Geometry of bastion fortifications magistral line: influences and development

Abstract

In the middle of the 15th century, the fortification system of medieval European cities underwent a transformation into a bastion fortification system. This system will dominate until the mid-19th century, experiencing its zenith by the late 17th century with the improvements of the French military engineer Vauban. A defining feature of this system is the distinct contour of the magistral line in the ground plan, with its geometry progressing from simplicity in the early stages, to comprehensive solutions that addresses dimensions of all elements within bastion fortifications, in the final ones. Our research encompasses the geometric principles used by four selected military engineers from this period (Marolois, de Ville, Pagan, Vauban) with the aim to identify the development of geometric procedures in the design of the magistral line. We explore the specifics and differences in their approaches and methodologies. Conducting a comparative analysis of the geometric principles employed by different builders, using the example of star fortresses, we follow the progression of theoretical framework and design aspirations underlying the key elements of the magistral line. We consequently consider whether the “Vauban style” is a spontaneous creation of a single genius, or an outcome that emerged from the collective contributions of numerous predecessors across Europe. This paper spotlights the principles of bastion fortress design based on geometry, offering a way to identify markers of influence in observed examples, and also to complement non-invasive methods of cultural heritage preservation with adapted 17th-century geometric procedures, making them more compatible with modern approaches.

Introduction

From the mid-fifteenth century, after the fall of Constantinople in 1453, through entire 16th and 17th centuries, the new gunpowder artillery, used in defense and attack alike, brought about a transformation of the fortified European town. The medieval defense tower gives way to the fortification system based on bastions and a whole range of interior and exterior interventions. The transformation occurred gradually across a wider area of Europe, where military conflicts alternated over an extended period as a direct consequence of the antagonism between major powers: the Holy Roman Empire, Spain, France, and England, as well as the ongoing conflict between the Catholic and Reformed Churches. Chronologically speaking, the first significant changes in the construction and design of fortifications took place during the Italian Wars (1494–1559), which began on the Italian Peninsula with the conflict between Charles VIII of France against the Holy Roman Empire, Spain, and the alliance of Italian states led by Pope Alexander VI^1,2. The Dutch War of Independence (1568–1648), also known as the 80 Years’ War, was significant for the development of bastion fortifications in the geographical area of the Dutch provinces^3,4. The expansion in redesigning existing fortresses and building new bastion fortifications occurred during the War of Spanish Succession (1713–1714), a broader conflict for European dominance between France and the Habsburgs, whose conclusion through the peace treaties of Utrecht (1713–1714) halted Louis XIV’s expansionist policies and established the balance of power as the basis of the international order^5,6.

At its core, the contour of bastion fortifications is defined by the shape of the magistral line, which was subjected to new geometric design principles. Do these principles differ across the conflict-ridden Europe or converge towards a universal style? Are the personal stamps of the master builders within a given area easily identifiable or tightly intertwined? These matters are seldom or insufficiently covered in the available literature. The aim of this paper is to highlight the significance of geometric principles in designing the outline of bastion fortresses—the main defensive line—as an identifying marker of the overlapping influences shaped by Europe’s turbulent history.

Sébastien Le Prestre, Marquis de Vauban (1633–1707), Marshal of France, and primarily a military engineer in the service of Louis XIV, known for his skill in designing fortifications and organizing attacks and sieges of existing ones, in widely accessible sources is often referred to as the “Father of Fortress”⁷. It is being emphasized that the fortresses were designed according to the “Vauban method”⁸ and it is repeatedly claimed that the fortresses were built in the “Vauban style”⁹. Despite such assertive claims, questions can be raised that challenge these deep-rooted opinions: is it truly about a style with distinctive features, and if so, is it closely associated with the name of a single designer? Alternatively, can we discuss a model and the application of certain building principles rather than adhering to a specific style? In this paper, through the analysis of the bastion fortification system’s development and the examination of geometric principles in shaping the magistral line, we trace the evolution of theoretical thought and practice in designing and constructing the bastion fortification outline until the end of the 17th century, aiming to answer these questions.

Characteristics and geometric features of the bastion fortifications

The development of artillery conditioned changes in the design of European medieval fortified cities, the shape of their ground plans and the walls themselves. The passive form of the medieval fortress, a legacy of Greco-Roman antiquity, had proved to be extremely vulnerable under a direct artillery fire, especially when attacked from points directly opposite to the line of the defensive wall. If invaders reached the foot of the fortification, they could strategically dig into or mine the foundations to bring down the wall. The “dead land,” the unprotected area in front of the walls, also posed a serious threat. All of these challenges forced engineers to devise more advanced defense systems, leading to the bastioned fortification system. In contrast to the tall medieval walls, the bastioned fortification system is a relatively low structure, featuring pronounced angular bastions, a polygonal magistral line, a deep trench, and a whole range of outbuildings. The new walls also increased in dead mass, enabling them to more effectively absorb the impact of artillery projectiles.

