De Sphæra & Cylindro On the Sphere and Cylinder LIB. I.

...cone, and an equal height: and that every cone is one-third of a cylinder having the same base as the cone, and an equal height: since indeed these things existed by nature in these figures, although before Eudoxus there had been many geometers not to be despised, it happened nevertheless that they were ignored by all, and perceived by no one. It will be permitted, however, to those who are able, to look into these matters. It would have been useful, and I would have wished that these things had been published while Conon was still alive: for we judge that he would have been especially suited to weigh these matters, and to offer an appropriate opinion concerning them; but judging it a good deed to share them also with other students and experts of mathematics, we send them to you, transcribing the demonstrations, concerning which it will be permitted to those who are versed in mathematics to look into. Farewell.

Instead of "of all" original: "τοῦ πάντων" I read "to be ignored by all" original: "ἀπὸ πάντων ἀγνοεῖσθαι".

Instead of "these" original: "τῆνων" I read "this" original: "τῆνον".

Instead of "to them" original: "αὐτῖς" I read "to others" original: "ἄλλοις".

---

First, both the Axioms and the Assumptions for their demonstrations are transcribed.

Definitions & Hypotheses.

I. There are certain curved lines in a plane that are bounded, whose ends, when joined by straight lines, either lie entirely on the same side of those lines, or have no part on the other side.

Scholium.

By the name of curved (or bent, καμπύλης γραμμῆς curved line) lines, is designated not only a line that is continuously curved throughout, but also one that is bent in any way; whether mixed from straight and curved, or composed entirely of straight lines. Just as the perimeter of any rectilinear figure, or any part of it that includes an angle, is a curved line. Since Eutocius rightly says:

"One must know, therefore, that he calls curved lines not simply those which are circular or conic, or those which have continuity without a break, but he names every line in a plane other than a straight line 'curved'; for he calls a line in a plane that is joined in any way, even if it consists of straight lines, &c."

Of such curves, he assumes some to lie, either entirely on the same side of the straight lines connecting their ends, or at least to have no part situated on the opposite side. Let the circular circumference ABC be an example, whose ends are connected by the straight line AC; it is clear that the entire line ABC is raised above the straight line AC on the sides.

Fig. 1.