Saturday 9 July 2022

On This Day in Math - July 9

Epitaph (by Kepler, for himself)
Mensus eram coelos, nunc Terrae metior umbras.
Mens coelestis erat, corporis umbra jacet.

I used to measure the Heavens, now I measure the shadows of Earth.
The mind belonged to Heaven, the body's shadow lies here.

The 190th day of the year; 190 is the largest number with only distinct prime Roman numeral palindrome factors that is a Roman numeral palindrome (190 = CXC = II * V * XIX). *Prime Curios

190 is a Harshad or Niven number divisible by the sum of its digits. In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-Harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver.

190 is the 19th triangular number, the sum of the first 19 integers. A nice problem relating triangular numbers to magic squares was asked in 1941 in the American Mathematical Monthly, posed by Royal Vale Heath, widely known for creating ingenious mathematical puzzles: "What is the smallest value of n for which the n^2 triangular numbers 0, 1, 3, 6, 10, . . . n^2(n^2 – 1)/2 can be arranged to form a magic square?" An explanation, and answer is in this blog by Ivars Peterson

190 is also a palindrome in base 4(2332) 190 is a Harshad or Niven number divisible by the sum of its digits. In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-Harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver.

190 is the sum of five consecutive squares, 190 = 4^2 + 5^2 + 6^2 + 7^2 + 8^2

190 is the 19th triangular number, the sum of the first 19 integers. A nice problem relating triangular numbers to magic squares was asked in 1941 in the American Mathematical Monthly, posed by Royal Vale Heath, widely known for creating ingenious mathematical puzzles: "What is the smallest value of n for which the n^2 triangular numbers 0, 1, 3, 6, 10, . . . n^2(n^2 – 1)/2 can be arranged to form a magic square?" An explanation, and answer is in this blog by Ivars Peterson

190 is a Happy Number. Summing the squares of the digits, and iterating, you eventually arrive at 1. It takes only four iterations.

190 = 121 + 49 + 16 +4 = 100+81+9

EVENTS

1595 Kepler gets the inspiration for his first model of the universe.. when the 23 year old Kepler, while teaching, made the first of a profound series of discoveries. Kepler fully elaborates this discovery in his Mysterium Cosmographicum, published less than a year later. It appeared to him that the respective radii of the orbits of the planets corresponded to the lengths determined by a specific sequence in which the five regular solids were placed within one another, with a sphere separating each solid from the other. The sphere (orbit) of Saturn enveloped a cube which in turn enveloped another sphere, the orbit of Jupiter. This circumscribed a tetrahedron, a sphere (the orbit of Mars), a dodecahedron, a sphere (the orbit of earth), an icosahedron, a sphere (the orbit of Venus), an octahedron, and the smallest sphere (the orbit of Mercury). The idea was the main theme of his Mysterium cosmographicum (1596). *Dave Richeson, Euler’s Gem
On that day, while standing at the blackboard drawing a geometrical figure for his class, Kepler had an epiphany. He believed it was a divine inspiration. Kepler had drawn a triangle with a circle circumscribed around it, which meant that each of the triangle's corners touched the rim of the circle. Then he inscribed another circle inside the triangle, which meant that the center of each side of the triangle touched the inner circle.
When Kepler stepped back and looked at what he had drawn, he realized with a shock that the ratios of the two circles were the same as the ratios of the orbits of Saturn and Jupiter. And with that realization, inspiration struck. Jupiter and Saturn were the outermost planets of the solar system, and the triangle was the simplest polygon. Kepler wondered whether you could fit the orbits of the other planets around other geometric figures, and tried his best inscribing circles in squares and pentagons. But the planetary orbits refused to fit.
Then Kepler had a second epiphany. The solar system was three dimensional – so why would he think that its governing pattern would be found in two dimensional figures? Kepler turned to three dimensional objects, and found his answer in the five perfect solids. A perfect solid is a three dimensional figure, such as a cube, whose sides are all identical. Conveniently for Kepler, there are only five perfect solids: the tetrahedron (which has four triangular sides), cube (six square sides), octahedron (eight triangular sides), dodecahedron (twelve pentagonal sides), and icosahedron (twenty triangular sides). Each perfect solid can be inscribed in and circumscribed around a sphere. *Spark Notes

