sicr. i.] DISSERTATION SECOND. 31 



shows the locus, on this hypothesis, to be always a conick 

 section ; and he distinguishes the cases in which it is a cir- 

 cle, an ellipsis, a parabola, or a hyperbola. It was an in- 

 stance of the most extensive investigation which had yet 

 been undertaken in geometry, though, to render it a com- 

 plete solution of the problem, much more detail was doubt- 

 less necessary. The investigation is extended to the cases 

 where the function, which remains the same, is of three, 

 four, or five dimensions, and where the locus is a line of a 

 higher order, though it may, in certain circumstances, be- 

 come a conick section. The lines given in position may 

 be more than four, or than any given number ; and the lines 

 drawn to them may either be perpendiculars, or lines mak- 

 ing given angles with them. The same analysis applies to 

 all the cases ; and this problem, therefore, afforded an ex- 

 cellent example of the use of algebra in the investigation of 

 geometrical propositions. The author takes notice of the 

 unwillingness of the ancients to transfer the language of 

 arithmetick into geometry, so that they were forced to 

 have recourse to very circuitous methods of expressing 

 those relations of quantity in which powers beyond the third 

 are introduced. Indeed, to deliver investigation from those 

 modes of expression which involve the composition of ra- 

 tios, and to substitute in their room the multiplication of the 

 numerical measures, is of itself a very great advantage, 

 arising from the introduction of algebra into geometry. 



In this book also, an ingenious method of drawing tan- 

 gents to curves is proposed by Descartes, as following from 

 his general principles, and it is an invention with which he 

 appears to have been particularly pleased. He says, " Nee 

 verebor dicere problema hoc non modo eorum, quae scio? 



here, as on many other occasions, a more general and more con 

 cise expression than could be otherwise obtained. 



