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

An other demonstration after Flussas.

Suppose that A be a rationall line, and let BD be a residuall line. And vpon the line BD apply the parallelogramme DT equall to the square of the line A (by the 45. of the first) making in breadth the line BT.* 1.1 Then I say that BT is a binominall line such a one as is required in the proposition. Forasmuch as BD is a residuall line, let the line cōueniently ioyned vnto it be GD. Wherfore the lines BG and GD are rationall commensurable in power onely.* 1.2 Vpon the rationall line BG apply the parallelogramme BI equall to the square of the line A and making in breadth the line BE. Wherefore the line BE is ra∣tionall and commensurable in length to the line BG (by the 20. of the tenth). Now forasmuch as the pa∣rallelogrammes BI and TD are equall (for

[illustration]

that they are eche equall to the square of the line A):* 1.3 therfore reciprokally (by the 14. of the sixth) as the line BT is to the line BE, so is the line BG to the line BD. Wherefore by con∣uersion of proportion (by the corrollary of the 19. of the fifth) as the line BT is to the line TE, so is the line BG to the line GD. As the line BG is to the line GD, so let the line TZ be to the line ZE by the corrollary of the 10. of the sixth. Wherefore by the 11. of the fifth the line BT is to the line TE, as the line TZ is to the line ZE. For either of them are as the line BG is to the line GD. Wherefore the residue BZ is to the residue ZT, as the whole BT is to the whole TE by the 19. of the fifth. Wherefore by the 11. of the fifth the line BZ is to the line ZT as the line ZT is to the line ZE. Wherfore the line TZ is the meane proportionall be∣twene the lines BZ and ZE. Wherefore the square of the first, namely, of the line BZ, is to the square of the second, namely, of the line ZT, as the first, namely, the line BZ, is to the third, namely, to the line ZE (by the corollary of the 20. of the sixth). And for that as the line BG is to the line GD, so is the line TZ to the line ZE: but as the line TZ is to the line ZE, so is the line BZ to the line ZT. Wherefore as the line BG is to the line GD, so is the line BZ to the line ZT (by the 11. of the fifth). Wherfore the lines BZ and ZT are commensurable in pow∣er onely, as also are the lines BG and GD (which are the names of the residuall line BD) by the 10. of this booke. Wherfore the right lines BZ and ZE are cōmensurable in length, for we haue proued that they are in the same proportion that the squares of the lines BZ and ZT are. And therefore (by the co∣rollary of the 15. of this booke) the residue BE (which is a rationall line) is commensurable in length vnto the same line BZ. Wherefore also the line BG (which is commensurable in length vnto the line BE) shall also be commensurable in length vnto the same line EZ (by the 12. of the tenth). And it is proued that the line RZ is to the line ZT commensurable in power onely. Wherefore the right lines BZ and ZT are rationall commensurable in power onely. Wherefore the whole line BT is a binomiall [ 1] line (by the 36. of this booke). And for that as the line BG is to the line GD, so is the line BZ to the line ZT: therefore alternately (by the 16. of the fifth) the line BG is to the line BZ, as the line GD is to the line ZT. But the line BG is commensurable in length vnto the line BZ. Wherefore (by the 10. of this booke) the line GD is commensurable in length vnto the line ZT. Wherefore the names BG and [ 2] GD of the residuall line BD are commensurable in length vnto the names BZ and ZT of the binomi∣al line BT: and the line BZ is to the line ZT in the same proportion that the line BG is to the line GD [ 3] as before it was more manifest. And that they are of one and the selfe same order is thus proued. If the [ 4] greater or lesse name of the residuall line, namely, the right lines BG or GD be cōmensurable in length to any rationall line put: the greater name also or lesse, namely, BZ or ZT shalbe commensurable in length to the same rationall line put by the 12. of this booke. And if neither of the names of the residu∣all line be commensurable in length vnto the rationall line put, neither of the names of the binomiall line shalbe commensurable in length vnto the same rationall line put (by the 13. of the tenth). And if the greater name BG be in power more then the lesse name by the square of a line commensurable in length vnto the line BG, the greater name also BZ shalbe in power more then the lesse by the square of a line commensurable in length vnto the line BZ. And if the one be in power more by the square of a line incommensurable in length, the other also shalbe in power more by the square of a line incommen∣surable in length by the 14. of this booke. The square therefore of a rationall line. &c. which was requi∣red to be proued.

Notes

• * 1.1

  An other de∣monstratiō af∣ter Flussas.
• * 1.2

  Construction.
• * 1.3

  Demons••ra∣tion.