Pal(l)adian Arithmetic as Revealed in the Palazzo Della Torre, Verona
Abstract
Room ratios in Palladio’s design for the Palazzo Della Torre mostly ignore his own canonical recommendations and none of the rooms exemplify his rules for room heights. Proportionately, however, the scheme, in plan and elevation, is a brilliant celebration of the cube root, just three years after Cardano published the solution to the cubic equation using methods passed to him by Tartaglia. Daniele Barbaro, Tartaglia and Cardano were all known to each other, and it seems most likely that Palladio would have taken a personal interest in the matter. The cube root that underpins the proportional scheme is Delian, that is, the cube root of 2 cited in Vitruvius. Palladio derives other roots of 2 in anticipation of the arithmetics which emerged in the early seventeenth century for the equal temperament musical scale. Of course, it must be understood that only rational convergents to the cube root of 2 are used. The relationship of room plan and elevation ratios in Palazzo Della Torre is illustrated by using the technique shown in Barbaro La Practica della Perspecttiva in which threedimensional objects are unfolded to make twodimensional “nets”, but figures are not used.
Keywords
Andrea Palladio Palazzo Della Torre Mathematical means Pythagorean arithmetic Renaissance architecture Doubling the cube Leonardo da Vinci Number theoryPalladio, in Book I, chapter XXI of his Four Books on Architecture, sets out
seven types of room that are the most beautiful and wellproportioned and turn out better: they can be made circular, though these are rare; or square; or their length will equal the diagonal of the square of the breadth; or a square and a third; or a square and a half; or a square and twothirds; or two squares (Palladio 1997, p. 57).
That is, he defines, apart from the circle, rectangles of ratios 1/1, √2/1, 4/3, 3/2, 5/3, and 2/1. In a previous publication, I have pointed out that between the extremes of 1/1 and 2/1; 4/3 is the harmonic mean, √2/1 is the geometric mean, 3/2 the arithmetic mean and 5/3 the contraharmonic mean (March 2003, p. 11).
These ratios have also been identified with musical intervals in just intonation: the unison, 1/1; perfect fourth, 4/3; augmented fourth/diminished fifth, √2/1: perfect fifth, 3/2; major sixth, 5/3; perfect octave, 2/1. Wittkower (1998) promoted this analogy. However, one interval that is noticeably missing from Palladio’s account is the major third, 5/4. This ratio is included in Serlio’s sevenpart canon where a square and a quarter replaces Palladio’s circle:
There are many rectangular proportions. I shall set down here, however, the seven principle ones which the architect can make use of for various things and can adapt to many situations—that which will not serve in one place could serve for another—since he will know how to use them (Hart and Hicks 1996, p. 30).
5/4 is not one of the eleven classical means together enumerated by Nicomachus and Pappus between the extremes 1 and 2 (Heath 1981, p. 87).
Also, in his Book I, chapter XXIII, Palladio sets out three methods to determine the heights of rooms: effectively the arithmetic, geometric, and harmonic means of their lengths and breadths. He concludes:
These heights are related to each other in the following way: the first is greater than the second and this is greater than the third; so we should make use of each of these heights depending on which one will turn out well to ensure that most of the rooms of different sizes have vaults of an equal height and those vaults will still be in proportion to them, so that they turn out to be beautiful to the eye and practical for the floor or pavement which will go above them because they will all end up on the same level. There are other heights for vaults which do not come under any rule, and the architect will make use of these according to his judgement and practical circumstances (Palladio 1997, pp. 58–59).

The arithmetic mean is (x + y)/2;

The geometric mean is √(xy);

The harmonic mean is 2xy/(x + y).

It is an elementary exercise to show ((x + y)/2) > √(xy) > (2xy/(x + y)).
It is known that a number lying between rational numbers p/q and p′/q′ is (p + p′)/(q + q′). Further, if p/q is a convergent value to √N, Nq/p will be a companion convergent since (p/q)·(Nq/p) = N.
