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 · Modules · Plans · Arches
The bay as a cube
Make each bay of an arcade as wide as its columns are high and as deep as it is wide, so that the loggia is a row of equal cubes of space.
bay width = column height = depth (Innocenti: ≈ 10 braccia ≈ 5.8 m)
Brunelleschi’s loggia for the Foundling Hospital in Florence is the first building designed wholly on the new principle that a single measure should govern every part. The column height, the distance between column axes and the depth of the loggia are equal, and the round arches rise from impost blocks above the capitals by half that measure. The unit of the design is therefore a cube, and the whole front is its repetition.
Nothing like this had been done in Gothic building, where proportion was usually worked out geometrically on the site from the plan. Here the modular idea is legible to anyone who walks along the loggia: the bays and vaults are all alike, and the eye can take in the rule at a glance. Surveys show some variation between bays, as one would expect of fifteenth-century building, so the figures are best read as the intended scheme.
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
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.
Give the villa garden porticoes for sun and for shade, grottoes made in imitation of nature with tufa and shells, evergreens planted in straight rows at equal distances, and hedges of box; set vines on columns over the walks.
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.
Keep every triglyph centred over a column and every intercolumniation equal, and finish the frieze at the corner with a half-metope half a module wide.
corner semi-metope = ½ M (Vitruvius); equal intercolumniations throughout
Cite this page Chris Paton, “The bay as a cube”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/brunelleschi-cubic-bay. Text, drawings and models licensed under CC BY-NC 4.0.