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 32. Theoreme. The 37. Proposition. If there be fower right lines proportionall: the Parallelipipedons descri∣bed of those lines, being like and in like sort described, shall be proportio∣nall. And i•• the Parallelipipedons described of them, being like and in like sort described, be proportionall: those right lines also shall be proportionall.

SVppose that these fower right lines AB, CD, EF, and GH, be proportionall, as AB is to CD, so let EF be to GH, and vpon the lines AB, CD, EF, and GH, describe these Parallelipipedons KA, LC, ME, and NG, being like and in like sort desc••••bed. Then I say, that as the solide KA is ••o the solide LC, so is the solide ME to the solide

[illustration]

NG.* 1.1 For forasmuch as the Pa∣rallelipipedon KA is like to the Parallelipipedon LC: therfore (by the 33. of the eleuenth) the solide KA is to the solide LC in treble proportion of that which the side AB is to the side CD••: and by the same reason the Parallelipipedon ME is to the Parallelipipedon NG in treble proportion of that which the side EF is to the side GH. Wherfore (by the 11. of the fift) as the Parallelipipedon KA is to the Parallelipipedon LC, so is the Parallelipipedon ME t•• the Parallelipipedon NG.

But now suppose, that as the Parallelipipedon KA is to

the Parallelipipedon LC, so is the Parallelipipedon ME to the Parallelipipedon NG. Then I say,* 1.2 that as the right line AB is to the right line CD, so is the right li•••• EF to the right line GH. For againe forasmuch as the solide KA is to the solide LC in treble proportion of that which the side AB is to the side CD, and the solide ME also is to the solide NG in treble proportion of that which the line EF is to the line GH, and as the solide KA is to the solide LC, so is the solide ME to the solide NG. Wherefore also as the line AB is to the line CD, so is the line EF to the line GH. If therefore there be fower right lines proportionall: the Parallelipipedons described of those lines, being like & in like sort described, shall be pro∣portionall. And if the Parallelipipedons described of them, and being like and in like sort de∣scribed, be proportionall: those right lines also shall be proportionall•• which was required to be proued.

Notes

• * 1.1

  Demonstrati∣on of the first part.
• * 1.2

  Demonstra∣tion of the second part, which is the conuerse of the first part.