ON THE DISCRIMINATION OF THE GENERAL CONIC 



By Prof. John Patrick Dalton, M.A., D.Sc. 



In the teaching of mathematics more than in that of any 

 other snliject we snffer from the influence of tradition. To 

 within a few decades ago the subject was an ordinary item in 

 the educational cnrricnhim, justifying its position by the mental 

 stimukis provided, and the conse(|uent sliarj)eniiip of tlie logical 

 faculty; but with modern developments of applied science there 

 has arisen a chiss of students — now forming the larger T'roportion 

 of the mathematical classes of the Universities — who are inter- 

 ested in the subject, becatise it afl:'ords powerftil methods of 

 solving the technical difficulties encountered in their work. On 

 these students methods of purely historical interest and discus- 

 sions of more or less metaphysical nature are wasted. A mathe- 

 matician, of course, cannot encourage the teacliing of certain 

 inadequate methods of comparatively recent origin unhappilv 

 termed " practical mathematics." for, if he is to be in a position 

 to use mathematical methods with certitude and facility, the 

 technician must be as rigorous as the theorist; but one should 

 aim at develo])ing the mathematical training of the technical 

 students upon as broad a l)asis as possible, and ensuring that his 

 methods are of wide applicability, while specialized processes of 

 juirel}- historical interest or of limited power should be relegated 

 to a stibordinate position. 



These considerations strike one rather forcibly in connection 

 with the study of Conic Sections. The importance of Conies is, 

 as a whole, somewhat overrated, and much time that is spent 

 studying ingenious corollaries to their fundamental properties 

 could be more tisefully employed otherwise. For the technical 

 man has to deal more frecjuently with transcendental curves, or 

 with algebraical functions of degree higher than the second; and, 

 moreover, pnx^esses which gave Plato pleasure in the early days 

 of geometry, or which delighted Des Cartes when he inaugurated 

 analysis, are not necessarily those best adapted to ])resent needs. 



Conies, like other curves, should be studied by means of 

 their slope, and. having once defined a differential coefficient, 

 tangent and normal properties ought to be deduced by its means, 

 and not by repeatedly proceeding to a limit. The object of the 

 present paper is to show how the discrimination of the general 

 conic may be eftected from consideration of its slope by methods 

 such as could l)e usefully and effectively employed in the dis- 

 cussion of higher curves. 



General Conic — Co-ordinates of Centre. — Students first 

 obtain an idea of the shapes of conies from the usual locus 

 definitions, and, in particular, their attention is drawn to the 

 double and single symmetry respectively of the central and non- 

 central curves. The general quadratic function — 



