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 44. Theoreme. The 62. Proposition. The square of a second bimediall line, applied vnto a rationall line: maketh the breadth or other side therof, a third binomiall lyne.

SVppose that AB be a second bimediall line, and let AB be supposed to be deuided into his partes in the point C, so that let AC be the greater part. And take a ra∣tionall line DE. And (by the 44. of the first) vnto the line DE apply the paralle∣logramme 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 third binomiall line. Let the selfe same construc∣tions be in this that were in the propositions next going before. And forasmuch as the line AB is a second bimediall line,* 1.1 and is deuided

[illustration]

into his partes in the point C,* 1.2 therfore (by the 38. of the tenth) the lines AC and CB are medials commensurable in power only, comprehēding a mediall superficies. Wher∣fore * 1.3 that which is made of the squares of the lines AC and CB added together, is mediall, and it is equall to the parallelo∣gramme DL by construction. Wherefore the parallelogramme DL is mediall, and is applied vnto the rationall line DE, wherfore (by the 22. of the tenth) the line MD is rationall and incommensurable in length to the line DE. And by the lyke reason also * 1.4 the line MG is rationall and incommensurable in length to the line ML, that is, to the line DE. Wherfore either of these lines DM and MG is rational, and incommensurable in length to the line DE. And foras∣much as the line AC is incommensurable in length to the line CB, but as the line AC is to the line CB, so (by the assumpt going before the 22. of the tenth) is the square of the line AC to that which is contained vnder the lines AC and CB. Wherfore the square of the line AC is inc••mmmensurable to that which is contayned vnder the lines AC and CB. Wherfore that * 1.5 that which is made of the squares of the lines AC and CB added together, is incommen∣surable to that which is contained vnder the lines AC and CB twise, that is, the parallelo∣gramme DL to the parallelogramme MF. Wherfore (by the first of the sixt, and 10. of the tenth) the line DM is incommensurable in length to the line MG. And they are proued both rationall,* 1.6 wherfore the whole line DG is a binomiall line by the definition in the 36. of the tenth. Now resteth to proue that it is a third binomiall line. As in the former propositions, so also in this may we conclude that the line DM is greater then the line MG, and that the line DK is commensurable in length to the line KM. And that that which is contained vnder the lines DK and KM is equall to the square of the line MN. Wherfore the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM, and neither of the lines DM nor MG is commensurable in length to the rational line DE. Wherfore (by the definition of a third binomi••ll line) the line DG is a third binomiall line: which was required to be proued.

¶Here follow certaine annotations by M. Dee, made vpon three places in the demonstration, which were not very euident to yong beginners.

† (The squares of the lines AC and C•• are medials (〈◊〉〈◊〉 i•• taught after the 21•• of this tenth) and ther∣••ore forasmuch as they are (by supposition) commēsurable th'one to the other: (by the 15. of the tēth)

the compound of them both is commensurable to ech part. But the partes are medials, therfore (by the co••ollary of the 23. of the tenth) the compound shall be mediall.

•• For that MX is equall (by construction) to that which is contayned vnder the lines AC and CB, which is proued mediall: therfore (by the corollary of the 23. of this tenth) MX is mediall, and ther∣fore (by the same corollary) his double MF is mediall. And it is applied to a rationall line, ML (beyng equall to D••) therfore by the 22. of the tenth, the line MG is rationall and incommensurable in length to ML, that is, to DE.

‡ Because the compound of the two squares (of the lines AC and C••) beyng commensurable one to the other, is also to eyther square (by the 15.) commensurable, therfore to the square of AC: But the square of AC is proued incommensurable to that which is contained vnder AC & CB once. Wherfore (by the 13. of the tenth) the compound of the two squares (of the lines AC and CB) is incommensu∣rable, to that which is cōtained vnder the lines AC and C•• once. But to that which is twise contained vnder the same lines AC and CB, the parallelogārme once contayned, is commensurable (for it is as 1. is to 2.) therfore that which is made of the squares of the lines AC and CB is incommensurable to the parallelogramme contained vnder AC and CB twise, by the sayd 13. of this tenth.

¶A Corollary.

* 1.7Hereby it is euident, that the squares made of the two partes of a second bi∣mediall line, composed, is a compound mediall, and that the same compound is in∣commensurable to the parallelogramme contayned vnder the two partes of the second bimediall lyne.

The proofe hereof, is in the first and third annotations here before annexed.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  †
• * 1.4

  *
• * 1.5

  ‡
• * 1.6

  DG, conclu∣ded a binomi∣all line.
• * 1.7

  A Corollary added by M. Dee.