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. Probleme. The 4. Proposition. Proportions in the least numbers how many soeuer beyng geuen, to finde out the least numbers in continuall proportion in the said proportions geuē.

SVppose that the proportions in the least numbers geuen, be A to B, C to D, and E to F. It is required to finde out the least numbers in continuall proportion, in the same proportion that A is to B, and that C is to D, and that E is to F. Take the least number whom B and C do measure, and let the same be G. And how often B measureth G, so many times let A measure H. And how oftē C measureth G, so many times let D measure K. Now E either measureth K or measureth it not.* 1.1 First let it measure it. And how often E measureth K, so many times let F measure L. And forasmuch as how often A measureth H, so many times doth B measure G: therfore by the 17. of the seuenth, as A is to B, so is H to G. And by the same reasō as C is to D•• so is •• to K, and moreouer as E is to F, so is K to L. Wherfore these numbers H, G, K, L, are in continuall proportion, and in the same proportion that A is to B, and that C is to D, and moreouer that E is to F. I say also that they are the least in those proportions••* 1.2 For if H, G, K, L, be not the least num∣bers in continuall proportion, and in the same proportions that A is to B, and C to D, and E to F, then are there some numbers less•• then H, G, K, L, in the same proportions that A is to B and C to D, and E to F, let those numbers be N, X, M, O. And forasmuch as A is to

B, so is N to X•• and A and B are the lest, but the least measure

[illustration]

those numbers that haue one and the same proportion with them equally, the greater, the greater, and the lesse the lesse, that is, the antecedent, the antecedent, & the consequent the consequent (by the 21. of the seuenth) therfore B measureth X. And by the same reason C also measureth X, wherfore B and C measure X. Wher∣fore the least number whom B and C measure, shall also by the 37 of the seuenth measure X. But the least number whome B and C measure is G. Wherefore G measureth X, the greater, the lesse, which is impossible. Wherfore there shall not be any lesse numbers then H, G, K, L, in continuall proportion, and in the same pro∣portions that A is to B, and C to D, and E to F.

But now suppose that E measure not K.* 1.3 And by the 36. of the seuenth, take the least number whome E and K measure, and let the same be M. And how often K measureth M, so often let either of these G and H measure either of these N and X.* 1.4 And how often E measureth M, so often let F measure O. And forasmuch as how often G measureth N, so often doth H measure X, therfore as •• is to G, so is X to N. But as H is to G, so is A to B Wherfore as A is to B, so is X to N. And by the same reason as C is to D, so is N to M. A∣gaine, forasmuch as how often E measureth M, so often F measureth O, therfore as E is to F so is M to O. Wherfore X, N, M, O, are in continuall proportion, and in the same proporti∣ons that A is to B, and C to D, and E to F. I say also that

[illustration]

they are the least in that proportion. For if X, N, M, O, be not the least in continuall proportion, and in the same pro∣portions that A is to B, and C to D, and E to F, then shall there be some numbers lesse then X, N, M, O, in continuall proportion, and in the same proportions that A is to B, and C to D, and E to F. Let the same be the numbers P, R, S, T. And for that as P is to R, so is A to B, and A and B are the least, but the least numbers measure those numbers that haue one & the same proportiō with them equally, the greater the greater, and the lesse the lesse, that is, the antece∣dent the antecedent, and the consequent the consequent by the 21. of the seuenth, therfore B measureth R. And by the same reason also C measureth R. Wherefore B and C mea∣sure R. Wherfore the least number wh••m B and C measure shall also measure R (by the 37. of the seuenth). But the lest number whom B and C measure is G, wherfore G measureth R. And as G is to R, so is K to S. Wherfore K measureth S. And E also measureth S. Wherefore E and K measure S. Wherfore the least number whom E and K measure, shall (by the selfe same) measure S. But the least number whom E and K measures M. Wherefore M measureth S the greater the lesse, which is impossible. Wherfore there are no numbers lesse then X, N, M, O, in continuall proportion, & in the same proportions that A is to B, and C to D, and E to F. Wherfore X, N, M, O are the least numbers in continuall proportion, and in the same proportions that A is to B, and C to D, and E to F: which was required to be done.

Notes

• * 1.1

  Two cases in this propositiō. The first case.
• * 1.2

  Demonstra∣tion leading to an absurditie.
• * 1.3

  The second case.
• * 1.4

  Demonstra∣tion.