The defense walls of bastioned fortress are lower, yet thicker, constructed out of locally available material, predominantly soil covered by stone or brick (which proved to be very effective in absorbing the impact of artillery bombardment). The reduced height of the defense wall meant digging deep and wide trenches to expose the invading army to interlocking defense fire, including the flanking fire from the bastions. The bank sloping from the fort (glacis) was mildly inclined so as to prevent the hostile artillery from hitting the main fortification wall directly. If the terrain allowed it, the trenches were filled with water, as illustrated by fortifications in (Fig. 1).

Fig. 1: Examples of bastioned fortifications with the the trenches filled with water.

a Bourtange, b Naarden, c Willemstad, Source: Goole Earth.

Exterior structures such as ravelins, redoubts, bonnettes, lunettes, tenailles, tenaillons, counterguards, crownworks, hornworks, curvettes, fausse brayes, scarps, cordons, banquettes, counterscarps¹⁰, provided additional protection against the enemy fire. The supplementary defense points comprised partially or fully detached citadelles, frequently found in fortification systems of large cities (Fig. 2).

Fig. 2: Examples of fortification systems of larger cities.

a The Citadel and Fortress City of Saint Martin de Ré⁴⁴ (b) The Citadel today, source Google Earth (c) The Citadel and Fortress City of Lille⁴⁵ (d) The Citadel today, source Google Earth.

The major characteristic of such fortification system is undoubtedly the bastion, the four-sided, spear-shaped addition to the defense wall. Bastions were thoughtfully designed not only to provide a full view of the battlefield, but also to enable defensive fire to cover all the fortification walls stretching to the next bastion, including the next bastion itself.

The first bastion is attributed to Francesco di Giorgio Martini (1439–1501). Martini is recognized as the “father of the bastion,” and his “Architettura civile e militare” in “Trattato di architettura civile e militare” (circa 1490, reissued 1841)¹¹ is frequently cited^12,13,14 as the source providing, if not the first, then the most comprehensive and detailed sketch of a corner bastion. Essentially, this sketch forms the core concept behind the design of the bastion’s contour. The angle of the bastion (i.e., salient angle) is always smaller than the interior angle of the basic polygon, allowing for a clear view from the face of one bastion onto the flank and face of the adjacent bastion. This provides the opportunity for additional defense of these sides of the fortress.

It should be noted that bastion with orillions¹² is associated with Giuliano and Antonio da Sangallo (c. 1445–1516).

The elements of a bastion are shown in (Fig. 3). Two adjacent bastions flank the curtain, whose outline follows the interior polygon, while gorges are positioned at its peaks. The distance between the neighboring bastions depends on the length of the defense line, correlating with the range of the defensive artillery¹⁵. The sides and fronts of all the bastions, together with their linking curtains, form the magistral line. The geometry of the magistral line served as the starting point in defining the shape and size of a fortress. It varied depending on the terrain upon which the fortress was built. Another property of bastion-type fortresses is that they were often erected on plains, given their geographical distribution, where the defense was the most difficult task. On the other hand, the flat relief offered more flexibility in the fortress’s form, no longer constrained by the morphology of the terrain. This led to the emergence of the star-fortress, both as a design and a philosophical concept, perfectly aligning with Renaissance ideals of the ideal city.

Fig. 3

Bastioned fortress elements: a bastion, b gorge, c curtain, d flank, e line of defense, f face, α central angle of a polygon, β interior angle of a polygon, γ angle of the bastion (salient angle), η exterior flanking angle, ι interior flanking angle, q magistral line, r external circle, s external polygon, t internal polygon.

Ideas about the ideal state (conceived by Plato and revived with Thomas More’s Utopia, 1516) produced consequently the idea of its physical embodiment through an ideal city, which must also have an ideal scheme of organization, diagrammatically resolved through circles, regular polygons and their concentricity¹⁶. Concrete considerations of ideal city plans, in their recognizable meaning, appear with the Italian Renaissance. The emblematic architects, the torchbearers of the ideal city concept are: Leon Battista Alberti (1404–1472)¹⁷ who in his treatise “De re aedificatoria” (On the Art of Building) advocated for harmonious proportions, symmetry, and the integration of urban and natural environments, Filarete (Antonio di Pietro Averlino) (c. 1400–1469)¹⁸ who in his “Trattato di Architettura” included designs for an Ideal City called Sforzinda and featured a star-shaped layout with geometrically planned streets and public spaces, or the aforementioned Francesco di Giorgio Martini (1439–1501) who in¹⁹ included plans for Ideal Cities emphasizing fortifications and orderly urban grids. So the star fortress is conceived.

