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 45. Theoreme. The 63. Proposition. The square of a greater line applied vnto a rationall line, maketh the breadth or other side a fourth binomiall line.

SVppose that the line AB be a greater line, and let it be supposed to be deuided into his partes in the point C, so that let AC be the greater part. And take a rationall line DE. And (by the 44. of the first) vnto the line DE, apply the parallelogramme DF equall to the square of the line AB, and making in breadth the line DG. Then I say, that the line DG is a fourth binomiall line. Let the selfe same construction be in this, that was in the former Propositions.* 1.1 And for∣asmuch

[illustration]

as the line AB is a greater line, & is deuided into his partes in the point C:* 1.2 therefore the lines AC and CB are incom∣mensurable in power, hauing that which is made of the squares of them added toge∣ther rationall, and the parallelogramme which is contayned vnder them, mediall. Now forasmuch as that which is made of the squares of the lines AC and CB added together is rationall, therefore the paralle∣logramme DL is rationall. Wherefore al∣so the line MD is rationall and commen∣surable [ 1] in lēgth to the line DE (by the 20. of this tenth). Againe forasmuch as that which is cōtained vnder the lines AC and CB twise is mediall, that is, the parallelogrāme

MF, and it is applied vnto the rationall line ML, therefore (by the ••2. of the tenth) the [ 2] line MG is rationall and incommensurable in length to the line DE, Therefore (by the 13. of the tenth) the line DM is incommensurable in length to the line MG. Wherefore the lines DM and MG are rationall commensurable in power onely. Wherfore the whole [ 3] line DG is a binomiall line. Now resteth to proue, that it is also a fourth binomiall line. Euen as in the former Propositions, so also in this may we conclude, that the line DM is greater then the line MC. And that that which is contayned vnder the lines DK and KM is equall to the square of the line MN. Now forasmuch as the square of the line AC is incommensurable to the square of the line CB, therefore the parallelogramme DH is incommensurable to the parallelogramme KL. Wherefore (by the 1. of the sixt, and 10. of the tenth) the line DK is incommensurable in length to the line KM. But if there be two vnequall right lines, and if vpon the greater be applied a pa∣rallelogramme equall to the fourth part of the square made of the lesse, and wanting in figure [ 4] by a square, and if also the parallelogramme thus applied deuide the line wherupon it is ap∣plied into partes incommensurable in length, the greater line s••all be in power more then the lesse, by the square of a line incōmensurable in length to the greater (by the 18. of the tenth). Wherefore the line DM is in power more then the line MG, by the square of a line incō∣mensurable in length to DM. And the lines DM and MG are proued to be rationall cō∣mensurable in power onely. And the line DM is commensurable in length to the rationall line geuen DE. Wherefore the line DG is a fourth binomiall line: which was required to be proued.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.