1.1 The curve of equilibrium

The theory of the chain, in the shape of a hanging collar, was proposed by Robert Hooke (1635–1703) at the end of his treatise A description of Helioscopes, and Some Other Instruments (1676). Hooke presented a solution that would be revealed as "Ut pendet continuun flexile, sic stabit contiguum rigidum inversum" [10]. The

Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century DOI: http://dx.doi.org/10.5772/intechopen.87060

awareness about the shape of the catenary was applied by Christopher Wren (1632–1723) in the dome of San Pablo (1675), with the collaboration of Robert Hooke in the design [11].

Simon Stevin (1548–1620), in De Beghinselen der Weeghconst (1586), had previously proven the law of equilibrium of a body on an inclined plane. There we can see a hanging cable which has the shape of a catenary [12]. Despite the evidence provided by the figure, there was not any mathematical approach to catenaries. That is why Jakob Bernoulli (1654–1705), in Acta Eruditorum (1690), issued a challenge to the mathematical community to solve this problem [13]. The solution was published in Acta Eruditorum (1691) by Johann Bernoulli (1667–1748) with the title "Solutio problematis funicaularii" [14] and also by Chistiaan Huygens (1629–1695) with the title "Dynastae Zulichemii, solutio problematis funicularii" [15].

The mathematic equation of the catenary would be formulated some years later by David Gregory (1659–1708) and published in the Philosophical Transactions of the Royal Society (1697). Gregory affirms that the catenary is the real shape of an arch, because if these can sustain themselves, it is because a catenary can be drawn in its section [16]. James Stirling (1692–1770), in the Lineae Tertii Ordinis Neutonianae (1717), compiled the ideas of the English school building a catenary with hanging spheres, to simulate the behavior of a constructive element [17]. This solution inspired the analysis by Giovanni Poleni (1683–1761) in the Memorie Istoriche della Gran Cupola del Tempio Vaticano (1748) [18], who developed a methodology similar to Stirling's, to understand the breaking of the vault of the San Pietro Basilica [19].

In Spain, the development and application of this theory take place in the context of the Mathematics Academy of Barcelona (1720). The work of Bernard Forest de Bélidor (1698–1761) is the main reference of the curve of equilibrium theory. In La science des ingénieurs dans la conduite des travaux de fortification et architecture civile (1729), Book II, Chap. III, Prop. V, Bélidor sets out the curve that must be given to a vault, so all its parts weigh the same and stand in equilibrium [20], and as a result its curve will have the shape of a catenary. And so he determines, for military constructions, up to five types of different vaults: rounded, tiers-point pointed, elliptical drawn as a segmental arch, the flat ones, and the derived forms of the catenary [21]. In addition, the work De la poussée des voûtes, (1729), by Pierre Couplet (+1743), mentions the chaînette, the hanging chain, as the best of all shapes for the construction of vaults. He also says that to build this vault, every part of the hanging rope has to be loaded with the proportional weight of the construction, so the resultant curve will be the one to be used [22].

### 2. The libraries of the military engineers of the eighteenth century

On 13 January 1710, King Philip V appointed Jorge Prosper Verboom (1665–1744) as engineer in chief. Verboom was a disciple of Sebastián Fernández de Medrano. Together with Alejandro de Retz (c. 1660–c. 1732), chief engineer of the Catalan region, Verboom was the link with the former academy in Brussels. Mateo Calabro was appointed head of the Mathematics Academy of Barcelona (1720–1738), and he had profound disagreements with Jorge Prosper Verboom with regard to the training program. Therefore, in 1738 Verboom offered the head position of the academy to Pedro de Lucuze y Ponce (1692–1779), who held that position until his death. Among other duties, the academy had to build a collection of scientific works which would be used as reference texts for military training. This bibliographic interest led Vicente García de la Huerta (1734–1787) to publish Bibliotheca Militar Española (1760), a collection of the most important military

Architecture. From then on, Martinell joined an exclusive group of disciples who

Otherwise, these concepts were introduced in the formation of architects through the Escuela Especial de Arquitectura de Madrid (1844). The work Traite Theorique et Pratique de L'art de bâtir (1802–1817) by Jean-Baptiste Rondelet (1742–1829) exposes the methodology to lay out catenary arches by means of the theory of the chain and another complicated procedure [7]. In addition, a treatise by John Millington (1779–1868) was also used in architecture schools. It was translated as Elementos de Arquitectura (1848) and contained Hooke's theory and the layout of the catenary [8]. Juan Torras i Guardiola (1827–1910) developed the scientific

The theory of the chain, in the shape of a hanging collar, was proposed by Robert Hooke (1635–1703) at the end of his treatise A description of Helioscopes, and Some Other Instruments (1676). Hooke presented a solution that would be revealed as "Ut pendet continuun flexile, sic stabit contiguum rigidum inversum" [10]. The

basis for the calculation of these structures in the Barcelona School of

Cesar Martinell sketch of transversal section El Pinell de Brai [COAC H101I-6-Reg 2502].

