Mathematics and the Palladian Villa

By John D. Breen

March 1999

The effect of mathematics on architecture seems to be at once obvious and oblique. The apparent significance may be deduced by noting the symmetry inherent in Renaissance structures and in the details of the arches and columns by which their facades are characterized. The interiors also possess a character distinct from that of the preceding middle ages, though perhaps not unlike that of antiquity. The rationale and inspiration for the dimensions employed, for the placement of structural elements, and for the design of interior rooms is less evident to the casual observer. Perhaps the Renaissance architect most conscientious of the principles of mathematics and by whom these principles were best applied was Andreas Palladio (1508-1580). This paper attempts to determine the ancient sources to which Palladio resorted when applying mathematical principles, and thereafter analyzes the application of these principles to the design of the Villa Emo.

Classical Origins

The Palladian influence on subsequent centuries is undeniable and may be confirmed by observing such disparate structures as St. Paul’s Cathedral in London, Stowe in Buckingham, and the Virginia State Capitol, the University of Virginia, and Monticello in Virginia. For his design of the latter two, Thomas Jefferson was inspired directly by, respectively, the Villa Rotunda and the incomplete Villa Trissino. To what, however, does one trace the origin of Palladio’s thought? The inescapable inspiration is that of the Greeks and of Rome, which Palladio acquired initially under the encouragement of the Humanist Gian Giorgio Trissino. Trissino motivated the young stonemason to educate himself in Latin literature, music, mathematics and architectural theory, notably that of Vitruvius. From his early analysis of architecture, he determined that "the ancient Romans, as in many other things, so in building well, vastly exceeded all those who have been since their time, [and] I proposed to myself Vitruvius for my master and guide." Such study, in correspondence with subsequent sojourns to Rome and zealous attention to ancient monuments rendered Palladio the quintessential and most influential classical architect of the Renaissance.

Perhaps the principal debt Palladio owes the ancients is that sentiment which prescribes for perfection in building the notions of mathematic exactitude and harmonic proportion. From the early Ionians to Pythagoras to Vitruvius, the mathematical conveyance of moderation, symmetry, and proportion was perceived not only as a desirable attribute of architecture, but as the preeminent basis upon which rested the structure of philosophy and of natural law. Palladio not only adopted the idea of mathematical symmetry and proportion, but clung to it with such tenacity that in certain structures he employed it beyond the realm of practical necessity. Nevertheless, such a conception corresponded perfectly with the Renaissance style and to the classical constructs by which they were inspired.

The Pythagoreans

Though the 6th century BC Milesian cosmologists pronounced a universal conflict of opposites to which the application of numbers might be a solution, the Pythagoreans of the following century were the progenitors of what might loosely be termed "mathematical philosophy". Aristotle relates that "the Pythagoreans, as they are called, devoted themselves to mathematics, they were the first to advance the study, and having been brought up in it, they thought its principles were the principles of all things."

Of particular consequence to the Pythagoreans was the realization that the musical intervals between notes on the lyre could be expressed numerically, pitch being a function of length, while intervals on the scale were convertible to ratios. Pythagoras observed that if a string of a particular tension is bisected, the pitch of the resultant string will be exactly one octave higher than that effected by plucking the string at its original length. Further, if the string is reduced to two-thirds or three-fourths its original length, the difference in pitch is, respectively, a fifth and a fourth. The mathematical expression of such a musical notation is therefore 1:2:3:4. As musical harmony depended on number, so the Pythagoreans reasoned did that of the universe. Again Aristotle:

since they saw that the attributes and the ratios of the musical scales were expressible in numbers; since then all other things seemed in their whole nature to be modeled after numbers, and numbers seemed to be the first things in the whole of nature, and the whole heaven to be a musical scale and a number.

This significance of the Pythagorean strain of musical harmony to Vitruvian (and consequently to Palladian) architecture is rendered apparent when Vitruvius declares that

the architect ought to understand [music] so that he may have knowledge of the canonical and mathematical theory, and besides be able to tune ballistae, catapultae, and scorpiones to the proper key. For to the right and left in the beams are the holes in the frames through which the strings twisted sinew are stretched by means of windlasses and bars, and these strings must not be clamped and made fast until they give the same correct note to the ear of the skilled workman.

Musical harmony as analog to the visual was a theme which resonated as much with Palladio as it had with his "master and guide." When Palladio suggested seven sets of the most beautiful and harmonious proportions to be used in the construction of rooms, he was echoing the Pythagorean musical scale. Of these, the lone exception to Pythagoras is the incommensurable proportion of the side of a unit square to its diagonal, or 1: square of 2.

