Tuesday, 11 August 2020

A Problem About Elevens, and Some Methods of "Casting Out Sevens"

Expanding an Archive blog from 2008 with some new insights:


Sometimes I enter the math contest they host at the Wild About Math blog site, and usually am among the people who get a correct answer, but haven't been the lucky name pulled from the hat yet. A recent problem about divisibility got me thinking about testing by divisibility by sevens again.
The problem was actually about divisibility by eleven, and asked
Consider all of the 6-digit numbers that one can construct using each of the digits between 1 and 6 inclusively exactly one time each. 123456 is such a number as is 346125. 112345 is not such a number since 1 is repeated and 6 is not used.

So.... How many of these 6-digit numbers are divisible by 11?

The answer, of course, is none.
Of course, is a danger word, like obviously, or trivially, in which we dismiss the idea that there is thinking involved.  If I were presenting this question to a class, and I have, I would say it differently. 

"None of these numbers are divisible by eleven, can you figure out why without test dividing any of them? "
 If you don't see it, don't worry, I'll spill the beans on how I would prove it down the page.  Just take a moment and try it yourself.

K
I
L
L
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G

T
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E

T
O

LET

YOU

THINK


The divisibility rule for eleven most commonly known is to take the digits abcdef and add every other one from the back, and then subtract the alternate ones.  So f-e+d-c+b-a  For example, 123456 would be  6-5+4-3+2-1=3.  Now they can only be divisible by eleven if the result is 0, or a multiple of 11.

So if we think about the numbers involved in all these numbers, they all have three odd numbers, and three even, so no matter how you split them, you'll have one group is odd, the other is even, so the difference is an odd number.  You can't get zero.  But can we get eleven?  Ummm, NO, because if we put all the big ones in one clump (6+5+4), and all the small ones in another (3+2+1), the difference is less than eleven.... so NONE of them are divisible by 11.

 My interest was piqued by a response by Jonathan, of the jd2718 blog made these observations about the 6! or 720 possible numbers formed with the six digits as described:

As a consolation, all 720 of them are multiples of 3.

Half of them (360) are even (multiples of 2)

One in 6 (120) are multiples of 5.

Eight in thirty (192) are multiples of 4.

None are multiples of 9.

Fourteen of 120 (84) are multiples of 8.

Multiples of 7? New puzzle, good place to stop.


It is Not surprising that Jonathon covered all the other one digit divisors but did NOT test seven. Seven is the one that books on mental math simply say "divide by seven"; but thinking about it, it shouldn’t be too hard
I wrote recently about a mental test for divisibility by seven but it did have one flaw. It worked fine to tell you a number was divisible by seven, but if the number was not divisible by seven it did not give the correct modulus. Casting out nines will always give you the correct modulus. The sum of the digits of 2134 is 10 which has a digit root of 1, so if you divide 2134 by 9 you get a remainder of one. My rule for seven didn't give the true remainder.

The remainder when you divide a number by seven (or any other number) is frequently called the modulus. I've mentioned this before, it is just a way to divide integers into sets. Odd numbers are all equivalent to 1 mod 2 (they all leave a remainder of one when you divide by 2) and even numbers are all equivalent to zero. If a number is equivalent to zero mod (something) that means it is divisible by that (something). One of the things that makes them effective is the simple rule that if a+b=c then for any modular index n(the number you are dividing by) c mod n = a mod n + b mod n. For example 25 mod 7 = 4.  We could have found that by breaking it into parts.  Since 25 = 20 + 5 we find 20 mod 7 = 6, and 5 mod 7 = 5... and the  6+5 = 11 which is equivalent to 4 mod 7. So now we have the basics.

Every increase of one in the units increases the modulus 1, and every increase in the tens increases the modulus three, (for example 21 mod 7 = 0; 31 mod 7 = 3,and 41 mod 7 = 6). I figured out that the rest would be an increase of two in the hundreds, six in the thousands (think minus one) four in the ten-thousands (minus three?) and five in the hundred-thousands (minus two….) so the sequence applied to our test number, 546231… and what a coincidence, if you multiply 5(-2)+4(-3)+6(-1)+2(2) +3(3) + 1(1) …YOU GET -14, which is zero mod 7, the sequence of moduli used in each place makes it a multiple of seven…(546231 = 0 [mod 7]…

 but wait, if you used the A-2B method in the blog  above, we would notice that 231 by itself is a multiple of seven, since 23 - 2(1) =21… and 546 is also. [54 - 2(6)=42]..   You could also apply that approach step by step, 54623-2*1=54621; and 5462-2(1) = 5460.  The zero we can throw away because 546 tens is not divisible by 7 if 546 is not, so we proceed to 54-2(6) = 42 and hey, we have a divisor by 7.  In each case we just combined two moduli to reduce the size of the number.

There is even a neat graphic approach that works the same way.  I came across a nice video of that at a site by Presh Talwalker called Mind Your Decisions.  Each step uses the same modulus increase approach as the method above, but challenge your students to figure out why it works (not an easy task).



To check other numbers, you can just write them out multiplied times the correct modulus for that place value and probably check it in your head, but you would have to check each one… so making up a sequence, 234561(this covers everything up to six digits, but if you have a super memory, figure out another period), we would think 2×5=10 (the first digit times the first modulus) drops to 3, plus 3(second digit)x4(second modulus) is 15 which drops to modulus of one, now add 4(6) to get 25 and drop to mod 4, then add 2×5 to get 14, and we are at mod 0 [so 234500 is divisible by seven] and we know that 61 is NOT divisible by seven, so we can be assured that 234561 == 61 == 5 mod 7… ok, not EASY, but certainly could be done sans calculator…. [footnote, for those of you who have just gone through a pre-calc class and somewhere along the way they taught you about vectors, you probably imagined that you would never run across another dot product in your life, but you just did. IF you think of the sequence of modular values as one vector (5,4,6,2,3,1) and the six-digits of the number as a second vector, then the dot product is an integer that has the same modulus as the original six digit number....come on... that's pretty cool use of vectors!!!]
It is easy enough to write/remember the modulus up to ten-thousands (6231) to make this pretty useful as a factoring tool if you really needed the correct modulus. If the number is 4723 we just think 4(6)=24--==; 3 + 7x2---==; still 3, + 2(3)=9 --==;2 and then + 3(1)= 5 so 6231 divided by seven leaves a remainder of five.

On This Day in Math - August 11


Boy observing Mural along the flood wall in Paducah, Ky

So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality.
~Albert Einstein


The 224th day of the year; 224 is the sum of the cubes of 4 consecutive integers:
224 = 23 + 33 + 43 + 53 and also 23 + 63

Cool thing about 224: 224 = 23+45+67+89 *Derek Orr ‏@Derektionary

Every number smaller than 224 can be expressed as the sum of distinct divisors of 224.


See More Math Facts for every Year Day here.



EVENTS

1124 "In the month of August on the 11th day, before the evening service, the Sun began to diminish and perished completely. Great fright and darkness everywhere. And the stars appeared and the Moon (sic). And the Sun began to augment and became full again and everyone in the town was very glad." Refers to a total solar eclipse in Novgorod, Russia, of 11 August 1124. From: Novorodskaya I Letopic. Quoted in Historical Eclipses and Earth's Rotation, by F Richard Stephenson, Cambridge University Press, 1997, page 391. *NSEC

1591 Kepler received his master’s degree from Tübingen and thereupon entered the process of practical preparation for either teaching or being a Protestant pastor. Halfway through his third year, however, an event occurred that completely altered the direction of his life. Georgius Stadius, teacher of mathematics at the Lutheran school in Graz, died; and the local authorities asked Tübingen for a replacement. Kepler was chosen; and although he protested abandoning his intention to became a clergyman, he set out on the career destined to immortalize his name. *www.encyclopedia.com *Thony Christie

In 1835, George B Airy began his 46-year reign as England's seventh Astronomer Royal. He was appointed in June of that year but seems to have assumed duty on August 11. At Greenwich he designed and installed the famous transit circle now named after him, used for timing the passage of stars across the meridian. It is the position of the Airy transit circle that defines the Greenwich meridian and since 1884 that meridian has been the basis of the world’s timekeeping and navigation. Hence Airy can be said to have brought the world’s clocks into step. *assorted sources

1859 Bernhard Riemann was made a corresponding member of the Berlin Academy based on his 1857 paper on Abelian Functions. It was the practice that newly selected members would submit an example of their recent research. Riemann submitted, "On the Number of Prime Numbers Less Than a Given Quantity." The paper contained his now famous Riemann Hypothesis that the non-trivial zeros of the Zeta function all have a real part of 1/2. It is the only paper he ever published on number theory.*John Derbyshire, Prime Obsession


1909 First SOS, The first well documented use of the SOS distress call is by the Arapahoe on August 11, 1909, when it suffered a broken shaft in the Atlantic Ocean, near Cape Hatteras, North Carolina. However, an article titled "Notable Achievements of Wireless" in the September, 1910 Modern Electrics suggests that an earlier SOS distress call was transmitted by the Cunard liner Slavonia, on June 10, 1909.
[The wireless operator aboard S.S. Arapahoe, T. D. Haubner, radioed for help. A few months later, Haubner on the S.S. Arapahoe received an SOS from the SS Iroquois, the second use of SOS in America.(*TIS)]
The first radio distress call to be adopted appears to have been "CQD", by the Marconi International Marine Communication Company​, for Marconi-operated shipboard stations. It was announced on January 7, 1904 by the company's "Circular 57" that "...on and after the 1st February, 1904, the call to be given by ships in distress or in any way requiring assistance shall be 'C.Q.D.'." ("CQ" was a general call to all stations; amateur or "ham" radio operators still use it today when soliciting a contact with any station that hears the call.)

