A beam is squeezed along its top and stretched along its bottom; stone resists squeezing well but stretching badly, so a stone lintel must be deep in proportion to its span and cannot be long. Doubling the span quadruples the stress from its own weight.
σ ≈ ¾ ρ g L² ⁄ d (self-weight) · + ≈ 0.43 ρ g L³ ⁄ d² (a wall above)
In a bonded masonry wall the courses above an opening corbel over it of their own accord, so the lintel carries roughly the triangle of wall with sides rising at 60° from its ends; everything above that triangle arches onto the jambs.
load ≈ equilateral triangle on the span: W ≈ 0.43 · L² · t · ρ g
Weight for weight, timber is many times stronger in bending than stone; long spans in trabeated buildings are therefore roofed with timber beams resting on stone walls and columns.
bending strength ÷ density: oak ≈ 70 · limestone ≈ 3 (kPa per kg ⁄ m³)
Cut a shaft with sixteen shallow flutes or facets, give it a square abacus and a low round base: the “proto-Doric” column of the Middle Kingdom rock tombs at Beni Hasan.
Plan the principal hall as a long rectangle entered on its short side through a porch with two columns between the side walls and a vestibule, and put a great round hearth in the centre of the hall, ringed by four columns carrying a raised roof.
Do not lay the stylobate dead level: raise it gently toward the middle of each side, for a perfectly flat platform looks to the eye as if it sagged.
Parthenon: rise ≈ 11 cm over 69.5 m (flanks), ≈ 6.5 cm over 30.9 m (fronts), about 1 : 600
Make the corner columns thicker than the rest by a fiftieth of their diameter, because, seen against the open sky, they look thinner than they are.
D corner = D × (1 + 1⁄50); Parthenon: 1.95 m against 1.905 m
Set the outer columns so that they incline slightly inward, toward the cella wall, the middle columns of the fronts plumb on their own axes and the corner columns leaning both ways.
Parthenon: lean ≈ 6–7 cm over 10.43 m (≈ 1 : 150); the axes would meet some 2 km above the ground
Give the shaft a continuous convex curve from base to necking, departing from a straight taper by only a few hundredths of a diameter, so that the column seems to flex under its load rather than to pinch.
Parthenon: greatest departure ≈ 1.7 cm on a shaft 9.6 m high (≈ 1⁄110 D)
A Doric frieze cannot obey all three of its rules at a corner: triglyphs over the column axes, a triglyph turning the corner, and metopes as broad as they are high. One of them must give way.
displacement of the corner triglyph = e − t⁄2 (e = half-depth of the architrave, t = triglyph width)
To end the frieze with a triglyph at the corner and keep every metope square, move the corner column inward: make the last intercolumniation narrower than the others by the amount the corner triglyph is displaced.
contraction c = e − t⁄2; Parthenon: 4.29 m normal, 3.68 m at the corners (c ≈ 0.61 m)
Let one simple ratio govern the whole: at the Parthenon the lower diameter is to the axial spacing, the width of the stylobate to its length, and the height of the order to the width of the front, all very nearly as 4 is to 9.
D : axial spacing ≈ 1.905 : 4.296 ≈ 4 : 9; stylobate 30.88 : 69.50 ≈ 4 : 9
Plan a Greek theatre by inscribing three squares in the orchestra circle; the side nearest the scene fixes the front of the stage (proscaenium), and the scene is set back to a line tangent to the circle.
proscaenium at r cos 45° ≈ 0.71 r from the centre; logeion 10–12 ft high
Build walls of squared stone in regular courses with the vertical joints of each course falling over the middle of the stones below: either with all courses of one height (isodomum) or with courses of alternating heights (pseudisodomum).
In a thick wall, do not build two separate skins with loose filling between them: lay stones lengthwise into the thickness at intervals, and set single stones right through from face to face, so that the wall acts as one mass.
Never let a vertical joint stand over another in the course below: interchange the direction of the stones, lengthwise and across the wall, so that every joint is covered and every stone rests on two.
overlap ≥ ¼ of a unit (brick: a quarter-brick lap)
Divide the height from floor to ceiling into three and a half parts and give two and a half to the opening; make a Doric door five and a half twelfths as wide as it is high (an Ionic one two-fifths), and narrow it slightly at the top.
h = 2½⁄3½ H; Doric w = 5½⁄12 h, Ionic w = h ÷ 2½; top narrowed by ⅓, ¼ or ⅛ of the jamb as h ≤ 16, ≤ 25, ≤ 30 ft
Every building must satisfy three demands at once: it must stand up (firmitas), serve its purpose (utilitas), and give pleasure to the eye (venustas).
architecture = firmitas + utilitas + venustas
Architecture depends on order (ordinatio), arrangement (dispositio), eurythmy, symmetry, propriety (decor) and economy (distributio).
