Take the section of the standard timber, the cai, as the module of the building: it is 15 fen high and 10 fen wide, and every member is dimensioned in fen.
1 cai = 15 fen (height) × 10 fen (width)
There are eight grades of cai, from 9 × 6 cun down to 4½ × 3 cun; use the first for great halls of nine or eleven bays and the smallest for pavilions and ceiling coffers.
grades 1–8: 9×6, 8¼×5½, 7½×5, 7.2×4.8, 6.6×4.4, 6×4, 5¼×3½, 4½×3 cun
Between two stacked arms leave a spacer of 6 fen, the zhi; a cai plus a zhi, 21 fen, is a “full timber” (zucai) and the vertical step of each tier of a bracket set.
zucai = 15 + 6 = 21 fen
Build the bracket set (puzuo) of blocks (dou) and arms (gong) crossing at right angles, each tier projecting no more than 30 fen beyond the one below; a set of n projecting tiers is called an (n + 3)-puzuo, from four to eight.
projection per tier ≤ 30 fen · rise per tier = 21 fen · puzuo = tiers + 3
Where the eaves must project far, replace the outer arms by slanting levers (ang) that run down and out beyond the bracket set and rise inward to be held down under an interior purlin.
Shape the underside of every arm’s end in four or five facets, and reduce the upper third of a spindle column by the same method, so that no member ends in a hard corner.
For a great hall, measure the distance between the front and back eave purlins and raise the ridge purlin above them by one third of it; for lesser halls, by about a quarter.
rise = ⅓ × depth between eave purlins (halls); ≈ ¼ (+ a small addition) for ting-tang halls
Draw a straight line from the ridge purlin to the eave purlin and lower the first purlin below the ridge from it by one tenth of the rise; from that point draw a new line to the eave purlin and lower the next purlin by one twentieth; continue, halving each time.
drop₁ = H⁄10 · drop₂ = H⁄20 · drop₃ = H⁄40 · …
Give each rafter span a fixed slope, increasing from the eaves to the ridge: half-pitch (5 jǔ) for the eave span, then about 6½ or 7, 7½, and 9 tenths for the span below the ridge.
rise of span k = run × (0.5, 0.65, 0.75, 0.9) for four spans
From the intermediate bracket set of the second bay the rafter ends advance progressively outward, so that at the corner the eave projects further (by 4 cun for a hall of one bay, 5 for three bays, 7 for five bays) and rises with the corner beam.
Give the hipped roof (wudian) to the highest halls, the hip-and-gable (xieshan) to the next, and the gabled roofs, overhanging (xuanshan) and flush (yingshan), to lesser buildings and houses; double the eaves for the very highest.
double wudian > double xieshan > wudian > xieshan > xuanshan > yingshan
Make the central bay (mingjian) the widest and let the bays diminish toward the ends; keep the eave columns no taller than the width of their bay.
Give a hall an odd number of bays across the front (three, five, seven, nine or eleven) so that the central axis passes through a bay and a doorway, never through a column.
Make each column a little taller than the one nearer the centre, so that the corner columns stand highest: by 2 cun per column outward from the central bay, giving 6 cun at the corners of a seven-bay hall and 1 chi 2 cun for thirteen bays.
corner rise = (bays − 1) cun · each step outward = 2 cun
Set the heads of the outer columns inward: on the front and back by 1 fen for every chi of height (1/100), on the ends by 8 li per chi (8/1000); the corner columns lean both ways.
lean = 1/100 of height (front) · 8/1000 (flanks)
Lay out the royal city as a square of nine li on each side, with three gates in each wall and, within, nine roads running north–south and nine east–west, each wide enough for nine chariots abreast.
side = 9 li · 3 gates × 4 sides · 9 jing + 9 wei · road width = 9 gui (tracks)
With the palace at the centre facing south, place the ancestral temple on its left (east) and the altar of soil and grain on its right (west); put the court in front and the market behind, each one fu (a hundred paces) square.
Find true north and south with a plumb-lined post and the shadows of sunrise and sunset, checked by the pole star; then turn the principal hall of every compound to face south.
Project the eaves from the centre of the eave purlin by 3½ chi for rafters of 3 cun diameter and 4 to 4½ chi for rafters of 5 cun; beyond them add flying rafters projecting six-tenths as far again.
eave projection ≈ 3½–4½ chi · flying rafters = 0.6 × eave projection
Make each storey of a pagoda narrower than the one below and derive all from the bay of the first storey; in the oldest Japanese pagodas the top storey is about half as wide as the first.
top storey ≈ ½ first storey (Hōryū-ji)
Run a single great pillar up the centre of the pagoda to carry the spire, but do not let the storeys rest on it: each storey stands on its own frame, stacked around the pillar.
Raise a thatched roof by a third and a tiled roof by a quarter: the rougher and more absorbent the covering, the steeper the roof must be.
thatch: rise = ⅓ · tile: rise = ¼
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°
Instead of a triangle, stack beams one above another, each shorter than the one below and raised on short posts, and lay a purlin on each beam-end; the heights of the purlins then set the section of the roof, which may be curved at will.
Combine a hipped lower roof with a gabled upper roof, the triangular gable set back from the end of the building; count one main ridge, four gable ridges and four hip ridges: nine in all.
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).
Make the avenues within the city nine chariot-tracks wide, the ring road that runs inside the wall seven tracks, and the roads in the country outside the gates five.
avenue (jing tu) 9 gui · ring road (huan tu) 7 gui · country road (ye tu) 5 gui · 1 gui = 8 chi
Set the ridge of the palace gates at five zhi, the corner towers of the palace at seven, and the corner towers of the city wall at nine, a zhi being one zhang (ten feet) in height.
palace gate : palace corner : city corner = 5 : 7 : 9 zhang (≈ 11.5 : 16 : 21 m)
Within the palace build nine chambers for the nine consorts; outside it build nine chambers where the nine ministers attend court; and divide the realm into nine parts for the nine ministers to govern.
Divide the wards between the main streets with narrow lanes running east–west, and line both sides of each lane with courtyard houses whose main halls face south.
great street 24 bu · small street 12 bu · lanes (hutong) between · lanes c. 70–80 m apart