16 Mr Taylor, On the history of geometrical continuity. [Oct. 25, 



centre; in the ellipse or hyperbola two, equidistant from the 

 centre : in the parabola one within the section, " alter vel extra vel 

 intra sectionem in axe fingendus est infinito intervallo a priore 

 remotus, adeo ut educta HG vel IG* ex Mo ccecofoco in quodcun- 

 que punctum sectionis G sit axi DK parallelos." 



In the circle the focus recedes as far as possible from the 

 nearest part of the circumference, in the ellipse somewhat less, in 

 the parabola much less ; whilst in the line-pair the " focus," as he 

 still calls it to complete the analogy, falls upon the line itself. 

 Thus in the two extreme cases of the circle and the line-pair 

 the two foci coincide. He then goes on to compare the latus 

 rectum and its intercept on the axis, or as he calls them the chorda 

 and sagitta, in the several sections, concluding with the case of the 

 line-pair, in which the chord coincides with its arc, "abusive sic 

 dicto, cum recta linea sit." But our geometrical expressions must be 

 subject to analogy, " plurimum namque amo analogias, fidelissimos 

 meos magistros, omnium naturae arcanorum conscios." And especial 

 regard is to be had to these analogies in geometry, since they com- 

 prise, in however paradoxical terms, an infinity of cases lying 

 between opposite extremes, "totamque rei alicujus essentiam lucu- 

 lenter ponunt ob oculos." 



(1) Hereupon be it remarked, that the principle of Analogy 

 on which he insists so fervently is the archetype of the principle 

 of Continuity. The one term expresses the inner resemblance of 

 contrasted figures A and B, which are connected by innumerable 

 intermediate forms; whilst the other expresses the possibility of 

 passing through those intermediate forms from A to B, without 

 any change per saltum. Geometry was not indebted to Algebra 

 for the suggestion of the law of continuity. 



(2) Having traced the transition from the line-pair to the 

 circle through the three standard forms of conies, he completes the 

 theory of the points henceforth named Foci by the discovery of 

 the " caecus focus " of the parabola, which is to be taken at infinity 

 on the axis either ivithout or within the curve. The parabola may 

 therefore be regarded indifferently as a hyperbola, having (rela- 

 tively to either of its branches) one external and one internal 

 focus, or as an ellipse, having both foci within the curve. 



(3) The further focus of the parabola being taken at infinity 

 on the axis in either direction, the two opposite extremities of 

 every infinite straight line are thus regarded as coincident or con- 

 secutive points — a conception which may be shewn to conduct 

 logically to the idea of imaginary points. 



* The figure indicates that the line from the further focus may be considered to 

 lie either within or without the parabola. 



