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

Flussas demonstrateth this proposition an otherway taking onely the bases of the solides, and that after this maner.

Take equall bases (which

[illustration]

yet for the surer vnderstan∣ding let be vtterly vnlike) namely, AEBF and ADCH, and let one of the sides of eche concurre in one & the same right line AED, & the bases being vpon one and the selfe same playne let there be supposed to be set vpon thē parallelipipedons vnder one & the selfe same altitude. Then I say that the solide set vpō the base AB is equal to the solide set vpon the base AH. By the poynt E draw vnto the line AC a

parallel line EG, which if it

[illustration]

fall without the base AB, produce the right line HC to the poynt I. Now foras∣much as AB and AH are pa∣rallelogrmaes, therefore by the 24. of this booke, the triangles ACI and EGL shall be equaliter the one to the other: and by the 4. of the first, they shal be equian∣gle and equall: and by the first definition of the sixth, and fourth Proposition of the same, they shall be like. Wherfore Prismes erected vppon those triangles and vnder the same altitude that the solides AB and AH a••e, shall be equall and like, by the 8. definition of this booke. For they are contayned vnder like playne superficieces equall both in multitude and mag¦nitude. Adde the solide set vpon the base ACLE common to them both. Wherefore the solide set vp∣pon the base AEGC, is equall to the solide set vpon the base AELI. And forasmuch as the superficie∣ces AEBF, and ADHC are equall (by supposition): and the part taken away AG is equall to the part taken away AL: therefore the residue BI shall be equall to the residue GD. Wherefore as AG is to GD as AL is to BI (namely, equalls to equalls). But as AG is to GD, so i•• the solide set vpon AG to the solide set vpon GD by the 25. of this booke, for it is cut by a playne superficies set vpon the line GE, which superficies is parallel to the opposite superficieces. Wherefore as AL is to BI, so is the solide set vpon AL to the solide set vpon BI. Wherefore (by the 11. of the fifth) as the solide set vpon AG (or vpon AL which is equall vnto it) is to the solide set vpon GD, so is the same solide set vpon AG or AL to the solide set vpon BI. Wherefore (by the 2. part of the 9. of the fifth) the solides set vpon GD and BI shall be equall. Vnto which solides if ye adde equall solides, namely, the solide set vpon AG to the solide set vpon GD, and the solide set vpon AL to the solide set vpon BI: the whole solides set vpon the base AH and vpon the base AB ••hall be equall. Wherefore Parallelipedons consisting vpon equall bases and being vnder one and the selfe same altitude are equall the one to the other: which was required to be proued.