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. 1.

Vpon Euclides third proposition demonstrated, it is made euident: that, of a line deuided by ex∣treame and meane proportion, if you produce the lesse segment, equally to the length of the greater: the line therby adioyned, together with the sayd lesse segment, make a new line deuided by extreame and middle proportion: Whose lesse segment, is the line adioyned.

For, if AB, be deuided by extreme and middell proportion in the point C, AC, being the greater segment, and CB be produced, from the poynt B, making a line, with CB, equall to AC, which let be CQ: and the line thereby adioyned, let be BQ: I say that CQ, is a line also deuided by an extreame and meane proportion, in the point B: and that BQ (the line adioyned) is the lesse segment. For by the thirde, it is proued, that halfe AC, (which, let be, CD) with CB, as one line, composed, hath his powre or square, quintuple to the powre of the

[illustration]

segment CD: Wherfore, by the second of this booke, the double of CD, is de∣uided by extreme and middell propor∣tion•• and the greater segment thereof, shalbe CB. But, by construction, CQ, is the double of CD, for it is equall to AC. Wherefore CQ is deuided by extreme and middle propor∣tion, in the point B: and the greater segment thereof shalbe, CB. Wherefore BQ, is the lesse segment, which is the line adioyned. Therefore, a line being deuided, by extreme and middell proportion, if the lesse segment, be produced equally to the length of the greater segment, the line thereby adioyned to∣gether with the sayd lesse segment, make a new line deuided, by extreme & meane proportion, who••e lesse segment, is the line adioyned. Which was to be demonstrated.