1 The Birth of Science - MSRI

284 10. Lost Science
Elements 212 and another definition (based on the invariance of the line with
respect to rotations that fix two points) that is missing in Euclid’s work
but appears in Definitions. 213 We can conclude that when Sextus reports
“descriptions” of basic geometric entities his source usually seems to be
the Definitions rather than Euclid. 214
As we have seen, Sextus knew that to avoid infinite regression one must
either eschew definitions altogether or accept that some entities must re- page 353
main undefined. 215 Since he could hardly have broached this subject without
taking into account Euclid’s Elements (a foundational work for all later
mathematical developments), this testimony of Sextus, too, suggests that
in the version(s) of the Elements known to him certain things were left undefined.
This would explain why on the subject of fundamental geometric
entities Sextus had to use the Definitions.
The definitions of fundamental geometric entities (point, line, straight
line, surface and plane) given in the Elements all closely follow passages
from the Definitions. We therefore narrow down our suspicions by conjecturing
that they are interpolations drawn from the Definitions of terms in
geometry.
What settles the issue for me is the definition of a straight line found in
the Elements, and quoted at the beginning of this section. This very murky
statement 219 seems to mean, if anything, that a straight line is “seen” in
the same way from all its points; that is, that there are rigid motions that page 354
leave the line invariant and take any point to any other. But this property
(which as we know also interested Apollonius 219a ) does not fully characterize
straight lines: in the plane it is also shared by the circumference. It
is strange that Euclid should not realize that a circumference, too, “lies
equally with respect to the points on itself”.
Now, the long paragraph on straight lines appearing in the Definitions
starts:
212 Sextus Empiricus, Adversus mathematicos, III, 94.
213 Sextus Empiricus, Adversus mathematicos, III, 98; Heronis Definitiones, 16, 21 – 18, 6 (ed. Heiberg).
214 This possibility could not have occurred to earlier scholars such as Heiberg and Heath, who
thought that Sextus Empiricus predated Heron (the correct dating for Heron having been established
later by Neugebauer). Note that if Knorr’s attribution of the Definitions to Diophantus becomes
established (note 210), the fact that Sextus seems to cite his work would help solve the
vexing question of Diophantus’ dating. (The argument generally used to assign Diophantus to the
third century A.D. goes back to Tannery, but its inconsistency has been demonstrated in [Knorr:
AS]. Knorr (op. cit., note 23) also considers the possibility that Diophantus was a source for Heron,
in particular for arithmetic terminology.)
215 See page 157.
219 Heath concludes his discussion of it with the words “the language is thus seen to be hopelessly
obscure” ([Euclid/Heath], vol. I, p. 167).
219a See page 87.
Revision: 1.11 Date: 2003/01/06 02:20:46