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 78. Theoreme. The 102. Proposition. The square of a lyne making with a mediall superficies, the whole superfi∣cies mediall applied to a rationall line, maketh the breadth or other side, a sixt residuall line.

SVppose that AB be a line making with a mediall superficies, the whole superficies mediall,* 1.1 and let CD be a rationall line. And vnto the line CD apply the paral∣lelogramme CE equall to the square of the line AB and making in breadth the line CF. Then I say that the line CF is a sixt residual line. For vnto the line AB let the line conueniently ioyned be BG. Wherfore the lines AG and BG are incommen∣surable in power hauing that which is made of their squares added together mediall, & that which is contained vnder them mediall, and moreouer that which is made of their squares added together is incommensurable to that which

[illustration]

is contained vnder them. Let the rest of the con∣struction be in this, as it was in the propositiōs go∣ing before.* 1.2 Wherfore the whole parallelogramme CL is mediall, (for it is equall to that which is made of the squares of the lines AG & GB added together, which is supposed to be mediall). Where∣fore the line CM is rationall and incommensura∣ble in length to the line CD: and in like manner the parallelogramme FL is mediall. Wherfore al∣so the line FM is rationall and incommensurable ••n length to the line CD. And forasmuch as that

which is made of the squares of the lines AG and GB added together, is incommensurable to that which is contained vnder the lines AG and GB twise, therefore the parallelogrāmes equall to them, namely, the parallelogrammes CL and FL are incommensurable the one to the other. Wherfore also the lines GM and FM are incommensurable in length, and they are both rational. Wherfore they are rationall cōmensurable in power only: Wherfore the line CF is a residuall line. I say moreouer that it is a sixt residuall line.* 1.3 Let the rest of the demonstra∣tion be as it was in the former propositions. And forasmuch as the lines AG and BG are in∣commensurable in power, therfore their squares, that is, the parallelogrammes which are e∣quall vnto them, namely, the parallelogrammes CH and KL are incommensurable the one to the other. Wherfore also the line CK is incommensurable in length to the line KM. Wher∣fore (by the 18. of the tenth) the line CM is in power more then the line FM by the square of a line incommensurable in length to the line CM. And neither of the lines CM nor FM is commensurable in length to the rationall line CD. Wherfore the line CF is a sixt residuall line. Wherfore the square of a line making with a medial superficies the whole superficies me∣diall applied to a rationall line, maketh the breadth or other side a sixt residuall line: which was required to be demonstrated.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  CF ••roued •• residuall.