An International Radiotelegraphic Convention, ... met in Berlin in 1906. This body signed an international agreement on November 3, 1906, with an effective date of July 1, 1908. An extensive collection of Service Regulations was included to supplement the Convention, and in particular Article XVI adopted Germany's Notzeichen distress signal as the international standard, stating: "Ships in distress shall use the following signal: · · · — — — · · · repeated at brief intervals". *Citizens Compendium

In 1999, the last total eclipse of the millennium occurred. Because it traveled across many populated areas, it was perhaps the most-watched eclipse of all time, seen by possibly 350 million people. Totality occurred first over the mid-Atlantic Ocean. The first land crossed by the moon's shadow was the Isles of Scilly, then the far south-west of England, in Cornwall. Although the sun was obscured by clouds there, a dramatic darkness fell, and the temperature dropped, during the totality lasting 1-min 30-sec. From there the path of totality tracked across Europe, India and Iran. In Egypt, Muslims were ordered by clerics to shut themselves away, but Jordan and Syria declared a national holiday.*TIS 



BIRTHS

1730 Charles Bossut(11 August 1730 – 14 January 1814) was a French mathematician who was famed for his textbooks which were widely used throughout France.*SAU

1829 Norman Macleod Ferrers; (11 August 1829 – 31 January 1903) John Venn wrote of him,
.. the Master, Dr Edwin Guest, invited Ferrers, who was by far the best mathematician amongst the fellows, to supply the place. His career was thus determined for the rest of his life. For many years head mathematical lecturer, he was one of the two tutors of the college from 1865. As lecturer he was extremely successful. Besides great natural powers in mathematics, he possessed an unusual capacity for vivid exposition. He was probably the best lecturer, in his subject, in the university of his day.
It was as a mathematician that Ferrers acquired fame outside the university. He made many contributions of importance to mathematical literature. His first book was "Solutions of the Cambridge Senate House Problems, 1848 - 51". In 1861 he published a treatise on "Trilinear Co-ordinates," of which subsequent editions appeared in 1866 and 1876. One of his early memoirs was on Sylvester's development of Poinsot's representation of the motion of a rigid body about a fixed point. The paper was read before the Royal Society in 1869, and published in their Transactions. In 1871 he edited at the request of the college the "Mathematical Writings of George Green" ... Ferrers's treatise on "Spherical Harmonics," published in 1877, presented many original features. His contributions to the "Quarterly Journal of Mathematics," of which he was an editor from 1855 to 1891, were numerous ... They range over such subjects as quadriplanar co-ordinates, Lagrange's equations and hydrodynamics. In 1881 he applied himself to study Kelvin's investigation of the law of distribution of electricity in equilibrium on an uninfluenced spherical bowl. In this he made the important addition of finding the potential at any point of space in zonal harmonics (1881).

Ferrers proved the proposition by Adams that "The number of modes of partitioning (n) into (m) parts is equal to the number of modes of partitioning (n) into parts, one of which is always m, and the others (m) or less than (m). " with a graphic transformation that is named for him. *SAU

1836 Cato Maximilian Guldberg (11 August 1836 – 14 January 1902) Norwegian chemist who, with his brother-in-law Peter Waage, formulated the law of mass action (1864), which details the effects of concentration, mass, and temperature on chemical reaction rates. The law states that the rate of a chemical change depends on the concentrations of the reactants. Thus for a reaction: A + B >> C the rate of reaction is proportional to [A][B], where [A] and [B] are concentrations. In 1870 Guldberg investigated the way in which the freezing point and vapor pressure of a pure liquid are lowered by a dissolved component. In 1890 he formulated Guldberg's law which relates boiling point and critical temperature.*TIS

1842 Enrico D'Ovidio (11 Aug 1842 in Campobasso, Italy - 21 March 1933 in Turin, Italy) D'Ovidio was to work for 46 years in the University of Turin. He was chairman of the Faculty of Science in 1879-80 and rector of the University between 1880 and 1885. Another spell as chairman of the Faculty of Science between 1893 and 1907 ended when he was appointed Commissioner of the Polytechnic of Turin.
Euclidean and noneuclidean geometry were the areas of special interest to D'Ovidio. He built on the geometric ideas which had been introduced by Lobachevsky, Bolyai, Riemann and Cayley. D'Ovidio's most important work is probably his paper of 1877 The fundamental metric functions in spaces of arbitrarily many dimensions with constant curvature.
D'Ovidio also worked on binary forms, conics and quadrics. He had two famous assistants, Peano (1880-83) and Corrado Segre (1883-84). D'Ovidio and Corrado Segre built an important school of geometry at Turin. *SAU

1895 Egon Sharpe Pearson, (Hampstead, 11 August 1895 – Midhurst, 12 June 1980) was the only son of Karl Pearson, and like his father, a leading British statistician.
He went to Winchester School and Trinity College, Cambridge, and succeeded his father as professor of statistics at University College London and as editor of the journal Biometrika.
Pearson is best known for development of the Neyman-Pearson lemma of statistical hypothesis testing.
He was President of the Royal Statistical Society in 1955–56, and was awarded its Guy Medal in Gold in 1955. He was awarded a CBE in 1946.
He was elected a Fellow of the Royal Society in Mar 1966. His candidacy citation read: "Known throughout the world as co-author of the Neyman-Pearson theory of testing statistical hypotheses, and responsible for many important contributions to problems of statistical inference and methodology, especially in the development and use of the likelihood ratio criterion. Has played a leading role in furthering the applications of statistical methods - for example, in industry, and also during and since the war, in the assessment and testing of weapons." *Wik

1912 Norman Levinson (August 11, 1912, Lynn, Massachusetts – October 10, 1975, Boston) set out to become an electrical engineer. Here he describes the events that led to his change of major:
I became acquainted with Wiener in September 1933, while still a student of electrical engineering, when I enrolled in his graduate course. It was at that time really a seminar course. At that level he was a most stimulating teacher. He would actually carry on his research at the blackboard. As soon as I displayed a slight comprehension of what he was doing, he handed me the manuscript of Paley-Wiener for revision. I found a gap in a proof and proved a lemma to set it right. Wiener thereupon sat down at his typewriter, typed my lemma, affixed my name and sent it off to a journal. A prominent professor does not often act as secretary for a young student. He convinced me to change my course from electrical engineering to mathematics.
Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, number theory, and signal processing. He worked closely with Norbert Wiener in his early career. He joined the faculty of the Massachusetts Institute of Technology in 1937. In 1954, he was awarded the Bôcher Memorial Prize of the American Mathematical Society. In 1974 he published a paper proving that more than a third of the zeros of the Riemann zeta function lie on the critical line, a result later improved to two fifths by Conrey.*SAU

1921 Tom Kilburn (11 August 1921 – 17 January 2001) British electrical engineer who wrote the computer program used to test the first stored-program computer, the Small-Scale Experimental Machine, SSEM, also known as "The Baby." First tested on 21 Jun 1948, the program took 52 minutes to run. The tiny experimental computer had no keyboard or printer, but it successfully tested a memory system developed at Manchester University in England. This system, based on a cathode-ray tube, was the first that could store programs, whereas previous electronic computers had to be rewired to execute each new problem.*TIS

1950 Steve Wozniak, (August 11, 1950; ) Apple inventor, is born
Wozniak and Jobs entered into business after Wozniak designed a single-board personal computer known as the Apple I. In 1976, with specifications in hand and an order for 100 machines at $500 each from the Byte Shop, he and Jobs founded the business.
While still studying at the University of California-Berkeley in 1972, Wozniak had shown his electronics skill as well as his sense of humor in building his blue box, a tone generator used to make free phone calls, which he sold in dormitories
Wozniak now teaches computer science to school children in his home town of Los Gatos, California. *CHM

