464 
CURVE. 
[785 
an order less than m; for instance, taking m = 2, if the hyperbola xy —1 = 0 be cut 
by the line y = /3, the resultant equation in x is fix — 1 = 0, and there is apparently 
only the intersection 
but the theorem is, in fact, true for every line 
whatever: a curve of the order m meets every line whatever in precisely m points. 
We have, in the case just referred to, to take account of a point at infinity on the 
line y = @) the two intersections are the point [x = ^, y = ^j, and the point at infinity 
on the line y = /3. 
It is moreover to be noticed that the points at infinity may be all or any of 
them imaginary, and that the points of intersection, whether finite or at infinity, real 
or imaginary, may coincide two or more of them together, and have to be counted 
accordingly ; to support the theorem in its universality, it is necessary to take account 
of these various circumstances. 
The foregoing notion of a point at infinity is a very important one in modern 
geometry; and we have also to consider the paradoxical statement that in plane 
geometry, or say as regards the plane, infinity is a right line. This admits of an easy 
illustration in solid geometry. If with a given centre of projection, by drawing from 
it lines to every point of a given line, we project the given line on a given plane, 
the projection is a line, i.e., this projection is the intersection of the given plane with 
the plane through the centre and the given line. Say the projection is always a 
line, then if the figure is such that the two planes are parallel, the projection is 
the intersection of the given plane by a parallel plane, or it is the system of points 
at infinity on the given plane, that is, these points at infinity are regarded as situate 
on a given line, the line infinity of the given plane*. 
Reverting to the purely plane theory, infinity is a line, related like any other 
right line to the curve, and thus intersecting it in m points, real or imaginary, distinct 
or coincident. 
Descartes in the Géométrie defined and considered the remarkable curves called 
after him ovals of Descartes, or simply Cartesians, which will be again referred to. 
The next important work, founded on the Géométrie, was Sir Isaac Newton’s Enumeratio 
linearum tertii ordinis (1706), establishing a classification of cubic curves founded chiefly 
on the nature of their infinite branches, which was in some details completed by 
Stirling, Murdoch, and Cramer; the work contains also the remarkable theorem (to be 
again referred to), that there are five kinds of cubic curves giving by their projections 
every cubic curve whatever. 
Various properties of curves in general, and of cubic curves, are established in 
Maclaurin’s memoir, “ De linearum geometricarum proprietatibus generalibus Tractatus ” 
(posthumous, say 1746, published in the 6th edition of his Algebra). We have in it 
a particular kind of correspondence of two points on a cubic curve, viz. two points 
correspond to each other when the tangents at the two points again meet the cubic 
in the same point. 
* More generally, in solid geometry infinity is a plane,—its intersection with any given plane being the 
right line which is the infinity of this given plane.