1714 Longitude Act receives royal assent. The Longitude Act offered rewards of up to £20,000 for a method of finding longitude at sea to within half a degree (equivalent to 2 minutes of time) after a six-week voyage to the West Indies. Smaller rewards were available for methods achieving lesser accuracy. The Act also nominated a number of Commissioners of Longitude, later known as the Board of Longitude, to assess submissions and decide on rewards. They included the Astronomer Royal at Greenwich and other scientific, maritime and political leaders. The Commissioners could also grant smaller sums of money to help bring promising ideas to sea-trial. *Richard Dunn ‏@Lordoflongitude

1743 Euler, in a letter to Goldbach, He had described to Goldbach a manor by which numbers of the form $n^2 + 1$ might be divisible. On Oct 28, 1752 Euler Published a paper listing the 161 numbers less than 15,000 for which $n^2+1$ is a prime. He also listed eight numbers for which $n^4 + 1$ is a prime; {1, 2, 4, 6, 16, 20, 24, and 34}. *L. E. Dickson, History of the Theory of Numbers

1814 Gauss made the 146th and last entry in his scientiﬁc diary. He observed a connection between biquadradic residues and the lemniscate functions. This has become the most famous entry in the diary as it led to the Weil conjectures. See Gray, Expositions Mathematicae, 2(1984), 97–130. *VFR

 Lemniscate of Bernoulli

1857 Weierstrass, in his inaugural speech at the Berlin Academy, stated that mathematics occupies an especially high place because only through its aid can a truly satisfying understanding of natural phenomena be obtained. *VFR

1912 A letter from U.S. Department of Agriculture, Central Office of the Weather Bureau, Washington, D.C. to Dr. Walker Abbe, in Cleveland, "Enclosing some papers which 'will seem childish' but 'present the one great need of our sciences -- a school & laboratory & class of enthusiastic physics-mathematics students specially devoted to the earth's atmosphere'. He adds 'May the time soon come when you may be called home to be the leader of such men'." *Imperial College & Science Museum Libraries

1917 This image of a solar flare taken from Mt Wilson observatory shows a solar flare 140,000 miles high. The white dot is image of the Earth drawn to scale. I came across this at a site called thegildedcentury .  You can see a streaming image of current Sun and Solar flares here

1941 British cryptologists break the secret code used by the German army to direct ground-to-air operations on the Eastern front.
British experts had already broken many of the Enigma codes for the Western front. Enigma was the Germans' most sophisticated coding machine, necessary to secretly transmitting information. The Enigma machine, invented in 1919 by Hugo Koch, a Dutchman, looked like a typewriter and was originally employed for business purposes. The Germany army adapted the machine for wartime use and considered its encoding system unbreakable. They were wrong. The Brits had broken their first Enigma code as early as the German invasion of Poland and had intercepted virtually every message sent through the occupation of Holland and France. Britain nicknamed the intercepted messages Ultra.
Now, with the German invasion of Russia, the Allies needed to be able to intercept coded messages transmitted on this second, Eastern, front. The first breakthrough occurred on July 9, regarding German ground-air operations, but various keys would continue to be broken by the Brits over the next year, each conveying information of higher secrecy and priority than the next. (For example, a series of decoded messages nicknamed "Weasel" proved extremely important in anticipating German anti-aircraft and antitank strategies against the Allies.) These decoded messages were regularly passed to the Soviet High Command regarding German troop movements and planned offensives, and back to London regarding the mass murder of Russian prisoners and Jewish concentration camp victims. *History.com

1953 France issued a stamp picturing Gaspard Monge. [Scott #279].
(Does anyone else think he looks like Marlon Brando??)

1981 Nintendo releases the arcade game Donkey Kong featuring the debut of Mario. *Wik

1982  Disney released Tron, the first mainstream film to use extensive computer-generated graphics and special effects. Starring Jeff Bridges, the film also had a computer-related plot in which a programmer is transported into a computer to fight a program called Master Control and replace it with the more reliable Tron system.

1996 With the satellite SOHO, they discover that solar flares causes sun quakes. *NSEC Scientists have shown for the first time that solar flares produce seismic waves in the Sun's interior that closely resemble those created by earthquakes on our planet. Dr. Alexander G. Kosovichev, a senior research scientist from Stanford University, and Dr. Valentina V. Zharkova from Glasgow (United Kingdom) University found the tell-tale seismic signature in data on the Sun's surface collected by the Michelson Doppler Imager onboard the Solar and Heliospheric Observatory (SOHO) spacecraft immediately following a moderate-sized flare on July 9, 1996. "Although the flare was a moderate one, it still released an immense amount of energy," said Dr. Craig Deforest, a researcher with the SOHO project. "The energy released is equal to completely covering the Earth's continents with a yard of dynamite and detonating it all at once."

