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. 217
5.
PERSPECTIVE AND CONTINUITY.
In Lemma XXII. (p. 213) Newton gives a construction made to illustrate his
algebraical transformation of an equation of any degree into another of the same degree.
After the proof that tangents remain tangents, he remarks that his demonstrations might
have been put together "more magis geometrico," but he aims at brevity. With this
Lemma should be read his Enumeratio Linearum Tertii Ordinis, where again he has
something to say about curves in general.
At the end of the preface to his Opticks Newton writes, And I have joined with
it another small Tract concerning the Curvilinear Figures of the Second Kind, which
was also written many Years ago, and made known to some Friends, who have solicited
the making it publick. He is referring to the Enumeratio above mentioned, in which
curves of the nth order are called curves of the (n - )th genus or kind, the straight
line in this way of speaking not being counted among curves. In this tract he gives
the theory of Perspective in space under the name Genesis Curvarum per Umbras, rays
from a luminous point being supposed to cast shadows of geometrical figures on to an
infinite plane. Thus, he says, the "Parabolie quinq; divergentes" generate by their
shadows all other cubic curves, and so from " Curvae qusedam simpliciores" of any
genus can be produced all the other curves of that genus.
Such genesis of curves by shadows may have been suggested to Newton by some
of Kepler's problemata obseruatoria (pp. 201, 203), in which he lets the sun shine through
a small aperture into a darkened room, and observes the diurnal course of its projection
on the floor. This varies with the latitude of the place, according to which the apparent
path of the sun itself in any day cuts or touches or does not meet the plane of the
horizon.
Thus Perspective as a modern method may be said to have originated with Kepler.
The idea of it was not altogether unknown to the ancients, but they were scarcely in
a position to put it to effective use, for this could not be done without a more or
less complete doctrine of Continuity, including especially the quasi-concurrence of parallels
at infinity. See AMGC, p. iv., and the writer's note on Perspective in vol. x. of the
Messenger of Mathenmatics (1881).
Newton's Lemma xxII. may have arisen from his genesis of curves by shadows.
Having seen how to connect varieties of the same order of curve graphically, he would
naturally seek to connect such curves algebraically; and this could obviously be done
by his transformation of coordinates from X, Y to x, y, with Xx and Yx/y constant.
Page 200. 21 quantumuis absurdis locutionibus] Poncelet used "ce qu'il appelle le
principe de continuity," which is Kepler's principle of Analogy under a new name. This
principle Kepler formulated in terms suitable to its later applications. Including normal
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