Notwithstanding its prevalence in art and architecture, this proportion confounded the Greeks and later mathematicians by virtue of the simple fact that square of 2 does not exist as a rational number. The Pythagoreans were therefore confronted with incommensurable magnitudes, i.e. distances on a line that could be correlated with no number, that were none the less (or, perhaps, therefore) willingly employed by contemporary and subsequent architects. That the diagonal of the unit square is not a manifest number implies a metaphysical relationship to which the Pythagoreans and future architects applied special significance. To them its appearance within the Squared Circle embodied the dialogue between the Heavens (the circle) and the Earth (the square).

Pythagoras further ascribed to music the notion of the Limit, a concept necessary to substantiate the Unlimited or Indeterminate from which the Milesian Anaximander believed all things originated. As the Limited lends form to the Unlimited, so it exemplifies music, in which proportion and harmony are numerically expressible. The Pythagoreans extended this idea further by claiming not only a numeric cosmic harmony, but that all things are numbers. The derivation of this notion proceeds from the conception that the numerical elements are the even and the odd (the former being unlimited, the latter limited) and that numbers may be regarded spatially. One is a point, two a line, three a surface and four a solid, so that all bodies comprise points or units in space which combined constitute a number.

The sacred nature of numbers is evinced by the tetraktys, a figure comprising ten points arranged so as to form a pyramid with one point at the pinnacle and four along the base:

.

. .

. . .

. . . .

The ten points are the sum of the one, two, three and four points which constitute each respective "row" of the pyramid. Moreover, numbers are either square or oblong, a distinction made apparent when one proceeds to add to one point successive odd numbers in the form of gnomons, thereby obtaining square numbers; or alternatively to append even numbered gnomons to two points to attain oblong numbers. The odd gnomes present a fixed (limited) quadratic shape, while the evens permit a continually changing (unlimited) rectangle. This juxtaposition of numbers with geometry explains how the Pythagoreans might regard all things as numbers. As points, lines, and surfaces compose all bodies, so all bodies are numbers.

Numbers to the Pythagoreans represented more than the material, though the rationale for many such representations can apparently be ascribed only to caprice. For instance, while the notion that justice is four may seem somewhat comprehensible, to understand why animation is six requires an elastic imagination. Because three is the first masculine number and two the first feminine, their sum, five, is therefore marriage. The Pythagoreans also employed numbers to aid in tending the soul, which to them had been harmoniously subdivided in manner consistent with the "lambda" configuration:

1

2_ 3

4__ 9

8 ___27

They believed that God created the soul, and from this whole he marked off one section, then another twice the size of the first, then another half again as large as the second and three times the first, next a fourth twice the size of the second, a fifth three times the third, a sixth eight times the first, and a seventh 27 times the first.

The effect of Pythagorean numerology is of considerable consequence to later art and architecture. Wittkower emphasizes number theory and numerology as a basis for Palladian proportion. He also notes that with harmonic proportion, the concept of the "perfect number" guided certain aspects of Palladio’s interior design. As Carl Gable relates,

The numbers six and ten were deemed perfect because they reflect the proportion of the human body in several dimensions, including front-to-back and side-to-side. In other words you would feel comfortable in a room that was in the ratio 6-to-10 because the room would have the same proportions as your body. Then, in a grammatical challenge, the number ‘16’ was deemed to be the ‘most perfect’ number, primarily because it was the sum of the other two.

An example of an arrangement of rooms with such perfect proportions may be found in the Villa Cornaro, in which the central living area is six repetitions of the module of the perfect room, all set within a square. The room proportion in this instance is 3:5 (identical to 6:10), with the actual width being 16 Vicentine feet, the most perfect of all numbers.

Plato and Aristotle

Respecting the philosophy of mathematics and its relationship to the universe, the heir of Pythagoras on the Greek mainland would no doubt be Plato. While mathematics for the Pythagoreans was one aspect of a religious creed which utilized scepticism and mysticism in response to the diminution of Ionian civilization, so it was for Plato a means by which to express his doctrine that all that exists is ultimately a single Being. He claimed that the diffusion of harmony and proportion throughout creation is among the "best, highest and fairest ways" in which the one God and the multiplicity of things are coexistent as a single reality. Plato in The Timaeus declares that "The best bond is the one that effects the closest unity between itself and the terms it is combining; and this is best done by a continued geometrical proportion." Harmonious creation was therefore a reflection of our harmonious relationship with the one true Being. This sentiment anticipates that of Palladio when he states that each part of a house must correspond to every other and to the whole.

