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

Here is the Assumpt (of the foregoing Proposition) confirmed.

Now let vs declare how as the line HF is to the line FE, so to make the line FK to the line EK. The line CD is greater then the line BD by supposition. Wherefore also the line HF is greater then the line FE (by alternate proportion, and the 14. of the fifth).* 1.1 From the line HF take away the line FL equall to the line FE. Wherefore the line remayning, name∣ly, HL, is lesse then the line HF, for the line HF

[illustration]

is equall to the lines HL & LF. As HL is to HF, so (by the 12. of the sixt) let FE be to FK.* 1.2 Wherfore by contrary proportion (by the Corolla∣ry of the 4. of the fifth) as HF is to HL, so is FK to FE. Wherefore by conuersion of pro∣portion (by the Corollary of the 19. of the fifth) as HF is to LF, that is, to the line equall vnto it, namely, to FE, so is the line FK to the line EK.

M. Dee, of this Assumpt, maketh (〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, that is, Acquisiuely,) a Probleme vniuersall, thus:

Two vnequall right lines being propounded, to adioyne vnto the lesse, a right line, which takē with [ 1] the lesse (as one right line) shall haue the same proportion, to the line adioyned, which, the greater of the two propounded, hath to the lesse.

The construction and demonstration hereof, is worde for worde to be taken, as it standeth here before: after these wordes: The line HF is greater then the line FE.

¶ A Corollary also noted by I. Dee.

It is therefore euident, that thus are three right lines (in our handling) in continuall proportion: [ 2] it is to weete, the greater, the lesse and the adioyned, make the first, the lesse with the adioyned, make the second: and the adioyned line is the third.

This is proued in the beginning of the demonstration, after the Assumpt vsed.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.