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
Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.
joints ⟂ intrados (radial to the centre)
An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.
H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)
Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.
Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.
An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.
In a row of arches the thrusts of neighbouring arches cancel over each intermediate pier, which may therefore be slender; at the ends of the row nothing balances them, so the outermost piers must be made wider or buttressed.
Over a doorway or window spanned by a lintel, turn an arch in the wall above so that the weight of the masonry is carried round the opening onto the jambs, leaving the lintel to carry only the small triangle of wall beneath the arch.
Divide the inside of a dome into coffers that shrink in step with the meridians as they rise, so that their perspective deepens the dome and draws the eye to the light of the lantern.
Give the flight of steps before a temple an odd number, so that one who starts with the right foot will also set the right foot first on the top.
Raise the central nave above the lean-to roofs of the aisles and pierce the upper walls with a row of windows, the clerestory, so that the middle of a wide hall is lit from high on both sides.
nave height ≈ 1½–2 × nave width; one window over each bay of the arcade
Edge the beds before the portico with box, clip box into figures of animals set facing one another, and in the garden beyond cut it into many shapes, even into the letters of the owner’s name.
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
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)
Keep every triglyph centred over a column and every intercolumniation equal, and finish the frieze at the corner with a half-metope half a module wide.
corner semi-metope = ½ M (Vitruvius); equal intercolumniations throughout
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).
Lay out a city’s streets so that they do not line up with the directions of the principal winds, which otherwise rush down them cold, violent and unhealthy.
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
Make the Tuscan column seven diameters high, diminish it by a quarter at the top, and give its base and capital each half a diameter.
column = 7 D; upper Ø = ¾ D; base = capital = ½ 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)
When orders are placed one above another, put the stronger below and the more delicate above (Doric under Ionic under Corinthian), and make each upper column a quarter shorter than the one beneath.
upper column = ¾ lower column; order sequence: Tuscan → Doric → Ionic → Corinthian → Composite
Set arches on piers and frame each bay with an engaged column or pilaster carrying a continuous entablature, so that the arch does the work and the order gives the measure.
arch height ≈ 2 × arch width (Vignola); impost at ⅔ of the column
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
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.
Compose the Roman or Composite capital from two rows of Corinthian leaves round the bell with a full Ionic capital (echinus and four diagonal volutes) above them.
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.
Do not leave a large circular building as a bare cylinder: steady it with features that break the curve and fix its axes: a portico, stairs, a projecting bay or a recess on the axis.
For a hall of assembly use the “D” or tribune plan, a half-circle closed by a straight side: it gives ample room to the audience, a central place to the speaker and scope for display in the architecture.
Give a portico three jobs: to shelter those who arrive, to mark the main entrance, and to give the building dignity and character with its depth of light and shade; proportion it, as a rule, like a temple front.
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
Set the columns of a round temple on a circle over a stepped base, carry the entablature round the circle, and cover the cella with a dome whose height is half the diameter of the whole.
dome height ≈ ½ diameter (Vitruvius IV.8, for the peripteral round temple)
Make the height of the rotunda to the top of its dome equal to its diameter, so that the interior would exactly contain a sphere resting on the floor: the drum rises to the centre of the sphere and the dome is its upper half.
Pantheon: Ø = height = 43.3 m (150 Roman feet); drum = ½ Ø
Divide the inner surface of the dome into sunk panels arranged in rings and meridians; let them diminish as they rise with the converging meridians, and stop them short of the crown so that a smooth zone surrounds the eye.
Pantheon: 5 rings × 28 coffers, each stepped in several recessions
Build a great concrete dome thickest and heaviest at its haunches and thinnest and lightest toward the crown, grading the aggregate from dense stone at the base to light volcanic pumice at the top.
Pantheon: thickness ≈ 6 m at the springing to ≈ 1.5 m at the oculus; aggregate travertine and tuff → brick → pumice and scoria
Expect a hemispherical dome to be in compression along its meridians everywhere but in tension round its horizontal rings below about 52° from the crown; restrain that lower zone with weight, thickness, rings or chains, or it will crack along the meridians.
thin hemisphere under its own weight: hoop force changes sign at ≈ 51.8° from the crown
Make the sweating room of a bath circular, as high to the springing of its dome as it is wide, with an opening at the crown closed by a bronze disc that can be raised and lowered on chains to regulate the heat.
height to springing = width; oculus with a movable bronze shield
Face a concrete wall either with small irregular stones set at random (opus incertum), which is strong though not handsome, or with small square-faced stones set diagonally in a net (opus reticulatum), which is handsome but, its joints running every way, more likely to crack.
Build a Roman wall as two facings of brick or small stone with a core of concrete poured between them in layers; tie the facings into the core with bonding courses of large tiles laid through the whole thickness at intervals.
bonding courses of bipedales (2 ft tiles) every 1–1.5 m of height
Mix lime with the volcanic ash of the Bay of Naples, and the mortar will bind rubble into a solid mass that hardens even under the sea.
