1 The Birth of Science - MSRI

10.14 The First Few Definitions in Euclid’s Elements 281
6.5, 198 but this is apparently contradicted by the first few definitions in
Book I of the Elements, of which we quote the first four: 199
A point is that which has no part.
A line is breadthless length.
The extremities of a line are points. 200
A straight line is a line that lies equally with respect to the points on
itself. 200a
Such definitions, and the next ones for a surface and a plane, are clearly
Platonist-essentialist, and fit better with the cultural climate of the imperial
age than with that of the early Hellenistic period: they find no parallel
in the works of Archimedes and Apollonius.
The Elements have reached us with interpolations from late Antiquity, 201
and it is not unreasonable to suspect that these definitions are instances page 350
thereof. One of the few definitions in the Elements methodologically analogous
to those just considered is found right at the beginning of Book VII:
An unit is that by virtue of which each of the things that exist is called
one. 201a
This “definition”, which is clearly of Platonist character, is attributed by
Iamblichus (circa 300 A.D.) to “more recent” writers ( ). 202 It is
possible, then, that similar interpolations of Platonist definitions by later
authors occurred for the fundamental geometric entities.
In seeking objective support for this suspicion, we can use documents
of two types: papyrus finds and the testimonia of authors who had access
to earlier versions of Euclid’s works than the ones available today.
Among the few papyri containing fragments of the Elements, only two
are relevant to the definitions from Book I. One, probably written by a
198
See particularly pp. 157 ff., 161.
199
The word “line” in English is ambiguous. In this section it will always mean a not-necessarilystraight
line (). We use “straight line” to translate .
200¨ ¨ (Heath
translation).
200a ¨ .
201
For example, Theon, in his commentary on Book III of the Almagest, mentions a theorem that
he demonstrated and inserted in his edition of the Elements. We also know that Heron wrote a
popularizing commentary to the Elements, and some passages that appear in all our manuscripts
of the Elements have been positively identified as interpolations from Heron’s commentary. For
instance, the Arabic commentator an-Nairīzī attributes proposition 12 of Book III to Heron (see
[Euclid/Heath], vol. II, pp. 28–29), and Proclus does the same regarding an alternative proof of
proposition 25 of Book I (see [Proclus/Friedlein], 346–347).
201a (Heath translation).
202
Iamblichus, In Nicomachi Arithmeticam introductionem, 11, 5 (ed. Pistelli).
Revision: 1.11 Date: 2003/01/06 02:20:46