before returning to England in 1768, they were both admitted to the American Society for Promoting Useful Knowledge, in Philadelphia.

Dixon sailed to Norway in 1769 with William Bayly to observe another transit of Venus. The two split up, with Dixon at Hammerfest Island and Bayly at North Cape, in order to minimize the possibility of inclement weather obstructing their measurements. Following their return to England in July, Dixon resumed his work as a surveyor in Durham. He died unmarried in Cockfield on 22 January 1779, and was buried in an unmarked grave in the Quaker cemetery in Staindrop.

Although he was recognized as a Quaker, he was not a very good one, dressing in a long red coat and occasionally drinking to excess. *Wik
1801 Sir. George Biddell Airy (27 July 1801 – 2 January 1892) born in Alnwick, England. *VFR English astronomer who became the seventh Astronomer Royal (1836-92). In his life he studied interference fringes in optics, made a mathematical study of the rainbow and computed the density of the Earth by swinging a pendulum at the top and bottom of a deep mine, determined the mass of the planet Jupiter and its period rotation, calculated the orbits of comets and cataloged stars. He designed corrective lenses for astigmatism (1825), the first that worked. His motivation was his own astigmatism. Airy had a long-standing battle with Babbage. In 1854, the conflict continued between the two during the battle of the incompatible railway gauges in England. Airy championed the railway narrow gauge and Babbage for the wide gauge. *TIS

1844 Ágoston Scholtz (27 July 1844 in Kotterbach, Zips district, Austro-Hungary (now Rudnany, Slovakia) - 6 May 1916 in Veszprém,) From 1871 he was a teacher of mathematics and natural philosophy at the Lutheranian Grammar School of Budapest which at that time had been upgraded to become a so called 'chief grammar school', namely one which offered eight years of teaching. This was precisely the school which later was attended by several famous mathematicians such as Johnny von Neumann and Eugene Wigner (or Jenó Pál Wigner as he was called at that time). Scholtz became the school director of the Lutheranian Grammar School in 1875. Unfortunately this excellent school was closed in 1952, and most of its equipment was lost. Due to the initiative and support of its former well-known students, among others Wigner, it was reopened in 1989 after being closed for thirty-seven years. Scholtz's field of research was projective geometry and theory of determinants. His results were recorded by Muir in his famous work The history of determinants *SAU

1848 Roland Baron von Eötvös (27 July 1848 – 8 April 1919) was a Hungarian physicist who studied at Heidelberg where he was taught by Kirchhoff, Helmholtz and Bunsen. Eötvös introduced the concept of molecular surface tension and published on capillarity (1876-86). For the rest of his life he concentrated on study of the Earth's gravitational field. He developed the Eötvös torsion balance, long unsurpassed in precision, which gave experimental proof that inertial mass and gravitational mass, to a high degree of accuracy, are equivalent - which later was a major principle of Albert Einstein.*TIS

1849 John Hopkinson (27 July 1849 – 27 August 1898) British physicist and electrical engineer who worked on the application of electricity and magnetism in devices like the dynamo and electromagnets. Hopkinson's law (the magnetic equivalent of Ohm's law) bears his name. In 1882, he patented his invention of the three-wire system (three phase) for electricity generation and distribution. He presented the principle the synchronous motors (1883), and designed electric generators with better efficiency. He also studied condensers and the phenomena of residual load. In his earlier career, he became (1872) engineering manager of Chance Brothers and Co., a glass manufacturer in Birmingham, where he studied lighthouse illumination, improving efficiency with flashing groups of lights.*TIS

1867 Derrick Norman Lehmer (27 July 1867, Somerset, Indiana, USA — 8 September 1938 in Berkeley, California, USA) was an American mathematician and number theorist.
In 1903, he presented a factorization of Jevons' number (8,616,460,799) at the San Francisco Section of the American Mathematical Society, December 19, 1903.
He published tables of prime numbers and prime factorizations, reaching 10,017,000 by 1909 (In Number Theory and Its History, Ore calls this the "best factor table now (1948) available"). He developed a variety of mechanical and electro-mechanical factoring and computational devices, such as the Lehmer sieve, built with his son Derrick Henry Lehmer.
He is also known for a reversible algorithm that assigns a Lehmer code to every permutation of size n. *SAU

1870 Bertram Borden Boltwood (July 27, 1870 Amherst, Massachusetts - August 15, 1927, Hancock Point, Maine) was an American chemist and physicist whose work on the radioactive decay of uranium and thorium was important in the development of the theory of isotopes. Boltwood studied the "radioactive series" whereby radioactive elements sequentially decay into other isotopes or elements. Since lead was always present in such ores, he concluded (1905) that lead must be the stable end product from their radioactive decay. Each decay proceeds at a characteristic rate. In 1907, he proposed that the ratio of original radioactive material to its decay products measured how long the process had been taking place. Thus the ore in the earth's crust could be dated, and give the age of the earth as 2.2 billion years.*TIS

1871 Ernest Friedrich Ferdinand Zermelo. (27 July 1871; Berlin, German Empire - 21 May 1953 (aged 81) Freiburg im Breisgau, West Germany) In 1904 he formulated the Axiom of Choice in Set Theory. Years later, when he refused to give the Nazi salute, he was threatened with dismissal from his univeristy position. In reply, he resigned. *VFR

DEATHS

1759 Pierre-Louis Moreau de Maupertuis (17 July 1698 – 27 July 1759) French mathematician, biologist, and astronomer. In 1732 he introduced Newton's theory of gravitation to France. He was a member of an expedition to Lapland in 1736 which set out to measure the length of a degree along the meridian. Maupertuis' measurements both verified Newton's predictions that the Earth would be an oblate speroid, and they corrected earlier results of Cassini. Maupertuis published on many topics including mathematics, geography, astronomy and cosmology. In 1744 he first enunciated the Principle of Least Action and he published it in Essai de cosmologie in 1850. Maupertuis hoped that the principle might unify the laws of the universe and combined it with an attempted proof of the existence of God.*TIS (he died in the home of Johann II Bernoulli. Johan Bernoulli (above) was born on the day Maupertuis died, but Johann II Bernoulli died on the Calendar date on which Maupertuis was born...)

