## Sunday 31 July 2022

### On This Day in Math - July 31

I advise my students to listen carefully the moment
they decide to take no more mathematics courses.
They might be able to hear the sound of closing doors.

~Caballero, James

The 212th day of the year; Besides being the Fahrenheit boiling point of water at sea level, 212 produces a prime of the form k10+k9+...+k2+k+1, when k=212. Edward Shore@edward_shore sent me a note:" That number would be 184,251,916,841,751,188,170,917.")
(students might explore different values of k, and different maximum exponents to produce primes..ie when k is 2, then 26 +25+...+22+2+1 is prime

The smallest even three-digit integer, abc, such that (abc)/(a*b*c) is also prime. [ie 212/(2*1*2)= 53 ]*Prime Curios

212 is a palindrome whose square is also a palindrome, 2122= 44944. It is the last year date for which this is true. It is also a palindrome in base 3(21212) with a copy of it's base 10 representation.

And I just learned from @fermatslibrary that 212 is in a palindromic approximation for π

666/212 = 3.141509... good for four decimal places.

EVENTS

1669 Lucasian professor Isaac Barrow sent John Collins a manuscript of Newton’s De analysi and thereby Newton’s anonymity began to dissolve. It was a summary of Newton’s work on the calculus and was written after Newton saw Nicholas Mercator’s Logarithmotechnia (1668).

$ln(1+x) = x -\frac{x^2}{2} + \frac{x^3}{3} -\frac{x^4}{4}+\cdots$

Newton wrote his paper in order that he would not lose credit for his work on inﬁnite series. Collins immediately recognized Newton’s genius. Although not published until 1711, this paper led to Newton’s appointment as Lucasian professor on 29 October 1669.*VFR

1730 Goldbach proves that Fermat numbers are pairwise coprime. (Fermat had said that the he thought the numbers of the form $2^{2^n} +1$ were all prime, although he could not prove it. The first five are (n=0...4) but Euler would prove the n=5 case was not prime by factoring it. No more primes have been found after n=4, but there is no proof there can not be more. I think this story, and Goldbach's discovery, make an interesting approach to proving the primes are infinite.) He claims that 1 is the only square among the triangular numbers *Euler Goldbach Correspondence

1744   Euler to Goldbach , "All around here chess is played passionately." He then mentions a certain strong local player he had been taking lessons from, then adds, "I am winning most games with him."  Master of us all in more ways than I knew.  *S. Strogatz

1790 The U.S. Patent Office issued its ﬁrst patent to Samuel Hopkins of Vermont for his “process for making pot and pearl ashes,” whatever they are. Since George Washington and Thomas Jefferson signed Hopkins’ patent, more than 4 million have been issued. *VFR In 1790, the first U.S. patent was granted to Samuel Hopkins of Vermont for a process for making potash and pearl ashes. Potash was important as an ingredient in soap and fertilizer. The patent was granted for a term of 14 years and signed by President George Washington, who had the previous month signed the first U.S. patent statute into law on 10 April 1790. Hopkins did not get Patent with a serial No.1 as thousands of patents were issued before the Patent Office began to number them. Congress had passed the Patent Act on 10 Apr 1790. Two other patents were granted that year - one for a new candle-making process and the other the flour-milling machinery of Oliver Evans. The next year, 1791, Samuel Hopkins also was granted the first Canadian patent.*TIS
 *C. Pickover

1810  Oliver Byrne, an eccentric British Victorian mathematician, was born July 31, 1810.  Byrne is best known for publishing an edition of the first six books of Euclid’s Elements (1847), in which he used primary colors to demonstrate some of the steps in the Euclidean proofs. The result was certainly one of the most attractive and appealing textbooks of geometry ever printed. We see above the title-page with its Pythagorean theorem vignette, and below a theorem that demonstrates, geometrically, that (a + b)2 = a2 +2ab + b2.   *Linda Hall Org

1943 Ireland issued—as its ﬁrst stamp with a mathematical theme—two stamps to celebrate the centenary of the discovery of Quaternions by Sir William Rowan Hamilton. [Scott #126-7]. *VFR

1851 Gauss witnessed the opening ceremonies when the newly constructed railway from Cassel reached Gottingen. *VFR

1990The U.S. government panel approved the use of gene therapy to treat human disease. Gene therapy uses DNA to treat disease, usually by replacing a faulty gene with a healthy copy. Recent clinical studies suggest this technique holds promise for the future treatment of Parkinson’s disease. *.rsc.org

2015 The second full moon this month (the other was on the 2nd). This only happens “Once in a blue moon”—and this is the origin of the phrase. Consequently, there were be thirteen full moons this year.  The last "blue moon" was in 1985, and the next is predicted in 2018.

