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 14. Probleme. The 49. Proposition. To finde out a second binomiall line.

TAke two numbers AC and CB, and let them be such that the number made of them both added together, namely,* 1.1 AB haue vnto BC that proportiō that a square number hath to a square number, and vnto the number CA let it not haue that proportion that a square number hath to a square number, as it was declared in the former proposition. Take also a ra••ionall line, and let the same be D, and vnto the line D let the line FG be commensurable in length Wherefore FG is a rationall line. And as the number CA is to the number AB so let the square of the line GF be to the square of the line FE (by the 6. of the tenth), Wherefore the square of the lin•• GF is commensurable to the square of the line FE. Wherfore also FE is a rationall line.* 1.2 And forasmuch as the nūber CA hath not vnto the number AB that proportiō that a square number hath to a square nūber, ther∣fore neither also the squar•• of the line GF

[illustration]

hath to the square of the line FE that propor∣tion that a square number hath to a square number. Wherefore the line GF is incom∣mensurable in length vnto the line FE (by the 9. of the tenth): wherefore the lines FG and FE are rationall commensurable in po∣wer onely. Wherefore the whole line EG is a binomiall line. I say moreouer that the lin•• EG is a second binomiall line. For for that by contrary proportion as the number BA is to the number AC, so is the square of the line EF to the square of the line FG. But the number BA is greater then the number AC, wherefore also the square of the line EF is greater then the square of the line FG. Vnto the square of the line EF, let the squares of the lin••s GF and H be equall. Now by conuersion (by the corollary of the 19. of of the fift) as the number AB is to the number BC, so is the square of the line EF to the square of the line H. But the number AB hath to the number BC that proportion that a square number hath to a square number. Wherefore the square of the line EF hath to the square of the line H, that proportion that a square num∣ber hath to a square number. Wherefore (by the 9. of the tenth) the line EF is commensu∣rable in length vnto the line H. Wherefore the line EF is in power more then the line FG by the square of a line commensurable in length vnto the line EF: and the lines

EF and FG are rationall commensurable in power onely, and FG being the lesse name ••s commensurable in length vnto the rationall line geuen, namely, to D. Wherefore EG is a se∣cond binomiall line: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.