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 19. Theoreme. The 22. Proposition. If vpon a rationall line be applied the square of a mediall line: the other side that maketh the breadth thereof shalbe rationall, and incommensura∣ble in length to the line wherupon the parallelograme is applied.

SVppose that A be a mediall line, and let BC be a line rationall, and vpon the line BC describe a rectangle parallelograme equall vnto the square of the line A, and let the same be BD making in breadth the line CD. Then I say that the line CD is rationall and incōmensurable in length vnto the line CB.* 1.1 For forasmuch as A is a mediall line, it containeth in power (by the 21. of the tenth) a rectangle parallelograme comprehen∣ded vnder rationall right lines commensurable in power onely. Suppose that is containe in power the parallelograme GF: and by supposition it also containeth in power the parallelo∣grame BD. Wherefore the parallelo∣grame

[illustration]

BD is equall vnto the parallelo∣grame GF: and it is also equiangle vn∣to it, for that they are ech rectāgle. But in parallelogrames equall and equiangle the sides which containe the equall an∣gles, are reciprocall (by the 14. of the sixt): Wherfore what proportiō the line BC hath to the line EG, the same hath the line EF to the line CD. Therefore (by the 22. of the sixt) as the square of the line BC is to the square of the line EG, so is the square of the line EF to the square of the line CD. But the square of the line BC is commensurable vnto the square of the line EG (by supposition). For either of them is rationall. Wherefore (by the the 10. of the tenth) the square of the line EF is commensurable vnto the square of the lin•• CD. But the square of the line EF is rationall. Wherefore the square of the line CD is likewise ratio∣nall. Wherefore the line CD is rational. And forasmuch as the line EF is inco••mensurable in length vnto the line EG (for they are supposed to be commensurable in power onely). But as the line EF is to the line EG, so (by the assumpt going before) is the square of the line EF to the parallelograme which is contained vnder the lines EF and EG. Wherefore (by the 10. of the tenth) the square of the line EF is incommensurable vnto the parallelograme which is contained vnder the lines FE and EG. But vnto the square of the line EF the square of the line CD is commensurable, for it is proued that ••ither of them is

a rationall lin••. And that which is contained vnder the lines DC and CB is commensurable vnto that which is contained vnder the lines FE and EG. For they are both equall to the square of the line A. Wherefore (by the 13. of the tenth) the square of the line CD is incom∣mensurable to that which is contained vnder the lines DC and CB. But as the square of the line CD is to that which is contained vnder the lines DC and CB, so (by the assumpt going before) is the line DC to the line CB. Wherefore the line DC is incommensurable in length vnto the line CB. Wherefore the line CD is rationall and incommensurable in length vnto the line CB. If therefore vpon a rationall line be applied the square of a mediall line, the o∣ther side that maketh the breadth thereof shalbe rationall, and incommensurable in length to the line whereupon the parallelogramme is applied: which was required to be proued.

A square is sayd to be applied vpon a line, when it, or a parallelograme equall vnto it, is applied vpon the sayd line.* 1.2 If vpon a rationall line geuen we will apply a rectangle parallelograme equall to the square of a mediall line geuen, and so of any line geuen, we must, by the 11. of the sixt, finde out the third line proportionall with the rationall line and the mediall line geuen: so yet that the rationall line be the first, and the mediall line geuen, (which containeth in power the square to be applied) be the second. For then the supe••ficies contained vnder the first and the third, shalbe equall to the square of the midle line, by the 17. of the sixt.

Notes

• * 1.1

  〈◊〉〈◊〉
• * 1.2

  How a square is sayde to be applied vppon a line.