PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 255 



does not reveal the essence of the matter : this criterion or test, 

 viz. shortness, is not that which affords either the best evidence or 

 the fundamental idea of the property defined. Consider for 

 instance any arc ACB of a circle, C being the middle point, and 

 M the middle of the chord. When the distance between M and 

 C is small, the difference in the shortness between ACB and AMB 

 is very much smaller. 1 Suppose that e were the smallest per- 

 ceptible difference in length, then the departure from straightness 

 would be no less than the relatively enormous quantity 



*-•(!&) (0) 



in which h denotes the distance MC, and S the length of the arc. 

 Or to state this conversely, if . h be the limit of perceptible 

 quantity, the difference of length 



6= W ( ° a) 



is quite beneath the limits of perception. This may be still more 

 strongly stated: if h be an infinitesimal quantity, then e is 

 infinitely smaller? and therefore at the limit 3 the criterion utterly 

 fails: that is to say, it is not a criterion for the determination of 

 extremely small curvature. 



It is evident therefore, that MC or h affords a better — and in the 

 limit an infinitely better — measure of straightness than the differ- 

 ence of the arc and line, i.e. than shortness, a result confirmed 

 alike by common sense, by the ordinary methods of testing 

 straightness, and by analytical theory. 



x The versed sine {i.e. the distance from chord to curve, MC = h) divided 

 by the difference between the arc (ACB = 8) and the chord (AMB = s) is : — 

 h _ 1 - cos \e _ _3 a _ _i_ 02 i Qi _ t v 



being the angle subtended by the arc ACB. As the arc approaches 

 straightness i e. as approaches zero, h becomes vastly better than 8 - s 

 as a measure of straightness and at the limit is infinitely better. The 

 above expression may be put in the form 



h 3s, 47i 3 , 38 , 28 h 2 , , 

 8^s= 81 < X + 1^ etc -> = 8S < l " 15 fli etC>) 

 which confirms the preceding statement, for as h becomes very small, the 

 ratio becomes very great, and when h is an infinitesimal, the ratio is 

 infinite. 



* The non-mathematical reader will see later some remarks concerning 

 the meaning of these expressions. 



