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 49. Theoreme. The 67. Proposition. A line commensurable in length to a bimediall line, is also a bimediall lyne and of the selfe same order.

SVppose that the line AB be a bimediall line, And vnto the line AB, let the lyne CD be commensurable in length. Then I say that the line CD is a bimediall line, and of the self order that the line AB is. Deuide the line AB into his partes in the point E.* 1.1 And forasmuch as the line AB is a bimediall line, and is deuided into his partes in the point E, therfore (by the 37. and 38. of the tenth) the lines AE and EB are medials commensurable in power onely, And (by the 12. of the sixt) as the line AB is to the line CD, so let the line AE be to the line CF.

[illustration]

Wherfore (by the 19. of the fift) the residue,* 1.2 name¦ly, the line EB is to the residue, namely, to the line FD, as the line AB is to the line CD. But the line AB is commensurable in length to the lyne CD. Wherfore the line AE is commensurable in length to the line CF, and the line EB to the line FD. Now the lines AE and EB are me∣diall, wherfore (by the 23. of the tenth) the lines CF and FD are also mediall. And for that as the line AE is to the line EB, so is the line CF to the line FD. But the lines AE and EB are commensurable in power onely, wherfore the lines CF and FD are also commensura∣ble in power onely. And it is proued that they are mediall. Wherfore the lyne CD is a bimedi∣all [ 1] line. I say also that it is of the selfe same order that the line AB is. For, for that as the line AE is to the line EB, so is the line CF to the line FD, but as the line CF is to FD, so is the square of the lyne CF to the parallelogramme contained vnder the lynes CF and FD, by the first of the sixt. Therfore as the line AE is to the line EB, so (by the 11. of the fift) is the square of the line CF to the parallelogramme contained vnder the lines CF and FD: but as AE is to EB, so by the 1. of the sixt, is the square of the line AE, to the parallelogramme contained vnder the lines AE and EB, therfore (by the 11. of the fift) as the square of the line AE is to that which is contained vnder the lines AE and EB, so is the square of the line CF to that which is contained vnder the lines CF and FD. Wherfore alternately (by the 16. of the fift) as the square of the line AE is to the square of the line CF, so is that which is contained vnder the lines AE and EB to that which is contained vnder the lines CF & FD. But the square of the line AE is commensurable to the square of the line CF, because AE and CF are commensurable in length. Wherfore that which is contained vnder the lines AE and EB in commensurable to that which is contained vnder the lines CF and FD. If therfore that which is contained vnder the lines AE and EB be rationall, that is, if the line

AB be a first bimediall line, that also which is contained vnder the lines CF and FD is ra∣tionall. Wherfore also the line CD is a first bimediall line. But if that which is contained vn∣der [ 3] the lines AE and EB be mediall, that is, if the line AB be a second bimediall line, that also which is contayned vnder the lines CF and FD is mediall: wherfore also the line CD is a second bimediall line. Wherfore the lines AB and CD are both of one and the selfe same order: which was required to be proued. [ 4]

¶ A Corollary added by Flussates: but first noted by P. Monta••reus.

A line commensurable in power onely to a bimediall line, is also a bimediall line, and of the selfe same order.

Suppose that AB be a bimediall line, either a first or a second, wherunto let the line GD be cōmen∣surable in power onely.* 1.3 Take also a rationall line EZ, vpon which (by the 45. of the first) apply a rectan∣gle parallelogramme equall to the square of the line AB, which let be EZFC, and let the rectangle parallelogramme CFIH be equall to the square of the line GD. And forasmuch as vpon the rationall line EZ is applyed a rectangle parallelogramme EF

