This study investigates the contributions made by Gaspard Monge and the students of his School to the stereotomy of vaulted systems in France between the eighteenth and nineteenth centuries. The complexity of the apparatuses and the generality of the proposed solutions express the extent of the contributions that descriptive geometry made to the applications that preceded it. First among these stereotomy, which, though in decline from an operational point of view, was considered fundamental in the schools that were then being founded. The ellipsoidal vault, the helicoidal apparatus and the arrière voussure de Marseille are expressions of the relationship between the operability of stereotomy and the theoretical speculations of descriptive geometry, which operates through the synthetic language of drawing. These applications make explicit a modus operandi, capable of resolving the problems of defining, representing and expressing the geometric properties of figures using the synthetic methods of descriptive geometry.
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For further information on Gaspard Monge’s role in descriptive geometry, see Vito Cardone’s studies (Cardone 2017).
The École du Génie de Mézières was founded in 1748; Monge taught stereotomy here from 1760. This School was followed in the coming years by the École Normale and the École Polytechnique, founded in 1794.
About the teaching at the École Royale du Génie de Mézières, see Sakarovitch (1995: 208–210).
This treatise (mémoire) was published by Theodore Olivier in Applications de géométrie descriptive aux ombres, à la perspective, à la gnomonique et aux engranages. In the first two notes Olivier specifies that this mémoire is part of a manuscripts collection of the library of the École d'application de l'artillierie et du Génie de Metz. This comes from the ancient École du Génie which was transferred from Mézières to Metz in 1793. Olivier hypothesizes that this essay, anonymous, was wrote between 1775 and 1780 for the education of the young officers of the Génie (Olivier 1847: 5–24). According to Bruno Belhoste this essey, antecedent to 1764, must be attributed to Chastillon (Belhoste 1990: 106–107; Sakarovitch 1998: 85).
There was nothing more clean for their [engineers] to provide this perfect knowledge of drawing than to have them take courses in stone and wood cutting; … regardless of the advantages which result from this study, relative to the constructions under the control of the engineering officers, it is easy to understand that, when we develop all the faces and know all the plane or solid angles of any stone used in a vault, trompe, etc., or of a frame used in an attic, a dome, a staircase, etc. … that, when we know how to form the representation of all these things to make them understand to others, we are able to represent them as if they were already executed, and to combine the different constructions to make them as perfect as they can be (translation by the authors).
The École du Génie de Mézières, together with the École des Ponts et Chaussées, was active until 1803. In the early years of the Republic, which followed the French Revolution, the École Normale was established, where Monge taught for 4 years. In 1794 the École Polytechnique, first called the École centrale des travaux publiques, was founded, and it gradually replaced the first two. In the Polytechnique, the teaching of descriptive geometry appeared for the first time. Monge taught there for a short time, entrusting Hachette with responsibility for teaching this new science in 1795 (Sakarovitch 1998: 220–227).
The first printed treatise on stereotomy was by Philibert de l’Orme in 1567. This work testifies to a wise use of the representation in plan and elevation. However, evidence of the use of this method can be seen in the previously published treatises on perspective, first of all that written by Piero della Francesca at the end of the fifteenth century.
From a long time and for a very long time, the art of projection was known to stereometers, and thus to stone-cutters and carpenters; but it is really since Monge that descriptive geometry has been recognized as a science, and it is to Monge's works that we owe it; because it was he who first demonstrated that, in what was called the art of projections, really resided a scientific method which made it possible to seek and demonstrate certain geometric truths, and thus all those relating to the form of figurative space (translation by the authors).
Loria says that, starting from Monge, physical models made with ropes, wood or metal, were used to illustrate the most complex geometric figures (Loria 1921: 122).
If I redid my work, which has the title Analysis applied to geometry,… I would write in two columns: in the first I would give the demonstrations by analysis; in the second, I would give the demonstrations by descriptive geometry, in other words, by the method of projections; and you would see perhaps… by reading this opening, that the advantage would almost always be on the side of the second column, for the clarity of the reasoning, the simplicity of the demonstration, and the ease of application of the theorems found in the various works of the engineers (translation by the authors).
The repertoire of the subjects studied is generally referable to that published by Jean Baptiste de la Rue in his Traité de la coupe des pierres of 1728, a particularly substantial manual work. The numerous cases described in the treatise were quite well known at the time, because they were proposed as exercises in the stereotomy courses of the École de Mézières.
The main curvatures will be taken up by Gauss for the theorization of the “Gaussian curvature” (product of the main curvatures) currently used today in mathematical NURBS modellers. The curvature at an individual point of a surface can be of three types: positive, negative and zero. This property makes it possible to classify surfaces into different types: surfaces with negative curvature (such as ruled surfaces); surfaces with positive curvature (such as the ellipsoid); surfaces with zero curvature (such as developable surfaces); finally surfaces with mixed curvature (such as the torus).
The umbilicuses are remarkable points of a surface where the main curvatures are indeterminate. A surface composed entirely of umbilicuses is the sphere; in every point of this surface the main curvatures are indeterminate.
Monge imagined creating the vault of the halls hosting the legislative assemblies as an application of the theory of lines of curvature on the surface of an ellipsoid, in iron and glass (Monge 1795: 162–163).
The theme of bias vaults recurs in all stereotomy treatises from de l’Orme onwards. Desargues would make it the subject of a dedicated treatment in his Brouillon projet dedicated to stonecutting.
On the question of bias vaults and England’s contributions to the solution of solving the problem, see Sakarovitch (1995).
For further information on the design of impost bearings, see the studies on ruled surfaces and the stereotomy of stone (Fallavollita and Salvatore 2012b).
There are several models of arrière voussure, but this in particular raises a descriptive geometric question that finds its scientific rigor in the theory of ruled surfaces.
For further details on the vault and the related theorems, see (Fallavollita 2008).
The other two tangent planes are along the generatrix line at the points of intersection with the second and third directrix.
The tangent planes to a ruled surface at the points of a non-singular generatrix form a sheaf having this generatrix as its axis, and projective to the dotted of the contact points (translation by the authors).
The properties of the ruled surfaces were studied and discovered by Hachette and published in his treatise of 1828. Years later, Gino Fano, grouped them under the heading “consequences of the Chasles theorem”, because of the theorem that still carries today its name, although it was Hachette who derived its properties (Fano 1935: 355–357).
Two ruled surfaces which have a common line and three common tangent planes in three points of this line, are tangent to each other in all the points of the common line (translation by the authors).
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Fallavollita, F., Salvatore, M. Stereotomy of Vaulted Systems of the École Polytechnique. Nexus Netw J 22, 831–851 (2020). https://doi.org/10.1007/s00004-020-00509-w
- Vaulted systems
- Descriptive geometry
- Ruled surfaces
- Developable surfaces
- Gaspard Monge