On This Day in Math - June 19

The more I see of men, the better I like my dog.

Blaise Pascal (over 350 years before Carrie Underwood)


EVENTS
240  In 240 BC, Eratosthenes, a Greek astronomer and mathematician, estimated the circumference of the earth. As the director of the great library of Alexandria, he read in a papyrus book that in Syene, approaching noon on the summer solstice, the longest day of the year, shadows of temple columns grew shorter. At noon, they were gone. The sun was directly overhead. However, a stick in Alexandria, far to the north, could cast a pronounced shadow. Thus, he realized that the surface of the Earth could not be flat. It must be curved. Not only that, but the greater the curvature, the greater the difference in the shadow lengths. By measurement on the ground and application of geometry, he calculated the circumference of the earth. *TIS

325 The early Christian church opened the council of Nicaea, which decided the rules for computing the date of Easter: The first Sunday after the first full moon on or after the vernal equinox *VFR 

In 1963, Soviet cosmonaut Valentina Tereshkova returned to Earth after spending nearly three days as the first woman in space. She had been interested in parachute jumping when she was young, and that expertise was one of the reasons she was picked for the cosmonaut program. She became the first person to be recruited without experience as a test pilot. On 16 Jun 1963, Tereshkova was launched into space aboard Vostok 6, and became the first woman to travel in space. Her radio name was "Chaika," Russian for "seagull." Her flight made 48 orbits of Earth. Tereshkova never made a second trip into space. She became an important member of the Communist Party and a representative of the Soviet government.*TIS

BIRTHS

1623 Blaise Pascal born in Ferrand, Auvergne, France.  He laid the foundation for the modern theory of probabilities. In hydrodynamics he formulated what came to be known as Pascal's law of pressure, and invented the syringe and hydraulic press. Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years (1642-45). The device, called the Pascaline, resembled a mechanical calculator of the 1940s. This, almost certainly, makes Pascal the second person to invent a mechanical calculator for Schickard had manufactured one in 1624. He died at the young age of 39 having been sickly and physically weak through life. Autopsy showed he had been born with a deformed skull.*TIS

1669 Leonty Magnitsky was a Russian teacher who wrote the first guide to mathematics published in Russia.*SAU

1771 Joseph Gergonne born. He came under the influence of Gaspard Monge, the Director of the new École Polytechnique in Paris. In 1810, in response to difficulties he encountered in trying to publish his work, Gergonne founded his own mathematics journal, officially named the Annales de mathématiques pures et appliquées but generally referred to as the Annales de Gergonne. The most common subject of articles in his journal was geometry, Gergonne's specialty. Over a period of 22 years, the Annales de Gergonne published about 200 articles by Gergonne himself, and other articles by many distinguished mathematicians, including Poncelet, Servois, Bobillier, Steiner, Plücker, Chasles, Brianchon, Dupin, Lamé, even Galois.
Gergonne was appointed to the chair of astronomy at the University of Montpellier in 1816. In 1830, he was appointed Rector of the University of Montpellier, at which time he ceased publishing his journal. He retired in 1844.
Gergonne was the first mathematician to employ the word polar. In a series of papers beginning in 1810, he discovered the principle of duality in projective geometry, by noticing that every theorem in the plane connecting points and lines corresponds to another theorem in which points and lines are interchanged, provided that the theorem embodied no metrical notions. In 1816, he devised an elegant solution to the problem of Apollonius: find a circle which touches three given circles.
In 1813, Gergonne wrote the prize-winning essay for the Bordeaux Academy, Methods of synthesis and analysis in mathematics, unpublished to this day and known only via a summary. The essay is very revealing of Gergonne's philosophical ideas. He called for the abandonment of the words analysis and synthesis, claiming they lacked clear meanings. Surprisingly for a geometer, he suggested that algebra is more important than geometry, at a time when algebra consisted almost entirely of the elementary algebra of the real field. He predicted that one day quasi-mechanical methods would be used to discover new results.
In 1815, Gergonne wrote the first paper on the optimal design of experiments for polynomial regression. According to S. M. Stigler, Gergonne is the pioneer of optimal design as well as response surface methodology.

