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. 213
Case 2. In the general case, HI being the given tangent and BCDP the given
points, draw HAI, ICPG, GBDH, and make the ratio compounded of
HA2/HD. HB; GB. GD/GP. GC; IC. IP/IA2;
a ratio of equality. Thus HA/IA is determined and the point of contact A is found
within or without HI.
This is Newton's solution briefly stated, and it is identical with the modern solution
by what is called Carnot's theorem. When A is found the two conics can be described
by the methods used in Case 1.
PROP. XXIV. PROB. XVI. To describe a conic through three given points and touching two given lines.
Given two points and two tangents, Newton proves that the chord of contact must
pass through one of two fixed points. This may be shewn as follows.
Let B, D be the given points and GH, GK the given tangents.
Take H and K in line with BD, and suppose BD and the chord of contact to
cross at R.
Then by the trilinear theorem, all the distances being measured along BD, we have
BR2/DR2 =- BH. BK/DH. DK.
Divide BD within and without at R in the ratio thus determined, and we have
two points through one of which the chord of contact must pass.
A third given point C taken with B or D determines two points S through one
of which the chord of contact must pass. Thus there are four possible positions of RS,
giving four solutions.
When RS is found the conic can be described as in the first case of Prop. xxIIm.
Imaginary Points. In the second case of Prop. xxIIm. and in Prop. xxiv. Newton
uses an auxiliary line which is supposed to cut the conic in points X and Y.
At- the end of Prop. xxiv. he remarks that the constructions given will be the
same whether the line XY cuts the trajectory or not. For the sake of brevity he
gives no special proofs for the case in which, as we should say, the points X and Y
are imaginary.
LEMMA XXII. Figuras in alias ejusdem generis figuras rutare.
Here Newton gives a method of homographic transformation, in which the loci of
points G, g correspond so that the coordinates X, Y of G and x, y of g are connected
by relations of the form,
OA. AB y OA.Ly
X X0