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.

218 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
and limiting forms of a figure under one definition, we are led to paradoxical ways of
speaking, "sine vsu, tantum ad analogian complendam" (p. 199. 5-6); as when we
think of a hyperbola as a sort of ellipse, and postulate imaginary elements in the one
analogous to what we see in the other.
Newton in some of his constructions virtually uses imaginary points (pp. 213, 216),
whether or not, like Boscovich, he thought definitely of geometrical figures as having
imaginary elements. To say that equations in x and y, which represent coordinates, may
have imaginary roots (Opticks, p. 151) is to say in effect that there are what may be
called imaginary points. Newton doubtless used equations for his own satisfaction in
some places where he does not fully explain his geometry. An equation representing
the locus described in Lemma XXI. (p. 211), is given in Prob. LIII. of the Arithmetica
Universalis (1707). By the method of Fluxions he discovered things which he gave to
the world proved "more magis geometrico." Thus he writes:
"At length in the winter between the years 1676 and 1677 I found the Proposition
that by a centrifugal force reciprocally as the square of the distance a Planet mu st
revolve in an Ellipsis about the center of the force placed in the lower umbilicus of
the Ellipsis and with a radius drawn to that center describe areas proportional to the
times......And this is the first instance upon record of any Proposition in the higher
Geometry found out by the method in dispute."
Two imaginary points the FocoIDS (AfGC, p. 281), or " Circular Points at Infinity,"
play a great part in modern geometry. Their existence may be proved in geometrical
form as follows.
Draw any circle in a given plane, and let sb and b' be the two points in which
it cuts the line Infinity. These will be the same for all circles in the plane.
For take points A, B on the circle subtending any angle a at the circumference;
and take any other two points a, b in the plane.
Then the angle AfB is equal to a, because b is on the circle; and the lines
fA, Ca are parallel, and likewise OB, Ob, because b is at infinity.
Therefore
Z acb= z ABB=a,
or any two lines through b may be regarded as intersecting at any angle.
Hence every circle in the plane passes through b, and similarly through 4'.
Conversely, a conic through O and b' is a circle.
The orthoptie locus of a curve of the nth class is of the degree n(n - 1-), since its
intersection with the line Infinity consists of b and J' taken ~n (n- 1) times.
From the equation
x2 + y2 - (x + iy) ( - iy) = 0
in rectangular coordinates it seems at first that ~c and 4' are indeterminate, because
x (or y) may have any direction. But the angles tan-l + i are indeterminate.