By midcentury the Welsh physician and mathematician Robert Recorde had computed the doubling of a cube with sides 3 feet as requiring sides “3 feet and 77/100 and 1/7 of 1/160” (Recorde 1969). Recorde was in the court of Edward VI during the time Barbaro was Venetian Ambassador from 1548 to 1551. The approach, like the result, was untidy. It seems evident that a similar method to that for the extraction of square roots might apply to the extraction of cube roots. It has been shown that 5/4 is less than the cube root of 2 and it is evident that 4/3 is greater. The cube root of 2 must lie between these two extremes. (5 + 4)/(4 + 3) = 9/7 is such a value and it is greater. The solution must lie between this upper value and the lower 5/4. Such a value is 14/11. This too is greater, so (5 + 14)/(4 + 11) = 19/15 is a better convergent, but still larger than required.
(5 + 19)/(4 + 15) = 24/19 is yet another improvement, and so on, mediating between upper and lower estimates. A modern reader may check on these values by resort to the decimal system, still half a century away from Palladio’s day. To the fifth decimal place: Open image in new window
Ratios used for floor plans in the Four Books are shown in bold (March 1998, p. 278). Early convergents happen to belong to the Palladian canon.
Room Proportions in Palazzo Della Torre
On the ground floor, on entry, the principle room (labelled [1] in Fig. 5) is P.30 × P.19 and P.24 high, (where P. is a piede vicentino). The next room [2] in the enfilade is P.19 × P.15, then [3] P.19 × P.11, then [4] P.19 × P. 19, and round the corner an undimensioned room, then across the entrance from the street [5] P.19 × P.17. All these rooms are ostensibly P.24 high. Apart from two square corner rooms, none of the remaining rooms conform to Palladio’s canon stated so clearly in the Four Books. On ascending the grand oval staircase—the type of which is attributed to Marc’Antonio Barbaro, in Book I, chapter XXVIII (Palladio 1997, p. 67)—the first rooms on arrival [6], on either side of the vestibule, are dimensioned P.22 ½ by P.18, a ratio of 5/4. Then up again is the great hall [7], spanning over the courtyard P.34 × P.32 and again P.24 high. In summary, each room is defined dimensionally by length L, width W, and height H. For comparison, the recommended largest and smallest heights given by Palladio’s method using the arithmetic and harmonic means of length and width are given, H_{A} and H_{G}: Open image in new window
It is seen that only room [1] satisfies the recommendation closely. Lower ceilings are suggested for rooms [2] to [5], especially room [3]. Room [6] is not given a ceiling height, but note that the harmonic mean is a whole number. The great hall exceeds the stated ceiling height by almost a third. However, Palladio’s practical advice is to level the ceilings for the sake of level floors above, and he appears to take the ceiling of the first room [1] as key. This is acceptable, it appears, in all the rooms except the smallest [3]. Here there is an external spiral staircase, and the fenestration indicates a possible mezzanine. Likewise, in room [1] the fenestration suggests the possibility of an open gallery at mezzanine level. It should also be noted that the Ionic columns are P.24 high.
The rational ratios of floor plans F, and the walls (long L and short S) are set out in parallel with their cube and square root proxies: Open image in new window
This ratio can also be expressed √9/√8, that is to say the geometric mean between the unison, 1/1, and the major second, 9/8.
