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 12. Probleme. The 35. Proposition. To finde out two right lines incommensurable in power, whose squares ad∣ded together, make a mediall superficies, and the parallelogramme contai∣ned vnder them, make also a mediall superficies, which parallelogramme moreouer, shall be incommensurable to the superficies made of the squares of those lines added together.

TAke (by the 32. of the tenth) two mediall lines AB and BC commensurable in power onely,* 1.1 comprehending a mediall superficies, so that let the line AB be in power more then the line BC by the square of a line incommensurable in length vnto the line AB. And vpon the line AB describe

[illustration]

a semicircle ADB, and let the rest of the construction be as it was in the two former propositions. And forasmuch as (by the 2 part of the 18. of the tenth) the line AF is in∣commensurable in length vnto the line FB,* 1.2 therfore the line AD is incommensurable in power vnto the line DB (by that which was demonstrated in the pro∣positiō going before). And forasmuch as the square of the line AB is mediall, therefore that also which is composed of the squares of the lines AD and DB (which squares are equall to the square of the line AB by the 47. of the first) is mediall.* 1.3 And forasmuch as that which is

contained vnder the lines AF and FB, is equall to either of the squares of the lines EB and FD, for by supposition the parallelogramme contained vnder the lines AF and FB is e∣quall to the square of the line EB, and the same parallelogramme is equall to the square of the line DF (by the third part of the first assumpt going before the 33. of the tēth). Wherfore the line BE is equall to the line DF. Wherfore the line BC is double to the line FD. Where∣fore that which is contained vnder the lines AB and BC is double to that which is contai∣ned vnder the lines AB and FD. Wherfore they are commensurable by the sixt of this boke: but that which is contained vnder the lines AB and BC is mediall by supposition. Wherfore also that which is contained vnder the lines AB and FD is mediall (by the corollary of the 23, of the tenth) but that which is contained vnder the lines AB and FD, is (by the fourth part of the first assumpt going before the 33. of the tenth) equall to that which is contained vnder the lines AD and DB:* 1.4 wherfore that which is contained vnder the lines AD and DB is also mediall. And forasmuch as the line AB is incommensurable in length vnto the line BC. But the line BC is commensurable in length vnto the line BE. Wherfore (by the 13•• of the tenth) the line AB is incommensurable in length vnto the line BE. Wherefore the square of the line AB is incommensurable to that which is contained vnder the lines AB and BE (by the first of the sixt and 10. of this booke) But vnto the square of the line AB are equall the squares of the lines AD and DB added together (by the 47. of the first): and vnto that which is contayned vnder the lines AB and BE, is equall that which is contai∣ned vnder the lines AB and FD, that is, which is contained vnder AD and DB. For the parallelogramme contained vnder the lines AB and FD is equall to the parallelogramme contained vnder the lines AD and DB (by the last part of the first assumpt going before the 33. of this tenth booke).* 1.5 Wherfore that which is composed of the squares of the lines AD and DB is incommensurable to that which is contained vnder the lines AB and DB. Where∣fore there are found out two right lines AD and DB incommensurable in power, whose squares added together, make a mediall superficies,* 1.6 and the parallelogramme contayned vn∣der them, make also a mediall superficies, which parallelogramme moreouer is incommensu∣rable to the superficies composed of the squares of those lines added together, which was re∣quired to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  The first part concluded.
• * 1.4

  The second part cōcluded.
• * 1.5

  The third part cōcluded.
• * 1.6

  The totall con¦clusion.