In contradistinction to Pythagoras, Plato considered natural bodies not to be numbers but, because they incorporate the irrational element of matter, merely to participate in numbers. The numerical triplets of which the ratio of right-angled isosceles and scalene triangles are composed contain, respectively, the irrational numbers square of 2 and square of 3, thereby expressing the Platonic contingency in natural objects. The architectural significance of square of 2 in one of the seven ideal shapes prescribed by Palladio may also derive from a certain sequence of fractions to which Plato alluded. Alternate fractions converge upwards and downwards to square of 2, so that the terms of the sequence in their original order are alternately greater and less than square of 2, while cooperatively converging toward the ultimate limit of square of 2.

In contrast to Plato, Aristotle argued against the theory that forms are numbers, stating that "mathematics has come to be the whole of philosophy for modern thinkers, though they say that it should be studied for the sake of other things". This criticism reverberates and is repeated throughout history, in disciplines ranging from architecture to economics. Aristotle questioned many aspects of Pythagorean-Platonic number mysticism and of the method by which it was employed. Of the Pythagoreans he asserted that

all the properties of numbers and scales which they could show to agree with the attributes and parts of the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth - the "counter-earth".

That numbers could be Platonic Forms, Beings or something intermediate between these and mathematics was to Aristotle incompatible with both reason and the use of numbers as arithmetic tools.

This is not to assert, however, that Aristotle was disinterested in composition or of no consequence to Renaissance architects. While he distanced himself from numerology, he did discuss in a general manner the technique of composition so that his conclusions could be applied to any class of thing. Any work, whether tragedy, musical composition or building, may be characterized by classical poetics (deriving from the Greek poiein, to make) and rhetoric, i.e. the extent to which it is complete, whole and perfect. Being constructed in accordance with classical rules demarcates a creation from its environment, and by virtue of its special order establishes it as a world within a world.

The internal and external sections of a work are distinguished by the schema of tripartition, which divides a building into two borders and an enclosed area, the beginning, middle and end of the "whole". Such a distinction anticipates the symmetry of Palladio, for contrary to music or tragedy, architecture may be observed from beginning-to-end as well as from end-to-beginning (left-to-right or right-to-left). The plan of a Palladian villa may be read in much the same fashion, as each side of the central axis is often a mirror of the other. As Palladio notes,

Beauty will result from the form and correspondence of the whole, with respect to the several parts with regard to each other, and of these again to the whole; that the structure may appear an entire and complete body, wherein each member agrees with the other, and all necessary to compose what you intend to form.

Palladio therefore adheres to Aristotle’s theory of non-contradiction by having each part of his buildings correspond to every other, accommodating the classical legend that the whole may be reconstructed from any remnant of its parts.

Frozen Music

Pythagoras, Plato and Aristotle were therefore of considerable influence to contemporary and subsequent architects, including Vitruvius and Palladio. The association of architecture with music is indisputable, and is evident in the design of the Parthenon. The plan of that structure is of a proportion 4:9, a relationship repeated in the ratio of the columns, and in that of the temple width to its height. Such marvelous harmony contributed to the Greek notion of architecture as "frozen music" , a concept adopted by the Roman masters whom Palladio so admired.

As the principal master to whom Palladio most frequently resorted, Vitruvius must of course be considered in any examination of ancient influence on Palladio’s design. In his discussion of the proportions of an atrium, Vitruvius reflects the influence of Pythagorean harmonics and anticipates four of Palladio’s seven most beautiful and harmonic proportions. Vitruvius recommends that

atriums are designed according to three classes. The first laid out by dividing the length into five parts and giving three parts to the width; the second, by dividing it into three parts and assigning two parts to the width; the third, by using the width to describe a square figure with equal sides, drawing a diagonal line in this square, and giving the atrium the length of this diagonal.

Vitruvius later directs that the "dining rooms ought to be twice as long as they are wide", and that "the height of all oblong rooms should be calculated by adding together the measured length and width, taking one half of this total, and using the result for the height." This passage anticipates not only Palladio’s prescribed proportions of 1:square of 2, 3:5, 2:3 and 1:2, but also the concept of the arithmetic mean which Palladio employed to a significant extent.