1956 Pierre-Louis Lions (August 11, 1956, ) French mathematician who was awarded the Fields Medal in 1994 for his work since the 1980's on partial differential equations. The sources of such equations are many - for example, physical, probabilistic or geometric and other diverse subareas - each studying different phenomena for different nonlinear partial differential equations by utterly different methods. Pierre-Louis Lions has been called unique in his ability to transcend these boundaries and to solve pressing problems throughout the field.*TIS



DEATHS

1464 Nicholas von Cusa died (1401 – August 11, 1464). We know his work in mathematics primarily because a home for the aged in Kues, which he generously endowed, has survived the ravages of time and war. Luckily his own manuscripts were housed there.*VFR
Nicholas is also considered by many to be a genius ahead of his time in the field of science. Nicolaus Copernicus, Galileo Galilei and Giordano Bruno were all aware of the writings of Cusanus as was Johannes Kepler (who called Cusanus 'divinely inspired' in the first paragraph of his first published work). Predating Kepler, Cusanus said that no perfect circle can exist in the universe (opposing the Aristotelean model, and also Copernicus' later assumption of circular orbits), thus opening the possibility for Kepler's model featuring elliptical orbits of the planets around the Sun. He also influenced Giordano Bruno by denying the finiteness of the universe and the Earth's exceptional position in it (being not the center of the universe, and in that regard equal in rank with the other stars). He was not, however, describing a scientifically verifiable theory of the universe: his beliefs (which proved uncannily accurate) were based almost entirely on his own personal numerological calculations and metaphysics.
Cusanus made important contributions to the field of mathematics by developing the concepts of the infinitesimal and of relative motion. He was the first to use concave lenses to correct myopia. His writings were essential for Leibniz's discovery of calculus as well as Cantor's later work on infinity. *Wik

1578 Pedro Nunes or Nunez (1502 – August 11, 1578) was a Portuguese scholar who worked in geometry, spherical trigonometry, algebra as well as geography, physics, and cosmology. *SAU He was the first to propose the idea of a loxodrome and was also the inventor of several measuring devices, including the nonius, named after his Latin surname. *Wik

1854 Macedonio Melloni, (11 April 1798 – 11 August 1854) Italian physicist who was the first to extensively research infrared radiation. Sir William Frederick Herschel discovered infrared radiation in 1800, but research stalled until the invention of a thermopile in 1830. That instrument was a series of strips of two different metals that produced electric current when one end was heated. Melloni improved the thermopile and used it to detect infrared radiation. In 1846, from an observation point high on Mount Vesuvius, he measured the slight heating effect of moonlight. He showed also that rock salt, being transparent to infrared, made suitable lenses and prisms to demonstrate the reflection, refraction, polarization and interference of infrared in the same manner as visible light.*TIS

1892 Enrico Betti (21 October 1823 – 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equations, giving early expositions of Galois theory. He also discovered Betti's theorem, a result in the theory of elasticity. *Wik

1955 Robert Williams Wood (May 2, 1868 – August 11, 1955) was an American experimental physicist. He photographed the reflection of sound waves in air, and investigated the physiological effects of high-frequency sound waves. The zone plate he devised could replace the objective lens of a telescope. He invented an improved diffraction grating, did research in spectroscopy, and extended the technique of Raman spectroscopy (a method to study matter using the light scattered by it.) He made photographs showing both infrared and ultraviolet radiation and was the first to photograph ultraviolet fluorescence. Wood was the first to observe the phenomenon of field emission in which charged particles are emitted from conductors in an electric field. *TIS
According to a post at Greg Ross' Futility Closet:
"How to clean a 40-foot spectrograph, from R.W. Wood’s Researches in Physical Optics, 1913:
The long tube was made by nailing eight-inch boards together, and was painted black on the inside. Some trouble was given by spiders, which built their webs at intervals along the tube, a difficulty which I surmounted by sending our pussy-cat through it, subsequently destroying the spiders with poisonous fumes.
This was the least of Wood’s exploits. Walter Bruno Gratzer, in Eurekas and Euphorias, writes that the physicist “would alarm the citizens of Baltimore by spitting into puddles on wet days, while surreptitiously dropping in a lump of metallic sodium, which would explode in a jet of yellow flame.”

1977 Sir Frederic Williams, (26 June 1911 Stockport – 11 August 1977 Manchester) British electrical and electronics engineer who, with Tom Kilburn, invented the Williams tube, a cathode-ray tube using the persistence of the image on the phosphor screen for data storage. This made possible the random access memory that launched the digital computer age. As the Chair in Electrotechnics at Manchester University, he incorporated this invention into the Mark I computer, the world's first stored-program digital electronic computer to be commercially produced during the early 1950's. *TIS

1995 Alonzo Church (June 14, 1903 – August 11, 1995) made important contributions to mathematical logic and theoretical computer science.*SAU He is best known for the lambda calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology, and the Church–Rosser theorem. *Wik

2003 Armand Borel (21 May 1923 –11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. *Wik



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

Monday, 10 August 2020

On This Day in Math - August 10




I have made this letter longer than usual, because I lack the time to make it short.
~Blaise Pascal

The 223rd day of the year; 223 is a prime number, and the sum of three consecutive primes (71 + 73 + 79), 223 and also the sum of seven consecutive primes (19 + 23 + 29 + 31 + 37 + 41 + 43)

Every number can be formed with no more than 36 fifth powers, except one, 223 is the only number that requires 37 fifth powers. This is related to Waring'a problem. In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers to the power of k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909.

Fans of Star Wars may know that this is sometimes called the Star Wars Droid Prime because it uses only the numbers in the names of R2-D2 and C3PO

An interesting note, if you take a prime number less than 223 and reverse it, some are prime, some have two factors, but 223 is the smallest prime whose reversal has three factors (322 = 2 * 7 * 23)


See More Math Facts for every Year Day here



EVENTS
1548 Tartaglia and Ferrari met in a mathematical debate in the church of Santa Maria del Giardino dei Minori Osservanti in Milan. In the presence of a distinguished audience, which included the governor of Milan as judge, they argued over a problem that Ferrari posed on 24 May 1547, and which Tartaglia could not resolve. The next day, Tartaglia left in disgrace for his native Brescia. *VFR  For details about this story you probably didn't know (I didn't) see this post from the Renaissance Mathematicus.

1675 Greenwich Observatory founded. *VFR The observatory was commissioned in 1675 by King Charles II, with the foundation stone being laid on 10 August. Charles II had appointed John Flamsteed as his first Astronomer Royal in March of that year and oversaw the laying of the foundation stone.  Rebekah Higgitt explains why this is "An auspicious day to found an observatory" at her Teleskopos blog.

1732 Pell equations are forever (mis)named. Euler writes to Goldbach and mentions the equation x2 -A y2 =1 and adds;
Dr. Pell, an Englishman, devised a unique way of solving problems of this kind, as shown in the works of Wallis."
Most historical authorities believe this was due to a cursory or misreading of Wallis' work in which Pell is often mentioned, but not in regard to this equation. "Pell has done it no other service than to set it forth again in a much read work," i.e., in the notes to the English translation which Brancker,4 in 1668, published of the "Teutschen Algebra" of Rahn. (Pell was a teacher of Rahn and much of the work is based on Pell's methods. Pell wsa also influential in the English translation of the work.) *Edward Everett Whitford, The Pell Equation

1792 Against his better judgment, Monge was forced into the Ministry of the Navy and the Colonies of France. *VFR

1846 The Smithsonian Institution established in Washington. Benjamin Peirce was on the five-member committee that drew up the program for its organization. *VFR In 1846, an Act of Congress signed by President James K. Polk established the Smithsonian Institution as a trust to administer the generous bequest of James Smithson, an amount over $500,000. In 1826, James Smithson, a British scientist, drew up his last will and testament, naming his nephew as beneficiary. Smithson stipulated that, should the nephew die without heirs (as he would in 1835), the estate should go “to the United States of America, to found at Washington, under the name of the Smithsonian Institution, an establishment for the increase and diffusion of knowledge among men.” The motives behind Smithson’s bequest remain mysterious; he had never traveled to the U.S. and seems to have had no correspondence with anyone there.*TIS

1897 German chemist Felix Hoffmann synthesized acetylsalicylic acid (ASA) in a stable form usable for medical applications. In 1899 it was marketed for the first time under the trade name Aspirin. Acetylsalicylic acid, the active ingredient of aspirin, was first discovered from the bark of the willow tree in 1763 by Edward Stone of Wadham College, University of Oxford. *http://scihi.org
1912 “Some time between August 10 and August 16 it became clear to Einstein that Riemannian geometry is the correct mathematical tool for what we now call general relativity.” [From Abraham Pais, Subtle is the Lord ... The Science and the Life of Albert Einstein, as quoted in the New York Times Book Review, Nov. 28, 1982, p. 9]*VFR