The room height in [3] is twice that of the width. It is suggested above that there could be a mezzanine. Within the system of proportioning revealed in the Palazzo Della Torre, above, a hypothetical height of P.15 is proposed for room [3], leaving a reasonable height of P.9 for the mezzanine including its floor structure. The long wall is then proportioned to the cube root of 2, 19/15, and the short wall to the ratio 15/11. This latter ratio is also to found in the floor plan of the Villa Rotunda. This ratio has a beautiful symmetry: the square root of three to the cube root of two. Open image in new window
That room [6] is the only rectangular room with a whole number geometric mean height, P.20, suggests that this might be explored further. Such a height would match the unmarked second storey room shown in section at the street entrances, which is less than P.24. The wall ratios are then 20/18 = 9/8 and 22 ½/20 = 10/9. The latter ratio is used later in the Olympic room of the Villa Barbaro at Maser. These ratios are associated with the major and minor tones of the then contemporary just intonation scale. A value in between these two is (10 + 9)/(9 + 8) = 19/17, the floor plan ratio of room [5]. 9/8 does not seem to be a ratio Palladio uses in his palazzi and villa plans in the Four Books. Nevertheless, in musical theory of the period it was a matter of dispute as to whether the tone might be divided into two equal parts, semitones (Palisca 1985, pp. 88–110). An approximation was accepted by some. They argued that doubling the tone 9/8 × 2 = 18/16, while 18/16 = (18/17) · (17/16), and that 18/17 was a minor semitone, 17/16 a major semitone. Twelve minor semitones just fall short of the octave. Twelve major semitones exceed the octave. Indeed, 18/17 was generally accepted by lutenists and luthiers for tuning purposes. It is noteworthy, that the arithmetic shown in this Palazzo preludes musicians’ quest for equal temperament later in the century, in which roots of 2—the cube root, in particular—played a key part. Musical intervals implicit in Palazzo Della Torre include: Open image in new window
Palladio also indicates two details: the window dimensions and the diameter of the Ionic columns. The window has dimensions 7 ¾ by 3 ½, a ratio of 31/14. This can be thought of as a square 14/14 and a rectangle 17/14. In turn, the ratio 17/14 may be derived as a composite using convergents already recognized above: Open image in new window
ANDREAS sums to 32, PALADIVS to 34 using digits only. These are the dimensions of the great hall. The floor area of the great hall is 34·32 = 1088. This happens to be the number of VITRVVIVS using digits, tens and hundreds. From the inside of one entrance to the other entrance—that is, the length of the whole courtyard—is P.132 (50 + 32 + 50). This is a number for PALLADIVS computed in triangular numbers—one of the not uncommon methods. Further, the first rooms [6] to be entered from the grand stair have a floor area of 405, the number of PALLADIVS using digits, tens and hundreds: Open image in new window
Pal(l)adio received his Latin name when he was with Count Gian Giorgio Trissino. In its time, it is not improbable that Trissino performed some alphanumeric computations to arrive at a name relating his protégé to Vitruvius. Wittkower draws attention to Giuseppe Gualdo:
Palladio’s contemporary, [who] wrote in his reliable life of the architect, that ‘when Trissino noticed that Palladio was a very spirited young man with much inclination for mathematics, he decided in order to cultivate his genius to explain Vitruvius to him, … (Wittkower 1998, p. 62).
Background
Palazzo Della Torre is no more. It was bombed during WWII in January 1945 (Zorzi 1965; Puppi 1975). Branko Mitrović (2004) provides an axonometric reconstruction of the scheme and argues convincingly for threedimensional analyses of Palladio’s architecture. Pythagorean arithmetics were a standard texts among humanists (March 1998, 2008). The eleven means of Nicomachus/Pappus are enumerated in (Heath 1981). The means are computed between the six ratios between a,b,c (a > b>c) where b is a mean and the ratios between positive differences A = b−c, B = a−c, C = a−b. Between the extremes a = 2/1 and c = 1/1 the means are, in the order presented in Heath (1981, p. 87): Open image in new window
Of Palladio’s canon, 3/2 appears three times, 4/3 twice, 5/3 and √2/1 once. Items 5 and 9 have means equal to the golden section. If the golden section had any aesthetic value at the time, surely this Pythagorean arithmetic relationship would have been noted and grasped.
The Palazzo Della Torre is assumed to have been planned in 1551, and was still under construction in 1568, at the death of Count Giovanni Battista Della Torre. In 1545 Cardano published Artis magnae sive de regulis algebraicis liber unus, or Ars magna (Cardano 1993): a book considered to be one of the great books of the Renaissance and a significant landmark in the history of mathematics. From 1535 onwards there had been a public rumpus over the authorship of the solution to the cubic equation; something Luca Pacioli, at the turn of the century, in his Summa de arithmetica, geometria, proportioni et proportionalità of 1494 had declared could not be done and was as impossible as squaring the circle. At root it was rivalry between two mathematicians, Niccolò Tartaglia in Venice and Girolamo Cardano in Milan. Tartaglia taught mathematics at Verona, Brescia and Venice. In the course of what Oystein Ore, in his foreword to the translation of Cardano’s Ars Magna, describes as ‘one of the most violent feuds in the history of science’ (Cardano 1993, p. ix), public notices, cartelli, were published over several months; in 1548 a contest was held in Santa Maria del Giardino dei Minori Osservanti, Milan; challenges were arbitrated by Don Ferrante di Gonzaga, governor of Milan; the victor by default, Cardano’s secretary, Ludovico Ferrari, was thought to have been announced and rewarded (Jayawardene 2008). The Venetian, defeated, slunk home. It was not a matter that interested persons could ignore, especially in the Venetian Republic. At midcentury, it would not be unreasonable to suggest that cubes and cubic roots were in the air among the numerate in the Republic, including Verona. Is it possible that proportionality in Palazzo Della Torre celebrates contemporary mathematical advances? Or at the very least, is it a paean to the Vitruvian story about the doubling of the altar at Delos? (Vitruvius 2009, p. 147).
