The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

¶ The 5. Theoreme. The 5. Proposition. Pyramids consisting vnder one and the selfe same altitude, hauing tri∣angles to their bases: are in that proportion the one to the other that their bases are.

SVppose that these two Pyramids, whose bases are the triangles ABC & DEF, and toppes the pointes G and H, be vnder equall altitudes. Then I say, that as the base ABC is to the base DEF, so is the pyramis ABCG to the pyramis DEFH. For if the pyramis ABCG be not to the pyramis DEFH, as the base ABC is to the base DEF, then as the base ABC is to the base DEF,* 1.1 so is the pyra∣mis ABCG to a solide, either lesse then the pyramis DEFH, or greater. First let it be to some lesse, and let the same be X. And (by the 3. of the twelfth) let the pyramis DEFH be deuided into two pyramids equall the one to the other, and like vnto the whole, and into two equall prismes. Now the two prismes are greater then the halfe of the whole pyramis.

And againe (by the same) let the

[illustration]

pyramids which are made of the diuision, be in like sort diuided, and do this continually, vntill there remaine some pyramids made of the pyramis DEFH, which are lesse then the excesse, whereby the pyramis DEFH excedeth the solide X. Let such pyramids be taken, and for ex∣ample sake, let those pyramids be DPRS, & STVH. Wherfore the prismes remayning which are in the pyramis DEFH, are greater then the solide X. De∣uide (by the Proposition next go∣ing before) the pyramis ABCG in like sort, & as many times as the pyramis DEFH ••s deuided. Wherefore (by the same) as the base ABC is to the base DEF, so are all the prismes which are in the pyramis AB∣CG, to all the prismes which are in the pyramis DEFH. But as the base ABC is to the base DEF, so is the pyramis ABCG to the solide X. Wherefore (by the 11. of the fift) as the pyramis ABCG, is to the solide X, so are the prismes which are in the pyramis ABCG, to the prismes which are in the pyramis DEFH. Wherefore alternately (by the 16. of the fift) as the pyramis ABCG is to the prismes which are in it, so is the solide X, to the prismes which are i•• the pyramis DEFH. But the pyramis ABCG is greater then the prismes which are in it. Wherefore also the solide X is greater then the prismes which are in the pyramis DEFH (by the 14. of the fift). But it is supposed to be lesse which is impossible. Wherefore as the base ABC is to the base DEF, so is not the pyramis ABCG to any solide lesse then the pyramis DEFH.

I say moreouer, that as the base ABC is to the base DEF, so is not the pyramis ABCG, [ 1] to any solide greater then the pyramis DEFH. For if it be possible, let it be vnto some greater, namely, to the solide X. Wherefore (by conuersion, by the Corollary of the 4. of the ••i••••) as the base DEF is to the base ABC, so is the solide X to the pyramis ABCG. But as the solide X is to the pyramis ABCG, so is the pyramis DEFH to some solide lesse then the pyramis ABCG, * 1.2 as we haue before proued. Wherefore also (by the 11. of the ••ift) as th•• base DEF is to the base ABC, so is the pyramis DEFH, to some solide lesse then the pyramis ABCG which thing we haue proued to be impossible. Wherfore as the base ABC is to the base DEF, so is not the pyramis ABCG to any solide greater then the pyramis DEFH: and it is als•• proued that it is not in that proportion to any lesse then the pyramis DEFH. Wherefore as the base ABC is to the base DEF, so is the pyramis ABCG to the pyramis DEFH. Wherefore pyramids consisting vnder one and the selfe same altitude, and hauing triangles to their bases, are in that proportion the one to the other, that their bases are: which was required to be demonstrated.

Notes

• * 1.1

  Demonstrati∣on leading to an impossibi∣litie.
• * 1.2

  In the As∣su••p•• ••••llow∣in•• the second ••rop••••ition of this b••••ke.