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 aduertisment added by Flussas.

By this meanes therefore, the diameter of a sphere being geuen, there shall be ge∣uen the side of euery one of the bodies inscribed. And forasmuch as three of those bo∣dies haue their sides commensurable in power onely, and not in length, vnto the dia∣meter geuen (for their powers are in the proportion of a square number to a number not square: wherefore they haue not the proportion of a square number to a square number, by the corollary of the 25. of the eight: wherefore also their sides are incom∣mensurabe in length by the 9. of the tenth): therefore it is sufficient to compare the powers and not the lengths of those sides the one to the other•• which powers are con∣tained in the power of the diameter: namely, from the power of the diameter, let there ble taken away the power of the cube, and there shall remayne the power of the Tetra∣hedron: and taking away the power of the Tetrahedron, there remayneth the power of the cube: and taking away from the power of the diameter halfe the power thereof, there shall be left the power of the side of the octohedron. But forasmuch as the sides of the dodecahedron and of the Icosahedron are proued to be irrationall (for the side of the Icosahedron is a lesse line, by the 16. of the thirtenth: and the side of the dedocahe∣dron is a residuall line, by the 17. of the same) therfore those sides are vnto the diame∣ter which is a rationall line set, incommensurable both in length and in power. Where∣fore their comparison can not be diffined or described by any proportion expressed by numbers, by the 8. of the tenth: neither can they be compared the one to the other: for irrational lines of diuers kindes are incōmēsurable the one to the other: for if they should be commensurable, they should be of one and the selfe same kinde, by the 103. and 105. of the tenth, which is impossible. Wherefore we seking to compare them to the power of the diameter, thought they could not be more aptly expressed, then by such proportions, which cutte that rationall power of the diameter according to their sides: namely, diuiding the power of the diameter by lines which haue that proportiō, that the greater segment hath to the lesse, to put the lesse segment to be the side of the Icosahedron: & deuiding the sayd power of the diameter by lines hauing the propor∣tion of the whole to the lesse segment, to expresse the side of the dodecahedron by the lesse segment: which thing may well be done betwene magnitudes incommensurable.