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)
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
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.
Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.
r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)
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.
A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.
round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2
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
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.
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
Gather the weight and thrust of the vaults onto a few points by ribs, carry them down piers and out across the aisles by flying buttresses to external buttress piers, and the wall between the supports carries almost nothing, so it can be opened into windows.
Divide the elevation of the nave into three storeys: a tall arcade opening into the aisles, a low triforium passage in the thickness of the wall against the aisle roof, and a clerestory of large windows reaching up into the vault.
Chartres: arcade ≈ clerestory, triforium a narrow band between
Carry the outward thrust of a high vault across the aisle roof by a half-arch of stone (the flyer) that leans against the clerestory wall at the level of the vault’s haunches and springs from a massive buttress pier outside the aisle, which takes the thrust down to the ground.
Load the top of a buttress pier with a heavy pinnacle: the added weight steepens the resultant of the flyer’s thrust and the pier’s own weight, keeping it well inside the base of the pier.
Draw a square the width of the pinnacle; inscribe a second square on the mid-points of its sides, and a third on the mid-points of the second. Turned back into line, the squares give the widths of the body, its panels and the spire; every height is a multiple of the first side.
widths: s, s⁄√2, s⁄2, s⁄(2√2) … · heights: n × s
To make a square of half the area of another, join the mid-points of its sides: the rotated square so drawn has exactly half the area. Repeat to halve again.
side of inscribed square = s ⁄ √2 · area = ½ s²
Proportion the cross-section of a church from the square: let the total height to the crown of the nave vault equal the total width, and take the heights of the aisles and the springing of the vaults at whole divisions of that square.
H = W · levels at k × (W ⁄ n)
Alternatively, fit the section to an equilateral triangle standing on the width, so that the height is √3⁄2 of the width, and take the levels at equal divisions of that height.
H = W · √3⁄2 ≈ 0.866 W · Milan: 96 braccia wide → 83.1, rounded to 84 = 6 × 14
A building’s form must follow the geometrical rules of the craft (scientia) as well as the experience of the builder (ars); neither is sufficient alone.
Proportion every member of a vaulted choir from its clear width: the wall a fixed fraction of the width (a tenth, as the rule is usually read), and buttresses, window jambs, mullions and ribs as simple multiples or fractions of the wall.
wall ≈ W ⁄ 10 · buttress, jambs, ribs = multiples of the wall
Instead of cutting separate openings through a plate of wall, fill the whole window arch with glass and divide it by slender stone bars (mullions below, and circles, foils and sub-arches above) set out with compasses from the centres of the window’s own arches.
Date a Gothic window by its tracery: openings pierced through a plate (twelfth century); geometric bar tracery of circles and foils (thirteenth); flowing curvilinear tracery of ogees and daggers (fourteenth); and, in England, perpendicular panel tracery with mullions running straight up into the arch (late fourteenth to sixteenth).
Design a rose by dividing the circle into equal sectors; fill each with the same traceried unit (a light with a cusped pointed head, an oculus above) radiating from a central hub, so that the whole is generated from one sector by rotation.
n petals · each sector 360° ⁄ n
End the choir in a semicircle of piers; wrap an aisle (the ambulatory) around it, and open a ring of chapels off the ambulatory, each with its own altar and windows, so that the east end steps down in tiers of roofs and buttresses.
Within a generation the nave vaults of the French cathedrals rose from about 37 m to 48 m while their spans stayed near 15 m: the ratio of height to width climbed from about 2.3 to over 3, until at Beauvais part of the choir vault fell.
Chartres ≈ 37 m · Reims ≈ 38 m · Amiens ≈ 42 m · Beauvais ≈ 48 m
Put the hall at the heart of the house and enter it at one end, by a passage running through the house from front door to back door, screened from the hall. Open the buttery, the pantry and, between them, the way to the kitchen off that passage, at the hall’s lower end; put the family’s parlour, and the great chamber over it, beyond the hall’s upper end, where the high table stands.
In a timber-framed house, let the floor joists of an upper storey run out beyond the wall below and carry the upper wall on their ends: the upper storey oversails the lower by a foot or two, gaining room upstairs and sheltering the wall beneath.
