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)
To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.
pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet
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.
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.
To set a round dome on a square bay, imagine a sphere passing through the four corners of the square; cut it by the four vertical planes of the sides and by the horizontal plane of their crowns. The four spherical triangles left in the corners are the pendentives; the circle at their top carries the dome.
sphere radius R = S⁄√2; arches = sphere ∩ side planes (radius S⁄2); ring at h₀ + S⁄2 with radius S⁄2
If the sphere that forms the pendentives is simply continued above the ring, the bay is covered by a single shallow dome, the sail vault or pendentive dome, which springs directly from the four corners.
crown = h₀ + S⁄√2 (lower than a hemisphere on the ring, h₀ + S)
Lift the dome on a cylindrical or polygonal drum pierced with windows between the ring of the pendentives and the springing of the dome, to light the centre of the church and to make the dome visible from afar.
drum height ≈ ⅓–1 × its radius (Byzantine); much taller in Russian churches
Give the villa garden porticoes for sun and for shade, grottoes made in imitation of nature with tufa and shells, evergreens planted in straight rows at equal distances, and hedges of box; set vines on columns over the walks.
Lay the ideal garden out on a circular plan in concentric rings, woods outside, then meadows and orchards, then gardens of knots and clipped evergreens divided by walks, with water and the fountain of the goddess at the centre.
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
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
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)
Keep the lowest third of the shaft cylindrical; above it, reduce the radius along a curve found by dividing a quarter-circle and the upper two-thirds of the shaft into the same number of equal parts.
r(k) = R · cos(α·k), cos α = r_top ⁄ R, k = 0…1 over the upper ⅔
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
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°)
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)
Divide the whole height into nineteen parts: four for the pedestal, twelve for the column, three for the entablature. Without a pedestal, divide it into fifteen, giving twelve to the column and three to the entablature.
pedestal = ⅓ column; entablature = ¼ column
Make the Tuscan column 14 modules high, the Doric 16, the Ionic 18, and the Corinthian and Composite 20, the module being half the lower diameter.
Tuscan 7 D · Doric 8 D · Ionic 9 D · Corinthian 10 D · Composite 10 D
Rank the orders in a sequence from the plainest and strongest to the richest and most slender (Tuscan, Doric, Ionic, Corinthian, Composite), each one diameter taller than the last.
6 · 7 · 8 · 9 · 10 D
Make the Tuscan column 7 diameters high, the Doric 8 with its base, the Ionic 9, the Corinthian 9½ and the Composite 10; give Tuscan and Doric an entablature of about a quarter of the column, the richer orders a fifth.
7 · 8 · 9 · 9½ · 10 D; entablature ≈ ¼ → ⅕
Grade the orders by slenderness (Tuscan 7½ diameters, Doric 8½, Ionic 8¾, Composite 9¾ and Corinthian 10), placing the Composite, the “Roman” order, below the Corinthian.
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.
Beauty is the reasoned harmony of all the parts within a body, so that nothing may be added, taken away or altered but for the worse.
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
In all architecture the principal ornament is the column: a part of the wall left standing and pierced through.
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
Make the height of a vaulted room the arithmetic, geometric or harmonic mean of its length and breadth.
arithmetic (a+b)/2 · geometric √(ab) · harmonic 2ab/(a+b)
Place rooms on the right of the entrance equal and corresponding to those on the left, so that the house is alike on both sides and the walls bear the roof equally.
Combine a central arched opening with two narrower flat-headed openings on either side, the arch springing from the entablature of the side openings.
side lights ≈ ½ central span; arch springs from the lintel of the sides
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.
Compose from the plan: base it on the elemental figures (circle, square, rectangle and their derivatives), arrange its parts symmetrically about axes, and develop section and elevation from it together.
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.
Place the staircase by three tests: its relation to the principal rooms it serves; keeping the main axial communications free of obstruction; and light, for itself and, as a well of light, for the corridors and vestibules around it.
Between two numbers a and b, find the arithmetic mean (a + b)/2, the geometric mean √(ab) and the harmonic mean 2ab/(a + b); use them to find a third dimension that belongs to the other two.
for 6 and 12: arithmetic 9 · geometric 8.49 · harmonic 8; always A ≥ G ≥ H, and G² = A × H
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
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
Swing the diagonal of a √2 rectangle to obtain a rectangle of 1 : √3, the proportion of an equilateral triangle’s height to half its base, used for setting out by triangles (ad triangulum).
