# The Ellipse and the Oval in the Design of Spanish Military Defence in the Eighteenth Century

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## Abstract

The Spanish military engineers built
several U-shaped strongholds during the eighteenth century, using the geometrical constructions of the *ellipsis et* *ovum*. In the project on paper, engineers used the oval because of its ease of layout, but they could use either of the two figures in the staking of the fortifications. In 1704 Vicente Tosca published a methodology for building infinite ovals from its main axes. The assessment of the staking of the artillery platform of San Jorge reveals that the trace can be based both on the ellipse as the oval shape, since they have negligible difference on the scale of construction. The *ellipsis et ovum* discussion had a linguistic aspect from the perspective of applied mathematics, since different geometrical approximations for these artillery platforms may converge.

## Keywords

Military architecture Spanish military architecture Fortifications century Ellipse Geometry Oval## Abbreviations

- AGS
Archivo General de Simancas

- MPD
Mapas, Planos y Dibujos

- SHM
Servicio Histórico Militar

## Introduction

During the War of the Spanish Succession (1701–1713), Philip V (1683–1746) created the Royal Corps of Military Engineers (1711) and the Academy of Mathematics of Barcelona (1720), introducing Enlightenment thought to Spain. Earlier, the Habsburg dynasty had established the Academy of Mathematics, regulating the powers of the King’s Engineers in Madrid (1582, 1612), followed by Brussels (1675) and Barcelona (1692).

The Spanish military treatises of the seventeenth and eighteenth century assume the universality conferred on mathematics by Cartesianism as a method for the investigation of reality, dealing with the causes and effects that perfect the world (León 1992). The engineer Diego Enrique de Villegas (d. 1651) defined military architecture as a science that teaches its students about all the possible types of siege, based on their strength or weakness. Its foundations lie in mathematics and it is one of the parts into which it is divided (De Villegas 1651: 5–9). The engineer Andrés Davila y Heredia (d. 1672) explained the hierarchy which some parts of Mathematics retain within the arts, and stated in (Davila 1672: 6–56) that arithmetic and geometry are the foundations for the others. Geometry is sovereign to such an extent that all its operations are considered for the use and success of the arts, because the architect is unable to work in any area without it. According to the engineer José Chafrión (1653–1698), speculative arithmetic considers the hidden properties of numbers and their practice and use (Chafrión 1693: 1).

*ellipsis et ovum*(Migliari 1995: 93–102), is a direct reference to the mathematical studies of conic sections, and their application in military architecture. The increased use of the string method for tracing the ellipse, as well as the dissemination of several methods for tracing ovals in the treatises on military architecture of the late sixteenth century, led to the discussion concerning the scale of one of these artillery platforms. The

*ellipsis et ovum*discussion had a linguistic aspect from the perspective of applied mathematics. At the scale of construction, the geometric approach followed for defensive constructions—whether the ellipse and the oval—made a negligible difference.

## Non-Polygonal Defensive Bastions

When laying out these fortifications, Bourbon engineers used either the measurements of the Castilian *vara*, as decreed by Philip II on 24 June 1568, or the *toesa*, decreed by Philip V on 4 July 1718, depending on the tradition they followed. The discrepancies between the work of some Spanish military engineers, and especially between the work in the Academy, where the vara was used, and standard practice, which used the toesa, led to the publication of a circular to the Captains General dated 14 July 1750. This stated that not only would the Castilian vara be used in the teaching of mathematics, but that it would also be used in all matters relating to the Army and Navy. However, the Royal Order by Fernando VII dated 14 February 1751 stated that all military facilities had to use the toesa. The debate on the unit of measurement used in teaching at the Academy of Mathematics and in standard practice led its director, Juan Martin Zermeno, to reconsider the discrepancies between the use of the vara and the toesa in 1768 (Lucuze 1773: 3–57).

The debate focused on the military engineers’ transfer of the project, which has a geometric structure, to an arithmetic metrology in the practice of fortification. The toesa used in the Tortosa region measured 194.90 cm, and was subdivided into six *pies* (about 32.48 cm). The Castilian vara, the length of which was equal to 83.59 cm, was subdivided two ways: into four *palmos* (about 20.90 cm) and into three *pies* (about 27.86 cm).

