DEFINITIONS
293
asks what is to be done if the interlocutor says he does not
know what colour is; what alternative definition is there?
Socrates replies that it will be admitted that in geometry
there are such things as what we call a surface or a solid,
and so on; from these examples we may learn what we mean
by figure; figure is that in which a solid ends, or figure is
the limit (or extremity, 7repay) of a solid. 1 Apart from
‘ figure ’ as form or shape, e. g. the round or straight, this
passage makes ‘ figure ’ practically equivalent to surface, and
we are reminded of the Pythagorean term for surface, \p°La,
colour or skin, which Aristotle similarly explains as xP™P a ,
colour, something inseparable from vrep ay, extremity. 2 In
Euclid of course opoy, limit or boundary, is defined as the
extremity (uepas) of a thing, while £ figure ’ is that which is
contained by one or more boundaries.
There is reason to believe, though we are not specifically
told, that the definition of a line as ‘ breadthless length ’
originated in the Platonic School, and Plato himself gives
a definition of a straight line as ‘that of which the middle
covers the ends ’ 3 (i. e. to an eye placed at either end and
looking along the straight line); this seems to me to be the
origin of the Euclidean definition ‘ a line which lie$ evenly
with the points on it which, I think, can only be an attempt
to express the sense of Plato’s definition in terms to which
a geometer could not take exception as travelling outside the
subject matter of geometry, i. e. in terms excluding any appeal
to vision. A point had been defined by the Pythagoreans as
a ‘ monad having position ’; Plato apparently objected to this
definition and substituted no other; for, according to Aristotle,
he regarded the genus of points as being a ‘geometrical
fiction calling a point the beginning of a line, and often using
the term ‘ indivisible lines ’ in the same sense. 4 Aristotle
points out that even indivisible lines must have extremities,
and therefore they do not help, while the definition of a point
as ‘ the extremity of a line ’ is unscientific. 5
The ‘round’ (aTpoyyvXov) or the circle is of course defined
as ‘ that in which the furthest points (rd ’¿(T\aTa) in all
1 Meno, 75 a-76 a. 2 Arist. De sensu, 439 a 31, &c.
3 Parmenides, 137 E, 4 Arist. Metaph. A. 9, 992 a 20.
5 Arist. Topics, vi. 4, 141 b 21.