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IF Three Lines are continually propor∣tional, the Solid Parallelepipedon made of them is equal to an equiangular paralle∣lepipedon, which hath all it sides equal to the middle Line.
If the Lines A, B, C, be continually proportional, the parallelepipedon FE made of them, that is to say, which hath the side FI equal to the Line A, and HE equal to B, HI equal to C; is equal to the equiangular Parallelepipe∣don KL, which hath its sides LM, MN, KN,
equal to the Line B. Let there be drawn from the Points H and N, the Lines HP, NQ, perpendicular to the Planes of the Bases; they shall be equal, seeing that the Solid Angles E and K are supposed equal, (in such manner that if they did penetrate each other, they would not surpass each other) and that the Lines EH, KN, are supposed equal. There∣fore the heights HP, NQ, are equal.
Demonstration. There is the same Ratio of A to B, or of FI to LM, as of B to C, or of LM to HI; so then the parallelogram FH comprehended under FI, IH, is equal to the paralle∣logram LN, comprehended under LM, MN, equal to B (by the 15th. of the 6th.) the Bases are thence equal. Now the heights HP, NQ, are so likewise; therefore (by the 31st.) the Parallelepi∣pedons are equal.