Beauty is concinnitas

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

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.

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)

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