Subject: Re: Pythagorean Theorem
From: RWW Taylor (RWTNTS@ritvax.isc.rit.edu)
Date: Wed Mar 22 2000 - 13:10:34 EST
Stefan Baratto writes:
> I am trying to find a proof of the converse of the Pythagorean Theorem that
> doesn't require the Pythagorean Theorem as a given. That is, if a triangle
> has sides whose lengths satisfy the relationship a^2 + b^2 = c^2 then it is
> necessarily a right triangle. The only proofs I could find (or deduce)
> require one to assume that if a triangle is right, then the sum of the
> squares of the lengths of the legs is equal to the square of the length of
> the hypotenuse.
The question is what degree of "purity" is desired here? Classically, there
was no mention of "squaring lengths", and squares were always thought of as
_areas_, really a different _data type_ (to use a contemporary term). Euclid
always talked about the "square on the hypotenuse, etc." and the conception was
an actual, constructed square. Figures could be compared by assuming certain
principles of area measure, but there was no necessary connection between
linear measure and area measure. A proof of the converse, as desired, that
sticks to this careful understanding would be the highest level of response,
but would likely need to be very subtle.
Right down to the middle ages, powers of quantities were conceived of as having
geometrical significance, and different powers could not logically be combined
by addition. Today, of course, all is number and we have no trouble forming
arbitrary polynomials at will. I expect that a proof of the desired theorem
would not be difficult to come up with in a coordinate geometry setting
(though I don't have one in mind just at the moment). Would such a proof be
satisfactory?
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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