A SQUARE CONICAL VAULT”

The squinches is a very versatile construction type in the field of stonework. It is defined in its most elementary case as the covering of a triangular area with a conical surface, but, nevertheless, there are an infinite number of variants increasing the complexity of the layout and its execution. The studied case in this paper is the development of the layout of the "square conical vault" (actually, a cantilevered vault), in the treatise on stonework by Tomás Vicente Tosca, published in 1727. This author's work reflects the tradition of the local area (which comes mainly from the brilliant episode of the 15th century in Valencia), but it is also part of the pre-illustrated cultural trends of the time. The traces that appear are the result of combining ancient graphic habits with the development, to a certain extent, already scientific, of geometry and mathematics. All this leads, in the case of the "conical-square vault", to the development of a fundamentally descriptive trace, result of a mentality that is more analytical than practical, but that does not manage to emancipate itself from the old building tradition, as this study makes evident, and also the detected graphic habits.


THE SQUINCH ON THE SPANISH STEREOTOMY
Within the ideology of the European stonemasonry, and particularly of the Hispanic one, the squinches are one of the basic types and with greater presence.Its origin is difficult to relate to a specific chronology (perhaps in the Byzantine or Roman period, or, for the stone masonry, the late medieval Armenian context) (López, Alonso, Calvo and Rabasa 2013, 555).In any case, this type has its origin in the resolution of square-planned dome spaces.This origin, on the construction point of view, will propitiate the variants that can be found.In construction terms, the transition from a square to a circular space is one of the most well known problems in historical construction, and affects brick and stone-cutting manufacturing alike (Palacios 2003, 23).We will focus, then, on the second case, and on those examples that follow established geometric patterns.
The two best known forms of passage between a square wall box and a circular tambour are the penditive and the squinch (Figure 3).The penditive consists of spherical surfaces, while the squinches are defined by cones.Also, we know that the penditive solves the dome problem with precision, while the squinch only guarantees the transition to the octagon.In many primitive cases, on temple crosses, four cantilevered domes are arranged forming an octagon, over which it is not difficult to cover with an approximately spherical shape.The construction experience, in these early cases, meant that, with hardly any geometric patterns, the construction itself solved the problems, often forming some inaccurate surfaces (figure 3).
The advantage of the squinch as an architectural type is the easier execution.While a strict penditive requires carving spherical surfaces, the squinch can be easily devised by conceiving voussoirs as if it were an arch (Palacios 2003, 25).In fact, tracing the geometry of a squinch is easier than tracing a penditive.Let us focus, then, on the Spanish stone-cutting, on the traces, and on the preserved manuscripts.The squinch, as well as the treatises as a whole, appeared at the beginning of the 16th century.This was the moment when the old guild system began to undergo transformations and individual knowledge began to be universal1 .
The corpus of Spanish manuscripts and treaties, on the other hand, traces a whole tradition and an evolution, from the oldest examples (Hernán Ruiz ca.1550) to the later ones (Tosca 1707(Tosca -1715)), already at the beginning of the so-called century of the Enlightenment.And throughout this whole, the tracing of squinches will be a recurrent feature, offering a range of examples and solutions (Rabasa 2000, 210).

