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 9. Probleme. The 9. Proposition. About a square geuen, to describe a circle.

SVppose that the square geuen be AB

[illustration]

CD. It is required about the square ABCD to describe a circle. Drawe right lines from A to C, and from D to B, & let them cut the one the other in the poynt E.* 1.1 And forasmuch as the lyne DA is equall vn∣to the lyne AB,* 1.2 and the line AC is common vnto them both, therfore these two lines DA and AC are equall vnto these two lynes BA and AC, the one to the other. And the base DC is equall vnto the base BC. Wherefore (by the 8. of the first) the angle DAC is equall vnto the angle BAC. Where∣fore the angle DAB is deuided into two equall partes by the line AC. And in

like sor•• may we proue that euery one of these angles ABC, BCD, and CDA is deuided into two equall partes by the right lines AC and DB. And foras∣much as the angle DAB is equall vnto the angle ABC, and of the angle DAB the angle EAB is the halfe; and of the angle ABC the angle EBA is the halfe: Therfore the angle EAB is equall vnto the angle EBA: wherfore (by the 6. of the first) the side EA is equall vnto the side EB. In like sorte may we proue that either of these right lines EA and EB is equall vnto either of these lines EC and ED. Wherfore these foure lines EA, EB, EC, and ED are e∣quall the one to the other. Wherfore making the centre E, and the space any of these lines EA, EB, EC, or ED. Describe a circle and it will passe by the pointes A, B, C, D, and shall be described about the square ABCD, as it is euident in the figure ABCD. Wherfore about a square geuē is described a cir∣cle: which was required to be done.

¶ A Proposition added by Pelitarius.

A square circumscribed about a circle, is double to the square inscribed in the same circle.

Suppose that the square ABCD be cir∣cumscribed about the circle EFGH,* 1.3 whose centre let be K. And let the poyntes of the

[illustration]

touches be E, F, G, H. And drawing these two diameters EG, and FH, and these right lines EF, FG, GH, and HB, there shall be inscri∣bed in the circle a square EFGH (by the sixt of this booke). Then I say, that the square ABCD, is double to the square EFGH. For forasmuch as the side AB of the greater square, is (by the 34. of the first) equall to FH, which is the diameter of the lesse square: but the square of FH is double to the square whose diameter it is, namely, to the square EFGH (by the 47. of the first). Wherefore also the square of AB which is ABCD, is double to the square EFGH: which was re∣quired to be proued.

Thys may also be demonstrated by the equalite of the triangles and squares contayned in the great squares.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  A Propos••tion added by Pe∣litarius.