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 11. Theoreme. The 14. Proposition. If there be sower right lines proportionall, and if the first be in power more then the second by the square of a right line commensurable in length vnto the first, the third also shalbe in power more then the fourth, by the square of a right line commensurable vnto the third. And if the first be in po∣wer more then the second by the square of a right line incommensu∣rable in length vnto the first, the third also shall be in power more then the fourth by the square of a right line incommensurable in length to the third.

SVppose that these foure right lines A, B, C, D, be proportionall. As A is to B, so let C be to D. And let A be in power more then B, by the square of the line E. And likewise let C be in power more then D, by the square of the line F.* 1.1 Then I say that if A be commensurable in length vnto the line E, C also shall be commen∣surable in length vnto the line F. And if A be incommen∣surable

[illustration]

in length to the line E, C also shall be incommen∣surable in length to the line F. For for that as A is to B, so is C to D, therefore as the square of the line A is to the square of the line B, so is the square of the line C to the square of the line D (by the 22. of the sixt). But by suppo∣sition vnto the square of the line A are equall, the squares o•• the lines E and B, and vnto the square of the line C are equall the squares of the of the lines D and F: Where∣fore as the squares of the lines E and B (which are equall to the square of the line A) are to the square of the line B, so are the squares of the lines D and F (which are equall to the square of the line C) to the square of the line D (by the seuenth of the fift). Wherfore (by the 17. of the fift) as the square of the line •• i•• to the square of the line B, so is the square of the line F to the square of the line D. Wherfore also as the line E is to the line ••, so is the line F to the line D (by the second parte of the 22. of the sixt) wherefore contrari∣wise (by the Corollary of the fourth of the fift) as B is to E so is D to F. But (by supposition)

••s A is to B, so is C to D, Wherfore of equallitie (by the 22. of the fift) a•• A is to ••, so is C is F. If therfore A be commensurable in length vnto E, C also shall be comm••nsurable in lēgth vnto F: and if it ba incommensurable in length vnto E, C also shalbe incommensurablel in length vnto F, by the 10. of this booke. If therfore there be foure right tlines proportionall, and if the first be in power more then the secondby the square of a right line commensurable in length vnto the first, the third also shall be in power more then the fourth, by the square of a right line commensurable in length vnto the third and if the first be in power more thē the second, by the square of a right line incōmensurable in length vnto the first, the third also shall be in power more the the fourth, by the square of a right line incommensurable in length to the third: which was required to be proued.

Note that the line A may be proued to be in proportion to the line E, as the line C is to the line F, by an other way, namely, by conuersion of proportion (of some•• as we haue before noted,* 1.2 called inuerse proportion) by the 19. of the fift. For, forasmuch as the foure lines A, B, C, D, are proportionall: ther∣fore (by the 22. of the sixt) their squares also are proportionall. And forasmuch as the antecedent, name∣ly, the square of the line A excedeth the consequent, namely, the square of the line B, by the square of the line E: and the other antecedent, namely, the square of the line C, excedeth the other consequent, namely, the square of the line D, by the square of the line F, therefore as the square of the line A is to the excesse, namely, to the square of the line E, so is ••he square of the line C to the excesse; namely, to the square of the line F. Wherefore (by the second part of the 22. of the sixt) as the line A is to the line E, so is the line C to the line F.

Notes

• * 1.1

  Demonstra∣tion.
• * 1.2

  An other way to proue that the lines A, E, C, F, are pro∣portionall.