A building’s form must follow the geometrical rules of the craft (scientia) as well as the experience of the builder (ars); neither is sufficient alone.
The Parisian master Jean Mignot, called to Milan in 1399, condemned the piers and buttresses as too weak. The Milanese replied, among other things, that pointed arches do not push on their buttresses; Mignot answered that ars sine scientia nihil est (1400). The Milanese retorted that scientia sine arte was nothing either.
Both were half right. Pointed arches do thrust, as the flyers of every French cathedral testify, but less than round ones; and the geometrical rules that Mignot invoked were themselves distilled experience. Modern structural analysis has shown that such rules, applied consistently, produce masonry that is safe at any scale.
Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.
joints ⟂ intrados (radial to the centre)
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)
A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.
free-hanging chain (catenary) ↔ ideal arch under its own weight
Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.
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
An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.
Cite this page Chris Paton, “Art without science is nothing”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/ars-sine-scientia. Text, drawings and models licensed under CC BY-NC 4.0.