GENERAL PRINCIPLES 29 



matical point may, then, be regarded as indivisible, but 

 only because there is nothing in it to divide. It cannot 

 be a real unit, for there is nothing to determine its unity. 

 We should have to conceive it as the unit of that whose 

 sole characteristic is to consist of units, to be a quantity. 

 For such is, strictly speaking, the nature of Descartes's 

 ' extension.' Thus, as Leibniz puts it, ' mathematical 

 points are exact ' [i. e. indivisible] ; * but they are only 

 modalities 1 ,' that is to say abstractions and not real 

 existences 2 . 



Now, while Leibniz regards the parts of Cartesian 

 extension as thus indivisible without being real, he 

 maintains on the other hand that the parts to which 

 Atomism reduces material substance are real only if 

 they are not indivisible. Their claim to be indivisible 

 rests upon the supposition that they are infinitely hard. 

 But hardness is a relative term. There is no absolute 

 hardness, as there is no absolute motion or rest. And 

 thus infinite hardness is a self-contradictory conception. 

 'By an atom,' says Leibniz, 'I understand a corpuscle, 



1 New System, n. 



2 Cf. Epistola ad Bernoullium (1698) (G. Math. iii. 535) : 'Indeed 

 many years ago I proved that a number or sum of all numbers 

 involves a contradiction, if it be taken as one whole. And the 

 same- is true of an absolutely greatest number and an absolutely 

 least number or an absolutely smallest fraction. . . . Now, just as 

 there is no (given) numerical element or smallest part of unity or 

 least among numbers, so there is no (given) least line or lineal 

 element ; for a line, as a unity, can be cut into parts or fractions. . . . 

 Suppose that in a line there are actually , , |, ^ ^, &c v a n <* 

 that all the terms of this series actually exist. You infer from 

 this that there is also an absolutely infinite term, but I think 

 nothing else follows from it than that there actually exists any 

 assignable finite fraction, however small. . . . And indeed I conceive 

 points, not as elements of a line, but as limits, or negations of 

 further progress, or as ends [termini] of a line.' Cf. Lettre a Foucher 

 (1693) (E. n8a; G. i. 416). 'As to indivisible points in the sense 

 of the mere extremities of a time or a line, we cannot conceive in 

 them new extremities, nor parts, whether actual or potential. 

 Thus points are neither large nor small, and no leap is needed to 

 pass them. Yet the continuous, though it has everywhere such 

 indivisible points, is not composed of them.' Cf. Explanation of the 

 New System, i, note. 



