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 4. Probleme. The 27. Proposition. To finde out mediall lines commensurable in power onely, contayning a rationall parallelogramme.

LEt there be put two rationall lines commensurable in power onely, namely, A and B.* 1.1 And (by the 13. of the six) take the meane proportionall betwene the lines A and B, and let the same line be C. And as the line A is to the line B, so (by the 12. of the sixt) let the line C be to the line D.* 1.2 And forasmuch as A and B are rationall lines commensurable in power onely, therfore (by the 21. of the tenth) that which is contay∣ned vnder the lines A and B, that is, the square of the line C. For the square of the line C is equall to the parallelogramme contayned vnder the lines A an•• B (by the 17. of the sixth) is mediall, ••herfore C also is a mediall line. And for that

[illustration]

as the line A is to the line B, so is the line C to the line D, therfore as the square of the line A is to the square of the lyne B, so is the square of the line C to the square of the line D (by the 22. of the sixth). But the squares of the lines A and B are commensurable, for the li••••s A and B a••e supposed to be rationall commē∣surable in power onely. Wherefore also the squares of the lines C and D are commensurable (by the 10. of the tenth) wherfore the lines C and D are commensu∣rable in power onely. And C is a mediall line. Wherfore (by the 23. of the tenth) D also is a mediall line. Wherfore C and D are mediall lynes commensurable in power onely. Now also I say that they contayne a rationall parallelogramme. For for that as the line A is to the line B, so is the line C to the line D: therfore alternately also (by the 16. of the fift) as the line A is to the line C, so is the lyne B to the lyne D. But as the lyne A is to the lyne C, so is the line C to the lyne B: wherfore as the line C is to the line B, so is the line B to the lyne D. Wherfore the parallelogrāme cōtayned vnder the lines C and D is equal to the square of the line B. But the square of the lyne B is rationall. Wherfore the parallelograme which is contayned vnder the lynes C and D is also rationall. Wherfore there are found out mediall lines commensura∣bl••

in pow••r on••ly contayning a rationall parallelogramme•• which 〈◊〉〈◊〉 required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonst••a∣tion.