ii4 SCIENCE IN GRECO-ROMAN ANTIQUITY 



this manner of thinking, supporting it by meta- 

 physical arguments. The mathematical sciences can- 

 not be founded on the unstable and changeful 

 phenomena of the sensible world ; for instance, the 

 aim of geometry is the knowledge of the eternal, and 

 hence it attracts the soul towards truth, and makes it 

 look upwards instead of downwards ; arithmetic like- 

 wise has the virtue of elevating the soul by compelling 

 it to reason about abstract numbers, without ever 

 suffering its calculations to revolve about visible or 

 tangible objects (Rep. 525 D). Thus there exists a 

 world of notions or ideas which is complete in itself, and 

 which has no need of support from the sensible world. 

 These notions or ideas maintain between themselves 

 immutable relations, the discovery of which is the 

 province of the human mind. 



On this point, all the Greek geometers, whether they 

 accept or reject the Platonic idealism, are in accord. 

 The figures about which we reason are not those per- 

 ceived by our senses. There does not exist in reality 

 any point which has no parts, any line without breadth, 

 or surface without thickness. The material figures aid 

 the imagination and thus are a help to the reasoning, 

 but they are only an accessory aid. What constitutes 

 the essential character of a geometrical figure, what 

 causes it to be a mathematical entity, is the connection, 

 defined once for all, of its component parts. Let us 

 take, for example, the circle. Having once postulated 

 the notions of straight line, distance, equal distance, we 

 create, so to speak, the circle ideally, declaring with 

 Euclid (Definition xv, Elements, I, p. 4) that a circle is a 

 plane figure, bounded by one line, and such that from 

 one interior point we can draw to this line straight 

 lines all equal to one another. The circle thus created 

 has no definite magnitude in the imagination, for it may 

 represent a microscopic surface just as well as a region 

 extended as far as desired into space. The definition 