1939 Albert Einstein “wrote” President F. D. Roosevelt that “Some recent work by E. Fermi and L. Szilard ... leads me to expect that the element uranium may be turned into a new and important source of energy in the immediate future. ... This new phenomenon would also lead to the construction of bombs, and it is conceivable—though much less certain—that extremely powerful bombs of a new type may be constructed.”
The letter, drafted by Fermi, Szilard, and Wigner and seems not to have actually been signed by Einstein until August 10, and was then given to Alexander Sachs, a confident of Roosevelt, who did not deliver it to him until October 30. Roosevelt quickly started the Manhattan Project. Einstein later regretted signing this letter. *(VFR & Brody & Brody); (the letter can be read at Letters of Note) They recognized the process could generate a lot of energy leading to power and possibly weapons. There was also concern the Nazi government of Germany was already searching for an atomic weapon. This letter would accomplish little more than the creation of a "Uranium Committee" with a budget of $6,000 to buy uranium and graphite for experiments.
Sir Fred Soddy's book, The Interpretation of Radium, inspired H G Wells to write The World Set Free in 1914, and he dedicated the novel to Soddy's book. Twenty years later, Wells' book set Leo Szilard to thinking about the possibility of Chain reactions, and how they might be used to create a bomb, leading to his getting a British patent on the idea in 1936. A few years later Szilard encouraged his friend, Albert Einstein , to write a letter to President Roosevelt about the potential for an atomic bomb. The prize-winning sc1802 Germain Henri Hess (August 7, 1802 – November 30, 1850)Swiss-born Russian chemist whose studies of heat in chemical reactions formed the foundation of thermochemistry. He formulated an empirical law, Hess's law of constant heat summation (1840), which states that the heat evolved or absorbed in a chemical process is the same whether the process takes place in one or in several steps. It is explained by thermodynamic theory, which holds that enthalpy is a state function. Chemists have made great use of the law of Hess in establishing the heats of formation of compounds which are not easily formed from their constituent elements. His early investigations concerned minerals and the natural gas found near Baku, and he also discovered the oxidation of sugars to yield saccharic acid.*TIS

ience-fiction writer, Frederik Pohl , talks about Szilard's epiphany in Chasing Science (pg 25),
".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."


BIRTHS
787 Albumasar (10 Aug 787, 9 Mar 886 at age 98)Persian astrologer, a.k.a. Abu Ma'shar al-Balkhi, or Ja'far ibn Muhammad, who was the leading astrologer of the Muslim world. He is known primarily for his theory that the world, created when the seven planets were in conjunction in the first degree of Aries, will come to an end at a like conjunction in the last degree of Pisces. *TIS

1602 Gilles Personne de Roberval born. *VFR(according to some, see August 9th)

1839 Aleksandr Grigorievich Stoletov (August 10, 1839 – May 27, 1896) was a Russian physicist, founder of electrical engineering, and professor in Moscow University. He was the brother of general Nikolai Stoletov. By the end of the 20th century his disciples had headed the chairs of Physics in five out of seven major universities in Russia.
His major contributions include pioneer work in the field of ferromagnetism and discovery of the laws and principles of the outer photoelectric effect.*Wik

1856 William Willett (10 August 1856 – 4 March 1915) English builder who invented Daylight Saving Time. He claimed he had the idea while taking an early summer morning ride in Petts Wood near to his home in Chislehurst, London. He observed that many blinds were still down, although there was already good daylight, yet many made no use of it. He used his wealth as a prominent home builder to campaign for a scheme of adjusting clocks with the season and published a pamphlet in 1907. His original idea was to make four weekly changes of 20-mins each, for a total of 80-mins. The first Daylight Saving Bill, proposing a single one hour at the change of season failed in 1908. After his death, the idea was adopted during WW I for wartime fuel savings. A memorial was erected in Petts Wood.*TIS A sun dial, right, Memorial was erected in the Pett"s Wood in his honor.

1859 Georg Alexander Pick (August 10, 1859 – July 26, 1942) was an Austrian mathematician. He died in the Theresienstadt concentration camp. Today he is best known for Pick's formula for determining the area of lattice polygons. He published it in an article in 1899; it was popularized when Hugo Dyonizy Steinhaus included it in the 1969 edition of Mathematical Snapshots. Pick headed the committee at the (then) German university of Prague which appointed Albert Einstein to a chair of mathematical physics in 1911. Pick introduced Einstein to the work of Italian mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita in the field of absolute differential calculus, which later in 1915 helped Einstein to successfully formulate General relativity.*Wik A really nice article about this theorem with references and interactive graphics is Found at Alexander Bogomolny's Cut The Knot web site.
Does Pick’s theorem generalise to 3 dimensions? In 1957, John Reeve delivered some bad news. Reeve tetrahedra have vertices at: (0,0,0), (1,0,0), (0,1,0), & (1,1,r) where r is a positive integer. All Reeve tetrahedra contain the same number of lattice points (just their four vertices). But their volumes are different. But then, in 1967 Eugene Erhart found a way to do it in 3 dimensions and more. He showed that all integral polytopes had an Ehrhart polynomial that encodes the relationship between the volume (and several other characteristics, it seems) of a polytope and the number of integer points the polytope contains. *Richard Elwes, Pick’s Theorem & Ehrhart Polynomials

1889 Charles Brace Darrow (August 10, 1889 – August 28, 1967) was an American inventor who designed the board game Monopoly. He had invented the game on 7 Mar 1933, though it was preceded by other real-estate board games. On 31 Dec 1935, a patent was issued for the game of Monopoly assigned to Parker Brothers, Inc., by Ch*arles Darrow of Pennsylvania (No. 2,026,082). The patent titled it a "Board Game Apparatus" and described it as "intended primarily to provide a game of barter, thus involving trading and bargaining" in which "much of the interest in the game lies in trading and in striking shrewd bargains." Illustrations included with the patent showed not only the playing board and pieces, cards, and the scrip money. *TIS

1901 Franco Dino Rasetti (August 10, 1901 – December 5, 2001) was an Italian scientist. Together with Enrico Fermi, he discovered key processes leading to nuclear fission. Rasetti refused to work on the Manhattan Project, however, on moral grounds.*Wik

1926 Carol (Vander Velde)Karp (10 August 1926, Forest Grove, Ottawa County, Michigan – 20 August 1972, Maryland) was born in Forest Grove, Michigan. She received her Ph.D. in 1959 from Southern California under the direction of Leon Henkin. She created the field of Infinitary Logics which studies logics such as Lω,ω which allowed for the conjunction and disjunction of countably many formulas. This work has become very important in modern logic. *VFR




DEATHS

1802 Franz Aepinus (December 13, 1724 – August 10, 1802) was a German scientist who did important work on electricity and magnetism. Aepinus' study of electricity and magnetism led to the publication of his book Tentamen theoriae electricitatis et magnetismi (An Attempt at a Theory of Electricity and Magnetism) in 1759. It was the first work to apply mathematics to the theory of electricity and magnetism and "... is one of the most original and important books in the history of electricity."
Euler was working at the Berlin Academy during the time that Aepinus worked there, and in fact Aepinus lived in Euler's house for the two years that he was in Berlin.
Other achievements of Aepinus include improvements to the microscope, and his demonstration of the effects of parallax in the transit of a planet across the Sun's disk (1764). However, except for his masterpiece on electricity and magnetism, his work was no better than competent.*SAU

1843 Robert Adrain (30 September 1775 – 10 August 1843) was a scientist and mathematician, considered one of the most brilliant mathematical minds of the time in America.. Irish born,  Adrain was an editor of and contributor to the Mathematical Correspondent, the first mathematical journal in the United States. Later he twice attempted to found his own journal, The Analyst, or, Mathematical Museum, but in both the 1808 and 1814 attempts it did not attract sufficient subscribers and quickly ceased publication. In 1825 he founded a somewhat more successful publication targeting a wider readership, The Mathematical Diary, which was published through 1832.He is chiefly remembered for his formulation of the method of least squares, published in 1808. Adrain certainly did not know of the work of C.F. Gauss on least squares (published 1809), although it is possible that he had read A.M. Legendre's article on the topic (published 1804).*Wik He was one of the first American mathematicians to do creative work. *VFR