BIRTHS

1704 Gabriel Cramer (31 July 1704 – 4 January 1752). He is best known for “Cramer’s Rule,” a method for solving systems of simultaneous linear equations using determinants. *VFR Gabriel Cramer (31 July 1704 – 4 January 1752) was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics. In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli. He published his best known work in his forties. This was his treatise on algebraic curves, "Introduction à l'analyse des lignes courbes algébriques", published in 1750. It contains the earliest demonstration that a curve of the n-th degree is determined by n(n + 3)/2 points on it, in general position. He edited the works of the two elder Bernoullis; and wrote on the physical cause of the spheroidal shape of the planets and the motion of their apsides (1730), and on Newton's treatment of cubic curves (1746). He was professor at Geneva, and died at Bagnols-sur-Cèze.*Wik

 *Geeks For Geeks

1712 Samuel König (July 31, 1712, Büdingen – August 21, 1757, Zuilenstein near Amerongen) was a German mathematician who is best remembered for his part in a dispute with Maupertuis over the Principle of Least Action.*SAU

1718 John Canton (31 July 1718 – 22 March 1772) British physicist and teacher, born Stroud, Gloucestershire. He made a number of minor discoveries in physics and chemistry. As a result of preparing artificial magnets in 1749 he was elected to the Royal Society. In 1762, he demonstrated that water was slightly compressible. He invented a number of devices in connection with electricity. His notable work, between 1756 and 1759, was to record that on days when the aurora borealis was particularly bright, a compass needle behaved with more irregularity than usual. Thus he was the first to record this as an electromagnetic phenomenon for what is now known to be a magnetic storm.*TIS

1790 The U.S. Patent Office issued its first patent to Samuel Hopkins of Vermont for his “process
for making pot and pearl ashes,” whatever they are. Since George Washington and Thomas
Jefferson signed Hopkins’ patent, more than 4 million have been issued. *VFR
Hopkins was a Philadelphia Quaker who later moved to New Jersey. He was living in Philadelphia when the patent was granted. The patent was signed by President Washington, Attorney General Randolph and Secretary of State Jefferson. The original document is in the collections of the Chicago Historical Society.

Potash was America’s first industrial chemical. It is an impure form of potassium carbonate mixed with other potassium salts. Until the 1860s it was solely derived from the ashes of hardwood trees and certain other plants. Potash was a leading industrial alkali from antiquity until the close of the nineteenth century, when it was finally abandoned for most uses in favor of soda (sodium carbonate). It was essential for making soap and glass, dyeing fabrics, baking, and making saltpeter for gunpowder. Today it is principal ingredient in fertilizers. *SUITER SWANTZ

1826 Daniel Friedrich Ernst Meisse mathematical work covers number theory, work on Möbius inversion and the theory of partitions as well as work on Bessel functions, asymptotic analysis, refraction of light and the three body problem. *SAU

1843 Friedrich Robert Helmert (July 31, 1843 – June 15, 1917) German geodesist and an important writer on the theory of errors.
From 1887 Helmert was professor of advanced geodesy at the University of Berlin and director of the Geodetic Institute.
Helmert received many honours. He was president of the global geodetic association of "Internationale Erdmessung", member of the Prussian Academy of Sciences in Berlin, was elected a member of the Royal Swedish Academy of Sciences in 1905, and recipient of some 25 German and foreign decorations. *TIA

1858 Richard Dixon Oldham (31 July 1858 – 15 July 1936) Irish geologist and seismologist who discovered evidence for the existence of the Earth's liquid core (1906). In studying seismograms of great 1897 Indian Earthquake he identified P (primary) and S (secondary) waves. It is interesting that he did not get a clue to the presence of the core from the S waves, which are actually incapable of being transmitted through the liquid of the outer core. (The liquid core does not transmit the shear wave energy released during an earthquake.) Rather he noted the existence of a shadow zone in which P waves from an earthquake in the opposite hemisphere of the earth failed to appear*TIS

