VOL. XXIir.] PHILOSOPHICAL TRANSACTIONS. 450 



fore called conic sections, have, from the earliest ages of geometry, engaged the 

 attention of mathematicians, on account of their extensive utility in the solution 

 of many problems, which w^ere incapable of being constructed by any possible 

 combination of right lines and circles, the magnitudes used in plane geometry. 

 The properties of these curves are become far more interesting within the last 2 

 centuries, since they have been found to be similar to those described by the 

 motions of the celestial bodies in the solar system. 



Two different methods have been taken by the writers who have treated of 

 their properties; the one, and the more ancient, is to deduce them from the 

 properties of the cone itself; the other is to consider the curves, as generated by 

 the constant motion of 1 or more straight lines moving in a given plane, by 

 certain laws. There are various methods of generating these curve lines in 

 piano ; one method will give some properties very easily ; but others, with much 

 trouble : while, by another mode of description, some properties may be readily 

 derived, which, by the former, were not so easily come at: so that it appears 

 there may be a manner of describing the curves similar to the conic sections, by 

 the motion of lines on a plane, which in general shall produce the most essential 

 properties, with the greatest facility. 



That excellent mathematician, the late Win. Jones, Esq. f.r.s. had drawn 

 up some papers on the description of these curves, or lines of the second kind, 

 very different from what he gave in his Synopsis Palmariorum Matheseos, pub- 

 lished in the year 1706; or from that of any other writer on this subject. A 

 copy of these papers he let Mr. R, take about the year 1740, who, though they 

 were in an unfinished state, thought them of too much consequence to be lost; 

 and therefore was desirous of preserving them in the Phil. Trans, in the manner 

 he at first transcribed ; though he is aware they might have been put into a form 

 more pleasing to the generality of readers : Mr. R. indeed annexed larger dia- 

 grams than what accompanied the author's copy, in order to render the lines 

 more distinct, as all the relations are to be represented in a single figure, of each 

 kind. Mr. R. then proceeds to state that Mr. Jones, having laid down a very- 

 simple method of describing these curves, seems to have been desirous of arriving 

 at their properties in as expeditious a way as he could contrive; and therefore he 

 has used the algebraic method, in general, of reducing his equations ; and on 

 some occasions has used the method of fluxions, to deduce some properties 

 chiefly relating to the tangents; and by a judicious use of these, he has very 

 much abridged the steps which otherwise he must have taken, to have deduced 

 the very great variety of relations he has obtained : these he intended to have 

 arranged in tables, whence an equation expressing the relation between any 3 

 or more lines of the conic sections, might be taken out as readily as a logarithm 

 out of their tables ; this he has only partly executed ; but it may easily be con- 



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