Introduction.

“Archimedes to Dositheus greeting.

In this book I have set forth and send you the proofs of the remaining theorems not included in what I sent you before, and also of some others discovered later which, though I had often tried to investigate them previously, I had failed to arrive at because I found their discovery attended with some difficulty. And this is why even the propositions themselves were not published with the rest. But afterwards, when I had studied them with greater care, I discovered what I had failed in before.

Now the remainder of the earlier theorems were propositions concerning the right-angled conoid [paraboloid of revolution]; but the discoveries which I have now added relate to an obtuseangled conoid [hyperboloid of revolution] and to spheroidal figures, some of which I call oblong (παραμάκεα) and others flat (ἐπιπλατέα).

I. Concerning the right-angled conoid it was laid down that, if a section of a right-angled cone [a parabola] be made to revolve about the diameter [axis] which remains fixed and return to the position from which it started, the figure comprehended by the section of the right-angled cone is called a right-angled conoid, and the diameter which has remained fixed is called its axis, while its vertex is the point in which the axis meets (ἅπτεται) the surface of the conoid.