Cut a shaft with sixteen shallow flutes or facets, give it a square abacus and a low round base: the “proto-Doric” column of the Middle Kingdom rock tombs at Beni Hasan.
Rules · Proportion · Modules · Theory
Commensurability
Let every part of a building be measured by the same unit and relate to the whole by simple whole-number ratios, so that one module generates all its dimensions.
e.g. 2 : 3, 3 : 4, 1 : 2: ratios of small whole numbers of modules
Vitruvius calls this symmetria: not mirror symmetry but the agreement of the parts with the whole through a common measure, as in a well-formed human body. In the orders the measure is the column diameter; in plans it may be a foot or a bay. Commensurable ratios can be written down and checked by anyone with a rule, which made them the natural language of builders’ contracts as well as of humanist theory.
The alternative tradition works with ratios that cannot be expressed in whole numbers (the diagonal of the square, the golden section) but that can be drawn with compass and straightedge (The diagonal of the square). Much of the history of proportion is the interplay of these two ways of thinking, arithmetic and geometric.
Related rules
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
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.
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.
Make the corner columns thicker than the rest by a fiftieth of their diameter, because, seen against the open sky, they look thinner than they are.
D corner = D × (1 + 1⁄50); Parthenon: 1.95 m against 1.905 m
Cite this page Chris Paton, “Commensurability”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/commensurability. Text, drawings and models licensed under CC BY-NC 4.0.