A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.
free-hanging chain (catenary) ↔ ideal arch under its own weight
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
Front a basilican church with two storeys of the order: the lower as wide as nave and chapels, the upper as wide as the nave alone, joined by scrolls and crowned by a pediment.
Il Gesù (reconstructed): lower ≈ 34 m wide, upper ≈ 20 m; scrolls fill the difference
Build up a façade in layers: let the wall advance in stages towards the central bay, double the pilasters or columns as it advances, and break the entablature forward over every support.
Mark the main door by setting a segmental pediment inside a larger triangular one, or by breaking the pediment open to hold a cartouche or statue.
To unify a façade of two or more storeys, run a single order of pilasters or columns through all of them, from the base to the main cornice.
To combine the unity of a central plan with the direction a church needs, plan the space as an oval, set across the axis to widen the view of the altar, or along it to lead the eye towards it.
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.
Let the wall of a façade curve in plan (concave, convex, concave) so that the front answers the street, the light and the space within, and the order bends with it.
Light the climax of a Baroque interior (the altar or the dome) from windows the visitor cannot see, so that it seems to glow of itself.
Proportion every step so that twice its height plus its depth equals the length of a natural pace on the level: two feet, or 24 pouces.
2 r + g = 24 pouces ≈ 65 cm (modern practice: 60–66 cm)
Hold that the beauty of the orders comes from fixed proportions that please by nature, as consonances do in music; where the ancients differ, the differences answer to circumstances such as distance and height.
Make the stair of a palace a processional space: one broad flight rising to a landing, then turning back in two flights either side to arrive at the principal floor beneath a single great vault.
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
Make the parterre oblong rather than square, since figures somewhat oblong look better on the ground than square ones; later practice made it as broad as the front of the house and about twice as long.
parterre oblong; later practice: width ≈ house front, length ≈ 2 × width
Proportion the width of an allée to its length: a walk of about 100 toises (some 200 m) should be 5 or 6 toises wide, one of 200 toises 7 or 8, and so on, so that a long walk does not seem to narrow to a slot in the distance.
100 toises long → 5–6 wide · 200 → 7–8 · 300 → 9–10 · 400 → 10–12 (1 toise ≈ 1.95 m)
Organise the whole garden on one axis from the centre of the house to the horizon; lay out the parts symmetrically about it, and use water, slopes and hidden changes of level so that the view along it reveals the garden by stages.
Avoid the ornaments that have no model in the hut: broken or curved pediments, pediments on the long side of a roof, orders piled storey upon storey on a church front, arches carried on columns, and twisted or cabled shafts.
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.
Use the elliptical plan chiefly for interiors, where the whole curve is seen at once; on the outside of any but a very large building its changing curvature looks like a distorted circle, and the circle is better.
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.
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.
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.
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.
Break the slope of the roof on all four sides: a steep lower slope, nearly a wall, which holds a storey of rooms lit by dormers, and a shallow upper slope, often invisible from the street.
lower slope ≈ 60–75° · upper slope ≈ 15–30°
Classify a roof by how its planes meet: one slope (shed), two meeting at a ridge (gable), four meeting at hips (hip, pyramid), a hip below a gable (hip-and-gable), two pitches each side (gambrel, mansard), or a curved surface (barrel, cone, dome).
Join the principal churches of the city by straight streets cut across hills and empty land, so that a pilgrim can walk from one to the next directly, and mark the end of every street with an obelisk visible from far away.
Where the main road enters the city, open a piazza inside the gate and send three straight streets from it into the town, fanning from a common point marked by an obelisk, with a pair of matching churches in the angles between them.
Before the mother church lay out an oval piazza enclosed by free-standing colonnades four columns deep, so that the church seems to embrace the faithful; set an obelisk at the centre and a fountain on each side along the cross axis, and join the oval to the church front by straight corridors.
Bring water back to the high, empty parts of the city by aqueduct; let it arrive at a monumental fountain and feed public fountains at the crossings of the new streets, so that people will come to live there.
Put the prince’s palace at the centre of a circle of thirty-two straight rays; give nine of them, between the wings of the palace, to the town, and send the other twenty-three through the forest as rides, so that every street and every ride ends on the palace tower.
32 rays at 11¼°: 9 streets + 23 forest rides
Lay the parterres next to the house, low and flat so that nothing blocks the view from its windows; set bosquets and groves beyond them, and lead long allées out toward the horizon, so that the garden always seems larger than it is.