1844 John Dalton, (6 September 1766 – 27 July 1844) English teacher who, from investigating the physical and chemical properties of matter, deduced an Atomic Theory (1803) whereby atoms of the same element are the same, but different from the atoms of any other element. In 1804, he stated his law of multiple proportions by which he related the ratios of the weights of the reactants to the proportions of elements in compounds. He set the atomic weight of hydrogen to be identically equal to one and developed a table of atomic weights for other elements. He was the first to measure the temperature change of air under compression, and in 1801 suggested that all gases could be liquified by high pressure and low temperature. Dalton recognised that the aurora borealis was an electrical phenomenon.*TIS

1931 Jacques Herbrand (12 February 1908 – 27 July 1931) was a French mathematician who died young but made contributions to mathematical logic.*SAU Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant. *Wik

1999 Aleksandr Danilovic Aleksandrov (4 Aug 1912 in Volyn, Ryazan, Russia
- 27 July 1999) approached the differential geometry of surfaces [by extending the notion of the objects studied], extending the class of regular convex surfaces to the class of all convex surfaces ... . In order to solve concrete problems Aleksandrov had to replace the Gaussian geometry of regular surfaces by a much more general theory. In the first place the intrinsic properties (i.e. those properties that appear as a result of measurements carried out on the surface) of an arbitrary convex surface had to be studied, and methods found for the proof of theorems on the connection between intrinsic and exterior properties of convex surfaces. Aleksandrov constructed a theory of intrinsic geometry of convex surfaces on that basis. Because of the depth of this theory, the importance of its applications and the breadth of its generality, Aleksandrov comes second only to Gauss in the history of the development of the theory of surfaces. *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

## Saturday, 26 July 2014

### Another Hugh Worthington Approximation for Right Triangles.

A while back I wrote about a nice hand calculation of the angles in a right triangle which appeared in an article, The Resolution of Triangles(1780), by Rev. Hugh Worthington. The article prompted some really nice math from several bloggers leading to an understanding that it was essentially a version of Taylor's method applied to the ArcTan function and converted to degrees. A good explanation is at the Theorem of the Day site.

So today I thought I would introduce another of Rev Worthington's hand calculation approaches for a right triangle, this one with the angles and one side given, to find another side. Worthington credits this method at a Mr. Henry Watson from Navigation new modelled (abt 1715). The Watson method is shortened into a shorter proportion, which I will give below, but I begin with Watson's method of using an "artificial number".

The method works two ways, to find a smaller angle if the Hypotenuse is known, and then to find the Hypotenuse if a leg is known.
I will illustrate first the method using a given leg.
First, find the compliment of the angle opposite the side given, and square it, then multiply the result by four. Now divide that amount by 300 + three times the compliment. Finally add that amount to the angle again. This is the "artificial number" which he says is sometimes called the natural radius (suggestions why?). I will call this artificial number N and the acute angle we use A.

Then $\frac{N}{hyp} = \frac{A}{a}$

Suppose we begin with a right triangle with a side a = 5, and A= 40o, B=50o and C is the right Angle.

We take the compliment of 40, and square it to get 2500. Four times this amount gives 10,000. Now we divide by 300+ 3 times the compliment, (450) to get 22 2/9 Finally we add back the 40 to get 62 2/9 for the artificial number.
So the proportion is $\frac{62 2/9}{ Hyp }= \frac{40}{5}$ ,   giving us a Hypotenuse of 7 7/9 . My calculator gives Sin-1 $(\frac{5}{7\frac{7}{9}})$ as 40.0052.

Which I'm calling pretty darn close.

To find a leg when the hypotenuse is the side given, we use the compliment of the angle opposite the side we seek to know, and proceed in similar fashion.   This time, for ease of computation, let's let the  Hypotenuse be 5, and keep the angles at 40 and 50.  We want to find the side opposite the forty degree angle, so 50 is still the compliment of the angle so we have the same artificial number.

Our proportion for this looks like $\frac{62 2/9}{ 5 }= \frac{40}{a}$   giving  a= 45/14 or 3.2142857....   Checking again, we get 5 * sin (40o = 3.21938 .... pretty close again.

The Rev Worthington gives the shortcut (modified for the letters used above) to directly calculate without the intermediate of the artificial number.

$\frac{57.3}{b} + \frac{3b}{1000} = \frac{hyp}{a}$

I'm hoping that any high school student worth their salt can figure out the presence of 57.3, but if not, see the comments of the previous post linked above.

So who was this mathematical minister, this preacher of trigonometric hand calculations? A note about his popularity as a preacher and a brief history is here, and includes a reference to this "Resolution of Triangles paper, and no others. Perhaps it was his sole excursion into math.