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 first Proposition added by Flussates.

To describe two rectiline figures equall and like vnto a rectiline figure geuen and in like sort situate, which shall haue also a proportion geuen.

Suppose that the rectiline figure geuen be ABH.* 1.1 And let the proportion geuen be the proportion of the lines GC and CD. And (by the 10. of this booke) deuide the line AB like vnto the line GD in the poynt E (so that as the line GC is to the line CD, so let the line AE be to the line EB). And vpon the line AB describe a semicircle AFB. And from the poynt E erect (by the 11. of the first) vnto the line AB a perpendicular line EF cutting the circumference in the poynt F. And draw these lines AF and FB. And vpon either of these lines describe rectiline figures like vnto the rectiline figure AHB and in like sort situate (by the 18. of the sixt): which let be AKF, & FIB. Then I say, that the rectiline figures AKF, and FIB,

[illustration]

haue the proportion geuē (namely, the propor∣tion of the line GC to the line CD) and are e∣quall to the rectiline figure geuen ABH vnto which they are described like and in like sort si∣tuate. For forasmuch as AFB is a semicircle, therefore the angle AFB is a right angle (by the 31. of the third) and FE is a perpendicular line.* 1.2 Wherefore (by the 8. of this booke) the triangles AFE and FBE are like both to the whole triangle AFB and also the one to the other. Wherefore (by the 4. of this booke) as the line AF is to the line FB, so is the line AE to the line EF, and the line EF to the line EB, which are sides cōtayning equall angles. Wher∣fore (by the 22. of this booke) as the rectiline fi∣gure described of the line AF is to the rectiline figure described of the line FB, so is the recti∣line figure described of the line AE to the recti∣line figure described of the line EF, the sayd rectiline figures being like and in like sort

••ituate. But as the rectiline figure described of the line AE being the ••irst, is to the rectiline ••igure described of the line EF being the second, so is the line AE the first, 10. the line ••B the third (by the 2. Corollary of the 20. of thys booke). Wherfore the recti∣line figure described of the line AF is to the rectiline figure described of the line FB, as the line A•• is to the line EB. But the line AE is to, the line EB (by construction) as the line GC is to the line CD. Wherefore (by the 11. of the fift) as the line GC is to the line CD, so is the r••••tiline ••igure described of the line AF to the rectiline ••igure described 〈◊〉〈◊〉 the line ••B, the sayd rectiline figures being like and in like sort described. But the 〈…〉〈…〉 described o•• the lines AF and FB, are equall to the rectiline ••igure d••••••••••bed o•• the line AB, vnto which they are (by construction) described lyke and in like sort situate. Wherefore there are described two rectiline figures AKF and FIB equ••ll and like vnto the rectiline figure geuen ABH and in like sort situate, and they ha••e also the one to the other the proportion geuen, namely, the proportion of the line GC to the line CD: which was required to be done.

Notes

• * 1.1

  Construction of the Pro∣bleme.
• * 1.2

  Demonstar∣tion of the same.