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 21. Probleme. The 87. Proposition. To finde out a third residuall line.

TAke rationall line, & let the same be A: and take three numbers E, B, C,* 1.1 and CD, not hauing the one to the other that proportion that a square nūber hath to a square number: and let the number BC haue to the number BD that proportion that a square number hath to a square number. And let the num∣ber BC be greater then the number CD. And as the number E is to the num∣ber BC, so let the square of the line A be to the square of

[illustration]

the line FG: and as the number BC is to the number CD, so let the square of the line FG be to the square of the line HG.* 1.2 Wherefore the square of the line A is com∣mensurable to the square of the line FG. But the square of the line A is rationall. Wherefore also the square of the line FG is rationall: wherefore the line FG is also rationall. And forasmuch as the number E hath not to the number BC that proportion that a square number•• hath to a square number, therefore neither also hath the square of the line A to the square of the line FG that proportion that a square number hath to a square number. Wherefore the line A is incom∣mensurable in length to the line FG. Againe for that as the number BC is to the number CD, so is the square of the line FG to the square of the line HG, therefore the square of the line FG is commensurable to the square of the line HG. But the square of the line FG is ra∣tionall. Wherefore also the square of the line HG is rationall. Wherefore also the line HG is rationall. And for that the number BC hath not to the number CD, that proportion that a square number hath to a square number, therefore neither also hath the square of the line ••G to the square of the line HG, that proportion that a square nūber hath to a square num∣ber. Wherefore the line FG is incommensurable in length to the line HG: and they are both rationall. Wherefore the lines FG & HG are rationall cōmensurable in power onely. Wher∣fore the line FH is a residuall line. I say moreouer, that it is a third residuall line. For for that as the number E is to the number BC, so is the square of the line A to the square of the line FG: and as the number ••C is to the number CD, so is the square of the line FG to the square of the line HG•• therefore by equalitie of proportion, as the number E is to the num∣ber CD, so is the square of the line A to the square of the line HG•• but the number E hath •••••• to the number CD that proportio•• that a square num••••r hath to a square number, ther∣fore

neither also hath the square of the line A to the square of the line HG that proportion that a square number hath to a square number, therefore the line A is incommensurable in length to the line HG. Wherefore neither of the lines FG and HG is commensurable in length to the rationall line A. And forasmuch as the

[illustration]

square of the line FG is greater then the square of the line HG (that the line FG is greater then the line HG it is mani••est, for by supposition the number BC is grea∣ter then the number CD) vnto the square of the line FG let the squares of the lines HG & K•• be equall. And for that as the nūber BC is to the number CD, so is the square of the line FG to the square of the line H•• ••••er∣fore (by conuersion of proportion) as the number BC is to the number BD, so is the square of the line FG to the square of the line K. But the nūber BC hath to the num∣ber BD that proportion that a square number hath to a square number. Wherefore the square of the line FG hath to the square of the line K that proportion that a square number hath to a square number. Wherefore the line FG is commensurable in length to the line K. Wherefore the line FG is in power more then the line HG, by the square of a line commen∣surable in length to the line FG, and neither of the lines FG and GH is commensurable in length to the rationall line A: when yet notwithstanding either of the lines FG and GH is rationall. Wherefore the line FH is a third residuall line. Wherefore there is found out a third residuall line: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.