2004 The day after a transit of Venus occurred as predicted by Horrocks using William Crabtree's measures of the size of Venus and the Earth-Sun distance, a commemorative street nameplate in memory of William Crabtree was unveiled at the junction of Lower Broughton Road and Priory Grove which marks the northern boundary of Crabtree Croft where he observed the transit. *Wik

2006 Tesla Memorial at Canadian side of Niagara Falls unveiled. The bronze statue by Les Drysdale depicts Tesla atop an AC motor, in commemoration of Tesla's engineering achievements at Niagara Falls. The marker at the site credits Tesla with the " "world's first hydroelectic power system". Since this power plant suppossedly "the first power reached Buffalo at midnight, November 16, 1896" this will surprise folks who are aware that Thomas Edison's electrical power plant in New York was in 1882. (What Tesla did was momentus, whether it was first or not).

BIRTHS

1819  Elias Howe, Jr., an American inventor, was born July 9, 1819. Working in a textile factory as a young man, and then as a mechanic and machinist, he got the idea of building a sewing machine when he was just 20 years old, began working seriously on the project when he was 24, and within two years, in 1845, he had a working model. His sewing machine was not the first in the United States.  When he received his patent in 1846, it was the fifth sewing machine patent granted by the U.S. Patent Office.  His machine used an eye-pointed needle, but Howe did not invent that, and it was not a part of his patent application.  His machine sewed with the cloth held vertically, and the curved needle flicked back and forth, sideways.  Others had machines with the cloth held horizontally and the needle moving vertically, up and down.  So why is Howe generally credited, even by historians of technology, with inventing the sewing machine? Primarily because his machine actually worked, unlike the many others that never got beyond the patent stage.  And because his machine used an ingenious back-and-forth shuttle, fed by a bobbin, that produced a lockstitch, rather than the chain-stitch produced by most of his predecessors.  A chain-stitched seam can unravel; a lockstitch will not.  Moreover, Howe understood all the little things that had to go right for a machine to sew effectively. He understood, for example, that the thread that feeds the needle must have its tension relaxed when the needle penetrates the cloth, so that a loop can form and the shuttle can pass through, and then the tension has to tighten when the needle withdraws, to set the knot.  His machine did all the things that were actually necessary to sew two pieces of cloth together, and it was the first to do so in a manner that could be commercialized.  *Linda Hall Org

1845 Birthdate of George Howard Darwin (9 July 1845 – Cambridge, 7 December 1912), ﬁfth child of the evolutionist Charles Darwin. After graduating second wrangler and Smith’s prizeman at Cambridge in 1866 he studied law before settling down to his life work in mathematical astronomy. He addressed the Fifth International Congress of Mathematicians at Cambridge in 1912 on his work on the three body problem. *VFR  He championed a theory (no longer accepted) that the Moon was once part of the Earth, in what is now the Pacific Ocean. His was the first mathematical analysis of the evolution of Earth's Moon. He suggested that since the effect of the tides has been to slow the Earth's rotation and to cause the Moon to recede from the Earth, then by extrapolating back 4.5 billion years ago the Moon and the Earth would have been very close, with a day being less than five hours. Before this time the two bodies would actually have been one, until the Moon was torn away from the Earth by powerful solar tides that would have deformed the Earth every 2.5 hours. *TIS

1855 Spiridon Gopcevic (July 9, 1855 – ? 1928) was a Serbian astronomer and historian. He is also known by his pen name of Leo Brenner. In 1893 he founded Manora Observatory on Mali Lošinj. This observatory was named for his wife, a wealthy Austrian noblewoman. At this observatory, Spiridon used the 17.5cm refractor telescope at the observatory to make observations of Mars, the rings of Saturn, and other planets. However he would eventually close the observatory in 1909 due to financial problems.
From 1899 until 1908 he was the founder and editor of the Astronomische Rundschau, a popular scientific journal. He spent several years in America before returning to Europe and editing an army journal in Berlin during the war. The circumstances of his death are somewhat uncertain, but he appears to have been impoverished.
The crater Brenner on the Moon was named after him (based on his nom de plume) by his friend Phillip Fauth. A new observatory was built on Mali Lošinj in 1993, and was named "Leo Brenner". *Today in Astronomy

