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 53. Theoreme. The 71. Proposition. If two superficieces, namely, a rationall and a mediall superficies be cōposed together, the line which contayneth in power the whole superficies, is one of these foure irrationall lines, either a binomial line, or a first bimediall lyne, or a greater lyne, or a lyne contayning in power a rationall and a mediall superficies.

SVppose that there be two superficieces AB and CD,* 1.1 of which let the superficies AB be rationall, & the superficies CD mediall. Thē I say that the lyne contay∣ning in power y^• whole superficies AD, is either a binomial line, or a first bimedi∣all line, or a greater line,* 1.2 or a line cōtayning in power a rationall & a mediall su∣perficies. For the superficies AB is either greater or lesse thē the superficies CD for they cā by no meanes be equall, whē as the one is rationall, and the other medial). First let it be greater,* 1.3 and take a rationall line EF. And (by the 44. of the first) vnto the line EF apply the paral∣lelogramme EG equal to the superficies AB, & ma∣king

[illustration]

in breadth the line EH, and to the same line EF, that is, to the line HG apply the parallelogramme HI equall to the superficies DC, and makyng in breadth the line HK.* 1.4 And forasmuch as the super∣ficies AB is rationall, and is equall to the parallelo∣gramme EG, therfore the parallelogramme EG is also rational, and it is applied vnto the rational line EF, making in breadth the line EH. Wherfore the line EH is rationall and commensurable in length to the line EF (by the 20. of the tenth). Againe for¦asmuch as the superficies CD is mediall, and is e∣qual

to the parallelogrāme HI, therfore the parallelogrāme HI is also mediall, and is appli∣ed vnto the rationall line EF, that is, vnto the lyne HG making in breadth the line HK. Wherfore the lyne HK is rationall and incommensurable in length to the line EF (by the 2•• of the tenth.) And forasmuch as the superficies CD is mediall, and the superficies AB is ra∣tionall, therfore the superficies AB is incommensurable to the superficies CD. Wherfore also the parallelogramme EG is incommensurable to the parallelogramme HI. But as the paral∣lelogramme GE is to the parallelogramme HI, so (by the 1. of the sixt) is the line EH to the lyne HK. Wherfore (by the 10. of the tenth) the line EH is incommensurable in length to the line HK, and they are both rationall. Wherfore the lines EH and HK are rationall com¦mensurable in power onely. Wherfore the whole line EK is a binomiall line, and is deuided into his names in the poynt H. And forasmuch as the super••icies AB is greater then the su∣perficies CD, but the superficies AB is equall to the parallelogramme EG, and the superficies CD to the parallelogramme HI. Wherfore the parallelogramme EG is greater then the pa∣rallelogramme HI. Wherfore the line EH is greater then the line HK. Wherfore the line EH is in power more then the line HK either by the square of a line commensurable in length to the lyne EH, or by the square of a lyne incommensurable in length to the lyne EH.* 1.5 First let it be in power more by the square of a lyne cōmēsurable in lēgth vnto the line EH. Now the greater name, namely, EH is commensurable in length to the rational line geuen EF, as it hath already bene proued. Wherfore the whole line

[illustration]

EK is a first binomiall lyne. And the lyne EF is a rationall lyne. But if a superficies be contayned vn∣der a rationall line, and a first binomiall lyne, the lyne that contayneth in power the same superficies, is (by the 54. of the tenth) a binomiall line. Wherefore the lyne containing in power the parallelogramme EI is a binomiall line. Wherefore also the line contai∣ning in power the superficies AD is a binomiall line.

But now let the lyne EH be in power more then the line HK by the square of a line incommensurable in length to the line EH:* 1.6 now the greater name that is, EH is commensurable in length to the rationall line geuen EF. Wher∣fore the line EK is afourth binomiall line. And the line EF is rationall. But if a superfi∣cies be contained vnder a rationall line and afourth binomiall line, the line that containeth in power the same superficies is (by the 57. of the tenth) irrational, and is a greater line. Wher¦fore the line which containeth in power the parallelogramme EI is a greater line. Wherefore also the line containing in power the superficies AD is a greater lyne.

* 1.7But now suppose that the superficies AB which is rationall, be lesse then the superficies CD which is mediall. Wherfore also the parallelogramme EG is lesse then the parallelogrāme HI. Wherfore also the line EH is lesse then the line HK. Now the line HK is in power more then the lyne EH either by the square of a line cōmensurable in length to the line HK, or by the square of a lyne incommensurable in length vnto the lyne HK. First let it be in power more by the square of a line commensurable in length vnto HK:* 1.8 now the lesse name, that is EH is commensurable in length to the rationall line geuen EF, as it was before proued. Wherfore the whole line EK is a second binomiall line. And the line EF is a rationall line. But if a superficies be contained vnder a rationall line and a second binomiall lyne, the lyne that contayneth in power the same superficies, is (by the 55. of the tenth) a first bimediall line. Wherfore the line which contayneth in power the parallelograme EI is a first bimediall line. Wherfore also the line that containeth in power the superficies AD is a first bimediall lyne.

* 1.9But now let the line HK be in power more then the line EH, by the square of a line in∣cōmensurable

in length to the lyne HK, now the lesse name, that is, EH is cōmensurable in length to the rationall lyne geuen EF. Wherfore the whole line EK is a fift binomiall lyne. And the lyne EF is rationall. But if a superficies be contayned vnder a rationall lyne, and a fift binomiall lyne, the line that contayneth in power the same superficies, is (by the 58. of the tenth) a line containing in power a rationall and a mediall. Wherefore the lyne that contay∣neth in power the parallelogramme EI is a line contayning in power a rationall and a medi∣all. Wherfore also the lyne that containeth in power the superficies AD is a lyne contayning in power a rationall and a mediall. If therfore a rationall and a mediall superficies be added together, the lyne which contayneth in power the whole superficies, is one of these foure irrati∣onall lines, namely, either a binomiall line, or a first bimediall line, or a greater lyne, or a lyne contayning in power a rationall and a mediall: which was required to be demonstrated.

Notes

• * 1.1

  The seuent•• Senary.
• * 1.2

  Construction.
• * 1.3

  Two cases in this propositi∣on, of which ech also hath his two cases.

  First case.
• * 1.4

  Demonstra∣tion.
• * 1.5

  First part of the first case.
• * 1.6

  Second part of the first case.
• * 1.7

  The second case.
• * 1.8

  First part of the secōd case.
• * 1.9

  Second part of the secōd case.