272 10. EUCLIDEAN GEOMETRY1.3. Pythagorean geometry. Euclid's geometry is an elaboration and system-
atization of the geometry that came from the Pythagoreans via Plato and Aristotle.
From Proclus and other later authors we have a glimpse of a fairly sophisticated
Pythagorean geometry, intertwined with mysticism. For example, Proclus reports
that the Pythagoreans regarded the right angle as ethically and aesthetically su-
perior to acute and obtuse angles, since it was "upright, uninclined to evil, and
inflexible." Right angles, he says, were referred to the "immaculate essences," while
the obtuse and acute angles were assigned to divinities responsible for changes in
things. The Pythagoreans had a bias in favor of the eternal over the changeable,
and they placed the right angle among the eternal things, since unlike acute and
obtuse angles, it cannot change without losing its character. In taking this view,
Proclus is being a strict Platonist; for Plato's ideal Forms were defined precisely
by their absoluteness; they were incapable of undergoing any change without losing
their identity.
Proclus mentions two topics of geometry as being Pythagorean in origin. One
is the theorem that the sum of the angles of a triangle is two right angles (Book 1,
Proposition 32). Since this statement is equivalent to Euclid's parallel postulate, it
is not clear what the discovery amounted to or how it was made.
The other topic mentioned by Proclus is a portion of Euclid's Book 6 that is not
generally taught any more, called application of areas. However, that topic had to
be preceded by the simpler topic of transformation of areas. In his Nine Symposium
Books^6 Plutarch called the transformation of areas "one of the most geometrical"
problems. He thought solving it was a greater achievement than discovering the
Pythagorean theorem and said that Pythagoras was led to make a sacrifice when
he solved the problem. The basic idea is to convert a figure having one shape to
another shape while preserving its area, as in Fig. 1. To describe the problem in a
different way: Given two geometric figures A and B, construct a third figure C the
same size as A and the same shape as B. One can imagine many reasons why this
problem would be attractive. If one could find, for example, a square equal to any
given figure, then comparing sizes would be simple, merely a matter of converting
all areas into squares and comparing the lengths of their sides. But why stop at that
point? Why not do as the Pythagoreans apparently did, and consider the general
problem of converting any shape into any other? For polygons this problem was
solved very early, and the solution appears in very elegant form as Proposition 25
of Euclid's Book 6.
Related to the transformation of areas is the problem of application of areas.
There are two such problems, both involving a given straight line segment AB and
a planar polygon Ã. The first problem is to construct a parallelogram equal to Ã
on part of the line segment AB in such a way that the parallelogram needed to
fill up a parallelogram on the entire base, called the defect, will have a prescribed
shape. This is the problem of application with defect, and the solution is given
in Proposition 28 of Book 6. The second application problem is to construct a
parallelogram equal to à on a base containing the line AB and such that the portion
of the parallelogram extending beyond AB (the excess) will have a prescribed shape.
This is the problem of application with excess, and the solution is Proposition 29
of Book 6. The construction for application with defect is shown in Fig. 2. This
(^6) The book is commonly known as Contrivial Questions. The Greek word symposion means
literally drinking together.