## Archimedes: Assumptions from Sphere and Cylinder

- Of all lines which have the same extremities the straight line is the
least.
- 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.
- Similarly, of surfaces which have the same extremities, if those extremities
are in a plane, the plane is the least in area.
- 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.
- 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.

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Last modified: 17 November 2003
Duncan J.
Melville

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