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 40. Theoreme. The 58. Proposition. If a superficies be contained vnder a rationall line and a fift binomiall line: the line which contayneth in power that superficies is irrationall, and is a line contayning in power a rationall and a mediall superficies.

SVppose that the superficies AG be contayned vnder the rationall line A••, and vnder a fift binomiall line AD•• ••nd let the same lin•• AD be supposed to be deuided into his names in the poynt E, so that let the line AE be the greater name. Then I say, that the line which contayneth in power the superficies AC is irrationall, and is a line contayning in power a rationall and a mediall superficies.* 1.1 Let the selfe same constructions be in this, that were in the foure Proposition next going before. And it is manifest, that the line MX contayneth in power the superfici•••• AG. Now testeth to proue that the line MX is a line contayning in power a rationall & a mediall superficies. For∣asmuch

as the line AG is incommensurable in length to the line GE, therefore (by the 1. of the sixt, and 10. of the tenth) the parallelogramme AH is incommensurable to the paralle∣logramme [ 2] HE, that is, the square of the line MN to the square of the line NX. Wherefore the lines MN and NX are incommensurable in power. And forasmuch as the line AD i•• a fif•• binomiall line, and his lesse name or part is the line ED, therefore the line ED is com∣mensurable

[illustration]

in length to the line AB. But the line AE is incommensurable in length to the line ED. Wherefore (by the 13. of the tenth) the line AB is incommensurable in length to the line AE. Wherefore the lines AB and AE are rationall commensurable in power onely. Wherefore (by the 21. of the tenth) the parallelogramme AK is mediall, that is, that which is composed of the squares of the lines MN & NX added together. And forasmuch [ 3] as the line DE is commensurable in length to the line AB, that is, to the line EK, but the line DE is commensurable in length to the line EF, wherefore (by the 12. of the tenth) the line EF is also commensurable in length to the line EK. And the line EK is rationall. Wherefore (by the 19. of the tenth) the parallelogramme EL, that is, the parallelogramme MR, which is contayned vnder the lines MN and NX is rationall. Wherefore the lines [ 4] MN and NX are incommensurable in power, hauing that which is composed of the squares of them added together, Mediall, and that which is contayned vnder them, Rationall. Wherefore (by the 40. of the tenth) the whole line MX is a line contayning in power a ra∣tionall and a mediall superficies, and it contayneth in power the superficies AC: which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion.