128 Dr. L. Silberstein : Farther Contribution* 



Thus for instance, iE A is normal to B, and it' C = A — B, we- 

 have, by the distributive law, 



C 2 = A 2 + B 2 



or 



,C, 2 = : A 2 -r- B| 2 , 

 the generalization of Pythagoras' theorem, which reads: the 

 squared number of projective unit steps contained in the 

 hypotenuse is equal to the sum of those contained in the 

 sides. 



The reader will find it a useful recreation to verify this graphically in 

 a convenient case, such as A\=3, (B =4, drawing" any ellipse as k 

 round O, constructing its polar T, and so on. If he draws the rather 

 numerous lines, necessary for the whole construction, with some care he 

 will find the number of unit steps contained in C to be hardly distin- 

 guishable from •">. The writer has actually constructed this case with 

 good success, his naive aim being originally to verify thus the analytic 

 result. 



More generally, for airy triangle with A, B and C = A — B 

 as vector sides, 



|C 2 = A 2 +,Bj 2 -2|Ai . ;B . ab. . . (9 A 



Many other examples of generalized theorems could at 

 once be given, but those quoted above will suffice for the 

 present. 



5. Angles, their arc-measure and trigonometric functions. — 

 We have incidentally called the pair a, b an " angle," in 

 particular a, a a zero angle and a, —a a straight angle. We 

 may now, without fear of a misunderstanding, call a, b, 

 L e. aOb or bOa, in the case when ab = 0, a right angle. In 

 general, for any a, b, or their equals constructed anywhere, 

 we can speak of the angle aOb. Still, it remains true that 

 we do not yet know what " equal angles" and "multiples of 

 an angle " mean, simply because we have not yet fixed the 

 meaning of these words. There is nothing to prevent us 

 from denoting the angle a Ob by a single letter, say #, provided 

 we keep in mind that hitherto this 6 is not itself a magnitude. 

 We shall define it presently as such. 



Meanwhile w T e can speak of the number ab, already 

 familiar to us. as a certain "indirect measure" or "function" 

 of the angle # = a, b. As such let us call it the cosine of 0, 

 writing 



ab= cos 6~ cos (a, b). 



We know from Section 4 that this number is represented by 

 the orthogonal projection N of b upon a, that is to say, that 



cos 6 = cos (a, b) = ± i 0N\ , . . . (10) 



