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)
Rules · Proportion · Geometry
The three means
Between two numbers a and b, find the arithmetic mean (a + b)/2, the geometric mean √(ab) and the harmonic mean 2ab/(a + b); use them to find a third dimension that belongs to the other two.
for 6 and 12: arithmetic 9 · geometric 8.49 · harmonic 8; always A ≥ G ≥ H, and G² = A × H
The three means come from Greek mathematics and were taught in the Middle Ages as part of arithmetic and music. The harmonic mean divides an interval so that the parts stand in the ratios of consonances: between 6 and 12 it gives 8, and 6 : 8 : 12 contains the fourth (6 : 8), the fifth (8 : 12) and the octave (6 : 12). Alberti explains how to find each mean, and Palladio uses them to give vaulted rooms a height derived from their length and breadth (The height of a vaulted room).
The geometric mean is the only one of the three that cannot usually be found by arithmetic with whole numbers (√(6 × 12) is irrational), which is why Palladio also gives compass constructions for it.
Related rules
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
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 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
If the sphere that forms the pendentives is simply continued above the ring, the bay is covered by a single shallow dome, the sail vault or pendentive dome, which springs directly from the four corners.
crown = h₀ + S⁄√2 (lower than a hemisphere on the ring, h₀ + S)
Lay the ideal garden out on a circular plan in concentric rings, woods outside, then meadows and orchards, then gardens of knots and clipped evergreens divided by walks, with water and the fountain of the goddess at the centre.
Cite this page Chris Paton, “The three means”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/three-means. Text, drawings and models licensed under CC BY-NC 4.0.