1818.] Mr. Herapath on the Law of Continuity. 211 



of saltatious change, or the saltation of states ; but I did not do 

 this with a new to destroy continuity altogether. I was too well 

 convinced of the existence of continuous or gradual change, in 

 most of the great operations of nature, to attempt a refutation of 

 it in toto. My efforts, therefore, at the first consideration, were 

 directed to confute the idea of its being a law, by proving that 

 an opposite principle was equally possible, if not equally probable. 

 But afterwards finding that its universality was acknowledged by 

 some of the most distinguished philosophers of modern times, 

 and considering that, as a principle in philosophy, it was of suffi- 

 cient importance to merit further attention, and that at present 

 I had only considered it metaphysically, I was induced to take 

 a different view of it, and to see what would be the consequence 

 of a mathematical investigation. The following is part of the 

 result of my reflections. 



Let the indefinite straight line N O 

 represent any indefinite portion of 

 time, and let the line M P be so 

 related to it, that the perpendicular 



distances of the parts of the line .. t _ a j — - - 



M P from the right line N O, shall °' DCBA M 



always be as the states of the body 



corresponding to the intersections of NO by the distances. 

 Any where in the line N O take the contiguous little parts 

 A B, CD; and let a b, c d, be the corresponding conti- 

 guous parts intercepted by the perpendicular distances ill 

 the line M P. Then since the extremes B, C, as well as 

 the extremes b, c, touch, but do not coincide, the distances 

 B b, C c, touch, but not coincide. And because the ex- 

 tremes b, c, have no interval between them, nor the extremes 

 B, C, the distances B b, C c, cannot have any difference, 

 and must, therefore, be equal. But it will not be the same 

 with the distances A a, B b, or C c, and D d ; for if the 

 state be variable, it may happen that all three shall be different. 

 Moreover, since A B, C D, are of indefinite lengths, the same 

 will hold good under every dimension of them, even if we sup- 



Sose them to dwindle into mere points. Hence, therefore, if 

 1 P be a continuous line, it will be impossible for any change 

 to take place without the intervention of time. A body will be 

 precisely in the same states at the end of one and the beginning 

 of the following instant ; but during the lapse of time, however 

 small it may be, a change may easily take place, and the body 

 be in different states. But even this change is not without 

 limits. For no part of M P can be perpendicular to N O, and, 

 therefore, the difference of any two distances A a, D b, must 

 always be less than a d. But with respect to the time A D, 

 the change may have any proportion between o and oo . 



'Ibis is strictly the law of continuity taken in its utmost 

 extent, from the consideration of which, by assigning certain 



o 2 



