142 DEVELOPMENT AND PURPOSE CHAP. 



any two points, however close we may endeavour to take 

 them. No point is next to any other, because, between 

 every two points there is always another, and that is to say, 

 there is an infinity. Now this account draws a just conclu- 

 sion from its hypothesis, but the hypothesis itself is open to 

 more than one interpretation. If we keep resolutely to 

 the conception of the point as devoid of magnitude, no 

 finite number of these zeros will lead us anywhere. But 

 this result seems to be falsely interpreted if it is taken to 

 mean that space is an assemblage of point-elements of 

 which there is actually, in the shortest possible line, an 

 infinitude. This conception would balance one fiction 

 with another. The true interpretation appears to be 

 rather that the point is the abstraction of position within 

 a continuum, and that no summation of such abstractions 

 will yield the continuum itself, but rather that in the 

 smallest possible quantum of the continuum the abstrac- 

 tion could be repeated in an infinitude of different relations. 

 With this conception I think we approach a genuine 

 intellectual reconstruction of the sense-percept of con- 

 tinuity. 



To reduce continuous magnitude to a form in which it 

 can be subject to calculation, ordinary thought breaks it 

 up into units, and the units readily become fictitious parts. 

 As continuous space is dissolved into points, so time is 

 conceived as a succession of instants, though there are no 

 instants and no breaks between the end of one time element 

 and the beginning of another. In the same way, motion 

 is regarded as the successive occupation of positions, 

 though the moving body, strictly speaking, occupies no 

 position. However short the time taken, it is moving 

 through space, not occupying a single position in space. 

 Now for the purpose of calculation, the error involved in 

 treating the moving body as occupying a position may be 

 made as small as we please. In the same way, motion in 

 a curve may be resolved into a series of motions in very 

 short straight lines enclosed at very wide angles, and the 

 error may be reduced below any assignable point, and 

 generally the rate of a continuous change may be treated 

 as the limit which we should arrive at by taking the differ- 



