*a priori*. In search of useful illustrations of this important claim, I took down from the shelf my old copy of Book I of Euclid's

*Elements*which, I see from the flyleaf, I bought used many years ago for two dollars.

When I was a boy in high school, Euclid had already been superceded. I am curious. Has anyone out there ever actually been called on in school to look at the

*Elements*? It is really rather elegant. Cartesian Geometry [named, of course, after Descartes] is vastly superior mathematically, but nowhere near so charming.
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The Elements is a wonderful book. The Heath edition is filled with a seemingly endless supply of footnotes that discuss the history of this and that. I've only made it completely through the first book. The next to the last theorem there is that when you have a right triangle, the sum of the squares of the sides is equal to the square of the hypotenuse, but the very last theorem is that when you have a triangle in which the sum of the squares of the sides equals the square of the hypotenuse, then you have a right triangle! Nice....

In the Boston Science Museum back when my boys were young they had a marvelous exhibit, consisting of a physical right triangle with boxes constructed on the three sides. The squares on the sides adjacent to the right angle were filled with colored liquid, and you could rotate the triangle with a handle so that the fluid would flow into the box on the hypoteneuse, just filling it, thus showing that the square on the hypoteneuse is equal to the sum of the squares on the adjacent sides. Nifty.

Bob, If you are interested in a contemporary mathematician's discussion of "The Elements," have a look at "Euclid: The Creation of Mathematics" by Benno Artmann. It is a fascinating and beautiful book.

Thank you, Bruce. I will.

Ahh, I shall have to look at the Artmann book. In an undergraduate course, long ago, I worked my way through Hilbert's Foundations of Geometry which, to put it baldly and shortly, Euclid redone to modern standards. Somewhere buried in a file drawer are my beautiful multi-colored renderings of Desargue's Theorem. In fact it was sort of the invention of modern axiomatic treatments--very important historically. (precursors, of course. Pasch famously.) But I doubt you'll want to digress that much.

I went to St John's College, Annapolis. Freshman math is devoted pretty much entirely to Euclid and Ptolemy. Then, in Sophomore Year, we read Apollonius's "Conics," and move on to Descartes' Geometry. I found that working my way to Descartes in this way gave me a much greater appreciation of his achievement in inventing analytic geometry. You really see the limits of the older approach, and then Descartes comes along and blows the roof off. Also, familiarity with Euclid and Apollonius makes it much easier to read Newton's Principia. Working my way through Euclid's first reliance on a reductio ad absurdum probably taught me more about logic than any other single exercise.

Bob, can you tell me, how can someone be a Kantian after the advent of non-Euclidean geometry, relativity theory, and quantum mechanics, each of which undoes some of the supposedly a priori elements of his philosophy? I'm not implying that you *are* a Kantian, but perhaps you could enlighten me.

In the very early 60's we spent a year in high school math studying Euclidean geometry proof by proof. We didn't use the Elements as a textbook, but our text incorporated Euclid's proofs.

Yeah, I think I would distinguish between three very different questions:

- Do most people (at least people who learn mathematics in the first place) learn geometry from Euclid nowadays?

- Do most people learn Euclidean geometry?

- Has Euclidean geometry been superseded by Cartesian analytic geometry?

My answers: no, yes, and no. The standard 9th grade geometry course, which is the first place mathematics students learn how to do proofs, is all about Euclidean geometry. But, for better or for worse, people don't usually learn Euclidean geometry from Euclid, and haven't for about a century, just as people don't usually learn Newtonian mechanics by reading Newton. It's entirely commonplace to go through a graduate education in physics or mathematics without ever reading Euclid or Newton.

I also wouldn't say that Cartesian analytics geometry is superior to Euclidean geometry, any more than I would say that differential geometry or algebraic geometry is superior to either of those things. They're different fields and address different questions.

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