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

Io. Dee his aduise vpon the Assumpt of the 6.

As concerning the making of the line DE, equall to the right line AB, verely the second of the first, without some farther consideration, is not properly enough alledged. And no wonder it is, for that in the former booke••, whatsoe•••••• ••a••h of lines

[illustration]

bene spoken, the same hath alway••s bene imagined to be in one onely playne superficies considered or execu∣ted. But here the perpendicular line AB, is not in the same playn•• superficies, that the right line DB is. Ther∣fore some other helpe must be put into the handes of young beginners, how to bring this probleme to exe∣cution: which is this, most playne and briefe. Vnder∣stand that BD the right line, is the common section of the playne superficies, wherein the perpendiculars AB and CD are, & of the other playne superficies, to which they are perpendiculars. The first of these (in my former demonstration of the 6)•• I noted by the playne superfi∣cies QR: and the other, I noted by the plaine superficies OP. Wherfore BD being a right line common to both the playne sup••rficieces QR & OP, therby the ponits B and D are cōmon to the playnes QR and OP. Now

from BD (sufficiently extended) cutte a right line equall to AB, (which suppose to be BF) by the third of the first, and orderly to BF make DE equall, by the 3. o•• the first, if DE be greater then BF. (Which alwayes you may cause so to be, by producing of DE sufficiently). Now forasmuch as BF by constructi∣on is cutte equall to AB, and DE also, by construction, put equ••ll to BF, therefore by the 1. common sentence, DE is put equall to AB: which was required to be done.

In like sort, if DE were a line geuen to whome AB

[illustration]

were to be cutte and made equall, first out of the line DB (su••••iciently produced) cutting of DG, equall to DE by the third of the first: and by the same 3. cutting from BA (sufficiently produced) BA, equall to DG: then is it euidēt, that to the right line DE, the perpēdicular line AB is put equall. And though this be easy to conceaue, yet I haue designed the figure accordingly, wherby you may instruct your imagination. Many such helpes are in this booke requisite, as well to enforme the young stu∣dentes therewith, as also to master the froward gayne∣sayer of our conclusion, or interrupter of our demon∣strations course.