UNITAS IV — DISTINCTIO V–VI: DE RADICIBUS CUBICIS ET DIMENSIONE CORPORUM

De Inventione Radicum Cubicarum, Mensuratione Pyramidum ac Sperarum, et Quinque Corporibus Regularibus

Fons Criticus: boncompagni:1862 (Scritti di Leonardo Pisano, Vol. II, pp. 148–202 · Cod. Urb. Lat. 292, fol. 92v–130v) · English Translation · Summary · GEOMETOR Notes

[Folio fol. 92v–99r | p. 148–159]

Distinctio V — Incipit Quinta de Radicibus Cubicis Inveniendis

Versio Anglica: Distinctio V — Cube Roots (pp. 148–159) · Commentarius: Notae ad Radices Cubicas

Explicit distinctio quarta; incipit quinta de radicibus cubicis inveniendis.

Quemadmodum in secunda distinctione ex compositione quadrati ostensa est inventio radicum quadratorum, ita ex sectione cubi $(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$ ostenditur inventio radicis cubice per ternarios figurarum, cum regula propinquationis in numeris non cubicis $a^3 + R$:

$$\sqrt[3]{a^3 + R} \approx a + \frac{R}{3a(a + 1) + 1}.$$

[Folio fol. 99v–121v | p. 159–190]

Distinctio VI (Pars I) — De Dimensione Columnarum, Pyramidum et Sperarum

Versio Anglica: Distinctio VI Part 1 — Solids & Spheres (pp. 159–190) · Commentarius: Notae ad Stereometriam

Pertica corporalis continet pedes cubicos $.CCXVI.$ ($216$). Ex undecimo et duodecimo libro Euclidis (euclid:elements) atque ex Archimenide (archimedes:sphere) demonstratur:

  • Columnae et Prismata: Mensurantur ex multiplicatione embadi basis in altitudinem (cathetum). [XII. 11]

  • Pyramides et Coni: Omnis pyramis est tertia pars prismatis sive columne habentis eandem basim et eandem altitudinem ($V = \frac{1}{3} B h$). [XII. 7, 10]

  • Pyramis Caput Abscisa: Mensuratur multiplicando tertiam partem catheti in aggregatum embadorum basis $B_1$, capitis $B_2$, et radicis producti unius in alterum ($\sqrt{B_1 B_2}$).

  • Spera: Omnis spera est due tertie partes ($\frac{2}{3}$) cilindri sibi circumscripti, cuius basis dyameter et altitudo equantur dyametro spere $d$; superficies vero spere quadrupla est maximo circulo in ea descripto ($V \approx \frac{11}{21} d^3$).

[Folio fol. 122r–130v | p. 190–202]

Distinctio VI (Pars II) — De Quinque Corporibus in Spera Descriptis

Versio Anglica: The Five Regular Polyhedra in Sphere Diameter 6 (pp. 190–202) · Commentarius: Notae ad Euclidis XIII et Paciolum

Invenire latera et corpulentias quinque corporum regularium tertijdecimi libri Euclidis in spera cuius dyameter sit perticarum $.VI.$. [XIII. 13–18]
Distinctio VI & Euclid XIII.18 — Five Regular Polyhedra in Sphere of Diameter 6 ( O A ): x**2 + y**2 - 9 = 0 [ A K ]: y = 0 / O C /: length = 3 / A C /: length = 3*sqrt(2) / K B /: length = 2*sqrt(2) / A B /: length = 2*sqrt(3) / B I /: length = 2*sqrt(6) / A N /: length = sqrt(18 - 6*sqrt(5)) / O M /: length = 3 / M I /: length = 3*sqrt(50 - 10*sqrt(5))/5 O = (0, 0) K = (-1, 0) A = (-3, 0) I = (3, 0) C = (0, 3) B = (-1, 2*sqrt(2)) M = (3*sqrt(5)/5, -6*sqrt(5)/5) N = (-3 + sqrt(12*(-2 + ((-2*sqrt(2) - 1)*(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1)) + (2 - sqrt(2))*sqrt(-(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1))**2 + 3*(2 - sqrt(2))**2 + 3*(-2*sqrt(2) - 1)**2))/((2 - sqrt(2))**2 + (-2*sqrt(2) - 1)**2) + sqrt(2))**2 + 12*(-2*sqrt(2) - 1 + ((2 - sqrt(2))*(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1)) - (-2*sqrt(2) - 1)*sqrt(-(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1))**2 + 3*(2 - sqrt(2))**2 + 3*(-2*sqrt(2) - 1)**2))/((2 - sqrt(2))**2 + (-2*sqrt(2) - 1)**2))**2)/6, sqrt(2)*sqrt(12*(-2 + ((-2*sqrt(2) - 1)*(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1)) + (2 - sqrt(2))*sqrt(-(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1))**2 + 3*(2 - sqrt(2))**2 + 3*(-2*sqrt(2) - 1)**2))/((2 - sqrt(2))**2 + (-2*sqrt(2) - 1)**2) + sqrt(2))**2 + 12*(-2*sqrt(2) - 1 + ((2 - sqrt(2))*(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1)) - (-2*sqrt(2) - 1)*sqrt(-(-6*sqrt(2) - 3 + (1 + 2*sqrt(2))*(2 - sqrt(2)) + (-2*sqrt(2) - 1)*(-sqrt(2) - 1))**2 + 3*(2 - sqrt(2))**2 + 3*(-2*sqrt(2) - 1)**2))/((2 - sqrt(2))**2 + (-2*sqrt(2) - 1)**2))**2)/6) O K A I C B M N
Semicirculus ( O A ) super dyametrum spere AI = 6 (centrum O, AK = 2): latus pyramidis / B I / (Q = 24), octaedri / A C / (Q = 18), cubi / A B / (Q = 12), dodecaedri / A N / (Q = 18 - rad. 180) et icosaedri / M I / (Q = 18 - rad. 64 4/5).