Lay the channel of an aqueduct with a steady fall of not less than a quarter of an inch in every hundred feet, cover it with a vault against the sun, and carry it through hills in tunnels and across valleys on arches.
minimum fall ≈ 1 : 4,800 (Morgan’s reading); Nîmes aqueduct: ≈ 12.6 m in ≈ 50 km ≈ 1 : 4,000
Cross a deep valley on superimposed tiers of arches, the piers of each tier standing directly over those below, the spans matched from tier to tier, and the smallest arches at the top where they carry only the channel.
Pont du Gard: tiers ≈ 22 m, 20 m and 7 m high; 6, 11 and 35 surviving arches; largest span ≈ 24.5 m
Draw the orchestra circle and inscribe in it four equilateral triangles at equal intervals; the side nearest the scene fixes the front of the scaena, a parallel through the centre fixes the front of the stage, and the angles on the auditorium side give the directions of the stairways.
12 vertices at 30°: 7 fix the stairs of the lower cavea, 5 fix the doors and passages of the scene; stage depth = r⁄2
Make the scaena twice as long as the orchestra is wide; raise its podium a twelfth of the orchestra diameter, its lower columns a quarter, their entablature a fifth of the columns; above, a parapet half the podium and columns a quarter shorter than those below.
L = 2D; podium D⁄12; columns D⁄4; entablature ⅕ col.; upper parapet ½ podium; upper columns ¾ lower
Roof the colonnade at the top of the seats at the same height as the top of the scene building, so that the voice rises evenly to the highest rows and is not checked on its way.
Give the atrium one of three shapes: width to length as 3 : 5, or 2 : 3, or a length equal to the diagonal of the square on the width; and make its height to the girders a quarter less than its width.
L = 5⁄3 W, 3⁄2 W, or √2 W; height to girders = ¾ W
Proportion the rooms round the atrium from it: the alae from its length (a third for atria of 30–40 feet, a quarter for 50–60), the tablinum from its width (two-thirds for a 20-foot atrium, half for 30–40 feet, two-fifths for 40–60), and the entrance passage from the tablinum.
alae = L⁄3 … L⁄5; tablinum = ⅔ W (20 ft), ½ W (30–40 ft), ⅖ W (40–60 ft); fauces = ⅔ or ½ tablinum
Leave an opening in the atrium roof not less than a quarter nor more than a third of the atrium’s width, with a length in proportion; beneath it set a basin to receive the rain.
compluvium width = ¼ to ⅓ W
Make the peristyle a third longer across than it is deep, its columns as high as the colonnades are wide, and the spaces between them not less than three nor more than four diameters.
width = 4⁄3 depth; column height = colonnade width; intercolumniation 3–4 D
Make a basilica between a third and a half as broad as it is long; make each aisle a third of the width of the central space, and the columns as high as the aisles are broad.
W = ⅓ to ½ L; aisle = ⅓ nave; column height = aisle width; upper columns ¼ smaller
Pierce a massive rectangular pier with one arch, or with a tall central arch between two lower ones; frame the openings with columns on pedestals carrying an entablature, and crown the whole with a high attic for the dedicatory inscription.
Arch of Titus: 13.5 × 15.4 m, one opening 5.4 × 8.3 m; Arch of Constantine: 25.9 × 21 m, openings 6.5 × 11.5 m and 3.4 × 7.4 m
Bury no one inside the city; keep bodies, urns and sarcophagi outside the walls, even the relics of apostles and martyrs. So the great shrines of the martyrs rise in the cemeteries along the roads, not in the town.
Repair the public buildings of the city before beginning new ones; and pull down the booths, sheds and houses that private people build into the public porticoes or against the walls of public buildings.
Size the forum to the population, neither cramped nor deserted, and make its breadth two-thirds of its length, so that the oblong suits the shows held in it.
breadth : length = 2 : 3
Put the temples of the city’s protecting gods (Jupiter, Juno and Minerva) on the highest ground; Mercury in the forum; Apollo and Bacchus by the theatre; Hercules by the circus; Mars and Venus outside the walls.
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.
Site the baths where it is warmest, turned away from the north and north-east, and light the hot and tepid rooms from the south-west, since people bathe from midday to evening.
A barrel (tunnel) vault is an arch extruded along its axis. It thrusts outward all along both sides, so it must rest on continuous thick walls, which can be pierced only sparingly, and only below its springing.
thrust per metre of wall ≈ constant along the whole length
Intersect two barrel vaults of equal height at right angles: the result is a groin vault, whose weight and thrust are gathered along the diagonal groins to the four corners, leaving the sides free for openings.
surface = max(barrel₁, barrel₂) · groins on the diagonals
Where the groin vault is the union of two barrels, the cloister (domical) vault is their intersection: four curved cells rise from all four walls and meet in valleys along the diagonals. It needs walls on every side.
surface = min(barrel₁, barrel₂) · valleys on the diagonals