The fourteenthcentury Aristotelian polymath Nicole Oresme established the use of fractional exponents in De proportionibus proportionum around the midfourteenth century (Oresme 1966). The notion could not have been unfamiliar two centuries later. In the presentation here, modern symbolism is used.
Mathematics in sixteenthcentury Italy was twofaced. One face turned towards the future as Cardano does in Ars Magna with his acceptance of the square roots of negative numbers before the later understanding of complex numbers (Rose 1975). The other face looked back and played to occult themes—hermetic, cabalistic, neoPlatonic, Pythagorean (Yates 1983; Copenhaver 1992; Allen 1994). Yet even a progressive like Cardano had his conservative side as an astrologer (Grafton 1999). The expulsion of Jews from Spain in 1492 led to a substantial migration to the Venetian Republic. Frances Yates (1933) tells of the Jewish influence on the Venetian friar, Francesco Giorgi. Both the Greek and Hebrew languages use their alphabets for numbers. That is to say they do not have separate symbols for numerals. It is not surprising that alphanumeric transformations are common in both (Heath 1921; Cajori 1993). Johann Reuchlin, in his De arte cabalistica of 1516 (Reuchlin 1983) had polished his Latin with Ermolao Barbaro (Geiger 1964), uncle to Daniele Barbaro, with whom Palladio was collaborating on the edition of Vitruvius. In 1531 Henry Cornelius Agrippa presented a ninesquare table to enable Latin words and names to be converted into numbers using the 23letter Latin alphabet (Agrippa 2009). It should also be noted that Arabic languages were alphanumeric. Venetians would have been familiar with this through trade with the Ottomans and North Africa (Ifrah 1985).
This paper has indicated one method of computing convergent rational values for roots. Others exist. The one chosen is derived from Fowler (1999). It was always possible to use square and cube tables with parallel columns, one with a simple value and the other with the values multiplied by N, the number of the root required. Thus, for N = 2 with cubes Open image in new window from which (4/3)^{3} is seen to exceed 2, while (5/4)^{3} falls short. It is also possible that among the secrets held by masons were root tables. In discussing proportionality in Frank Lloyd Wright’s early work, I drew attention to a carpenter’s manual containing exactly such tables with the same convergents used above (March 1995; Anonymous 1899). Pal(l)adio, born Andrea di Pietro della Gondola, had been trained in the trades.
