Four Books on Conics (Commandino 1566)

BOOK I OF THE CONICS.

Whence "parabola"

through the axis, if it is parallel to the side ac of the triangle. But if the cone is obtuse-angled, as in the second figure, meaning the angle bac is obtuse and the angle aef is a right angle, the angles bac and aef together will be greater than two right angles. The line def will not meet the side ac on the side where f is, but rather where a and e are, by producing ca to d. Therefore, the cutting plane will make a section on the surface of the cone called a hyperbola, from the Greek ἀπὸ τοῦ ὑπερβάλλειν from "to exceed". This is because the angles bac and aef exceed two right angles, or because the line def exceeds the vertex of the cone, and it meets the line ca extended. But if the cone is acute-angled, meaning the angle bac is acute, the angles bac and aef will be less than two right angles. The lines ef and ac, when produced, will eventually meet at some point, for we can extend the cone further. There will thus be a section on the surface called an ellipsis ellipse, from διὰ τὸ ἐλλείπειν θυὸ ὀρθαῖς τὰς προειρημένας γωνίας because the aforementioned angles fall short of two right angles, or because the ellipse is a certain diminished circle.

The ancients, in this manner, placed the cutting plane through def at right angles to the side ab of the triangle passing through the axis of the cone, and by using different cones, they created a specific section for each. But Apollonius, by positioning the cone, makes the right and scalene cones create different sections through the intersection of the same plane. Let the cutting plane, as in the same figures, be kel. The common section of this plane and the base of the cone is the line kl. The common section again of the same plane and the triangle abc is ef, which is also called the diameter of the section. Therefore, in all sections, he places the line kl at right angles to the base of the triangle abc. Verily, if ef is equidistant to ac, a parabola is formed by the section kel on the surface of the cone. If, however, it meets the side ac outside the vertex of the cone, as at d, the section kel becomes a hyperbola. If it meets inside, the section becomes an ellipsis ellipse, which he also calls a thyreos shield.

Generally, therefore, the diameter of a parabola is equidistant to one side of the triangle; the diameter of a hyperbola and an ellipsis meets it. The diameter of the hyperbola meets it toward the side of the cone's vertex, while that of the ellipsis toward the side of the base. Furthermore, one must know that the parabola and the hyperbola are of the number of those things that increase to infinity; the ellipsis does not. It turns entirely into itself, just like a circle.

Since there are many editions, as Apollonius himself writes in his letter, I judged it best to collect the more manifest points from the many that occurred. I have kept his own words so that an easier path to these matters might be open to those reading, but I have explained the different methods of demonstration separately in commentaries, as is fitting. Therefore, in the letter, he says that the first four books contain the elements of this discipline. The first of these encompasses the generations of the three conic sections, and those called opposite, and also their principal accidents, that is, whatever occurs to them in the first generation. It also has other consequences. The second book treats those things which pertain to diameters, to the axes of the sections, and to those lines which do not meet the section, which are called by the Greeks ἀσύμπτωτοι asymptotes. Then he discusses other things that bring both general and necessary utility to determinations. A determination, however, is twofold, as is manifest: one is after the exposition, it signifies...

A large diagram depicts the geometry of the cone. Three separate views show a cone being intersected by a plane to form the three conic sections: the parabola, the hyperbola, and the ellipse. Each figure identifies the base of the cone, the axis, and the specific lines of intersection (labeled a through l). Decorative floral engravings adorn the right margin.

Whence "hyperbola"

Whence "ellipsis"

How a parabola is made. Hyperbola. Ellipsis.

The parabola and hyperbola increase to infinity.

Asymptotes. Twofold determination.