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. 211
3.
CURVARUM DESCRIPTIO ORGANICA.
LEMMA XX. AB, AC are given chords and P a given point of a conic. Through
P draw parallels to AC, AB forming with them a parallelogram PQAS; and across PQ,
PS draw CRD, BDT to any sixth point D of the conic. Then will PR/PT be a constant ratio, and conversely.
Case 1. The constancy of PR/PT is deduced from the four-line theorem proved in
Lemma xvII. Interchanging P, D in this paragraph only, let A, B, C, D in the figure
of Prop. A be fixed and P variable. Through D draw a parallel to AC meeting CP
in r, and a parallel to AB meeting BP in t. Then, CD being given, Dr varies as
PR/RC, and therefore as PR/PS; and, BD being given, Dt varies as PT/TB, and
therefore as PT/PQ. Therefore Dr/Dt is a constant ratio.
In like manner, with P fixed and D variable as in Lemma xx., PR/PT (p. 206) is
a constant ratio.
Hence the lne RT is given in direction. See Prop. xxII. and Prop. xxIII., where
Pt/Pr is made equal to PT/PR by drawing tr parallel to TR, "actâ recta tr ipsi TR
parallelâ" Hence, K being the position of D found by drawing RT through C (p. 206),
it follows that RT is parallel to CK. Thus Prop. A is in fact Lemma xx.
LEMMA XXI. Take a triangle BPC, and let angles equal to its angles at B and
G turn about those points as poles, one pair of the ines or bars containing the angles
p
intersecting at M on a fixed line or director which cuts BC in N. Then the other
pair will cross at a point D lying on a conic through B and C.
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