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.

DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 205
~uvres de Desargues I. 103, states confidently that Desargues could not have held that
"es gebe nur einen Unendlichkeitspunkt einer Gerade." "Auch in I. 105...darf man
jenen modernen Sinn nicht hineinlesen." But the oneness of opposite infinities followed
simply and logically from a first principle of Desargues, that every two straight lines,
including parallels, have or are to be regarded as having one common point and one
only. A writer of his insight must have come to this conclusion, even if the paradox
had not been held by Kepler, Briggs, and we know not how many others, before Desargues
wrote.
In Poudra's QEuvres de Desargues, I. 210, under the head Traité des Coniques, we
read, "Nombrils, point brulans, foyers.-C'est à dire que les deux points comme Q et
P sont les points nommés nombrils, brulans, ou foyers de la figure, au suiet desquel
il y a beaucoup à dire." Desargues must have learned directly or indirectly from the
work in which Kepler propounded his new theory of these points, first called by him
the Foci (foyers), including the modern doctrine of real points at infinity.
(B)
NEWTON.
In the fifth section of the first book of the Principia, entitled Inventio orbium ubi
umbilicus neuter datur, the determination of conic orbits from data not including a focus,
Newton proves the property of the Locus ad quatuor lineas of which no geometrical
demonstration was extant, shews how to describe conics by rotating angles and otherwise, and solves the six cases of the problem to determine a conic of which n points
and 5 - n tangents are given. Two more problems, each with its Lemma prefixed,
complete the section, which ends with the words, " Hactenus de orbibus inveniendis.
Superest ut motus corporum in orbibus inventis determinemus."
The following pages contain a summary of the greater part of the section, with
suggestions for the simplification of some of its contents and a few additional constructions and propositions. The Lemmas and Propositions of the Principia are quoted
by their Roman numerals.
1.
THE CONIC THROUGII FIVE POINTS.
PROP. A. Given five points of a conic to find a sixth.
Let A, B, C, D, P be given points of a conic. Through P draw PTSO parallel
to BA across BD, AC, CD. It is required to find the point K in which it meets the
conic again.