1915 Henry Gwyn Jeffreys (23 November 1887 – 10 August 1915) Moseley English physicist who experimentally demonstrated that the major properties of an element are determined by the atomic number, not by the atomic weight, and firmly established the relationship between atomic number and the charge of the atomic nucleus. He began his research under Ernest Rutherford while serving as lecturer at the Univ. of Manchester. Using X-ray photographic techniques, he determined a mathematical relation between the radiation wavelength and the atomic numbers of the emitting elements. Moseley obtained several quantitative relationships from which he predicted the existence of three missing elements (numbers 43, 61, and 75) in the periodic table, all of which were subsequently identified. Moseley was killed in action during WW I.*TIS
When World War I broke out in Western Europe, Moseley left his research work at the University of Oxford behind to volunteer for the Royal Engineers of the British Army. Moseley was assigned to the force of British Empire soldiers that invaded the region of Gallipoli, Turkey, in April 1915, as a telecommunications officer. Moseley was shot and killed during the Battle of Gallipoli on August 10, 1915, at the age of just 27. Some prominent authors have speculated that Moseley could have been awarded the Nobel Prize in Physics in 1916, had he not died in the service of the British Army.*Wik

1929 Pierre Joseph Louis Fatou (28 February 1878 – 10 August 1929) was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory. Fatou continued his mathematical explorations and studied iterative and recursive processes such as z == z2+C . Fatou was particularly interested in the case where Z0 = 0, which was later analysed with computers by Benoît Mandelbrot to generate graphical representations of the behaviour of this series for each point, c, in the complex plane – now popularly called the Mandelbrot set.
Fatou wrote many papers developing a Fundamental theory of iteration in 1917, which he published in the December 1917 part of Comptes Rendus. His findings were very similar to those of Gaston Maurice Julia, who submitted a paper to the Académie des Sciences in Paris for their 1918 Grand Prix on the subject of iteration from a global point of view. Their work is now commonly referred to as the generalised Fatou–Julia theorem.*Wik

1945 Robert Hutchings Goddard (October 5, 1882 – August 10, 1945) American professor, physicist and inventor, "father of modern rocketry". From age 17 Goddard was interested in rockets (1899) and by 1908 he conducted static tests with small solid-fuel rockets. He developed mathematical theory of rocket propulsion (1912) and proved that rockets would functioned in a vacuum for space flight (1915). During WW I, Goddard developed rocket weapons. He wrote A Method of Reaching Extreme Altitudes, in 1919. Over the following two decades he produced a number of large liquid-fuel rockets at his shop and rocket range at Roswell, N.M. During WW II he developed rocket-assisted takeoff of Navy carrier planes and variable-thrust liquid-fuel rocket motors. At the time of his death Goddard held 214 patents in rocketry.*TIS Goddard is buried at Hope Cemetery in Worcester, Massachusetts, his birthplace.


1960 Oswald Veblen, (June 24, 1880 – August 10, 1960) a world famous geometer, died in Brookline, MA, at age 80. *VFR American mathematician who made important contributions in early topology, and in projective and differential geometry - work which found applications in atomic physics and the theory of relativity. In 1905, he proved the Jordan curve theorem, which states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside". Although it may seem obvious in its statement, it is a very difficult theorem to prove. During WW II, he was involved in overseeing the work that produced the pioneering ENIAC electronic digital computer. His name is commemorated by the American Mathematical Society's Oswald Veblen Prize. Awarded every five years, it is the most prestigious award in recognition of outstanding research in geometry.*TIS

1981 Jack Carl Kiefer (January 25, 1924 – August 10, 1981) Kiefer's main research area was the optimal design of experiments, and about half his 100 publications dealt with that topic. However he also wrote papers on a whole variety of topics in mathematical statistics including decision theory, inventory theory, stochastic approximation, queuing theory, nonparametric inference, estimation, sequential analysis, and conditional inference. His first paper Almost subminimax and biased minimax procedures written jointly with his fellow graduate student at Columbia, Peter Frank, was published in 1951. A paper Sequential minimax search for a maximum which Kiefer published in the Proceedings of the American Mathematical Society in 1953 was based on his master's thesis. The method of search proposed in the paper, namely the Fibonacci Search, became a widely used tool. *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

Sunday, 9 August 2020

On This Day in Math - August 9



You don't understand anything until you learn it more than one way.
— Marvin Minsky
The 222nd day of the year; 222 is called a sphenic (Greek for wedge) number. They have three distinct prime factors. 30= 2x3x5 is the smallest sphenic number. Can you find two consecutive numbers that are both sphenic numbers?

More??
222 = (3!)3 + (2!)2 + (1!)1 + (0!)0 *Derek Orr ‏@Derektionary

222 is the number of lattices on 10 unlabeled nodes.
and 222 is the sum of consecutive primes.
See More Math Facts for every Year Day here.



EVENTS
0975 "The Sun was eclipsed . . . . Some people say that it was entirely total. During the hours mao and ch'en (some time between 5 and 9 h) it was all gone. It was the colour of ink and without light. All the birds flew about in confusion and the various stars were all visible. There was a general amnesty (on account of the eclipse)." From: Nihon Kiryaku. "At the hour ch'en (7-9 h), the Sun was eclipsed; it was completely total. All under heaven became entirely dark and the stars were all vis ible."
From: Fuso Ryakki. "The Sun was eclipsed; it was all gone. It was like ink and without light. The stars were all visible (or: stars were visible in the daytime)." From: Hyaku Rensho. These three Japanese quotations refer to a total solar eclipse of 9 August AD 975. Quoted in Historical Eclipses and Earth's Rotation, by F Richard Stephenson, Cambridge University Press, 1997, pages 267 and 268. *NSEC

1207 An educational institution is founded for the study of the works of Bhaskara II, an Indian mathematician and astronomer.*VFR
Bhaskara II was rightly achieved an outstanding reputation for his remarkable contribution. In 1207 an educational institution was set up to study Bhaskaracharya's ("Bhaskara the teacher")works. A medieval inscription in an Indian temple reads: Triumphant is the illustrious Bhaskaracharya whose feats are revered by both the wise and the learned. A poet endowed with fame and religious merit, he is like the crest on a peacock. It is from this quotation that the title of Joseph's book comes. *SAU


1654 Fermat to Carcavi "Monsieur, I was overjoyed to have had the same thoughts as those of M. Pascal, for I greatly admire his genius and I believe him to be capable of solving any problem he attempts. The friendship he offers is so dear to me and so precious that I shall not scruple to take advantage of it in publishing an edition of my Treatises. If it does not shock you, you could both help in bringing out this edition, and I suggest that you should be the editors: you could clarify or augment what seems too brief and thus relieve me of a care which my work prevents me from taking. I would like this volume to appear without my name even, leaving to you the choice of designation which would indicate the author, whom you could qualify simply as a friend." *York Univ Hist of Stats.

1658 Simon Douw obtains a patent for a pendulum clock that will draw Huygen’s attack in Horologium. “Today, no clock by Simon Douw is known; I find that most curious, it is as if he has been excised from history, deliberately. Dutch Court papers described Douw as "City clockmaker of Rotterdam... a master in the art of great tower, domestic or office clocks", ("en meester in de kunst van groote Toorn, Camer ofte Comptoirwerken"). Yet his mechanical insights. his escapement, also his drive mechanisms, are best, and now only, revealed by his Patent Grant on August 9th, 1658, and by the evidence and judgement in a claim and counterclaim started in the Provinces of Holland and West Friesland, but then referred to the Court of The Netherlands in October 1658, with a Judgement by Consent on December 5th, 1658. And that case went entirely in Douw's favour, against the highly favoured joint Complainants Huygens and Coster.
In itself, that is remarkable. Huygens, the Noble patrician, the most famous Dutch scientist, and the self-professed inventor of the pendulum clock, who had in the course of this trial published "Horologium", was forced by the judges to settle the case rather than face unfavourable verdict; also to concede Consent; also one-third Royalties to Douw. It would have been a crushing humiliation for Huygens, the seed of his libels. Subsequently, the Lower Court of Holland, Zeeland and Friesland confirmed to Douw, on December 16th and 19th 1658, their Upper Court's judgement by consent”. * From Keith Piggott

1895 Percival Lowell, convinced about intelligent beings on Mars, published an article in The Atlantic Monthly about how lucky they are to have low gravity. "LUCK OF THE BEING WHO LIVES ON MARS; He Can Do More Work Much More Easily than Man on Earth." *HT Paul Halpern‏ @phalpern

1898 Rudolf Diesel patents an internal combustion engine in the US (filed July 15, 1895.) "My invention has reference to improvements in apparatus for regulating the fuel supply in slow-combustion motors" *Google.com

1975 To display Mexican-Lebanese friendship, Mexico issued a stamp of the Teacher’s Monument in Mexico City by I. Naffa al Rozzi, which shows Cadmus, a mythical Phoenician, teaching the alphabet.



