Proportion and Number

A building can be too big or too small for its purpose, but it can also be out of proportion: a door too narrow for the wall it pierces, a room too high for its floor, a column too slender for the load it seems to carry. Proportion is the relation of sizes to one another rather than to a ruler, and it is the part of architecture that has most often been expressed in numbers. Every tradition in this book has had its own way of fixing proportions (the Egyptian seked, the Chinese cai module, the Japanese kiwari), and this chapter follows the one that dominated European architecture from antiquity to the twentieth century.

Two ideas run through it, sometimes in agreement and sometimes in conflict. One is arithmetic: choose a unit and make every dimension a whole number of units, so that all the parts are related through simple ratios such as 2 : 3. The other is geometric: construct the proportions with compass and straightedge, as the diagonal of a square or the division of a line in extreme and mean ratio, even when the result cannot be written as a fraction. Renaissance architects used both; modern writers have often confused them.

The module and commensurability

Vitruvius begins his account of temples with the human body. A well-formed man, he says, is proportioned so that his face is a tenth of his height, his foot a sixth, his forearm a quarter; and a temple should be proportioned in the same way, every part being a definite fraction of the whole. The Greek word he uses is symmetria, which meant not mirror symmetry but commensurability, the measuring of all parts by a common unit (Commensurability). In the orders that unit is the diameter of the column, the module, and the whole of Part II of this book depends on it.

℟Commensurability

Let every part of a building be measured by the same unit and relate to the whole by simple whole-number ratios, so that one module generates all its dimensions.

e.g. 2 : 3, 3 : 4, 1 : 2: ratios of small whole numbers of modules

The practical advantage of a module is obvious. A designer who fixes one dimension (the column diameter, the width of a bay, the side of a floor tile) can derive every other from it by multiplication and division, and a builder can check the result with a measuring rod. Brunelleschi’s loggia for the Foundling Hospital in Florence is the first Renaissance building planned in this way: its bays are as wide as the columns are high and as deep as they are wide, so that the whole arcade is a row of equal cubes of space (The bay as a cube).

Music made visible

The Renaissance explained why some ratios please more than others by appeal to music. According to a story told throughout antiquity, Pythagoras noticed that strings whose lengths stand in the ratios 1 : 2, 2 : 3 and 3 : 4 sound together in harmony (the octave, the fifth and the fourth), while other ratios clash. If the ear delights in these numbers, reasoned the humanists, it is because they express an order built into the universe, and the eye should delight in them too. Alberti made the argument explicit in Book IX of On the Art of Building, recommending that the plans of rooms be drawn from the musical consonances and their compounds (Ratios from the musical consonances).

℟Ratios from the musical consonances

Proportion rooms and façades by the same ratios that make musical consonances: 1:2 (octave), 2:3 (fifth), 3:4 (fourth), and their compounds.

short plans 1:1, 2:3, 3:4 · medium 1:2, 4:9, 9:16 · long 1:3, 3:8, 1:4

Alberti grouped plans into short, medium and long. A short plan might be a square (1 : 1), a fifth (2 : 3) or a fourth (3 : 4); a medium plan the octave (1 : 2) or compound ratios such as 4 : 9 (two fifths, 4 : 6 : 9); a long plan the octave-and-a-fifth (1 : 3), 3 : 8 or the double octave (1 : 4). The figure below shows the musical rectangles among others in a gallery of proportions. Whether one hears a fifth when looking at a 2 : 3 room is doubtful, and later critics said so; but the theory gave architects a small, memorable set of ratios that could be combined without arithmetic confusion, which is its real merit.

Palladio’s seven rooms

Palladio gives the most famous list. In Book I of the Four Books (1570) he names seven shapes for rooms as the most beautiful and best proportioned: round; square; the length equal to the diagonal of the square; a square and a third; a square and a half; a square and two-thirds; and two squares (The seven most beautiful room shapes). Five of these are musical ratios; one, the diagonal of the square, is irrational; and one, the circle, has no ratio at all. The list is practical rather than doctrinaire (Palladio’s own plans use other shapes too), but it shows how naturally a sixteenth-century architect thought of rooms as ratios of breadth to length.