*Principios de Fortificación*written by Pedro de Lucuze (1692–1779) (1772) (Fig. 3), where it is described as the most common type occurring in the batteries in a fortress on the seashore, or the banks of a navigable river. The curvature of the bastion facilitates direct cannon shots in any direction. The entrance to the enclosure is located in the gullet or mouth of the U, forming a small fortified front to defend the door from the flanks. The door is located in the middle of the retaining wall, and protected by a small pit (Lucuze 1772: 96–97, Fig. 53).

*Etliche underricht zu befestigung der Stett, Schlosz, und flecken*(1527), in the

*Geschützrondellen*, rotundas with artillery, and the

*Pastey*, a term for a bastion. To understand the U-shaped defensive arrangement, an obligatory reference is Chap. 5, ‘Creating the powder artillery defense platform’ (Dürer 1527) (Fig. 4a).

Curved shapes are very rarely used in military treatises, and limited to small defences. These include those by Diego González de Medina Barba (González de Medina Barba 1599: 200) (Fig. 4b), Gabrio Busca (c. 1540–1605) with semicircular elements (Busca 1601: 224), Jean Errard Bar-le-Duc (1554–1610) (Errard de Bar-le-Duc 1604: 109–130), and those of the fortress and Castle of Lecco (Chafrión 1687: 31–32).

## The Eighteenth-Century Defences on the Coast of Tortosa

^{1}The layout of the project (Fig. 5a) was expressed in

*toesas*and was performed using an oval, with the axis set at 12

*toesas*. The minor radius is 22

*pies*and the centre of the major radius is 18 feet from the axis, so that the oval is inscribed in a rectangle of 72 × 26.8

*pies*(12 × 4.46 toesas). The metrological design of the ellipse would allow a rectangle of 72 × 27

*pies*, with a focus located 9

*pies*from the axis of the fortification (Fig. 6).

*toesas*. The measure of the front axis was 12.046

*toesas*(23.49 m) by 4.456

*toesas*(8.69 m). With these methods, and having a header with a metrology of 12

*toesas*and 4.5

*toesas*, there was a tendency to assume that the battery that was laid out by an ellipse with major axes of 12 and 4.5

*toesas*. The commensurability of the measure led to a manual delineation of the battery layout using the so-called gardener’s or string method (see the “Appendix I: The Precedents for the Ellipse and the Conic Sections in the Literature” below), for which it was necessary to construct an ellipsograph.

^{2}The document contains two orthogonal projections of the ground plans: “Explicación, Plano bajo…” (Explanation, ground floor) containing thirteen rooms and “Plano alto…” (Upper floor) with another three, and the “Perfil A–B” (orthogonal cross-section A–B). The scale used for the plan shows 50 Spanish

*varas*(19.6 cm, about 1:220), while the scale used for the cross-section is 30 Spanish

*varas*(23.3 cm, about 1:110). The design has a ground plan layout with a metrological base of a width of 46

*varas*, constructed by means of an oval (Fig. 8). The major axis is 15

*varas*from the head of the counterguard, the minor radius is 9

*varas*from the central axis, the major axis is 16

*varas*from that, and the rise is 16

*varas*and 1.5

*palmos*. If the plan is laid out on the ground with a precise metrology of 43 × 47

*varas*, the ellipse of 46 × 32

*varas*would have a focus located at about 5

*varas*from the axis of the fortification.

## The Metrological Foundations of Batteries

Different strategies for the layout of ovals can be found in the plans of Salobreña, 1722 (AGS: MPD, 59, 032); Marbella, 1732 (AGS: MPD, 39, 065); Campo de Gibraltar, 1750 (AGS: MPD, 56, 038); and los Alfaques, 1779 (AGS: MPD, 08, 130). In some cases, the ovals’ major axis was divided into three equal parts, in others it was performed a metrological solution, and in other cases the layout is similar to the basket arch layout. The main drawback of this type of oval is that it is difficult to translate into the construction work, as it is necessary to perform several operations in order to lay out the centers.

