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 19. Theoreme. The 26. Proposition. If from a parallelogramme be taken away a parallelograme like vnto the whole and in like sorte set, hauing also an angle common with it, then is the parallelogramme about one and the selfe same dimecient with the whole.

SVppose that there be a parallelogramme ABCD, and from the paral∣lelogramme ABCD, take away a parallelogramme AF like vnto the parallelogramme ABCD, and in like sort situate, hauing also the an∣gle DAB common with it. Then I say, that the parallelogrammes ABCD and AF are both about one and the self same * 1.1 dimecient AFC, that is, that the dimecient AFC of the whole parallelogramme ABCD passeth by the angle F of the parallelogramme AF, and is common to either of the parallelogrammes. For if AC do not passe by the point F, then if it be possible let it passe by some o∣ther point, as AHC doth. Now then the dimetient AHC shall cut eyther the side GF or the side EF of y^e parallelogramme AF. Let it cut y^e side GF in the point H. And (by the 31. of the first) by the point H let there be drawen to ei∣ther of these lines AD and BC a parallel line HK wherfore GK is a paralle∣logramme, and is about one and the selfe same

[illustration]

dimetient with y^e parallelogramme ABCD. And forasmuch as y^e parallelogrammes ABCD and GK are about one and the self same dimecient, therfore (by the 24. of the sixth) the parallelogramme ABCD is like vnto the parallelogramme GK. Wherfore as the line DA is to the line AB so is the line GA to the line AK (by the conuersion of the first definition of the sixth) And for that the pa∣rallelogrammes ABCD, and EG are (by supposition) like, therfore as the line DA is to the lyne AB so is the line GA to the line AE. Wherfore the line GA hath one and the selfe proportion to either of these lines AK and AE. Wherfore (by the 9. of the fifth) the line AK is e∣quall vnto y^e line AE, namely, y^e lesse to y^e greater, which is impossible. The selfe same inconuenience also will follow, if you put the dimetient AC to cut the side FE. Wherfore AC the dimetie••t of the whole parallelogramme ABCD pas∣seth by the angle and poynt F. And therfore the parallelogramme AEFG is a∣bout one and the selfe same dimetient with the whole parallelogramme ABCD. Wherfore if from a parallelogramme be taken away a parallelograme lyke vnto the whole, and in lyke sorte situate, hauing also an angle common with it, then is that parallelogramme about one and the selfe same dimetient with the whole: which was required to be proued.

¶ An other demonstration after Flussates, which proueth this proposition affirmatiuely.

From the parallelogramme ABGD let there be taken away the parallelogramme AEZK like and in like sorte situate with the whole parallelogramme ABGD, and ha∣uing also the angle A common with the whole parallelogramme.* 1.2 Then I say that both their diameters, namely, AZ and AZG do make one and the selfe same right line. De∣uide the sides AB and B•• into two equall partes in the pointes C and F (by the 10. of

the first.) And drawe a line from C to F.

[illustration]

Wherefore the line CF is a parallel to the right line AG (by the corollary added by Campane after the 29. of the first) Wher∣fore the angles BAG and BCF are equall (by the 29. of the first): but the angle EA∣Z is equall vnto the angle BAG (by reasō the parallelogrammes are supposed to be like) wherefore the same angle EAZ is e∣quall to the angle BCF, namely, the out∣ward angle to the inward and opposite an∣gle. Wherfore (by the 28. of the first) the lines AZ and CF are parallel lynes. Now then the lines AZ and AG being parallels to one and the selfe same lyne, namely, to CF do concurre in the point A. Wherefore they are set directly the one to the other, so that they both make one right line (by that which was added in the ende of the 30. proposition of the first) wherfore the parallelo∣grammes ABGD, and AEZK are about one and the selfe fame dimetient: which was required to be proued.

Notes

• * 1.1

  By the di∣metiēt is vnderstand here the dime∣tient which is ••rawen from the angle which is com∣mon to them both to the op∣posite angle.

  Demonstra∣tion leading to an absurditie.
• * 1.2

  An other way after Flussates.