SQUINCH LAYOUT AND ITS MAIN VARIANTS
Before focusing on the case of Tosca, it is convenient to briefly develop the tracing system of a French horn and its main variants, using examples from historical authors.
We will begin with Vandelvira (ca.1600), perhaps the most representative author of the Hispanic tradition, and one of those who treat the issue of the squinch with greater extension.
In fact, his manuscript preserved in the Library of the Escuela Técnica Superior de Madrid, begins by developing the horns and their variants in great detail2 (Vandelvira ca.1600, folios 7r-17v).
Vandelvira starts developing a simple squinch on right-angled walls.The author proposes to simply outline the plan, and to place the center and the arch that define the cone.Then, he divides the arch and places the joints on the plan.Once this simple process is done, the approximate development of a portion of the cone is obtained, finding the true dimension of the "triangular" face that defines one of the pieces (all equal).And finally, we obtain the angle ("saltarregla") between the intrados and the wall.In other words, the tracisist obtains the basic cutting patterns that a stonemason needs to be able to execute the piece, not worrying on developing theoretical nature questions (figure 4)3 .
The first of the complex variants is the oblique or deviated squinch.We would also recall those examples of squinches whose cantilevered edges are outward or inward.That is to say, the squinch cone does not end in a straight vertical wall, but in a curved or mixtilinear surface outwards or inwards, which complicates the layout considerably.In fact, the author, by drawing simple templates to carve pieces, solves (maybe unconsciously) complex geometrical intersections between conical and extruded (cylindrical) surfaces (figure 5).In addition to these examples, we should mention a paradigmatic case, by Philibert de l'Orme (1567, 89-98): the squinch with an irregular perimeter.In this case, the intersection of the cone with a vertically extruded shape is taken to an extreme (figure 5), but the goal of this case, as complex as it may seem, continues to be essentially practical, linked to construction proposes  (Fernández 2003, 13).
The complete title of this is Compendio Mathemático, which contains all the main subjects of the sciences, that deal with quantity, and is composed of 9 volumes whose subject matter is varied, and not only architectural.However, Volume V is especially interesting: Civil Architecture, Stonework, Military Architecture, Pyrotechnics and Artillery4 .
In this section of the Compendium, the author develops questions related to stonemasonry, typical of the Hispanic tradition, and focuses essentially on the resolution of geometric traces.Even so, in this brief compilation of problems, a significant difference can be seen compared to previous works.The difference is the focus of the text, which no longer seems to pose constructive problems, but rather approaches of a mathematical type.
The 15th treatise (the stonework part) begins by developing the arches in their different variants (arches and capials), continues with the squinches, with the vaults (spherical or groin), and ends by developing some stairs examples.All this, then, is structured in different propositions (numbered), that increase in difficulty.
Although the focus of the text is fundamentally theoretical, it is interesting to note the inference of some local models, i.e.Valencian ones.This is the case, in the vaults, of the quotation to the dome of Valencia Cathedral, and although it does not name it, in the set of squinches, the analogy of a square conical vault with the squinch built in the courtyard of the Palace of the Generalidad of Valencia.This last case is discussed below.

THE "SQUARE CONE VAULT"
Under the title of Proposition II (Tosca 1727, 185) Drawing a square conical vault, Tosca develops the layout of a conical squinch (a circular cone) in cantilever, with a square plan.
...The one we are now outlining has a square floor plan, and that is why we call it a square: it is very useful in many cases, because it can be used to load an angle of a construction ... The author, apart from the text, encloses a figure with a series of drawings (figures 1, 6), which should be exhibited together.
First of all, the author defines the plan of the squinch , its cantilever, and establishes a "virtual" plane at 45º, in which to place the directional arch of the cone.This arch is divided, and the joints are determined in the plan.
Then the author develops several questions in a complex way: The obtaining in real dimension of the fronts of the cantilevered part of the penditive, the inclination on the horizontal plane of the different joints, and finally the approximate development of the intrados of the cone in real magnitude, by pieces.
In other words, the proposed graphic resolution is analogous to that proposed in previous treaties, but with one exception: the text makes no reference to the constructive process.To illustrate this point, I quote the end of the proposition: ...to whom will be given the curved concavity with the truss rule, or ordinary baffle, cut according to the front of the fundamental arch (Tosca 1727, 188).This is all that is said regarding the construction.

THE SQUINCH OF THE PALAU DE LA GENERALITAT
As mentioned above, the cantilevered squinch is a unique variation, which is not often seen.Even so, there are some interesting examples: the Philibert de l'Orme squinch built in Toulouse (Palaces 2003, 44), and the horn from the Palace of the Generalidad, from the first years of the 16th century, which, if it is not the oldest preserved example, is one of the earliest (Zaragozá, Marín y Navarro 2019, 19).
Its construction is documented in the year 1530, by the master mason Pere Real and the stonemason Joan Navarro.Moreover, it does not only is cantilevered, but also supports a spiral staircase, about 15 meters high (made of brick) (figures 2, 7).
This case has been surveyed using laser scanner technology5 , and has been carefully analyzed from a set of selected points of the resulting cloud (on the figures, in red).The conical geometry of the surface of the intrados, the plan of the walls at an angle, slightly more than 90º, and the apparent descent of the cantilevered part have been corroborated.
Then, a restitution of the layout has been made based on the gathered data, and following another nearby treaty source: the stonemasonry manuscript by Joseph Gelabert (1653).This work, written in Mallorca at a somewhat late date, includes numerous cases of the Aragonese tradition, and specifically proposes a squinch very similar to that one in Valencia: the Pitxina radona de tres peñades (page 108v, 109r) (Rabasa 2011, 294).
The trace is not different from other sources, except for the shape of the cantilevered element, bevelled, which is practically the same as in the Valencian case, and the similar (almost identical) (figure 8).

