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 .20. Probleme. Octaedrons side giuen, to searche out all his conteyned bo∣dies, sides, diameters and axes.

DOdecaedron within this body may not aptly be described, for as much as only eight of his solide angles touche the superficies of this body béeing situate in the centers of octaedrons bases, the other 12 falling quite within this body, not touching his super∣ficies on any parte. But Tetraedron may be inscribed and all his solide angles placed in the centers of octaedrons triangles. Likewise the cubes solide angles take their place in the centers of his bases, only Icosaedron hath his situation somwhat more straunge and intricate, euery of his 12 angles resting in a side of octaedron, and diuide his 12 sydes by extreame and meane proportion. Now to attain the diameters axes &c. Peruse these rules ensuing.

For Tetraedron.

Augment octaedrons side by 2, and diuide by 3, the quotient is tetrae∣drons side, the square therof augmented by 3, and parted by 2, yeldeth his diameter, which diuided by 6▪ declareth the axis.

The square of octaedrons side multiplied by 2 and diuided by 3 produ∣ceth this Cubes diameters square, which againe diuided by 3, sheweth his sides square, half the side is the axis.

For Icosaedron.

Diuide the side giuen by extreme and meane proportion, the square of the lesser parte double, and from the product extract the roote quadrate, so haue yée the inscribed Icosaedrons side, deduct one of those former founde parts of the giuen syde from the other, and square the difference, for that added to the square of Icosaedrons side bringeth the square of his dime∣tient. Nowe for the axis ye may deduct the third part of Icosaedrons sides square from the square of his semidimetiente, the roote quadrate of the remainder is his ••athetus.

Examples of Tetraedron.

Octaedrons side giuen 1▪ increased by 2, and diuided by 3, maketh ⅔, the Te∣traedrons side, the square therof being ⅘ augmēted by 3 and parted by 2, yeldeth ⅔√{powerof2}⅔ is his diameter, which diuided by 6, bringeth √{powerof2}1/54 the axis.

Of the Cube.

The square of Octaedrons side 1, augmented by 2, and diuided by 3, produ∣ceth ⅔ the diameters square, that again diuided by 3, yeldeth 2/9, √{powerof2}2/9 is the cu∣bes side, whose halfe being √{powerof2}1/18 is the axis.

Of Icosaedron.

Octaedrons side 1 diuided by extreme and meane proportion, the greater part •• √{powerof2}5/4—½, the lesser 3/2—√{powerof2}5/••▪ this latter partes square doubled is 7, —√45, the roote quadrate vniuersall therof is Icosaedrons side. Againe by subtrac∣tion of the partes ye shall find the difference √{powerof2}5—2, the square therof added to the square of Octaedrons side giuen, maketh 10, —√{powerof2}80. I cōclude √{powerof2} vni. 10, —√{powerof2}80 the Icosaedrons diameter. Finally for the axis I deduct 7/3—√{powerof2}5 the thirde parte of Icosaedrons sides square, from 5/2—√{powerof2}5, the semidi∣metient of Icosaedrons containing sphere, there remayneth ⅙√{powerof2}⅙ is the axis.

The comprehen∣ding Octaedrons side 1

• ...Tedraedrons
  • Syde ⅔
  • Diameter √{powerof2}⅔
  • Axis, √{powerof2}1/54
• ...Hexaedrons
  • Syde √{powerof2}2/9.
  • Diameter √{powerof2}⅔.
  • Axis √{powerof2}1/18.
• ...Icosaedrons
  • Syde √{powerof2} vni. 7—√{powerof2}45.
  • Diameter √{powerof2} vni. 10—√{powerof2}80.
  • Axis √{powerof2}⅙.

Or thus vvithout ayde of numbers, admit AB the Octaedrons side giuen, thereon (makyng C the medietie a center) I describe the semicircle ADB, and from C, I erect the perpendicular CD, dravving lines from D to AB, the semidiameter CB, I diuide in 3 equall partes at E and I, vppon I, I errere the perpendicular IK, concurring vvith the circumference in K, from E, I dravv EM paralele to DB, cutting AD in M, as vvas taughte in the firste booke of this treatise: againe by the first probleme I diuide AB by extreme and mean proportion in F, and thervpon I raise the perpendicular FG, cros∣sing AD in G, and from F to D, I extend a streyght line▪ finally I dravve the ryght lyne KB, cuttyng of a thirde parte at H, and thus is the fygure fully fynyshed, contaynyng all these bodies, sydes, Diameters, and Axes: For AE is Tetraedrons syde, KB hys Semidiameter, HB his axis, AM

the cubes side, KB his semidiameter, MD his axes, AG Icosaedrons side, F D his semidiameter, and KB his axes.

[illustration]

Theoremes of Octaedrons inscribed regular bodyes. 1.

THe conteynyng Octaedrons side beareth proportion to the side of his con∣teyned tetraedron, as 3 to 2.

The seconde theoreme.

Octaedrons side, to his inscribed cubes side, beareth the proportion of √{powerof2}4 ½ vnto 1.

The third Theoreme.

The side of tetraedron, to the side of the cube, is as 2 vnto √{powerof2}2.

The fourth Theoreme.

Octaedrons conteyning dimetient, is triple in power to his inscribed Icosae∣drons axis.

The fifth theoreme.

The conteyaing octaedrons diameter is triple to the conteyned cubes side.

The sixte Theoreme.

The axes of these 3 inscribed bodies are proportionall, and the cubes axes it the meane proportion betwene the other two.

The seuenth Theoreme.

The side of Octaedron diuided by extreme and mean proportion, that quan∣titie which is double in power to the lesser part, is equall to the inscribed Icosae∣drons side.

The eight theoreme.

Icosaedrons diameter is equall in power to the side of the conteyning Octae∣dron, and the d••fference or excesse of Octaedrons sides partes being deuided in extreme and meane proportion.

The ninth Theoreme.

Octaedrons dimetient diuided by extreame and meane proportion maketh his lesser segment the inscribed Icosaedrons side.

The tenth theoreme.

Octaedrons side being rationall, Icosaedrons side is an Apotome proportioned to the Octaedrons side, as √{powerof2} vniuers. 7—√{powerof2}45 vnto an vnitie.