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

A Corollary added by Campane.

Hereby it is 〈…〉〈…〉 from the grea•••••• 〈◊〉〈◊〉 of a line deuided by an extreame & meane proportion, be 〈◊〉〈◊〉 away 〈◊〉〈◊〉 segment: the sayd great a segment shall be deuided by an extreame and meane proportion, and the greater segment thereof shall be the line taken away.

As let the line ••••, be deuided by an extreame and mean•• proportion, in the point C. And le•• the 〈…〉〈…〉 line 〈…〉〈…〉 A. D. I say that A∣C is also deuided by an extreame and mean•• proportion in the point D, and that his greater portion is DC. For, by the definiti∣no (of a line so deuided) AB, is to AC, as AC is to CB. But as AC is to CB, so is AC to DC, by the 7. of the 〈…〉〈…〉 is equall to CB) wherefore, by the 11. of the fifth, as AB is to AC, so is AC to CD: and therefore by the 19. of the fifth, as AB is to AC, so is the residue CB, to the residue AD. But CB is to AD, as DC is to AD (by the 7. of the fifth) for DC is by construction equall to C••. Wherefore, 〈…〉〈…〉 so by the definition of A line de¦uided by an extreame and meane 〈…〉〈…〉 the point D, to be deuided, by an ex∣treame and meane proportion: which was to be proued.

Two Corollaries (added by M. Dee) following chiefely vpon the veritie, and demonstration of his Additions, vnto the 3. porposition annexed, and partely vpon this fifth, by Euclide demonstrated.

Notes

• This is before demonstrated most euidently and briefly by M. Dee, after the 3. proposi∣tion.