Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

204 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
circle, and any point of the hyperbola from B and the other circle. When C is at
infinity on either side of D the circle about it becomes rectilinear. Hence any point P
of the parabola is equidistant from B and the perpendicular DF to DA. This is expressed by Briggs as follows:
A
c
"Si A sit vertex sectionis, et B, C foci, et AB, AD aequales, et centro C, radio
CD describatur peripheria: quodlibet punctum sectionis eandem servabit distantiam a
foco B et dicta peripheria. Eruntque...in Parabola (cui focus alter deest, vel distat
infinite, et idcirco recta DF vicem obtinet peripheriae) PB, FP aequales."
The writer comprehended and accepted Kepler's way of looking at parallels as lines to
or from a point at infinity in one direction or its opposite.
DESARGUES.
The famous geometer Desargues worked on the lines of Kepler, and is now commonly
credited with the authorship of some of the ideas of his predecessor.
Poncelet in the first edition of his Traité des Propriétés Projectives des Figures
(1822) writes with reference to a letter of Descartes, "On voit aussi, dans cette lettre,
que Desargues avait coutume de considérer les systèmes de droites parallèles comme
concourant à l'infini, et qu'il leur appliquait le même raisonnement" (p. xxxix.). Chasles
on the Porisms of Euclid refers to this remark of Poncelet. In his Aperçu Historique
(p. 56, 1875) he writes that Kepler " introduisit, le premier, l'usage de l'infini dans la
Géométrie," but really with reference only "aux méthodes infinitésimales." The saying
that Kepler introduced the use of the infinite into geometry has been repeated by
other writers unacquainted with his doctrine of the infinitely great.
Dr Moritz Cantor in his Vorlesungen iber Geschichte der Mathematik writes under
the head of Girard Desargues (1593-1662), "Wir ntissen einige wesentliche Dinge
hervorheben und darunter zunachst die Anwendung des Unendlichen in der Geometrie...Auch Kepler hat 1615, Cavalieri 1635 in Druckwerken, deren Besprechung
uns obliegen wird, wenn wir von den Anfangen der Infinitesimalrechnung reden, den
gleichen Gedanken zu nie geahnten Folgerungen ausgebeutet, aber bei Desargues waren
es ganz andere Unendlichkeitsbetrachtungen als bei diesen Vorgangern" (II. 619, 1892). He
goes on to say that Desargues regarded parallels as meeting at infinity, and thus in
effect that Kepler did not so regard them. Cantor (p. 620 n.), referring to Poudra's