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.

212 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
For inflect PR to CD, making the angle CPR equal to the constant angle CNJM.
Then PCR, N/CMI are similar triangles, and
PR/NM = PC/NC.
Inflect PT to BD, making the angle BPT equal to the constant angle BNM. Then
PBT, NBM are similar triangles, and
PT/NM= PB/NB.
Therefore PR varies as PT, and by Lemma xx., PR and PT being on fixed lines,
the locus of (CR, BT) is a conic through B and C, and conversely.
The lengths PR, PT in the figure, which differs somewhat from Newton's, are as
the perpendiculars from NY to PB, PC.
Given four points B, C, D, P, an infinity of conics can thus be drawn through
them, for the given point D determines only one point M of the director. Given a
fifth point of the conic, the director is determined, and one conic only can be described.
To draw the tangent BT at B, make D coincide with B. See Prop. xxII., Corol. 1.
In other words, make the angle NC]M equal to the angle PCB, and then the angle
MBT equal to the angle PBC.
To find the directions of the axes. If the arms BM, CM be made constantly parallel,
the intersection D of the others will trace a circle through B and C. This will cut
the conic again at the two points found by making the parallel arms successively
coincident with BC and parallel to the director. Four points common to the circle and
the conic having been found, the axes must be parallel to the bisectors of the angles
between a pair of chords joining them. For Newton's construction see Prop. xxvii.
Scholium (p. 216).
PROP. B. If two angles AOB, AoB of given magnitudes turn about poles O, w,
and if the intersection A traces a curve of the nth order, the intersection B will in
general trace a curve of the 2nth order.
For a given position of the arm OB there are n positions of A and therefore n
of B. When OB is in the position Ow all the B's coincide with w, which is therefore
an n-fold point on the locus of B, as is also the point 0; and since any line through
O (or co) meets the locus of B in n other points, the locus is of the order 2?n.
4.
INVENTIO ORBIUM.
PROP. XXII. PROB. XIV. To describe the conic through five points. This is done by
Lemma xx., and again by Lemma xxI.
PROP. XXIII. PROB. XV. To describe a conic through four points and touching a
given line.
Case 1. When one of the points is the point of contact the construction is effected
as in Prop. xxII.