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 2. Theoreme. The 2. Proposition. If the first be equemultiplex to the second as the third is to the fourth, and if the fifth also be equemultiplex to the second as the sixt is to the fourth: then shall the first and the fifth compo∣sed together be equemultiplex to the second, as the third and the sixt composed together is to the fourth.

SVppose that there be sixe quantities, of which let AB be the first, C the second, DE the third, F the fourth, BG the fifth, & EH the sixt: and suppose that the first, AB, be equemultiplex vnto the second, C, as the third, DE, is to the fourth, F: and let the fift, BG, be equemultiplex vnto the second, C, as the sixt, EH, is to the fourth, F. Then I say, that the first and the fifth composed together, which let be AG, is equemultiplex vnto the second, C, as the third and sixt composed together, which let be DH, is to the fourth, F. For forasmuch as AB is equemultiplex to C,* 1.1 as DE is

[illustration]

to F, therefore how many magnitudes there are in AB equall vnto C, so many magnitudes are there in DE equall vnto F: and by the same reason how ma∣ny there are in BG equall vnto C, so many also are there in EH equall vnto F. Wherefore how many there are in the whole AG equall vnto C, so many are there in the whole DH equall vnto F. Wherefore how multiplex AG is vnto C, so multiplex is DH vnto F. Wherefore the first and the fifth composed

together, namely, AG is equemultiplex vnto the second C, as the third and the sixte composed together, namely, DH, is to the fourth F. If therfore the first be equemultiplex to the second as the third is to the fourth, and if the fifth also be equemultiplex to the second as the sixt is to the fourth: then shall the first & the fifth cōposed together, be equemultiplex to the second, as the third and the sixt composed together•• is to the fourth: which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion••