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 7. Theoreme. The 9. Proposition. If two numbers be prime the one to the other, and if betwene them shall fall numbers in continuall proportion: how many numbers in continuall pro∣portion fall betwene them, so many also shall fall in continuall proportion betwene either of those numbers and vnitie.

SVppose that there be two numbers prime the one to the other A and B: and let there fall betwene them in continuall proportion these numbers C and D: and let E be vnitie. Then I say, that how many numbers in continuall proportion fall betwene A and B, so many also shall fall in continuall proportion betwene A and vnitie E:* 1.1 and likewise betwene B and vnitie E. Take (by the 35. of the seuenth) the two least numbers that are in the same proportion that A, C, D, B, are: and let the same be F and G: and then take three of the least nūbers that are in the same proportion that A, C, D, B, are: and let the same be H, K, L: and so alwaies in order one more, vntill the multitude of them be equall to the multitude of these numbers A, C, D, B: and those being so taken let them be M, N, X, O. Now it is manifest, that F multiplying him selfe produced H, and multiplying N produced M.* 1.2 And G multiplying him selfe produced L, and multiplying L produced O. And forasmuch as M, N, X, O, are (by supposition) the least of all numbers that haue the same proportion with G, F: and A, C, D, B, are (by the first of the eight) the least of all numbers that haue the same proportion with G, F: and the multitude of these numbers M, N, X, O, is equall to the multitude of these numbers A, ••, D, B: therefore euery one of these numbers M, N, X, O, is equall to euery one of these numbers A, C, D, B. Wherefore M is equall vnto

[illustration]

A, and O is eq••all vnto B. And forasmuch as F multiplying him selfe produced H: therfore •• measureth H by those vnities which are in F: and vnitie E measureth F by those vnities which are i•• F: wherfore (by the 15. of the seuenth) vnitie E, so many times measureth the number F, as F measureth H•• wherefore as vnitie E is to the number F, so is F to H. Againe forasmuch as F multiplying N produced M, therfore H measureth M by those vnities which are in F. And vnitie E measureth F by th••se vnities which are in F: wherefore (by the self same) vnitie E so many times measureth F, as H measureth M. Wherefore as vnitie E is to the numbers F, so is H to M. But it is proued, that as vnitie E is to the number F, so is F to H: wherefore as vnitie E is to the number F, so is F to H, and H to M. But M is equall vnto A•• wherefore as vnitie E is to the number F, so is F to H, & H to A. And by the same reason as vnitie E is to the number G, so is G to L and L to B. Wherefore how many num∣bers fall in continuall proportion betwene A and B: so many numbers also in continuall pro∣portion fall there betwene vnitie E and the number A, and likewise betwene vnitie E

and the number ••: which was required to be demonstrated.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.