References
 Agrippa, Henry Cornelius. 2009. Three Books of Occult Philosophy (1531). Trans. James Freake, ed. Donald Tyson. Woodbury MN: Llewellyn Publications.Google Scholar
 Allen, Michael J. B. 1994. Nuptial Arithmetic. London: University of California Press.Google Scholar
 Anonymous. 1899. Steel Square and its Uses: a Complete UptoDate Encyclopedia on the Practical Uses of the Steel Square, Showing How It Can Be Used by the Carpenter in his Daily Work. River Forest: Duke HillGoogle Scholar
 Barbaro, Daniele. 1567. I Dieci Libri dell’architettura di M. Viruvio tradotti e commentati da Mons. Daniel Barbaro eletto Patriarca d’Aquileia, da lui riveduti & ampliati; & hora in piu commoda forma ridotti. Venice: Francesco de’ Franceschi Senese e Giovanni Chrieger Alemano Compagni. Latin edition: M. Vitruvvii Pollionis De Architectura Libri Decem cum commentariis, Venice, 1567.Google Scholar
 Cajori, Florian. 1993. A History of Mathematical Notation (1928). New York: Dover.Google Scholar
 Cardano, Girolamo. 1993. Ars Magna or the Rules of Algebra (1545). T. Richard Witmer, trans. New York: Dover.Google Scholar
 Copenhaver, Brian P. 1992. Hermetica. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
 Fowler, David. 1999. The Mathematics of Plato’s Academy: A New Reconstruction. Oxford: Clarendon Press.zbMATHGoogle Scholar
 Geiger, L. 1964. Johann Reuchlin sein Leben und seine Werke. Nieuwkoop: De Graaf.Google Scholar
 Grafton, Anthony. 1999. Cardano’s Cosmos: The Worlds and Works of a Renaissance Astrologer. Cambridge MA: Harvard University Press.Google Scholar
 Hart, Vaughan, and Peter Hicks. 1996. Sebastiano Serlio on Architecture, vol. I. New Haven: Yale University Press.Google Scholar
 Heath, Thomas. 1981. A History of Greek Mathematics (1921). New York: Dover.Google Scholar
 Ifrah, Georges. 1985. From One to Zero. New York: Viking Penguin Inc.zbMATHGoogle Scholar
 Jayawardene, S. A. 2008. Ferrari, Ludovico. Complete Dictionary of Scientific Biography. http://www.encyclopedia.com. Retrieved 6 Oct. 2014.
 March, Lionel. 1995. Sources of characteristic spatial relations in Frank Lloyd Wright’s decorative designs. In: Frank Lloyd Wright: The Phoenix Papers, ed. Paul K. Zygas, 12–49. Tucson: University of Arizona Press.Google Scholar
 March, Lionel. 1998. Architectonics of Humanism: Essays on Number in Architecture. London: Academy Editions.Google Scholar
 March, Lionel. 2003. Foreword. In: Silvio Belli, On Ratio and Proportion: the common properties of quantity, trans. and eds. Stephen R. Wassell and Kim Williams, 7–13. Florence. Kim Williams Books.Google Scholar
 March, Lionel. 2008. Palladio, Pythagoreanism and Renaissance Mathematics. Nexus Network Journal 10, 2: 227–243.CrossRefGoogle Scholar
 Mitrović, Branko. 2004. Learning from Palladio. New York: W. W. Norton & Company.Google Scholar
 Oresme, Nicole. 1966. De proportionibus proportionum and Ad pauca respicientes (ca.1350). Edward Grant, ed. and trans. Madison: The University of Wisconsin Press.Google Scholar
 Palisca, Claude V. 1985. Humanism in Italian Renaissance Musical Thought. New Haven: Yale University Press.Google Scholar
 Palladio, Andrea. 1997. The Four Books of Architecture. Robert Tavernor and Richard Schofield, trans. Cambridge MA: MIT Press.Google Scholar
 Puppi, Lionello. 1975. Andrea Palladio. London. Phaidon Press Limited.Google Scholar
 Recorde, Robert. 1969. The Whetstone of Witte (1557). New York: Da Capo Press.Google Scholar
 Reuchlin, Johann. 1983. De arte cabalistica (1516). Trans. Martin and Sarah Goodman. New York: Arabis Books.Google Scholar
 Reti, Ladislao ed. 1974. The Unknown Leonardo. New York: Abradale Press.Google Scholar
 Rose, Paul Lawrence. 1975. The Italian Renaissance of Mathematics. Geneva: Librairie Droz.Google Scholar
 Vitruvius. 2009. On Architecture. Richard Schofield, trans. London: Penguin Classics.Google Scholar
 Wittkower, Rudolf. 1998. Architectural Principles in the Age of Humanism (1949). London: Academy Editions.Google Scholar
 Yates, Frances A. 1983. The Occult Philosophy in the Elizabethan Age. London: ARK.Google Scholar
 Zorzi, Giangiorgio. 1965. Le opere pubbliche e i palazzi privati di Andrea Palladio. Venice: Neri Pozza.Google Scholar