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 Theoreme. 2.

What right line so euer, being deuided into two partes, hath those his two partes, proportionall, to the two segmentes of a line deuided by extreame and meane proportion: is also it selfe deuided by an ex∣treame and meane proportion: and those his two partes, are his two segments, of the sayd proportion.

Suppose, AB, to be a line deuided by an extreame and meane proportion in the point C, and AC to be the greater segment. Suppose also the right line DE, to be deuided into two partes, in the point F: and that the part DF, is to FE, as the segment AC, is to CB: or DF, to be, to AC, as FE is to CB. For so these pa••tes are proportionall, to the sayd segmentes. I say now, that DE is also deuided by an extreame and meane proportion in the point F. And that DF, FE, are his segmentes of the sayd pro∣portion. For, seing, as AC, is to CB: so is DF, to FE: (by supposition). Therfore, as AC, and CB (which is AB) are to CB: so is DF, and FE, (which is DE) to FE: by the 18. of the fifth. Wherefore (alternate∣ly) as AB is to DE: so is CB, to FE. And

[illustration]

therefore, the residue AC, is to the residue DF, as AB is to DE, by the fifth of the fift. And then alternately, AC is to AB, as DE, is to DF. Now therefore backward, AB is to AC, as DE is to DF. But as AB is to A∣C, so is AC to CB: by the third definition of the sixth booke. Wherefore DE is to DF, as AC is to CB: by the 11. of the fifth. And by suppositi∣on, as AC is to CB, so is DF to FE: wherefore by the 11. of the fifth, as DE is to DF: so is DF to FE. Wherefore by the 3. definition of the sixth, DE is deuided by an extreame and meane proportion, in the point F. Wherefore DF, and FE are the segmentes of the sayd proportion. Therefore, what right line so euer, being deuided into two partes, hath those his two partes, proportionall to the two seg∣mentes of a line deuided by extreame and meane proportion•• is also it selfe deuided by an extreme and meane proportion, and those his two partes are his two segmentes, of the sayd proportion•• which was requisite to be demonstrated.

Note.

Many wayes, these two Theoremes, may be demonstrated: which I leaue to the exercise of young studentes. But vtterly to want these two Theoremes, and their demonstrations: in so principall a line, or rather the chiefe piller of Euclides Geometricall pallace,* 1.1 was hetherto, (and so would remayne) a great disgrace. Also I thinke it good to note vnto you, what we meant, by one onely poynt. We m••••••••, that the quantities of the two segmentes, can not be altered, the whole line being once geuen. And though, from either end of the whole line, the greater segment may begin:* 1.2 And so as it were the point of secti∣on may seeme to be altered: yet with vs, that is no alteration: forasmuch as the quantities of the seg∣mentes, remayne all one. I meane, the quantitie of the greater segment, is all one: at which end so euer it be taken: And therefore, likewise the quantitie of the lesse segment is all one, &c. The like confidera∣tion may be had in Euclides tenth booke, in the Binomiall lines. &c.

Io••n Dee. 1569. Decemb. 18.

Notes

• * 1.1

  The chie••e line in all Euclides Geometrie.
• * 1.2

  What is ment here by, A secti∣on in one onely poi••t.