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

Resolution of the 3. Theoreme.

Suppose that a certayne right line AB, be diuided by an extreame, and meane propor∣tion in the point C•• and let the greater segment thereof be the line AC, and let the halfe of the line AC, be the line CD. Then I say that the square of the BD is quintuple to the square of the line CD. For forasmuch as the square of the line BD, is quintuple to the square of the line CD. But the square of the line DB, is that which is contayned vnder the lines AB, and BC, together with the square of the line DC (by the 6. of the second). Wherefore that which is contayned vnder the lines AB, and BC,

[illustration]

together with the square of the line DC, i•• quintuple to the square of the line DC. Wherefore, that which is contayned vnder the lines AB, and BC, is quadruple to the square of the line DC. But vnto that which is contayned vnder the lines AB, and BC, is equall the square of the line AC: for the line AB, is diuided by an extreame and meane proportion in the point C. Wherefore the square of the line AC is quadruple to the square of the line DC: and so is it in deede, for the line AC is double to the line DC.