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 24. Theoreme. The 29. Proposition. Parallelipipedons consisting vpon one and the selfe same base, and vn∣der one and the selfe same altitude, whose standing lines are in the selfe same right lines, are equall the one to the other.

SVppose that that these parallelipipedons CM and CN doo consist vpon one and the selfe same base, namely AB, and let them be vnder one and the selfe same altitude, whose standing lines, that is, the fower sides of eche solide which fall vpon the base, as the lines AF, CD, BH, LM of the solide CM, and the lines CE, BK, AG, and LN

[illustration]

of the solide CN, let be in the selfe same right lines or paral∣lel lines FN, DK. Then I say that the solide CM is equall to the solide CN. For forasmuch as either of these superficieces CBDH, CBEK is a paralle∣logramme, therefore (by the 34. of the first) the line CB is e∣quall to either of these lines DH and EK. Wherefore also the line DH is equall to the line EK. Take away EH which is common to them both, where∣fore the residue namely DE is equall to the residue HK. Wherfore also the triangle DCE is equall to the triangle HKB. And the parallelogramme DG is equall to the parallelo∣gramme HN. And by the same reason the triangle AGF is equall to the triangle MLN, and the parallelogramme CF is equall to the parallelogramme BM. But the parallelogrāme CG is equall to the parallelogramme BN, by the 24. of the tenth for they are opposite the one to the other. Wherefore the prisme contayned vnder the two triangles FAG and DCE and vnder the three parrallelogrāmes AD, DG, and CG is equall to the prisme cōtayned vnder the two triangles MLN and HBK, and vnder the three parallelogrāmes, that is, BM, NH, and BN. Put that solide common to them both, whose base is the parallelogrāme AB, and the opposite side vnto the base is the superficies GEHM. Wherefore the whole pa∣rallelipipedon CM is equall to the whole parallelipipedon CN: Wherfore parallelipipedons consisting vpon one and the selfe same base, and vnder one and the selfe same altitude, whose standing lines are in the selfe same right lines, are equall the one to the other: which was re∣qui••ed to be demonstrated.

Although this demonstration

[illustration]

be not hard to a good imaginati∣on to conceaue by the former fi∣gure (which yet by M. Dee•• refor∣ming is much better then the figure of this proposition commonly des∣cribed in other copyes both greake and lattin): yet for the ease of those which are young beginners in thys matter of solides, I haue here set an other figure whose forme if it be described vpon pasted paper, with the like letters to euery line as they be here put, and then if ye finely cut not thorough but as it were halfe way the three lines LA, NMGF, and KHED, & so folde it accordingly, & compare it with the demonstratiō, it will geue great light thereunto.

Stāding lines are called those fower right lines of euery parallelipipedon which ioyne together the angles of the vpper and nether bases of the same body. Which according to the diuersitie of the angles of the solides, may either be perpendicular vpon the base, or fall obliquely. And forasmuch as in thys proposition and in the next proposition following, the solides compared together are supposed to haue one and the selfe same base, it is manifest that the standing lines are in respect of the lower base in the selfe same parallel lines, namely, in the two sides of the lower base. But because there are put two solides vpon one and the selfe same base, and vnder one and the selfe same altitude, the two vpper bases of the solides may be diuersly placed. For forasmuch as they are equall and like (by the 24. of this booke) either they may be placed betwene the selfe same parallel lines: and thē the standing lines are in either solide sayd to be in the selfe same parallel lines, or right lines: namely, when the two sides of eche of the vpper bases are contayned in the selfe same parallel lines: but contrariwise if those two sides of the vp∣per bases be not contayned in the selfe same parallel or right lines, neither shal the standing lines which are ioyned to those sides be sayd to be in the selfe same parallel or right lines. And therefore the stan∣ding lines are sayd to be in the selfe same right lines, when the sides of the vpper bases, at the least two of the sides are contayned in the selfe same right lines: which thing is required in the supposition of this 29, proposition. But the standing lines are sayd not to be in the selfe same right lines, when none of the two sides of the vpper bases are contayned in the selfe same right lines, which thing the next propositi∣on following supposeth.

Notes

• Looke at the end of the de∣monstratio•• what is vn∣derstanded by stāding lines.
• Iohn Dee his figure.

  By this figure it app••ar••th why ••uch Pris¦mes were cal∣led ••••edges: of 〈◊〉〈◊〉 v••ry shape of a wedge, as is the solide DEFGA∣C. &c.
• Stāding lines.