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 .16. Probleme. The diameter of any sphere knovven, to searche out the sides, Axes and contayning or contayned circles semidiameters of all suche bodyes regulare as are therein to be described, both Arithmeti∣cally and Geometrically. (Book 16)

COnsidering that in any Sphere all the regulare bodyes maye be described as it is by Euclide sufficiently demon∣strated, and that both the proportion of their sides Axes and perpendiculars to the spheres dimetient, and also the maner of working in euery of them is differēt, I thynk it best to adioyn seueral rules for euery of their operations, & first for y^e in∣uention

of their sides peruse the preceptes following: square the spheres diameter, and from the producte deducte ⅓, the roote quad•••••••• of the re∣mainder is the Tetraedrons side: Double the square of the Spheres se∣midiameter, so haue ye the square of Octaedrons side, diuide the Diame∣ters square by 3, the roote quadrate of your quotient is the Cubes side, For the Icosaedrons side ye shall diuide the Diameters square by 5, the rote of the quotient note, and supposing the same a circles semidiameter, by the second Probleme searche out the circles pentagonall Corde, for that is the Icosaedrons side. Finally, for the side of Dodecaedron, ye shal diuide the square of the spheres Diameter by 3, and from the producte extracte the quadrate roote, this roote diuided by extreame and meane proportion, (as ye were taught in the firste Probleme) yéeldeth for hys greater parte the Dodecaedrons side. Nowe the sides of euery regulare body thus founde, for their perpendiculares, Axes, and Semidiameters of their contayning and contayned circles, ye shall resorte for Tetrae∣dron to the sixte Probleme, for Octaedron to the seauenth, for Hexaedrō to the eight, and so in order for the reste, wher ye are taughte by the side to searche out all those forenamed lines, and so shall ye finde the exacte quantitie of all perpendiculares Axes and semidiameters of any regu∣lare body in that sphere contayned.

Example.

It vvould be ouer tedious to shevve the calculation for finding of euery particular line, vvherfore I shall only giue examples of the sides, for that the practize of their rules hath not bene yet set forth in any former Probleme. Admit the spheres Diameter that shall comprehende these regulare bodies 10, the square thereof diuided by 3, yeldeth 33 ⅓ vvhich deducted from 100, leueth 66 ⅔, the quadrate rote thereof is the inscribed Tetraedrons side. The square of the spheres semid. is 25, vvhich doubled maketh 50, the Zēzike rote thereof, is the Octaedrons side. 100 diuided agayne by 3, yeeldeth as before 33 ⅓, the roote square thereof is the Cubes side, vvhich diuided by extreame and meane proportion (as vvas taught in the firste Probleme) maketh the greater segment √{powerof2}41 ⅔—√{powerof2}8 ⅓, so much I affirme the contayned Do∣decaedrons side. Novv, for the side of Icosaedron I diuide 100 by 5, thereof ariseth 20, vvhose roote quadrate admitted a side Hexagonall, his corres∣pondente Corde Pentagonall by the seconde Probleme ye shall finde

√{powerof2} v 50—√{powerof2}500, the sides thus knovvne for the Axis, Semidiameters, and other lines, 〈…〉〈…〉 vse the same supputation that you did in those 5 Pro∣blemes past, vvhere ye vvere taught by the sides knovvne to attaine all the other lines, and for more plainnesse I shal at the end of this chapiter adioyne a Table containing the true quantitie of all the rest, vvhich ye may vse in stead of an Example to direct you, if happely you erre in your supputations, and for the farther satisfaction of such as seeke to reach beyond the commō sort, and vvill not content them selues vvith bare rules and preceptes, vnlesse they may also conceiue some grounde and reason of their vvorkings, I haue thought good to euery of these Problemes ensuing, to adioyne his peculiare figure, vvith meanes Geometricall (no regarde had to Irrationall numbers vvithout aide of Arithmeticall supputation) to searche out the sides, Diame∣ters, and Axis, of al the regular bodies inscribed or circumscribed of spheres, by knovvledge of their Diameters, or mutually cōferred together by knovv∣ledge of some side, according to the tenure of the Chapiter vvherin they are placed. And although breuitie (vvhich in this trifeling treatise I haue chefely affected) compell me not to stay in making demonstration of euery rule and Theoreme, yet the very construction of the figures vvell vvayed and confer∣red vvith Euclides 5 last bookes of Solides, vvill geue sufficient light to the in∣genious both to vnderstand the cause of these, and to inuent many mo vvher∣of there is no ende.

Geometrically vvithout aide of Arithmeticall calculation, to attaine the quantitie of all these forenamed lines ye shall thus vvorke.

