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

3. Similarly, there are certain finite surfaces, not themselves lying in a plane, but having their endpoints in a plane, such that they either lie entirely on the same side of the plane in which their endpoints lie, or have no part on the other side.

4. I call such surfaces concave in the same direction, in which, if any two points are taken, the straight lines between the points either all fall on the same side of the surface, or some fall on the same side, some upon the surface itself, and 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 contained by the surface of the cone and the surface of the sphere within the cone.

6. I call a solid rhombus a double-cone figure, when two cones having the same base have their vertices on opposite sides of the plane of the base, such that their axes lie in a straight line, the solid figure composed of both cones.

I assume these things:

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

2. Of the other lines, if they are in a plane