STATEMENT OF LEIBNIZ S PHILOSOPHY 79 



formed by an addition of independent integers, such as 

 i + i + i, &c., or even 1 + 2+3, & c -> that is to say, by an 

 addition not conditioned by any special law. But there 

 are certain numerical series in which the terms are not 

 mutually indifferent (nor immediately reducible to a set 

 of mutually indifferent terms), but are arranged, or rather 

 proceed from one another, according to. a definite law, 

 which law is of such a kind that, although it never 

 brings the series actually to an end, it results in the sum 

 of the series approaching more and more nearly to some 

 finite quantity. Accordingly it is held that, if the series 

 be regarded as consisting of an infinite number of terms, 

 the difference between the sum of its terms and the finite 

 quantity will be infinitely little, and therefore practically 

 negligible. 



This ' practically negligible ' is the keystone of the 

 bridge between algebraic quantity and geometrical, 

 physical, or any other kind of relation. Strictly speaking, 

 if the series be regarded as a pure sum, and therefore 

 ultimately analyzable into an addition of homogeneous 

 units (i + i + i, &c., or n + n + n, &c.), the finitude of its 

 sum is incompatible with its having an infinite number 

 of terms. It is only inasmuch as the series is regarded, 

 not as a merely quantitative unity, but as a unity deter- 

 mined by a characteristic law or principle, that we are 

 entitled to disregard the ' infinitely little ' difference 

 between the sum of its terms and the finite quantity. 

 There can be no absolute * infinitely little ' in mere 

 quantity. The ' infinitely little' here considered is 

 * infinitely little ' as determined by the law or character 

 of the particular series. That is to say, we are certain 

 that the law of the series holds unchangeably, however 

 far the process of analysis may be carried ; and we have 

 thus inferential certainty regarding the result of the 

 analysis (the equation of the sum of the terms to the 

 whole finite quantity), even although we may be unable 

 actually to count each one of the terms. It is the law 