BIRTHS

1537 Franciscus Barocius (9 August 1537 – 23 November 1604) born. In 1560 he published the first important translation of Proclus’ commentary on the first book of Euclid’s Elements. In 1587 he was brought before the Inquisition on charges of sorcery, more particularly of having caused a torrential rainstorm in Crete. *VFR

1602 Gilles de Roberval (August 9, 1602 – October 27, 1675)(His date of birth is given as 8th, 9th and 10th in various sources) was a French scientist who developed powerful methods in the early study of integration.*SAU Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or infinitesimals, which would today be solved by calculus. He worked on the quadrature of surfaces and the cubature of solids, which he accomplished, in some of the simpler cases, by an original method which he called the "Method of Indivisibles"; but he lost much of the credit of the discovery as he kept his method for his own use, while Bonaventura Cavalieri published a similar method which he independently invented.
Another of Roberval’s discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions. (The limacon was named by Roberval in 1650 when he used it as an example of his methods of drawing tangents.)
He also discovered a method of deriving one curve from another, by means of which finite areas can be obtained equal to the areas between certain curves and their asymptotes. To these curves, which were also applied to effect some quadratures, Evangelista Torricelli gave the name "Robervallian lines."
(He also wrote a) work on the system of the universe, in which he supports the Copernican heliocentric system and attributes a mutual attraction to all particles of matter. *Wik
 I was recently informed (2018) by Vincent Panteloni that in France, the balance scale, "We refer to such a scale by saying 'une balance de Roberval'".

1757 Thomas Telford (9 August 1757 Glendinning, Westerkirk, Eskdale, Dumfriesshire, Scotland - 2 September 1834 (aged 77) 24 Abingdon Street, Westminster, London) He is the founder of modern bridge construction, his crowning achievement being the Menai suspension bridge in Wales. Do you know the shape of the cables on a suspension bridge? *VFR

1776 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. 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

1819 Jonathan Homer Lane (August 9, 1819, Geneseo, New York – May 3, 1880, Washington D.C.) U.S. astrophysicist who was the first to investigate mathematically the Sun as a gaseous body. His work demonstrated the interrelationships of pressure, temperature, and density inside the Sun and was fundamental to the emergence of modern theories of stellar evolution.*TIS

1911 William Alfred "Willie" Fowler (August 9, 1911 – March 14, 1995)
American astrophysicist. He should not be confused with the British astronomer Alfred Fowler.
Fowler won the Henry Norris Russell Lectureship of the American Astronomical Society in 1963, the Eddington Medal in 1978, the Bruce Medal in 1979, and the Nobel Prize for Physics in 1983 for his theoretical and experimental studies of the nuclear reactions of importance in the formation of the chemical elements in the universe (shared with Subrahmanyan Chandrasekhar). *TIA

1919 Leona Woods (August 9, 1919 – November 10, 1986), later known as Leona Woods Marshall and Leona Woods Marshall Libby, was an American physicist who helped build the first nuclear reactor and the first atomic bomb.
At age 23, she was the youngest and only female member of the team which built and experimented with the world's first nuclear reactor (then called a pile ), Chicago Pile-1, in a project led by her mentor Enrico Fermi. In particular, Woods was instrumental in the construction and then utilization of geiger counters for analysis during experimentation. She was the only woman present when the reactor went critical. She worked with Fermi on the Manhattan Project, and, together with her first husband John Marshall, she subsequently helped solve the problem of xenon poisoning at the Hanford plutonium production site, and supervised the construction and operation of Hanford's plutonium production reactors.
After the war, she became a fellow at Fermi's Institute for Nuclear Studies. She later worked at the Institute for Advanced Studies in Princeton, New Jersey, the Brookhaven National Laboratory, and New York University, where she became a professor in 1962. Her research involved high-energy physics, astrophysics and cosmology. In 1966 she divorced Marshall and married Nobel laureate Willard Libby. She became a professor at the University of Colorado, and a staff member at RAND Corporation. In later life she became interested in ecological and environmental issues, and she devised a method of using the isotope ratios in tree rings to study climate change. She was a strong advocate of food irradiation as a means of killing harmful bacteria. *Wik

1927 Marvin Minsky (August 9, 1927 - January 24, 2016 (aged 88)) Biochemist and the founder of the MIT Artificial Intelligence Project. Marvin Minsky has made many contributions to AI, cognitive psychology, mathematics, computational linguistics, robotics, and optics. He holds several patents, including those for the first neural-network simulator (SNARC, 1951), the first head-mounted graphical display, the first confocal scanning microscope, and the LOGO "turtle" device. His other inventions include mechanical hands and the "Muse" synthesizer for musical variations (with E. Fredkin). In recent years he has worked chiefly on imparting to machines the human capacity for commonsense reasoning. *TIS He died in Boston of a cerebral hemorrhage .

1940 Linda Goldway Keen (8 August 1940- ) In addition to studying Riemann surfaces, Keen has worked in hyperbolic geometry, Kleinian groups and Fuchsian groups, complex analysis, and hyperbolic dynamics. In the field of hyperbolic geometry, she is known for the Collar lemma.
Keen has worked at the Institute for Advanced Study, Hunter College, University of California at Berkeley, Columbia University, Boston University, Princeton University, and the Massachusetts Institute of Technology, as well as at various mathematical institutes in Europe and South America. After her initial appointment in 1965, in 1974 Keen was promoted to Full Professor at Lehman College and the CUNY Graduate Center.
Keen served as president of the Association for Women in Mathematics during 1985-1986 and as vice-president of the American Mathematical Society during 1992-1995. She served on the Board of Trustees of the American Mathematical Society from 1999-2009 and as Associate Treasurer from 2009-2011. In 1975, she presented an AMS invited address and in 1989 she presented an MAA joint invited address. In 1993 she was selected as a Noether Lecturer. *Wik

1943 Jacques Lewiner, (9 August, 1943 - ) is a French physicist and inventor. He is Professor and Honorary Scientific Director of École supérieure de physique et de chimie industrielles de la ville de Paris (ESPCI ParisTech).
His works have been devoted to electrical insulators and particularly electrets, instrumentation and sensors, for instance in medical imaging, or on the improvement of telecommunication networks.
He has filed a large number of patent applications leading to industrial development, either through licenses granted to industrial companies or through start-up companies often created with former students or researchers. He has participated in the creation of various technology oriented start up companies, for instance Inventel, specializing in Telecommunications, Finsécur which develops and markets fire detection systems, Sculpteo which is an online 3D printing platform, Roowin in the field of chemical synthesis and Cynove in embedded electronics devices. Most of these companies have experienced a strong growth. For instance Inventel, which was the French leader for multimedia gateways was bought by Thomson SA in 2005.
Lewiner is laureate of the French Academy of Sciences in 1990, Knight in the National Order of the Legion of Honor, member of the French Academy of Technologies since 2005 and Honorary Fellow of the Technion. *Wik


DEATHS

1932 John Charles Fields died (May 14, 1863 - August 9, 1932). In his will he left funds for an international medal for contributions to mathematics. The International Congress of Mathematicians in Zurich in 1932 adopted the proposal, and the first Fields Medals were awarded at the Oslo Congress in 1936 to Lars Ahlfors, age 29 of Harvard, and Jesse Douglas, age 39 of Massachusetts Institute of Technology. *VFR It became the most prestigious award for mathematicians, often referred to as the equivalent of a Nobel Prize for mathematicians. As a professor at the University of Toronto, he had worked to bring the International Congress of Mathematicians to Toronto (1924). The Congress was so successful that afterward there was a surplus of about\( $2,500\) which Fields, as chairman of the organizing committee, proposed be used to fund two medals to be awarded at each of future Congresses. This was approved on 24 Feb 1931. He died the following year, leaving \($47,000\) as additional funding for the medals, which have been awarded since 1936.*TIS


1969 Cecil Frank Powell (5 December 1903 – 9 August 1969) British physicist and winner of the Nobel Prize for Physics in 1950 for his development of the photographic method of studying nuclear processes and for the resulting discovery of the pion (pi-meson), a heavy subatomic particle. The pion proved to be the hypothetical particle proposed in 1935 by Yukawa Hideki of Japan in his theory.*TIS

1994 Helena Rasiowa (June 20, 1917 – August 9, 1994) worked in algebraic logic and the mathematical foundations of computer science.*SAU

2006 James Alfred Van Allen (September 7, 1914 – August 9, 2006) American physicist who discovered the Earth's magnetosphere, two toroidal zones of radiation due to trapped charged particles encircling the Earth (also known as the Van Allen radiation belts). During WWII he gained experience miniaturizing electronics, such as in the proximity fuse of a missile. After the war, he studied cosmic radiation, taking advantage of the unused German stock of V2 rockets launched into the outer regions of the atmosphere, carrying research devices using radio to relay back the data gathered. He was also involved in the early U.S. space program, and he had radiation measuring instruments on the first U.S. satellite, Explorer 1, launched 31 Jan 1958 with follow-up carried out by satellites Explorer 3 and 4 later the same year.*TIS

2007 Graham Robert Allan (August 13, 1936 Southgate, London - August 9, 2007 (aged 70)) was an English mathematician, specializing in Banach algebras. He was a reader in functional analysis and vice-master of Churchill College at Cambridge University. *Wik