Give a pointed building a steep roof, the gable close to an equilateral triangle, which throws off rain and snow and completes the vertical character of the design.
pitch ≈ 55–60° (equilateral gable = 60°)
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
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
Where one nave vault spans two bays of the arcade, give the principal piers, which receive the vault, a heavier compound section with shafts running up the wall, and make the intermediate supports lighter: columns or simpler piers.
A–B–A (pier, column, pier) · or A–B–B–A in Saxon churches
Run a shaft or pilaster from the floor, up the face of the pier and the wall above, to the roof or vault, so that the long nave wall is divided into a sequence of vertical bays, each containing one arch of the arcade and the windows above it.
Pitch a roof no lower than its covering allows (roughly 45° for thatch, 35° for plain tiles, 20–25° for slate, 15–20° for interlocking tiles, a few degrees for sheet metal) and steeper where snow lies long or rain is heavy.
thatch ≥ 45° · plain tile ≥ 35° · slate ≥ 22½° · pantile ≥ 17½° · metal ≥ 5° · membrane ≈ 1–2°
Over the vaults of a northern church raise a timber roof of about 55–60°, its section close to an equilateral triangle, so that snow slides off and the slates or tiles overlap generously.
equilateral roof: pitch = 60°, rafter = span
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.
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).
Build the monastery, if possible, so that everything it needs (water, a mill, a garden and the various crafts) is inside the enclosure, so that the monks need not wander outside, which does their souls no good; and set a wise old monk at the gate, with a cell beside it.
Let all the monks sleep in one place if their number allows, each in his own bed, clothed and ready to rise, with a light burning until morning; place the dormitory where they can go straight to the church for the night office.
Receive every guest as Christ, the poor and pilgrims with the greatest care; give the abbot and the guests a kitchen of their own, so that guests arriving at unexpected hours do not disturb the brothers; lodge them in a guest house apart, in the care of a God-fearing brother.
Give the sick a room set apart and a brother to serve them, with baths when they need them; give the novices a place of their own where they study, eat and sleep under a senior.
Lay out a new town with the cord on open ground: straight streets crossing at right angles, running through from gate to gate, blocks of equal size between them, and a square in the middle; then, when the town can afford it, enclose the rectangle with a wall.
Give every settler a house plot of the same size in the town, at the same yearly ground rent, with a garden plot and a share of fields outside; so the blocks are divided into narrow lots of equal frontage, each house on the street with its yard behind.
Leave one block of the grid open as the market square; surround it with houses standing on arcades, so that the market may be held under cover on all four sides; put the market hall in the middle, with the standard measures.
Give the church a block of its own next to the market square, near a corner of it, so that the market keeps its open ground and the church its churchyard.
Do not build party walls: leave a narrow gap between neighbouring houses, to carry off rain and waste and to stop fire running along the row.
Two semicircular barrels of different spans rise to different heights. To groin-vault an oblong bay the narrower barrel must be stilted or flattened, and the groins then wander out of plane: the difficulty that pushed builders towards ribs and pointed arches.
Build the arches of the vault first (transverse, wall and diagonal ribs) on light centring, then fill the triangular cells between them with thin webs of stone laid course by course from rib to rib.
Over an oblong bay, make the diagonal ribs semicircular and strike the transverse and wall arches as pointed arches of the heights needed to bring every crown nearly level with the central boss.
diagonal: r = ½ diagonal · transverse & wall: pointed, rise ≈ half-diagonal
Cover two bays of the nave with a single square vault divided into six cells by its diagonals and an extra transverse rib through the centre. The main piers carry three ribs each, the intermediate piers only one, so the supports may alternate strong and weak.
Raise the crown of each bay well above the tops of its boundary arches, so that the vault becomes a shallow dome on ribs whose thrust falls more steeply onto the walls.
crown ≈ half-diagonal above the springing; boundary arches ≈ half-span
Add to the diagonal ribs further ribs (tiercerons) that rise from the same springers to points on the ridge rib, so that each springer sends up a sheaf of ribs and the vault reads as a palm.
Add short ribs (liernes) that spring from neither the springers nor the central boss but run from rib to rib, linking them into star and net patterns across the vault.
Make each springer the foot of an inverted half-cone (a conoid) whose ribs all have the same curve and are cut across by horizontal rings; where the conoids stop, fill the centre with flat panels.