Admit the Diameter of the comprehending sphere geuen AB, vvhich ye shall as vvas taught in the first booke diuide in tvvo equal partes at C, and in three at E, AE being a third part, vppon either of those sections errear Per∣pendiculars, and (describing a Semicircle vpon the Diameter) note their in∣tersections vvith FD, dravving lines from either of them to AB, so is AF the Cubes side, AD Octaedrons side, FB the side of Tatraedron, AF di∣uided by extreame and meane proportion (as ye vver taught in the first Pro∣bleme at G) maketh AG the Dodecaedrons side, vvhich extended out to H, ye shall make FH equall to FG, dravving the right line HB, and from F extend a Paralele to HB, till it crosse the Diameter in I, erecting there∣vppon the Perpendicular IK, so is the Cord KB the inscribed Icosaedrons side, IL is a third part of IB, ML a Paralele to IK, MB is the Diame∣ter of Icosaedrons basis containing circle, vvhose medietie MN is the Dia∣meter of the contained circle, the halfe therof MS parted by extreame and

meane proportion in V, so as SV be the greater segment, VN vvil be the Semidiameter of Dodecaedrons contained circle, and NB the Semidimeti∣ent of his containing circle, NC the Axis both of Icosaedron and Dodeca∣edron, VB is the Perpendicular of Dodecaedrons basis, and MA his So∣lides altitude, BS the perpendicular of Icosaedrons basis, and MA like∣vvise his Solides altitude, AF (the inscribed cubes side) is also Dodecae∣drons basis line Diagonal, EF is the greater Semidiameter of Tetraedrons base, and EP his medietie the lesser Semidiameter, EC Tetraedrons Axis, EB his Perpendicular or altitude, FB the cubes line Diagonall, OB his medietie the greater Semidiameter of the cubes base, OC the lesse Semidia∣meter, and Hexaedrons axis Octaedrons containing circles Semidimetient, OF the Sediameter of his contained circle RO, his Axis CO, and AF his altitude. Thus haue ye Geometrically in one figure the exacte quantities and proportions of all the regular bodyes sides, Diameters, Axes, Perpendi∣culars, and lines Diagonall, vvhereby ye may also be able bothe to conceiue some reason of such rules as are past, or Theoremes that shall ensue. And also inuent diuers meanes to abreuiate suche painful calculation as by the former rules ye shalbe forced to enter into, vvhile ye laboure vvith irrationall num∣ber to searche out the hidden proportions of these vnknovvne lines, as by proofe the industrious vvill soone perceiue.

[illustration]

The comprehen∣ding Spheres Di+metient geuen AB 10

• ...Tetraedrons
  • Syde, BF √{powerof2}66 ⅔
  • Basis greater semid, FE, √{powerof2}22 2/••
  • Basis lesse semid. PE. √{powerof2}5 5/••
  • Axis, EC, 1 ⅔
  • Altitude, EB, 6 ⅔
• ...Hexaedrons
  • Syde, FA, √{powerof2}33 ⅓
  • Basis greater Semid. OF, √{powerof2}16 2/••
  • Basis lesse semid. OC, √{powerof2}8 ⅓
  • Axis CO, √{powerof2}8 ⅓
  • Altitude AF, √{powerof2}33 ⅓
• ...Octaedrons
  • Syde AD, √{powerof2}50
  • Basis greater semid. OF, √{powerof2}16 ⅔
  • Basis lesse semid. OR √{powerof2}4 ⅓
  • Axis QF, √{powerof2}8 ⅓
  • Altitude AF √{powerof2}33 ⅓
• ...Icosaedrons
  • Syde KB, √{powerof2} v. 50—√{powerof2}500
  • Basis greater sem. MN, √{powerof2} v. 16 ⅔—√{powerof2}55 5/••
  • Basis lesse semid. N.S √{powerof2} v. 4 ⅙—√{powerof2}3 17/36
  • Axis, CN, √{powerof2} v. 8 ⅓+√{powerof2}55 5/9
  • Altitude MA, √{powerof2} v. 33 ⅓+√{powerof2}888 8/••
• ...dodecaedrōs
  • Syde AG, √{powerof2}41 ⅔+√{powerof2}8 ⅓
  • Basis greater semi. NB, √{powerof2} v. 16 ⅔—√{powerof2}55 5/9
  • Basis lesse semid. NV, √{powerof2} v. 4 ⅙+√{powerof2}3 17/36
  • Axis CN, √{powerof2} v. ⅓+√{powerof2}55 5/9
  • Altitude MA, √{powerof2} v. 33 ⅓+√{powerof2}888 8/9

Theoremes of the Regular bodyes in one containing sphere described. Theoreme first.

THe cōtaining spheres diameters square to the square of the inscribed Tetrae∣drons side is as 3 to 2.

The second theoreme.

The spheres Dimetient is in power double to Octaedrons side.