√3 ≈ 1.732; height of an equilateral triangle of side 2 = √3
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
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.
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
Make each bay of an arcade as wide as its columns are high and as deep as it is wide, so that the loggia is a row of equal cubes of space.
bay width = column height = depth (Innocenti: ≈ 10 braccia ≈ 5.8 m)
Cover a square bay with a hemisphere whose diameter is the bay’s diagonal, cut off by the four vertical planes of its sides; the cut edges are then the four semicircular arches of the bay.
vault radius = ½ × diagonal = s / √2; the side arches are semicircles of span s
Draw the curve of a pointed dome with arcs whose radius is four-fifths of the span, their centres standing on the springing line one-fifth of the span in from the opposite side.
r = ⅘ S; centre at ⅕ S from the far springing; rise ≈ 0.74 S
Build a great dome as two shells (a thick inner one that carries, a thin outer one that protects and gives the silhouette) joined by ribs and locked at the top by a ring under the lantern.
Florence: inner shell ≈ 2 m at the base; outer ≈ 0.9 m; 8 angle ribs + 16 intermediate ribs
Lay the brickwork of a large dome in horizontal rings, each ring locked by bricks set on end in a herringbone pattern, so that every stage of the work stands by itself without a timber frame beneath it.
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)
Where the purpose allows, plan the church about a single centre (a circle, a square or a Greek cross) so that every part stands in the same relation to the middle, under a dome.
Where a church has a tall nave and low aisles, give the front a lower storey as wide as the whole church and an upper storey as wide as the nave, and join the two with great scrolls.
Lay out the ideal city as a regular star inscribed in a circle (two squares turned 45° to one another), with gates in the re-entrant angles and towers at the points.
Sforzinda: 8 points, 8 gates, 16 radial streets
Run the principal streets straight from the central square to the gates, so that every gate is visible from the centre and every quarter lies on a direct route to it; set secondary squares along the radials.
Give a fortified town a polygonal wall with an angled bastion at every corner, so that each face of every bastion and every stretch of curtain wall can be swept by fire from its neighbours.
Palmanova: a nine-sided trace with nine bastions and three gates
Where an old structure fixes bays of unequal width, keep the central arch of every Serliana the same and let the flat-headed side openings take up the differences.
Raise the principal rooms of a house on a basement storey that holds the kitchens, cellars and service rooms, and reach them by an external stair; put the lesser rooms above them.
Place the stair where it does not spoil the rooms and yet is easily found; light it well; and make the steps no higher than half a foot and the treads at least a foot deep.
riser ≤ ½ ft (≈ 17 cm); going ≥ 1 ft (≈ 35 cm) in Vicentine feet
Where space is short, wind the stair round a newel or an open well; measure the going on the walking line, about halfway along the tread, where it should be as generous as on a straight flight.
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
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
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
To roof a wide span, join each pair of rafters to a tie beam at their feet so that the three form a triangle that cannot spread; hang the tie beam from the apex by a king post, or by two queen posts on larger spans.
At the centre of the city lay out the main piazza as a rectangle twice as long as it is wide, with the cathedral at one end and the prince’s palace at the other, and put a merchants’ square and a market square beside it.
Sforzinda: main piazza 300 × 150 braccia (about 175 × 90 m)
Partway along the sixteen streets that run from the centre to the gates and towers, open a secondary square in each, and give half of them parish churches and the other half markets, so that every quarter has its church and its market close at hand.
Make a public square twice as long as it is wide, surround it with porticoes, and keep the buildings round it in proportion to the open space: about a third of its breadth high, and never less than a sixth.
length = 2 × breadth; height of the buildings = ⅓ to ⅙ of the breadth
Set the principal temple where it can be seen, standing free in its own open space, and raise it on a podium reached by steps, so that it stands above the ordinary business of the town.
In a great and powerful city make the main streets broad and straight, lined with porticoes and with houses of the same height and design, so that the street itself becomes an ornament; in a small town let the streets wind instead.
Give a new city a regular polygonal outline with a bastion at every angle, put the principal piazza at the centre with the chief church and the seat of government on it, and distribute lesser squares for the markets through the quarters.
Line the principal street with the houses of great persons (palaces with regular fronts, cornices and pediments) receding to a monument, as in the stage set for tragedy; leave the humbler houses, shops and taverns of the comic scene to the side streets.
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