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 2. Probleme. The 14. Proposition. Vnto a rectiline figure geuen, to make a square equall.

SVppose that the rectiline figure geuen be A. It is required to make a square equall vnto the rectiline figure A.* 1.1 Make (by the 45. of y^e first) vnto the rectiline figure A an equall rectangle parallelogramme BCDE. Now if y^e line BE be equall vnto the line ED, then is y^e thyng done whiche was required: for vnto the

[illustration]

rectiline figure A is made an equal square BD. But if not, one of these lines BE & is ED the greater. Let BE be the grea∣ter, and let it be produced vnto y^e point F. And (by the 3. of the first) put vnto ED an equall line EF. And (by the 10. of the first) deuide the line BF into two equall partes in the point G. And making the centre the point G, and the space GB or GF describe a semicircle BHF. And (by the 2. peticion) extend the line DE vnto y^e point H.* 1.2 And (by the 1. peticion) draw a line from G to H. And forasmuch as the right line FB is deuided into two equall partes in the point G, and into two vnequal partes in the point E, therfore (by the 5. of the second) the rectangle figure comprehended vnder the lines BE and EF together with the square which is made of the line EG, is equall to the square which is made of the line GF. But the line GF is equall vnto the line GH. VVherfore the rectangle figure comprehended vnder the lines BE and EF together with the square which is made of the line GE is equall to square which is made of the line GH. But vnto the square which is made of the line GH are equall the squares whiche are made of the lines HE and GE (by the 47. of the first.) VVherfore y^t which is contained vnder y^e lines BE and EF together with y^e square which is made of GE is equall to y^e squares which are made of HE and GE. Take away the square of the line EG common to them both. VVherfore the rectangle figure contained vnder the lines BE & EF is equall to the square which is made of the line EH. But that whiche is contained vnder the lines BE and EF is the parallelogramme BD, for the line EF is equall vnto the line ED. VVherfore the parallelogramme BD is equall to y^e square whiche is made of the line HE. But the parallelograme BD is equall vnto the rectiline figure A. VVherfore y^e rectiline figure A is equall to the square which is made of y^e line HE. VVherfore vnto the rectiline figure geuen A, is made an equall square described of the line EH: which was required to be done.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstratiō