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 22. Probleme. The 88. Proposition. To finde out a fourth residuall line.

TAke a rationall line and let the same be A: and vnto it let the line BG be commensurable in length. Wherefore the line BG is rationall. And take two numbers DF and FE,* 1.1 and let them be such that the whole number, namely, DE haue to neither of the numbers DF and FE that proportion that a square number hath to a square

[illustration]

number. And as the number DE is to the number EF, so let the square of the line BG be to the square of the line GC: wherefore the square of the line BG is com∣mensurable to the square of the line GC, wherefore also the square of the line GC is rationall, and the line GC is also rationall. And for that the number DE hath not the number EF that proportion that a square number hath to a square number,* 1.2 therefore the line BG is incom∣mensurable in length to the line GC. And they are both rationall: wherefore the line BC is a residuall line. I say moreouer that it is a fourth residuall line. For forasmuch as the square of the line BG is greater then the square of the line GC, vnto the square of the line BG let the squares of the lines CG and H be equall. And for that as the number DE is to the number EF, so is the square of the line BG to the square of the line GC, therefore by con∣uersion of proportion as the number DE is to the number DF, so is the square of the line BG to the square of the line H. But the numbers DE and DF haue not the one to the other that porportion that a square number hath to a square number. Wherefore the line BG is incommensurable in length to the line H. Wherefore the line BG is in power more then

the line GC by the square of a line incommensurable in length so the line BG: and the whole line BG is commensurable in length to the rationall line A, Wherefore the line BC is a fourth residuall line. Wherefore there is founde out a fourth residuall line: which was re∣quired to be doone.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion