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

¶ An other Probleme added by Pelitarius.

Betwene two rectiline Superficieces, to finde out a meane superficies proportionall.

* 1.1Suppose that the two superficieces be A and B, betwene which it is required to place a meane superficies proportionall. Reduce the sayd two rectiline figures A and B

vnto two like parallelogrāmes (by the 18. of this booke) or if you thinke good reduce eyther of them to a square, (by the last of the second). And let the said two parallelo∣grammes like the one to the other and equall to the superficieces A and B, be CDEF and FGHK. And let the angles F in either of them be equall, which two angles let be placed in such sort, that the two parallelogrammes ED and HG may be about one and the selfe same dimetient CK (which is done by putting the right lines EF and FG in such sort that they both make one right line, namely,

[illustration]

EG). And make cōplete the parallelogrāme CLK M. Then I say, that either of the supplements FL & FM is a meane proportionall betwene the superficieces CF & FK, that is, betwene the superficieces A and B: name∣ly, as the superficies HG is to the superficies FL, so is the same superficies FL to the superficies ED. For by this 24. Proposition the line HF is to the line FD, as the line GF is to the line FE. But (by the first of this booke) as the line HF is to the line FD, so is the su∣perficies HG to the superficies FL: and as the line GF is to the line FE, so also (by the same) is the superficies FL to the superficies ED. Wherfore (by the 11. of the fift) as the superficies HG is to the superficies FL, so is the same superficies FL to the superficies ED: which was required to be done.

Notes

• * 1.1

  Another ad∣dition of Pelitarius.