A column is narrower at the top than at the bottom, but the taller the column the less it should be diminished, because distance already makes its top look smaller.
upper Ø : lower Ø = 5⁄6 (≤15 ft) · 11⁄13 (15–20 ft) · 6⁄7 (20–30 ft) · 13⁄15 (30–40 ft) · 7⁄8 (40–50 ft)
Columns may be spaced 1½ diameters apart (pycnostyle), 2 (systyle), 2¼ (eustyle), 3 (diastyle) or more (araeostyle). The eustyle, with a wider central bay of 3 diameters, is the best for use and beauty.
clear space = 1½ · 2 · 2¼ · 3 · >3 D
In a peripteral temple the flanks have twice as many intercolumniations as the front: six columns in front means eleven along each side, counting the corner columns twice.
side columns = 2n − 1 (Vitruvius) · 2n + 1 (Greek practice)
Divide the whole front of the cornice into nine parts; give one of them to the height of the tympanum, measured at its centre.
tympanum height = cornice width ⁄ 9 (≈ 12½°)
Divide the front of a Doric temple into modules; the column’s thickness is two modules, its height with the capital fourteen, the capital one, the architrave one, and the triglyphs one and a half high and one wide.
M = ½ D; column = 14 M; capital = 1 M; architrave = 1 M; triglyph = 1 × 1½ M; metope = 1½ × 1½ M
Set a triglyph over the axis of every column and over the middle of every intercolumniation (or two, in a wider bay); make the metopes between them as wide as they are high.
axis spacing = n × (triglyph + metope) = n × 2½ M
The Corinthian capital is as high as the column is thick; its abacus is one seventh of that height, and the rest is divided in three: the lower leaves, the upper leaves, and the stalks from which the volutes spring.
capital = 1 D; abacus = ⅐; leaves : leaves : volutes = 1 : 1 : 1; abacus diagonal = 2 D
An order consists of a column (base, shaft and capital) carrying an entablature (architrave, frieze and cornice), optionally raised on a pedestal (plinth, die and cap). Each part is proportioned to the lower diameter of the column.
order = [pedestal] + column(base, shaft, capital) + entablature(architrave, frieze, cornice)
Build every profile from a small set of elements (fillet, torus, astragal, scotia, ovolo, cavetto, cyma recta and cyma reversa), alternating curved with flat and light with shadow.
Use the orders according to their character: the Doric for strength and gravity, the Ionic for a middle dignity, the Corinthian for delicacy and splendour.
Read the age of a Doric order from its proportions: the oldest columns are little more than four diameters high, the finest Greek ones five and a half to six and a half, the Roman seven and a half to eight; as the column rose, the architrave grew lower.
archaic ≈ 4 D → classical 5½–6½ D → Roman 7½–8 D
Where the architrave runs on over the end of a wall, widen the wall’s end into an anta at least as broad as the architrave, so that the beam has a full bearing and the wall looks able to take it, even if the wall itself is thinner.
Do not enlarge or reduce an order by one set of ratios: design its members for the size at which it will be built, the material and the distance from which it is seen. In a large order make the parts relatively finer, in a small one relatively bolder.
Draw the Ionic volute seven eye-diameters high and six wide, with the eye four diameters below the top; lay three turns about centres set out in the eye so that each winding is half as wide as the one outside it and the spiral runs into the top of the eye.
height = 7 e, width = 6 e; eye centre 4 e below the top; radii from the eye centre (e = 16 parts): 60, 52, 44, 36, 30, 26, 22, 18, 15, 13, 11, 9 → 8
Space the eggs from centre to centre at the height of the moulding; make each egg as wide as the space between two eggs, and divide that space into three, giving the middle part to the dart and the outer parts to the shell that encloses each egg.
pitch = height; egg width = gap; gap = shell + dart + shell (⅓ each)
When a window or door narrows towards the top and has no cornice, let the ends of the lintel run out beyond the jambs as ‘ears’ (crossettes), so that the head is firmly marked.
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
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.
Put a pediment only where the roof actually ends in a gable, on the short sides of a building, never along its length or stacked one above another, because the pediment represents the gable of the roof.
Divide the city into equal rectangular blocks by straight streets crossing at right angles, and assign whole blocks to the public buildings; but let walls follow the ground, not the grid.
Keep a free agora for public life, clear of merchandise and near the magistrates, the temples and the gymnasium; put buying and selling in a separate market place convenient for goods.
Make the Greek agora square, surround it with spacious colonnades, double if possible, with close-set columns and architraves of stone or marble, and walks above on an upper floor.
Put the court on the south side of the house and the principal rooms, fronted by a porch, on its north side facing south, so that the winter sun enters the porch and the summer sun passes over the roof; build the north side high and the south low.
Set the temples where they can be seen from the whole city and approached apart from its traffic: on a raised, commanding site above the agora.
Build the palaestra as a square or oblong court with colonnades all round, the side facing south doubled, and rooms for bathing, oiling and teaching opening off them; beside it lay out a running track with covered and open lanes.
Project the towers beyond the line of the wall, no further apart than a bowshot, so that attackers are exposed on their flanks; bring the roads to the gates obliquely, so that an enemy’s unshielded right side faces the wall.
Build the theatre on a healthy site and do not let its curved auditorium face south, where the sun would heat the enclosed air and harm the audience sitting through the plays.