A Beautiful Polynomial Trick

I was just reminded of a beautiful polynomial relationship in Dan Kalman's wonderful, Unusual Mathematical Excursions, Polynoimia and Related Realms (below).It is not that I'm so clever I had figured this out on my own, but Prof. Dan had showed me this a while before.

If you are teaching quadratics, or polynomials in general right now; and even if you aren't, here is a beautiful little polynomial puzzle that will dazzle your students, and even may teach them something.
Ask one to pick a quadratic or cubic (The idea works with any polynomial but calculations get a little freaky with too many terms) with all positive integer coefficients.  Then tell them if they answer two polynomial evaluations of their function, you will tell them the coefficients.  I will use a cubic to keep my life simple:  suppose they pick f(x)= 2x³ + 3x²+5x+4.

You first ask them to tell you f(1).  This will tell you the sum of the coefficients, in this case 14.  Now ask them to tell you f(14).  When they do, you can convert their answer to base 14 and get the coefficients of the polynomial.   This turns out to be the incredibly large number 6150.  Now to earn their wows, you need to turn this into a base 14 number, but beautifully, 6150 base ten works out to be 2(14³) + 3(14²) + 5(15) + 4.  And the coefficients of the powers are the coefficients of the original polynomial.  Now dividing by 14 and recording the successive remainders gives us the coefficients.....
  6150 / 14 is 439 with rem. of 4. This first remainder is the constant term of the original polynomial.

439/14 is 31 R 5, so 5 becomes the linear term.

31/14 is 2 R 3 , so 3 becomes the quadratic term

and 0/14 is 0 R 2 so our last, and leading, coefficient is 2.

Student's who find this method of converting to base 14 (or any other base) mysterious can be shown the relationship between it and synthetic division by the intermediate of Horner's Method of writing  a polynomial Ax³+Bx² + Cx + D in the form ((Ax+B)x+C)x+D with which we can convert any number from any base, x, to it's decimal equivalent.   It should be quickly obvious that to go back our first remainder will be D, with a quotient of (Ax+B)x+C . 

There is much mystery in this method for the young who are less experienced with dividing polynomials, and I would follow it with the decimal approach shown farther below for young students.  For more mature students, I think it would be wise to show them the decimal method (Which they will usually pick up quickly as trivial), but then show them how to convert the coefficients from the base f(1) and encourage them to prove that it is always true with some simple algebra. 

Before I show the decimal method, here is the simple explanation of why the above works. 

If we start with a cubic, Ax³+ Bx² +Cx +D  and let f(1) = p, then f(p) will be 

Ap³+ Bp² +Cp +D with all of A, B, C and D less than p (since p was the sum of all the coefficients and hence no messy carry over problems) and in base p, the coefficients will be the same as the originals.  The unknowing student gives you this sum in base ten, and you simply convert to base p by repeated division of each quotient by p. 

If prefer you can use a base ten approach that also works.  After you get f(1) and know the sum of the coefficients, you can pick the smallest power of 10 that is greater than f(1).  In the above cubic example we could use 100 since that is greater than f(1).  Now when we ask for f(100) you will get 2030504.  This would work even if one or more of the original coefficients had two digits as long as f(1) was smaller than 100.  My preference was to always tell them that all the coefficients had to be a single digit, and then f(100) would also work.  With that example I could have every student pick his own example of a cubic or quadratic (or for that matter any thing up to a tenth power) and enter them as functions on their calculators, then when they evaluate f(100) they instantly realize their number has been revealed.  For newer algebra students just studying polynomials, this helps them to start to see why the arithmetic they learn with polynomials is so like the arithmetic they learned in elementary school.  

Professor Kalman's book is full of beautiful and interesting ideas regarding polynomials.  I think it would be an excellent addition to the library of any teacher of HS math.  

The image at top is what is commonly called a zeotrope, a device that produces the illusion of a moving image by showing different pictures in rapid succession. It was invented by William George Horner in 1834, for whom Horner's method is named, (but which he did NOT invent and which was used 600 years earlier, by the Chinese mathematician Qin Jiushao). Horner called it 'Daedalum' that translates to 'The wheel of the devil'. The American developer William F. Lincoln names his the 'Zoetrope', which means 'wheel of life'.

On This Day in Math - March 25

*http://agutie.homestead.com/

Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.
~Girolamo Cardano

The 84th day of the year, with nine points equally placed around a circle there are 84 different triangles using three of these points as vertices. Are any of them right triangles?

EVENTS

1539 Tartaglia tells Cardano about his method of solving cubic equations and Cardano signs an oath to keep the method secret, according to Tartaglia. *B L van der Waerden, History of Algebra

"Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccolo Tartaglia of Brescia gave Niccolo occasion to discover it. He [Tartaglia] gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in [various] forms. This was very difficult." Cardano in Ars Magna (Basel, 1545)

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1655 Christiaan Huygens was the first to discover a moon of Saturn, when he viewed Titan (the largest and easiest to see) on 25 Mar 1655. However, the moon wasn't named until almost two centuries later when Sir John Herschel, discoverer of Uranus, assigned names to the seven moons of Saturn that were known at that time. Saturn's largest moon was named simply "Titan," since the word means "one that is great in size, importance, or achievement." *TIS

