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 25. Theoreme. The 30. Proposition. Parallelipipedons consisting vpon one and the selfe same base, and vnder the selfe same altitude, whose standing lines are not in the selfe same right lines, are equall the one to the other.

SVppose that these Parallelipipedons CM and CN, do consist vpon one and the selfe same base, namely, AB, and vnder one and the selfe same altitude, whose standing lines, namely, the lines AF, CD, BH, and LM, of the Parallelipipedon CM, and the standing lines AG, CH, BK, and LN, of the Parallelipipedon CN, let not be in the selfe same right lines. Then I say, that the Parallelipipedon CM, is equall to the Parallelipipedon CN. Forasmuch as the vpper superficieces FH & ••K, of the former Parallelipipedons, are in one and the selfe same superficies (by reason they are supposed to be of one and the selfe same alti∣tude):* 1.1 Extend the lines FD and MH, till they concurre with the lines N•• and KE (suffi∣ciently both waies extended: for in diuers cases their concurse is diuers). Let ••D extended, meete with NG, and cut it in the point X: and with KE in the point P. Let likewise MH extended, meete with NG (sufficiently produced) in the point O, and with KE in the point R. And drawe these right lines AX, LO, CP, and BR. Now (by the 29. of the eleuenth) the solide CM, whose base is the parallelogramme ACBL, and the parallelogramme opposite vn∣to

it is FDHM, is equall to the solide CO, whose base is ACBL and the opposite side the parallelogramme XPRO,* 1.2 for they consiste vpon one and the selfe same base, namely, vpon the parallelogrāme ACBL, whose standing lines AF, AX, LM, LO, CD, CP, BH, and ••K,

[illustration]

* 1.3 are in the selfe same right lines FP and MR. But the solide CO, whose base is the paralle∣logramme ACBL, and the opposite superficies vnto it is XPRO, is equall to the solide CN, whose base is the parallelogramme ACBL, and the opposite superficies vnto it is the superfi∣cies G••KN, for they are vpon one and the selfe same base, namely, ACBL, and their stan∣ding lines AG, AX, CF, CP, LN, LO, BK, and BR, are in the selfe same right lines NX, and PK. Wherefore also the solide CM, is equall to the solide CN. Wherefore Parallelipi∣pedons consisting vpon one and the selfe same base, and vnder the selfe same altitude, whose standing lines are not in the selfe same right lines, are equall the one to the other: which was required to be proued.

This demonstra∣tion

[illustration]

is somwhat har∣der then the former to conceaue by the fi∣gure described in the plaine (and yet before M. Dee inuented that figure which is placed for it, it was much harder) by reason one solide is contained in an other. And there∣fore for the clerer light therof, describe vpō pasted paper this figure here put with the like letters and finely cut the lines AC, CB, EG, BL, EPK, ROHM, and folde it accordingly that euery line may exactly agree with his correspondent lyne (which obseruing the letters of euery line

ye shall easily do) and so cōpare your figure with the demonstration, and it will make it very plaine vnto you.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  I. Dees figure.