FOM: Sum of the first n positive integers

[Calvin Jongsma]

>>> Mark Steiner <marksa at vms.huji.ac.il> 11/08/99 03:38PM  wrote: >>>
. . .
As a result of this idea, I actually made a "discovery," the only
mathematical idea I have ever had.  Namely, most of the usual proofs of
the Pythagorean theorem are not explanatory in the above sense.  But
there is one given by Polya (he states that it's the one Euclid really
had in mind), which is that if we draw the altitude of the triangle to
the hypotenuse as base, this decomposes the triangle into two triangles
(I and II) which are similar to each other and to the entire triangle
(call it III).  This property uniquely characterizes the right
triangle.  Now consider triangles I, II, III as constructed on the three
sides of the triangle; I+II=III.  By considerations of similarity, three
squares constructed on the sides of the triangle will also sum, qed.
. . .

I'm not sure I follow just how similarity considerations generate the conclusion, and I don't want to sidetrack the discussion, but I would like attach a historical note having some foundational content.  There is an old proof of the Pythagorean Theorem using the similarity and proportionality of the three triangles, and this may satisfy an explanatory criterion, but it also assumes a theory of ratio and proportion.  In the case of the Greeks and Euclid, it seems that the discovery of incommensurability made an early proof of the Pythagorean Theorem along these lines invalid.  Consequently, the proof in Euclid (I.47) avoids such a similarity approach.  In fact, similarity is left for book VI, after Eudoxus' theory of ratio and proportion for continuous magnitudes has been developed in book V.  These concepts seem to have been considered too complicated to serve as an explanation for the Pythagorean Theorem.

>From the balmy Midwest,
Calvin Jongsma