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 16. Theoreme. The 22. Proposition. If there be foure right lines proportionall, the rectiline figures also described vpon them beyng lyke, and in like sorte situate, shall be proportional. And if the rectiline figures vppon them described be proportional, those right lynes also shall be pro∣portionall.

SVppose there be foure right lines AB, CD, EF, and GH, and as AB is to CD, so let EF be to GH. And vpon the lines AB and CD (by the 1••. of the sixth) let there be described two rectiline figures KAB, and LCD like the one to the other, and in like sort situate. And vpon the lynes EF and GH (by the same) let there be described also two rectiline figures MF and NH like the one to the other, and in like sorte situate. Then I say that as the ••igure KAB is

[illustration]

to the figure LCD, so is the figure MF to the figure NH. Vnto the lines AB and CD (by the 11. of the sixth) make a third lyne in propor∣tion, namely, O: and vnto the lines EF and GH in like sort make a third lyne in a line proportion, namely, P. And for that as the line AB is to the line CD, so is the line EF to the line GH, but as the line CD is to the line O, so is the line GH to the lyne P. Wherfore of equality (by the 22. of the fifth) as the lyne AB is vnto the line O, so is the lyne EF to the line P. But as the line AB is to the line O, so is the figure KAB to the figure LCD (by the second corollary of the 20. of the sixth). And as the line EF is to the lyne P, so is the figure M F to the figure NH. Where∣fore (by the 11. of the fifth) as the figure KAB is to the figure LCD, so is the fi∣gure M F to the figure NH.

But now suppose that as the figure KAB is to the figure LCD, so is the figure M F to the figure NH, then I say that as the line AB is to the line CD, so is the line EF to the line GH. As the line AB is to the lyne CD, so (by the 1••. of the sixth) let the lyne EF be to the lyne QR, and vpon the lyne QR (by the 18. of the sixth) describe vnto either of these figures MF and NH a like fi∣gure, and in like sort situate SR. Now forasmuch as the lyne AB is to the lyne CD, so is the lyne EF to the line QR, and vpon the lines AB and CD are de∣scribed two figures lyke, and in like sort situate KAB and LCD, and vpon the lines EF and QR are described also two figures like, and in like sort situate MF and SR, therfore as the figure KAB is to the figure LCD, so is the figure MF to the figure SR: wherfore also (by the 11. of the fifth) as the figure MF is to the figure SR, so is the figure MF to the figure NH, wherfore the figure

M F hath to either of these figures NH, and SR one and the same proportion, wherfore by the 9. of the fifth, the figure NH is equal vnto the figure SR. And it is vnto it like, and in like sort situate. But in like and equall rectiline figures beyng in like sort situate, the sides of like proportion on which they are described are equall. Wherfore y^e line GH is equall vnto the line QR. And because as the lyne AB is to the line CD, so is the line EF to the line QR, but the line QR is equall vnto the line GH, therfore as the line AB is to he line CD, so is the line EF to the line GH.

[illustration]

If therefore there be foure right lines proportionall, the rectiline figures al∣so described vpon them beyng like and in lyke sort situate shall be proportionall And if the rectiline figures vpon them described beyng like and in like sort situ∣ate be proportionall, those right lines also shall be proportional: which was required to be proued.

An Assumpt.

And now that in like and equall figures, being in like sort situate, the sides of like proportion are also equall (which thing was before in this proposition taken as graunted) may thus be proued. Suppose y^t the rectiline figures NH and SR be equall and like, and as HG is to GN, so let RQ be to QS, and let GH and QR be sides of like proportion. Then I say that the side RQ is equall vnto the side GH. For if they be vnequall, the one of them is greater then the other, let the side RQ be greater then the side HG. And for that as the line RQ is to the line QS, so is the line HG to the line GN, and alternately also (by the 16. of the fifth) as the line RQ is to the line HG, so is the line QS, to the lyne GN, but the line RQ is greater then the line HG. Wherfore also the line QS is grea¦ter then y^e line GN. Wherefore also y^e figure RS is greater then the figure HN but (by supposition) it is equall vnto it, which is impossible. Wherfore y^e line QR is not greater then y^e line GH. In like sorte also may we proue that it is not lesse then it, wherfore it is equall vnto it: which was required to be proued.

Flussates demonstrateth this second part more briefly, by the first corollary of the ••0. of this boke, thus. Forasmuch as the rectiline figures are by supposition in one and the same proportion, and the same pro∣portion is double to the proportion of the sides AB to CD, and EF to GH (by the foresaid corollary) the proportion also of the sides shall be one and the selfe same (by the 7. common sentence) namely, the line AB shall be vnto the line CD as the line EF is to the line GH.

Notes

• The first part of this proposition.
• The second part which is the conuerse of the first.
• Note that this is proued in the assumpt following.
• An Assumpt.
• An other demō∣stration of the second part af∣ter Flussates.