274 10. EUCLIDEAN GEOMETRYhave a unique solution. Were the additional conditions imposed simply to make the
problem determinate? Some historians have speculated that there was a further
motive.
In the particular case when the excess or defect is a square, these problems
amount to finding two unknown lengths given their sum and product (application
with deficiency) or given their difference and product (application with excess). In
modern terms these two problems amount to quadratic equations. Some authors
have argued that this "geometric algebra" was a natural response to the discovery of
incommensurable magnitudes, described in Chapter 7, indeed a logically necessary
response. On this point, however, many others disagree. Gray, for example, says
that, while the discovery of incommensurables did point out a contradiction in a
naive approach to ratios, "it did not provoke a foundational crisis." Nor did it force
the Pythagoreans to recast algebra as geometry:
There is no logical necessity about it. It would be quite possible
to persevere with an arithmetic of natural numbers to which was
adjoined such new quantities as, say, arose in the solution of equa-
tions. There is nothing more intelligible about a geometric seg-
ment than a root of an equation, unless you have already acquired
a geometric habit of thought. Rather than turning from algebra
to geometry, I suggest that the Greeks were already committed to
geometry. [Gray. 1989, p. 16]1.4. Challenges to Pythagoreanism: unsolved problems
this much was known to the early Pythagoreans, we can easily guess what problems
they would have been trying to solve. Having learned how to convert any polygon
to a square of equal area, they would naturally want to do the same with circles and
sectors and segments of circles. This problem was known as quadrature (squaring) of
the. circle. Also, having solved the transformation problems for a plane, they would
want to solve the analogous problems for solid figures, in other words, to convert
a polyhedron to a cube of equal volume. Finding the cube would be interpreted
as finding the length of its side. Now, the secret of solving the planar problem
was to triangulate a polygon, construct a square equal to each triangle, then add
the squares to get bigger squares using the Pythagorean theorem. By analogy, the
three-dimensional program would be to cut a polyhedron into tetrahedra, convert
any tetrahedron into a cube of equal volume, then find a way of adding cubes
analogous to the Pythagorean theorem for adding squares. The natural first step
of this program (as we imagine it to have been) was to construct a cube equal to
the double of a given cube, the problem of doubling the cube, just as we imagined
that doubling a square may have led to the Pythagorean theorem.
Yet another example of such a problem is that of dividing an arc (or angle)
into equal parts. If we suppose that the Pythagoreans knew how to bisect arcs
(Proposition 9 of Book 1 of the Elements) and how to divide a line into any number
of equal parts (Proposition 9 of Book 6), this asymmetry between their two basic
figures—lines and circles—would very likely have been regarded as a challenge. The
first step in this problem would have been to divide an arc into three equal parts,
the problem of trisection of the angle.
The three problems just listed were mentioned by later commentators as an
important challenge to all geometers. To solve them, geometers had to enlarge
their set of basic objects beyond lines and planes. They were rather conservative in