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 .17. Probleme. Arithmetically and geometrically to search out all the sides, diameters, perpendiculars, and lines Diagonall, vvith the bases semidiame∣ters, of all suche regular bodies as shall circumscribe or cōprehend any sphere vvhose dimetient is knovvn. (Book 17)

SEing these bodies as it is demonstrated by Euclide, ar•• of sutche vniforme composition that they will bothe re∣ceiue an inward sphere touching with his conuex super∣ficies euery of their bases centers, and also an outward sphere inclosing and with his concaue peripherye tou∣ching euery of their angles, for resolution of this Pro∣blem it shall be requisite to shewe how the diameter of the containing sphere may be found by knowledge of the cōtayned spheres dimetient: whiche doone, by the proposition laste past, the sides, and semidi∣ameters, may consequently be knowne. But bycause these 5 bodies being described without one sphere, are not also inuironed of one, but haue thrée seuerall differente comprehendyng spheres, the largest enuironyng Te∣traedron, the nexte encompassing the cube and Octaedron, and the laste in∣closing Icosaedron and dodecaedron (for this as it is by demonstration ap∣proued of Euclide, so is it also apparant by these former problemes) it sée∣meth therefore requisite to prescribe thrée seueral rules for y^e inquiring out of their Dimetientes. And first for Tetraedrons Diameter, ye shall only increase the diameter geuen by 3, the producte is youre desire. And for the spheres dimetiente that contayneth the cube and Octaedron, ye shall di∣uide the diameter geuen by √{powerof2}••/3 the quotiente or resultyng summe is likewise your demaunde. Or diuide the same diameter by √{powerof2} vniuers. ⅓—√{powerof2}8/90, So haue yée the spheres dimetiente that comprehendeth Dodecaedron and Icosaedron.

Example.

Admit the diameter of the sphere vvheron these bodies shall be described 10, this augmented by 3, maketh 30, the diameter of tetraedron: Againe the square of 10 augmented by 3, bringeth 300, so is √{powerof2}300, the diameter of octaedron and the cube: I diuide 100 by ⅓—√{powerof2}4/45 therof ariseth 1500, —√{powerof2}1800000, I cōclude therfore √{powerof2} v. 1500—√{powerof2}1800000, the diameter of the sphere that shal include the dodecaedron & Icosaedron, vvhose inscri∣bed spheres diameter is 10, and proceedyng by the laste probleme yee shall fynde the dodecaedrons syde √{powerof2} v. 1250.—√{powerof2}1512500, Icosaedrons side √{powerof2} vniuers. 1050—√{powerof2}1012500, and so foorth of al the other solides sides, and semidiameters: for consideryng theyr operation is nothynge differente from that vvas shevved in the 16 probleme, it vvere in vaine here agayne to make therof a superfluous recitall.

Geometrically to perfourme the same.

Vpon AB the diameter giuen of the sphere (vvhich al these regular bodies shal circūscribe) describe the semicircle, AKB, and vpon the same center C, describe the semicircle DNE, hauing his diameter DE triple to AB the dia¦meter giuen, and frō A, errere the perpendicular AF, vvhich diuided in tvvo equall parts at ω, maketh A ω the lesse semidiameter of Tetraedrons base, and AF the greater: from F extende cordes to ED, EF is the tetraedrons side, and EA his perpendiculare. Novve yf ye fixe one foote of the com∣passe in C (and openyng the other to ω) describe the semicircle HLI, it vvill touche the medietie of FE at G, vvhereby ye haue HI the diameter of Hexaedron and octaedron, HL, the side of Octaedron, AB the side of the cube, A ω the semidimetient of Octaedrons basis conteyning circle, AK, the semidiameter of the Cubes conteynyng circle, AC the se∣midimetient of the cubes inscribed circle, and AM the medietie of AK is the octaedrons basis lesser semidimetiente. Thus haue you founde the diameters, sides, and circular semidiameters, of these firste 3 bodyes: for the other tvvo ye shall thus proceede, from N as ye vvere taughte in the firste booke, dravve NO paralele to CE equall to the medietie thereof, and (couplyng CO togyther vvith a strayght lyne) from P vvhere it cut∣teth the greatest semicircles circūference let fal the perpendicular PQ, and vpon S (leauing SQ equal to 23 part of QE) erect the perpēdicular SR, thē

from R and P, extende straight lines to C & E, and to the middle of RE ex∣tēd the right line CT, cutting the left semicircle in φ, from thēce dravv a pa∣ralele to RE, cutting RC, CE in π, x, and (making Cx a semidiameter) des∣cribe the semicircle β π x crossing CP in ρ, frō thēce to x dravv the right line x ρ, for that is the circumscribing Icosaedrons side, and φ x is the greter semi∣dimetient of his base, vvx the lesser, β x the diameter of Icosaedrōs compre∣hēding sphere: This done from R dravv a paralele to the semicircles diame∣ter, c••ossing the greatest circumferēce in Y, fro thence to the end of the gre∣test dimetient, dravv the line DY, from vvhose medietie θ, dravv a straight line to the center C, then as vvas taught in the first booke, cut D θ in half at μ, and diuide D μ in extreme & mean proportion at λ by the first probleme, from these points λ, FY, to the cēter C extend right lines, cutting the cir∣cumference of the last described semicircle in the points α z, from thē to β, dravv lines again, and diuide the cord α β in extreme and mean proportion at γ: Likevvise β z is diuided in halfe at δ, being both the contact of the least circūference, and also the intersection made vvith θ C, but vvhere λ C con∣reth vvith β z place this letter ε, so is β γ the circūscribing dodecaedrōs side β x his dimetient, β δ the semidiameter of his conteyning circle, ε δ the les∣ser semidiameter of the dodecaedrons pentagonall basis, and α β his line dia∣gonall. Thus haue ye in one figure all the sides and diameters both circular and spherall of all suche regulare solides as comprehende or circumscribe the assigned sphere.

