- HELLENISTIC GEOMETRY 323
prism required the least material to enclose a given volume, out of all the possible
prisms whose base would tile the plane.^3
Analysis, locus problems, and Pappus' theorem. Book 7 of the Synagoge is a trea-
sure trove of fascinating information about Greek geometry for several reasons.
First, Pappus describes the kinds of techniques used to carry on the research that
was current at the time. He lists a number of books of this analysis and tells who
wrote them and what their contents were, in general terms, thereby providing valu-
able historical information. What he means by analysis, as opposed to synthesis,
is a kind of algebraic reasoning in geometry. As he puts it, when a construction is
to be made or a relation is to be proved, one imagines the problem to have been
solved and then deduces consequences connecting the result with known principles,
after which the process is reversed and a proof can be synthesized. This process
amounts to thinking about objects not yet determined in terms of properties that
they must have; when applied to numbers, that process is algebra.
Second, Book 7 also contains a general discussion of locus problems, such as we
have already encountered in Apollonius' Conies. This discussion exerted a strong
influence on the development of geometry in seventeenth-century France.
Proposition 81 of Euclid's Data, discussed above, inspired Pappus to create a
very general proposition about plane loci. Referring to the points of intersection of
a set of lines, he writes:
To combine these discoveries in a single proposition, we have writ-
ten the following. If three points are fixed on one line... and all
the others except one are confined to given lines, then that last one
is also confined to a given line. This is asserted only for four lines,
no more than two of which intersect in the same point. It is not
known whether this assertion is true for every number.
Pappus could not have known that he had provided the essential principle by
which a famous theorem of projective geometry known as Desargues' theorem (see
Section 2 of Chapter 12) was to be proved 1400 years later. Desargues certainly
knew the work of Pappus, but may not have made the connection with this theorem.
The connection was pointed out by van der Waerden (1963, p. 287).
Pappus discusses the three- and four-line locus for which the mathematical
machinery is found in Book 3 of Apollonius' Conies. For these cases the locus is
always one of the three conic sections. Pappus mentions that the two-line locus
is a planar problem; that is, the solution is a line or circle. He says that a point
satisfying the conditions of the locus to five or six lines is confined to a definite
curve (a curve "given in position" as the Greeks said), but that this curve is "not
yet familiar, and is merely called a curve." The curve is defined by the condition
that the rectangular parallelepiped spanned by the lines drawn from a point to
three fixed lines bears a fixed ratio to the corresponding parallelepiped spanned by
the lines drawn to three other fixed lines. In our terms, this locus is a cubic curve.
(^3) If one is looking for mathematical explanations of this shape, it would be simpler to start with
the assumption that the body of a bee is approximately a cylinder, so that the cells should be
approximately cylinders. Now one cylinder can be tightly packed with six adjacent cylinders of
the same size. If the cylinders are flexible and there is pressure on them, they will flatten into
hexagonal prisms.