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.

210 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
The distances of any point on a conic from the tangents at fixed points A, B,
C, D being denoted by a, b, c, d respectively, its distances from AB, BC, CD, DA vary
as mean proportionals to ab, bc, cd, da.
Hence obviously the four-line theorem, AB. CD =k. BC. DA. Apollonius, who claims
to have solved the Locus ad tres et quatuor lineas completely, may very well have deduced
the four-line theorem from the three-line theorem in this way.
The Lemmas and Propositions quoted below by number are Newton's, whose proofs
and diagrams in lib. I. sect. v. of the Principia should be referred to.
LEMMA XVII. Case 1. A C, BD being given parallel chords of a conic, through any
point P of the curve draw the chord PK parallel to AC and crossing AB, CD in Q, R;
and a parallel to AB meeting AC, BD in S, T. Then PQ. QK/AQ. QB is a constant ratio.
But, the intercepts PR, QK being equal, the rectangle PQ. PR is equal to PQ. QK,
and therefore varies as AQ. QB or PS. PT.
Thus Newton's proof for this case is the same as that of Apollonius for the threeline theorem, which it includes, since the parallels AC, BD may be supposed to coalesce.
In Case 2, with the help of Case 1, the theorem is shewn to hold when AC, BD
are not parallel. In this general case Newton does not use the point K, which might
have been found by drawing the parallel to RT through B. This construction leads to
the proof of his Lemma xvII. in Prop. A, Cor. 10. The proof in question is given by
Messrs J. J. Milne and R. F. Davis in their Geometrical Conics, followed by a corollary
in which Lemma xx. is deduced from Lemma xvII., as by Newton.
LEMMA XVIII. Conversely, the locus of a point P such that PQ. PR/PS.PT is
constant is a conic section.
Corol. The trilinear theorem is deduced as a limiting case.
Scholium. The term conic section includes the line-pair and the circle. For a
trapezium may be substituted a re-entrant quadrilateral; and one or two of the points
A, B, C, D may be at infinity.
LEMMA XIX. Any line being drawn through A, the point P in which it meets
the locus again is determined.
Corol. 1. The tangent at a given point is drawn.
Corol. 2. It is then shewn how to find a pair of conjugate diameters, and the
different species of conics belonging to the locus are discriminated.
At the end it is said, with tacit allusion to the algebraic proof of the quadrilinear theorem by Descartes, "Atque ita problematis veterum de quatuor lineis ab Euclide
incoepti & ab Appollonio continuati non calculus, sed compositio geometrica, qualem
veteres quserebant, in hoc corollario exhibetur."