The masters who raised the medieval cathedrals had no structural calculations, nor scale drawings as we understand them today. They did, however, possess precise knowledge, handed down in the masons' lodges: a set of geometric procedures that allowed every measurement of a building to be derived from a single starting dimension. The two best-known methods bear Latin names: ad quadratum, «according to the square», and ad triangulum, «according to the triangle».
Designing without numbers
To understand these methods we must set aside a modern habit. We think in measurements: a nave so many metres wide, a pillar so many metres high. The medieval builder thought in ratios: once a fundamental dimension was fixed, for instance the width of the choir, all the others were derived from it with compass and square.
This system had considerable practical advantages. Units of measurement varied from city to city, but a geometric ratio holds everywhere. A master could transfer a design to a distant building site carrying not a table of numbers but a sequence of gestures.
The ad quadratum method
The ad quadratum procedure is based on rotating the square. You start with a square and inscribe a second one, rotated by forty-five degrees, with its corners at the midpoints of the first square's sides. Then a third inside the second, and so on.
Each square has a side that relates to the previous one as 1 to the square root of 2. This yields a series of decreasing dimensions, all linked to one another, which builders used to set the thickness of walls, the width of bases, and the tapering of pinnacles and spires.
A precious document is the booklet on the correct construction of pinnacles published in 1486 by Mathes Roriczer, master of Regensburg Cathedral. It teaches precisely how to derive the plan and elevation of a Gothic pinnacle from rotated squares, with a series of practical step-by-step instructions. It is one of the rare occasions on which the knowledge of the lodges was put in writing.
The ad triangulum method
The ad triangulum method uses the equilateral triangle instead. Once the width of a façade or section is fixed, an equilateral triangle is built upon it, and its apex determines the height of the building or of one of its parts.
The result is a greater vertical thrust than with the square: the height of an equilateral triangle is about 87% of its side, but by stacking several triangles very tall and slender proportions can be obtained.
| Method | Base figure | Key ratio | Architectural effect |
|---|---|---|---|
| Ad quadratum | rotated square | side and diagonal, 1 : √2 | solidity, balance, regular tapering |
| Ad triangulum | equilateral triangle | side and height, 2 : √3 | vertical thrust, upward tension |
The dispute over Milan Cathedral
The liveliest evidence of these methods comes from the building site of Milan Cathedral, begun in 1386. In the years that followed, the cathedral's works council summoned Lombard and foreign masters to decide the height of the naves, and the discussions were heated.
Two positions faced each other: building ad quadratum, with a height equal to the width, or ad triangulum, with proportions derived from the equilateral triangle. A mathematician, Gabriele Stornaloco, was also consulted, and he proposed a triangular scheme with rounded measurements to make it workable on site. In the end a solution lower than the full triangle prevailed.
In 1400 a French master, Jean Mignot, harshly criticised the Milanese choices and uttered a phrase that became famous: ars sine scientia nihil est, art without science is nothing. The local masters replied by reversing the motto: science without art is nothing. That exchange sums up the tension between geometric theory and practical experience that runs through all sacred architecture.
Milan Cathedral also enjoyed an editorial afterlife: in the edition of Vitruvius edited by Cesare Cesariano and printed in Como in 1521, sections of the cathedral appear laid out on grids of equilateral triangles.
The symbolic meaning
For medieval people the choice of a figure was not merely technical. The square, with its four sides, recalled the earth, stability, and the heavenly Jerusalem described as a square city. The triangle evoked the Trinity and the movement of the soul towards God.
A building constructed according to the square or the triangle thus became a discourse in stone: its proportions said something about the order of the world. In the tradition of Studio Esoterico Cornelio, this coincidence between a building rule and a spiritual meaning is considered the heart of sacred geometry: not added decoration, but structure that carries meaning.
How to try the two methods
Anyone wishing to understand these procedures can draw them on a sheet of paper:
- draw a ten-centimetre square and inscribe three rotated squares one inside the other;
- on a ten-centimetre segment build an equilateral triangle and stack two more, with their bases on the same line;
- compare the two figures: you will notice the difference in character between the stillness of the square and the thrust of the triangle.
In short
- Medieval builders designed by geometric ratios, deriving every measurement from an initial dimension.
- The ad quadratum method uses rotated squares and the ratio between side and diagonal.
- The ad triangulum method uses the equilateral triangle and produces more slender proportions.
- The disputes over Milan Cathedral show the clash between the two methods and between theory and practice.
- The chosen figures also carried symbolic value: the square for the earth, the triangle for the soaring towards the divine.