The geometric design of star fortresses reflects its own purpose, since the shape of the fortress is a product of its own internal logic, where its functional geometry inherently serves military purposes. The efficiency and beauty of the structure stem solely from its design and functionality. Its symmetrical, precise design accompanied with external elements reflects the mindset of that era, where exuberance of form was axiomatic. Although originated from the concept of the Renaissance ideal city, the star-shaped fortress found its perfect place in the subsequent Baroque paradigm, which emphasized grandeur, complexity, and dynamic forms. Star forts, with their intricate, symmetrical, and radiating patterns, with heir geometric complexity and ornamental precision embodies aesthetics where functional and decorative elements seamlessly merge.

In star fortresses, the magistral line follows the geometrical pattern of regular polygons. The magistral line of a star fortresses based on the regular polygon must be inscribed in circle, which is then divided by the desired number of strongholds (bastions). The position of these bastions pinpoints the vertices of the exterior base polygon.

Understanding the geometry of the magistral line remains essential even in contemporary contexts, going beyond historical considerations and proving particularly important in matters related to the reconstruction and revitalization of fortresses.

The development of conceptual and geometrical postulates of bastion fortresses design

The first theoretical underpinnings of fortification reorganization are associated with urban space management in Renaissance Italy^20,21,22. In the second half of the 15th century fortification design equally attracted artists, architects, political philosophers and war strategists, not infrequently united in a single person—the Renaissance author. Regular polygons with bastions at their vertices were a geometrical solution that enabled the establishment of the active field of defense for the entire magistral line. The growing Neoplatonic influence on theoretical postulates (geometry), accompanied by a boost in iron production, spurred the development of offensive systems (artillery weapons) first and then those of defense (bastioned fortresses). The dissemination of these new ideas was facilitated by the development of printing technology, which provided theoretical writings primarily based on the practical experience of master builders. Learning was further refined with the establishment of a network of publishers, military academies, and a system of patronage of military architects. Military engineers introduced mathematics and geometry to fortification design and construction, which polished the theoretical postulates of the Renaissance architects. To put it in Derek Denman’s words: “The military engineer sought to emulate pure geometric form, equating mathematically defined order with effective defense”²³.

The development of the bastion fortification system, from a chronological perspective, closely followed the major military conflicts of the late 15th century and throughout the 16th and 17th centuries. It began in Italy and, through the Netherlands, France, Spain, and Austria, eventually spread beyond the borders of Europe into the colonial territories of the European powers under study. With this in mind, the evolution of the idea of a new fortification system in this paper is examined through three schools: the Italian, Dutch, and French, focusing on their influence on the development of the bastioned fortification system, while not diminishing the influence of other European powers (Spain, Portugal, Austria, Russia) on the development of the bastion fortification system. The reform of fortifications started in present-day Italy, the Netherlands is where the earthwork rampart became part of the fortification system, and France serves as an example of a state that, through the efforts of its military engineers, gathered and enhanced existing knowledge, and through the successful organization of military schools, applied this knowledge across its broad sphere of influence.

Italian school of fortress design: introducing the “Italian model”

The endeavors of first Italian Renaissance authors were summed up in Leonardo da Vinci’s and Giuliano da Sangallo’s discussions of architecture, where they explored how its geometrical shape might increase the fortress’ defense potential against enemy fire. Francesco di Giorgio Martini in ref. ¹⁹ offers guidance on how to reduce the total height of a fortress by building walls from a deep, wide trench, to make them less vulnerable to artillery²⁴. Instead of the medieval vertical defense method, he proposed horizontal or lateral wall defense. Just as importantly, he advocated the position that effective defense depends on the general plan of a fortress rather than on the thickness of its walls²⁵. The foregoing sets the scene for the development of bastioned fortress.