1863 George Abram Miller (31 July 1863 – 10 February 1951) was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance.
Miller was born in Lynnville, Lehigh County, Pennsylvania, and died in Urbana, Illinois.*Wik

1923 Joseph B. Keller (born July 31, 1923, Paterson, New Jersey) is an American mathematician who specializes in applied mathematics. He is best known for his work on the "Geometrical Theory of Diffraction" (GTD).
He worked on the application of mathematics to problems in science and engineering, such as wave propagation. He contributed to the Einstein-Brillouin-Keller method for computing eigenvalues in quantum mechanical systems.
In 1988 he was awarded the U.S. National Medal of Science, and in 1997 he was awarded the Wolf Prize by the Israel-based Wolf Foundation. In 1996, he was awarded the Nemmers Prize in Mathematics.*Wik

1927 F. E. Browder born. Worked in Nonlinear monotone operators and convex sets in Banach spaces. and more.

1943 Ireland issued—as its first stamp with a mathematical theme—two stamps to celebrate the centenary of the discovery of Quaternions by Sir William Rowan Hamilton. [Scott #126-7].*VFR

------------------------------------------------------------------------------------------------------------------
1945 John O'Connor (31st July 1945 in Luton, Bedfordshire, England.- )
Lists his Research interests A lapsed topologist, I am interested in Computational Algebra.
I am interested in the History of Mathematics and at present am supervising two research students in this area. * His Personal web page

DEATHS

1726 Nikolaus II Bernoulli died (February 6, 1695, Basel, Switzerland – July 31, 1726, St. Petersburg, Russia). *VFR Nicolaus(II) Bernoulli was the favourite of three sons of Johann Bernoulli. He made important mathematical contributions to the problem of trajectories while working on the mathematical arguments behind the dispute between Newton and Leibniz.*SAU

1784 Denis Diderot died. (October 5, 1713 – July 31, 1784) was a French philosopher, art critic, and writer. He was a prominent persona during the Enlightenment and is best-known for serving as co-founder and chief editor of and contributor to the Encyclopédie. *Wik

1896 Ludwig Christian Wiener (7 December 1826 Darmstadt – 31 July 1896 Karlsruhe) was a German mathematician, physicist and philosopher, known for his explanation of Brownian motion , which identified him as a skillful experimenter. He mainly dealt with geometry.*Wik

1913 John Milne (30 December 1850 – 31 July 1913) English seismologist who invented the horizontal pendulum seismograph (1894) and was one of the European scientists that helped organize the seismic survey of Japan in the last half of the 1800's. Milne conducted experiments on the propagation of elastic waves from artificial sources, and building construction. He spent 20 years in Japan, until 1895, when a fire destroyed his property, and he returned home to the Isle of Wight. He set up a new laboratory and persuaded the Royal Society to fund initially 20 earthquake observatories around the world, equipped with his seismographs. By 1900, Milne seismographs were established on all of the inhabited continents and he was recognized as the world's leading seismologist. He died of Bright's disease*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

## Saturday 30 July 2022

### On This Day in Math - July 30

I have created a new universe from nothing.

~Janos Bolyai

The 211th day of the year; 211 is a primorial prime,(a prime that is one more, or one less than a primorial  can you find the next larger (or smaller) primorial prime?

211 is also the sum of three consecutive primes (67 + 71 + 73)...

There are 211 primes on a 24-hour digital clock. (00:00 - 23:59) *Derek Orr @ Derektionary

211 is the 4th** Euclid number: 1 + product of the first n primes.(after Euclid's method of proving the primes are infinite. most Euclid numbers, unlike 211, are not themselves prime, but are divisible by a prime different than any of the primes in the product n#) (**some would call it the fifth since Euclid seemed to consider 1 as a unit as similar to the primes.)
EVENTS

1738 Euler sends a letter to John Bernoulli with the solution to a question from Danial Bernoulli regarding isoperimetric curves, particularly the  one for which the integral of rm gave a maximum or minimum.

