350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
365 
45. In the division into genera, Newton distinguishes the Hyperbolas into the 
redundant and defective, and the redundant hyperbolas into those for which the 
asymptotes form a triangle, and those for which the asymptotes meet in a point. The 
redundant hyperbolas with asymptotes forming a triangle are distinguished according as 
they have no diameter, a single diameter, or three diameters. The like distinction 
might have been, but is not, made as to the redundant hyperbolas with asymptotes 
meeting in a point; these are in fact included in a single genus; but the distinction 
presents itself in the species of that genus. As to the defective hyperbolas, Newton 
attends only to the real asymptote; and the only distinction is according as they have 
no diameter or a real diameter. The Parabolic Hyperbolas are in like manner divided 
according as they have no diameter or a diameter. The Central Hyperbolisms, according 
as the parallel asymptotes are real or imaginary, are the hyperbolisms of the hyperbola 
or of the ellipse. The hyperbolisms of the hyperbola form a single genus. Each of 
the Hyperbolisms might have been distinguished according as there is no diameter or 
a single diameter; this distinction appears in the species. The Trident Curve, the 
Divergent Parabolas, and the Cubical Parabola, form each a single genus. 
46. We have thus the following Table of the Newtonian genera: I show in it 
the species in each genus, retaining Newton’s numbers, and distinguishing by the 
numbers 10', 13', 22', 22" the four species added by Stirling, and by 56' and 56" the 
two species added by Murdoch or Cramer: I show also the division of genus 4, 
according to the number of diameters; and I also show the five species of curves 
having a centre. 
Table of the Newtonian Genera. 
1. Redundant Hyperbolas with asymptotes forming a triangle, and without a 
diameter. 
Sp. 1, 2, 3, 4, 5, 6, 7, 8, 9. 
2. Redundant Hyperbolas with asymptotes forming a triangle, and with a single 
diameter. 
Sp. 10, 10', 11, 12, 13, 13', 14, 15, 16, 17, 18, 19, 20, 21. 
3. Redundant Hyperbolas with asymptotes forming a triangle, and with three 
diameters. 
Sp. 22, 22', 22", 23. 
4. Redundant Hyperbola with asymptotes meeting in a point. 
Without a diameter, Sp. 24, 25, 26, 27. With one diameter, Sp. 28, 29, 30, 31. 
With three diameters, Sp. 32. Sp. 27 has a centre. 
5. Defective Hyperbolas without a diameter. 
Sp. 33, 34, 35, 36, 37, 38. Sp. 38 has a centre.