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 2. Probleme. The 10. Proposition. To deuide a right line geuē not deuided, like vnto a right line geuen beyng deuided.

SVppose that the right line geuen not deuided be AB, and the right lyne geuen being deuided, let be AC. It is required to deuide y^e line AB which is not deuided like vnto the line AC which is deuided.* 1.1 Suppose the lyne AC be deuided in the pointes D and E, & let y^• lines AB & AC so be put, that they make an angle at all aduentures, and draw a line from B to C, and by the pointes D and E draw vnto the line BC (by the 31. of the first) two parallel lines DF and EG: and by the point D vnto

[illustration]

the line AB (by the same) draw a parallel line DHK. Wherfore either of these figure•• FH and HB are parallelogrammes. Wherfore the line DH is equall vnto the line FG, and the line HK is equall vnto the line GB.* 1.2 And be∣cause to one of the sides of the triangle DKC, namely, to the side KC is drawn a paral∣lel line HE, therefore the line CE (by th•• 2. of the sixt) is in proportion vnto the line ED as the line KH is to the line HD: but the line KH is equall vnto the line BG, and the line HD is equal vnto the line GF. Wher¦fore (by the 11. of the fift) as CE is vnto ED, so is BG to GF. Agayne because to one of the sides of the triangle AGE, namely, to GE is drawn a parallel lyne FD, therfore the line ED (by the 2. of the sixth) is in proportion vnto the lyne DA, as the line GF is to the line FA. And it is already proued that as CE is to ED, so is BG to GF. VVherfore as CE is to ED, so is BG to GF, and as ED is to DA, so is GF to FA. VVherfore the right line geuen not deuided, name¦ly, AB is deuided like vnto the right line geuen being deuided, which is AC: which was required to be done.

¶ A Corollary out of Flussates.* 1.3

By this Proposition we may deuide any right line geuen, accordyng to the pro∣portion

of any right lynes geuen. For let those right lynes hauyng proportion be ioyned together directly, that they may make all one right lyne, and then ioyne them to the lyne geuen anglewise. And so proceede as in the proposition, where you see that the right line geuen AB is deuided into the right lynes AF, FG and GB which haue the selfe same proportion that the right lines AD, DE, and EC haue.

By this and the former proposition also may a right line geuen be easily deui∣ded into what partes so euer you will name.* 1.4 As if you will deuide the line AB in∣to three equall partes, let the lyne DE be made equall to the lyne AD, and the lyne EC made equall to the same by the third of the first. And then vsing the selfe same maner of construction that was before: the lyne AB shall be deuided into three equall partes. And so of any kynde of partes whatsoeuer.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  A Corollary out of Flussates.
• * 1.4

  By this and the former propo••ition may a right line be deui∣ded into what partes soeuer you will.