to Non-Metrical Vector Algebra. 135 



we have, by the generalized Pythagorean theorem, 



I ds=\\ V 1 + {dyldxY . dx , 

 ji I h 



with # 2 + ?/ 2 = l as the conic equation. But taking a as the 

 usaxis o£ orthogonal coordinates f, 77, we have again 



f'Hf 



J« iji 



Vl + {dv/dg)* . d£ 



P4V=1. 



Thus the measures of the two angles are equal. In short, 

 the common arc-measure of both, and therefore of all 

 (concave) angles having the cosine ab = cosO = x is given by 



f*x dx 



which is the well-known function cos -1 x or arc cos x. From 

 this the converse theorem : if ab = 0i m then ab = lnv, follows 

 at once. In particular, if ab = 0, we shall have, as the arc- 

 measure of all right angles, 



Or 



1 Vl 



dx 77- 



, v "' 



where ir is the familiar transcendental number 3' 14 ... . 

 Thus also the circumference of the unit conic k is seen to be 

 2-7T, i. e. to contain Itt projective unit steps or — as I should 

 like to propose to call them — 2tt staudtians (the projective 

 scale being due to George von Staudt). This simple result 

 holds, of course, independently of the particular shape of the 

 ellipse adopted as standard conic and of the position of O 

 inside it. 



We need not be surprised by this result, which is but one 

 instance of the reappearance of familiar geometrical formula?. 

 For, as was already mentioned, the validity of the distributive 

 law of multiplication of vectors will carry with itself a 

 formal and numerical identity with all euclidean formula?, 

 the numbers of inches or of cms. being replaced by the 

 numbers of staudtians. That the same thing should hold 

 for curved arcs (considered as limits of polygons) as well as 

 for tensors of vectors, or "sizes" of straight segments, 

 might have been expected at once. 



In order to make the angle liable of assuming positive 

 and negative values it is enough to prescribe a sense of 

 circulation round tc as the positive one, and the opposite as 

 the negative one, in much the same way as in the familiar 

 treatment of angles as ;i rotations." 