The third theoreme.

The spheres Diameter is in power triple to the Cubes side.

The fourth theoreme.

The spheres Dimetient being rationall, Icosaedrons side is suche an irra∣rationall

as Euclide calleth Minor, and beareth proportion to the Diameter, 〈◊〉〈◊〉 √{powerof2} vni. ½—√{powerof2}1/20 to 1.

The fifth theoreme.

The spheres Dimetient rationall, Dodecaedrons side is an irrationall Apo∣tome, retaining proportion to the dimetient, as √{powerof2}5/12—√{powerof2}1/12 vnto 1.

The sixth theoreme.

Tetraedrons Axis, is a sixth part of his spheres dimetient.

The seuenth theoreme.

Octaedron and the Cubes Axis are equall, and triple in power to the Axis of Tetraedron.

The eight theoreme.

The Axis of Icosaedron and Dodecaedron are equall, either of them being such an irrationall as Euclide calleth Maior, the spheres dimetient admitted ra∣tionall.

The nynth Theoreme.

Tetraedrons side, is the meane proportionall betweene his Perpendicular and his diameter.

The tenth Theoreme.

Octaedrons containing circles semidimetient is meane in proportion betweene his inscribed circles semidiameter and Tetraedrons side.

The eleuenth Theoreme.

The Hexaedrons side is a meane proportional betwene Tetraedrons side, and Octaedrons basis containing circles semidiameter.

The tvvelfth theoreme.

The cubes side being rational, the dodecaedrons side is an irrational Apotome, bearing proportion to the side of the Cube as √{powerof2}1 ¼—½ vnto 1.

The thirtenth theoreme.

Octaedrons side being rationall, the side of Icosaedron is such an irrationall as Euclide nameth Minor, and retaineth proportion to Octaedrons side as √{powerof2} vn. 1—√{powerof2}⅕ to an vnitie.

The fourtenth theoreme.

Octaedrons side being rationall, the Dodecaedrons side is an irrationall Apo∣tome, retaining proportion therunto as √{powerof2} v. 1—√{powerof2}5/9 vnto an vnitie.

The fiftenth theoreme.

The semidiameter of Dodecaedrons bases containing circle, hathe the same proportion to the Cubes side, that Icosaedrons side hath to the Diameter of his containing sphere.

The 16 theoreme.

Hexaedrons side beyng rationall, the conteyned Dodecaedrons circles semi∣diameter is an irrationall named by Euclide Maior, hauyng proportion to the Cubes side, as √{powerof2} v. ⅛+√{powerof2}••/320 vnto 1.

The 17 theoreme.

Tetraedrons side admitted rationall, the conteyned circles semidiameter of Icosaedron is an irrationall Minor, bearing proportion to the Tetraedrons side as √{powerof2} vniuers. ⅛—√{powerof2}1/320 vnto √{powerof2}2.

The 18 Theoreme.

Octaedrons conteyned circles diameter to the dimetient of dodecaedrons con∣teyned circle, hath proportion as 1 to √{powerof2} v. 1+√{powerof2}⅕.

The 19 theoreme.

Dodecaedrons conteyned circles semidiameter beeing an irrationall Maior, Icosaedrons conteyned circles semidimetient shall be an irrationall Minor, bea∣ring proportion as √{powerof2} v. 1+√{powerof2}⅕ to √{powerof2} vni. 1—√{powerof2}⅕.

The 20 Theoreme.

Dodecaedrons side, to octaedrons axis, retaynethe sutche proportion, as the greater part of a line diuided by extreame and meane proportion, to the medie∣tye of the whole.

The 21 Theoreme.

If from the square of dodecaedrons dimetient, ye subtracte the square of his side, the roote of the remaynder (diuided by extreme and meane proportion) maketh his greater part the Cubes side, and his lesse the dodecaedrons side.

The 22 theoreme.

A right lyne equall in power to the diameter and semidiameter of dodeca∣edrons basis conteyning circle, reteineth the same proportion to their conteyning spheres diameter, that dodecaedrons side doth to the side of Icosaedron.

The 23 theoreme.

Dodecaedrons side reteineth the same proportion to the side of the cube, that the semidiameter of dodecaedrons conteyning circle dothe to the diameter of his conteyned circle.

The 24 theoreme.

Dodecaedrons basis internall circles semidiameter diuided by extreme and meane proportion, maketh his greater part the Icosaedrons inscribed circles se∣midimetient.

The 25 theoreme.

If two lines equall in power to the spheres diameter retaine the proportion of

a lyne (diuided by extreme and meane proportion) to his greater part, the lesser of those lines is Icosaedrons side: but if their proportion bee as the whole to the lesser part, then is that lesse part the side of dodecaedron.