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.

206 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
By a property of conics and by similar triangles, if AB, CD meet in I,
OK. OP/OC. OD = IA. IB/I. ID= OS. OT/OC. OD.
H
Therefore OK. OP = OS. OT,
which determines K when the other points are given.
Inflect PR to CD parallel to A C. Then the point K is found by drawing CK
pacrallel to RT, and PRT, SCK are similar triangles.
Cor. 1. To determine the conic through five given points A, B, C, D, P. Having found
K, we find H where PR meets the conic again in like manner, nainely by drawing BH
parallel to TR. Having two pairs of parallel chords, we can draw their diameters and
find the centre. This with either pair of the parallel chords determines the conic, if
the pair be unequal. If they be equal, we can use the parallel chord through D in
lieu of one of them. Giveln five points A, B, C, D, E, two pairs of parallel chords
can also be determined as in Prob. LV. of the Arithmetica Universalis. Let AC, BE cross
in H. Inflect DI to A C parallel to BE, and EK to DI parallel to A C. Then, in
order that ID, EK may meet the conic again in F, G, we must have wTith Newton's
notation for rectangles and proportions,
AHC. BHE:: AIC.FID:: EKG. FKD.
Cor. 2. To determine the conic touching lines IB, ID at B, D and passing through
P. Supposing AC, BD in the figure to coalesce, find K as in the general case, and