Archimedes: Assumptions from Sphere and Cylinder

1. Of all lines which have the same extremities the straight line is the least.
2. Of other lines in a plane and having the same extremities, any two such are unequal whenever both are concave in the same direction and one of them is either wholly included between the other and the straight line which has the same extremities with it, or is partly included by, and is partly in common with, the other; and that line which is included is the lesser of the two.
3. Similarly, of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least in area.
4. Of other surfaces with the same extremities, the extremities being in a plane, any two such are unequal whenever both are concave in the same direction and one surface is either wholly included between the other and the plane which has the same extremities with it, or is partly included by, and partly common with, the other; and that surface which is included is the lesser of the two areas.
5. Further, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with it and with one another.