TRACING THE CANTILEVERED SQUINCH
Firstly, some basic metric questions have been analysed: the angle formed by the walls (not exactly straight), the shape of the floor plan, and the analysis of the upper joints in order to determine the settlement or movements of the work over the centuries, since it bears a high load.In fact, it has been proven that there is a descent of the squinch in its cantilevered part (about 17 cm), but it is difficult to specify if this occurred or not after the assembly of the squinch (figure 7).
The reconstruction of the layout, according to the different handwritten sources (and especially the works of Gelabert and Tosca), begins with the definition of the plan, and the directional arch of the cone.Then the cutting is established, the joints are placed, and the development of the surface of the intrados begins (figure 9).
If we are strict in following Gelabert's indications, the drawing of the bevelled shape of the squinch's peak is especially delicate, having to obtain the distance in true dimension from the centre of the bevel to the start of the cone.
Another peculiarity observed by Gelabert (not so in Tosca), is the simplification of some lines, which should be broken (or curved) in the course of the development, and are straight in the line.
It is assumed that these simplifications are not correctable on the construction of the squinch itself.

SIZE AND CONSTRUCTION
We find very interesting to follow the indications given by Gelabert for carving the pieces that make up the squinch.This process necessarily begins with the lower pieces, those of the bases, whose templates and angles are taken from the tracing.
Once the bases have been made, each piece on top (as the squinch is closed) is carved taking measurements of the previous one, that is, the voussoir on which it will rest (figure 10).The last piece will be the keystone, which also has the shape of a bevel, and its shape will be finished in place, ensuring the continuity of the front of the squinch.We should also raise some questions that do not appear in the treaties, such as that of the formwork in the construction process.Nowadays we are used to thinking of the assembly of these works as a complex system of supports, but this surely did not take place.Most probably, the necessary formwork for the construction of the squinch was only an arch, semicircular, corresponding to the cone's guideline.The rest of the parts could easily be fastened during assembly with some auxiliary supports.

TOSCA'S APPROACH AS A SCIENTIFIC PRACTICE
We can now affirm that the Valencia squinch is a model that fits well with the manuscripts and treaties.It can also be said that the indications given by the treatists, especially Gelabert, are useful for practical use.On the other hand, it is clear that a geometrical trace must be drawn up to provide a solution to the architectural problem.Without this graphic approach, however succinct it may be, it is impossible to materialise a case of such complexity.However, returning to Tosca, it is worth putting some relevant questions on the table: In general terms, the graphic part (perhaps the one most closely linked to tradition itself) is clearly in line with architectural needs.But the same does not happen with the supporting text, which rarely refers to the processes of carving and construction, and which rather seems to describe a mathematical problem.
In other words, Tosca's treatise seeks to justify or convert the art of the tracist into a kind of regulated science.The approach, moreover, is not particular to this author, nor does it belong to the local or national tradition, but can be detected in other contemporary authors, such as Guarino Guarini (Euclides Adauctus, 1671) (Bianchini 2008, 17), who also raises cases from the stoneworking tradition (possibly Hispanic) from an almost strictly mathematical point of view.Furthermore, there is a circumstance that should be highlighted in this article, and that is the fact that Tosca refers to the squinch as a conical vault, and Guarini, in a contemporary way, is talking in his works about vaults that are sometimes composed of conical forms.

CONCLUSION
The main conclusion to be highlighted in this paper is the double and joint connection of Tosca's work with the building tradition, and with the pre-illustrated scientific world.This means that a compendium of traces, typical of a 16th-18th century Tracist master, wants to be justified or supported by a theoretical discourse of a rigorously mathematical nature.This, moreover, is not a local fact, but rather one that is typical of the Europe of that time.
On the other hand, the article emphasizes the importance of the horn of the Palace of the Generality of Valencia, which, because of its execution, stereotomic boast, structural and early chronology, must be considered one of the key pieces in the stereotomy, at least Hispanic.And, in fact, it is probably one of the pieces on which Tosca based itself to develop the layout shown here.
Catedral de Valencia, y aunque no lo nombra, en el grupo de las trompas, la analogía de bóveda cónica cuadrada con la trompa construida en el patio del Palacio de la Generalidad de Valencia.De este último caso hablamos a continuación.

Fig. 06 .
Fig. 06.Details of the square conical vault (Tosca 1727).a -plan, b -front of the horn, c -obtaining the inclinations, d -development of the surface of the intrados.

Fig. 07 .
Fig. 07.Squinch of the palace of the Generalitat Valenciana.Survey by the authors.