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.

216 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
Corol. 2. The anharmonic tangent-property of conics.
A sixth tangent eq is drawn, and it is shewn that
KQ: Qq =Me:Ee.
Thus the four tangents LK, EQ, eq, LM determine equal cross ratios on the
tangents IK and IM.
Corol. 3. A tangent quadrilateral being given, the locus of the centre of the conic
is the straight line which bisects its diagonals.
PROP. XXVII. Hence, five tangents being given, two tetrads of them give two
lines through the centre. The parallel tangents can then be drawn, their points of
contact found by Lemma xxiv., and the conic described by Prop. xxII.
Scholium. The preceding problems include cases in which the centre or an asymptote
is given. For an asymptote is a tangent at infinity, and the centre with one point
or tangent determines another point or tangent.
To find the axes and foci of a conic described by Lemma xxi. Set the arms
BP, CP (which by their intersection described the conic) parallel and let them so
rotate. The intersection X of the other arms of the two angles will then describe a
circle through B, C. Draw its diameter KL crossing the director at right angles in
H. When X is at K, then CP is parallel to the major or minor axis according as
KH is less or greater than LH; and when X is at L, then CP is parallel to the
other axis. Hence when the centre is given the axes are given, and the foci can be
found.
Newton does not explain his construction for the directions of the axes, which has the
appearance of having been first made for the hyperbola, and then stated for the ellipse
also as having imaginary points at infinity. Le Seur and Jacquier, in their annotated
edition of the Principia, having explained the construction for the case of the hyperbola
by means of its asymptotes, or tangents "ad distantiam infinitam," merely remark in
conclusion that it applies also to the parabola into which the hyperbola is changed when
the intersections of the director with the circle coalesce, and to the ellipse into which
the parabola is turned when the director passes outside the circle*.
The squares of the axes are as KH to LH. Hence a trajectory of given species or
eccentricity can easily be described through four given points. Conversely a trapezium of
given species, "si casus quidam impossibiles excipiantur," can be inscribed in a given conic.
There are also other lemmas by the help of which trajectories of given species
can be described when points and tangents are given. For example, the middle point
of a chord drawn through a fixed point to a conic traces a similar and similarly
situated conic. "Sed propero ad magis utilia."
* Their words are, "Superior autem constructio non Ellipsi in quam vertitur parabola, dum recta MN extra
solum hyperbole convenit, sed & parabol in quam hyper- circulum transit," the points M and mn being the interbola mutatur, dum puncta m, M coeunt; atque etiam sections of the director MN and the circle.