1846 Antonio Abetti (19 Jun 1846, 20 Feb 1928 at age 81) Italian astronomer who was an authority on minor planets. At first a civil engineer, he became an astronomer at the University of Padua (1868-93), with an interest in positional astronomy and made many observations of small planets, comets and star occultations. In 1874, Abetti went to Muddapur, Bengal, to observe the transit of Venus across the sun's disk where his use of a spectroscope was the first use of this kind. Later, he became director at the Arcetri Observatory and Professor of astronomy at the University of Florence (1894-1921). The observatory had been founded by G. B. Donati in 1872, and Abetti equipped it with a new telescope that he had built in the workshops at Padua. He was active after retirement, until his death, and was followed by his son Giorgio.*TIS

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1851 Silvanus P. Thomson born. In 1910 he published Calculus Made Easy, which was published anonymously until after his death in 1916. It is still in print. *VFR  He was a noted physicist and engineer, and a celebrated teacher and writer on electricity and magnetism. He also wrote popular biographies of Faraday and Lord Kelvin. At his death he was professor at City and Guilds Technical College at Finsbury (London). Thompson’s particular gift was in his ability to communicate difficult scientific concepts in a clear and interesting manner. He attended and lectured at the Royal Institution giving the Christmas lectures in 1896 on Light, Visible and Invisible with an account of Röntgen Light. He was an impressive lecturer and the radiologist AE Barclay said that: “None who heard him could forget the vividness of the word-pictures he placed before them.”

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DEATHS

1945 Stefan Mazurkiewicz, one of the founders of Fundamenta Mathematicae, died.

Credits:
*VFR = V Frederick Rickey, USMA
*TIS= Today in Science History
*Wik = Wikipedia
*SAU=St Andrews Univ. Math History

The B-2 Theorem

In my B2 Pre-calc class today, we re-discovered a theorem about Pascal's triangle that I had not known. It began, appropriately enough, with a question Jacob C. asked about dealing cards from a standard deck; "How many 13 card hands can be dealt that contain exactly two suits. " As we were working through the problem, I began by attacking the somewhat easier problem, "how many hands can be dealt with only hearts and diamonds, but at least one of each." We began writing out the possibilities of 12 hearts, 1 diamond plus 11 hearts two diamonds ..etc To make life easy, for this short while let’s let (n,r) mean “n choose r” , the combinations of n things taken r at a time. So we needed to find (13,1)(13,12) for the first part, 12 hearts and 1 diamond. Then we needed to add on (13,2)(13,11) + (13,3)(13,10)…. And all the way down to (13,1)(13,12). One of the clever ones quickly realized that each of these pairs were just the same number due to the symmetry of Pascal's triangle, and so we were really looking for (13,1)² + (13,2)²... etc. While some of the kids were adding these on their calculators, I wrote out several lines of the arithmetic triangle and began to write the sums of squares on the right....

As I wrote the totals of each row, 1, 2, 6, 20, 70.. it struck me that they were all the center number of an even numbered row, (2n,n). I remembered them from working with Catalan’s Numbers (another cool pattern that shows up in Pascal’s triangle). About the time the first students were coming up with an answer, I asked them to check (26, 13) and compare it to the answer they got for the actual squares of the thirteenth row…

Close, but not right, was the reply.... huh??? … , oh yeah, we had avoided the case of (13,0) and (13,13) because we wanted to ignore the case where all were hears or all were diamonds, so the answer to our mini-problem was (26,13) - 2; and the only thing needed to solve the original problem was to multiply by 6, to account for all the ways we could pick two suits to be in the hand out of the four possible suits.

When I showed them the result, and we checked a couple of more cases to be more sure, I admitted that I had never seen this theorem. One kid suggests it should be a test question… I countered with, “and extra credit for the person who comes up with the best name for it. Several played to my ego, “Ballew’s theorem, of course!” but then they thought they might deserve partial credit, and hence the name, B-2 theorem, at the top.

Unfortunately, we were not the first to stumble across this little gem. I haven’t had time to chase it down fully, but it may actually date back to the Chinese around the 12th century. So fame and fortune will have to wait, but when you walk in the footsteps of greatness, you’re taking pretty big steps; so congratulations class, I’m proud of you, and it will always be the B-2 theorem when I teach it. Dennis was going to send me a class picture we took on his phone, so if it turns out, I will add that later,

While I was searching for the history of the sum of the binomial coefficients, I came across another place where the triangle is related to squares. One of those theorems we teach when we get to sequence and series in high school is the sum of the integers, 1 + 2 + 3 + … + n, and the sum of the squares of the integers, 1² + 2² + 3² + ….. + n². Usually we present the formula for this last without proof since it occurs before they are introduced to inductive proofs. As I was researching I came across this neat little relation to the arithmetic triangles. To find the sum of the squares of the first ten integers, just go down to 10 at (10,1) and turn right and follow the diagonal down two numbers to (12,3) and add this to the number on the diagonal above it (11,3), the sum of 220 + 165 = 385 which is the same as 1² + 2² + … + 10²

In general you can find

And I think they can accept that as evidence, at least until we get to inductive proofs.