THE GEOMETRY OF MOTION 227 



the same manner as from mean speed we reached a con- 

 ception of actual speed. This process of reaching a Hmit 

 in conception, which cannot be really attained in perception, 

 is so important that we will again consider it for this special 

 case, in order that the reader may have little difficulty 

 henceforth in discovering and discussing such limits for 

 himself Let us accordingly suppose the distances be- 

 tween the points P^ P.„ P^, . . . Pg plotted off (Fig. 16) 

 down a vertical line as in the time-chart of Fig. 10 (p. 

 213). Along the horizontal line P^M^ instead of assuming 

 units of length to represent units of time, let them repre- 

 sent units of angle,^ and let the number of units taken 

 from Pj represent successively the number of units of angle 

 between the tangents P.,L„ P.^L,,, P^L^, etc., in Fig. 1 4 (p. 

 225), and the tangent to the curve at P^. Thus let P^M^ 

 represent the angle between the tangents at P^ and at P^ ; 

 PjMj. that between the tangents at P^ and at P., and so on. 

 Now draw in Fig. 16 vertical lines through the points M.„ 

 M3, etc., and horizontal lines through the points P^, P^, etc., 

 and suppose these lines pair and pair to meet in the 

 points Q^, Q3, etc. We have then a series of points Q, 

 which increase in number as we increase the points P in 

 Fig. 14, and in conception ultimately give us the curve 

 marked in Fig. 16 by the continuous line. The diagram 

 thus obtained is a chart of the bending or curvature in 

 Fig. 14. For, the mean curvature in the length P^P,^ is 

 the ratio of the angle L^NL. to the length P^P^ in Fig. 

 14, or, what is the same thing, the ratio of the number of 



1 According to Euclid iii. 29 and vi. 33, the angles at the centre of a 

 circle which stand on equal arcs are themselves equal ; if we double or treble 

 the arc we must double or treble the angle ; the arc is thus seen to be a fit 

 measure of the angle. Further (Clifford's Conimoii Sense of the Exact Sciences, 

 pp. 123-5), the arcs of different circles subtending equal angles at their 

 respective centres are easily shown to be in the ratio of their radii. If, there- 

 fore, we take as our standard circle for measuring angles the circle whose 

 radius is the unit of length, its arc c for any given angle will be to the arc a of 

 a circle of radius r subtending the same angle in the ratio 6f i to r, or in the 

 form of a proportion, c : a : : 1 : r, whence it follows that c = ajr or the 

 circular measure c of any angle is the ratio of the arc a subtended by this angle 

 at the centre of any circle to the radius r of this circle. The unit of angle in 

 circular measure will therefore be one for which a equals r, or which subtends 

 an arc equal to the radius. This unit is termed a radian, and is generally 

 used in theoretical investigations. 



