354
[350
350.
ON THE CLASSIFICATION OF CUBIC CURVES.
[From the Transactions of the Cambridge Philosophical Society, vol. xi. Part i. (1866),
pp. 81—128. Read April 18, 1864.]
The notion of a curve of a given order may be considered as arising from Descartes’
invention of his method of coordinates; and one of the earliest applications of the
method was made by Sir Isaac Newton in the Enumeratio linearum tertii Ordinis (1706),
a work worthy of its author, and which opened a new field of geometrical science. The
classification is according to the nature of the infinite branches; there are fourteen
genera containing together seventy-two species, but four species were added by Stirling
in his Linece tertii Ordinis Neiutoniance; sive Illustratio &c. (1717), and two more by
Murdoch or Cramer( 1 ), making in all seventy-eight species. A new classification was
made by Pliicker in his System der Analytischen Geometrie, 1835 ; this is likewise
according to the nature of the infinite branches, but after his six head divisions, and
some subordinate divisions thereof, Pliicker establishes the divisions called Groups, which
have nothing analogous to them in the Newtonian theory; there are sixty-one groups,
and the total number of species is 219.
The present Memoir contains an exposition of the foregoing classifications, and of
the principles on which they are founded, in so far as relates to the superior divisions
of the two classifications: and in particular I develope more completely than was done
by Pliicker the theory of the division into groups. I do not however consider otherwise
than very slightly the ultimate division into species.
The above-mentioned work of Newton contains, under the heading “Genesis Curva-
rum per Umbras,” the remarkable theorem that the curves of the third order may all
of them be considered as the shadows of the five Divergent Parabolas; I reserve for
a separate Memoir the whole series of considerations to which this theorem gives rise.
1 The two additional species are, I believe, first mentioned in Murdoch’s Genesis Curvarum per Umbras
«(1746), but one of them is there ascribed to Cramer.
