CONICS I.

First Definitions.

If a straight line, drawn from a point to the circumference of a circle—which is not in the same plane as the point—is extended in both directions, and while the point remains fixed, the line is moved around the circumference of the circle until it returns to the same position from which it began to move, I call the surface generated by the straight line, which consists of two surfaces lying vertically opposite to each other, each of which increases to infinity as the generating straight line is extended to infinity, a conic surface a double cone surface; its vertex is the fixed point, and its axis is the straight line drawn through the point and the center of the circle.

A cone a solid of revolution is the figure contained by the circle and the conic surface between the vertex and the circumference of the circle; the vertex of the cone is the point which is also the vertex of the surface, the axis is the straight line drawn from the vertex to the center of the circle, and the base is the circle.

Of cones, I call those right orthogonal whose axes have right angles to the bases, and scalene oblique those whose axes do not have right angles to the bases.

For any curved line, which is in one plane, I call a diameter a bisecting line a straight line which, drawn from the curved line, bisects all straight lines drawn within the line that are parallel to a certain straight line. The vertex of the line is the endpoint of the straight line which is at the curve, and each of the parallels is said to be drawn ordinally parallel to the diameter onto the diameter.