1859 Bernhard Reimann is appointed full professor at Gottingen, succeeding his two former teachers, Gauss and Dirichlet. He also is allowed to occupy Gauss' apartments at the observatory. *John Derbyshire, Prime Obsession, pg 135

In 1898, Corn Flakes were invented by William Kellogg. At Battle Creek Sanitarium, Sanitarium superintendent, Dr. John Harvey Kellogg and Will Keith Kellogg, his younger brother and business manager, invented many grain-based foods, including a coffee substitute, a type of granola, and peanut butter to provide patients a strict nutritious diet. In 1894 they unintentionally invented a flaked cereal process based on wheat. By 1898, W.K. Kellogg had developed the first flaked corn cereal. Patients enjoyed the cereals and wanted more to take home. In 1906, the Battle Creek Toaster Corn Flake Company was founded by W.K. Kellogg.*TIS

1907 The Axiom of Choice is usually given as created by Zermelo in 1908, presumably because that was the year it appeared in Mathematische Annalen, but the date on the actual paper is "Chesières, 30 July 1907.". The paper contains, "AXIOM VI. (Axiom of choice). If T is a set whose elements all are sets that are different from 0 and mutually disjoint, its union "union of T" includes at least one subset S1 having one and only one element in common with each element of T." [The original German read "Axiom der Auswahl".]
Ernst Zermelo used the Axiom of Choice to prove that every set can be well-ordered on a paper of 1904, but did not use the name "Axiom of Choice". *Jeff Miler, Earliest Known Uses of Some of the Words of Mathematics

1918 Richard Courant sat down with Ferdinand Springer and signed a contract for the series of books now famous as the “Yellow Series.” *Constance Reid, Courant in Gottingen and New York, p. 72

1971 Apollo 15 mission became the fourth mission to land on the moon when the Falcon lunar lander touched down. This mission allowed the astronauts to spend more time on the surface of the moon. The lander stayed three days on the surface and the crew conducted over 18 hours of outside work. They also were aided for the first time by a lunar rover vehicle.*Science Today

BIRTHS

1857 Thorstein Bunde Veblen, (July 30, 1857 – August 3, 1929) was an American economist and sociologist, and a leader of the so-called institutional economics movement. Besides his technical work he was a popular and witty critic of capitalism, as shown by his best known book The Theory of the Leisure Class (1899).

1863 Henry Ford (July 30, 1863 – April 7, 1947) American inventor and car manufacturer, born in Dearborn, Mich. Ford first experimented with internal combustion engines while he was an engineer with the Edison Illuminating Company. He completed his first useful gas motor on 24 Dec 1893. The Quadricycle, he designed made its first road test on 4 Jun 1896. In 1903 the Ford Motor Company was incorporated. By 1908, Ford was manufacturing the low cost, reliable Model T, while continuing to revolutionize his industry. Ford introduced precision manufactured parts designed to be standardized and interchangeable parts. In 1913, production was increased using a continuous moving assembly line. By 1918, half of all cars in America were Model T's.*TIS

1878 Joel Stebbins (July 30, 1878 – March 16, 1966) was an American astronomer who pioneered photoelectric photometry in astronomy.
He earned his Ph.D at the University of California. He was director of University of Illinois observatory from 1903 to 1922 and the Washburn Observatory at the University of Wisconsin-Madison from 1922 to 1948. After 1948, Stebbins continued his research at Lick Observatory until his final retirement in 1958.
Stebbins brought photoelectric photometry from its infancy in the early 1900s to a mature technique by the 1950s, when it succeeded photography as the primary method of photometry. Stebbins used the new technique to investigate eclipsing binaries, the reddening of starlight by interstellar dust, colors of galaxies, and variable stars.
Stebbins received the following awards:

Rumford Prize of the American Academy of Arts and Sciences (1913)
Henry Draper Medal of the National Academy of Sciences (1915)
Bruce Medal of the Astronomical Society of the Pacific (1941)
Gold Medal of the Royal Astronomical Society (1950)
Henry Norris Russell Lectureship of the American Astronomical Society (1956)

The Lunar crater Stebbins and the asteroid 2300 Stebbins are named in his honor. *TIA

1887 Felix Andries Vening Meinesz (The Hague July 30, 1887 - Amersfoort August 10, 1966) was a Dutch geophysicist and geodesist who was known for his measurements of gravity at sea for which he devised the Vening Meinesz pendulum apparatus with comparable accuracy as on land. Starting in 1923 he conducted several global gravity surveys on voyages on submarines, particularly to and in the Indonesian Archipelago. He detected strong gravity anomaly belts running parallel to the Indonesian deep sea trenches. He explained these Meinesz belts as sites of downbuckling of the Earth's crust. He introduced the concept of regional isostasy taking flexure of an elastic crust into account. He also contributed to physical geodesy: The Vening Meinesz formula connects the deviation of the vertical from the plumbline to gravity anomalies. *TIS

