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 .18. Probleme. The side of any Tetraedron giuen, to finde the sides, Diameters, and Axes, of all such regulare bodyes as maye therein bee described. (Book 18)

HAuing heretofore at large set foorthe by Problemes sundrie wayes (the sides of these bodyes gyuen) to finde the semidiameters of their contayning and con∣tayned circles, the diameters of their comprehending and comprehended spheres, with their contentes su∣perficiall and solide: hauing also by Theoremes she∣wed manifolde diuersitie of proportions rational and surde of these bodyes, their Superficies and lines compared with theirs comprehending and contayned spheres, there remayneth only nowe to conferre these bodyes mutually inscribed or circumscribed one with an other, and to search out by the side of any one knowen, the sides and dia∣meters both circulare and spherall, with the capacities superficiall and solide of all such bodyes as may within or without the same bodye be de∣scribed, I shall therefore first beginne with Tetraedron, and so procéede with the reste. Tetraedron receyueth only Octaedron and Icosaedron, for the Cube and Dodecaedron cannot possibly therein be so placed, that all their angles at one instante might exactly touche his superficies, the Tetraedrons side therfore giuen parted in two equall portions, either medietie is the inscribed Octaedrons side: Likewise the medietie of Tetraedrōs sides square, is y^e square of Octaedrons diameter which diui∣ded by 12, produceth a nūber, whose quadrate roote is y^e Octaedrons Axis.

For the inscribed Icosaedrō ye shall diuide the medietie of Tetraedrons side by extreame and meane proportion, and double the square of the lesse portion, the quadrate roote of the producte is the side. Or deducte one of these portions from the other, & adde the square of the remaynes medietie to the square of Octaedrons sides medietie, the rote quadrate of the producte doubled, is the Icosaedrons Diameter: Or if from the laste producte, ye abate the thirde parte of Icosaedrons sides square, the roote quadrate of the remayne is the Icosaedrons Axis.

Example.

Tetraedrons side supposed, Octaedrons side is ½, √{powerof2}½ his Diameter, vvhose square diuided by 12 bringeth 1/24, the roote being √{powerof2}1/24, is the Oc∣taedrons Axis. Likevvise for Icosaedron the medietie of Tetraedrons side diuided in extreame and meane proportion by the first Probleme, maketh the lesser portion ••/4—√{powerof2}5/16, the square hereof doubled, hath for his roote √{powerof2} vni. 14/8—√{powerof2}180/64, so much is the inscribed Icosaedrons side. Agayne, the difference of Tetraedrons sides medieties Portions diuided by ex∣treame and meane proportion is √{powerof2}20/16—1, the square of halfe this diffe∣rence is 9/16—√{powerof2}80/256, vvhich added to the square of Octaedrons sides me∣dietie, produceth ⅝—√{powerof2}5/16, the roote thereof doubled, is √{powerof2} v. 5/2—√{powerof2}5 the true quātitie of Icosaedrōs dimetiēt. Novv by subtracting 7/12—√{powerof2}45/144 the third parte of Icosaedrons sides square, from ⅝—√{powerof2}5/16, the laste pro∣ducte vvhose roote ye doubled to make the Diameter youre remainder vvil be this number 1/24+√{powerof2}45/144—√{powerof2}5/16, vvhose roote vniuersall is the inscri∣bed Icosaedrons Axis.

The con∣tayning Tetrae∣drons side 1

• ...Tetraedrons
  • Diameter. √{powerof2}3/2
  • Axis, √{powerof2}1/24
• ...The inscribed Octaedrons
  • Syde ½
  • Diameter √{powerof2}½
  • Axis, √{powerof2}1/24
• ...The cōtayned Icosaedrons
  • Syde √{powerof2} vni. 7/4—√{powerof2}45/16
  • Diameter √{powerof2} vni. 5/2—√{powerof2}5
  • Axis, √{powerof2}1/24.

Or thus geometrically without respect of number.

Admit AB the Tetraedrons side giuen, thereon I describe the semicir∣cle ACB, erecting the perpendiculare DC, and drawing the line AC, that Corde AC is the inscribed Octaedrons dimetiente, and AD his side. Now diuide DC by extreame and meane proportion as was taught in the firste probleme at E, and from E erect the perpendiculare EF, cutting AC in F, conclude FC the inscribed Icosaedrons side: againe parte AC in halfe at H, and from H to E extende a straight line, for that shal be the Icosaedrons semidiameter. For the Axis ye shall cut of from DB his twelfth parte, as ye were taught in the first booke: suppose it BI, vpon I erecte a perpendicu∣lare, cutting the circumference in K, so is the Corde KB the inscribed Ico∣saedron and Octaedrons Axis. As for their bases circulare semidiameters they are founde as was taught in the fifth and sixtenth problemes, foras∣much as both their sides and dimetientes are knowen.

[illustration]

Theoremes of Tetraedrons inscribed solides. 1.

Tetraedrons side is double to the side of his contayned Octaedron.

The second Theoreme.

Octaedrons diameter is a meane proportionall betweene his side and the side of his contayning Tetraedron.

The thirde Theoreme.

If any equilater triangle be so described and situate within the Tetraedrons equiangle triangulare base, that the angles of the inscribed triangle diuide the sides of the base by extreame and meane proportion, the medietie of this inscri∣bed triangles side, is the side of this Icosaedron.

The fourth Theoreme.

Tetraedrons side diuided by extreame and meane proportion, his lesser parte

is double in power to this Icosaedrons side.

The fifth Theoreme.

The dimetient of Octaedron diuided by extreame and meane proportion ma∣keth his lesser portion the Icosaedrons side.

The sixth Theoreme.

Icosaedrons diameter is equall in power with Octaedrons side, and the excesse or difference of his two partes, the one deducted fro the other.

The seuenth Theoreme.

The side of Tetraedron beeing rationall, his inscribed Icosaedrons side is an Apotome, bearing proportion to the Tetraedrons sides, as √{powerof2} v. 1 ¾—√{powerof2}••••/16 vnto 1.

The eight theoreme.

The Axes of these inscribed bodies Octaedron and Icosaedron are equall.

The ninth theoreme.

The side of Tetraedron beeing rationall, the contayned bodies axes are ratio∣nall, in power commensurable, and beare proportion to the side as √{powerof2}24 to 1.

The tenth theoreme.

Octaedrons diameter beeing rationall, Icosaedrons dimetient is an irrationall Minor, proportioned to the dimetiente of Octaedron as √{powerof2} v. 5—√{powerof2}20 vnto an vnitie.