Kitap 11
33
Porism. From this it is manifest that, if four straight lines be ltcontinuouslygt proportional, as the first is to the fourth, so will a parallelepipedal solid on the first be to the similar and similarly described parallelepipedal solid on the second, inasmuch as the first has to the fourth the ratio triplicate of that which it has to the second.
PROPOSITION 34.
34
In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal.
34
Let AB, CD be equal parallelepipedal solids; I say that in the parallelepipedal solids AB, CD the bases are reciprocally proportional to the heights, that is, as the base EH is to the base NQ, so is the height of the solid CD to the height of the solid AB.
34
First, let the sides which stand up, namely AG, EF, LB, HK, CM, NO, PD, QR, be at right angles to their bases; I say that, as the base EH is to the base NQ, so is CM to AG.
34
If now the base EH is equal to the base NQ, while the solid AB is also equal to the solid CD, CM will also be equal to AG.
34
For parallelepipedal solids of the same height are to one another as the bases; [XI. 32] and, as the base EH is to NQ, so will CM be to AG, and it is manifest that in the parallelepipedal solids AB, CD the bases are reciprocally proportional to the heights.
34
Next, let the base EH not be equal to the base NQ, but let EH be greater.
34
Now the solid AB is equal to the solid CD; therefore CM is also greater than AG.
34
Let then CT be made equal to AG, and let the parallelepipedal solid VC be completed on NQ as base and with CT as height.
34
Now, since the solid AB is equal to the solid CD, and CV is outside them, while equals have to the same the same ratio, [V. 7] therefore, as the solid AB is to the solid CV, so is the solid CD to the solid CV.