1888 Vladimir Zworykin (July 29 [O.S. July 17] 1888 – July 29, 1982) was born in Russia. After emigrating to Pittsburgh, Zworykin took a job at Westinghouse Electric Corp., where in 1923 he filed a patent for the iconoscope, the first television transmission tube and a technology that was to become of interest to early computer designers. With a later invention, the kinescope, Zworykin was able to create the first all-electric television system. Zworykin took the technology to RCA in 1929, where he continued his work and earned the title "father of television.*CMH

DEATHS

1762 William Braikenridge (1700; 30 July 1762 in London, England) was an English clergyman who worked on geometry and discovered independently many of the same results as Maclaurin.*SAU

1832 French chemist John Antoine Chaptal He authored the first book on industrial chemistry, and coined the name "nitrogen". Chaptal also helped improve the technology used to manufacture sulfuric acid, saltpetre for gunpowder, beetroot sugar and wine, amongst other things. *RSC.Org

1978 Rufus Bowen (23 February 1947 - 30 July 1978) worked on dynamical systems. Rufus died of a cerebral hemorrhage at the age of 31. *SAU
In 1970, Bowen completed his doctorate in Mathematics at Berkeley under Stephen Smale, and joined the faculty as assistant professor in that year. At this time he began calling himself Rufus, the nickname he had been given because of his red hair and beard.  He was an invited speaker at the 1974 International Mathematical Conference in Vancouver, British Columbia.He was promoted to full professorship in 1977.
Bowen's mature work dealt with dynamical systems theory, a field which Smale, Bowen's dissertation advisor, explored and broadened in the 1960s.

1985 Julia Robinson (December 8, 1919 – July 30, 1985) died of leukemia. After receiving her Ph.D. in 1948 under the direction of Alfred Tarski, she began work on Hilbert’s tenth problem, the problem which occupied most of her professional life.*VFR She also worked on computability, decision problems and non-standard models of arithmetic. *SAU Her sister was Constance Reid who wrote biographies of several mathematicians and several popular math books.

2002 Dr. Lyle B. Borst, (Nov 24, 1912 - July 30, 2002) was a nuclear physicist who helped build Brookhaven National Laboratory's nuclear reactor and was an early member of the Manhattan Project.
In 1950, Dr. Borst led the construction of the Brookhaven Graphite Research Reactor, which was the largest and most powerful reactor in the country and the first to be built solely for research and other peacetime uses of atomic energy.
Within the first nine months of operating the reactor, Dr. Borst announced that it had produced a new type of radioactive iodine, which is used in treating thyroid cancer.
In 1952, based on studies of new types of atomic nuclei created in the reactor, Dr. Borst helped explain the mystery behind giant stars, known as supernovae, that burst with the energy of billions of atomic bombs and flare for several years with the brilliance of several million suns.
Dr. Borst found that beryllium 7, an isotope of beryllium that does not occur naturally on earth, is formed in supernovae by the fusion of two helium nuclei. The fusion takes place after the star has used up its hydrogen supply. This reaction absorbs huge quantities of energy, causing the star to collapse in the greatest cosmic explosion known. *NY Times obit.

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

## Friday 29 July 2022

### 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

There are only two triangular numbers that are the product of consecutive integers that are year days.  210 = 14 x 15.  The other is much smaller and should be easier to find.  (Well???)

210 is also the product of three consecutive integers, 5 x 6 x 7 = 210.  There are only six of these in total, and three of them are year days.  210 is again the largest of them.

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.
See More Math Facts for every Year Day here.

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

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

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.

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.[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

## Thursday 28 July 2022

### The Tower of Hanoi And two clever solutions

Back awhile, in a blog about Fibonacci, I mentioned that Edouard Lucas had created the "Tower of Hanoi" game and received comments and mail from people who thought I must be mistaken because the game was "really old". Turns out, it really isn't, but just the creation of a master mathematical story teller. Here are some notes about the man, and the history of the Towers of Hanoi from my Math Words Etymology page.

Also, you can find a java applet to play the game at this site... and if you've never done it (where HAVE you been?) don't start with all 12 discs, that takes 4095 moves to solve (see below).

