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 13. Theoreme. The 15. Proposition. If vnitie measure any number, and an other number do so many times mea∣sure an other number: vnitie also shall alternately so many times measure the third number, as the second doth the fourth.

SVppose that vnitie A do measure the number BC: and let an other nūber D so many times measure some other nūber, namely, EF. Then I say, that alternate∣ly, vnitie A shall so many times measure the number D, as the number BC doth measure the number EF.* 1.1 For forasmuch as vnitie A doth so many times mea∣sure BC, as D doth EF: therefore how many vnities there are in BC,

[illustration]

so many numbers are there in EF equall vnto D. Deuide (I say) BC in∣to the vnities which are in it, that is, into BG, GH, and HC. And deuide likewise EF into the numbers equall vnto D, that is, into EK, KL, and LF. Now then the multitude of these BG, GH, and HC, is equall vnto the multitude of these EK, KL, LF.* 1.2 And forasmuch as these vnities BG, GH, and HC, are equall the one to the other, and these numbers EK, KL, & LF, are also equall the one to the other, and the multitude of the vnities BG, GH, and HC, are equall vnto the multitude of the numbers EK, KL, & LF: therefore as vnitie BG is to the num∣ber EK, so is vnitie GH to the number KL, and also vnitie HC to the number LF. Wher∣fore

(by the 12•• of the seue••th) as one of the antecede••t•••• ••s to one of the

[illustration]

consequentes, so are all the anteceden••es to all the consequentes: Wherfore as vnitie BG is to the number EK, so is the number BC to the number EF. But vnitie BG is equall vnto vnitie A, and the number EK to the number D. VVherefore (by the 7. common sentence) as vnitie A is to the number D, so is the number BC to the number EF. VVherefore vnitie A measureth the nū∣ber D, so many times, as BC measureth EF (by the 21 definition of this booke): which was required to be proued.

Notes

• * 1.1

  Co••str••ctio••.
• * 1.2

  Demonstra∣tion.