1 The Birth of Science - MSRI

176 6. The Hellenistic Scientific Method
exact and approximate area formulas. But when the result obtained is consciously
kept internal to the theory, that is, when it must be applied, often
indirectly, to a variety of problems not known a priori, a mathematician’s
rigor becomes essential.
Thus, in the absence of direct information, a good way to get an idea of
the breadth of applications of mathematics in a particular historical moment
is to inspect its level of rigor.
In the preface to Book IV of his fundamental treatise on conic sections, page 224
Apollonius writes:
Moreover, apart from these uses, they [some theorems of Conon of
Samos] are worthy of being accepted for the demonstrations’ sake
alone, in the same way that we accept many other things in mathematics
for this reason and no other. 89
There is no question that Apollonius is sincere (and that he is right!), but
the need to justify the value of pure science is so characteristic of civilizations
where science is the locomotive of technology that this quote by itself
would be enough to document the existence of applied mathematics in
Hellenistic times. Had there been no applied mathematics, Euclid’s quip
and Apollonius’ apologetics would be unthinkable: no one would defend
in such stark terms the value of “pure” mathematics unless to distinguish
it proudly from existing, and well-known, applied mathematics. In fact,
the same contrast is seen in modern times. While Galileo had to rack his
brains to invent practical applications that might persuade the Venetian
government to increase his stipend, once physics took on a prime role in
technological development it won the luxury of producing “theoretical
physicists” apparently uninterested in potential applications of their own
research.
Apollonius himself is generally thought of as the quintessential pure
mathematician. But note that this impression comes primarily from the
sifting of his works carried out by later generations. It is known that he
wrote books on astronomy and one on catoptrics, but all these have been
lost, and of the only work that was partially preserved in Greek, 90 the
treatise on conics, we have lost the eighth and last book, which likely was
devoted to applications of the theory. 91
89 Apollonius, Conics, preface to Book IV.
90 One other work by Apollonius has come down to us, in Arabic translation. It studies the problem
of finding a line whose intersections with two fixed half-lines form segments having a fixed
ratio.
91 This is the implication of a remark in the preface to Book VII of the Conics, where Apollonius
says that the theorems contained therein (on diameters of conics) were applicable to “problems of
many types” and that examples of such applications would be given in Book VIII.
Revision: 1.7 Date: 2002/09/14 23:17:37