260 Mr DAVIES on the Equations of Loci 



the Geometry of Three Dimensions, beyond its simplest elements, 

 as well as the most interesting and important portions of the doc- 

 trine of plane loci itself. I only wish to express, that the direct 

 labours of DESCARTES on this subject are of very little value, if of 

 any whatever ; and this will be at once conceded, if we consult the 

 Jesuit RABUEL'S Commentary upon his Geometria, published 

 nearly a century after the original work, and containing the most 

 amplified attempt to illustrate this particular part of the treatise 

 that had then been made. The writings, too, of LEIBNITZ and 

 the two BERNOULLIS may be searched in vain for any attempts to 

 lay down new methods, or to develope and apply that of DES- 

 CARTES for the general discussion of curve surfaces, or of curves 

 any how situated in space. Indeed, we cannot but be struck 

 with the very small number of investigations of this kind at- 

 tempted by them ; and even those few might be said to have 

 arisen out of the rivalry between the old geometry and the new. 

 The singular facility possessed by these illustrious men, of de- 

 vising methods of inquiry of modifying those already in exist- 

 ence, to accomplish the purpose they had in view and their de- 

 votion to every thing that could set the powers of the calculus in 

 a more advantageous light, these would certainly have led them 

 to the consideration of the subject, had the method of DESCARTES 

 been adequately stated, and its adaptation to any kind of calculus 

 been even tolerably obvious. JAMES BERNOULLI did, indeed, 

 throw out a casual observation in the Leipsic Acts, that an equation 

 between three variables represented a curve surface, as an equation 

 between two variables represented a plane curve ; but the remark 

 seems to have suggested no further inquiries, and so far as JAMES 

 BERNOUILLI was concerned, the subject dropped at this point. 

 CLAIRAULT, by an independent train of inquiry, and whilst yet 

 a minor, was conducted to the same result ; but, pushing the in- 

 vestigation one step further, he was led to consider every curve 

 line in space as the intersection of two curve surfaces, or as a line 