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1792 D’Alembert wrote: “I would like to see our friend Condorcet, who assuredly has great talent and wisdom, express himself in another manner.” Reading Condorcet’s mathematical works is a thankless task, for the notation is inconsistent, the expression of ideas often imprecise and obscure, and the proofs labored. Perhaps this helps explain why he is not a well known mathematician. [DSB 3, 384]*VFR

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1822 Gauss reveals plans to contact aliens: Gauss wrote of the heliotrope's potential (an instrument invented by Gauss in 1821 that uses a mirror to reflect sunlight over great distances) as a celestial signaling device in a March 25, 1822 letter to Heinrich Olbers, by which he reveals a belief and interest in finding a method to contact extraterrestrial life: "With 100 separate mirrors, each of 16 square feet, used conjointly, one would be able to send good heliotrope-light to the moon.... This would be a discovery even greater than that of America, if we could get in touch with our neighbors on the moon."
Some have suggested Gauss may have also proposed the constructing an immense right triangle and three squares on the surface of the earth to signal to aliens from the Moon or Mars. See more on that story here.*Wik

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In 1857, Frederick Laggenheim took the first photograph of a solar eclipse. This is often reported but seems not to be the first. I found a note on Wikipedia that "The first correctly-exposed photograph of the solar corona was made during the total phase of the solar eclipse of 28 July 1851 by a local daguerreotypist named Berkowski at the Royal Observatory in Königsberg, Prussia (now Kalinigrad in Russia). Berkowski, whose first name was never published, observed at the Royal Observatory. A small 6-cm refracting telescope was attached to the 15.8-cm Fraunhofer heliometer and a 84-second exposure was taken shortly after the beginning of totality. *Wik

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In 1903, The Times newspaper reported that the French physicist, Pierre Curie assisted by Marie Curie, communicated to the Academy of Sciences that the recently discovered Radium “possesses the extraordinary property of continuously emitting heat, without combustion, without chemical change of any kind, and without any change to its molecular structure, which remains spectroscopically identical after many months of continuous emission of heat ... such that the pure Radium salt would melt more than its own weight of ice every hour ... A small tube containing Radium, if kept in contact with the skin for some hours ... produces an open sore, by destroying the epidermis and the true skin beneath ... and cause the death of living things whose nerve centres do not lie deep enough to be shielded from their influence.” *TIS

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1992 Excel 4.0 Spreadsheet Software Released: Microsoft Corporation releases its Excel 4.0 spreadsheet program. Excel was one in a long line of practical applications that Microsoft and other companies developed for personal computers, making them more appealing to home and office users. The earliest commercial computerized spreadsheet was VisiCalc, written by Ed Frankston and Dan Bricklin and released for the Apple II personal computer in 1979.*CHM

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BIRTHS

1538 Christopher Clavius (March 25, 1538 – February 6, 1612) was a German Jesuit mathematician and astronomer who was the main architect of the modern Gregorian calendar. In his last years he was probably the most respected astronomer in Europe and his textbooks were used for astronomical education for over fifty years in Europe and even in more remote lands (on account of being used by missionaries). As an astronomer Clavius held strictly to the geocentric model of the solar system, in which all the heavens rotate about the Earth. Though he opposed the heliocentric model of Copernicus, he recognized problems with the orthodox model. He was treated with great respect by Galileo, who visited him in 1611 and discussed the new observations being made with the telescope; Clavius had by that time accepted the new discoveries as genuine, though he retained doubts about the reality of the mountains on the Moon. Later, a large crater on the Moon was named in his honour.*Wik
Called the Euclid of the sixteenth-century, born in the German town of Bamberg, the see of the prince-bishop of Franconia. He was also the leader of the Gregorian calendar reform. Perhaps his greatest contribution was as an educational reformer.
In his Astrolabium (Rome,1593) he uses a dot to separate whole numbers from decimal fractions, but it would be 20 more years before the decimal point would be widely accepted. Carl Boyer mentions "the Jesuit friend of Kepler" who was the first to use the decimal point with a clear idea of its significance. In the same work, Clavius originated a way of dividing a scale for precise measurements. His idea was adopted by Vernier 42 years later.
In his Algebra (Rome, 1608) Clavius was the first to use parenthesis to express aggregation and the first to use a symbol for an unknown quantity. Other innovations were also seen in the symbols attributed to him by Florian Cajori such as the radical sign, plus and minus signs.

Clavius proposed a proof that there can be no more than three dimensions in geometry, based on the fact that only three concurrent lines can be drawn from a point so that they are mutually perpendicular. He discovered and proved a theorem for a regular polygon with an odd number of sides which two centuries later enabled Carl Friedrich Gauss to construct a 17-sided polygon by ruler and compass.
In hisTriangula sphaerica (Mainz 1611) Clavius summarized all contemporary knowledge of plane and spherical trigonometry. His prostlaphaeresis , the grandparent of logarithms, relied on the sine of the sum and differences of numbers. In this way he was able to substitute addition and subtraction for multiplication, by solving the identity with which we are familiar today: 2 sin x sin y = cos(x-y)-cos(x+y). D. E. Smith gives the details of the proof and emphasizes the impact Clavius' work had on the discovery of logarithms. Smith also underlines the modesty of Clavius in generously giving to one of his contemporaries more credit than is due for his own prostlaphaeresis . *Joseph MacDonnell, S.J., Fairfield Univ webpage
Some really nice detail about Clavius is at Renaissance Mathematicus

