324 11. POST-EUCLIDEAN GEOMETRY
Third, in connection with the extension of these locus problems, Pappus con-
siders the locus to more than six lines and says that a point satisfying the corre-
sponding conditions is confined to a definite curve. This step was important, since
it proposed the possibility that a curve could be determined by certain conditions
without being explicitly constructible. Moreover, it forced Pappus to go beyond the
usual geometric interpretation of products of lines as rectangles, thus pushing the
same boundary that Heron had gone through. Noting that "nothing is subtended
by more than three dimensions," he continues:
It is true that some of our recent predecessors have agreed among
themselves to interpret such things, but they have not made a
meaningful clear definition in saying that what is subtended by
certain things is multiplied by the square on one line or the rec-
tangle on others. But these things can be stated and proved using
the composition of ratios.
It appears that Pappus was on the very threshold of the creation of the modern
concept of a real number as a ratio of lines. Why did he not cross that threshold?
The main reason was probably the cumbersome Euclidean definition of a composite
ratio, discussed in Chapter 10. But there was a further reason: he wasn't interested
in foundational questions. He made no attempt to prove or justify the parallel
postulate, for example. And that brings us to the fourth attraction of Book 7. In
that book Pappus investigated some very interesting problems, which he preferred
to foundational questions. After concluding his discussion of the locus problems, he
implies that he is merely reporting what other people, who are interested in them,
have claimed. "But," he says,
after proving results that are much stronger and promise many
applications,.. .to show that I do not come boasting and empty-
handed. .. I offer my readers the following: The ratio of rotated
bodies is the composite of the ratio of the areas rotated and the
ratio of straight lines drawn similarly [at the same angle] from
their centers of gravity to the axes of rotation. And the ratio of
incompletely rotated bodies is the composite of the ratio of the ar-
eas rotated and the ratio of the arcs described by their centers of
gravity.
Pappus does not say how he discovered these results, nor does he give the
proof. The proof would have been fairly easy, given that he had read Archimedes'
Quadrature of the Parabola, in which the method of exhaustion is used. For the
first theorem it would have been sufficient to compute the volume generated by
revolving a right triangle with one leg parallel to the axis of rotation, and in that
case the volume could be computed by subtracting the volume of a cylinder from
the volume of a frustum of a cone. If the theorem is true for two nonoverlapping
areas, it is easily seen to be true for the union of those areas. Pappus could then
have applied the method of exhaustion to get the general result. The second result
is an immediate application of the Eudoxan theory of proportion, since the volume
generated is obviously in direct proportion to the angle of rotation, as are the
arcs traversed by individual points. The modern theorem that is called Pappus'
theorem asserts that the volume of a solid of revolution is equal to the product of
