| If a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis. | |
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Let the cylinder AD be cut by the plane GH parallel to the opposite planes AB and CD. Let the plane GH meet the axis at the point K.
I say that the cylinder BG is to the cylinder GD as the axis EK is to the axis KF. | |
| Produce the axis EF in both directions to the points L and M. Set out any number whatever of axes EN and NL equal to the axis EK, and any number whatever FO and OM equal to FK. Construct the cylinder PW on the axis LM with the circles PQ and VW as bases. | |
| Carry the planes through the points N and O parallel to AB and CD and to the bases of the cylinder PW, and let them produce the circles RS and TU about the centers N, O. | |
| Then, since the axes LN, NE, and EK equal one another, therefore the cylinders QR, RB, and BG are to one another as their bases. | XII.11 |
| But the bases are equal, therefore the cylinders QR, RB, and BG also equal one another. | |
| Since then the axes LN, NE, and EK equal one another, and the cylinders QR, RB, and BG also equal one another, and the multitude of the former equals the multitude of the latter, therefore, the multiple the axis KL is of the axis EK is the same multiple the cylinder QG is of the cylinder GB. | |
| For the same reason, the multiple the axis MK is of the axis KF is the same multiple the cylinder WG is of the cylinder GD. | |
| And, if the axis KL equals the axis KM, then the cylinder QG also equals the cylinder GW; if the axis is greater than the axis, then the cylinder is also greater than the cylinder; and if less, less. Thus, there being four magnitudes, the axes EK and KF and the cylinders BG and GD, there have been taken equimultiples of the axis EK and of the cylinder BG, namely the axis LK and the cylinder QG, and equimultiples of the axis KF and of the cylinder GD, namely the axis KM and the cylinder GW, and it has been proved that, if the axis KL is in excess of the axis KM, the cylinder QG is also in excess of the cylinder GW; if equal, equal; and if less, less. Therefore the axis EK is to the axis KF as the cylinder BG is to the cylinder GD. | V.Def.5 |
| Therefore, if a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis. | |
| Q.E.D. | |
Book XII Introduction - Proposition XII.12 - Proposition XII.14.