259 ) 



On the Equations of Loci traced upon the Surface of the Sphere, 

 as expressed by Spherical Co-ordinates. By THOMAS STE- 

 PHENS DAVIES, Esq. F. R.S.ED. F.R.A.S. 



(Read 16th January 1832J 



1 HE modern system of analytical geometry of three dimensions 

 originated with CLAIRAULT, and received its final form from the 

 hands of MONGE. DESCARTES, it is true, had remarked, that the 

 orthogonal projections of a curve anyhow situated in space, upon 

 two given rectangular planes, determined the magnitude, species, 

 and position of that curve ; but this is, in fact, only an appro- 

 priation to scientific purposes of a principle which must have 

 been employed from the earliest period of architectural delinea- 

 tion the orthography and ichnography, or the ground-plan and 

 section of the system of represented lines. Had DESCARTES, 

 however, done more than make the suggestion had he pointed 

 out the particular aspect under which it could have been ren- 

 dered available to geometrical research had he furnished a suit- 

 able notation and methods of investigation and, finally, had he 

 given a few examples, calculated to render his analytical pro- 

 cesses intelligible to other mathematicians ; then, indeed, this 

 branch of science would have owed him deeper obligations than 

 it can now be said to do. Still I do not wish to be understood 

 to undervalue the labours of that extraordinary man, indirectly 

 at least, upon this subject ; for it is certain, that, though he did 

 succeed in placing the inquiry in its true position, yet it is ulti- 

 mately to his method of treating plane loci, by means of indeter- 

 minate algebraical equations, that we owe every thing we know in 



