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

3. Similarly, there are certain finite surfaces, not themselves positioned in a plane, but having their endpoints positioned in a plane, which are either entirely on the same side of that plane in which they have their endpoints positioned, or certainly have nothing positioned on the other side.

4. I call such surfaces concave in the same direction, in which, when two points are taken, the straight lines joining the points either all fall on the same side of the surface, or some fall on the same side, some on the surfaces themselves, but none on the other side.

5. I call a solid sector a portion of a sphere, when a cone cuts a sphere, having its vertex at the center of the sphere, the figure which is contained by the surface of the cone and that part of the surface of the sphere which falls within the cone.

6. I call a solid rhombus a double-cone figure, when two cones having the same base have their vertices positioned on either side of the plane in which the base is, such that their axes are situated in a direct line, a solid figure composed of both cones.

I postulate these things:

1. Of all lines having the same endpoints, the straight line is the shortest.

2. Of the other lines, if they are positioned in a plane and have the same endpoints...