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 8. Probleme. The 28. Proposition. Vpon a right line geuen, to apply a parallelogramme equall to a rectiline figure geuen, & wanting in figure by a paralle∣logramme like vnto a parallelogrāme geuen. Now it beho∣ueth that the rectiline figure geuen, whereunto the parallelo∣grāme applied must be equall, be not greater thē that paralle∣logramme, which so is applied vpon the halfe lyne, that the defectes shall be like, namely, the defect of the parallelogrāme applied vpon the halfe line, and the defect of the parallelo∣gramme to be applied (whose defect is required to be like vn∣to the parallelogramme geuen).

SVppose the right line geuen to be AB, and let the rectiline figure ge∣uen wherunto is required to apply vpon the right line AB an equall rectiline figure be C, which figure C, let not be greater then that pa∣rallelogrāme which is so applied vpon the halfe line, that the defectes shall be like, namely, the defect of the parallelogramme applied vpon the halfe line, and the defect of the parallelogramme to be applied (whose defect is requi∣red to be like vnto the parallelogramme geuē). And let the figure whereunto the defect or want of the parallelogramme is

[illustration]

required to be like D. Now it is requi∣red vpō y^e right line geuē AB, to describe vnto the rectiline figure geuen C, an e∣qual parallelogramme wanting in figure by a parallelogramme like vnto D.* 1.1 Let the line AB (by the 10. of the first) be deuided into two equall partes in y^e point E. And (by the 18. of the sixth) vppon the line ••B describe a rectiline figure EBFG like vnto the parallelogramme D and in like sort situate, which shall also be a parallelograme. And make complete the parallelogramme AG.* 1.2 Now then the parallelogramme AG is either equal vnto the rectiline figure C, or greater then it by supposition. If the parallelo∣gramme AG be equal vnto the rectiline figure C, then is that done which we ••ought for.* 1.3 For then vpō the right line AB is described vnto the rectiline figure

geuen C an equal parallelogramme AG wanting in figure by the parallelograme GB,* 1.4 which is like vnto the parallelogramme D. But if AG be not equal vnto C then is AG greater then C, but AG is equall vnto GB (by the first of the sixt). Wherfore also GB is greater then C. Take the excesse of the rectiline figure BG aboue the rectiline figure C (by that which Pelitarius addeth after the 4••. of the first) And vnto that excesse (by the 15. of the sixt) describe an equall recti∣line figure KLMN like and in like sort situate vnto the rectiline figure D. But the rectiline figure D is like vnto the rectiline GB, wherfore also the recti∣line figure KLMN is like vnto the rectiline figure GB (by the 25. of the sixt) Now then let the sides KL and GE be sides of like proportion, let also y^e sides LM and GF be sides of like proportion. And forasmuch as the parallelogrāme GB is equal vnto the figures C and KM, therfore the parallelogramme GB is greater then the parallelogramme KM. Wherefore also the side GE is greater then the side KL, and the side GF is greater then the side LM, vnto the side KL put an equall line GO (by the 2. of the first) and likewise vnto the side LM put an equall line GP. And make perfect the parallelogramme OGPX. Wherfore the parallelogramme GX is equal & like vnto the parallelogramme KM. But the parallelogramme KM is lyke vnto the parallelogramme GB. Wherfore also the parallelogramme GX is like vnto the parallelogramme GB. Wherfore the parallelogrammes GX and GB are (by the 26. of the sixt) about one and the self same dimecient. Let their dimecient be GB, and make complete the figure. Now forasmuch as the parallelogramme BG is equall vnto the recti∣line figure C, and vnto the parallelogramme KM, and the parallelogramme GX, which is part of the parallelogramme GB, is equal vnto KM. Wherfore the Gnomon remayning YQV is equall vnto the rectiline figure remayning, name∣ly, to C. And forasmuch as the supplement PR is equall vnto the supplement OS, put the parallelogramme XB common vnto them both. Wherfore the whole parallelogramme PB is equall vnto the whole parallelograme OB. But the pa∣rallelogramme OB is equal vnto the parallelogramme TE by the 1. of the sixt, (for the side AE is equal vnto the side EB) Wherfore the parallelogramme TE is equal vnto the parallelogramme PB. Put the parallelogramme OS com∣mon to them both. Wherfore the whole parallelogramme TS is equall vnto the whole gnomon YQV. But it is proued that the gnomon YQV is equal vnto the rectiline figure C. Wherfore also the parallelogramme TS is equal vnto the re∣ctiline figure C. Wherfore vpon the right line geuen AB is applied a parallelo∣gramme TS equal vnto the rectiline figure geuen C, and wanting in figure by a parallelogramme XB which is like vnto the parallelogramme geuen D, for the parallelogramme XB is like vnto the parallelogramme GX: which was re∣quired to be done.

¶ A Corollary added by Flussates.

Hereby it is manifest, that if vpon a right line be applied a parallelogramme wantyng

in figure by a square,* 1.5 the parallelogramme ap∣plied

[illustration]

shall be equall to the rectangle figure which is contayned vnder the segments of the line geuen which are made by the application. For the rest of the line is equall to the o∣ther side of the parallelogramme applied. For that they are sides of one & the selfe same square, as the parallelogramme AG is contained vnder the lines AD and DB, or DG which is equall to DB.

Notes

• * 1.1

  Construction.
• * 1.2

  Two cases in this Proposi∣tion.
• * 1.3

  The first case.
• * 1.4

  The second case.
• * 1.5

  A Corollary ad∣ded by Flussa∣tes, and is put of Theon as an assumpt be••ore the 17. proposi∣tion of the tēth booke: which ••or that it follo∣weth of this proposition I thought it not amisse here to place.