## Monday 29 July 2024

### On This Day in Math - July 29

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

~Ronald Fisher

The 210th Day of the Year
210 is the last year day that is a Primorial, 210 = 7# = 7*5*3*2. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.*Wikipedia Of course that means it is the smallest number that is the product of four distinct primes, and the only such year date.

210 is a Harshad (joy-giver) number, divisible by the sum of its digits.  In fact, it is a multiple Harshad number since 210/3 = 70, which is also a Harshad number.

(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

There are an infinite number of numbers that appear six or more times in Pascal's Arithmetic Triangle, but only three of them; 1, 120, and 210 are year dates.

7! hours is 210 days.

The Combination of ten things taken four at a time is 210. It is also C(21,2)

13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210, the sum of eight consecutive primes

210 is the 20th Triangular number, the sum of the integers from 1 - 20.

210 is the last year date which is both a triangular number and the product of consecutive numbers, 14 x 15.  It is also the last to be the product of three consecutive numbers,  5 x 6 X 7.

Three different ways to make a 3x3 magic square with a magic constant of 210, Take the classic 3x3 and multiply each term by 14,
56 126 28
42 70 98
112 14 84

Or with consecutive integers starting at 76

69 74 67
68 70 72
73 66 71

Or maybe with increments of five

65 90 55
60 70 80
85 50 75

The magic is in the middle, all else stems from there.

210 in binary is a balanced number, with the same numbers of ones and zeros, and reading from left to right the zeros never outnumber the ones.

The sum of the squares of the divisors of 12, is 210.

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.”

Cajori shows the symbol as a raised dot. However, according to Margherita Barile, consulting Gerhardt's edition of Leibniz's Mathematische Schriften (G. Olms, 1971), the dot is never raised, but is located at the bottom of the line. She writes that the non-raised dot as a symbol for multiplication appears in all the letters of 1698, and earlier, and, according to the same edition, it already appears in a letter by Johann Bernoulli to Leibniz dated September, 2nd 1694.

The dot was used earlier by Thomas Harriot (1560-1621) in Analyticae Praxis ad Aequationes Algebraicas Resolvendas, which was published posthumously in 1631, and by Thomas Gibson in 1655 in Syntaxis mathematica. However Cajori says, "it is doubtful whether Harriot or Gibson meant these dots for multiplication. They are introduced without explanation. It is much more probable that these dots, which were placed after numerical coefficients, are survivals of the dots habitually used in old manuscripts and in early printed books to separate or mark off numbers appearing in the running text" (Cajori vol. 1, page 268).

However, Scott (page 128) writes that Harriot was "in the habit of using the dot to denote multiplication." And Eves (page 231) writes, "Although Harriot on occasion used the dot for multiplication, this symbol was not prominently used until Leibniz adopted it."

The colon (:) was used in 1633 in a text entitled Johnson Arithmetik; In two Bookes (2nd ed.: London, 1633). However Johnson only used the symbol to indicate fractions (for example three-fourths was written 3:4); he did not use the symbol for division "dissociated from the idea of a fraction" (Cajori vol. 1, page 276).

Gottfried Wilhelm Leibniz (1646-1716) used : for both ratio and division in 1684 in the Acta eruditorum .

Use of decimal point and comma around the world

Blue - decimal point, Lt Gr - comma,  Dk Green - both, Red - Arabic decimal separator, Gray - no data

 *Wik

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.)

The first schoolhouse west of the Alleghenies was built by a band of Moravian missionaries that had come to Ohio to establish a community to minister to the Lenape (Delaware) Indians. The band was led by David Zeisberger who believed everyone had the right to an education. He translated the Bible in the Lenape language and opened the Christian school to teach white and native children alike. School was taught in German, the Moravian native language, and the Lenape languague.

In colonial times, most schools did not teach boys and girls together. Girls from prosperious families went to seperate schools that taught home-making skills. Public schools didn't allow girls to attend. Puritans believed in teaching girls how to read so they could learn Scripture, but that was as far as their formal education would get. Educated girls were considered to not be suitable wives. Schools where blacks and native Americans attended with white children were unheard of although there were some Quaker and missionary schools that taught black and native Americans.

The Schoenbrunn School bucked all of these colonial traditions. In Moravian schools, blacks, native Americans, and girls were taught together with white boys. The Moravians believed that all children should receive an education so they could study the Bible and minister to others. Schoenbrunn School was one of the first public schools in the United States to do this.

 HHHistory.com

On this day in 1808, François Arago *escaped* from prison in Mallorca where he had been imprisoned as a spy, and started his journey back to France carrying his logbook of measurements of the meridian. After some misadventures he reached France 11 months later.

On 3 September 1806 Arago and Biot set out for Spain. They continued the task which Méchain had been undertaking on his final expedition and by 1808 they were on Mallorca, an important point which allowed the Paris meridian to be continued south of Barcelona. They had been operating in Spain at an extremely difficult time, given that they were French. Napoleon had turned his attention towards Spain and Portugal in 1807 and marched his armies through Spain to Portugal in October 1807. They conquered Portugal and occupied parts of Spain. In May 1808 Napoleon declared his brother Joseph Bonaparte as Spanish ruler and the War of Independence began. Biot and Arago must have looked extremely suspicious; two Frenchmen with sophisticated measuring instruments working on Spanish territory. Biot fled back to France but Arago remained on Mallorca, disguised as a Spaniard, trying to complete his measurements which he had recorded in a logbook. However lighting of fires on the top of Mount Galatzo was pretty suspicious so he was arrested as a spy and put in prison.