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1798 Christoph Gudermann (March 25, 1798, September 25, 1852) was born in Vienenburg. He was the son of a school teacher and became a teacher himself after studying at the University of Göttingen, where his advisor was Karl Friedrich Gauss. He began his teaching career in Kleve and then transferred to a school in Münster.
He is most known today for being the teacher of Karl Weierstrass, who took Gudermann's course in elliptic functions, 1839–1840, the first to be taught in any institute. Weierstrass was greatly influenced by this course, which marked the direction of his own research.
Gudermann originated the concept of uniform convergence, in an 1838 paper on elliptic functions, but only observed it informally, neither formalizing it nor using it in his proofs. Instead, Weierstrass elaborated and applied uniform convergence.
His researches into spherical geometry and special functions focused on particular cases, so that he did not receive the credit given to those who published more general works. The Gudermannian function, or hyperbolic amplitude, is named after him.Gudermann died in Münster. *Wik

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1833 (Henry Charles) Fleeming Jenkin (25 Mar 1833; 12 Jun 1885 at age 52) British engineer noted for his work in establishing units of electrical measurement. After earning an M.A. (1851), he worked for the next 10 years with engineering firms engaged in the design and manufacture of submarine telegraph cables and equipment for laying them. In 1861 his friend William Thomson (later Lord Kelvin) procured Jenkin's appointment as reporter for the Committee of Electrical Standards of the British Association for the Advancement of Science. He helped compile and publish reports that established the ohm as the absolute unit of electrical resistance and described methods for precise resistance measurements. *TIS

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1859 Samuil Shatunovsky (25 March 1859 – 27 March 1929) was a Russian mathematician. focused on several topics in mathematical analysis and algebra, such as group theory, number theory and geometry. Independently from Hilbert, he developed a similar axiomatic theory and applied it in geometry, algebra, Galois theory and analysis.[1] However, most of his activity was devoted to teaching at Odessa University and writing associated books and study materials.*Wik

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1865 Pierre-Ernest Weiss (25 Mar 1865, 24 Oct 1940) French physicist who investigated magnetism and determined the Weiss magneton unit of magnetic moment. Weiss's chief work was on ferromagnetism. Hypothesizing a molecular magnetic field acting on individual atomic magnetic moments, he was able to construct mathematical descriptions of ferromagnetic behaviour, including an explanation of such magnetocaloric phenomena as the Curie point. His theory succeeded also in predicting a discontinuity in the specific heat of a ferromagnetic substance at the Curie point and suggested that spontaneous magnetization could occur in such materials; the latter phenomenon was later found to occur in very small regions known as Weiss domains. His major published work was Le magnetisme ( 1926).*TIS

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1923 Kenneth Linn Franklin (25 Mar 1923, ) American astronomer who discovered that the giant planet Jupiter emits radio waves (1955). Dr. Bernard F. Burke and Franklin, astronomers at the Carnegie Institution in Washington, were scanning the sky for radio waves from galaxies. By chance, they found a radio signal that resembled short bursts of static, similar to interference by lightning on home radios. After weeks of study, finding the signals were periodic, four minutes earlier each day, they pin-pointed Jupiter as the source. Never before had radio sounds from a planet in our solar system been detected. Later it was discovered that the radio waves were circularly polarized, so a magnetic field was involved.*TIS

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1939 Richard Alfred Tapia (March 25, 1939 - ) is a renowned American mathematician and champion of under-represented minorities in the sciences. In recognition of his broad contributions, in 2005, Tapia was named "University Professor" at Rice University in Houston, Texas, the University's highest academic title. The honor has been bestowed on only six professors in Rice's ninety-nine year history. On September 28, 2011, President Barack Obama announced that Tapia was among twelve scientists to be awarded the National Medal of Science, the top award the United States offers its researchers. Tapia is currently the Maxfield and Oshman Professor of Engineering; Associate Director of Graduate Studies, Office of Research and Graduate Studies; and Director of the Center for Excellence and Equity in Education at Rice University.
Tapia's mathematical research is focused on mathematical optimization and iterative methods for nonlinear problems. His current research is in the area of algorithms for constrained optimization and interior point methods for linear and nonlinear programming.*Wik

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DEATHS

1818 Caspar Wessel (8 Jun 174525 Mar 1818 at age 72) was a Norwegian mathematician who invented a geometric way of representing complex numbers which pre-dated Argand. *SAU
His fundamental paper, Om directionens analytiske betegning, was published in 1799 by the Royal Danish Academy of Sciences and Letters. Since it was in Danish, it passed almost unnoticed, and the same results were later independently found by Argand and Gauss.
One of the more prominent ideas presented in "On the Analytical Representation of Direction" was that of vectors. Even though this wasn't Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added if we unite them in such a way that the second line begins where the first one ends and then pass a straight line from the first to the last point of the united lines. This line is the sum of the united lines". This is the same idea as used today when summing vectors. Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognized. His paper was re-issued in French translation in 1899, and in English in 1999 as On the analytic representation of direction (ed. J. Lützen et al.).*Wik

