Four Books on Conics (Commandino 1566) (1566) - Page 24

6. of the 6th.
4. of the 6th.
cor. 20. of the 6th.

proportional: the triangle a b m will be similar to the triangle m n b: and angle b a m is equal to angle n m b. But of the triangles a m f and m b f, the angle f a m is equal to the angle f m b: and the angle at f is common to both. Therefore, the remaining angle is equal to the remaining, and triangle will be similar to triangle. Wherefore, as a f is to f m, so is f m to f b. As, therefore, the first is to the third f b, so is the square to the square of f m.

THE FIRST DEFINITIONS.

1 If a straight line is joined from some point to the circumference of a circle, which is not in the same plane in which the point is, and is produced in both directions: and while the point remains, it is turned around the circumference of the circle until it returns to the place from which it began to move: I call the surface described by the straight line, which consists of two surfaces fitted to each other at the vertex, each of which is increased to infinity, namely, the straight line which describes it being produced to infinity, a conical surface. 2 The remaining point is its vertex. 3 The axis is the straight line which is drawn through the point and the center of the circle. 4 I call a cone the figure contained by the circle and the conical surface, which is intercepted between the vertex and the circumference of the circle. 5 The vertex of the cone is the point which is also the vertex of the conical surface. 6 The axis is the straight line which is led from the vertex to the center of the circle. 7 The base is the circle itself. 8 I call those right cones which have axes at right angles to their bases. 9 But scalene cones are those which do not have axes at right angles to their bases. 10 I call the diameter of any curved line existing in one plane the straight line which, when drawn from the curved line, bisects all lines which are drawn within it equidistant to a certain line. 11 The vertex of the line is the endpoint of the straight line which is on the line itself. 12 Each of the equidistant lines is said to be applied orderly (ordinate) to the diameter. 13 Similarly, concerning two curved lines existing in one plane, I call the transverse diameter the straight line which bisects all lines drawn in both of them equidistant to a certain line. 14 The vertices of the lines are the endpoints of the diameter which are on the lines themselves. 15 I call the right diameter that which is placed between two lines and bisects all lines drawn equidistant to a certain straight line and intercepted between them. 16 Each of the equidistant lines is said to be applied orderly to the diameter. 17 I call the conjugate diameters of a curved line and of two curved lines the straight lines of which each is a diameter and bisects lines equidistant to the other. 18 The axis of a curved line and of two curved lines is the straight line which, being the diameter of the curved line or of the two curved lines, cuts the equidistant lines at right angles. 19 The conjugate axes of a curved line and of two curved lines are the straight lines which, being conjugate diameters, cut the lines equidistant to them at right angles.

B
C