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 84. Theoreme. The 108. Proposition. If from a rationall superficies be taken away a medialt superficies, the line which containeth in power the superficies remayning, is one of these two ir∣rationall lines, namely, either a residuall line, or a lesse line.

SVppose that BC be a rationall superficies,* 1.1 and from it take away a mediall su∣perficies, namely, BD. Then I say, that the line which containeth in power the superficies remayning, namely, the superficies EC, is one of ••hese two irrationall lines, namely, either a residuall line, or a lesse line. Take a rationall line FG. And vpon FG describe (by the 44. of the first) a rectangle parallelogramme GH equall to the super••icies BC.* 1.2 And from the parallelogramme GH take away the parallelogramme GK equall to the superficies BD.* 1.3 Wherefore (by the third common

[illustration]

sentence) the super••icies remayning, namely, EC, is equall to the parallelogramme remayning, namely, to LH. And foras∣much as BC is rationall, and BD is mediall, and BC is equall to the parallelogramme GH, and BD to the parallelogramme GK: therefore GH is rationall, and GK is mediall: and the parallelogramme GH is applied vnto the rationall line FG. Wherefore (by the ••0. of the tenth) the line FH is rationall and commensurable in length to the line FG. And the paral∣lelogramme GK is also applied vnto the ra••ionall line FG. Wherefore (by the 22. of the tenth) the line FK is rationall and incommensurable in length to the line FG. Wherefore (by the Assumpt of the 12. of the tenth) the line FH is incommen∣surable in length to the line FK. And they are both rationall. Wherefore the lines FH and FK are rationall commensurable in power onely. Wherefore the line KH is a residuall line: and the line conueniently ••oy••ed vnto it is KF. Now the line FH is in power more th••n the line KF, either by the square of a line commensurable in lengt•• to the line FH, or by the square of a line incommensurable in length to the line FH. First let it be in power more then the line FK, by the square of a line commensurable in length to the line FH, and the whole line FH is commensurable in length to the rationall line put, namely, to FG. Wherefore the line KH is a first residuall line. But if a superficies be contained vnder a rationall line, and a first residuall line, the line that containeth in power that superficies, is (by the 91. of the tenth) a residuall line. Wherefore the line which containeth in power LH, that is, the super∣ficies EC, is a residuall line. But if the line HF be in power more then the line FK, by the square of a line incommensurable in length to the line FH, and the whole line FH is com∣mensurable in length to the rationall line geuen FG. Wherefore the line KH is a fourth residuall line. But a line containing in power a superficies contained vnder a rationall line and a fourth residuall line, as a lesse line (by the 94. of the tenth). Wherefore the line that containeth in power the superficies LH, that is, the superficies EC, is a lesse line. If there∣fore from a rationall superficies be taken away a mediall super••icies, the line which containeth

in power the superficies remayning, is one of these two irrationall lines, namely, either a resi∣duall line, or a lesse line: which was required to be proued.

Notes

• * 1.1

  Seuenth Se∣nary.
• * 1.2

  Constraction.
• * 1.3

  Demonstra∣tion.