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 11. Probleme. The 34. Proposition. To finde out two right lines inc••••mensurable in power, whose squares ad∣ded together make a mediall superficies, and the parallelogramme contay∣ned vnder them, make a rationall superficies.

TAke (by the 31. of the tenth) two mediall lines AB and BC, commensurable in power onely, comprehending a rationall superficies,* 1.1 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 describe vpon the line AB a semicircle ADB. And by the 10. of the first, deuide the line BC

[illustration]

vnto two equall partes in the point E. And (by the 28. of the sixt) vpon the line AB apply a parallelogramme equall to the square of the line BE, and wantyng in figure by a square, and let that paral∣lelogramme be that which is contayned vnder the lines AF and FB. Wherfore the line AF is incommensurable in length vnto the line FB (by the 2. part of the 18. of the tenth.) And from the point F vnto the right line AB, raise vp (by the 11. of the first) a per∣pendiculer line FD, and draw lines from A to D, and from D to B.* 1.2 And forasmuch as the line AF is incommensurable vnto the line FB: but (by the second assumpt going before the 33. of the tenth) as the line AF is to the line FB, so is the parallelogramme contayned vnder the lines BA and AF, to the parallelogramme contained vnder the lines BA and BF, wher∣fore (by the tenth of the tenth) that which is contained vnder the lines BA and AF is in∣commensurable to that which is contayned vnder the lines AB and BF: but that which is

contained vnder the lines BA and AF is equall to the square of the line AD, and that which is contained vnder the lines AB and BF is also equall to the square of the line DB (by the second part of the first assumpt going before the 33. of the tēth) wherfore the square of the line AD is incōmensurable to the

[illustration]

square of the line DB. Wherefore the lines AD and DB are incommensura∣ble in power. And forasmuch as the square of the line AB is mediall, there∣fore also the superficies made of the squares of the lines AD and DB ad∣ded together is mediall.* 1.3 For the squares of the lines AD and DB are (by the 47. of the first) equall to the square of the line AB. And forasmuch as the line BC is double to the line FD (as it was proued in the pro∣position going before) therefore the parallelogramme contained vnder the lines AB and and BC is double to the parallelogramme contained vnder the lines AB and FD (by the third assumpt going before the 33. proposition) wherefore it is also commensurable vnto it (by the sixt of the tenth) But that which is contained vnder the lines AB and BC is suppo∣sed to be rationall. Wherfore that which is contained vnder the lines AB and FD is also ra∣tionall. But that which is contained vnder the lines AB and FD, is equall to that which is contained vnder the lines AD and DB (by the last part of the first assumpt going before the 33. of the tenth) Wherfore that which is contayned vnder the lines AD and DB is also ra∣tionall.* 1.4 Wherefore there are ••ound out two right lines AD and DB incommensurable in power, whose squares added together, make a mediall superficies, and the parallelogramme cō∣tayned vnder them, make a rationall superficies: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  The first part of the deter∣mination con∣cluded.
• * 1.4

  The second part cōcluded.

  The totall conclusion.