364 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
Neiutons Classification. Article Nos. 42 to 46. 
42. Newton establishes in the first instance the following four cases; viz. the 
equation of a cubic curve is one of the forms 
I. 
II. 
III. 
IY. 
xy 3 -F ey = a« 3 + bai 2 + cx + d, 
xy = act? + ba? + cx + d, 
y 2 = ax 2 + bar + cx + d, 
y = ax 3 + bx 2 + cx + d. 
It is not, I think, necessary to reproduce here the very interesting reasoning by 
means of which this most important step in the classification was effected. 
43. Starting from the four cases, Newton obtains his 14 genera, viz. Case I gives 
11 genera, and Cases II, III, IY give each a single genus. But these genera group 
themselves as follows, viz. 1, 2, 3, 4, 5, 6 are Hyperbolas ; 7 and 8, Parabolic Hyper 
bolas ; 9 and 10, Central Hyperbolisms; 11, Parabolic Hyperbolisms; 12, the Trident 
Curve; 13, the Divergent Parabolas; and 14, the Cubical Parabola. And the equations 
are as follows : 
the Hyperbolas xy 2 + ey = 
the Parabolic Hyperbolas xy 2 + ey = 
the Central Hyperbolisms xy 2 + ey = 
the Parabolic Hyperbolisms xy 2 + ey = 
the Divergent Parabolas y n - = 
xy 2 + ey = ax? + bx? + cx + d, 
xy 2 + ey = bx 2 + cx + d, 
cx + d, 
d, 
y 2 = ax? + bx 2 + cx + d, 
xy = ax 3 + bx? + cx 4- d, 
y = ax? + bx 2 + cx + d, 
the Trident Curve xy 
the Cubical Parabola y 
where it is to be understood that the highest expressed power on the right-hand side 
of each equation does not vanish. 
44. In these equations the axes x = 0, y = 0 are not for the most part lines 
precisely determined in relation to the curve, but it is easy to see as well analytically 
as geometrically how by a proper transformation of the equations they may be brought 
into forms such as those previously obtained, in which the several lines p = 0, &c., 
stand in a determinate relation to the curve. Thus, taking the equation of the 
Cubical Parabola, this may be written 
what is the same 
thing, y’ — ax' 3 . Or geometrically, we see that x = 0 is a line completely determined 
as to its direction, it is in fact a line through the cusp at infinity; but that y = 0 is 
an arbitrary line in regard to the curve; taking for y = 0 the tangent at the inflexion, 
and for x — 0 the line from the inflexion to the cusp at infinity, then the curve must 
pass through the point (x = 0, y = 0), and y= 0 must give a threefold value of x\ the 
equation thus is y = ax?. And so in other cases.