For Martinell, Gaudí was a much more interesting lesson of life and architecture than most of the teachings given at university. Gaudí's words became architectural when the very statement of the scientific truth and the procedures he himself invented were able to explain problems and geometric concepts which remained unclear in the classrooms of the school of architecture. Antoni Gaudí's theory of structures relies on the strength of geometry, in particular on the strength of the parabolic and catenary shapes. The construction technique used by Cèsar Martinell stems from the methods applied by Gaudí to calculate the geometric shapes of vaults and arches [5] (Figure 1). Other architects have written about this view; see, for instance, the lecture entitled La fábrica de ladrillo en la construcción catalana (1900), by Josep Domènech i Estapà (1858–1917). In this lecture it is claimed that the parabolic and catenary shapes are the lines of equilibrium in a system of evenly distributed loads, where the parabolic shape relates to the horizontal projection and

learned a way of doing architecture aside from university teachings [4].

the catenary shape relates to the arch length [6].

Architecture (1875) [9].

Military Engineering

Figure 1.

28

1.1 The curve of equilibrium

engineering treatises which had been written between the sixteenth and the eighteenth centuries [23].

Some of the most relevant texts available for military engineers in the libraries are L'architecture des voûtes (1643) by François Derand (1588–1644) (Figure 2), in the library of Jorge Prosper Verboom (1665–1744) [24], and the Treatise on

Stereotomy by Abraham Bosse (1604–1676), in addition to La pratique du trait à preuve de M. des Argues Lyonnois pour la coupe des pierres en Architecture (1643), in

Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century

Another reference work for engineers is Compendio Mathematico (1707–1715) by Tomás Vicente Tosca (1651–1723). Treatise XVI in volume V, which is divided into six parts, deals with military architecture (1712). It is available in the academy's library, in Verboom's library (+1744), in Cermeño's library (+ 1790), and in Hermosilla's library (+1776). [26] This treatise deals with arch and vault dimensioning, as well as their collapse mechanisms. It claims that the perfect shape of an arch is a mixed one, made up by the intrados of a round arch and the extrados of a pointed arch. Furthermore, Vicente Tosca improves the three-centered arch or basket-handle arch (arco apaynelado or arco carpanel), since in Book II, Prop. III, he establishes for the first time the geometrical construction of ovals, which are

Yet another work of reference in the libraries of the military engineers was Traité des Ponts (1716) by Hubert Gautier (1660–1737) in the libraries of Aylmer (+1788), Aedo Espinosa (+1787), Roncali (+1794), and Cermeño (+1790) [28].The second edition (issued in 1723) includes the famous statement "Ut pondera libra, sic aedificia architectura," referring to the difference in thrust between a round arch (which tilts the balance) and a pointed arch [29]. This edition also includes an additional dissertation: Augmenté d'une Dissertation sur les Culées, Piles, Voussoirs, et Poussées des Ponts. The 1728 edition includes a revised and enlarged version of Dissertation sur l'Epaisseur des Culées des Ponts, sur la Largeur des Piles, sur la Portée des Voussoirs, sur l'Erfort & la Pesanteur des Arches à differens

The Oeuvres de Monsieur Maroitte (1717), by Edme Mariotte (1620–1684), was also used for teaching purposes. The most interesting part is Volume 2, which includes Traité du mouvement des eaux et des autres corps fluides, divisé en V parties; this work is in Verboom's library (+1744) and in Cermeño's library (+1790) [31],

However, the main references for the Spanish military engineers were definetly

Mathématique (1725), La science des ingénieurs (1729), and the first (1737) and second (1739) volumes of his Architecture hydraulique (1739), in the Library of Verboom (+1744), Espinosa (+1787), Burgo (+1788), Cermeño (+1790), and Juan Miguel de Roncali (+1794). In Nouveau cours de Mathématique (1725), Bernard Forest de Bélidor discusses a practical application of masonry mechanics to the construction of gunpowder magazines [32]. Bélidor calculates the abutment for a barrel vault and for a third-point arch. He includes a table summarizing the size of the pieds droits depending on their curvature and their location, specifying as well if they are supporting the basement floor or the roof. In La science des ingénieurs dans la conduite

the works of Bernard Forest of Bélidor (1698–1761): the Nouveau cours de

des travaux de fortification et architecture civile (1729) (Figure 2), Book II, Chap. III. Prop V, Bélidor establishes the curvature that a vault should have so that all its parts have the same weight and are well balanced (the result is a curve with the shape of a catenary). Bélidor differentiates five vault topologies in military constructions: barrel vaults, third-point vaults, surbased vaults with an elliptical profile, flat vaults, and vaults with a shape that results from the chain [33].

