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

An other construction and demons••••ation after Proclus.

Suppose that the right line geuen be AB:* 1.1 & let the point in it geuen be A, & let the rectiline angle geuē be CDE. It is required vpō the right line geuē AB, & to the point in it geuē A, to make a rectiline angle equal to the rectiline angle geuē CDE. Drawe a line frō C to E. And produce the line AB on either side to the points F and G. And vn∣to

the line CD, put the line FA equal, & vnto the line DE let the line AB be equal, & vn∣to the line EC put the line BG equal. And making the cētre the point A, & the space AF, describe a circle KF. And agayne making the centre the point B and th•• space BG de∣scribe

[illustration]

an other circle ••L: which shal of necessitie cut the one the other, as we haue be∣••ore proued. Let them cut the one the other in the pointes M & N. And draw these right lin••s AN, AM, BN, and BM. And forasmuch as FA is equall to AM•• and also to AN (by the definition of a circle) but CD is equall to FA, wherfore the lines AM and AN are ech•• equall to the line DC. Ag••yne forasmuch as BG, is equall to BM, and to BN, and BG is equall to CE: therfore either of these lines BM and BN is equall to the line CE. But the line BA is equall to the line DE. Wherfore these two lines BA & AM, are equall to these two lines D E and DC, the one to the other, and the base BM is equal to the base CE. Wherfore (by the 8. proposition) the angle MAB, is equall to the angle at the point D. And by the same reason the angle NAB, is equall to the same angle at the point D. Wherfore vpon the right line geuen AB, and to the point in it geuen A, is de∣scribed a rectiline angle on either side of the line AB: namely, on one side the rectiline angle NAB, and on the other side the rectiline angle MAB, either of which is equall to the rectiline angle geuen CDE: which was required to be done.

Notes

• * 1.1

  An other con∣struction and de¦monstration af¦ter Proclus.