CHAPTER LIV¶
On the Body of Seventy-Two Bases (Septuaginta Duarum Basium), Its Origin, and Its Architectural Use in Cupolas and Vaults (Plates XXXIX–XL).
Continuing further along this path toward the sphere, we arrive at the celebrated Body of Seventy-Two Bases (Septuaginta Duarum Basium), depicted in solid and hollow perspective in Plates XXXIX and XL, which is of supreme utility in architecture for the coffered vaults of round temples, tribunes, and cupolas.
1. Generation of the Seventy-Two-Base Body (Plates XXXIX–XL)
Plate XXXIX (Folio 0269, Septuagintaduarum Basium Planum Solidu[m]) and Plate XL (Folio 0271, Septuagintaduarum Basium Vacuum): How is this body generated in the sphere? By the method of the seventeenth of the twelfth book (which is the fourteenth of the twelfth in Campanus), where our philosopher teaches how to inscribe a polyhedral solid inside the outer of two concentric spheres by dividing the great circle of the sphere into equal parts (for instance into 12 or 16 or 24 equal arcs) and drawing parallel latitudinal circles (paralleli) and longitudinal meridians (meridiani) from pole to pole!
Specifically, in the 72-base body:
Dividing the equatorial circle into 12 equal parts (dodecagon) and stepping in 3 latitudinal zones toward each pole produces:
48 trapezoidal or quadrangular faces (48 basi quadrilatere / d’elmuariffe, arranged in 4 latitudinal rings of 12 quadrangles each: 24 in the upper hemisphere and 24 in the lower hemisphere); and
24 isosceles triangular faces (24 basi triangulari, 12 meeting around the upper pole and 12 around the lower pole, closing the polar caps);
making \(48 + 24 =\) 72 plane bases in all! (Or in the hollowed ring form known in Tuscany as the mazzocchio, the open faceted torus of quadrangular coffers.)
2. Architectural Application: The Pantheon in Rome, Santa Maria delle Grazie, and Bramante’s Sacristy of San Satiro in Milan
This geometric generation is the very foundation of all round and polygonal vaulted architecture (tribune, cupole, e volte a botta o a crosera):
The Pantheon in Rome (La Rotonda): Thus the ancient Romans constructed the stupendous vault of that most famous temple called the Pantheon (today Sancta Maria Rotonda in Rome), originally dedicated to Cybele, Mother of all the Gods: its hemispherical dome is divided by meridians and parallels into stepped rings of quadrangular coffers (quadri / lacunari) diminishing proportionally toward the central oculus at the pole, exactly according to the zones of this 72-base body!
Milan — Santa Maria delle Grazie and San Satiro: And in our own day in your renowned city of Milan, we see this same geometric art practiced in the magnificent tribune of Santa Maria delle Grazie (constructed under Your Ducal Highness’s pious patronage) and in the admirable octagonal Sacristy of San Satiro designed by Maestro Donato Bramante, where the coffered vaults, pilasters, and perspective reliefs follow these exact circular and polygonal subdivisions.
Ludovico Sforza’s Heraldic Devices (*Imprese*) and Pythagoras’s Angle of Justice (*Angulus Iustitiae*): In the coffers and friezes of such vaults are carved roses, foliage, and Your Ducal Highness’s own famous heraldic devices (imprese)—the brush (la scopetta) and the axe (la cetta)—symbolizing how Your Highness cleanses and prunes away all vice and superfluous disorder through Justice (Iustitia). And well did the ancient philosophers, above all Pythagoras (as Vitruvius also recalls), call the right angle (\(90^{\circ}\), formed by the carpenter’s square, la squadra) the Angle of Justice (Angulus Iustitiae): for just as the right angle inclines neither to the acute (deficiency) nor to the obtuse (excess), standing invariable and upright between both extremes, so true Justice gives to every person what is rightfully theirs without tilting to either side!