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1995 James S(amuel) Coleman (12 May 1926, 25 Mar 1995 at age 68) was a U.S. sociologist, a pioneer in mathematical sociology whose studies strongly influenced education policy. In the early 1950s, he was as a chemical engineer with Eastman-Kodak Co. in Rochester, N.Y. He then changed direction, fascinated with sociology and social problems. In 1966, he presented a report to the U.S. Congress which concluded that poor black children did better academically in integrated, middle-class schools. His findings provided the sociological underpinnings for widespread busing of students to achieve racial balance in schools. In 1975, Coleman rescinded his support of busing, concluding that it had encouraged the deterioration of public schools by encouraging white flight to avoid integration. (We can never control the law of unintended consequences)*TIS

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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

On This Day in Math - March 24

Harrison H1

But mathematics is the sister, as well as the servant, of the arts and is touched by the same madness and genius.
~Marston Morse

The 83rd day of the year; 83 is the smallest prime number which is the sum of a prime number of consecutive prime numbers in a prime number of different ways, i.e., 23 + 29 + 31 = 11 + 13 + 17 + 19 + 23. *Prime Curios (Whew! say that three times in a hurry)

EVENTS

1789 Throughout his life, Jefferson was avid to keep up with the mathematical world, and to spread knowledge about it to others. How deeply he explored mathematics depended obviously on what else was happening in his life at the time, but he was always keen to pass on what he had learned to his correspondents. Staying in Paris in 1789 he was eager to pass on information about the latest work by Lagrange In a letter to Harvard President Joseph Willard on March 24, 1789 he writes, "A very remarkeable work is the 'Mechanique Analytique' of La Grange in 4to. He is allowed to be the greatest mathematician now living, and his personal worth is equal to his science. The object of his work is to reduce all the principles of Mechanics to the single one of the Equilibrium, and to give a simple formula applicable to them all. The subject is treated in the Algebraic method, without diagrams to assist the conception. My present occupation not permitting me to read any thing which requires a long and undisturbed attention, I am not able to give you the character of this work from my own examination. It has been received with great approbation in Europe." *John Fauval, Lecture at Univ of Va.

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1899 Ren´e Louis Baire defended his doctoral thesis on the theory of functions of a real variable. He was influential in introducing transfinite set theory into analysis. *VFR

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1930 Planet X  was officially named Pluto on March 24, 1930:  On the nights of Jan 23 and 30th of January, 1930, Tombaugh found a planet in the images that he thought was the Planet X. "The discovery made front page news around the world. The Lowell Observatory, who had the right to name the new object, received over 1000 suggestions, from "Atlas" to "Zymal". Tombaugh urged Slipher to suggest a name for the new object quickly before someone else did. Name suggestions poured in from all over the world. Constance Lowell proposed Zeus, then Lowell, and finally her own first name. These suggestions were disregarded.
The name "Pluto" was proposed by Venetia Burney (later Venetia Phair), an eleven-year-old schoolgirl in Oxford, England. Venetia was interested in classical mythology as well as astronomy, and considered the name, one of the alternate names of Hades, the Greek god of the Underworld, appropriate for such a presumably dark and cold world. She suggested it in a conversation with her grandfather Falconer Madan, a former librarian of Oxford University's Bodleian Library. Madan passed the name to Professor Herbert Hall Turner, who then cabled it to colleagues in America. *Wik

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1959 TI Demonstrates Integrated Circuit Invented by Jack Kilby:
Texas Instruments demonstrates the first integrated circuit. Its inventor, Jack Kilby (b. Nov 8, 1923), created the device to prove that resistors and capacitors could exist on the same piece of semiconductor material. His circuit consisted of a sliver of germanium with five components linked by wires. It was Fairchild's Robert Noyce, however, who filed for a patent within months of Kilby and who made the IC a commercially-viable technology. Both men are credited as co-inventors of the IC.*CHM

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BIRTHS 1693 John Harrison (24 Mar 1693; 24 Mar 1776 at age 83)
English horologist who invented the first practical marine chronometer, which enabled navigators to compute accurately their longitude at sea. He was prompted to begin this work after a huge reward was offered by the British government for new navigational tools to avoid further disasters at sea. John Harrison took on the scientific and academic establishment of his time and won the longitude prize through extraordinary mechanical insight, talent and determination. *TIS [The Dictionary of Scientific Biographies  shows an uncertainty in the date of birth as 24 Mar(?) 1693.]

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1809 Joseph Liouville   (24 Mar 1809, 8 Sep 1882) French mathematician who discovered transcendental numbers (those which are not the roots of algebraic equations having rational coefficients), and that there are infinitely many of them. He also did work in real and complex analysis, number theory, and differential geometry. His name is remembered in the Sturm-Liouville theory of differential equations that generalizes Joseph Fourier's ideas, and are important in mathematical physics. He studied celestial mechanics. Liouville founded in 1836, and edited for nearly four decades, the Journal de Mathématique which remains a leading French mathematical publication. He edited and published (1843) the manuscripts left behind upon the untimely death of Evariste Galois 22 years earlier.*TIS Liouville was one of Lord Kelvin's mathematical heroes, and he once stopped a lecture in Glascow to ask his students, "Do you know what a mathematician is?" He then wrote on the blackboard the equation

and, pointing to the board stated, A mathematician is one to whom that  is as obvious as twice two are four is to you.  Liouville was a  mathematician." *Walter Gratzer, Eurekas and Euphorias, pg 21

