232 On the Theory of Angular Geometry. [No. 123. 



magnitude is; All these definitions then fail in strictly fulfilling their 

 object. Each has been in turn severely criticised by following reviewers, 

 anxious to establish the validity of the most infallible of all — their own. 

 But all agree in this; that they are descriptions of different characteristics 

 of the same idea. If from any one we can obtain a definite conception 

 of what is intended, we immediately perceive that all are sufficiently 

 correct to recall it to our minds. All agree in understanding angular 

 quantity to be " something," or if the expression be too bold, " that" 

 which lies between two intersecting straight lines. All of them agree 

 further in considering, that for purposes of comparison as to magnitude, 

 angles must be estimated crossways, or by the width between the lines, 

 and not with any reference whatever to the longitudinal extension in 

 the direction of the sides. 



Now, if we analyse the various definitions of an angle in this manner, 

 it is, I think, impossible to come to any other conclusion than that an 

 angle is the plane surface between two lines ; of a peculiar nature, 

 partly bounded and partly unlimited, whose value could consequently be 

 only estimated by reference to the bounded direction, that is, the width 

 between the sides. And the neatest and shortest mode of expressing 

 this will apparently best solve our difficulty, as it connects axomatically 

 an explicit definition with the working one. 



The first place in which I believe this idea was embodied, was Ber- 

 trand's celebrated solution of the difficulty in the theory of parallels. 

 The principle of that demonstration is as follows : Any angle, however 

 small, can by repeated reduplication be made to exceed any given angle 

 however great, but the band of unlimited space between two parallel 

 lines, though repeated ever so often, will never fill up that given angle. 

 Hence an inter-parallel space is less than any assignable angle in value, 

 and therefore a line which cuts one of two parallels, must also cut 

 the other, otherwise the angle which it makes with the one it does cut, 

 would be wholly contained within the inter-parellel space, and be less 

 than it. The stress of this demonstration evidently rests on the com- 

 parison of surfaces, and it is surprising that its extreme elegance did not 

 lead Geometers earlier to seek the solution of the problem in that di- 

 rection. The truth is, that the new "unlimited spaces" were treated 

 as interlopers in the science of figure, and the demonstration rejected, as 

 " wearing only the semblance of geometrical accuracy." 