The Lucas sequence is similar to the Fibonacci sequence. The Lucas sequence is given by {1, 3, 4, 7, 11, 18, ...} . Each term is the sum of the two previous numbers, as in the Fibonacci sequence. Just as in the Fibonacci sequence, the limit of the ratio of consecutive terms is the Golden Ratio. The Lucas numbers can also be constructed from the Fibonacci numbers by the function Ln = Fn-1 + Fn+1, thus the fifth Lucas number, 11, is the sum of the fourth and sixth fibonacci numbers (3+8).

The sequence is named for Edouard Lucas, a French mathematician of the later half of the nineteenth century. He used his sequence and the Fibonacci sequence to develop techniques for testing for prime numbers. Lucas is also remembered for his unusual death, caused by a waiter dropping a plate which shattered sending a piece of plate into his neck. Lucas died several days later from a deadly inflamation of the skin and subcutaneous tissue caused by streptococcus. The disease, officially listed as erysipelas (from the Greek for "red skin") was more commonly known as "Saint Anthony's Fire".

Lucas was also the creator of a popular puzzle called The Tower of Hanoi in 1883. You can see the original box cover above. Note that the author on the box cover is Professor N. Claus de Siam, an anagram of Lucas d' Amiens (his home). The professors college, Li-Sou-Stian, is also an anagram for "Lycee Saint-Louis" where Lucas worked.

France was building an Empire in Indochina (the peninsula stretching from Burma to Viet Nam and Malaysia) and the "mysterious East" was a very fashionable topic. Lucas created a legend (some say he embellished an existing one, but I can find no earlier record of one) of monks working to move 64 gold disks from one of three diamond points to another after which the world would end. The solution for a tower of n disks taks 2n -1 moves, so the game often had less than the 64 disks of the legend. Solving the 64 disks at one move a second would require 18,446,744,073,709,551,615 seconds, which at 31,536,000 seconds a year would take 584 Billion years. (and you thought Monopoly took a long time to finish).  The reference in his instructions to Buddhist monks in a temple in Bernares(Varanasi),  India seems, even now, to make people believe there was such an activity taking place.  Varanasi is considered the holiest of the seven sacred cities (Sapta Puri) in Hinduism, and Jainism, and is important to Buddhism because it was in nearby Sarnath that Buddha gave his first teaching after attaining enlightenment, in which he taught the four noble truths and the teachings associated with it. There is a Buddhist temple there with many relics of the Buddha, but so far as I can find, no monks moving golden disks on needles.

Students/teachers interested in further explorations of the history and math of the famous game should visit the work of Paul K Stockmeyer who maintains the page with the cover illustration mentioned above, and his Papers and bibliography on the Tower of Hanoi problem.

Lucas developed several other mathematical games of his on, including the well known children's pastime of dots and boxes (which he called  La Pipopipette), which on large boards is still essentially unsolved, I believe.  He also (probably) invented a Mancala type game called Tchuka Ruma.

Lucas is also remembered for suffering an unusual death.  At a banquet in 1891  a waiter dropped a dining plate and one of the pieces cut Lucas on the neck and cheek. Within a week he was dead from what was called the "Holy Fire" or St Anthony's Fire, a form of septicemia.

A while after I wrote the above, I learned a little more, and so:

Just browsing through Wikipedia, and they show a solution to the Towers of Hanoi puzzle that I had never seen using a ruler as a solution key.

If you have been off planet for the last 130 years and don't know the Towers problem, you can play online here. You might try that first, and set the number of discs to 6 so that it matches the solution shown below.

And for those who know the game but just want to see how a ruler is used, here is the graphic.

For any move, just move the disc whose size compares to the marks on the ruler. For instance the first five marks on a ruler marked in 32nds would be 1/32, 1/16/ 3/32, 1/8, 5/32.... The denominators tell you which disk to move. The largest denominator (smallest scale) goes with the smallest disc, etc. If you then apply two fundamentals of any solution, always move the smallest disc From rod A, to B to C and back to A in a cycle, and never put a bigger disc on a smaller one, then you have a solution... That's easier than Gray codes isn't it.

Why have I never encountered this before? The connection was made in 1956 by Donald W. Crow, in relation to traversing the vertices of a cube in n-dimensions[ D. W. Crowe, The n-dimensional cube and the tower of Hanoi, Amer. Math. Monthly, 63 (1956), 29-30.]

POSTSCRIPT:::: For another really insightful solution (maybe the best of them all) See the comment by Jeffo....Thanks guy, why don't I see ideas like that?