Sit dyameter spere $AI = 6$ super lineam AK circa centrum O in circulo (OA), sitque $AK = 2$ tertia pars ipsius $AI$, et erigantur perpendiculares KB et OC. [XIII. 18]

  1. Pyramis Quatuor Basium Triangularium (Tetraedrum): Latus eius est BI, cuius quadratum est $AI \cdot KI = 24$ ($e_4 = \sqrt{24}$), cathetus $4$, et corpulentia $\sqrt{192}$ ($8\sqrt{3}$). [XIII. 13]

  2. Cubus Sex Basium Quadratarum: Latus eius est AB, cuius quadratum est $AI \cdot AK = 12$ ($e_6 = \sqrt{12}$), et corpulentia $\sqrt{1{,}728}$ ($24\sqrt{3}$). [XIII. 15]

  3. Octaedrum Octo Basium Triangularium: Latus eius est AC, cuius quadratum est $AO^2 + OC^2 = 18$ ($e_8 = \sqrt{18}$), et corpulentia ex duabus pyramidibus quadratis est numerus rationalis $.XXXVI.$ ($36$). [XIII. 14]

[Folio fol. 125v–130v | p. 196–202]

Modus Dividendi Lineam Media et Extrema Proportione: Dodecaedrum et Ycosaedrum

Versio Anglica: Extreme and Mean Ratio: Dodecahedron & Icosahedron (pp. 196–202) · Commentarius: Notae ad Cordam Auream

Distinctio VI — Modus Dividendi Lineam Media et Extrema Proportione ( E M ): y**2 + (x - 2*sqrt(3))**2 - 3 = 0 ( C E ): (x - 2*sqrt(3))**2 + (y - sqrt(3))**2 - 3 = 0 ( B D ): x**2 + y**2 - (-2*sqrt(3) + 2*sqrt(15)/5)**2 - (-sqrt(3) + sqrt(15)/5)**2 = 0 [ B G ]: y = 0 [ B D ]: -sqrt(3)*x + 2*sqrt(3)*y = 0 / E C /: length = sqrt(3) / A B /: length = sqrt(18 - 6*sqrt(5)) / A E /: length = sqrt(18 - 6*sqrt(5)) / B G /: length = -sqrt(3) + sqrt(15) M = (sqrt(3), 0) B = (0, 0) E = (2*sqrt(3), 0) C = (2*sqrt(3), sqrt(3)) D = (-2*sqrt(15)/5 + 2*sqrt(3), -sqrt(15)/5 + sqrt(3)) G = (-sqrt(3) + sqrt(15), 0) A = (sqrt(3), sqrt(3)*sqrt(5 - 2*sqrt(5))) M B E C D G A
Divisio corde pentagoni BE = rad. 12 media et extrema proportione per perpendicularem / E C / = rad. 3, circulum ( C E ) et arcum ( B D ), efficiens latus dodecaedri / B G / = / A B / = / A E / = rad. 15 minus rad. 3.

  • Modus Dividendi Lineam Media et Extrema Proportione (p. 196): Quia corda pentagoni dodecaedri est latus cubi $BE = \sqrt{12}$, dividitur $BE$ media et extrema proportione [VI. 30, XIII. 17] secando eam per equalia in M, erigendo per circulum (EM) cathetum EC $= \sqrt{3}$, copulando BC ($= \sqrt{15}$), auferendo per circulum (CE) portionem $CD = \sqrt{3}$ in D, et transferendo $BD$ per circulum (BD) in G.

  • Dodecaedrum ($.XII.$ Basium Pentagonalium): Maior portio BG (sive AN in figura spere) et latera pentagoni AB ac AE in ABE sunt invicem equalia, linea irrationalis que dicitur recisum:

    $$e_{12} = \sqrt{15} - \sqrt{3}, \qquad e_{12}^2 = 18 - \sqrt{180}, \qquad af^2 = 6 - \sqrt{7\frac{1}{5}}, \qquad hf^2 = 3 + \sqrt{7\frac{1}{5}}.$$

  • Ycosaedrum ($.XX.$ Basium Triangularium): Per radium OM invenitur corda MI [XIII. 16], cuius quadratum est $e_{20}^2 = 18 - \sqrt{64\frac{4}{5}}$; et secundum librum quartumdecimum (Hypsicles) proportio corpulentie dodecaedri ad corpulentiam ycosaedri in eadem spera contentorum est sicut latus cubi ($e_6$) ad latus ycosaedri ($e_{20}$).

Explicit distinctio sexta in dimensione corporum.