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 3. Probleme. The 11. Proposition. Vnto a right line first set and geuen (which is called a rationall line) to finde out two right lines incommensurable, the one in length onely, and the other in length and also in power.

SVppose that the right line first set and geuen, which is called a rationall line of purpose be A. It is required vnto the said line A, to finde out two right lines in∣commensurable, the one in length onely, the other both in length and in power.* 1.1 Take (by that which was added after the 9. proposition of this booke) two num∣bers B and C, not hauing that proportion the one to the other, that a square number hath to a square number, that is, let them not be like plaine numbers (for like plaine numbers by the 26. of the eight haue that proportion the one to the other that a square number hath to a square number). And as the number B is to the number C, so let the square of the line A be vnto the square of an other line, namely, of D (how to do this was taught in the assumpt put before the 6. proposition of this booke.) Wherfore the square of the line A, is vnto the square of the line D commensurable (by the sixt of the tenth.)

[illustration]

And forasmuch as the number B hath not vnto the num¦ber C, that proportion that a square number hath to a square nūber, therfore the square of the line A hath not vnto the square of y^e line D, that proportiō that a square number hath to a nūber. Wherfore by the 9. of the tenth, the line A is vnto the line D incommensurable in length onely. And so is found out the first line, namely, D incom∣mensurable in length onely to the line geuen A.* 1.2 Agayne take (by the 13. of the sixt) the meane proportionall be∣twene the lines A and D, and let the same be E. Wherfore as the line A is to the line D, so is the square of the line A to the square of the line E (by the Corollary of the 20. of the sixt). But the line A is vnto the line D incommen∣surable in length. Wherfore also the square of the line A is vnto the square of the line E incommensurable by the second part of the former proposition. Now forasmuch as the square of the line A is incōmē∣surable to the square of the line E, it followeth (by the definition of incommensurable lynes) that the line A is incommensurable in power to the line E. Wherfore vnto the right line ge∣uen, and first set, A, which is a rationall line, and which is supposed to haue such diuisions and so many partes as ye list to conceyue in minde, as in this example 11, whereunto, as was de∣clared in the 5. definition of this booke, may be compared infinite other lines, either commen∣surable

or incommensurable, is found out the line D incommensurable in length onely. Wher¦fore the line D is rationall (by the sixt definitiō of this booke) for that it is incommensura∣ble in length onely to the line A, which is the first line set, and is by suppositiō rational. There is also found out the line E, which is vnto the same line A incommensurable, not onely in length but also in power, which line E compared to the rationall line A, is by the definition irrationall. For Euclide alwayes calleth those lines irrationall, which are incommensurable both in length and in power to the line first set, and by supposition rationall.

Notes

• * 1.1

  To finde out the first line incommensu∣rable in length onely to the line geuen.
• * 1.2

  To finde out the second line incommensu∣rable both in length and in power to the line geuen.