210 Mr IL'Kipath on the Law of Continuity. [March, 



form one straight line precisely equal in length to both together, 

 and the extreme points B, C, have not the least distance between 

 them, these points do not, therefore, coincide. 



They are still distinct, each the extreme of its — ■ „ , ' .,, 



own, but not of the other line. For if the two 

 points B, C, coincide, the lines cannot be said to touch, but 

 overlap, and the whole length A D is not the sum of A B and 

 C D, but one point less.* 



Some may hence infer, that if there be no lapse of time 

 between the first and second states, and none between the 

 second and third, there will be none between the first and third ; 

 and, therefore, none between the first and any other state ; and, 

 consequently, that a body may be in any number of different 

 states, or undergo any number of changes, without the smallest 

 lapse of time, which is manifestly absurd. This, however, is 

 by no means a correct inference. A body, according to our 

 ideas, may be in two different states, unlimited with respect to 

 their difference, without the lapse of time ; but it cannot be in 

 more than two, unless time intervene. For notwithstanding there 

 may be no interval of time between the first and second moments, 

 there is evidently a lapse of one whole moment between the 

 first and third, of two between the first and fourth, and so on 

 between the first and any other. And since these moments are 

 not absolutely nothing, but supposed to be so small that the body 

 is in one state only in each, it follows, that the number of 

 moments from any given point of time expresses the number of 

 states in which the body has been ; and, vice versa, the number 

 of states expresses the number of moments in which the changes 

 have been effected, but not accurately the length of time ; it 

 being possible for these moments to be unequal among them- 

 selves. If the chunge of state be uniform, that is, if equal 

 changes be made in equal times, it is probable that these 

 moments are equal ; but if the change be not uniform, or un- 

 equal changes be made in equal times, it is likely that the 

 moments are unequal. 



It was thus I at first endeavoured to establish the possibility 



* It will, perhaps, he said, that I here depart from the common idea of a point, 

 and that 1 am giving extension to lhat which has ever heen admitted to have none. 

 To this I reply that if a point be considered to be absolutely nothing, I am 

 certainly of a different opinion. By a point, I have uniformly understood that 

 which is indefinitely small, or which has no assignable dimensions; and in this 

 sense it is always, I believe, taken by mathematicians, though defined to be that 

 which has neither length, breadth, nor thickness; but assignable, determinate, or 

 some such word, appears to me to be always implied. If, according to my ideas, 

 a point be indefinitely multiplied, it will make up aline, and a line so multiplied 

 will make up a superficies, and a superficies a solid. Thus a superficies I imagine 

 to be an indefinitely small part of a solid, a line an indefinitely small part of a 

 superficies, and a point an indefinitely small part of a line. And this is the rela- 

 tion I conceive to exist between these four things; but there is no such relation 

 between a point and nothing : for nothing taken an infinitely infinite number of 

 times, will still be nothing ; it can never be so multiplied as to amount to what I 

 mean by a poigt. 



