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EQual Parallelepipedons have their Bases and height Reciprocal, and those which have their Bases and height Recipro∣cal, are equal.
If the Parallepipedons AB, CD, are equal, they shall have their Bases and height reciprocal; that is to say, there shall be the same Ratio of the Base AE to the Base CF, as of the height CH to the height AG. Having made
CI equal to AG, draw the Plane IK parallel to the Base CF.
Demonstration. The Parallelepipedon AB, hath the same Ratio to CK of the same height, as hath the Base AE to CF, (by the 32d.) Now as AB to CK, so is CD to the same CK, seeing that AB and CD are equal; and as CD is to CK, which hath the same Base, so is the height CH to the height CI (by the Coroll. of the 32d.) Wherefore, as the Base AE is to the Base CF, so is the height CH to the height CI or AG.
I further add, that if there be the same Ratio of AE to CF, as of the height CH to the height AG; the Solids AB, CD, shall be equal.
Demonstration. There is the same Ratio of AB to CK, of the same height, as of the Base AE to the Base CF, (by the 32d) there is also the same Ratio of the height CH to the height CI or AG, as of CD to CK; we suppose that the Ratio of AE to CF, is the same of that of CH to CI or AG; so then there shall be the same Ratio of the Solid AB to the Solid CK, as of the Solid CD to the same Solid CK. Therefore (by the 8th. of the 5th.) the Solids AB, CD, are equal.
THis reciprocation of Bases, ren∣dreth these Solids easie to be mea∣sured; it hath also a kind of Analogy with the Sixteenth Proposition of the Sixth Book which beareth this, that parallelo∣grams equiangled and equal, have their Sides reciprocal, and it Demonstrateth also as well the practice of the Rule of Three.