During the first half of the 16th century Italian authors still prefer fortifications made of stone, and consider earthen ramparts as a temporary solution, which may be due both to the long-standing tradition and the characteristics of a given location. Giovan Battista della Valle di Venafro (1470–1550) was one of the first to share the military experience from northern Europe and promote earthen ramparts strengthened by mutually defensible bastions²⁵. This evidently substantiates the claim that the concept of earthen ramparts originated in the north of Europe, which coincides with the advances in iron production (and artillery, in consequence) and corresponds to the characteristics of the terrain where stone is in short supply, yet water and earthen materials are abundant. However, it was Italian builders whose writings got printed in the second half of the 16th century, to be later reprinted in France, and it was Italian builders who got various military engagements across Europe, which explains why the phrase “Italian model” was coined to describe this novel fortification system.

The new system required exhaustive knowledge of geometry, engineering, artillery, defensive and offensive tactics, and was too complicated for an ordinary architect, which is why qualified military engineers took over the design and construction of fortresses. At the same time, they applied themselves to theoretical thought and methodological improvements. Torn between the roles of builders, theoreticians and soldiers, many left their work unpublished after a torturous life of a warrior²⁶.

Giacomo Lanteri (c. 1530–1601) in ref. ²⁷ was the first to treat fortification planning and modeling as a purely abstract geometrical problem, holding that military architecture was part of mathematical sciences, which is of particular importance for the purpose of this paper. His followers Galasso Alghisi (1523–1573) and Marcaurelio da Pasino (c. 1510–1559) transfered this fortification design concept to Flanders, setting up the stage for the development of the 17th century Dutch school of fortification²⁵. This eventually closes the circle and brings the theoretical development of fortifications exactly back in the region where the first earthen ramparts with bastions were built.

Meanwhile, there are developments of the experience-based theoretical concept that the fortification must be adjusted to the terrain. The first advocates of this idea were Antonio Lupicini (c. 1530–c. 1606) and Jacopo Fusto Castriotto (1510–c. 1563) who went to France to become the General Superintendant of Fortresses of the Kingdom.

To fully grasp the development of bastion fortification system and the underlying philosophy behind the geometric solutions during that period, one must also consider the Francesco de Marchi’s (1504–1576) treatise Della architettura militare written between 1542 and 1565. Horst de la Croix in ref. ²⁵ points out that this treatise establishes a concept where elements of bastion fortifications are developed outside the magistral line, dependent on its geometrical pattern, which actually introduces new strategies for both defense and attack: moving the defense away from the magistral line, and basing the attack on an elaborate network of access trenches and parallels. He then concludes regarding de Marchi’s attack concept: A century later, with slight modifications, it became known as the “Vauban system”²⁵.

In the aftermath of the seizing and despoiling of Rome in 1527 at the hands of Charles V’s forces, a multitude of Italian artists and architects found refuge within the realms of European courts²⁸. This indicates that Italian influence had a substantial impact on the theoretical concepts and practical techniques employed by authors in the northern regions of Europe, including Germany and the Netherlands.

The first treatises regarding fortifications in Germany emerged through the works of Albrecht Dürer, yet it was Daniel Speckle of Strasbourg who rose to prominence as the most esteemed engineer and influential author in the domain of military architecture. In the middle of the 16th century, Germany, Austria, England, and France witnessed a transition where domestic constructors yielded to the expertise of Italian engineers, either through replacement or concurrent employment.

Dutch school of fortress design: edifying geometry

From 1540 onwards, as a result of Charles V’s campaign, there was a continuous work on fortifications in the Netherlands. In addition to the already active Dutch engineers (Keldermans family of Malines, Jean de Terremonde, Willem van Noort Sebastian, Jacques van Noyen), Italian engineers were hired, including notable figures such as Donato de Buoni Pellezuoli, Alessandro Pasqualini, Marco di Verona, Battista Grimaldi, and Aurelio de Pasino. In the second half of the 16th century, turbulent political-historical events in northern Europe and the constant need for improving existing fortifications and constructing new ones gave rise to a series of local military engineer-builders of fortifications (Germain Le Febvre, Pieter Timmermans, Abraham Andriessen, Hans Duyck, Leon de Futere, Joost Janszoon, Cornelis Bloemaert). The most prominent engineer was Adriaan Anthoniszoon, who, from 1580 onwards, received support from three distinguished engineers in different parts of the Netherlands: Jacob Kemp, Johan van Rijswijck, and Joost Mattheus²⁹.

In 1594, Simon Stevin published De Sterctenbouwing³⁰, the first book on fortifications in the Dutch language^29,31. The need for educating engineers led to the establishment of the School of Engineers as a department of Leiden University in the year 1600. Simultaneously, the study of mathematics at Franeker University expanded to include lessons in fortification, taught by Adriaan Metius, a son of Adriaan Anhoniszoon.

