Archimedis Opera cum Apollonii Conicorum Libri IIII

Let the arc A G be bisected at B, and let the tangent K G L be drawn; and let K M, L N be erected parallel to the axis of the cylinder, and let M N be connected: and it is clear that parallelogram A E F C + B E F D symbol: less than parallelogram A K M C + K L N M + B L N D (because, a in 13 hujus. ^a as before, A E + B E symbol: less than A K + K L + B L). Let the excess be X, not less, first, than the segments A K G, B L G, C M H, D N H. And because the surface composed of parallelograms A K M C, K L N M, B L N D & trapezoids A B L K, C D N M symbol: less than cylindrical surface A G B D H C + segments A G B, C H D (the parallelogram A B D C being the common limit), by subtracting the common segments A G B, C H D, the remaining parallelograms ^a A K M C, K L N M, B L N D + segments A K G, B L G, C M H, D H N symbol: less than cylindrical surface A G B D H C. Therefore, all the more are parallelograms A K M C, K L N M, B L N D + X (^c that is, parallelograms A E F C, B D F G) symbol: less than cylindrical surface A G B D H C. c hyp. Q.E.D.

If X should be less than the said segments, let the arcs A G, B G be bisected, and let tangents be drawn, ^d until the segments become smaller than X; and the demonstration will proceed in a similar manner as before.

These things having been demonstrated, it is clear from the aforesaid,
1. That if a pyramid is inscribed in an Isosceles cone, the surface of the pyramid, excluding the base, is less than the surface of the cone, the base also being removed.
For each of the triangles containing the pyramid is less than each of the conical surfaces which they intercept and subtend. Therefore those together are less than these together, that is, the surface of the pyramid is less than the surface of the cone.
2. And that if a pyramid is circumscribed about an Isosceles cone, the surface of the pyramid, except for the base, is greater than the surface of the cone, with the base also excluded.
3. Likewise, it appears from what has been shown that if a prism is inscribed in a right cylinder, the surface of the prism composed of parallelograms is less than the surface of the cylinder, without the base.
For each parallelogram of the prism is less than the cylindrical surface which it cuts off.
4. And that if a prism is circumscribed about a right cylinder, the surface of the prism, consisting of parallelograms, is greater than the surface of the cylinder, with the base set aside.

Thus far he has prepared useful lemmas for the following demonstrations; he now proceeds to the principal Theorems.