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

Saturday, 8 August 2020

Not All Math Symbols are Equal, or How "=" Became Ubiquitous

Another from the Archives:2008


I have a friend named Dave Refro who writes and edits questions for one of those high stakes tests that is used for admission into certain graduate programs and uses his job as an excuse for his fascination with archiving old math journal articles. Some folks garden, Dave archives. He spends hours pouring through journals and abstracts and fits together articles with a common theme. If you read almost any math discussion on line, you will probably have come across one of Dave's responses to a question with numerous links to how the question was addressed, discussed, and argued over through history.
Fortunatly for me, Dave sometimes finds an article that he thinks might be of interest to me, and when he gets a stack of them, I get a big present in the mail and my wife knows I will be taking my meals in the den for a few days. In a stack of journal articles he sent recently, (THANKS Dave!) there was one particularly interesting article by Florian Cajori from 1923. In the article Cajori points out two interesting things about the equal sign that every one uses; and that is one of the interesting things he points out, is that EVERYONE uses it. Even in 1923, it was one of the most ubiquitous math symbols in the world and today there are still only about four math symbols that you could write and they would not only be understood, but written exactly the same way whether you found yourself in darkest Africa, the Far East, or downtown Los Angeles. It seems like the perfect symbol, and as Robert Recorde said when he created the symbol in 1557 in his "Whetstone of Witte", the use of "a pair of paralleles, or Gemowe(twin, from the same root as Gemini) lines of one length ... bicause noe 2 thynges can be moare equalle." In fact, Recorde's equal sign had much longer lines than is common today, sort of like == but longer.

Indeed, one wonders why it hadn't been thought of years before, and assume that it immediately became the most common of mathematical notations....ahhh, but not so. The other thing Cajori commented on that I think would surprise young students is that it took over a hundred years for the symbol to become accepted. So what symbol did mathematicians use before the good old == signs? Well, many of them used nothing. The early development of algebra occurred with a very rhetorical approach. When people wanted to write 7x+5 = 26, they would say," the product of seven and some quantity when added to five will equal twenty-six." Ok, they probably said it in Latin, and sometimes they did write numbers in place of the words for numbers, but for equals, they often wrote out the Latin aequales or some variation of it. Frequently they used abbreviations instead of full words and so "p" would stand for plus and "m" for minus...and they would shorten aequales as "aeq" or just "ae". By the time that Recorde had his inspirational stroke, lots of other people had decided THEY had a really good symbol. A pair of vertical lines, ||, was used by Xylander (Wilhelm Holzman) in his translation of Diophantus, Arithmetica only a few years later, and Regiomontanus had used a single horizontal line for equality almost a century earlier. Descartes used the symbol below in hisGeometrie, which was probably drawn from the "ae" abbreviation for aequalis. and Johann Caramuel used equal lines where we would use a decimal point, so Pi would be 3==1415 etc.
 Descartes symbol became a popular competitor on the continent, finding favor with Huygens and the Bernoulli's, while many of the he English mathematicians, Wallis, Barrow, and Newton, followed Recorde's lead. Others used the "gemowe" lines of Recorde for other meanings, Descartes used it to mean +/- in his

So what brought the divided world into a common accord? It took a brand new idea, a revolutionary idea, the calculus. As if by divine providence, the two great minds that created the calculus, almost in unison, tended to publish their versions with a common symbol for equality, Recorde's "gemowe" lines. They disagreed on almost every other symbol they used, but in the last half of the 1600's and the early 1700's the = sign rose to world dominance. In Cajori's words, "The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption."
If you can get your students to understand how long and difficult it is to get mathematicians to accept a symbol, perhaps they will not be too surprised if their College Prof goes into a rant when they use the symbol "ln" for the natural log... and if they accept that the symbol exists (honest, they don't all accept it's use), I can't begin to imagine how they will react if you pronounce it differently than they would. My advice to students; wait for them to say it first!

On This Day in Math - August 8




This result is too beautiful to be false;
it is more important to have beauty in one's equations
than to have them fit experiment.
Paul A M Dirac


The 221st day of the year; 221 the sum of consecutive prime numbers in two different ways 221 = (37 + 41 + 43 + 47 + 53) = (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41)

221221 + 122 is prime, it is the only known number greater than one with this property.

221 is the number of 7-vertex Hamiltonian planar graphs ( a graph that allows a closed path that visits each node exactly once.)
See More Math Facts for every Year Day here.



EVENTS

1576 Laying of the cornerstone of Tycho Brahe’s observatory on the island of Hveen. *VFR

1667. John Evelyn records a visit to Secretary of Royal Society held in Tower of London; "Visited Mr. Oldenburg, a close prisoner in the Tower, being suspected of writing intelligence. I had an order from Lord Arlington, Secretary of State, which caused me to be admitted. This gentleman was secretary to our Society, and I am confident will prove an innocent person." *Diary of John Evelyn
It seems to have been some rash political comment in Oldenburg's own letters during the Dutch war in 1667 which led to his imprisonment for two months in the Tower of London. But he was soon rehabilitated, and resumed his tireless work for the Royal Society, continuing to manage its voluminous correspondence until his death in 1677.

1786 Standards for the decimal system of money established (in the USA). *VFR The first really popular English language arithmetic by an American born author was in 1788 when Nicholas Pike published A New and Complete System of Arithmetic, which he said was "Composed for the use of the citizens of the United States". Well, patriotism probably won't hurt sells in a new country. Pikes book carried endorsements from several noted persons, including the governor of Massachusetts, James Bowdoin, and Yale President Ezra Stiles. The book even included a copy of the Act of Congress of 1786 which created the U. S. Federal Money System with denominations of mills (1/1000 of a dollar), cents, dimes, dollars, and Eagles (ten dollars). With all this emphasis on the new USA, it seems strange that none of the problems in the book involved the new American money, but instead were based on the English system. [The 2nd edition, in 1797, includes in the (very long) title; "adapted to the Federal Currency by Nathaniel Lord, A.M.;Boston]"

1802  We have William Herschel's Diary to thank for the exact date, unfortunately, he only paraphrased the words that passed between Napoleon and Laplace, so we have to sort out the various reports of those who were not present.  My favorite version is told by Thomas Levenson in his "The Hunt for Vulcan..." It goes like this:
Napoleon took a moment during the brief peace of 1802 to engage in a bit of intellectual banter.  He entertained a few savants - Sir William Herschel himself, the distinguished physicist Count Rumford, his minister of the interior - a chemist by profession - Jean Antoine Chaptal, and Laplace.  After engaging politenesses with Herschel, the First Consul next turned to Laplace, who had just published the third volume of Celestial Mecahanics.  Released from matters of state, Napoleon delighted in putting awkard questions to his guests, and so he told his mathematical friend that he had read Newton, and saw that his great book had mentioned God often.  But "I have perused yours, but failed to find his name even once."  Why is that, he asked? 

In the grand tradition of this story, Laplace is reported to have replied, "I have no need of that hypothesis."

In 1854, metal bullet cartridges were patented by Smith & Wesson.*TIS Prior to this, cartridges were formed from paper. In 1776 a schoolmaster in Vienna named Felkel completed a factor table to 408,000 that was intended to be part of a larger work to reach several million. The tables were published at the expense of the Austrian government in the hope that subscriptions would pay for the cost. When the subscriptions failed to meet expectations, the printed volumes were supposedly used for cartridge paper. *Oystein Ore.. Number Theorey and Its History, pg 54

1876 Thomas Alva Edison, of Menlo Park, New Jersey, obtained patent #180,857 for a “method of preparing autographic stencils for printing,” the first mimeograph machine. *VFR Only those of us who have been in the classroom a loooong time will remember these handy machines (and their intoxicating aroma). (addendum; Charles Wells has advised me "The solvent in mimeo ink was castor oil.")

1900 Hilbert delivers his address to the International Congress of Mathematicians. Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics. Hilbert presented only ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the Second International Congress of Mathematicians, speaking on 8 August in the Sorbonne.
"Who of us would not be glad to lift the veil behind which the future is hidden..."
He had provided an extract of the speech (most unusual at that time) in French before hand for those who were not fluent in French.

1931 George Birkhoff publishes “A Set of Postulates for Plane Geometry Based on Scale and Protractor" in Annals of Mathematics. The system has undefined elements of point and line, and undefined relations of distance and angle. (pb)
in 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms. Birkhoff's axiom system was utilized in the secondary-school text Basic Geometry (first edition, 1940) *Wik

1975 The term global warming was probably first used in its modern sense on 8 August 1975 in a science paper by Wally Broecker in the journal Science called "Are we on the brink of a pronounced global warming?". Broecker's choice of words was new and represented a significant recognition that the climate was warming; previously the phrasing used by scientists was "inadvertent climate modification," because while it was recognized humans could change the climate, no one was sure which direction it was going. The National Academy of Sciences first used global warming in a 1979 paper called the Charney Report, which said: "if carbon dioxide continues to increase, [we find] no reason to doubt that climate changes will result and no reason to believe that these changes will be negligible." The report made a distinction between referring to surface temperature changes as global warming, while referring to other changes caused by increased CO2 as climate change.

