120 Dr/Roget on a Violation of the Laiso of Continuity. 



Not content with establishing the universality of this law by 

 induction, Boscovich has endeavoured to prove it by other 

 arguments, of a still more positive nature, derived from abs- 

 tract considerations. This universality, he says, arises ne- 

 cessarily from the very nature of continuity. The limit which 

 joins the precedent and the consequent of any thing is com- 

 mon to both, and is therefore indivisible. Thus, a superficies 

 separating two solids has no thickness, and is that in which a 

 transition from the one to the other occurs ; a line, dividing 

 two portions of a continued superficies, has no breadth ; a 

 point, discriminating two segments of a continued line, has no 

 dimension of any kind. So it is with regard to time: for the 

 limit of two consecutive portions is common to both and in- 

 divisible; and as every change of a variable quantity from one 

 magnitude to another must be made in time, so every change 

 must participate in the continuity of time. But to every mo- 

 ment of time, a certain magnitude of the variable quantity cor- 

 responds, and the limit of two moments of time is common 

 and indivisible ; therefore the limits of two magnitudes, corre- 

 sponding to these two moments, must, in like manner, be com- 

 mon and indivisible. Moreover, it is impossible for any quan- 

 tity to have two magnitudes at the same time, and when con- 

 tinually varying, that it shall have the same magnitude at dif- 

 ferent times ; much more impossible, therefore, that in the li- 

 mit of two moments of time it shall have two magnitudes, the 

 one course pending to the precedent, and the other to the 

 consequent moment; or shall not have gone through the in- 

 termediate magnitudes in the intermediate moments of time. 

 For the same reason, a body cannot have two velocities at the 

 same time, and therefore cannot have two velocities in the li- 

 mit common to two moments of time ; and when continually 

 changing its velocity, cannot have the same velocity in different 

 moments of time, but must go through all the intermediate ve- 

 locities in the intermediate moments of time. Hence, then, 

 in passing from the magnitude 8 to the magnitude 12, the 

 variable quantity passes through the magnitudes 9, 10, 11. 

 The increase or diminution of temperature, for example, goes 

 on gradually ; the mercury in the thermometer rises or falls 

 progressively, passing through every intermediate degree from 

 one point of the scale to another. Now as this reasoning is 

 unaffected by any considerations of the hardness, softness, 

 elasticity, or other physical property of bodies, the universality 

 of the law, as resulting from the nature of continuity, is a 

 truth independent of such considerations. From these argu- 

 ments, which I have given as stated by his able expositor 

 Professor Robison, Boscovich has concluded that the law of 



continuity 