Between 1612 and 1619, the military advisor to the Dutch Republic was Samuel Marolois, the first to present the geometric principles for constructing polygonal plans in the Dutch bastion fortification system. His treatise on fortifications, printed between 1614 and 1616 as part of Opera Mathematica, laid the foundation for the architectural work of Menno van Coehoorn in the second half of the 17th century. Through translations into French, German, Latin, and English, it influenced the development of fortifications across wider European territories.

French School and Vauban style: striving for a comprehensive geometric solution

Following the prevalence of Italian authors in fortification construction within France, towards the end of the 16th century, an initiative led by King Henry IV resulted in the formulation of the first French treatise on fortifications, La Fortification reduicte en art et demonstrée, written by Jean Errard de Bar-le-Duc. This work aspired to ground fortification practices in scientific principles³². Jean Errard, well-versed in mathematics and geometry, laid the theoretical foundations that inspired the work of many later military engineers, including the highly gifted theorist Blaise François Pagan. Pagan, a prominent figure in the field of military engineering in France during his time, was the founder of the first school on fortification in the country and also a mentor to Vauban. Another notable figure is Antoine de Ville, who played a significant role as one of the main military engineers of that era. Serving the Prince of Savoy, the Republic of Venice, and Cardinal Richelieu, he not only designed numerous fortresses (see Table 1) but also authored a theoretical treatise “Les fortifications” (1629), focusing on fortress construction.

Table 1 The chronology of bastioned fortress: the pivotal builders and their most important work

Lastly, we cannot overlook the contributions of Vauban, who not only inherited previous doctrines but, according to various sources^8,9,25,33, made the greatest contribution to the development of theoretical principles and practical knowledge regarding the defense and offense of bastioned fortresses within the national framework. His impact is comparable to that of his contemporary, Menno van Coehoorn (1641–1704), who made similar strides in the Netherlands.

Primarily, he built or adapted a large number of fortresses, as he had the backing of a strong centralized state, Louis XIV’s absolute trust and a network of schools belonging to the Direction of Royal Buildings within the Royal Academy of Architecture which trained military engineers and printed and distributed writings promoting what certain authors term the “Vauban style” of fortification construction. France trained generations of engineers which Vauban streamlined to apply and develop new defensive and offensive techniques, which in consequence make him famous. On the other hand, the other European monarchies, grouped in more or less stable alliances had to rely on the receding group of overworked and demotivated cosmopolitan military experts³⁴.

During his long reign, Louis XIV institutionalized the glorification of French symbols and triumphs, be they civil or military. By supporting the institutions such as the Petite Académie and, from 1691, the Académie des Inscriptions he promoted the academic models of his time, indirectly aggrandizing himself. This is how Vauban, the man behind the greatest engineering enterprises of the era, gained recognition outside France.

Authors who engaged in the theoretical definition of new fortress construction principles, often did not have the opportunity to practically implement their theoretical ideas through fortress construction. Therefore, there was an imbalance in practice between theorists and practitioners.

Through the chronological overview presented, we observe that the construction of bastion fortresses began in Italy, aligning with the initial theoretical writings. Simultaneously, in northern Europe (Netherlands), engineers also recognized the need for a change in existing construction procedures, limited by the building material in their surroundings. It is challenging to precisely define whose influence was stronger, but due to the nature of the military careers of these engineers, who often changed their places of engagement and tasks, they spread their knowledge and absorbed others’, resulting in a frequent intertwining of these influences. The import of technology occurred through conquests and the adoption of new positive achievements, as the same fortress, due to turbulent historical events in Europe at that time, passed from one ruler to another.

In (Table 1) we give a chronological review—the evolution of the fortresses construction through the display of the milestone fortresses and builders in whose realization they participated as military engineers, from the mid-15th century to Vauban.

The development of geometrical principles for constructing the magistral line

The first systematically constructed magistral line of the bastion fortification was the work of Antonio da Sangallo the Younger (1484–1546), on Fortezza da Basso from 1533³⁵. The specialist writings on fortifications, military tactics and ballistics appear at the end of the 16th century, and they primarily rely on the principles of Euclidean geometry¹². In his work Nova Scientia Inventa: Disciplinae Mathematicae loquuntur, Niccolò Tartaglia (1499–1557) begins his discussion of projectile motion with a chapter on geometry. Giacomo Lanteri in ref. ²⁷ examines fortification planning based on the principles of Eucledean geometry. He stated that an architect, an expert in geometry and mathematics, could easily surpass a soldier gifted with practical abilities, because only by combining theory and practice can he deal with the complex issue of establishing effective defense³⁶.

