STATEMENT OF LEIBNIZ'S PHILOSOPHY 8 1 



Geometry reduces the discontinuity of Synthetic Geo- 

 metry to the relative continuity of number, or quantity 

 of homogeneous units. But number as a sum of finite 

 units (even though it may take the form of an infinite 

 series) is still to some extent discontinuous. It may, 

 however, be made continuous by regarding its elements 

 not as finite units, but as t infinitesimals ' or infinitely 

 little quantities. In other words, any numerical unit we 

 may choose to employ may be subdivided infinitely, and 

 thus every finite number may be regarded as the sum of 

 an infinite series of infinitely small terms. This is the 

 basis of the Infinitesimal Calculus as originally conceived 

 by Leibniz. It may be otherwise expressed by saying 

 that the series of finite numbers or quantities is ulti- 

 mately to be expressed, not as a series of terms which 

 grow by finite increments (like i-f-(i + i) + (i + i+i) 

 &c.), but as a series whose terms flow into one another, 

 their differences being infinitely small. That is to say, 

 any variable magnitude must be regarded as increasing or 

 diminishing by infinitely small increments or decrements. 

 The work of the Calculus is to determine the relations 

 between unknown quantities or magnitudes, not by 

 considering them merely as fixed wholes and directly 

 finding equations between them, but indirectly, by treat- 

 ing the quantities as variables or as growing, and in the 

 first place finding equations between their elements or 

 differences *. 



1 From one point of view it may be regarded as the solving of the 

 problem of Achilles and the tortoise. Cf. Lettre a M. Foucher (1693) 

 (E. 118 a; G. i. 416) : 'As to indivisibles, in the sense of the mere 

 extremities of a time or of a line, we cannot conceive new extremi- 

 ties, nor actual nor potential parts in them. Thus points are 

 neither large nor small, and no leap is needed to pass them. Yet 

 the continuous, although it everywhere has such indivisibles, is 

 not composed of them, as the objections of sceptics seem to suppose. 

 There is, in my opinion, nothing insurmountable in these objections, 

 as will be found if they are put into strict form. Father Gregory 

 of St. Vincent has excellently shown, by the Calculus of infinite 

 divisibility, the place where Achilles should overtake the tortoise 

 which starts before him, according to the proportion of their 

 velocities. Thus Geometry dissipates these apparent difficulties.' 







