A geometrical practise, named Pantometria diuided into three bookes, longimetra, planimetra, and stereometria, containing rules manifolde for mensuration of all lines, superficies and solides: with sundry straunge conclusions both by instrument and without, and also by perspectiue glasses, to set forth the true description or exact plat of an whole region: framed by Leonard Digges gentleman, lately finished by Thomas Digges his sonne. Who hathe also thereunto adioyned a mathematicall treatise of the fiue regulare Platonicall bodies, and their Metamorphosis or transformation into fiue other equilater vniforme solides Geometricall, of his owne inuention, hitherto not mentioned of by any geometricians.

The .23. Probleme. The side, Diameter or Axis of any regulare bodye knovven, to searche out all those forenamed lines in any regulare bodye that shall include or circumscribe that proponed solide. (Book 23)

ALthough this question of circumscribing bodyes mighte be diuided into 5 Chapters, and in euery one as many dif∣ferente rules, preceptes, and Theoremes taught, as was in the former of bodyes inscribed, yet for breuitie sake, I thinke beste to remitte the more ample handeling here∣of to the ingeniouse studente, who comparing the rules, and well way∣ing the Theoremes alreadie giuen, shall easely applie them to this purpose, and inuente many m••e perhappes of greater facilitie and no lesse certaintie: and in this Chapter I will only open one way, leauing a large fielde for others to inuent and exercise them selues in at plea∣sure. It shall therefore be requisite when the side, Diameter or Axis of any regulare bodye is proponed, to consider by the fifte Problemes

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.

The 3 theoreme.

Tetraedrons conteyning Octaedrons side to the side of his inscribed Octae∣dron is triple.

The 4 Theoreme.

Tetradrons side being rationall, his encompassing Icosaedrons side is an Apo∣tome, and triple to tetraedrons inscribed Icosaedrons side.

The 5 theoreme.

Tetraedrons side being rationall, his circumscribing dodecaedrons side is an Apotome of the 6 order, proportioned to the side as √{powerof2} vni. ¾—√{powerof2}5/16 vn∣to an vnitie.

The 6 Theoreme.

Hexaedron hath only 3 circumscribing regular bodies, for no tetraedron may be so placed about a cube but that his superficies shal either cut or not touch some of the Hexaedrons angles.

The 7 Theoreme.

Hexaedrons conteyning Octaedrons side is triple to his conteyned Octae∣drons side, and to the Hexaedrons side it beareth sutche proportion, as tetrae∣drons diameter to his containyng circles semidimetient.

The 8 theoreme.

Hexaedrons side being rationall, his comprehending Icosaedrons side, is an irrationall Minor, proportioned to the cubes side, as √{powerof2} v. 31 ½√{powerof2}911 ¼ vnto 1.

The 9 theoreme.

Hexaedrons side being rational, his comprehending Dodecaedrons side is an Apotome of the sixt he order proportioned to the side as √{powerof2} v. 1 ½—√{powerof2}¼ to an vnity.

The 10 Theoreme.

Hexaedrons Dimetient is meane proportionall betwene his side and the Di∣metiente of his contayninge Octaedron.

The 11 theoreme.

Octaedron may be comprehended of all the other regular bodies, and his side is the medietie of his encompassing tetraedrons side.

The 12 Theoreme.

Octaedrons externall cubes side is equall to his diameter, and double in po∣wer to his side.

The 13 Theoreme.

Octaedrons side being rationall, his including Icosaedrons side is an Apoto∣me of the 6 order, hauing proportion to Octaedrons side, as √{powerof2} vniuer. 3—√5 vnto 1.

The 14 theoreme.

Octaedrons side being rationall, his enuironing dodecaedrons side is an Apo∣tome,

proportioned to the Octaedrons side, as √{powerof2} vni. 7—√{powerof2}45 vnto 1.

The 15 theoreme.

Octaedrons comprehending Dodecaedrons side is equall to his conteyned I∣cosaedrons side.

The 16 Theoreme.

Icosaedron may be comprehended perfectely of all the other regulare solides, and his contayning tetraedrons side is triple to the side of his contained tetrae∣dron.

The 17 theoreme.

Icosaedrons side being rationall, his comprehendyng cubes side is a Binomy, and the Icosaedrons Diameter in power equall to them bothe.

The 18 theoreme.

Icosaedrons side rational his encompassing Octaedrons side is a Binomye and equall to the medietie of his externall Tetraedrons side.

The 19 theoreme.

Icosaedrons side rationall, his comprehending dodecaedrons side is an irrati∣onal Apotome, bearing proportion to the side as √{powerof2} v 17 ½—√{powerof2}281 ¼ vn∣to 1.

The 20 theoreme.

Icosaedrons comprehendyng cubes side is double in power to his compre∣hended Octaedrons side.

The 21 theoreme.

Dodecaedron can perfectly be comprehended of no regulare bodie saue onely of Icosaedron, whose side reteyneth sutche proportion to the side of Dodecae∣dron, as √{powerof2} vni. 13 ½—√101 ¼ vnto an vnitie.

The 22 theoreme.

Dodecaedron and Icosaedron hauyng equall sides, the cubes side that con∣teyneth Icosaedron is equall to the cubes side conteyned of dodecaedron.

The 23 theoreme.

Dodecaedron and octaedron hauing equal and rationall sides, the side of octa∣drons comprehending Icosaedron is an Apotome of the same order, that Dode∣caedrons inscribed Tetraedrons side is a Binomye, and their names or compo∣ning quantities equall.

The 24 theoreme.

Dodecaedron and the cube hauyng equall sides, whether they be rationall or surd, the dodecaedrons conteyning Icosaedrons side is triple to the cubes com∣prehending dodecaedrons side.

The 25 theoreme.

Dodecaedron and Tetraedron hauyng equall Sydes, whether they

be rationall or irrationall, the dodecaedrons conteyning Icosaedrons side to the Tetraedrons comprehending Dodecaedrons side, reteyneth sutch proportion, as Tetraedrons diameter doth to the lesse semidiameter of his basis.