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 18. Probleme. The 53. Proposition. To finde out a sixt binomiall line.

TAke two numbers AC & CB, and let thē be such that the number which is made of them both added together,* 1.1 namely, AB, haue to neither of the numbers AC or CB that proportion that a

[illustration]

square nūber hath to a square number. Take also any other number which is not a square number, and let the same be D. And let not the num∣ber D haue to any one of these num∣bers AB and AC that proportion that a square number hath to a square nūber. Let there be put moreouer a rationall line,

and let the same be E. And as the number D is to the number AB, so let the square of the line E be to the square of FG. Wherefore (by the 6. of the tenth) the line E is commensurable in power to the line FG, & the line E is rationall. Wherfore also the line FG is rationall. And for that the number D hath not ••o the number AB that proportion that a square nūber hath to a square number, therefore neither also shall the square of the line E haue to the square of the line FG that proportion that a square number hath to a square number. Wherefore the line FG is incommensurable in length to the line E. Againe, as the number BA is to the number AC, so let the square of the line FG be to the square of the line GH.* 1.2 Wherefore (by the 6. of the tenth) the square of the line FG is commensurable to the square of the line GH. And the square of the line FG is rationall. Wherefore the square of the line GH is also ra∣tionall. Wherefore also the line GH is rationall. And for that the number AB hath not to the number AC, that proportion that a square number hath to a square number: there••ore neither also hath the square of the line FG to the square of the line GH, that proportion that a square number hath to a square number. Wherefore the line FG is incommensurable in length to the line GH. Wherefore the lines FG and GH are rationall commensurable in power onely. Wherefore the whole line FH is a binomiall line. I say moreouer, that it is a sixt binomiall line. For for that as the number D is to the number AB, so is the square of the line E to the square of the line FG. And as the number BA is to the number AC, so is the square of the line FG to the square of the line GH. Wherefore of equalitie (by the 22. of the fift) as the number D is to the number AC, so is the square of the line E to the square of the line GH. But the number D hath not to the nūber AC that proportion that a square number hath to a square number. Wherefore neither also hath the square of the line E to the square of the line GH that proportion that a square number hath to a square number. Wher∣fore the line E is incommensurable in length to the line GH. And it is already proued, that the line FG is also incommensurable in length to the line E. Wherefore either of these lines FG and GH is incommensurable in length to the line E. And for that as the number ••A is to the number AC; so is the square of the line FG to the square of the line GH: therfore the square of the line FG is greater then the square of the line GH. Vnto the square of the line FG, let the squares of the lines GH and K be equall. Wherefore by euersion of propor∣tion, as the number AB is to the number BC, so is the square of the line FG to the square of the line K. But the number AB hath not to the number BC that proportion that a square number hath to a square number. Wherefore neither also hath the square of the line FG to the square of the line K that proportion that a square number hath to a square number. Wherefore the line FG is incommensurable in length vnto the line K. Wherefore the line FG is in power more then the line GH, by the square of a line incommensurable in length to it. And the lines FG and GH are rationall commensurable in power onely. And neither of the lines FG & GH is commensurable in length to the rationall line geuen, namely, to E. Wherefore the line FH is a sixt binomiall line: which was required to be found out.

¶A Corollary added out of Flussates.

By the 6. form••r Proposi•••••••••• it i•• manifest, ho•• 〈◊〉〈◊〉 divide any right line geuen into the names of euery one of the six•• foresayd binomiall lines.* 1.3 For if it be required to deuide a right line ge∣uen into a first binomiall line, then by the 48•• of this booke finde out a first binomiall line. And this right line being so found out deuided into his names, you may by the 10. of the sixt, deuide the right line geuen in like sort. And so in the other fiue following.

Although I here note vnto you this Corollary out of 〈…〉〈…〉, in very conscience and of gratefull ••inde•• I am enforced to certifie you, that, i•• any yeare••, before the trauailes of Flussas (vpō Eu••li•••••• Geo∣metricall Elementes) were published, the order how to deuide, not onely the 6. Binomiall lines into their names, but also to adde to the 6. Resid••••ls their due partes: ••nd f••rthermore to deuide all the o∣ther

irrational•• lines (of this tenth booke) into the partes distinct, of which they are composed: with many other straunge conclusions Mathematicall, to the better vnderstanding of this tenth booke and o∣ther Mathematicall bookes, most necessary, were by M. Iohn Dee inuented and demonstrated:* 1.4 as in his booke, whose title is Tyrocinium Mathematicum (dedicated to Petru•• Nonnius, An. 1559.) may at large appeare. Where also is one new arte, with sundry particular pointes, whereby the Mathematicall Sci∣ences, greatly may be enriched. Which his booke, I hope, God will one day allowe him opportunitie to publishe: with diuers other his Mathematicall and Metaphysicall labours and inuentions.

