Four Books on Conics (Commandino 1566)

EUTOCIUS.

Approaching the definitions, Apollonius teaches the generation of the conical surface, not a definition which declares what the thing is; although it will be possible for those who wish to do so to infer the definition from the generation itself. We, however, shall bring light to the things said by Apollonius by means of figures.

A

If from some point to the circumference of a circle: etc.

Let there be a circle a b, whose center is c; and some point d elevated above it; and let the line d b be joined and produced to infinity in both directions to the points e and f. If, therefore, the straight line d b is moved around the circumference of the circle a b until the point b is restored to the place from which it began to move, it will describe a certain surface, which indeed consists of two surfaces touching each other at point d. I call this a conical surface; which also increases to infinity, when the straight line d b, describing it, is produced to infinity. He calls the point d the vertex of the surface, and the line d c the axis. He calls a cone the figure contained by the circle a b and that surface which the line d b alone describes; the vertex of the cone is the point d; the axis is d c; and the base is the circle a b. But if d c is perpendicular to the circle, he calls it a right cone; if otherwise, a scalene cone.

A geometric diagram shows a double cone where two cones meet at a single vertex labeled 'd'. The upper cone is smaller, and the lower cone features a circular base labeled 'a c b'. Lines 'e' and 'f' extend through the vertex. Floral woodcuts frame the diagram.

A scalene cone will be described when a line is erected from the center of the circle which is not perpendicular to the plane of the circle; and from a point of the line which is elevated, a straight line is drawn to the circumference of the circle; and while the point remains, it is turned around it. For the figure contained therein will be a scalene cone. It is therefore evident that the line carried around in the revolution sometimes becomes greater, sometimes smaller, and sometimes equal, in relation to different points of the circle. This, however, we shall demonstrate in this way.

A second geometric diagram illustrates a scalene cone where the axis 'd c' is not perpendicular to the base 'a c b'. The vertex is 'd', and lines 'e' and 'f' pass through it. Decorative floral elements appear to the right.

If straight lines are drawn from the vertex of a scalene cone to the base, one will be the minimum and one the maximum, but two only on either side of the minimum and maximum will be equal to each other. And that which is closer to the minimum will always be smaller than that which is farther away.

Let there be a scalene cone, whose base is the circle a b c and vertex is point d. And since the line drawn from the vertex of a scalene cone perpendicular to the subject plane either falls on the circumference of the circle a b c, or outside, or inside. Let it first fall upon the circumference itself, as it appears in the first figure, which let be d e; and having taken the center of the circle k, let the line e k be drawn from e to k and produced to b. Moreover, let b d be joined; and on either side of point e, let two equal circumferences f e and e g be taken; and likewise on either side of b, let two other equal ones, a b and b c, be taken; and let f e, e g, d f, d g, a e, e c, a b, b c, d a, and d c be joined. Since, therefore, the straight line e f is equal to e g (for they subtend equal circumferences), and d e is common and at right angles, the base d f will be equal to the base d g. Again, since the circumference a b is equal to the circumference b c, and b e is the diameter of the circle, the remaining e f a will be equal to the remaining e g c. Wherefore the straight line a e is also equal to e c. But d e is common to both and at right angles. Therefore, the base a d is equal to the base d c. Similarly, those that are equally distant from d e or d b will be demonstrated to be equal to each other. Again, since the triangle is e d f and the angle d e f is a right angle, the line d f will be greater than d e. And since the straight line a e is greater than f e, because the circumference e f a is also greater than

29. of the third
4. of the first
18. of the first