The Conics of Apollonius

theorems, which will be useful both for the compositions of solid loci and for their determinations; many of which are beautiful and new. Considering these things, we noticed that the method of composing the locus to three and four lines was not provided by Euclid, but only a certain fragment of it, and even this not very successfully. For it was not possible for that composition to be correctly completed without those things which were discovered by us. The fourth book sets forth in how many ways the sections of cones can meet one another and the circumferences of circles, and many other things for a fuller doctrine, none of which has been handed down to memory by those who were before us; [it treats] how the section of a cone, the circumference of a circle, and opposite sections may meet opposite sections at how many points. The remaining 4 books pertain to a more abundant knowledge. For indeed, the fifth deals in great part with Minima and Maxima. The sixth [deals] with equal and similar conic sections. The seventh contains theorems which have the power of determination. The eighth [contains] determined conic problems. But truly, all these having been published, it is permitted to everyone who happens upon them in reading to judge according to the opinion of their own mind. Farewell.

BOOK I.

Prop. I.

Fig. 5.

Straight lines (AG) which are drawn from the vertex (A) of a conical surface to points (G) that are on the surface will be on the surface itself.

a 1. def. 1. of this [book].

For since points A and G are on the conical surface, the straight line describing it will pass through points A and G. Thus it is clear that the straight line AG exists on the conical surface.

Coroll. 1. From this it is evident that if a straight line is drawn from the vertex to any point of those which are within the surface, it will fall within; and if to any of those which are outside, it will fall outside the surface.