1 The Birth of Science - MSRI

286 10. Lost Science
Points and lines are each defined twice in the Elements (Definitions 1
and 3 for points, 2 and 6 for lines). Even this duplication, which amounts
to a clear logical incongruence, is easy to explain in the scenario discussed
above. In the Definitions, the notion of a point was explained with a list
of many characterizations, and likewise the line. It is then understandable
that the abridger, faced with the task of choosing one sentence to be kept
as a “definition”, should sometimes have hesitated and chosen two just in
case. page 356
Significant support for our thesis can be found in the very preamble to
the Definitions, which starts:
In describing [] and summarizing for you, illustrious Dionysius,
as concisely as possible, the technical terms presupposed in
the foundations of geometry [ ¥
], I will lay down the beginnings and general structure
according to the teachings of Euclid, the author of the Elements
of geometric theory. 226
This wording makes sense if we assume that one of the author’s goals
was to illustrate the geometrical entities left undefined by Euclid, that is,
the “technical terms presupposed in the foundations of geometry”. The
fact that the author considers his own work as preliminary to the Elements
provides strong support to the conjecture that either he or later editors
prepended extracts from the Definitions to the Euclidean text; the tradition page 357
could hardly have failed to merge the texts at some later point.
This conjectural reconstruction is consistent with other available testimonia,
in that no author who cites Euclid attributes to him the definitions
we are considering, all the way down to late Antiquity. 226a This silence
is significant if we consider it together with the testimonia of several ancient
authors who quote definitions of fundamental geometrical entities,
none of which come from the Elements. We have already examined the
passages in Sextus Empiricus; the case is similar with Plutarch, for whom
the straight line is still characterized by being the shortest line between
two points, and not by the definition that we now find in the Elements. 226b
Again when defining a point Plutarch does not use the Elements: he says
226 [Heron: OO], vol. IV, p. 14.
226a I have checked the Thesaurus Linguae Grecae for all passages containing Euclid’s name, and all
authors not included in the TLG corpus who to my knowledge were interested in mathematical
definitions.
226b Plutarch, Platonicae quaestiones, 1003E; De Pythiae oraculis, 408F.
Revision: 1.11 Date: 2003/01/06 02:20:46