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1835 Josef Stefan (24 Mar 1835, 7 Jan 1893) Austrian physicist who proposed a law of radiation (1879) stating that the amount of energy radiated per second from a black body is proportional to the fourth power of its absolute temperature. (A black body is a theoretical object that absorbs all radiation that falls on it.) This law is known as Stefan's law or the Stefan-Bolzmann law. He also studied electricity, the kinetic theory of gases and hydrodynamics.*TIS

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1848 Jules Tannery (March 24, 1848 – December 11, 1910) was a French mathematician who notably studied under Charles Hermite and was the PhD advisor of Jacques Hadamard.
He discovered a surface of the fourth order of which all the geodesic lines are algebraic. He was not an inventor, however, but essentially a critic and methodologist. He once remarked, "Mathematicians are so used to their symbols and have so much fun playing with them, that it is sometimes necessary to take their toys away from them in order to oblige them to think."
He notably influenced Paul Painlevé, Jules Drach, and Émile Borel to take up science.
His efforts were mainly directed to the study of the mathematical foundations and of the philosophical ideas implied in mathematical thinking.*Wik

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1892 Harold Calvin Marston Morse (24 March 1892 in Waterville, Maine, USA - 22 June 1977 in Princeton, New Jersey, USA) developed variational theory in the large with applications to equilibrium problems in mathematical physics, a theory which is now called Morse theory and forms a vital role in global analysis*SAU

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1893 Walter Baade (24 Mar 1893; 25 Jun 1960 at age 67) German-American astronomer who, with Fritz Zwicky, proposed that supernovae could produce cosmic rays and neutron stars (1934), and Baade made extensive studies of the Crab Nebula and its central star. During WW II blackouts of the Los Angeles area Baade used the 100-inch Hooker telescope to resolve stars in the central region of the Andromeda Galaxy for the first time. This led to his definition of two stellar populations, to the realization that there were two kinds of Cepheid variable stars, and from there to a doubling of the assumed scale of the universe. Baade and Rudolph Minkowski identified and took spectrograms of optical counterparts of many of the first-discovered radio sources, including Cygnus A and Cassiopeia A. *TIS

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1941 Joseph H. Taylor Jr. (24 Mar 1941,   )American radio astronomer and physicist who, with Russell A. Hulse, was the corecipient of the 1993 Nobel Prize for Physics for their joint discovery of the first binary pulsar (1974). This unique phenomenon, two stars orbiting each other - one of them giving off regular radio-frequency "beeps" - has been important as a deep space proving ground for Einstein's general theory of relativity. Their research group at Princeton used the 1,000 foot radio telescope at Arecibo, Puerto Rico, the largest and most sensitive in the world for catching radio waves from space. *TIS

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1776 John Harrison (24 Mar 1693; 24 Mar 1776 at age 83)
English horologist who invented the first practical marine chronometer, which enabled navigators to compute accurately their longitude at sea. He was prompted to begin this work after a huge reward was offered by the British government for new navigational tools to avoid further disasters at sea. John Harrison took on the scientific and academic establishment of his time and won the longitude prize through extraordinary mechanical insight, talent and determination. *TIS
Dava Sobel's book, below, is excellent.

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1948 Alice Chang (24 March 1948 in Ci-an, China)is a Chinese American mathematician specializing in aspects of mathematical analysis ranging from harmonic analysis and partial differential equations to differential geometry. She is a professor of mathematics and chair of the department at Princeton University.*Wik

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DEATHS

1956 Christine Mary Hamill (July 24, 1923 – March 24, 1956) was an English mathematician who specialized in group theory and finite geometry. After receiving her Ph.D. at the University of Cambridge in 1951, she was appointed to a lectureship in the University of Sheffield and later was appointed lecturer in the University College, Ibadan, Nigeria.*Wik

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1956 Sir Edmund Taylor Whittaker (24 Oct 1873; March 24 1956) English mathematician who made pioneering contributions to the area of the special functions, which is of particular interest in mathematical physics. Whittaker is best known work is in analysis, in particular numerical analysis, but he also worked on celestial mechanics and the history of applied mathematics and physics. He wrote papers on algebraic functions and automorphic functions. His results in partial differential equations (described as most sensational by Watson) included a general solution of the Laplace equation in three dimensions in a particular form and the solution of the wave equation. On the applied side of mathematics he was interested in relativity theory and he also worked on electromagnetic theory. *TIS

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1995 (Noël) Joseph (Terence Montgomery) Needham (9 Dec 1900, 24 Mar 1995 at age 94)was an English biochemist, embryologist, and historian of science who wrote and edited the landmark history Science and Civilization in China, a remarkable multivolume study of nearly every branch of Chinese medicine, science, and technology over some 25 centuries. As head of the British Scientific Mission in China (1942-46) he worked to assure adequate liaison between Chinese scientists and technologists and their colleagues in the West. As an historian of science and technology he wanted to break through the parochial, Europe-centred views of most of his colleagues by disclosing the achievements of traditional China and the contributions made by China leading up to the scientific revolution. *TIS

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1976 Francis Dominic Murnaghan (4 Aug 1893 in Omagh, Co. Tyrone, Ireland- 24 March 1976 in Baltimore, Maryland, USA) was an Irish mathematician, former head of the mathematics department at Johns Hopkins University. His name is attached to developments in group theory and mathematics applied to continuum mechanics (Murnaghan and Birch–Murnaghan equations of state).*SAU

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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

On This Day in Math - March 23

model of third Rhine bridge at Schaffhauser *TIS

Science does not have a moral dimension. It is like a knife. If you give it to a surgeon or a murderer, each will use it differently.
~Wernher von Braun

The 82nd day of the year; 82 is the sum of the 10th(8+2) prime and the 16th(8x2) prime. It is the smallest number with this property. Can you find the next?

