Professor Young's Analytical Geometry, 603 



We might at once urge that by the same rule practical utility would 

 become the standard by which we should measure the extent to 

 which natural philosophy itself ought to be pursued. 



Of the two methods by which the properties of the conic sections 

 and surfaces of the second order may be demonstrated, (the method 

 of coordinates and the methods of pure geometry,) the prevailing 

 prejudice is in favour of the former. That the coordinate method 

 harmonizes much better with the general methods of inquiry which 

 are found most advantageous in physical research, no one can deny ; 

 but in any other point of view the superiority of this method is very 

 doubtful. As an intellectual exercise there can be no question re- 

 specting the superiority of the ancient methods : as an instrument of 

 investigation of the properties of curves and surfaces of the second 

 order, the method of transversals (especially when we adopt Chasles's 

 application of Prop. 129. lib. vii. of the Mathematical Collections of 

 Pappus,) is by far the most ready and effective, as well as the most 

 general and comprehensive. On the other hand, if the investiga- 

 tion of these properties be viewed in reference to exercising the 

 student in the modes of investigation that he will be most frequently 

 required to employ in physical research, the geometry of coordinates 

 should be the sole method he should employ. Every judicious stu- 

 dent will, however, acquaint himself in some degree with the other 

 methods, — fashionable as it is in our day and in this country, (but in 

 this country only,) to treat such modes of research as antiquated 

 and unworthy of notice. 



The systematic introduction of surfaces of the second order into 

 an elementary course of study is comparatively recent ; for, though 

 surfaces of revolution were considered by the Greeks, only very few 

 of their properties had been investigated prior to the time of Monge, 

 Hachette, and their disciples of the Polytechnic School. So assi- 

 duously, however, have these surfaces been studied since, that their 

 known properties in their most general form are as numerous, (or 

 perhaps more numerous) as even those which relate to lines of the 

 same order ; and no treatise on analytical geometry can be considered 

 complete without the introduction of a considerable number of these. 

 Nor are they mere subjects of speculative curiosity even in themselves. 

 They have a direct bearing on several physical subjects of inquiry. 

 Their great utility, however, arises from the exercise they alFord to 

 the student in the discussion of phaenomena taking place in space 

 of three dimensions — that is, in short, of nearly all the phaenomena 

 of inorganic nature. Very few motions take place in one fixed plane, 

 and even these can be studied only in reference to some other plane or 

 planes. They are always, or almost always, in curves of double cur- 

 vature, or upon curve surfaces ; and when not so, the lines or planes 

 in which they take place cannot be contemplated and determined 

 but by means of some other lines and planes in which the component 

 forces act. To speak familiarly, they are resultants to be deter- 

 mined — not composants actually given. 



The attachment to the coordinate method of geometrical research 

 in reference to its ulterior physical application, is not a mere preju- 



