traced upon the Surface of the Sphere. 267 



of all spherical loci. It enters into discussions respecting these 

 in the same way as the straight line does into the geometry of 

 rectilinear co-ordinates ; that is, as its generatrix, and as the 

 simplest line that can be traced upon the surface : and hence, 

 whatever might be the class of problems that should form the 

 principal subject of our investigation, it appeared certain that 

 no general system could be devised, which did not include 

 the equation of the circle as one of its most important ele- 

 ments. That this may be the case, it became necessary to 

 assign the equations of circles, great and less, subject to every 

 variety of defining conditions. The expressions here given, are, 

 it is true, sometimes long, but they always involve symmetrical 

 data symmetrically in the result : and on that account are pre- 

 ferable, both for their generality of application, and their greater 

 elegance as analytical formulas. Those cases where greatest sim- 

 plicity is required are obtained at once, by giving suitable values 

 to the data of the problem ; or they may be so transformed by 

 taking suitable origin and direction of the co-ordinate axes. 

 There is, besides, but little difficulty in discovering how this is 

 to be done ; and I have not, therefore, dwelt upon it in treat- 

 ing of the circle. In one or two cases where the discovery 

 of a truth was the object in view, rather than the discovery of a 

 general formula, I have not scrupled to adopt this shorter pro- 

 cess ; and in others, the inquiry would have given results so 

 complex as to be incapable of reduction, had I not taken that 

 method of simplification. This, however, is only identical with 

 our common practice in the geometry of rectilinear co-ordi- 

 nates, and if we urge it as a difficulty against one method, it ap- 

 plies with great force against the other. I have no doubt, then, 

 that when we have become as familiar with this mode of inquiry 

 as with the other, we shall find it equally simple, easy, and intel- 

 ligible, as that is to us now. 



In the next place, I have given formulae of transformation of 