EVENTS

4BC A lunar eclipse may have coincided with the death of one of the most notorious kings of all time. Historian Flavius Josephus notes that an eclipse of the Moon preceded the death of the biblical king Herod. Three eclipses fit the bill as occurring in the right time frame and being visible from the Middle East, but the favorite contender is the March 23, 4 B.C. rising total lunar eclipse that may have marked the demise of Herod.*12 Famous Eclipses in History

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*fotoart

In 1840, Englishman J.W. Draper took the first successful photo of the Moon. He made a daguerreotype, a precursor of the modern photograph.*TIS

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1881 J J Sylvester writes to Arthur Cayley to announce, "I believe that I have proved Gordan's Theorem, and can assign a superior limit to the number of fundamental invariants." His proof was founded on the prospect that a certain sequence increased without bound. By October he knew that it did not.

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1920 When Professor Johann Palisan of the University of Vienna discoved an asteroid on this date, he chose to name it after Herbert Hoover, who would later become the President of the USA. As head of the American Relief Administration, after WWI, Hoover organized shipments of food for millions of starving people in Central Europe. In Finnish it is common to use the word “hoover” with the meaning, “to help”. (at least it was, can someone tell me if this is still so) It seems strange that today most history books in his home country preserve his name in the term “Hooverville’s” for hobo camps and shanty towns where displaced peoples lived during the American Great Depression. *Wik

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In 1950, the U.N. World Meteorological Organization was established. *TIS The World Meteorological Organization (WMO) is a specialized agency of the United Nations. It is the UN system's authoritative voice on the state and behaviour of the Earth's atmosphere, its interaction with the oceans, the climate it produces and the resulting distribution of water resources. *WMO webpage

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1981 The March 23 issue carried the first mention of Rubik's Cube in Time Magazine. *Mark Longridge, A Rubik's Cube Chronology
Rubik had begun distribution in Hungary in 1977, and by early 1979 Mathematical Intelligencer carried an introduction in of "The Hungarian Magic Cube" by David Singmaster with the note, "A new mathematical toy has been slowly becoming available in western Europe and is becoming more popular than the Soma Cube, Instant Insanity, and may well surpass the popularity of Mastermind or Sam Loyd's Fifteen puzzle."

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In 1989, fusion at room temperature was claimed by Martin Fleischmann and Stan Pons, two Utah electrochemists. They believed they had sustained a controlled nuclear fusion reaction in a bench-top fusion percolator made up of two electrodes with heavy water which generated up to 100 per cent more energy than they put in. There were sporadic sightings of excess heat, which Fleischmann said cannot be accounted for by chemistry alone. However, the idea of cold fusion was discredited because leading scientists were unable to replicate the work and found no hallmarks of nuclear processes, especially none of the subatomic particles called neutrons. Their tantalizing promise of a limitless supply of cheap energy were invalid.*TIS

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2012 Sonia Kovalevsky Day.. "Her birthday is January 15, but today gets to be her day. Cathy O'Neil (aka Mathbabe) started Sonia Kovalevsky Day at Barnard College in 2006, and it sounds like it's been going strong ever since." *Sue VanHattum at MathMama Writes blog.

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BIRTHS

1709 Hans Ulrich Grubenmann (23 Mar 1709; 24 Jan 1783 at age 73)  Swiss carpenter, who with his brother Johannes, built a bridge (1758) over the Limmat River at the town of Wettingen, near Zürich, that is believed to be the first timber bridge to employ a true arch in its design. The brothers' ingenious combination of the arch and truss principles made it possible to construct bridges longer and better than ever before. They constructed churches as well as other bridges. *TIS

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1754 (Jurij)Georg Freiherr von Vega (23 Mar 1754 in Zagorica, Ljubljana, Slovenia - 26 Sept 1802 in Vienna, Austria) wrote about artillery but he is best remembered for his tables of logarithms and trigonometric functions. Vega calculated π to 140 places, a record which stood for over 50 years. This appears in a paper which he published in 1789.
In September 1802 Vega was reported missing. A search was unsuccessful until his body was found in the Danube near Vienna. The official cause of death was an accident but many suspect that he was murdered. *SAU

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1795 Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. Holmboe was hired as a mathematics teacher at the Christiania Cathedral School in 1818, where he met the future renowned mathematician Niels Henrik Abel. Holmboe's lasting impact on mathematics worldwide has been said to be his tutoring of Abel, both in school and privately. The two became friends and remained so until Abel's early death. Holmboe moved to the Royal Frederick University in 1826, where he worked until his own death in 1850.
Holmboe's significant impact on mathematics in the fledgling Norway was his textbook in two volumes for secondary schools. It was widely used, but faced competition from Christopher Hansteen's alternative offering, sparking what may have been Norway's first debate about school textbooks. *Wik

