traced upon the Surface of the Sphere. 265 



tivated. But the more general form which loci, situated in space, 

 had assumed, caused this to be looked upon as an unimportant, 

 or at least so subordinate, a method, as to be unworthy of culti- 

 vation ; and it was therefore taken up by no one competent to 

 complete the task which EULER had left ready to be entered on. 

 The beautiful theorem of LEXELL, however, which assigns the 

 locus of the vertical angle whose base and area are given, would, 

 we should have been led to expect, have recalled attention to the 

 subject : but it did not ; and even to the present day, amongst 

 the continental geometers, no traces of a system of spherical co- 

 ordinates are to be found. It is the more singular, inasmuch as 

 several spherical loci, which are known by other methods to be 

 circles, have already been determined to be so by trigonometrical 

 considerations. Yet none of these methods are capable of showing 

 a priori what the locus is ; they only shoiv whether the suspicion 

 previously entertained (for whatever reason) of their being circles, 

 be or be not true. We may refer to the demonstrations given by 

 LAGRANGE and LEGENDRE to LEXELL'S theorem, as specimens, 

 these being amongst the most elegant of this class. The method 

 of spherical co-ordinates has been for some years familiar to my 

 mind, and I have occasionally used it in the occasional inquiries 

 which presented an opportunity of applying it, and generally with 

 complete success : and, indeed, it seemed so natural and obvious 

 a method of proceeding, that I could not, till after considerable 

 search and frequent consultation of my mathematical friends, 

 persuade myself that it had not been already discussed, and 

 its principles and application properly detailed. The near ap- 

 proach to it in several of the discussions concerning the rhumb- 

 line (especially that of Dr HUTTON), and more particularly the 

 method of expressing the angle which it made with the meridian, 

 rendered me so doubtful on this point, that I did not venture to 

 state, in my paper on the Hour-Lines, that the general method 

 of investigating spherical loci, of which the Hectimoria furnished 



