332 On the Fundamental Principles of Mathematics. 



the origin outward being still positive and the contrary neg- 

 ative, &c, in the case of any such axis ; all the six axes being 

 moreover in the same plane. What has thus been extended from 

 the case of three axes thus situated to more than three, may in like 

 manner be extended to any number of axes however great. 

 Hence direction outward must be regarded as positive on any 

 and every straight line drawn from the same origin, in the same 

 plane, and the contrary direction must be regarded as negative, 



the origin. 



ifter that manner, on the contrary side of 



cessity of regarding c 

 measured outward fj 



the foundation ft 



ft 



? 



it is regarded as extending across or beyond that centre, or pole. 



(18.) The doctrine of "imaginary quantities" would be next 

 in order; but this, of itself, would furnish matter for an entire 

 dissertation ; if it were even advisable to enter upon the consid- 

 eration of a subject, so much and so often discussed.* It may 

 not however be amiss to advert to one or two results of anal- 

 ysis which seem to admit of explanation, by a reference to the 

 principle, that imaginary quantities, occurring in a geometrical 

 investigation, may sometimes have a possible existence out of the 

 plane of reference. Two equations first discovered by Euler, 



x+/r\ — xyri 2\/f -%f^\ 



e -e e +e 

 sin. x— 2~~/f t anc * cos - x= 2 ~ J 



when transformed, by substituting for the real arc x, the imag- 

 inary arc x%/-\, give, respectively, 



X 



X x 



sin. (x\/ 





Here the cosine is real, though the sine and the arc are both 

 imaginary. This seems to arise from the fact that the cosine 



effect 



v/ 



It, therefore, has a real value : while the sine and arc, being both 

 out of the plane of the axes, are imaginary. This being admit- 

 ted, the secant of the real arc (i. e., the arc whose cosine has this 

 real value) will =l-i- by the value of the cosine; while the 

 secant of the imaginary arc having the same cosine must, it 

 would seem, =;/7r_f- by the value of that cosine; unless the 

 imaginary arc were reduced to the limit of 0°, or 180°, or 360 

 &c. ; when it would be terminated in the plane of the axes : 

 when also its sine must = 0. 



o 



See among others, the " Essai" of M, Faure, (already 