Arago managed to persuade the commander of the prison that he was a scientist, not a spy, and the commander agreed to give Arago a chance to escape. He did so on 29 July 1808 and, still carrying his precious logbook, managed to find a fishing boat heading for Algiers, which he boarded. Reaching Algiers on 3 August he went to the French consul who supplied him with a forged Austrian passport and by 16 August he was on a boat heading to Marseille. This might have been a remarkable adventure had it ended at that point, but more drama was to come. The boat on which Arago was sailing was captured while on its way to France by a Spanish warship and he was back in captivity again. Arago was held in a Spanish prison in Roses but after only a short spell the Spanish decided to send their prisoners to Palamos since the French armies were advancing through Spain. However Arago was lucky and, having been recognized by the authorities, was released an put on another boat for Marseille on 28 November.

It was not to be, however, for again Arago failed to reach his homeland. A storm blew the boat back to Bougie on the north African coast where he was captured by Muslims. After further adventures during which he persuaded his captors that he wished to convert to Islam to obtain favorable treatment, he was allowed to return to Algiers which he did overland, arriving there on 25 December. A new local leader in Algiers was opposed to the French and Arago found himself in prison waiting to be shipped off to a penal colony. However the French consul again came to his rescue and, on 21 June 1809, Arago was put, for the third time, on a ship bound for Marseille. This time he reached his destination without mishap and on 2 July 1809 he was standing on French soil.

 Arago's grave in the Père Lachaise cemetery in Paris *SAU

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

Hill was president of Antioch College from 1860 to 1862 until the Civil War forced the college to shut down; he then held the presidency of Harvard University from 1862 to 1868. *PB Notes

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

 Vulcan in a lithographic map from 1846 *Wik

In 1890, Laroy Sunderland Starrett received a U.S. patent for his micrometer screw guage (No. 433,311), which is the form still familiar and indispensible to any machinist or person measuring small objects in a physics lab. Since 1881, he had worked to improve the micrometers then existing. His design had a vernier scale, a smaller head, a locking device and a small knob extending from the barrel to enable quick rotation while closing the initial gap to the inserted object being measured. By 1899, his micrometer was available in a one-inch size for \$6.50, including a leather case. He took out many other patents for the tools he invented or improved, and established a substantial tool manufacturing business. *TiS

Yep, had a hand full of these in my day.Couldn't afford them today.

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

2015 On July 29, 2015, a 15th type of pentagon that would tile the plane was announced by Casey Mann, Jennifer McLoud, and David Von Derau of the University of Washington Bothell. In 1918, K. Reinhardt discovered five different families of convex pentagons that could tile the plane. This was the complete list until 1968, when Richard Kershner wrote about three more families of tiling pentagons. Martin Gardner wrote about the complete list of eight tiling pentagons in 1975, and then got a message from Richard James III about another type. Martin updated the readers of Mathematical Games, but then got a message from a housewife with no mathematical training, Marjorie Rice, who found four more families of tiling pentagons. In 1985, Rolf Stein found a convex pentagon that can tile the plane. Now, there is one more. *Wolfram

 *guardian.com

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

Three Nobel Prize winners in 1962: John Bardeen, Isidor Rabi, and Werner Heisenberg (left to right); the occasion is unknown (Wikimedia commons

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

1917 Henry Albert Howard Boot (29 July 1917 – 8 February 1983) was an English physicist who with Sir John Randall and James Sayers developed the cavity magnetron, which was one of the keys to the Allied victory in the Second World War. *Wik

“Harry” Boot was an English physicist who worked with John Randall developing the cavity magnetron, the microwave-generating device used in radar. This made a major contribution to winning WWII. Earlier magnetrons made in the 1920s gave low power output. By Feb 1940, advances by Randall and Boot in the design of the small-sized cavity magnetron, produced centimeter wavelengths at much higher power, which allowed radar to detect smaller objects. In turn, this more compact equipment with a smaller antenna permitted easy mobile installation of high-resolution radar in aircraft. *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.

1994 Dorothy Mary Hodgkin OM FRS (12 May 1910 – 29 July 1994), known professionally as Dorothy Crowfoot Hodgkin or simply Dorothy Hodgkin, was a British biochemist who developed protein crystallography, for which she won the Nobel Prize in Chemistry in 1964.
She advanced the technique of X-ray crystallography, a method used to determine the three-dimensional structures of biomolecules. Among her most influential discoveries are the confirmation of the structure of penicillin that Ernst Boris Chain and Edward Abraham had previously surmised, and then the structure of vitamin B12, for which she became the third woman to win the Nobel Prize in Chemistry.
In 1969, after 35 years of work and five years after winning the Nobel Prize, Hodgkin was able to decipher the structure of insulin. X-ray crystallography became a widely used tool and was critical in later determining the structures of many biological molecules where knowledge of structure is critical to an understanding of function. She is regarded as one of the pioneer scientists in the field of X-ray crystallography studies of biomolecules. *Wik

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. In addition to his formal results in mathematics, he was "deeply involved in [a] struggle against the votaries of Darwinism," 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

His most famous result is his proof, joint with John G. Thompson, of the Feit–Thompson theorem that all finite groups of odd order are solvable. At the time it was written, it was probably the most complicated and difficult mathematical proof ever completed. He wrote almost a hundred other papers, mostly on finite group theory, character theory (in particular introducing the concept of a coherent set of characters), and modular representation theory. Another regular theme in his research was the study of linear groups of small degree, that is, finite groups of matrices in low dimensions. It was often the case that, while the conclusions concerned groups of complex matrices, the techniques employed were from modular representation theory.

He also wrote the books:The representation theory of finite groups and Characters of finite groups, which are now standard references on character theory, including treatments of modular representations and modular characters.

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