last past what bodies may conteyne or circumscribe the same, and resor∣ting to theyr peculiar chapters, serche out the numbers appropriate to the conteyning and conteyned solides sides, axes and diameters, which found by thrée quantities knowne, vsing the rule of proportion, ye may readi∣ly finde the fourth, as by the example shall more plainly appere.
Example.
Suppose the side of a tetraedron giuen 10, for the sides, diameters and axes I repaire first to the 19 probleme, vvhere I find the conteining cubes side, be∣ing 1, the conteined tetraedrons side √{powerof2}2, saying therfore by the rule of pro∣portion √{powerof2}2, the conteined tetraedrons side foūd heretofore in the 19 pro∣bleme, giueth 1 for his conteining cubes side, vvhat yeldeth 10 the side giuen your fourth proportional number vvill be √{powerof2}50, the containing cubes side. Likevvise in the 22 probleme, I find the conteining dodecaedrons side being 1, the cōtained tetraedrons side √{powerof2} v. 3+√{powerof2}5. augmenting therfore 10, the side giuen by tetraedrons side in that probleme found, and diuiding by √{powerof2} vni. 3+√5, your quotient vvill be √{powerof2}62 ½—√{powerof2}12 ½, so mutch conclude the side of a dodecaedron that shal conteyn or comprehend this tetraedron, vvhose side is 10. In like maner may ye serche out the other sides, diameters, and axes, of all the comprehending bodies, vvhereof I leaue to giue any far∣ther examples, these tvvo being sufficient to the ingenious to proceede vvith lyke order in the rest.
But for sutch as not contented with one kinde of working wil delighte themselues in the diuersitie of rules and kyndes of calculation, I haue thought good to adioyne these Theoremes ensuing, which wel wayed and compared with sutch as are already past, shal yeld matter abundantly for the inuention of many mo conclusions and strange operations, than hither∣to hath ben vsed or published by any.
Theoremes of these bodies mutually circumscribed and confer∣red vvith their inscribed regular bodies. 1.
TEtraedron may be conteyned or circumscribed of all the other foure regular bodies, and his side being rationall, his containing Octaedrons side is also ra∣tionall, proportioned thervnto, as 3 to 2.
The 2 theoreme.
Tetraedrons comprehending cubes side is equall to the dimetient of his in∣scribed Octaedron.