[illustration]

The diame∣ter of the in∣ternall giuen Sphere 10

• ...Tetrae∣drons
  • Diameter DE, 30
  • Syde EF, √{powerof2}600
  • Contayning circles sedimiameter, AF, √{powerof2}200
  • Inscribed circles semidimetient A ω √{powerof2}50
  • Perpendiculare or Altitude, AE 20
• ...Hexae∣drons
  • Diameter HI √{powerof2}300
  • Syde AB, 10
  • Basis greater semidiameter, AK, √{powerof2}50
  • Basis lesser Semidimetient, AC 5
  • Basis line Diagonall, √{powerof2}200
  • Altitude AB 10
• ...Octae∣drons
  • Diameter HI √{powerof2}300
  • Syde HL, √{powerof2}150
  • Basis greater semid. AK √{powerof2}50
  • Basis lesse semidiameter, AM, √{powerof2}25/2
  • Perpendicular AB, 10
• ...Icosae∣drons
  • Diameter β x √{powerof2} v. 1500—√{powerof2}1800000
  • Syde ρ x. √{powerof2} viii. 1050—√{powerof2}1012500
  • Basis greter semidiamet. ω φ √{powerof2} v. 350—√{powerof2}112500
  • Basis lesse semidiameter φ w, √{powerof2} v. 87 ½, —√{powerof2}7031 ¼
  • Altitude 10
• ...Dodeca∣edrons
  • Diameter β x √{powerof2} v. 1500—√{powerof2}1800000
  • Syde γ β √{powerof2} vni. 1250—√{powerof2}1512500
  • Basis greater semid. δ β—√{powerof2} v. 350—√{powerof2}112500
  • Basis lesse semidiameter ε δ √{powerof2} v. 37 ½—√{powerof2}781 ¼
  • Altitude AB, 10

Theoremes of the internall Spheres circumscribing Solides. 1

TEtraedrons diameter is rationall and triple to the contayned Spheres dia∣meter, the internall Spheres diameter supposed rationall.

The 2 Theoreme.

Tetraedrons sides square retayneth the same proportion to the square of the internall spheres Diameter, that his contayning spheres Diameter doth to his Axis.

The 3 Theoreme.

Tetraedrons Basis contayning circles semidiameter, is double in power to the contayned spheres Diameter.

The 4 Theoreme.

The square of Tetraedrons inscribed circles semidiameter to his Axis, hol∣deth the same proportion in power, that his perpendiculare dothe to his inscribed Spheres diameter.

The 5 Theoreme.

Tetraedrons side, is double to Octaedrons side, one Sphere beeing contayned of them both.

The 6 Theoreme.

The square of Tetraedrons side, to the square of the Cubes side, retayneth the same proportion, that Tetraedrons comprehending spheres Diameter dothe to his Axis.

The 7 Theoreme.

The diameter of Tetraedrons inscribed circle, is equall to the Diameter of Octaedrons contayning circle.

The 8 Theoreme.

Tetraedrons diameter to the diameter of the sphere that comprehendeth Octaedron and the Cube, hath the proportion of 3 to his quadrate roote.

The 9 Theoreme.

Octaedron and the Cubes comprehending spheres diameter beeing rationall, the diameter of Icosaedron and Dodecaedrons contayning sphere is an irratio∣nal Minor, bearing proportion to the forenamed diameter, as √{powerof2} v. 5—√{powerof2}20 vnto 1.

The 10 Theoreme.

Icosaedron and Dodecaedrons axis beeing rationall, their comprehending spheres dimetiente, shall bee an irrationall Minor, bearing proportion thervnto, as √{powerof2} vni. 60—√{powerof2}2880 vnto an vnitie.

The 11 theoreme.

Icosaedrons axis beeing rationall, his side is an irrationall Apotome propor∣tioned therevnto, as √{powerof2} v. 42—√{powerof2}1620 vnto an vnitie.

The 12 theoreme.

Dodecaedrons Axis rationall, his side is a surde Minor proportioned to the Axis, as √{powerof2} v. 50—√{powerof2}2420 vnto an vnitie.

The 13 Theoreme.

Icosaedrons side being an irrationall Apotome of the sixte order, Dodecae∣drons side is an irrationall Minor, retayning suche Proportion thereto, as √{powerof2} v. 50—√{powerof2}2420 vnto √{powerof2}27—√{powerof2}15.

The 14 Theoreme.

Icosaedrons axis being rational, his basis cōtayning circles semidiameter is an Apotome of the first order bearing proportiō to the Axis, as 3—√{powerof2}5 vnto an vnitie.

The 15 Theoreme.

Dodecaedrons axis being rationall, his Pētagonall basis greater semidiameter is an Apotome of the first order proportionall to his side, as √{powerof2} v 7—√{powerof2}45 to √{powerof2} v. 25—√{powerof2}605.