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.

202 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
being produced tends more and more to the form of its "Asymptote" (line 12). Parabolas are all similar and differ only in "quantity" (line 19).
He then goes on to speak of certain remarkable points related to the sections
which had NO NAME (line 21). The lines from them to any point of the section make
equal angles with the tangent. He will call them FocI (line 27). He would have
called them centres if that term had not been already appropriated. The circle has
one focus, at the centre: the ellipse has two, equidistant from the centre, and more
remote as the curve is more acute. In the parabola one is within the curve, while
the other may be regarded as either without or within it, so that a line hg or ig
drawn from that "cecus focus" to any point of the curve is parallel to the axis
(line 35).
PAGE 94.
In the hyperbola the focus external to either branch is the nearer to its internal
focus as the hyperbola is more obtuse. In the straight line (or line-pair), to speak in
an unusual way merely to complete the analogy, the foci fall upon the line itself.
Thus in the extreme limiting cases of the circle and the line-pair, the foci come together
at a point, which in the one is as far as possible from the nearest point of the circumference and in the other is on the line itself. In the intermediate case of the parabola
the foci are infinitely distant from one another (line 12): in the ellipse and the
hyperbola on either side of it they are a finite distance apart.
PAGE 95.
The line mn through the focus, i.e. the latus rectum, is called the chord, and br
or dk or es the sagitta (line 6). In the next line BF is a misprint for BP. The
lengths of the sagitta and the chord are compared ill the five sections, and it is said
that in the line-pair the one vanishes and the other becomes infinite (line 15), whereas,
if e be the eccentricity, they are in the finite ratio 1/2 (1 + e), and vanish together.
Kepler commends the principle of analogy in glowing terms, saying that he dearly loves
analogies, his most trusty teachers and conversant with all the secrets of nature
(line 19). Analogy leads us to comprise in one definition extreme limiting forms, from
the one of which we pass to the other by continuous variation through an infinity of
intermediate cases.
In the next paragraph Kepler shews how to describe an arc of a hyperbola by means
of threads fixed at the foci, the difference of the focal distances of a point on the curve
being constant. An ellipse is described more easily (line 33), with one thread.
PAGE 96.
In line 1 "AC duplicata" is inaccurate, the length of the thread being ac+cb. He is
shewing how to describe an ellipse by means of a thread fixed at the foci a and b, the
point c being a vertex. Having given his construction for this curve without the
troublesome compasses (line 6), he goes on to the parabola. To his grief he was long
unable to describe this analogously. At length he thought of the construction in the
text, in which adg represents a string of constant length ec +ca fixed at the focus a.