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

A Probleme. 4.

Two circles being geuen, to finde one circle equall to them both.

Suppose the two circles geuē, haue their diameters A•• & CD. I say that a circle must be ••ound equall to the two circles whose diameters are A•• and CD: vnto the

[illustration]

line A••,* 1.1 at the point A, erect a perpendicular line A••: from which (sufficiently produced) cut a line equall to CD, which let be AF. By the first peticion draw from F to •• a right line: so is FA•• made a tri∣angle rectangle. I say now that a circle whose diameter is F••, is equal to the two circles whose diameters are A•• and ••D.* 1.2 For by the 47. of the first, the square of F•• is equall to the squares of A•• & AF. Which AF is (by construction) equall to CD•• wherefore the square of F•• is equall to the squares of AB and CD. But circles are one to an other, as the squares of their diameters are one to the other, by this second of the twelueth. Therefore the circle whose diameter is •••• is equall to the circles whose diameters are A•• and CD. Therefore two circles being geuen we haue found a circle equall to them both. Which was required to be done.

A Corollary. 1.

Hereby it is made euident, that in all triangles rectangle, the circles, semicircles, quadrants, o•• any other portions of circles described vpon the subtendent line, is equall to the two circles, semicircles, quadrants, or any two other like portions of circles, described on the two lines comprehending the right angle, like to like being compared.

For like partes haue that proportion betwene them selues, that their whole magnitudes haue, of which they are like partes, by the 15. of the fifth. But of the whole circles, in the former probleme it is euident: and therefore in the fornamed like portions of circl••s, it is a true consequent.

A Corollary. 2.

By the former probleme, it is also manifest, vnto circles three, fower, fiue, or to how many so euer one will geue, one circle may be geuen equall.

For if first, to any two, by the former probleme, you finde one equall, and then vnto your found circle and the third of the geuen circles, as two geuen circles, finde one other circle equall, and then to that second found circle, and to the fourth of the first geuen circles•• as two circles, one new circle be found equall, and so proceede till you haue once cuppled orderly, euery one of your propoūded circles (except the first and second already doone) with the new circle thus found for so the last found circle is equall to all the first geuen circles. If ye doubt, or sufficiently vnderstand me not: helpe your selfe by the discourse and demonstration of the last proposition in the second booke, and also of the 31. in the sixth booke.

Notes

• * 1.1

  Cons••ruction.
• * 1.2

  Demonstra∣tion.