Four Books on Conics (Commandino 1566)

The line a c is therefore the least, and b c is the greatest of all those that are drawn from point c to the circumference of the circle a b.

marginal note: First definition of Apollonius.
A geometric diagram shows a cone with vertex c and base circle a-e-b-f. A line c-d extends outside the base, and lines c-a and c-b connect the vertex to the diameter of the circle.

If a straight line joined from any point to the circumference of a circle, which is not in the same plane in which the point is, be produced in both directions, etc.

Apollonius appropriately added, "be produced in both directions," when he teaches the generation of each cone. For if the cone were isosceles, it would be produced in vain, because the straight line that is rotated perpetually touches the circumference of the circle, since the point remaining from it is always at an equal distance. But since the cone can also be scalene, in which, as has now been demonstrated, the greatest and least side is found, he necessarily added that, so that the line which is the least is understood to be increased until it becomes equal to the greatest, and therefore always touches the circumference of the circle.

LEMMA II.

LET there be a line a b c, and a c given in position; all the perpendiculars drawn from the line a b c to a c are such that the square of each of them is equal to the rectangle contained by the parts of the base which are cut by it. I say that a b c is the circumference of a circle, and the line a c is its diameter.

A semi-circle diagram with diameter a-c and center k. Perpendicular lines d-f, b-g, and e-h are drawn from the arc to the diameter. Points are labeled a, f, d, k, b, g, e, h, c.

For let the perpendiculars d f, b g, and e h be drawn from the points d, b, and e. Therefore, the square of d f is equal to the rectangle a f c, and the square of b g is equal to the rectangle a g c, and the square of e h is equal to the rectangle a h c. Let a c be bisected at k, and let d k, k b, and k e be joined. Therefore, marginal note: 3rd of the second [book]. since the rectangle a f c together with the square of f k is equal to the square of a k, and the square of d f is equal to a f c, marginal note: 47th of the first [book]. the square of d f together with f k—which is the square of d k—will be equal to the square of a k. Wherefore, the line a k is equal to k d. Similarly, we shall show that each of the lines b k and e k is equal to a k or k c. marginal note: 15th definition of the first [book]. Therefore, a b c is the circumference of a circle around the center k, that is, around the diameter a c.

LEMMA III.

Two geometric figures showing triangles and parallel lines, illustrating ratios of segments.

LET there be three equidistant lines a b, c d, e f; and let two straight lines a g f c and b g e d be drawn into them. I say that as the rectangle which is made from a b and e f is to the square of c d, so is the rectangle a g f to the square of g c.

marginal note: Lemma in the 22nd of the tenth [book], 4th of the sixth [book].
For since as the line a b is to f e—that is, as the rectangle of a b and f e is to the square of f e—so is the line a g to g f—that is, the rectangle a g f to the square of g f—it will be that the rectangle of a b and f e...