Ramble, Part 18: Polymath

[stoutfellow]

Blaise Pascal deserves a fuller treatment than I'm going to give him. He wrote on philosophy (Penséés and Lettres écrites à un provincial) and on physics (Concerning the Vacuum, On the Weight of the Mass of the Air, and others) as well as mathematics. He was also credited with inventing the one-wheel barrow, so much more maneuverable than its two-wheeled predecessor, and it was apparently he who came up with the idea of the omnibus as a form of cheap public transportation. (The first omnibus line began soon after in Paris, at a cost of five sous a ride.)

Even within mathematics, his contributions were varied. I previously mentioned his discoveries in projective geometry. Working together with Pierre Fermat, he laid the foundations for probability theory, resolving the "problem of the points". (Briefly: if two gamblers are playing a game on, say, a best-of-nine basis, but must break off when the score is 3-2, how should the pot be divided? Earlier discussions had failed to notice significant questions of probability, but Pascal and Fermat resolved those issues and each came up with a procedure for fair division.) But, of course, he is most famous in connection with "Pascal's Triangle", which I will discuss further under the cut.

Let me first point out that the name, "Pascal's Triangle", is a misnomer. (I should probably write a post on the history of mathematical misnamings...) The triangle was known to the Chinese centuries earlier, and followed the usual trail westwards through India and the Muslim lands to Europe. Even within Europe, versions of the triangle are on record from as much as a century before the time of Pascal. He himself did not lay claim to priority, much less the name; he referred to it as the "arithmetic triangle", which is perhaps a more appropriate name.

Nonetheless, it was Pascal who brought the arithmetic triangle within the purview of Western-style mathematics, in his Treatise on the Arithmetical Triangle. In that paper, he laid out the key properties of the triangle in definition-theorem-proof style. In one of those theorems, he took one of those steps that, in retrospect, seem obvious, but which no one before him had explicitly done. The content of the theorem is not of great importance here; it is a statement concerning the rows of the triangle - or, rather, concerning individual triangles, seen by Pascal as each having a specified size. In the course of the proof, he says the following:

Although there is an infinity of cases, the demonstration can be briefly given by means of the two following lemmas:
The first, which is self-evident, that this equality is found in the first triangle [details omitted];
The second, that if an arithmetical triangle is found which has this equality [details omitted], I say that the following triangle will have the same property.
Whence it follows that all the arithmetical triangles have this equality. For the first has it by the first lemma, and it is even still evident in the second; therefore by the second lemma the next triangle will have it too, and consequently the next, and so on to infinity.

This is, it appears, the first explicit appearance of the Principle of Mathematical Induction anywhere. It was not until the late nineteenth century that it was realized how central a role induction must play in mathematics, but Blaise Pascal was the first to explicitly lay it out as a method of proof.

There's an odd coincidence, in all of this, in connection with Fermat, but that will have to wait until the next Ramble post.

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Comments

[countrycousin.livejournal.com]

Pascal's Triangle, the binomial theorem, and the formulas for the coefficients were a magical revelation of my youth. Thanks.

Re: rite of spring. New icon forthcoming?

[stoutfellow.livejournal.com]

New icon forthcoming?

'Fraid not. My scanner ceased to function some time ago. One of these days I'll pony up for a paid account and add some more icons, but they probably won't be derived from real pictures.

Hey, wait a moment. I could probably swipe my (summer-morph) picture off the departmental website...

[countrycousin.livejournal.com]

So you could ;<) Sorry if this is tardy response - main computer has been down some time. One side effect - this is replacing all the default icons, retroactively warming the February cold snap.

Looks good.

[stoutfellow.livejournal.com]

Hmm. I could go obsessive and fix them all, or I could rule that my picture is to be my current picture, not my picture at the time of posting. Decisions, decisions...