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1827 Pierre Simon, Marquis de Laplace (23 Mar 1749, 5 Mar 1827 at age 78) was a French mathematician, physicist, statistician and astronomer known for his mathematical analysis of the stability of the solar system (1773), alleviating Isaac Newton's concerns about perturbations between planets. He took an exact approach to science. He developed an explanation of surface tension of a liquid in terms of inter-molecular attractions, investigated capillary action and the speed of sound. He assisted Antoine Lavoisier (1783) investigating specific heat and heats of combustion, initiating the science of thermochemistry. He believed the solar system formed from a collapsing nebula. He contributed to the mathematics of probability and calculus, in which a differential equation is known by his name, and was involved in establishing the metric system.*TIS His last words were, “What we know is very slight; what we don’t know is immense.” *Eves, Revisited, 319◦

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1829 Norman Robert Pogson (23 Mar 1829; 23 Jun 1891 at age 62) English astronomer who devised the magnitude scale of the brightness of stars (1850) now in use. He divided the classical scale in which a first magnitude star is one hundred times brighter than a sixth magnitude star using five integer steps. Each step represents a fifth-root of 100 (about 2.512) increase in brightness. The Sun's magnitude on this scale is -26.91, whereby negative numbers denote objects brighter than first magnitude. Sirius is magnitude -1.58, Aldebaran is 1 and the faintest star detected is 30. His interest in astronomy began in his youth; by age 18 he had calculated orbits for two comets. He discovered 8 asteroids, 21 new variable stars and compiled a massive star catalogue. In 1860 he moved to India for the remainder of his life's work.*TIS

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1837 Richard Anthony Proctor (23 Mar 1837, 12 Sep 1888) English astronomer who first suggested (1873) that meteor impacts caused lunar craters, rather than volcanic action. He studied the motion of stars, their distribution, and their relation to the nebulae. In 1867 he prepared a map of the surface of Mars on which he named continents, seas, bays and straits (in the same manner that Riccioli used on his map of the moon). However, he did not perceive "canals" on the surface, which later Schiaparelli identified. Proctor participated in expeditions of 1874 and 1882 to observe the transit of Venus. He was very successful popularizing astronomy by his writings in books, periodicals, and lectures he gave as far abroad as Australia and America (where he stayed after 1881).*TIS

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1855 Franklin H(enry) Giddings (23 Mar 1855; 11 Jun 1931 at age 76) American sociologist, one of the first in the United States to turn sociology from a branch of philosophy into a research science dependent on statistics. He was noted for his doctrine of the "consciousness of kind," which he derived from Adam Smith's conception of "sympathy," or shared moral reactions. His explanation of social phenomena was based this doctrine - his theory that each person has an innate sense of belonging to particular social groups. He encouraged statistical studies in sociology. *TIS

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1862 Eduard Study (23 March 1862 in Coburg, Germany - 6 Jan 1930 in Bonn, Germany)Study became a leader in the geometry of complex numbers. He reformulated, independently of Severi, the fundamental principles of enumerative geometry due to Schubert. He also worked on invariant theory helping to develop a symbolic notation. In 1923 he published important work on real and complex algebras of low dimension publishing these results. Study's contribution is summarized by W Burau as follows, "... Study demonstrated what he considered to be a thorough treatment of a problem. ... With Corrado Segre, Study was one of the leading pioneers in the geometry of complex numbers. ... Adept in the methods of invariant theory ... Study, employing the identities of the theory, sought to demonstrate that geometric theorems are independent of coordinates. ... Study was the first to investigate systematically all algebras possessing up to four generators over R and C. "
Other areas which Study worked on were straight lines in elliptic space, with his student at Bonn J L Coolidge, and he simplified the method of differential operators. In 1903 he published Geometrie der Dynamen which considered euclidean kinematics and the mechanics of rigid bodies. *SAU

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1882 Amalie Emmy Noether (23 Mar 1882; 14 Apr 1935 at age 53) German mathematician best known for her contributions to abstract algebra, in particular, her study of chain conditions on ideals of rings. In theoretical physics, she produced Noether's Theorem, which proves a relationship between symmetries in physics and conservation principles. This basic result in the general theory of relativity was praised by Einstein. It was her work in the theory of invariants which led to formulations for several concepts of Einstein's general theory of relativity. For her obituary in The New York Times, Albert Einstein wrote: “Fraulein Noether was the most significant mathematical genius thus far produced since the higher education of women began. *TIS Someone once described her as the daughter of the mathematician Max Noether. To this Edumund Landau replied “Max Noether was the father of Emmy Noether. Emmy is the origin of coordinates in the Noether family.” *H. Eves, Introduction to the History of Mathematics
Emmy Noether’s house in Erlangen is in a blog at The Renaissance Mathematicus

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1897 John Lighton Synge (March 23, 1897–March 30, 1995) was an Irish mathematician and physicist. Synge made outstanding contributions to different fields of work including classical mechanics, general mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical networks, mathematical methods, differential geometry, and Einstein's theory of relativity. He studied an extensive range of mathematical physics problems, but his best known work revolved around using geometrical methods in general relativity.
He was one of the first physicists to seriously study the interior of a black hole, and is sometimes credited with anticipating the discovery of the structure of the Schwarzschild vacuum (a black hole).
He also created the game of Vish in which players compete to find circularity (vicious circles) in dictionary definitions. *Wik

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1907 Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes. *SAU