1911 John Archibald Wheeler (July 9, 1911 – April 13, 2008) was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission. One of the later collaborators of Albert Einstein, he tried to achieve Einstein's vision of a unified field theory. He is also known for having coined the terms black holequantum foam and wormhole and the phrase "it from bit". For most of his career, Wheeler was a professor at Princeton University and was influential in mentoring a generation of physicists who made notable contributions to quantum mechanics and gravitation.*Wik  I love this Wheeler quote, "In any field, find the strangest thing and then explore it."  Thanks to

1925 Bernice (Trimble) Steadman (July 9, 1925, Rudyard, Michigan – March 18, 2015, Traverse City, Michigan) was an American aviator and businesswoman. She was one of thirteen women chosen to take the same tests as the astronauts of the Mercury 7 during the early 1960's. The group later became known as the Mercury 13. However, Steadman and the other twelve women in the program were denied the opportunity to become astronauts due to their gender.[Steadman, a professional pilot, later co-founded the International Women's Air & Space Museum in Ohio during the 1980's.

Bernice Steadman died at her home in Traverse City, Michigan, on March 18, 2015, at the age of 89 following a lengthy battle with Alzheimer's disease. *Wik

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1931 Valentina Mikhailovna Borok (9 July 1931, Kharkiv, Ukraine, USSR–4 February 2004, Haifa, Israel) was a Soviet Ukrainian mathematician. She is mainly known for her work on partial differential equations.*Wik

DEATHS

1856 Count Amedeo Avogadro (9 August 1776, Turin, Piedmont – 9 July 1856) Italian chemist and physicist who found that at the same temperature and pressure equal volumes of all perfect gases contain the same number of particles,  known as Avogadro's Law (1811) leading to the Avogadro's constant being 6.022 x 1023 units per mole of a substance. He realized the particules could be either atoms, or more often, combinations of atoms, for which he coined the word "molecule." This explained Gay-Lussac's law of combining volumes (1809). Further, Avogadro determined from the electrolysis of water that it contained molecules formed from two hydrogen atoms for each atom of oxygen, by which the individual oxygen atom was 16 times heavier than one hydrogen atom (not 8 times as suggested earlier by Dalton.) The Italian, Romano Amadeo Carlo Avogadro, had suggested [in 1811] that all gases have the same number of molecules in a given volume. Loschmidt figured out [in 1865] how many molecules that would be which is the number now known as Avogadro's constant. John D. Cook suggested that maybe it should be called Loschmidt's constant, and pointed out three interesting coincidences involving Avogadro's Constant:
NA is approximately 24! (i.e., 24 factorial.)
The mass of the earth is approximately 10 NA kilograms.
The number of stars in the observable universe is 0.5 NA.
*John D. Cook, The Endeavour Blog

1953  Henri Padé (December 17, 1863 – July 9, 1953) made important contributions to the theory of continued fractions *SAU A Padé approximant is the "best" approximation of a function by a rational function of given order – under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed by Henri Padé, but goes back to Georg Frobenius who introduced the idea and investigated the features of rational approximations of power series.
The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. *Wik

1980  Arend Heyting (May 9, 1898 – July 9, 1980) is important in the development of intuitionistic logic and algebra. (try saying that three times really fast)*SAU Heyting gave the first formal development of intuitionistic logic in order to codify Brouwer's way of doing mathematics. The inclusion of Brouwer's name in the Brouwer–Heyting–Kolmogorov interpretation is largely honorific, as Brouwer was opposed in principle to the formalisation of certain intuitionistic principles (and went as far as calling Heyting's work a "sterile exercise").*Wik

1984 Edna Ernestine Kramer Lassar (May 11, 1902 - July 9, 1984) died of pneumonia at her home in Manhattan. She graduated magna cum laude from Hunter College and earned a Ph.D. at Columbia in 1930. She is best known for her book The Nature and Growth of Modern Mathematics, which is an excellent exposition of much of modern mathematics. It is very accessible to undergraduates. For more about her, see Women of Mathematics. A Biobibliographic Sourcebook (1987), edited by Louise S. Grinstein and Paul J. Campbell. *VFR

1996 Douglas George Chapman (March 20, 1920 - July 9, 1996)was a Canadian-born U.S. mathematical statistician  and an expert on wildlife statistics. He was one of the scientific advisors to the International Whaling Commission that warned in the 1960s that the number of whales being taken by the whaling industry was far in excess of what the population could stand, and proposed annual fin whale catch quotas that would permit the depleted populations of this species to recover. His later research on fish farming expanded to include mollusk aquaculture and he directed a program to develop quantitative methods to aid in the management of fisheries resources.*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