432 



SCIENCE. 



[N. S. Vol. XI. No. 272. 



mined upon it so that every one of them is 

 separated from Ahy B and C. 



In the third place, the poissible points so deter- 

 minable on that line cannot be distinguished 

 from one another by being put into one-to-one 

 correspondence with any system of ' assignable 

 quantities.' For such assignable quantities 

 form a collection whose multitude is exceeded 

 by that of another collection, namely, the col- 

 lection of all possible collections of those ' as- 

 signable quantities.' But points are, by our 

 postulate, determinable on the line in excess of 

 that or of any other multitude. Now, those 

 who say that two different points on a line 

 must be at a finite distance from one another, 

 virtually assert that the points are distinguish- 

 able by corresponding (in a one-to-one corre- 

 spondence) to different individuals of a system 

 of ' assignable quantities.' This system is a col- 

 lection of individual quantities of very moder- 

 ate multitude, being no more than the multi- 

 tude of all possible collections of integral 

 numbers. For by those ' assignable quantities ' 

 are meant those toward which the values of 

 fractions can indefinitely approximate. Accord- 

 ing to my postulate, which involves no contra- 

 diction, a line may be so conceived that its 

 points are not so distinguishable and conse- 

 quently can be at infinitesimal distances. 



Since, according to this conception, any mul- 

 titude of points whatever are determinable on 

 the line (not, of course, by us, but of their 

 own nature), and since there is no maximum 

 multitude, it follows that the points cannot be 

 regarded as constituent parts of the line, exist- 

 ing on it Ijy virtue of the line's existence. For 

 if they were so, they would form a collection ; 

 and there would be a multitude greater than 

 that of the points determinable on a line. We 

 must, therefore, conceive that there are only 

 so many points on the line as have been marked, 

 or otherwise determined, vipon it. Thdse do 

 form a collection ; but ever a greater collection 

 remains determinable upon the line. All the 

 determinable points cannot form a collection, 

 since, by the postulate, if they did, the multi- 

 tude of that collection would not be less than 

 another multitude. The explanation of their 

 not forming a collection is that all the deter- 

 minable points are not individuals, distinct, 



each from all the rest. For individuals can 

 only be distinct from one another in three ways : 

 First, by acts of reaction, immediate or mediate, 

 upon one another ; second, by having per se 

 different qualities ; and third, by being in one- 

 to-one correspondence to individuals that are 

 distinct from one another in one of the first 

 two ways. Now the points on a line not yet 

 actually determined are mere potentialities, 

 and, as such, cannot react upon one another 

 actually ; and, per se, they are all exactly 

 alike ; and they cannot be in one-to-one corre- 

 spondence to any collection, since the multitude 

 of that collection would require to be a maxi- 

 mum multitude. Consequently, all the possible 

 points are not distinct from one another ; al- 

 though any possible multitude of points, once 

 determined, become so distinct by the act of 

 determination. It may be asked, " If the 

 totality of the points determinable on a line 

 does not constitute a collection, what shall we 

 call it ? " The answer is plain : the possibility 

 of determining more than any given multitude 

 of points, or, in other words, the fact that there 

 is room for any multitude at every part of the 

 line, makes it conh'nMOMS. Every point actually 

 marked upon it breaks its continuity, in one 

 sense. 



Not only is this view admissible without any 

 violation of logic, but I find — though I cannot 

 ask the space to explain this here — that it forms 

 a basis for the differential calculus preferable, 

 perhaps, at any rate, quite as clear, as the 

 doctrine of limits. But this is not all. The 

 subject of topical geometry has remained in a 

 backward state because, as I apprehend, nobody 

 has found a way of reasoning about it with 

 demonstrative rigor. But the above conception 

 of a line leads to a definition of continuity very 

 similar to that of Kant. Although Kant con- 

 fuses continuity with infinite divisibility, yet it 

 is noticeable that he always defines a continuum 

 as that of which every part (not every echter 

 Theil) has itself parts. This is a very different 

 thing from infinite divisibility, since it implies 

 that the continuum is not composed of points, 

 as, for example, the system of rational frac- 

 tions, though infinitely divisible, is composed 

 of the individual fractions. If we define a 

 continuum as that every part of which can be 



