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.