Re: "Euclid's Fourteenth Book

David Ewan Kahana wrote:
Robert Carnegie wrote:
David Ewan Kahana wrote:
The essential feature of the Pythagorean theorem is not in fact
that the sum of the squares on the sides is equal to the
square on the hypotenuse. The theorem is true in fact, for
the areas of any similar figures drawn on the three sides
of a right triangle. This leads to one of the most beautiful
proofs of the theorem, in fact, in which the figures on the
sides are taken to be right triangles similar to the original
right triangle itself. It's then evident that the areas of the
two triangles on the sides themselves add to the area
of the original triangle, by dropping the altitude from the right
angle to they hypotenuse.


Given that areas of plane figures scale as the square of any
linear dimension of those figures, you then have the theorem.
This argument as I learned it, is due to G. Polya, but it was likely
already known in Euclid's time.

Cartesian coordinates as desCartes wrote them down, are
by definition flat space coordinates, and the Pythagorean
theorem is an automatic property of the scaling properties
of the coordinate system.

I'm not sure if I knew the argument of Polya. I must try it out.

Algebraically, the relationship between the lengths is described, with
the matter of the areas of figures constructed upon the lines almost an
afterthought, but maybe that's a distorted Descartian perspective.

It's a flat Descartian perspective ;->

The distance measure, which embodies the Pythagorean
theorem, and the scaling law for area with the linear dimensions
of a figure are valid for arbitrary sized figures are basic
characteristics of a space without curvature.

If you add curvature the Pythagorean theorem holds

(i should say a variant of the Pythagorean theorem
holds locally, of course, since if one postulated a
more general form of metric, the eigenvalues
of the metric tensor would appear in the distance
relation.)

only locally, for infinitesimal right triangles. Curvature
can be added in the analytic approach to geometry
by specifying a different distance metric on the
coordinates.

From the purely geometric, axiomatic point of view, you
can also construct curved geometries by modifying the fifth
postulate. The standard tiling proof of the Pythagorean
theorem, which is really based on the idea that triangles
can be moved around arbitrarily in the space, without
changing their areas, then fails.

The Pythagorean theorem, the curvature, and the
metric of the space are all intimately related, and
the theorem is thus inextricably woven into the
Cartesian description of the flat plane.

David

.

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