Global warming became more widely popular after 23 June, 1988 when NASA climate scientist James Hansen used the term in a testimony to Congress. He said: "global warming has reached a level such that we can ascribe with a high degree of confidence a cause and effect relationship between the greenhouse effect and the observed warming." His testimony was widely reported and afterward global warming was commonly used by the press and in public discourse. *Wik

1977 Derek T. Whiteside received the Sarton Medal, the highest honor that the History of Science Society can bestow, for his editorship of The Mathematical Papers of Isaac Newton. In delivering the award Richard S. Westfall said “Before Tom began, Newton’s mathematics was largely a land of myth and fable.” In his 25 years work on the papers, Whiteside has changed all that. *VFR

2013 Duck!!! Oops too late, The new radar images show the asteroid 2005 WK4 as it passed Earth at a safe distance of 1.93 million miles (3.1 million kilometers), which is about 8.2 times the distance between Earth and the moon. The images revealed the large asteroid to be between 660 and 980 feet across (200 to 300 meters), *NASA


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

1802 Germain Henri Hess (August 8, 1802 – November 30, 1850)Swiss-born Russian chemist whose studies of heat in chemical reactions formed the foundation of thermochemistry. He formulated an empirical law, Hess's law of constant heat summation (1840), which states that the heat evolved or absorbed in a chemical process is the same whether the process takes place in one or in several steps. It is explained by thermodynamic theory, which holds that enthalpy is a state function. Chemists have made great use of the law of Hess in establishing the heats of formation of compounds which are not easily formed from their constituent elements. His early investigations concerned minerals and the natural gas found near Baku, and he also discovered the oxidation of sugars to yield saccharic acid.*TIS


1901 Ernest Orlando Lawrence (August 8, 1901 – August 27, 1958) American physicist who was awarded the 1939 Nobel Prize for Physics for his invention of the cyclotron, the first device for the production of high energy particles. His first device, built in 1930 used a 10-cm magnet. He accelerated particles within a cyclinder at high vacuum between the poles of an electromagnetic to confine the beam to a spiral path while a high A.C. voltage increased the particle energy. Larger models built later created 8 x 104 eV beams. By colliding particles with atomic nuclei, he produced new elements and artificial radioactivity. By 1940, he had created plutonium and neptunium. He extended the use of atomic radiation into the fields of biology and medicine. Element 103 was named Lawrencium as a tribute to him.*TIS

1902 Paul Adrien Maurice Dirac, OM, FRS (8 August 1902 – 20 October 1984) English theoretical physicist known for his work in quantum mechanics and for his theory of the spinning electron. In 1933 he shared the Nobel Prize for Physics with the Austrian physicist Erwin Schrödinger. *TIS One of my favorite Dirac anecdotes (of which there are many)
Dirac was watching Anya Kapitza knitting while he was talking physics with Peter Kapitza. A couple of hours after he left, Dirac rushed back, very excited. "You know, Anya," he said, "watching the way you were making this sweater I got interested in the topological aspect of the problem. I found that there is another way of doing it and that there are only two possible ways. One is the one you were using; another is like that. . . . " And he demonstrated the other way, using his long fingers. His newly discovered "other way," Anya informed him, is well known to women and is none other than "purling."

1931 Sir Roger Penrose, British mathematician and theoretical physicist who in the 1960s calculated many of the basic features of black holes.*TIS A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tiles are considered aperiodic tiles. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.*Wik

1952 Roman Juszkiewicz (born 8 August 1952, died 28 January 2012) is a Polish astrophysicist whose work is concerned with fundamental issues of cosmology.
Juszkiewicz's scientific interests include the theory of gravitational instability, origins of the large-scale structure, microwave background radiation and Big Bang nucleosynthesis. He wrote nearly one hundred research papers, mostly in the area of cosmology. Calculated results based on observed motions of pairs of galaxies, obtained in 2000 by Roman Juszkiewicz and the group led by him, aimed at estimating the amount of dark matter in the Universe, were confirmed by the recently published data from the South Pole's ACBAR detector. *Wik



DEATHS

1555 Oronce Fine (20 December 1494 - 8 August 1555) was a French mathematician who published a major work on mathematics and astronomy.*SAU Although primarily a popularizer, Fine was one of the most prolific authors of mathematical books of his age. He worked in a wide range of mathematical fields, including practical geometry, arithmetic, optics, gnomonics, astronomy, and instrumentalism.
He gave the value of pi to be (22 2/9)/7 in 1544. Later, he gave 47/15 and, in De rebus mathematicis (1556), he gave 3 11/78. *Wik

1853 Josef-Maria Hoëné de Wronski (23 August 1776 – 9 August 1853)wrote on the philosophy of mathematics. *SAU He wrote exclusively in French, desirous that his ideas, of whose immortality he was convinced, should be accessible to all; he worked, he said, "through France for Poland." He published over a hundred works, and left many more in manuscript. When dying in the seventy-fifth year of his life, he exclaimed: "God Almighty, there's still so much more I wanted to say!"
In science, Hoene-Wroński set himself maximal tasks: the complete reform of philosophy and of mathematics, astronomy, technology. He not only elaborated a system of philosophy, but applications to politics, history, economics, law, psychology, music, pedagogy. It was his aspiration to reform human knowledge in an "absolute, that is, ultimate" manner.
Though during his lifetime nearly all his work was dismissed as nonsense, some of it has come in later years to be seen in a more favorable light. Although nearly all his grandiose claims were in fact unfounded, his mathematical work contains flashes of deep insight and many important intermediary results. Most significant was his work on series. He had strongly criticized Lagrange's use of infinite series, introducing instead a novel series expansion for a function. His criticisms of Lagrange were for the most part unfounded, but the coefficients in Wroński's new series were found to be important after his death, forming the determinants now known as the Wronskians (the name was given them by Thomas Muir in 1882).
The level of Wroński's scientific and scholarly accomplishments, and the amplitude of his objectives, placed Wroński in the first rank of European metaphysicians in the early 19th century. But the abstractness, formalism and obscurity of his thought, the difficulty of his language, his boundless self-assurance, his uncompromising judgments of others—alienated. He was perhaps the most original of the Polish metaphysicians, but others were more representative of the Polish outlook. *Wik

1989 Taro Morishima (22 April 1903 in Wakayama, Japan - 8 Aug 1989 in Tokyo, Japan)...His passion was algebraic number theory and he had a particular love of Fermat's Last Theorem. His first paper on Fermat's Last Theorem was published in the Proceedings of the Imperial Academy of Japan in 1928. It was the first of 12 papers written in German with the title Über die Fermatsche Vermutung, with 10 of these papers being in the Proceedings of the Imperial Academy of Japan between the years 1928 and 1935. By 1935 he had published a total of 16 papers, the other 6 being: Über den Fermatschen Quotienten (1931); On recent results about Fermat's last Theorem (Japanese) (1932); Über die Einheiten und Idealklassen des Galoisschen Zahlkörpers und die Theorie der Kreiskörper der l-ten Einheitswurzein (1933); and Über die Theorie der Kreiskörper der l-ten Einheitswurzein (1935). He also published a monograph Fermat's Problem (1934) in Japanese. All his papers are full of good ideas but they are extremely difficult to read since Morishima did not present enough detail.
Morishima's high research activity seems to have greatly lessened after 1935. Although difficulties relating to World War II​ and the difficult years in Japan following the war were partly responsible, nevertheless it does appear that he had already reduced his research activities. He did publish the book Higher Algebra in 1940 (in Japanese) but this and one further paper on Fermat's Last Theorem (in 1952) was all in published in the 30 years between 1935 and 1965.*SAU

1996 Sir Nevill F(rancis) Mott (30 September 1905 – 8 August 1996) English physicist who shared (with P.W. Anderson and J.H. Van Vleck of the U.S.) the 1977 Nobel Prize for Physics for his independent researches on the magnetic and electrical properties of amorphous semiconductors. Whereas the electric properties of crystals are described by the Band Theory - which compares the conductivity of metals, semiconductors, and insulators - a famous exception is provided by nickel oxide. According to band theory, nickel oxide ought to be a metallic conductor but in reality is an insulator. Mott refined the theory to include electron-electron interaction and explained so-called Mott transitions, by which some metals become insulators as the electron density decreases by separating the atoms from each other in some convenient way.*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