</.Y 



order to investigate the properties of a curve it is sufficient to select any 

 characteristic geometric property, as a definition, ami to expn- u 

 equation by means of the (current) coordinate^ ..t any point on i In- 

 curve; i.e., to translate the detinition into the laiitfua#- ot ana 

 geometry the equation so obtained contains implicitly every property 

 of the curve, and any particular property can !> deduced from it l>v 

 01, li miry algebra. 



While the earlier geometry is an admirable instrument for intellectual 

 training, and while it frequently affords an elegant deinon>tiat 

 some proposition the truth of which is already known, it requires a 

 special procedure for each individual problem; on the other hand, 

 analytic geometry lays down a few simple rules by which any property 

 can l>e at once proved. It is incomparably more potent than the 

 geometry of the ancients for all purposes of research. 



NOTK B 



Construction of any conic, given directrix, focus, and eccentricity. Let 

 - the directrix, F the focus, and e the eccentricity of a conic 

 (ci. 1'art 1, Art. 48), to plot the curve. 



CONSTRUCTION: Draw ZFX perpendicular to /XT), and ZW*o that, 

 if a = / XZ\V, tana = e. Now draw /-'A' p-rjHMidicular to /. 

 '/, It a; /i ; then R is a point of the conic; it is the end of the latu- in turn. 



Bisect the right angles at F by fW, and /'//,. interact int: Z II' in //, 

 and II r and draw //, I and //,/T perpendicular to ZX\ then A and A' 

 are points on the curve; they are the vertices of the conic. 



