Four Books on Conics (Commandino 1566)

equal. In the same way, all those which are equally distant from d b or d h will be demonstrated to be equal to each other. Therefore, since point e is taken on the diameter of the circle a b c g, which is not the center of the circle, e b will be the maximum and e h the minimum; and that closer to e b is always smaller than that which is farther away. Therefore, e h is smaller than e f. But e d is common and at right angles. Therefore, the base d h is smaller than the base d f. Again, since e f is smaller than e a, with e d being common and at right angles, the base d f will be smaller than the base d a. By the same reasoning, the base d a will be shown to be smaller than d b. Since, therefore, d h is smaller than d f, and d f than d a, and d a than d b, d h will be the minimum and d b the maximum; and that closer to d h is always smaller than that which is farther away.

7. of the third

A geometric diagram shows a cone with a circular base labeled a, b, c, g. A vertical axis extends from vertex d to the base. Various lines (e, f, h, k) are drawn within the base and connecting to the vertex. A decorative floral ornament is placed to the left of the cone.

I call the straight line... the diameter of every curved line existing in one plane etc.

He said "in one plane" because of the helices of a cylinder and sphere, for these are not in one plane. What he says is of this kind. Let there be a curved line a b c; and in it parallel lines a c, d e, f g, h k; and from point b, let a straight line b l be drawn which bisects these parallel lines. The line b l, he says, I call the diameter of the line a b c, and point b the vertex. Each of the lines a c, d e, f g, h k is said to be applied ordinatim ordered/parallel to the line b l. But if b l bisects the parallel lines at right angles, it is called the axis.

Similarly also of two curved lines, etc.

Two side-by-side diagrams of ellipses are presented. Each ellipse contains parallel chords (labeled with letters like f, g, h, k, d, e) intersected by a central diameter line (b-l). The left ellipse is oriented vertically, the right one slightly tilted.

For if we understand the lines a b, and within them parallel lines c d, e f, g h, k l, m n, x o; and the diameter a b produced in both directions, which bisects the parallel lines, I call a b the transverse diameter; the points a and b are the vertices of the lines; and c d, e f, g h, k l, m n, x o are said to be applied ordinatim to the diameter a b. But if it bisects at right angles, it will be called the transverse axis. If, however, a straight line such as p r drawn through the lines c x, e m, g k, h l, f n, d o bisects those parallel to a b, it is called the straight diameter. Each of the lines c x, e m, g k, h l, f n, d o is applied ordinatim to the diameter p r. If it bisects at right angles, it will be called the straight axis. But if the straight lines a b and p r bisect the parallel lines, they are called conjugate diameters. And if they bisect at right angles, they will be called conjugate axes.

Two rectangular frames containing hyperbolic curves are shown. The left frame shows two opposing curves with intersecting diameters and chords. The right frame shows a similar construction with different labeling. A floral ornament separates the two frames.