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

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have the same endpoints, such lines are unequal when both are concave in the same direction, and either one is entirely enclosed by the other and the straight line having the same endpoints as it, or it is partly enclosed and shares some common parts, and the enclosed one is the lesser.

3. Similarly, for surfaces having the same endpoints, if the endpoints lie in a plane, the flat surface is the lesser.

4. For other surfaces having the same endpoints, if the endpoints are in a plane, such surfaces are unequal when both are concave in the same direction, and either one surface is entirely enclosed by the other and the plane having the same endpoints as it, or it is partly enclosed and shares some common parts, and the enclosed one is the lesser.

5. Furthermore, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the lesser by such a magnitude that, when added to itself, it is possible for it to exceed any magnitude of those which are said to be in a ratio to one another. This is a foundational statement of the Archimedean property, ensuring that magnitudes can always be compared through multiples.

With these things assumed, if a polygon is inscribed in a circle, it is clear that the perimeter of the inscribed polygon is less than the circumference of the circle; for each of the sides of the polygon is less than the arc of the circle cut off by the same side.