CONICORUM LIBER I.

5. Similarly, for two curved lines placed in one plane, I call a transverse diameter a diagonal bisector a straight line which, intersecting those two lines, bisects all straight lines drawn in each line parallel to a certain straight line. I call the vertices of the lines the endpoints of the diameter positioned on the line, and I call the straight line that lies between the two lines and bisects all straight lines drawn parallel to a certain straight line, and intercepted between the lines, the diameter. Each of the parallels is said to be drawn ordinally parallel to the diameter to the diameter.

6. I call conjugate diameters paired bisectors of a curved line and of two curved lines those straight lines, of which each is a diameter and bisects the lines parallel to the other.

7. I call the axis principal diameter of a curved line and of two curved lines a straight line which is a diameter of the line or lines and cuts the parallels at right angles.

8. I call conjugate axes paired principal diameters of a curved line and of two curved lines those straight lines which are conjugate diameters and cut each other's parallels at right angles.

I.

Straight lines drawn from the vertex of a conic surface to points on the surface lie within the surface.

A woodcut depicts three geometric diagrams of double cones meeting at a common vertex A. The first diagram shows a cone with base circle Γ E and a line segment A Γ B extending from the vertex through the base. The second diagram shows a similar cone with points B, Z, Γ, E labeled. The third diagram shows a cone with points B, Γ, A, Z, E labeled, illustrating the extension of lines through the vertex to form a double cone.

Let there be a conic surface whose vertex is point A, and let some point B be taken on the conic surface, and let some straight line AΓB be drawn. I say that the straight line AΓB is in the surface.