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. 203
The horizontal line is a fixed ordinate, c is the vertex and d any point of the locus. His
construction assumes a case of the theorem that the sum or difference of the distances of
a point on the parabola from the focus and a fixed perpendicular to the axis is constant.
In conclusion he refers to later passages for applications of his theory of the conic
sections. See cap. v. De modo visionis, and cap. XI. prob. 22-23 (p. 375 sq.).
THE CONVERGENCE OF PARALLELS.
Vitellio, as we have seen, had proved that rays of the sun impinging equidistantly
from (i.e. parallel to) the axis upon a concave reflector of the form of a paraboloid of
revolution would all be reflected to a certain point on the axis, whereat consequently
"ignem est possibile accendi." Hence in different languages the name "burning point"
for what Kepler called Focus, in a parabola or other conic.
It would appear that the idea of the meeting of parallels at infinity came from
the observed fact that solar rays received upon a reflector may practically be regarded
as parallel. Moreover it was obvious that the distance, estimated on an infinitely
remote transversal, between "equidistant" lines would subtend a vanishing angle at an
assumed point of observation. Kepler does not say that his doctrine of parallels is
altogether new and strange, when he writes at the end of page 93, "adeo ut...", so
that lines from the point h (or i) are parallel,-as if that would be allowed to follow
from its being infinitely distant. But it was perhaps a new and original suggestion
that h and i at infinity were the same point.
Kepler states expressly that he gave the name FocI to certain points related to the
conic sections which had previously "no name." With their new name he associated his
new views about the points themselves, and bis doctrines of Continuity (under the name
Analogy) and Parallelism, which would soon have become known, and would after a time
have been taken up by competent mathematicians.
An abstract of the passage now quoted at length from Kepler's Paralipomena ad
Vitellionem was given by the writer in The Ancient and Modern Geometry of Conics*,
published early in 1881, and previously in a note read in 1880 to the Cambridge
Philosophical Society (Proceedings, vol. IV. 14-17, 1883), both of which have been referred
to by Professor Gino Loria in his writings on the history of geometry.
HENRY BRIGGS.
Frisch (II. 405 sq.) quotes a letter of Henry Briggs to Kepler dated, Merton
College, Oxford, "10 Cal. Martiis 1625," which suggests improvements in the Paralipomena ad Vitellionem. In this letter Briggs gives the following construction. Draw a
line CBADC, and suppose an ellipse, a parabola and a hyperbola to have B for focus
and A for their nearer vertex. Let CC be the other foci of the ellipse and the
hyperbola. Make AD equal to AB, and with centres CC and radius in each case equal
to CD describe circles. Then any point of the ellipse is equidistant from B and one
* Th1e Ancient and Modern Geometry of Conics is hereinafter referred to as AMGC.
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