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 89. Theoreme. The 113. Proposition. The square of a rational line applied vnto a residuall, maketh the breadth or other side a binomial line, whose names are commensurable to the names of the residuall line, and in the selfe same proportion: and moreouer that binomiall line is in the selfe same order of binomiall lynes, that the residual line is of residuall lynes.

SVppose that A be a rationall line, and BD a residuall line. And vnto the square of the line A let that which is contained vnder the lines BD and KH be equal. Wherfore the square of the rationall line A applied vnto the residuall line BD maketh the breadth or other side KH. Then I say that the line KH is a binomi∣all line, whose names are commensurable to

[illustration]

the names of the residuall line BD, and in the selfe same proportion, and that the line KH is in the selfe same order of binomiall lines, that the line BD is of residuall lines. Vnto the line BD let the line conueniently ioyned be DC.* 1.1 Wherfore the lines BC and DC are rationall commensurable in power onely. And vnto the square of the line A let the parallelogramme contained vnder the lines BC and G be equall. But the square of the line A is rationall.* 1.2 Wherfore the parallelogramme contained vnder the lines BC and G is also ra∣tionall. Wherfore also the line G is rationall and commensurable in length to the line BC (by the 20. of the tenth). Now forasmuch as the parallelogramme contained vnder the lines BC and G is equall to that which is contained vnder the lines BD and KH, therfore (by the 16. of the sixt) as the line BC is to the line BD, so i•• the line KH to the line G. But the line BC is greater then the line BD. Wherfore also the line KH is greater then the line G. Vnto the line G l••t the line KE be equall. Wherfore the line KE is rationall and cōmensurable in length to the line BC, as also the line G was (by the 12. of the tenth) And for that as BC is to BD, so is KH to KE. Wherfore by ••duersion of proportiō (by the corollary of y^e 19. of the fift) as BC is to DC, so is KH, to EH,* 1.3 ••KH into EH so let the line FH be to the line EF (how this is to be done, we will decare at the ende of this demonstration). Wherfore the residue KF is to the residue FH, as the whole KH is to the whole HE (by the 19. of the fift) that is, as the line BC is to the line CD. But the lines BC and CD are commensurable in power onely. Where∣fore also the lines KF and FH are commensurable in power only. And for that as KH is to HE, so is KF to FH, but as KH is to HE, so is also HF to FE, therfore as KF is to FH, so is FH to FE. Wherfore (by the corollary of the 19. of the sixt) as the first is to the third, so is the square of the first, to the square of the second. Wherefore as KF is to FE, so is the square of the line KF to the square of the line FH, but these squares are commensurable, for the lines KF and FH are commensurable in power. Wherfore the lines KF and FE are commensurable in length. Wherfore (by the second part of the 15. of the tenth) the lines KE and EF are commensurable in length. Wherfore (by the same) the lines KF and FE are commensurable in length. But the line KE is rationall and commensurable in length to the line BC: wherefore the line KF is also rationall and commensurable in lēgth to the line BC. And for that as the line BC is to the line CD, so it KF to EH, therfore alternately. (by the 16. of the fift) as BC is to KF, so is CD to FH. But the line BC is commensurable in length

to the line KF. Wherfore the line CD is commensurable in length to the line FH. But the line CD is rationall. Wherfore also the line FH is rationall. And the lines BC and CD are rationall commensurable in power onely. Wherfore the lines KF and FH are rationall com∣mensurable [ 1] in power onely. Wherfore the line KH is a binomiall line, whose names are com∣mensurable [ 2] to the names of the residuall line, and in the same proportion. I say moreouer that [ 3] it is a binomiall of the selfe same order of binomial lines, that the line BD is of residual lines. [ 4] For if the line BC be in power more then the line CD by the square of a line commensurable in length to the line BC, the line KF is also in power more then the line FH by the square of a line commensurable in length to the

[illustration]

line KF (by the 14. of the tenth). And if the line BC be commensurable in length to the rationall line put, the line KF is also (by the 12. of the tenth) commensurable in length to the rationall line, and so the [ 1] lyne BD is a first residuall lyne, and the line KH is in like sort a first binomiall line. If the line CD be commensurable in length to the rational line, the line FH is also commen∣surable [ 2] in length to the same line, and so the line BD is a second residuall line, and the line KH a second binomiall line. And if neither of the lines BC nor CD be commensurable in length to the rationall line, neither also of the lines KF nor FH is commensurable in length [ 3] to the same, and so the line BD is a third residuall line, and the line KH a third binomiall line. But if the line BC be in power more then the line CD by the square of a line incommen¦surable in length to the line BC, the line KF is in power more thē the line FH by the square of a line incommensurable in length to the line KF (by the 14. of the tenth) And if the line BC be commensurable in length to the rationall line put, the line KF is also commensura∣ble in length to the same line, and so the line BD is a fourth residuall line, and the line KH [ 4] a fourth binomiall line. And if the line CD be cōmēsurable in lēgth to the rational line, the [ 5] line FH is also cōmēsurable in lēgth to the same, & so the line BD is a fift residuall line, & the line KH a fift binomiall line. And if neither of the lines BC nor CD be commensurable in length to the rationall line, neither also of the lines KF nor FH is commensurable in [ 6] length to the same, and so the line BD is a sixt residuall line, and the line KH is a sixt bino∣miall line. Wherfore KH is a binomiall line, whose names KF and FH are commensurable to the names of the residuall line BD, namely, to BC and CD, and in the selfe same propor∣tion, and the binomiall line KH is in the selfe same order of binomiall lines, that the residu∣all BD, is of residuall lines. Wherefore the square of a rationall line applied vnto a residuall line, maketh the breadth or other side a binomiall line, whose names are commensurable to the names of the residuall line, and in the selfe same proportion, and moreouer the binomiall line is in the selfe same order of binomiall lines, that the residuall line is of residuall lines: which was required to be demonstrated.

The Assumpt confirmed.

Now let vs

[illustration]

declare how, as the line KH is to the line EH, so to make the line HF to the line FE. Adde vnto the line KH directly a line equall to HE, and let the whole line be KL, and (by the tenth of the sixt) let the line HE be deuided as the whole line KL is deuided in the point H: let the line HE be so deuided in the point F. Wher∣fore

as the line KH is to the line HL, that is, to the line HE, so is the line HF to the line FE.

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.* 1.4 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.* 1.5 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):* 1.6 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

  Construction.
• * 1.2

  Demons••ra∣tion.
• * 1.3

  An Assumpt.
• * 1.4

  An other de∣monstratiō af∣ter Flussas.
• * 1.5

  Construction.
• * 1.6

  Demons••ra∣tion.