1880.] Mr Taylor, On the history of geometrical continuity. 15 



in this case infinite. For the earliest trace of a focus of the 

 parabola we refer to proposition 238 of the seventh book of the 

 Collectio of Pappus (p. 1013 ed. Hultsch), where the property of 

 the focus, directrix, and determining ratio is given ; but still the 

 difficulty which presented itself to Apollonius is not in any direct 

 way surmounted. 



The foci long continued to be spoken of as the points arising 

 from the " application," puncta ex applications facta, with reference 

 to the above-mentioned construction of Apollonius. At a later 

 period they were called umbilici, foci, and occasionally poles. 

 Some time back I was engaged in an attempt to trace the origin 

 of the name " focus " of a conic, not finding any correct informa- 

 tion about its earliest use in the historical works with which I 

 was acquainted. At length I lighted upon a work in which it 

 was written, that the points in question, although sufficiently de- 

 fined by their properties, had nomen nullum, and the name foci 

 was accordingly proposed, with reference to their optical or re- 

 flexional property in relation to the conic. The writer was Kepler. 

 I had thus come to the end of my investigation, and not only so, 

 but had found much more than I was then in search of; for in the 

 same passage in which he gives to the points described by the 

 periphrasis puncta ex applicatione facta their new name of Foci, 

 he clearly and decisively lays down the law of Continuity, the 

 vital principle of the modern geometry. 



The work of Kepler entitled Ad Vitellionem* paralipomena 

 quibus Astronomia} pars Optica traditur (Francofurti, 160-1) con- 

 tains a short discussion De Coni Sectionibus (cap. iv. § 4, pp. 92 — 6) 

 from the point of view of analogy or continuity. The section of a 

 cone by a plane "aut est Recta, aut Circulus, aut Parabole aut 

 Hyperbole aut Ellipsis." Of all hyperbolas "obtusissima est linea 

 recta, acutissima parabole ;" and of all ellipses " acutissima est 

 parabole, obtusissima circulus." The parabola is thus intermediate 

 in its nature to the hyperbola and "recta" (or line pair) on the 

 one hand, and the closed curves the ellipse and the circle on the 

 other; "infinita enim & ipsa est, sed finitionem ex altera parte 

 affectat." He then goes on to speak of certain points related to 

 the sections, " qure definitionem certam habent, nomen nullum, nisi 

 pro nomine definitionem aut proprietatem aliquam usurpes." The 

 lines from these points to any point on the curve make equal 

 angles with the tangent thereat : " Nos lucis causa & oculis in 

 Mechanicam intentis ea puncta Focos appellabimus.' He would 

 have called them centres if that term had not been already appro- 

 priated. In the circle there is one focus, coincident with the 



* Optica" Thesaurus. Alhazem Arabis libri vn. Item Vitellionis libri x. 

 (Basil. 1572). 