At the turn of the century, Galileo Galilei, at the time being a professor of mathematics at the Universty in Padova, starts his treatise on military architecture Trattato di fortificazione by explaining the geometrical methods for solving the problem of defense wall planning. In the beginning of the 17th century Jean Errard de Bar-le-Duc (1554–1610), a military engineer for Henri IV, in La fortification redvicte en art et demonstree emphasizes that soldiers should be taught geometry, as its formal methods are the most efficient means of presenting plans and strategies.

In the 17th century, books (treatises) on fortifications were primarily crafted for military engineers tasked with devising strategies for both defense and attack. Providing the basic concept and system of fortification design and construction, these had both theoretical and practical significance. The theoretical part involves the knowledge of mathematics, particularly geometry. Geometry often extended over several introductory chapters and was presented as a means of improved architectural space management. The practical aspects of such writings dealt with how to solve problems when designing various fortification lines, as part of the general fortification design, to make allowance for the new military technology. These works evidently strive to present universally applicable geometrical principles in fortification design and organization. Their confidence in geometry was grounded in the practically proven effectiveness of precise design, representing an advancement over previous artisan approaches.

Analysis of geometric principles on the example of four builders

To show the development of geometrical principles of shaping the magistral line of bastioned 17th century fortifications we will present the works of Samuel Marolois, Antoine de Ville, Blaise François Pagan, and Sébastien Le Prestre—Marquis de Vauban. Marolois was a military engineer significant for the development of the Dutch fortification system, much like Pagan in the French tradition. As contemporaries, their contributions provide a compelling basis for comparison. De Ville and Pagan, as predecessors to Vauban, allow for tracing the evolution of geometric principles in the construction of the magistral line, highlighting the progression that impacted Vauban’s own innovations. Therefore, we have focused on them. All of them had rich military experience which they applied when formulating their principles of fortification design and construction.

We focus on five key elements of the magistral line: curtain (c), flank (d), bastion face (f), side of the base polygon: exterior (s) or interior (t), and bastion angle (γ), which we analyze on the example of star fortress.

Samuel Marolois

Samuel Marolois (1572–1627), the Dutch mathematician and military engineer is a prominent figure of the Old Dutch school of military architecture design and construction (together with aforementioned Menno van Coehoorn). His book Fortification ou Architecture militaire tant offensive que defensive³⁷, contained the comments by mathematician Albert Girard (1595–1632). In his theoretical works, he also explored the problems of perspective, a subject he detailed in Perspective contenant la théorie, practique et instruction fondamentale d’icelle, coauthored by Hans Vredeman de Vries (1527–c. 1607).

Marolois’ geometric design principle for the magistral line shaping³⁷ relies on predefined values for three of its specific elements: the curtain length (c) remains constant for any polygonal base spanning from a square to a dodecagon, also the length (f) of the bastion face, and the flank forming angle CAB (δ). The flank (d) and curtain create a 90° angle. The bastion angle (γ) is 15° larger than half of the interior angle (β) of the polygonal base. Thus, if the number of sides of the base polygon is n, and the central angle of the polygon α = 2π/n, then the angle γ can be expressed as:

$${\rm{\gamma }}={\rm{\pi }}\left(\frac{7}{12}-\frac{1}{n}\right)$$

(1)

Thus, based on the fixed dimensions c, f, and δ, and the variable number n, all other dimensions of the magistral line can be constructed. However, Marolois did not provide the general formula; instead, he described the procedure.

The formerly defined conditions which determine linear and angular measures of the magistral line, while the calculation of remaining elements’ lengths stems from the values previously established. To facilitate the selection of specific values, Marolois presented a table with recommended dimensions of the magistral line, covering polygonal bases from square to dodecagon (Table 2). This table reveals two systems. In the second system, in contrast to the first, the bastion angle (γ) is equal to two-thirds of the interior angle (β) of the polygon. Thus, the formula for the second system is:

$${{\rm{\gamma }}}_{b}=\frac{2}{3}{\rm{\pi }}\left(1-\frac{2}{n}\right)$$

(2)

Table 2 Dimensions of fortress magistral line according to Samuel MaroloisFootnote

. Remake of the Table des Dimensions de fortification³⁷.

Marolois actually does not offer an operable geometric construction. Instead, he presents a scheme detailing the composition of the magistral line, where the dimensions are predefined and tabulated. We present the Marolois construction according to the first system, for bases n = 5, n = 6 and n = 8 in (Fig. 4).

Fig. 4

Construction of magistral line fragments according to Samuel Marolois’ method.

