**GEOMETRICAL CONTINUITY.** In a report of the Institute prefixed to Jean Victor Poncelet’s *Traité des propriétés projectives*
*des figures* (Paris, 1822), it is said that he employed “ce
qu’il appelle le principe de continuité.” The law or principle
thus named by him had, he tells us, been tacitly assumed as
axiomatic by “les plus savans géomètres.” It had in fact been
enunciated as “lex continuationis,” and “la loi de la continuité,”
by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously
under another name by Johann Kepler in cap. iv. 4 of his *Ad*
*Vitellionem paralipomena quibus astronomiae pars optica traditur*
(Francofurti, 1604). Of sections of the cone, he says, there are
five species from the “recta linea” or line-pair to the circle.
From the line-pair we pass through an infinity of hyperbolas to
the parabola, and thence through an infinity of ellipses to the
circle. Related to the sections are certain remarkable points
which have no name. Kepler calls them foci. The circle has
one focus at the centre, an ellipse or hyperbola two foci equidistant
from the centre. The parabola has one focus within it,
and another, the “caecus focus,” which may be imagined to be
*at infinity* on the axis *within or without the curve*. The line from it
to any point of the section is parallel to the axis. To carry out
the analogy we must speak paradoxically, and say that the line-pair
likewise has foci, which in this case coalesce as in the circle
and fall upon the lines themselves; for our geometrical terms
should be subject to analogy. Kepler dearly loves analogies, his
most trusty teachers, acquainted with all the secrets of nature,
“*omnium naturae arcanorum conscios*.” And they are to be
especially regarded in geometry as, by the use of “however
absurd expressions,” classing extreme limiting forms with an
infinity of intermediate cases, and placing the whole essence of a
thing clearly before the eyes.

Here, then, we find formulated by Kepler the doctrine of the
concurrence of parallels at a single point at infinity and the
principle of continuity (under the name analogy) in relation to the
infinitely great. Such conceptions so strikingly propounded in
a famous work could not escape the notice of contemporary
mathematicians. Henry Briggs, in a letter to Kepler from
Merton College, Oxford, dated “10 Cal. Martiis 1625,” suggests
improvements in the *Ad Vitellionem paralipomena*, and gives
the following construction: Draw a line CBADC, and let an
ellipse, a parabola, and a hyperbola have B and A for focus and
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 the focus B and one circle, and any point of
the hyperbola from the focus B and the other circle. Any point
P of the parabola, in which the second focus is missing or infinitely
distant, is equidistant from the focus B and the line
through D which we call the directrix, this taking the place of
either circle when its centre C is at infinity, and every line CP
being then parallel to the axis. Thus Briggs, and we know not
how many “savans géomètres” who have left no record, had
already taken up the new doctrine in geometry in its author’s
lifetime. Six years after Kepler’s death in 1630 Girard Desargues,
“the Monge of his age,” brought out the first of his remarkable
works founded on the same principles, a short tract entitled
*Méthode universelle de mettre en perspective les objets donnés*
*réellement ou en devis* (Paris, 1636); but “Le privilége étoit de
1630.” (Poudra, *Œuvres de Des.*, i. 55). Kepler as a modern
geometer is best known by his *New Stereometry of Wine Casks*
(Lincii, 1615), in which he replaces the circuitous Archimedean
method of exhaustion by a direct “royal road” of infinitesimals,
treating a vanishing arc as a straight line and regarding a curve
as made up of a succession of short chords. Some 2000 years
previously one Antipho, probably the well-known opponent of
Socrates, has regarded a circle in like manner as the limiting
form of a many-sided inscribed rectilinear figure. Antipho’s
notion was rejected by the men of his day as unsound, and when
reproduced by Kepler it was again stoutly opposed as incapable
of any sort of geometrical demonstration—not altogether without
reason, for it rested on an assumed law of continuity rather
than on palpable proof.

To complete the theory of continuity, the one thing needful
was the idea of imaginary points implied in the algebraical
geometry of René Descartes, in which equations between variables
representing co-ordinates were found often to have imaginary
roots. Newton, in his two sections on “Inventio orbium”
(*Principia* i. 4, 5), shows in his brief way that he is familiar with
the principles of modern geometry. In two propositions he uses
an auxiliary line which is supposed to cut the conic in X and Y,
but, as he remarks at the end of the second (prop. 24), it may not
cut it at all. For the sake of brevity he passes on at once with the
observation that the required constructions are evident from the
case in which the line cuts the trajectory. In the scholium
appended to prop. 27, after saying that an asymptote is a tangent
at infinity, he gives an unexplained general construction for the
axes of a conic, which seems to imply that it has asymptotes.
In all such cases, having equations to his loci in the background,
he may have thought of elements of the figure as passing into the
imaginary state in such manner as not to vitiate conclusions
arrived at on the hypothesis of their reality.

Roger Joseph Boscovich, a careful student of Newton’s works,
has a full and thorough discussion of geometrical continuity in
the third and last volume of his *Elementa universae matheseos*
(ed. prim. Venet, 1757), which contains *Sectionum conicarum*
*elementa nova quadam methodo concinnata et dissertationem de*
*transformatione locorum geometricorum, ubi de continuitatis*
*lege, et de quibusdam infiniti mysteriis*. His first principle is
that all varieties of a defined locus have the same properties, so
that what is demonstrable of one should be demonstrable in like
manner of all, although some artifice may be required to bring
out the underlying analogy between them. The opposite
extremities of an infinite straight line, he says, are to be regarded
as joined, as if the line were a circle having its centre at the
infinity on either side of it. This leads up to the idea of a *veluti*
*plus quam infinita extensio*, a line-circle containing, as we say,
the line infinity. Change from the real to the imaginary state is
contingent upon the passage of some element of a figure through
zero or infinity and never takes place *per saltum*. Lines being
some positive and some negative, there must be negative rectangles
and negative squares, such as those of the exterior
diameters of a hyperbola. Boscovich’s first principle was that
of Kepler, by whose *quantumvis absurdis locutionibus* the boldest
applications of it are covered, as when we say with Poncelet
that all concentric circles in a plane touch one another in two
imaginary fixed points at infinity. In G. K. Ch. von Staudt’s
*Geometrie der Lage and Beiträge zur G. der L.* (Nürnberg, 1847,
1856–1860) the geometry of position, including the extension of
the field of pure geometry to the infinite and the imaginary, is
presented as an independent science, “welche des Messens nicht
bedarf.” (See Geometry: *Projective*.)

Ocular illusions due to distance, such as Roger Bacon notices
in the *Opus majus* (i. 126, ii. 108, 497; Oxford, 1897), lead up to
or illustrate the mathematical uses of the infinite and its reciprocal
the infinitesimal. Specious objections can, of course, be
made to the anomalies of the law of continuity, but they are
inherent in the higher geometry, which has taught us so much
of the “secrets of nature.” Kepler’s excursus on the “analogy”
between the conic sections hereinbefore referred to is given at
length in an article on “The Geometry of Kepler and Newton”
in vol. xviii. of the *Transactions of the Cambridge Philosophical*
*Society* (1900). It had been generally overlooked, until attention
was called to it by the present writer in a note read in 1880 (*Proc.*
*C.P.S.* iv. 14-17), and shortly afterwards in *The Ancient and*
*Modern Geometry of Conics, with Historical Notes and Prolegomena*
(Cambridge 1881).
(C. T.*)