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

by geometricians, who nevertheless were very numerous and most distinguished before Eudoxus, yet remained ignored and understood by no one. It will be possible for all who are capable, however, to examine these discoveries of mine. Certainly, these should have been published while Conon was alive; for I believe he, more than others, could have understood these things and provided a fitting judgment regarding them. But, thinking that I would be doing something worthwhile if I shared these with students of mathematics, I have sent you the proofs I have written down, which those skilled in mathematics may examine. Farewell.

First, the postulates are proposed, along with those things I have assumed for the proofs of my discoveries.

DEFINITIONS.

1. There are certain finite curved lines in a plane which are either entirely on the same side of the straight lines joining their endpoints, or have nothing positioned on the other side.

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

This page contains editorial notes regarding the transmission of the text through various manuscripts and later commentators such as Rivaltus and Simson.