Antoine de Ville

Antoine de Ville (1596–1658), a Toulouse-born military engineer, started his military career in France rather early, and played an important role in the La Rochelle siege in 1624. Since 1626 he was engaged in the Netherlands in the battalions of Charles Emmanuel I, the Duke of Savoy. Between 1630 and 1635, Antoine de Ville serves the Republic of Venice in Italy, and also works on the construction of the fortresses in Istria, Dalmatia and Slavonia (regions in today’s Croatia). He returns to France in 1635, where he continues his work. He shared his theoretical and practical knowledge of fortification design and construction in several books, most famous of which³⁸ became the focus of our research. In the initial sections, he explains the geometrical principles behind the design of the magistral line of regular-based fortifications. His system encompasses polygonal bases ranging from hexagon to dodecagon, excluding fortifications with square and pentagonal bases.²

Antoine de Ville’s geometrical construction of the magistral line elements starts from the side of the interior basic polygon (marked as HR in Fig. 5), whose length is fixed at 180 pas³. The polygon side (t = HR) is divided into six equal parts, where the first and the last sixth determine the starting point of the bastions’ flanks. The bastion’s flanks (d = EF) is perpendicular to the curtain line (c = FC) with the fixed length of 30 pas (MQ = HQ = HF = EF = 30 pas). The extension of the radius (OH) includes the vertex point of the bastion (A) and the intersection point (I) of the diagonals within the bastion’s deltoid. Hence, the line segments HA and EM form the right angle, as deltoid’s diagonals. The angle of the bastion (γ = EAM) is also the right angle. The intersection of the extension line of the bastion’s face (f = AE) and the curtain FC is point B. The line of defense is the line segment AC, while the angle CDB⁴ is the right angle, again. In this system, the dimensions of the internal polygon side (t), curtain (c), the flank (d), and the bastion angle (γ) remain constant and independent of the number (n) of the basic polygon sides. The length of the bastion face (f) is the only variable element. Along with this construction, de Ville, alike Marolois, provides a detailed table, practically, a ready-made template for practical application with carefully calculated measurements for individual elements of the magistral line, including the line of fire.

Fig. 5

Construction of magistral line fragment according to Antoine de Ville.

In (Table 3) we provide measurements of magistral line elements according to Antoine de Ville. These measurements are derived through trigonometric calculations, which are founded on geometric construction principles. The considered polygonal bases range from a hexagon to a dodecagon.⁵

Table 3 Values of magistral line elements according to A. de Ville’s system

Blaise François Pagan

Blaise François Pagan (1603–1665) displayed exceptional skills in both defense and attack capturing the attention of Louis XIII. His military career ended in 1642 due to his illness. Between 1642 and 1665 he wrote treatises on geometry, astronomy and bastioned fortifications. In Paris he published³⁹ which greatly influenced generations of military engineers even years after his retirement. He left his mark in Portuguese colonies in South America, and on Malta, to which he was invited in 1645.

Blaise Francois Pagan, similar the two engineers considered earlier, proposes predefined dimensions of elements and angles of the magistral line, now determined by the length of the side (s) of the fortification’s external polygon³⁹. He defines the length of magistral line segments for three selected fortification groups, where groups are formed according to the size of the initial external polygon (AB = 200 toises, AB = 180 toises and AB = 160 toises).

Having the assigned length of the external polygon’s side AB = s (e.g., of 200 toises), he constructs the perpendicular line CD, also of a predefined length (30 toises), from the midpoint of the side AB. Using predefined values again, of both the bastion face (f) and the radius CM = CN, all other elements, such as positions of the points E and F, the length of the curtain (c = MN), and the angles of the bastion (γ), are determined by following the simple geometric construction illustrated in (Fig. 6). In (Fig. 6) we depict the geometric determination of segments of the magistral line according to Pagan for fortresses with polygonal bases of n = 5, n = 6, and n = 8.

Fig. 6

Construction of magistral line fragment according to Blaise Francois Pagan, applying the values from (Table 4).

It should be emphasized that the measurements Pagan proposed refer to fortifications whose basis range from pentagonal to dodecagonal, while those with square bases were treated separately. Consequently, fortifications with bases from n = 5 to n = 12 of the same side length of the polygonal fortification base have identical lengths of magistral line elements. In other words, the length of magistral line elements depends on the side length (s) of the external polygon of the fortification rather than the polygonal base itself. The only element that is variable and depends on (n) is the bastion angle (γ) which changes significantly with the increase of (n), as the other elements of the magistral line are arranged in a rigid disposition. Pagan did not tabulate the measurements of the magistral line, but gave them descriptively. We provide a simplified table based on his descriptions in (Table 4).