In an arithmetic mean, the relationship of the width to height to length is as 2:3:4 or, what amounts to the same thing, adding the length to the width and halving the result to obtain the height. Palladio recommended such a procedure as one of three preferred means by which to establish the height of rooms. The other two are the geometric mean and the harmonic mean, the former deriving from Vitruvius, the latter from Plato. The geometric mean is one in which the first amount is in proportion to the second as the second is to the third, or as Palladio instructed regarding determination of the height, "multiply the lesser extreme with the greater; because the square root of the number which will proceed from said multiplication will be the height we seek." This statement echoes that of Vitruvius wherein he mandates the height of the Cyzicene to be one and one half times the width. The harmonic mean is that exceeding one extreme and being exceeded by the other by the same fraction. Palladio furnishes as an example a room with a length of 12’ and a width of 6’, so that the mean distance of 8’ represents the height. The height therefore exceeds the width by 2’, or a third of the smaller extreme, and the length exceeds the height by a distance one-third the length. The Platonic inspiration for such an arrangement may be found in The Timeaus, wherein Plato describes the composition of the soul:

Next, he (God) filled in the double and treble intervals by cutting off further sections and inserting them in the gaps, so there were two mean terms in each interval, one exceeding the extreme and being exceeded by the other by the same fraction of the extremes; the other exceeding and being exceeded by the same numerical amount.

As the Greeks believed music to be geometry transmogrified to sound, so the Renaissance architects conceived of their creations as geometry metamorphosed into spatial units. While Alberti was the first Renaissance architect to both revive this concept and commit it to paper, Palladio was among its foremost practitioners.

The Villa Emo

Palladio designed for Leonardo Emo the villa which bears his name and which was completed in 1556 to serve as a country retreat and a working farmhouse, primarily for the production of maize. The Villa Emo is significant in that all the dimensions appearing in the plans are harmonic numbers, i.e. those any combination of which can be related to musical ratios. Moreover, Villa Emo is "the only perfectly harmonic scheme in Book II which bears any comparison with the real thing", the variations arising in the construction of the side wings. The villa is therefore exemplary of the Palladian tendency to transfer musical harmony to architectural proportion.

With respect to Villa Emo, Wittkower remarks that "a thorough acquaintance with Renaissance ideas is often necessary to understand the legitimacy of the ratios given by Palladio." He goes on to acquaint the reader with some of these ideas by presenting certain of the ratios in the villa. The 16x27 portico and 27x27 hall, for instance, are framed by rooms of 16x16, 12x16, and 16x27. The ratio 16:27 may be understood by complying with Alberti’s instruction to relate it as 16:24:27, or a fifth and a major tone, while the compound ratio 12:27 can be produced from 12:24:27, an octave and a major tone. Ratios of a similar order may be found in the wings, as 12 is there impressed between 24 and 48 so as to generate 2:1:4. The entire building appears, in the words of Wittkower, as "a spatial orchestration of the consonant terms 12, 16, 24, 27, 48."

The orchestration encompasses not merely the plan, but the elevation of the structure. A narrow vestibule succeeds the entrance loggia, and is of approximately half the height of the latter. The hall with which the vestibule communicates is of a vertical development equal to the loggia from which the hall proceeds. The hall connects lower rooms of identical proportion on the north and south sides of the building, reinforcing Palladian references to ancient notions symmetry and proportion. With its design being one exclusively of harmonic numbers that combine to generate ratios of a similar description, the Villa Emo is notable as one of only five such Palladian villas to have been constructed in strict accordance with the mathematical and harmonic principles of the ancients.

Conclusion

Renaissance architects were influenced to a considerable extent by the philosophy and mathematical principles of the ancient Greeks and Romans. The mathematical philosophy and numerology of the Pythagoreans were perhaps the earliest inspiration for the harmonic proportion prevalent throughout Renaissance architecture in general and that of Palladio in particular. The Pythagoreans adopted numerology as the foundation upon which the principles of the universe were founded and about which it revolved. Pythagorean numerology was adopted by Plato as a means by which to explain the machinations of the universe as well as the composition of the soul. While distancing himself from certain aspects of Pythagorean number mysticism, Aristotle nonetheless constructed a theory of composition that influenced Renaissance architects no less than the artists of his day. Inspired by these Greek writers, Vitruvius was the direct ancient influence on Palladio, and his instruction on the ideal proportion of rooms were echoed by those of the latter architect. The Villa Emo is of a design entirely composed of harmonic numbers. In this respect, this structure exemplifies the influence on Palladio of ancient mathematical philosophy.