Dr. Roget on a Violation of the Law of Continuity. 203 



On another occasion I shall beg to add the results of a few 

 barometrical measurements, and furnish some geological no- 

 tices. 

 Leeds, Dec. 26, 1827. John Nixon. 



XXXII. On an Apparent Violation of the Law of Continuity. 

 By P. M. Roget, M.D. F.R.S* 



[Concluded from p. 121.] 



nPHE severest test to which the validity of this mode of 

 T reasoning can be put, is to apply it to the purely mathe- 

 matical relations of space and quantity, to which, if the law 

 of continuity have any necessary existence in the nature of 

 things, it ought to apply with the greatest rigour. We find, 

 accordingly, that all the changes of magnitude in those quan- 

 tities of which the value is dependent on that of certain other 

 quantities, accompany corresponding changes in these latter 

 quantities in a manner strictly conformable with the law of 

 continuity. The same exact correspondence also obtains 

 with regard to the geometric relations of space, such as lines, 

 angles, surfaces and volumes. Thus, all the trigonometrical 

 lines which are functions of the arcs of circles, undergo regu- 

 lar and continuous changes, corresponding to the regular and 

 continuous increase or diminution of the arcs to which they 

 are related. Some, as the sine and cosine, pass from every 

 intermediate magnitude between nothing and a certain limit, 

 which they do not exceed, but to which, when the arc in- 

 creases uniformly, their last approximations are made with 

 extreme slowness. Arrived at that limit, they again begin to 

 decrease ; but at first with extreme slowness, like bodies that 

 are set in motion, from a state of rest, by the action of a 

 moving force. Their progress of diminution is by degrees 

 accelerated ; and is again retarded only when they are pre- 

 paring, as it were, to undergo some other change. By the 

 time they arrive at any of their states, either of maximum or 

 minimum, their rate or velocities of increase or diminution 

 have gradually been extinguished, and reduced to nothing. 

 Some of these lines, as the tangent for example, have no li- 



* The former portion of this article was inadvertently reprinted in our 

 last Number, without having been previously submitted to the author; in 

 consequence of which, several inaccuracies in the original (now noticed in 

 p. 20b*,) were left uncorrected ; and the division having been made in an im- 

 proper place, the last paragraph of the former portion is here repeated. — 

 Edit. 



mits 



