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 19. Theoreme. The 21. Proposition. The left numbers in any proportion, measure any other nūbers hauing the same proportion equally, the greater the greater, & the lesse the lesse.

SVppose that CD & EF be the least numbers that haue one & the same propor∣tion with the numbers A and B.* 1.1 Then I say, that the number CD so many times measureth the number A, as the number EF measureth the number B. For for∣asmuch as by supposition CD is to EF, as A is to B, and CD and EF are also supposed to be lesse then A and B: therefore CD and EF are either part or partes of A and B (by the 4. of this booke, and by the 21. definition of the same). But they are not partes. For if it be possible, let CD be partes of A. VVherfore EF is the selfe same partes of B, that CD is of A. Wherefore how many partes of A there are in CD, so many partes are there of B in EF. Deuide CD into the partes of A, that is, into CG and GD. And likewise deuide EF into the partes of B, that is, into EH and HF. Now then the multitude

[illustration]

of these CG and GD, is equall vnto the multitude of these EH & HF. And forasmuch as CG and GD are numbers equall the one to the other, and these numbers EH and HF are also equall the one to the other, and the multitude of these CG and GD, is equall to the multitude of these EH and HF: therefore as CG is to EH, so is GD to HF. Wherefore (by the 12. of the seuenth) as one of the antecedentes is to one of the consequentes, so are all the antecedentes to all the consequentes. Wherefore as CG is to EH, so is CD to EF. Wher∣fore CG and DH ••re in the selfe same proportion that CD and EF are, being also lesse then CD and EF: which is impossible. For CD and EF are supposed to be the least that haue 〈◊〉〈◊〉 and the same proportion with them. Wherefore CD is not partes of A: wherefore it is a part. Wherefore EF is of B ••he selfe same part, that CD is of A. Wherefore CD so many 〈◊〉〈◊〉 measureth A as EF doth B•• which was req••ired to be demonstrated.

Notes

• * 1.1

  Demonstra∣tion leading to an impossibi∣lity.