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1912 Wernher Magnus Maximilian von Braun (23 Mar 1912; 16 Jun 1977 at age 65) was a German-American rocket engineer who was one of the most important developers of rockets and their evolution to applications in space exploration. His interest began as a teenager in Germany, and during WW II he led the development of the deadly V–2 ballistic missile for the Nazis (which role remains controversial). After war, he was taken to use his knowledge to produce rockets for the U.S. Army. In 1960, he transferred to the newly formed NASA and became director of Marshall Space Flight Center and chief architect of the Saturn V launch vehicle used to put men on the moon. His contributions include the Explorer satellites; Jupiter, Pershing, Redstone and Saturn rockets, and Skylab. *TIS "My experiences with science led me to God. They challenge science to prove the existence of God. But must we really light a candle to see the sun? "

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1928 Computer Pioneer Jean Sammet Is Born :
Jean Sammet, an early pioneer of computing, is born in New York. Sammet attended Mount Holyoke College and the University of Illinois, where she launched a teaching career. Trained in math, she moved into industry in 1961, developing the language FORMAC at IBM. The language was the first commonly used language for manipulating non-numeric algebraic expressions. She also wrote one of the classic histories of programming languages in her book, "Programming Languages: History and Fundamentals." *CHM

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DEATHS

1924 Thomas Corwin Mendenhall (4 Oct 1841, 23 Mar 1924 at age 82) American physicist and meteorologist who was the first to propose the use of a ring pendulum for measuring absolute gravity. From 1889 to 1894 he served both as Director of the U.S. Coast and Geodetic Survey and also Superintendent of the U.S. Standard Weights and Measures where he oversaw the shift in the fundamental standards of the U.S. from the English yard and pound to the international meter and kilogram. Mendenhall devised a quarter second's pendulum for gravity measurements and instituted improvements in the measurement of base lines with wire tapes, in the construction of instruments for precise leveling and in the methods used in triangulation and gravity work, and developed a comprehensive plan for the study of terrestrial magnetism. *TIS

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1945 Sir (William) Napier Shaw (4 Mar 1854; 23 Mar 1945 at age 90) was an English meteorologist who applied his training in mathematics. He studied the upper atmosphere, using instruments carried by kites and high-altitude balloons. He measured (1906) the movement of air in two anti-cyclones, finding descent rates of 350 and 450 metres per day. He calculated the reduction in pressure due to a certain depression to correspond to the removal of two million million tons of air. He introduced the millibar unit for measurement of air pressure (1000 millibar = 1 bar = 1 standard atmosphere) and the tephigram to illustrate the temperature of a vertical profile of the atmosphere. He also co-authored an early work on atmospheric polluiton, The Smoke Problem of Great Cities (1925).*TIS

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1946 Gilbert Newton Lewis (23 Oct 1875, 23 Mar 1946 at age 70) American chemist who collaborated with Irving Langmuir in developing an atomic theory. He developed a theory of valency, which introduced the covalent bond (c. 1916), whereby a chemical combination is made between two atoms by the sharing of a pair of electrons, one contributed from each atom. This was part of his more general octet theory, published in Valence and the Structure of Atoms and Molecules (1923). Lewis visualized the electrons in an atom as being arranged in concentric cubes. The sharing of these electrons he illustrated in the Lewis dot diagrams familiar to chemistry students. He generalized the concept of acids and bases now known as Lewis acids and Lewis bases. *TIS

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Max Mason (26 Oct 1877; 23 Mar 1961) American mathematical physicist, educator, and science administrator. During World War I he invented several devices for submarine detection - several generations of the Navy's "M," or multiple-tube, passive submarine sensors. This apparatus focused sound to ascertain its source. To determine the direction from which the sound came, the operator needed only to seek the maximum output on his earphones by turning a dial. The final device had a range of 3 miles. Mason's special interest and contributions lay in mathematics (differential equations, calculus of variations), physics (electromagnetic theory), invention (acoustical compensators, submarine-detection devices), and the administration of universities and foundations. *TIS

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1963 Thoralf Skolem,(23 May 1887 in Sandsvaer, Buskerud, Norway - 23 March 1963 in Oslo, Norway) number theorist and logician. At the International Congress of Mathematicians in Cambridge in 1950 he said “We ought not to regard all that is written in the traditional textbooks as something sacred.” It was this attitude that earlier allowed him to discover that the real numbers could have countable models, a fact known as Skolem’s paradox. *Wik This Norwegian logician was the first to introduce non-standard models of the natural numbers. *VFR

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1979 Ivo Lah (AKA Ivan Lah; September 5, 1896 Štrukljeva vas near Cerknica, Austria-Hungary, now Slovenia,– March 23, 1979, Ljubljana, SFR Yugoslavia, now Slovenia) was a Slovenian mathematician and actuary, best known for his discovery of the Lah numbers in 1955. His scientific bibliography contains about 120 items covering a wide spectrum of topics from Mathematics to Statistics, Demographics, etc. For instance one can find 10 items in Maths Reviews, and 19 items in Zentralblatt für Mathematik. His most important mathematical result, published in 1955, is the so-called "Lah identity" where he shows how the rising powers can be expressed in terms of falling powers. The reviewer of his paper was a leading combinatorialist of that time, John Riordan. *Wik Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers.*Wik

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2007 Paul Joseph Cohen (2 April 1934 in Long Branch, New Jersey, USA- 23 March 2007 in Palo Alto, California, USA) Cohen used a technique called forcing to prove the independence in set theory of the axiom of choice and of the generalized continuum hypothesis. *SAU

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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