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



mits to their increase of magnitude, and pass through every 

 possible linear dimension from nothing to infinity, while esti- 

 mated in one certain direction, characterized by the positive 

 sign. From positive infinity, the transition is made at once 

 to negative infinity, and the subsequent changes consequent 

 upon the further enlargement of the arc, consist in the gra- 

 dual diminution of the negative tangent; that is, of the line 

 estimated in a direction opposite to that which was assumed 

 as the measure of the original, or positive tangent. This ne- 

 gative tangent passes again, by the continuation of the same 

 process, into the primitive tangent, after having been again 

 reduced to nothing. 



By considering, in like manner, the generation and the 

 course of curves of every kind, whether produced by the in- 

 tersection of lines, moving according to certain laws, or of 

 points whose motions are limited to certain conditions with 

 relation to other lines or curves, we shall find that the same 

 law of continuity obtains, both with respect to the changes of 

 magnitude and the changes of direction. No mathematical 

 quantity changes from positive to negative, or from negative 

 to positive, without either becoming infinite, or being reduced 

 to nothing ; and it passes through every possible intermediate 

 degree of magnitude, in its progress from the one to the other. 

 The change from increase to diminution, or the reverse, is 

 never abruptly made ; but is always effected by a transition 

 through a condition of momentary quiescence, which is the 

 limit of opposite kinds of changes, and at which instant no 

 actual change takes place. 



Such being the universal law which is observed by lines 

 generated according to any given geometric conditions, we 

 should naturally be led, from analogy, to expect that the mo- 

 tions of points, having certain geometric relations to given 

 curves, would also be governed by the same law. Let us 

 take, for example, a small portion of any curve. This portion 

 will have, what is termed, a certain radius of curvature, and 

 the furthest extremity of this radius will be the centre of cur- 

 vature. In proportion as the curvature is diminished, this 

 radius will increase in length ; and the centre of curvature 

 will be removed to a greater distance. Its motion during this 

 gradual change will be along a right line, perpendicular to 

 the tangent of the curve at the point of contact. As the curve 

 in the change we are supposing it to undergo, approaches 

 more and more to a straight line, the centre will recede with 

 great rapidity ; till at length, when all curvature is lost, it 

 may be regarded as removed to an infinite distance ; and at 



this 