2.1 Gunpowder warehouses in the military architecture treatises

Military architecture treatises of José Cassani (1673–1750), developed at the end of the seventeenth and eighteenth centuries, make reference to the construction of these warehouses, especially if they have an element of high resistance, such as

the library of the Barcelona Academy [25].

DOI: http://dx.doi.org/10.5772/intechopen.87060

defined by the length of their two main axes [27].

previously published by de La Hire (1686).

surbaissemens. [30].

31

Figure 2. La science des ingénieurs (1729) of Bernard Forest de Bélidor.

Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century DOI: http://dx.doi.org/10.5772/intechopen.87060

Stereotomy by Abraham Bosse (1604–1676), in addition to La pratique du trait à preuve de M. des Argues Lyonnois pour la coupe des pierres en Architecture (1643), in the library of the Barcelona Academy [25].

Another reference work for engineers is Compendio Mathematico (1707–1715) by Tomás Vicente Tosca (1651–1723). Treatise XVI in volume V, which is divided into six parts, deals with military architecture (1712). It is available in the academy's library, in Verboom's library (+1744), in Cermeño's library (+ 1790), and in Hermosilla's library (+1776). [26] This treatise deals with arch and vault dimensioning, as well as their collapse mechanisms. It claims that the perfect shape of an arch is a mixed one, made up by the intrados of a round arch and the extrados of a pointed arch. Furthermore, Vicente Tosca improves the three-centered arch or basket-handle arch (arco apaynelado or arco carpanel), since in Book II, Prop. III, he establishes for the first time the geometrical construction of ovals, which are defined by the length of their two main axes [27].

Yet another work of reference in the libraries of the military engineers was Traité des Ponts (1716) by Hubert Gautier (1660–1737) in the libraries of Aylmer (+1788), Aedo Espinosa (+1787), Roncali (+1794), and Cermeño (+1790) [28].The second edition (issued in 1723) includes the famous statement "Ut pondera libra, sic aedificia architectura," referring to the difference in thrust between a round arch (which tilts the balance) and a pointed arch [29]. This edition also includes an additional dissertation: Augmenté d'une Dissertation sur les Culées, Piles, Voussoirs, et Poussées des Ponts. The 1728 edition includes a revised and enlarged version of Dissertation sur l'Epaisseur des Culées des Ponts, sur la Largeur des Piles, sur la Portée des Voussoirs, sur l'Erfort & la Pesanteur des Arches à differens surbaissemens. [30].

The Oeuvres de Monsieur Maroitte (1717), by Edme Mariotte (1620–1684), was also used for teaching purposes. The most interesting part is Volume 2, which includes Traité du mouvement des eaux et des autres corps fluides, divisé en V parties; this work is in Verboom's library (+1744) and in Cermeño's library (+1790) [31], previously published by de La Hire (1686).

However, the main references for the Spanish military engineers were definetly the works of Bernard Forest of Bélidor (1698–1761): the Nouveau cours de Mathématique (1725), La science des ingénieurs (1729), and the first (1737) and second (1739) volumes of his Architecture hydraulique (1739), in the Library of Verboom (+1744), Espinosa (+1787), Burgo (+1788), Cermeño (+1790), and Juan Miguel de Roncali (+1794). In Nouveau cours de Mathématique (1725), Bernard Forest de Bélidor discusses a practical application of masonry mechanics to the construction of gunpowder magazines [32]. Bélidor calculates the abutment for a barrel vault and for a third-point arch. He includes a table summarizing the size of the pieds droits depending on their curvature and their location, specifying as well if they are supporting the basement floor or the roof. In La science des ingénieurs dans la conduite des travaux de fortification et architecture civile (1729) (Figure 2), Book II, Chap. III. Prop V, Bélidor establishes the curvature that a vault should have so that all its parts have the same weight and are well balanced (the result is a curve with the shape of a catenary). Bélidor differentiates five vault topologies in military constructions: barrel vaults, third-point vaults, surbased vaults with an elliptical profile, flat vaults, and vaults with a shape that results from the chain [33].

#### 2.1 Gunpowder warehouses in the military architecture treatises

Military architecture treatises of José Cassani (1673–1750), developed at the end of the seventeenth and eighteenth centuries, make reference to the construction of these warehouses, especially if they have an element of high resistance, such as

engineering treatises which had been written between the sixteenth and the

Some of the most relevant texts available for military engineers in the libraries are L'architecture des voûtes (1643) by François Derand (1588–1644) (Figure 2), in the library of Jorge Prosper Verboom (1665–1744) [24], and the Treatise on

eighteenth centuries [23].

Military Engineering

Figure 2.

30

La science des ingénieurs (1729) of Bernard Forest de Bélidor.
