Scamozzi’s hierarchy

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

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°)