¶An Assumpt.

Is a right line be deuided into two partes how soeuer: the rectangle paral∣lelogramme contayned vnder both the partes, is the meane proportionall betwene the squares of the same parts. And the rectangle parallelogramme contained vnder the whole line and one of the partes, is the meane pro∣portionall betwene the square of the whole line and the square of the sayd part.

* 1.5Suppose that there be two squares AB and BC, and let the lines DB and BE so be put that they both make one right line. Wherefore (by the 14. of the first) the lines FB and BG make also both one right line. And make perfect the parallelogramme AC. Then I say, that the rectangle parallelogramme DG is the meane proportionall betwene the squares AB and BC: and moreouer, that the parallelogramme DC is the meane proportionall betwene the squares AC and CB. First the parallelogramme AG is a square. For forasmuch as the line DB is equall to the line BF, and the line BE vnto the line BG, therfore the whole line DE is equall to the whole line FG. But the line DE is equall to either of these lines AH & KC, and the line FG is equall to either of these lines AK and HC (by the 34. of the first).* 1.6 Wher∣fore the parallelogrāme AC is equilater, it is also rectan∣gle

[illustration]

(by the 29. of the first). Wherefore (by the 46. of the first) the parallelogrāme AC is a square. Now for that as the line FB is to the line BG, so is the line DB to the line BE. But as y^e line FB is to the line BG, so (by the 1. of the sixt) is the parallelogrāme AB, which is the square of the line DB, to the parallelogramme DG, and as the line DB is to the line BE, so is the same parallelogrāme DG to the parallelogramme BC, which is the square of the line BE. Wherefore as the square AB is to the pa∣rallelogramme DG, so is the same parallelogramme DG to the square BC. Wherefore the parallelogramme DG is the meane proportionall betwene the squares AB and BC. I say moreouer, that the paral∣lelogramme DC is the meane proportionall betwene the squares AC and CB. For for that as the line AD is to the line DK, so is the line KG to the line GC (for they are ech equall to eche). Wherefore by composition (by the 18. of the fift) as the line AK is to the line KD, so is the line KC to the line CG. But as the line AK is to the line KD, so is the square of the line AK, which is the square AC, to the parallelogramme cōtayned vnder the lines AK and KD, which is the parallelogramme CD: and as the line KC is to the line CG, so also is the parallelogramme DC to the square of the line GC, which is the square BC. Wherefore as the square AC is to the parallelogramme DC, so is the parallelogrāme DC to the square BC. Wherefore the parallelogramme DC is the meane proportionall betwene the squares AC and BC: which was required to be demonstrated.

¶An Assumpt.

Magnitudes that are meane proportionalls betwene the selfe same or e∣quall magnitudes, are also equall the one to the other.

Suppose that there be three magnitudes A, B, C.* 1.7

[illustration]

And as A is to B, so let B be to C. And likewise as the same magnitude A is to D, so let D be to the same magnitude C. Then I say that B and D are equall the one the other. For the proportion of A vnto C is double to that proportion which A hath to B (by the 10. definition of the fift) and likewise the selfe same proportion of A to C is (by the same definition) double to that proportion which A hath to D. But magnitudes whose equemultiplices are either equall or the selfe same, are also equall. Wherefore as A is to B, so is A to D. Wherefore (by the 9. of the fift) B and D are equall the one to the other. So shall if also be if there be other magnitudes equal to A and C, namely, E and F, betwene which let the magnitude D be the meane proportionall.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  A Corollary added by Flussates.
• * 1.4

  M. Dee his booke called Ty••••c••ni••m Mathemati∣cum.
• * 1.5

  This Assumpt as was before noted, f••ll••w∣eth most ••ri••f∣ly without farther demon¦stration of the 25. of this booke.
• * 1.6

  Demonstra∣tion.
• * 1.7

  An Assumpt.