℟The seven most beautiful room shapes

Make rooms round, square, or with a length equal to the diagonal of the square, to a square and a third, a square and a half, a square and two-thirds, or two squares.

circle · 1:1 · 1:√2 · 3:4 · 2:3 · 3:5 · 1:2

The height of a room was a third dimension to be found. For rooms with flat timber ceilings Palladio simply makes the height equal to the breadth. For vaulted rooms, which need more height to look well, he gives a rule: a square room should be vaulted to its breadth plus a third, and an oblong room to a height found from its length and breadth by one of three means (The height of a vaulted room).

The three means are old mathematical ideas. The arithmetic mean of two numbers is their average: between 6 and 12 it is 9. The geometric mean is the number that stands to the larger as the smaller stands to it: between 6 and 12 it is √72, about 8.49, a value that cannot be written as a fraction. The harmonic mean is 2ab/(a + b): between 6 and 12 it is 8, and the sequence 6 : 8 : 12 contains the fourth, the fifth and the octave, which is why it is called harmonic. For any two different numbers the arithmetic mean is the largest and the harmonic the smallest, with the geometric between them; the table gives the three heights for Palladio’s oblong rooms at a common breadth of 6.

The differences between the three means are small: for a room of two squares, nine units against eight. That is a warning against reading too much precision into proportional rules. They were ways of reaching a good result reliably, not formulas whose third decimal place mattered, and Palladio himself gives geometric constructions for each mean so that the builder can find them with a compass on the site.

In 1949 the art historian Rudolf Wittkower argued that Palladio went further, linking the rooms of whole villas into proportional systems in which the dimensions of one room reappear in the next and all the ratios are musical (Linked rooms in a plan). His book changed the way a generation of architects thought about proportion. Later surveys showed that the villas as built often depart from the numbers printed in the Four Books, and that Palladio tidied his figures for publication. The ideal was real; so was the compromise.

℟Linked rooms in a plan

Let the rooms of a house share dimensions, so that the length of one is the breadth of the next and each stands to its neighbours in the ratios of the scale; the whole plan then forms one proportional system.

e.g. rooms of 12 × 12, 12 × 18 and 18 × 30 feet, sharing 12 and 18

Geometry: the diagonal and the triangle

Palladio’s one irrational room, the square extended by its own diagonal, belongs to the other tradition, which fixes proportions by construction rather than by number. Swing the diagonal of a square down to its base line and you have a rectangle of 1 : √2 (The diagonal of the square). Medieval masons set out whole façades this way, rotating squares within squares (ad quadratum), or used the equilateral triangle, whose height stands to half its base as √3 to 1 (ad triangulum); the builders of Milan Cathedral argued in the 1390s over which system should govern its height (The √3 rectangle and the equilateral triangle).

Geometric proportions have one great practical virtue: they need no arithmetic and no standard unit. A mason with a cord and two pegs can lay out a √2 rectangle at any size, and the result will be exact however long his foot is. They also have properties that numbers lack. A √2 rectangle cut in half makes two smaller rectangles of the same shape (the reason it was chosen for the international A-series of paper sizes), and the golden rectangle, cut by a square, leaves another golden rectangle.

℟The diagonal of the square

To make a rectangle of 1 : √2, draw a square and swing its diagonal down to extend one side; the result cannot be measured in whole units of the square, but it can be drawn exactly.

√2 ≈ 1.414; halving a 1 : √2 rectangle gives two more of the same shape

The golden section

Divide a line so that the whole stands to the larger part as the larger part to the smaller, and the ratio is φ, about 1.618 (The golden section). Euclid defined this “extreme and mean ratio” around 300 BC because he needed it to construct the regular pentagon; in 1509 the mathematician Luca Pacioli published a book on it, De divina proportione, with drawings of polyhedra by Leonardo da Vinci. The golden section has remarkable properties: its square is itself plus one, its reciprocal is itself minus one, and a golden rectangle, stripped of a square, leaves a smaller golden rectangle, so that the process can continue for ever and quarter-circles through the successive squares trace a spiral.