[illustration]

equall to the square of a first bimediall line, therefore the other side therof, namely, EC, is a second bino∣miall line, by the 61. of this booke. And forasmuch as by supposition the squares of the lines AB & GD are commensurable, therefore the parallelogrammes EF and CI (which are equall vnto them) are also commensurable. And therefore by the 1. of the sixt, the lines EC and CH are commensurable in length. But the line EC is a second binomiall line. Where∣fore the line CH is also a second binomiall line, by the 66. of this booke. And forasmuch as the super∣ficies CI is contayned vnder a rationall line EZ or CF, and a second binomiall line CH, therefore the line which contayneth it in power, namely, the line GD is a first bimediall line, by the 55. of this booke. And so is the line GD in the selfe same order of bi∣mediall lines that the line AB is. The like demonstration also will serue if the line AB be supposed to b•• a second bimediall line. For so shall it make the breadth EC a third binomiall line whereunto the line CH shall be commensurable in length, and therefore CH also shall be a third binomiall line, by meanes whereof the line which contayneth in power the superficies CI, namely, the line GD shall also be a second bimediall line. Wherefore a line commensurable either in length, or in power onely to a bimediall line, is also a bimediall line of the selfe same order.

But so is it not of necessitie in binomiall lines, for if their powers onely be commensurable, it fol∣loweth not of necessitie that they are binomialls of one and the selfe same order,* 1.4 but they are eche bi∣nomialls eyther of the three first kindes, or of the three last. As for example. Suppose that AB be a first binomiall line, whose greater name let be AG, and vnto AB let the DZ be cōmēsurable in po∣wer onely. Then I say, that the line DZ is not of the selfe

[illustration]

same order that the line AB is. For if it be possible, let the line DZ be of the selfe same order that the line AB is. Whe••efore the line DZ may in like sort be deuided as the line AB is, by that which hath bene demonstrated in the 66. Proposition of this booke•• let it be so deuided in the poynt E. Wherefore it can not be so deuided in any other poynt, by the 42•• of this booke. And for that the line AB •••• to the line DZ, as the line AG is to the line DE, but the lines AG & DE, namely, the greater names, are com∣mensurable in length the one to the other (by the 10. of this booke) for that they are commensurable in length to 〈◊〉〈◊〉 and the selfe same rationall line, by the first definition of binomiall lines. Wherefore the lines AB and DZ are commensurable in length, by the 13. of this booke. But by supposition they are commensurable in power onely: which is impossible.

The selfe same demonstration also will serue, if we suppose the line AB to be a second binomial line: for the lesse names GB and EZ being commensurable in length to one and the selfe same ratio∣nall line, shall also be commensurable in length the one to the other. And therefore the lines AB and DZ which are in the selfe same proportion with them, shall also be commensurable in length the one to the other: which is contrary to the supposition. Farther, if the squares of the lines AB and DZ be applyed vnto the rationall line CF, namely, the parallelogrammes CT and HL, they shall make the breadthes CH and HK first binomiall lines, of what order soeuer the lines AB & DZ (whose squares were applyed vnto the rational line) are, (by the 60. of this

[illustration]

booke). Wherefore it is manifest, that vnder a rationall line and a first binomiall line, are confusedly contayned all the powers of binomiall lines (by the 54. of this booke). Wherfore the onely commensuration of the powers doth not of necessitie bryng forth one and the selfe same order of binomiall lines. The selfe same thyng also may be pro∣ued, if the lines AB and DZ be supposed to be a fourth or fifth binomiall line, whose powers onely are conmmensu∣rable, namely, that they shall as the first bring forth bino∣miall lines of diuers orders. Now forasmuch as the pow∣ers of the lines AG and GB, and DE and EZ are com∣mensurable & proportionall, it is manifest, that if the line AG be in power more then the line GB by the square of a line commensurable in length vnto AG, the line DE also shall be in power more then the line EZ by the square of a line commensurable in length vnto the line DE (by the 16. of this booke). And so shall the two lines AB and DZ be eche, of the three first binomiall lines. But if the line AG be in power more then the line GB by the square of a line incommensurable in length vnto the line AG, the line DE shall also be in pow••r 〈◊〉〈◊〉 then the line EZ by the square of a line incomensurable in length vnto the line DE, by the self•• same P••oposition. And so shall eche of the lines AB and DZ be of the three last binomiall lines. But why it is not so in the third and sixt binomiall lines, the reason is: For that in them neither of the name•• is commensurable in length to the rationall line put FC.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  A Coroll••ry add••d by Flussetes.
• * 1.4

  Note.