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.

DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 201
igitur graphio C & manu altera E difcede equalibus interuallis
à linea AE. fic vt manus altera & fili caput femper in EF maneat,
filumque DG femper ipfi AE parallelon; via CD. quam Graphio 2o
fignaueris, erit Parabole.
Dixi hec de feétionibus conicis tanto libentiùs, quôd non
tantùm hic dimenfio refrationum id requirebat, fed etiam infra in Anatome oculi vfus earum apparebit. Tum etiam inter
problemata obferuatoria mentio earum erit facienda duobus 25
locis. Denique ad proeftantiffima optica machinamenta, ad penfilem in aëre ftatuendam imaginem, ad imagines proportionaliter augendas, ad ignes incendendos, ad infinite comburen- Machinamléta Optica
dum, confideratio earum plan eft neceffaria. Porte.
The headlines of the edition quoted are Ioannis Kepleri and Paralipom. in
Vitellionemn up to page 221, and afterwards Ioannis Kepleri and Astronomico Pars Optica.
PAGE 92.
Kepler begins by saying that rays frori the centre of a sphere do not become
parallel after reflexion from its inner surface, but converge to the centre. Some other
surface then had to be sought which would reflect all rays from some point into
parallels. Vitellio in lib. IX. 39-44, in part supplying what was lacking in Apollonius,
had shewn that the paraboloid of revolution was of the required form. But the subject
of the Conic Sections presented difficulties because it had not been much studied.
Kepler therefore-pardon a geometer-proposed to discourse somewhat "mechanically,
analogically and popularly" about them.
Vitellio or Vitello (Witelo) had proved that at any point of a parabola the tangent
makes equal angles with a parallel to the axis and the line from the point to a
certain fixed point on the axis. Rays of the sun impinging equidistantly from the
axis upon the concavity of a reflecting paraboloid of revolution would therefore all be
reflected through a fixed point on the axis, and fire might so be kindled thereat.
Of cones right or scalene there are five species of sections (line 24), the right
line or line-pair, the circle, parabola, hyperbola and ellipse. From the line-pair we
pass through an infinity of hyperbolas to the parabola, and thence through an infinity
of ellipses to the circle. Of all hyperbolas the most obtuse is the line-pair, the most
acute the parabola. Of all ellipses the most acute is the parabola, the most obtuse
the circle.
PAGE 93.
The parabola is of the nature partly of the infinite sections and partly of the finite,
to which it is intermediate. As it is produced it does not spread out its arms in
direction like the hyperbola, but contracts them and brings them nearer to parallelism,
"semper plus quidein complectens at semper minus appetens" (line 5). The hyperbola
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