℟The golden section

To divide a line in extreme and mean ratio, make the whole stand to the larger part as the larger part to the smaller; the ratio is φ = (1 + √5)/2 ≈ 1.618. Construct the golden rectangle by swinging the line from the midpoint of a square’s side to its far corner.

φ = (1 + √5)/2 ≈ 1.618; φ² = φ + 1; 1/φ = φ − 1 ≈ 0.618

Here honesty is needed. A large popular literature claims that the Great Pyramid, the Parthenon, Notre-Dame and the façades of the Renaissance were designed on the golden section, and that it is the secret of beauty in nature and art. Almost none of this survives scrutiny (Measure before you believe). No ancient or medieval text on architecture prescribes φ; Vitruvius does not mention it; Pacioli himself, in the section of his book devoted to architecture, recommends Vitruvius’s simple ratios. The “discoveries” are usually made by drawing rectangles on photographs and choosing which edges to measure, and they tend to ignore the fact that ordinary whole-number ratios such as 3 : 5 (1.667) and 5 : 8 (1.6) are indistinguishable from φ within the accuracy of such measurements. Even the name “golden section” was coined only in the 1830s.

℟Measure before you believe

Treat claims that a building embodies the golden section with suspicion unless the designer’s own writings or drawings say so and careful measurement agrees; a ratio near 1.6 is found somewhere in almost anything.

Custom or nature?

The deepest challenge to proportional theory came not from measurement but from argument. In the 1670s and 1680s Claude Perrault (physician, scientist, translator of Vitruvius and one of the designers credited with the east front of the Louvre) pointed out that the ancient buildings which everyone admired use different proportions for the same order, and concluded that the exact ratios cannot be what makes them beautiful. There are, he said, two kinds of beauty: positive beauty, which pleases everyone for evident reasons (rich materials, fine workmanship, size, symmetry), and arbitrary beauty, which pleases because we are used to it, and to which the proportions of the orders belong (Positive and arbitrary beauty).

℟Positive and arbitrary beauty

Distinguish positive beauty, which pleases everyone for evident reasons (fine materials, size, precise workmanship, symmetry), from arbitrary beauty, which pleases through custom, including the exact proportions of the orders.

His colleague at the Royal Academy, François Blondel, answered that proportions please by nature, as musical harmonies do, and that the ancients varied them only to suit circumstances such as the distance from which a building is seen (Proportion is natural, like harmony). The quarrel was never settled, and it is still worth having. The orders compared across five authorities in Part II show how much the canon varied; and yet the variations stay within a narrow band, which suggests that both men were partly right.

The Modulor

In the twentieth century proportion returned in a new guise. Le Corbusier, who had absorbed the golden-section literature of his youth, developed during the 1940s a measure he called the Modulor (The Modulor). It starts from a man 183 cm tall with his arm raised to 226 cm, and derives from these two heights two series of dimensions, each term the previous one multiplied by φ. He used it to size everything from the height of a room to the width of a window in the Unité d’habitation at Marseille (1947–52). The Modulor is the golden section adopted openly, as a designer’s tool, rather than discovered after the fact; its story belongs to modernism, and it shows that the Renaissance dream of a single harmonic measure for body and building did not die with the orders.

℟The Modulor

Take the height of a man (183 cm) and of his raised hand (226 cm) and divide each by the golden section, again and again, to obtain two interlocking series of dimensions for every part of a building.

red series … 27 · 43 · 70 · 113 · 183 cm; blue series … 53 · 86 · 140 · 226 cm (each ≈ previous × φ)

The long history suggests a modest answer to the question of what proportion is for. A proportional system gives a designer a small number of related dimensions to work with, so that decisions are consistent and parts can be repeated and fitted together; it lets a builder check the work; and it gives the finished building a quality of order that people can sense without measuring it. The numbers themselves, musical, irrational or golden, matter less than the discipline of using some numbers consistently. That discipline is visible in the figures of this chapter and in the chapters that follow.