Notices respecting New Books. 313 



(p. 5). On turning to p. 93, the reason for adopting this unusual 

 form of statement becomes apparent ; and as the point is of some in- 

 terest we will let Mr. Watson speak for himself: — 



" When we speak of one straight line as being equal to another, 

 or more accurately as being equal in length to another, we have a 

 very exact and precise idea of what is meant, viz. that the one line 

 can be applied to the other so that the extremities and every inter- 

 mediate point of the one may coincide with the extremities and some 

 intermediate point of the other. So, again, when we speak of one 

 circular arc as being equal to another circular arc of equal radius, we 

 mean that the one arc can be applied to the other so that the extre- 

 mities and every intermediate point of the one may coincide with the 

 extremities and some intermediate point of the other. But our ideas 

 of equality of length are not confined to lines which may be super- 

 posed upon one another in this manner, and in practice we speak 

 familiarly of the lengths of curves or of circular arcs, meaning thereby 

 the lengths of straight lines equal in length to these curves or cir- 

 cular arcs, although a straight line and the arc of a circle cannot of 

 course be superposed one upon the other. It is well therefore to 

 have an accurate definition of the length of a curved line, which we 

 will now proceed to investigate." 



The whole of Mr. Watson's remarks on this point will well repay 

 perusal. It might not occur to the reader that there is any difficulty 

 about the conception of the length of a curved line. He might say 

 that we are made familiar, by our everyday experience, with threads 

 that are flexible and sensibly inextensible, and that in consequence 

 we have no more difficulty in forming the conception of a perfectly 

 flexible and inextensible line, than in forming the conception of a line 

 devoid of breadth or thickness. Moreover we are familiar with such 

 phenomena as the unwinding of cotton from a reel, or the paying 

 out of a rope. Such experiences, he might say, suggest the notion 

 of a curved line pulled straight (or rectified), and that these concep- 

 tions could be easily embodied in a definition. In the present work 

 the question is thus treated : the nature of a rolling motion is first 

 explained, and the result embodied in the following definition: — " If a 

 straight line be made to roll upon a curved line, the length of the 

 straight line between its first and last points of *contact with the 

 curved line is defined to be the length of the curved line between its 

 first and last points of contact with the straight line." 



The only objection to this way of looking at the matter is that 

 the notion of a rolling motion is less familiar than that of a flexible 

 line pulled straight. What, however, is of most importance from 

 the mathematician's point of view, is how the result of the recti- 

 fication is to be determined ; and this, after the necessary explana- 

 tions, Mr. Watson states thus : — " If a number of points be taken 

 between the first and last points on any finite curved line, and the 



k chords between each pair of points in succession be drawn, and if the 

 number of such points be indefinitely increased, and the length of 

 each chord be indefinitely diminished, the ultimate sum of the lengths 

 of these chords is the length of the curved line." In this, we sup- 



