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.

214 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
By this method, it is remarked, convergent lines can be transformed into parallels;
and when a problem has been solved in the simplified figure, this can be retransformed
into the original figure. In the solution of "solid problems" one of two conics can be
changed into a' circle. In the solution of "plane problems" a line and a conic can be
made a line and a circle.
PROP. XXV. PROB. XVII. To describe a conic through two given points and touching three given lines.
Transform the given tangents and the line through the given points into the sides
of a parallelogram.
Let these sides be hci, idk, kcl, Ibah, where a, b correspond to the given points
and c, d, e are the points of contact.
Take m, n mean proportionals to ha, hb and la, lb.
Then hc/m = ic/id = ke/kd = le/n,
and each of these ratios is equal to the given ratio of hi + kl, the sum of the
antecedents, to z + n + ki, the sum of the consequents. Thus the points of contact are
determined.
It may be remarked that this case is the reciprocal of Prob. xvI. Given two
points B, D and two tangents GH, GK, the pole of BD must lie on one of two fixed
lines. A third tangent being given, we can thus find four positions of the pole of BD.
Having then five tangents and the points of contact of two of them, we can trace
the four conics in various ways.
PROP. XXVI. PROB. XVIII. To describe a conic through a given point and touching
four given lines.
Newton's solution is in effect as follows. Let P be the given point, and let two
diagonals of the quadrilateral formed by the four tangents meet in O. Draw OPo to
the third diagonal, and take Q a harmonic conjugate to P with respect to O, e.
Then Q is on the conic, and the case is reduced to that of Prop. xxv.
He transforms the given tangents into the sides of a tangent parallelogram; finds
the centre O; and finds Q the other end of the diameter PO. In the retransformed
figure Q would therefore be found by the previous construction.
PROP. XXVII. PROB. XIX. To describe the conic touching five given lines.
This is led up to by three Lernmmas, one of which, with a transformation as in
Prop. xxv. or Prop. xxvi., would have sufficed for the solution of the problem.
LEMMA XXIV. Corol. 2. Using the figure of Lemma xxv., let AMF, BQI be
parallel tangents to a conic; A, B their points of contact; FQ, IM any third and
fourth tangents.