The art and science of stereotomy can be seen as the fulcrum between the worlds of the architect - civil or military - and the mason. For the architect it was the practical end of an abstract geometry which encompassed buildings and landscapes; for the mason the theoretical end of the labour of cutting stone.

Stereotomy is the cutting of stones in three dimensions based on drawings of the design in two dimensions, to enable stones to be cut in a yard and then placed on site where cutting would be time-consuming and dangerous. The link between two and three dimensions was geometry, particularly descriptive geometry - the translation between two and three dimensions, and projective geometry - the translation of one shape on to another, which, in stereotomy, means complex shapes like splayed vaults. Beyond this practical requirement we may see an enjoyment of geometry for its own sake, especially in ornamental staircases and elaborate vaults. The cutting of stone obviously was an esoteric skill of masons, part of their guild knowledge [2], but in the renaissance both architecture and masonry became the subjects of discourses open to all educated people.

The Vauban Citadelle at Besançon (1668-1711) may be considered a textbook illustration of stereotomy. It is one of twelve groups of fortifications by the French military engineer Sébastien Le Prestre de Vauban rightly listed as a World Heritage Site by UNESCO.[3] These fortifications formed a defensive ring around France and exhibited designs adapted to flat plains, such as Neuf-Brisach, where manipulation of landscape was part of the architectural strategy, or mountainous terrain such as Besançon, almost impregnable by itself.

The reason that the Vauban Citadelle at Besançon may be considered a textbook illustration of stereotomy is because stone cut to form vaults and embrasures may be seen everywhere and these obviously follow textbooks of stereotomy such as Amédée-François Frézier's La Théorie et la Pratique de la Coupe des Pierres (Strasbourg and Paris 1737). This particular textbook was not published until after Vauban's death, but it follows a development of textbooks of stereotomy contemporaneous with those on renaissance buildings themselves. We can be sure it was intended to be of practical use: it was dedicated to the marquis d'Asfeld, director-general of fortifications in France.

I did come across some curious features in the Citadelle: what could only be called fan vaults in some passageways. It may be they were rounded as a practical gesture, to reduce damage, or it may be these represented a continuation of masons' detailing where they would not interfere with the operation of the fortress. Fan vaults were one of the great achievements of masons' architecture during the Gothic period in England, culminating in King's College Chapel, Cambridge (1446-1515). French Gothic masons, on the other hand, almost always used quadripartite groin vaults, formed by the intersection of two ribbed vaults.

Projective geometry had several immediate applications for Vauban. It could be used to predict the flight of artillery shells, in offence or defence. As a result it could be used to define the geometry of fortifications to counter them or project them. Projective geometry was the foundation of optics so had practical applications in the newly founded telescopes and microscopes. These could be used for accurate surveying - telescopes for long distances and microscopes for vernier scales. And of course it formed the basis of stereotomy.

The first known French treatise on stereotomy was Le premier tome de l'architecture (1567) by Philibert de l'Orme, whose work may be seen at the Château de Chenonceau (1555). There was a revolution in descriptive and projective geometry in the early 17th century which dramatically improved the precision, and increased the scope, of stereotomy as may be seen in later works.

Girard Desargues, a mathematician and architect, wrote treatises on descriptive geometry, projective geometry, and conics: Manière universelle de Mr Desargues, pour pratiquer la perspective par petit-pied, comme le géométral [Universal method of Mr Desargues, of practicing perspective by geometrical figures] (Paris 1647) and Brouillon project d'une atteinte aux événements des rencontres du Cône avec un Plan [Rough draft for an essay on the results of taking plane sections of a cone] (Paris 1648), regarded as the foundation of projective geometry.

Desargues was also a practicing architect: he published a work on stereotomy - La pratique du trait a preuves de Mr. Desargues lyonnois pour la coupe des pierres en l'architecture [The practical treatise of Mr. Desargues, of Lyon, on the cutting of stone in architecture] (1643) and designed the Hôtel Roland (c. 1648) in Paris, with an oval vestibule and unusual diagonal staircase.[5] This was considered important enough at the time to be illustrated in two major catalogues of French architecture, the Petit Marot in 1654 and the Grand Marot in 1670.

It is a strange feature of the architecture of the renaissance that some mathematicians could become celebrated architects apparently out of nowhere - Christopher Wren (St. Paul's Cathedral, London, 1669-1708) and Claude Perrault (the East Front of the Louvre, Paris, 1667-74), for example. Doubtless their knowledge of stereotomy allowed them to communicate with masons. Masons did not have the education to design large complex buildings but several celebrated architects started the careers as masons - Philibert de l'Orme, Andrea Palladio and Nicholas Hawksmoor, for example. Doubtless their training in stereotomy gave them a substantial basis for practicing architecture.

The practical applications of projective and descriptive geometry were obvious. They enabled the relationship between drawing and reality. There was another connection however: de l'Orme's Le premier tome de l'architecture and Frézier's La Théorie et la Pratique de la Coupe des Pierres moved between highly abstract geometrical concepts, stereotomy and the Orders of classical architecture. They seemed to belong to the same intellectual world of architecture. Why the academic world of conics, the practical world of the cutting of stone, and the aesthetic world of the Orders should be linked so closely now seems obscure and evidence of the distance between their world and ours. One thing that linked them was disciplined drawing in 2 and 3 dimensions, and that loss was serious.