Wednesday, June 27, 2012

Mathematical misnomers


Mathematicians don’t always get their history correct. Maybe that’s why they went into math. Because of this, there are numerous examples of results that have gone by the name of someone other than the original discoverer. Stephen Stigler, author of several books on the often neglected topic of the  history of statistics has gone so far as to suggest (with his tongue firmly planted in his cheek) “Stigler’s Law of Eponymy”: “No scientific discovery is named after its original discoverer.”
Incidentally, Stigler’s Law was first postulated by Robert Merton.
Famous example
The Pythagorean theorem is perhaps the most well known mathematical misnomer. Although is it likely that Pythagoras first proved the theorem, the theorem was known by the Babylonians long before Pythagoras. There is a tablet from Mesopotamia that lists fifteen different Pythagorean triplets. This tablet was inscribed at least a millennia before the birth of Pythagoras.
Pascal’s triangle is yet another case where the proper person has not received full credit. Chu Shih-Chieh tabulated the triangle to show the binomial coefficients in 1303. Pascal got a triangle and a programming language named after him. Contrary to popular belief, Chu Shih-Chieh did not have a dog breed named after him.
This triangle appeared in print in Europe in the Rechnung by Peter Apian in 1527, nearly a hundred years before Pascal’s birth. Peter Apian was a bit luckier than Chu Shih-Chieh. A crater on the moon and a minor planet were named after him. Although Apian was into maps and astronomy and all that, he paid little attention to either his crater or his planet. To the best of my knowledge, he never planned a summer holiday to either.
Our number system (that is, the way we write numbers) has commonly been referred to as the Arabic number system. The Encarta encyclopedia calls our number system the Arabic numbers. Rouse Ball1 refers to our number system as Arabic. This is unfortunately a misnomer.
The number system we use today did make it to Europe from Arabia, but the concepts and symbols are entirely due to the Hindus. Hence it is more appropriate to call our numbers the Hindu system of numerals. It was first introduced to Europe through the translation of a book by Mohammed ibn-Musa al-Kwarizmi, and from him has become known as the Arabic system.
The hapless De Moivre
Ah, poor DeMoivre. So often misunderstood.
A statistician calls the curve the “normal curve”, a teacher calls it the “bell curve”, and a mathematician refers to it as the “Gaussian”. The discovery of this curve rightfully belongs to De Moivre, however. It was De Moivre who first described this as a probability distribution.
De Moivre was also cheated out of recognition for a formula to approximate the factorial. The formula
is known as Stirling’s formula, although De Moivre discovered it earlier. This formula isn’t that well known under either name. I was rather taken by it when I was in high school. I found it in a book somewhere. Perhaps if this formula became part of the normal high school curricula, there would be more people like me?
Ironically, the one relation that bears De Moivre’s name is one that he never explicitly stated. The famous De Moivre’s theorem states that
From reviewing his work, it is certain that De Moivre was aware of this magical result. I, personally, am not aware of any historian that came right out an asked the guy, "Hey Abraham, you ever hear of De Moivre's theorem?" I am guessing that this is yet another historical event that will never be completely resolved.
Cauchy
The Cauchy distribution and its odd properties as a statistical distribution were first written about by Poisson, almost thirty years before Cauchy did any work with this function.
The Cauchy distribution, itself a misnomer, is another name for the mathematical curve known as the witch of Agnesi. A more proper name for this might be the “curve of Grandi”. I will write more about the multiple times this was invented in a future blogpost.
Cauchy did a great deal to make other mathematicians aware the danger of a divergent series. He certainly popularized the ratio test for convergence of a series. Laplace immediately rushed home to recheck his series in Mécanique Céleste after hearing Cauchy give a paper.
Reading Boyer2 however, it is rather difficult to say just who deserves the credit for discovery of what:
“… the familiar ratio test … more frequently is known as Cauchy’s test, despite the fact that it had given by Waring as early as 1776.” [p. 458]
"Gauss as early as 1812, for example, used the ratio test to show [convergence of a hypergeometric series]… This test seems to have been first used long before, in England, by Edward Waring although it generally bears the name d’Alembert or, more occasionally, that of Cauchy.
"[Another convergence test] has come to be known as Cauchy’s criterion, but it was known earlier by Bolzano (and possibly still earlier by Euler). [p. 517]"
Cauchy first used the matrix of partial derivatives that has come to be known as the Jacobian.
Various other misnomers
“Euler’s formula” is the name given to the beautiful relation v + f = e + 2, where v is the number of vertices in a polyhedra, f is the number of faces, and e is the number of edges. As prolific as Euler was, he did not invent the formula. René Descartes first discovered this relationship in 1619. Euler was not born until 1707.
Cartesian coordinates were named after Rene Descartes, but were first used by Nicole Oresme, who lived in the 14th century. Oresme did have a crater on the moon named after him, though. Any guess as to whether he discovered the crater?
Pell's equation, y = ax2 + 1, where a is a non-square integer, was first studied by Brahmagupta  and Bhaskara. Its complete theory was worked out by Lagrange, not Pell.

I am still looking for an un-named discovery to put my name on. Any suggestions?
1)      Rouse Ball, W. W., A Sort Account of the History of Mathematics, 1960, Dover Publications, (unabridged and unaltered republication of the 1908 4th edition)
2)      Boyer, Carl B., A History of Mathematics, 2nd Edition, 1991, John Wiley and Sons

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