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 17. Probleme. The 52. Proposition. To finde out a fift binomiall lyne.

TAke two numbers AC and CB, and let them be such, that the number AB haue to neither of the numbers AC or CB that proportion that a square num∣ber hath to a square number,* 1.1 as in the former proposition. And take a ratio∣nall line and let the same be D. And vnto the line D let the line FG be com∣mensurable in length. Wherfore the line FG is rationall. And as the number CA is to the number AB, so let the square

[illustration]

of the line GF be to the square of the line EF.* 1.2 Wherefore the square of the line GF is commensurable to the square of the line FE. Wherefore also the line FE is ratio∣nall. An•• for that the number CA hath not to the number AB that proportiō that a square number hath to a square num∣ber, ther••ore neither also hath the square of the line GF to the square of the line FE that proportion that a square number hath to a square number. Wherfore (by the 9. of the tenth) the line GF is incommensurable in length to the line FE. Wherfore the lines EF and FG are rationall commensurable in power only. Wherfore the whole line EG is a binomiall line. I say moreouer that it is a fift binomiall line. For for that as the number CA is to the number AB, so is the square of the line GF to the square of the line FE, therfore contrariwise, 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. Wherfore 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 lines FG and H be equall. Wherfore by conuersiō (by the corollary of the 19. of the fift) as the nūber AB is to the num∣ber BC•• so is the square of the line EF to the square of the line H. But the nūber AB hath not to the number BC that proportiō that a square number hath to a square number. Wherefore neither also hath the square of the line EF to the square of the line H that proportion that a square number hath to a square number. Wherfore (by the 9. of the tenth) the line EF is in∣commensurable in length to the line H. Wherfore the line EF is in power more then the line FG by the square of a line incommensurable in length vnto it. And the lines EF and FG are rationall commensurable in power onely. And the line FG being the lesse name, is com∣mensurable in length to the rationall line geuen, namely, to D. Wherfore the whole line EG is a fift binomiall line: which was required to be found out.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.