The translation of the U-shaped work depended on two basic issues: (1) the ease of its layout and (2) the level of commensurability of the gauge of the U. If the dimensions of the axes are commensurable, it can be solved by the construction of either the ellipse or of the oval. The construction of the ellipse, using string, attributed to Anthemius of Tralles (c. 474–c. 558), was described by Cataneo (1567) and Bachot (1587). The design of the ellipse in the work is immediate. With the two axes, the foci are determined and the ellipse is laid out continuously, unlike the oval, in which the centre has to be changed. The foci require a compass operation at the end of the minor axis on the major axis of the battery. The difficulty with the figure lies in the construction of concentric ellipses, equidistant from main edges, as the focus changes position on the major axis.

One of the most influential figures in the theoretical training of Spanish military engineers, Tomás Vicente Tosca i Mascó (1651–1723) (Camara 2005: 133–158), provided instructions for constructing the oval when two axes are given (Tosca i Mascó 1707: I, prop. XV). The apparent difficulty of tracing the oval posed by the suppression of measures to find the centres was alleviated by his method. The tracing initially placed the centre of the minor arc on the major axis. This first measurement could be perfectly metrological, while the second centre of the oval, located on the minor axis, can be constructed by a simple squaring operation.

## Conclusion

The geometric study of the plans of the U-shaped batteries of San Jorge and Los Alfaques concludes that they are laid out using Serlio’s methods for ovals and their derivatives. The small-scale delineation of the platforms uses concentric ovals with two centers, as shown by the pin pricks in the paper left by the compasses used by the engineers.

Both represent a similar degree of difficulty in the layout on the ground. Two geometric operations must be performed in order to lay out the wall of the “U” using an ellipse, and thereby determine the foci of the concentric ellipses. If this is done using a Tosca oval, the centre of the minor radius is set on the major axis, meaning that two geometric operations are also required in order to lay out the other centre. The difficulty in laying out the ellipse and the oval for staking on the ground is very similar.

If we construct an ellipse and an oval with an area equivalent to 320.70 m^{2}, for the Fort of San Jorge, with axes of 23.49 and 17.38 m, the equivalent oval has the centre of the minor radius located 7.12 m from the flank of the Fort. The theoretical point is located 3.00 cm from the wall of the courtyard of the Fort.

An extraordinary approximation of both figures is obtained with the construction of the Tosca oval for San Jorge, and by placing a centre of the radius on the alignment of the courtyard wall. In fact, if the two perimeters are compared, in the flanks area, the oval tends to the extrados surface of the ellipse, with a difference of 6.02 cm. Meanwhile, in the central area of the perimeter, the oval tends towards the intrados of the ellipse, with a difference of 4.09 cm. The order of measurement is close to the margin of error of 1 % established in the survey of the Fort carried out in 1984. At that time it was determined that the perimeter was laid out by an ellipse. After the new studies, the Tosca oval equivalent, which has a minor radius located on the wall of the central courtyard, allows us to hypothesize this second solution, with the perimeter laid out using an oval.

Although the equations in the figures are mathematically very different, the formal parameterization of the tracing of the U-shaped battery of San Jorge, could be both an oval and an ellipse. The margin of error in any tracing for both figures can be perfectly absorbed in both hypotheses.

There is a fundamental difference between the two figures, derived from the science of optics of the Enlightenment. The two figures are very different in terms of receiving impact and disrupting the thrust of the projectile. If the impact is perpendicular during the descent of the parabola, the disruption of the oval figure tends to be univectorial, passing through any of the three centres of the oval. In the ellipse, the impact tends to decompose into two vectors passing through the foci located on the major axis.

## Footnotes

- 1.
The documents in SHM (9250) include: “Plano del puerto de San Jorge, situado en la marina. Jurisdicción de la plaza de Tortosa” and “Plano del Fuerte de San Jorge” by López Sopeña (1740); “Plano y perfil del repuesto de pólvora del Fuerte de San Jorge a la plaza de Tortosa” and “Plano del Fuerte destacado de San Jorge en Tarragona” probably made by Marcos Serstevens (1750); and “Plano del Fuerte de San Jorge en la costa del gobierno de Tortosa y el último que se encuentra yendo hacia Barcelona” by López Sopeña (1772).

- 2.
“Proyecto de una de las dos baterías que S. M. manda se erijan una en la punta del Franc y otra a la costa opuesta en el puerto de los Alfaques” (AGS, Secretary of War, Legajos, 03327). It was signed by Francisco Llovet, in Barcelona on April 30, 1779 (AGS: MPD, 08, 130).

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