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 33. Theoreme. The 45. Proposition. A greater line is in one poynt onely deuided into his names.

LEt AB being a greater line be deuided into his names in the poynt C, so that let the lines AC and CB be rationall incommensurable in power, hauing that which is composed of the squares of the lines AC and CB rationall, and that which is con∣tained vnder the lines AC and CB mediall.* 1.1 Then I say that the line AB can not in any o∣ther poynt then in C be deuided into his names. For

[illustration]

if it be possible, let it be deuided into his names in the poynt D, so that let AD and DB be lines incommē∣surable in power, hauing that which is composed of the squares of the lines AD and DB rationall, and that which is contayned vnder the lines AD and DB mediall. Now forasmuch as how much the squares of the lines AC and CB di••••er from the squares of the lines AD and DB, so much differeth that which is contained vnder the lines AD and DB twise from that which is contained vnder the lines AC and CB twise, by those thinges which haue be••e sayd in the demonstration of the 42. propositi∣on. But the squares of the lines AC and CB exceede the squares of the lines AD and DB by a rationall super••icies (for they are either of them rationall). Wherfore that which is containe•• ••••der the lines AD and DB twise exceedeth that which is contained vnder the lines AC and CB twise by a rationall super••icies: when as either of them is a mediall super∣s••••••es. Which is impossible (by the 26. of the tenth). Wherefore a greater line is in one poynt on••ly deuided into his names: which was required to be proued.

Notes

• * 1.1

  Demonstrati∣on leading to an impossibi∣••••••e.