1818.] Mr. Herapath on the Law of Continuity. 213 



assume at a greater distance, to be the interior extremity 

 of the part c d, corresponding to the part C D subsequent 

 to A B. And because the extremes B, C, touch without 

 coinciding, and the distances B 6, C c, are both perpendi- 

 cular to N O, these distances must touch in every part as far 

 as the extremity b without coinciding. And since the ex- 

 tremity b is contained within the distance B b, or does not 

 extend beyond it, this extremity itself must touch, but cannot 

 intersect, or even penetrate the distance C c. Hence if M P 

 be not a continued line, but composed of parts at various 

 distances from N O, and the state of the body be nevertheless 

 always as the distances of these parts from the right line N O, it 

 is possible for a body to be in one state at the end of one 

 moment and in a different in the beginning of the following ; 

 that is, to be in two different states without the lapse of time. 

 And because the distance between the parts is not confined to 

 limits, the difference of the states is likewise not confined to 

 limits, but may be of any magnitude whatever. Nor can the state 

 of the body pass through any of the intermediate degrees between 

 B b and C c. For the state is not, in strictness, though in many 

 eases it may safely be supposed to be, represented by a variable 

 line or distance carried uniformly along the straight line, and 

 which no sooner clears itself from the extremity b, than by a kind 

 of spring it extends itself to the other part M, but by fixed and 

 immutable distances, connecting every part of the one line with 

 the corresponding parts of the other : so that the body is no 

 sooner out of the state B b, than it is in the state C c , the change 

 being, if the expression may be used, a perfect saltatious 

 saltation. 



It would be a mere waste of time now to attempt to show 

 wherein we think the reasoning of Leibnitz defective. Our 

 ideas of the subject must be apparent to every one who has atten- 

 tively read what we have written ; but one remark it may not, 

 perhaps, be unnecessary to make, respecting the relation between 

 the state and time ; namely, that it appears from what we have 

 said, that, in all cases where continuity takes place, the state is 

 some function of the time ; but in other cases the state cannot 

 generally be a function of the time, unless the function be as 

 saltatious as the state. I am, dear Sir, very sincerely, 



Your humble servant, 



Knotolchill House, near Bristol, Oct. 30, 1817. J. HeRAPATH. 



