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 71. Theoreme. The 95. Proposition. If a superficies be contained vnder a rationall line and a fift residual line: the line that cōtayneth in power the same superficies, is a line making with a rationall superficies, the whole superficies mediall.

SVppose that there be a superficies AB contained vnder a rationall line AG and a fift residuall line AD. Thē I say that the line that cōtaineth in power y^e sup••r∣ficies AB, is a line making with a rationall superficies the whole superficies me∣diall. For vnto the line AD let the line DG be ioyned, which shal be cōmēs••••a∣ble in lēgth to the rational line AC. And let the rest of the constructiō be as in the propositio•• next going before.* 1.1 And forasmuch as the line AG is incōmensurable in lēgth to the line AC and they are both rationall, therfore the parallelogrāme AK is medial. Againe forasmuch as the line DG is rationall

[illustration]

and commensurable in length to the lyne AC, theref••re the parallelo∣gramme DK is ratio∣nall. Vnto the parallel••∣gramme AI describe an equall square LM; and vnto the parallelograme •••• describe an equall square N••, and as in 〈◊〉〈◊〉 proposition next go∣ing

before, so also in this may we proue, that the line LN containeth in power the superficies AB.* 1.2 I say moreouer that that line LN is a line making with a rationall superficies the whole superficies mediall. For forasmuch as the parallelogramme AK is mediall, therefore that which is equall vnto it, namely, that which is made of the squares of the lines LO and ON added together is also mediall. Againe forasmuch as the parallelogramme DK is ra∣tionall, therfore that which is equall vnto it, namely, that which is contained vnder the lines LO and ON twise, is also rationall. And forasmuch as the line AF is incommensurable in length to the line FG, therfore (by the 1. of the sixt, & 10. of the tenth) the parallelogrāme A•• is incommensurable to the parallelogramme FK, wherfore also the square of the lyne LO is incommensurable to the square of the line ON. Wherfore the lines LO and ON are in∣commensurable in power hauing that which is made of their squares added together medi∣all, and that which is contayned vnder them twise rationall. Wherfore the line LN is that irrationall line which is called a lyne making with a rationall super••icies the whole superfi∣cies mediall, and it contayneth in power the superficies AB. Wherfore the line contayning in power the super••icies AB, is a line making with a rationall superficies the whole superficies mediall. If therfore a superficies be contayned vnder a rationall lyne & a fift residuall line, the line that contayneth in power the same superficies, is a line makyng with a rationall su∣perficies the whole super••icies mediall: which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion.
• * 1.2

  The line LN.