Archimedis Opera Omnia cum Commentariis Eutocii, Vol. I (Heiberg 1880) (1880) - Page 24

to be ignored by all and understood by no one. It will be possible for those who are able to examine these matters. Therefore, these things ought to have been published while Conon was still alive. For we assume that he, above all others, would have been able to understand these things and offer an appropriate judgment regarding them. But, judging that it is right to share them with those who are devoted to the mathematical sciences, I am sending you the proofs I have written down, so that it may be possible for those who are engaged in mathematics to examine them. Farewell.

The axioms and the assumptions for their proofs are written down first.

AXIOMS.

1. There are certain finite curved lines in a plane, which either lie entirely on the same side of the straight lines joining their endpoints, or have no part on the other side.

2. I call such a line concave in the same direction, in which, if any two points are taken, the straight lines between the points either fall entirely on the same side of the line, or some fall on the same side and some upon the line itself, but none on the other side.