Jeffo said...

If the rods are placed in a circular arrangement instead of linear, then a correct solution will involve always moving the smallest disk one rod clockwise every other move. The alternate moves are forced.

### 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 (after the town where he lived) 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 209, 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
[The earliest use of Colombian number I can find is by B. Recaman (1974). "Problem E2408". Amer. Math. Monthly 81. Would love to know if there are earlier uses.]

209 is the maximum number of pieces that can be made by cutting an annulus with 19 straight cuts.

As far as I know, there are only two three digit numbers so that abc^2 = uvwxyz and uvw+xyz = abc.  These are called three digit Kaprekar numbers.  209 is involved in each case.  The two numbers are 297^2 = 88209, with 88 + 209 = 297 and 703^2 = 494209 where 494 + 209 = 703

See More Math Facts for every Year Day here

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

1984 The town of Eighty-four Pennsylvania celebrated it's centennial on this day.

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

 maser components at amhistorymuseum HT toBen Gross ‏@bhgross144
1915 Charles Hard Townes (July 28, 1915 – January 27, 2015) was an American Nobel Prize-winning physicist and educator. Townes was known for his work on the theory and application of the maser, on which he got the fundamental patent, and other work in quantum electronics connected with both maser and laser devices. He shared the Nobel Prize in Physics in 1964 with Nikolay Basov and Alexander Prokhorov.
In a career that spanned six decades, Dr. Townes developed radar bombing systems and navigation devices during World War II, advised presidents and government commissions on lunar landings and the MX missile system, verified Einstein’s cosmological theories, discovered ammonia molecules at the center of the Milky Way, and created an atomic clock that measured time to within one second in 300 years. He died at the age of 99 in Berkeley, California*Wik *NY Times

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

## Wednesday 27 July 2022

### 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.

208 is the number of paths from (0,0) to (7,7) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps.

208 = 2^4 x 13  and if you play the four-fours game, 208 = 4^4 - 4! - 4!

208 can be written as the sum of two squares in only one way, 12^2 + 8^2

208 is an abundant number, the proper divisors total 226(more than 208)
.
See more Math Facts for Every Year Day here

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

1861 The Athenaeum magazine carried a review of Charles Dodgson's pamphlet entitled The Formula of Plane Trigonometry in which he suggested new symbols for the six basic trig functions. The reviewer was not convinced.

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

1905 A Karl Pearson letter appears in Nature asking for assistance on a problem"of considerable interest” about random walks (based on a question in a letter he had received fro m Sir Ronald Ross, who  had discovered mosquitoes as the source of malaria spreading, without mentioning him by name),  Two days later Lord Rayleigh wrote the periodical to inform them he had solved the problem and posted results in 1880 in Phil. Mag..  Pearson's response launched the common name for random walk used for many years, Drunkards Walk,  "the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!” *Jordan Ellenberg , Shape

1936 Einstein writes to John Tate, editor of the Physical Review angrily withdrawing a paper that he had submitted for publication but had been rejected after peer review. Einstein and Rosen's paper claimed that gravitational waves did not exist. It was Einstein who introduced gravitational waves in his theory of general relativity in 1916, within a few months of finding the correct form of the field equations for it. However by 1936 he had changed his mind, and wrote to his friend, Max Born, "Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist,..."
Later he would submit the paper again, but then drastically revise the conclusions before publication. Einstein simply explained why “fundamental” changes in the paper were required because the “consequences” of the equations derived in the paper had previously been incorrectly inferred. The referee of the paper, it is now known, was relativist Howard Percy Robertson. He was on sabbatical at Caltech. When he returned to Princeton he struck up a friendship with Einstein’s then newly arrived assistant Infeld. Robertson then convinced Infield of the problems with the paper he had re-submitted, and after Infield talked to Einstein, the paper was revised. It seems that Einstein had never read the referee's comments.
*physicstoday

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

2007 Ralph Asher Alpher's belated recognition for his work on the "Big Bang" process. In 2005 Alpher was awarded the National Medal of Science. The citation for the award reads "For his unprecedented work in the areas of nucleosynthesis, for the prediction that universe expansion leaves behind background radiation, and for providing the model for the Big Bang theory." The medal was presented to his son Dr. Victor S. Alpher on July 27, 2007 by President George W. Bush, as his father could not travel to receive the award. *Wik

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