But it is more accurate and preciser cause to take the halfe of the spheare.
11. Spheares have a trebled reason of their diameters.
So before it was told you; That circles were one to another, as the squares of their diameters were one to another, because they were like plaines: And the diameters in circles were, as now they are in spheares, the homologall sides. Therefore seeing that spheres are figures alike, and of treble dimension, they have a trebled reason of their diameters.
12. The five ordinate bodies are inscribed into the same spheare, by the conversion of a semicicle having for the diameter, in a tetrahedrum, a right line of value
sesquialter unto the side of the said tetrahedrum; in the other foure ordinate bodies, the diagony of the same orginate.
The Adscription of ordinate plaine bodies is unto a spheare, as before the Adscription plaine surfaces was into a circle; of a triangle, I meane, and ordinate triangulate, as Quadrangle, Quinquangle, Sexangle, Decangle, and Quindecangle. But indeed the Geometer hath both inscribed and circumscribed those plaine figures within a circle. But these five ordinate bodies, and over and above the Polyhedrum the Stereometer hath onely inscribed within the spheare. The Polyhedrum we have passed over, and we purpose onely to touch the other ordinate bodies.
13 Out of the reason of the axeltree of the sphearicall the sides of the tetraedrum, cube, octahedrum and dodecahedrum are found out.
The axeltree in the three first bodies is rationall unto the side, as was manifested in the former. For it is of the sesquialter valew unto the side of the tetrahedrum; of treble, to the side of the cube: Of double, to the side of the Octahedrum. Therefore if the axis ae, be cut by a double reason in i: And the perpendicular io, be knit to a, and e, shall be the side of the tetrahedrum; and oe, of the cube, as was manifest by the [10 e viij], and [25 iiij]: And the greater segment of the side of the cube proportionally cut, is by the [24 e, xxv].
If the same axis be cut into two halfes, as in u: And the perpendicular uy, be erected: And y, and a, be knit together, the same ya, thus knitting them, shall be the side of the Octahedrum, as is manifest in like manner, by the said [10 e, viij], and [25 e iiij].