tanPMN = 



traced upon the Surface of the Sphere. 383 



J!_ I 



*. 4 J_ 

 ~ 



Now, as n is diminished, tan PMN approximates towards _E^L^ 



sin iltj> 



and at the limit we have actually 



rm<rvr 



tanPMN= 



putting d<f> and dQ for sin< and sin dB. Hence, in reference to the case of 

 ultimate approximation of PM and PN (that is, when they are coalescing, 

 as at M), we have 



.(7.) 



which is the same result as we obtained as that which has been long well 

 known by those geometers who have treated of the length of the loxodrome *, 

 and which has been used in my paper (xvi. 2.) 



* So far as I have been able to discover, this formula is due to EULEB, who employed 

 it in a paper on " Spherical Trigonometry, by the method of maxima and minima" in the 

 Berlin Memoirs for 1753, p. 226. It is very remarkable how near EULER approached, in 

 the passage here referred to, to the method of spherical co-ordinates, such as we have here 

 developed ; and yet he does not seem to have entertained the slightest notion that such a 

 method of investigation was capable of general application. Had it once occurred to his 

 mind, there is no doubt that Spherical Geometry would have been in a much more ad- 

 vanced state than it now is. Its principles have been fully developed, and its practice 

 rendered familiar as a branch of elementary study. 