Table 4 Dimensions of magistral line elements according to B. F. Pagan’s system

Sébastien Le Prestre—Marquis de Vauban

Sébastien Le Prestre—Marquis de Vauban (1633–1707), a marshal of France and military engineer serving Louis XIV, was renowned for his skilled fortification design and clever offense and defense tactics. Vauban undoubtedly stands out as the preeminent figure among them, to the extent that the entire architectural approach to fortress construction characterized by the meticulous adherence to geometric principles in delineating the magistral line has been named after him, as previously noted. He defines the rules to determine linear and angular parameters for the external polygon circumscribed about the magistral line, ranging from a square to a dodecagon but does not completely exclude the fortifications with other bases⁴⁰. The Vauban’s magistral line shaping system is based on a carefully calculated geometrical construction, whose procedure is illustrated in (Fig. 7).

Fig. 7

Construction of the fragment of the magistral line of a regular fortification with the hexagonal basis according to Vauban⁴⁰.

According to his method, the side AB (s) of the external polygon is divided into eight segments for a quadrangle, seven segments for a pentagon and six segments for a hexagon and larger polygons. After the division, one segment of the thus divided polygon’s edge (e.g., s/6) is transferred to the perpendicular line from the midpoint D of the length AB, giving the point C. Thus, the line segments AC and BC are obtained, which determine the directions of the lines of defense. The length of the bastion face (f = AE = BF) is always two sevenths of AB.

$$f=\frac{2}{7}s$$

(3)

The intersection of the arches (r = AF = BE) and the defense lines provides the endpoints N and M of bastion flanks (EM = FN) and the length of the curtain MN.

Thus, similar to Pagan, Vauban establishes a geometric dependence among the elements (s, f, c, and d) of the magistral line, forming a rigid and universal assembly, unaffected by the number of the polygon sides (n). We find his construction more complex than Pagan’s, because it requires different divisions for different bases and, in addition, it even requires dividing the base twice (e.g., into 6 and then into 7 parts). This, in turn, makes the magistral line itself constructible solely on the basis of one assigned measure (s), aligning the methodology closely with synthetic geometry principles. Moreover, this is carried out with special care to achieve optimal results, and this is exactly what his attention to detail reflects.

However, in his proposal for measurements, he presented a concise table for polygons from n = 4 to n = 12, specifying the lengths of the bastion flank (f), the radius CD, and the radius (R) of the circle circumscribed around the base polygon (Table 5). He noted that the results in the table were obtained for a polygon side of 180 toises.

Table 5 Dimensions of magistral line elements according to Vauban’s system⁴⁰

The magistral line shaping thus evolves from a simplistic geometry in Samuel Marolois’ system which applies predefined values, to Vauban’s meticulously calculated geometrical construction, which fully defines the magistral line elements. Marquis Vauban was the last in the line of great master builders, and he served the powerful King Louis XIV in times of wars, so it is little wonder that his name became a trademark for the bastioned fortification system.

The concept of bastioned fortification evolved independently of administrative and national boundaries. The geometrical principles for magistral line are so precise that they are still used today as a method of choice in bastioned fortification reconstruction. By analyzing the geometric proportions and angular values of the existing preserved elements of the fortifications, we can determine the type of geometric principles used in the construction of the magistral line, and indirectly the author by whose method the fortification was designed^41,42,43.

Discussion

All constructions presented in the paper pertain to the design of ideal fortresses, specifically star fortresses over a regular polygonal base, because these examples provide most illustrative insights into the fundamental principles, similarities, and differences in geometric procedures. Through a comparative analysis of four key builders of the observed period when the construction of bastion fortresses flourished, we note:

In the system developed by Marolois, the shape of the bastion changes with the number of sides (n) of the internal polygonal base, as the bastion angle (γ) and the flank length (d) vary in dependence on it. Despite being a proficient connoisseur of geometry, as evidenced by his elaborate publications covering remarkably broad areas of what will be known as “descriptive geometry” in the centuries to follow, he does not provide a comprehensive formulation of the proportions and dimensions of the bastion in relation to the curtain (c) or a side length of the polygon (s or t). Instead, he offers a simplified guide through pre-defined measures.

He bases his calculations on the lengths of the bastion face (f) and the curtain (c), including also the predefined value of the flank-forming angle δ, not on the side length of the base polygon inscribed or circumscribed around the magistral line. As the number of polygon sides (n) increases, the length of the polygon’s side (s), along with the flank length (d), bastion angle (γ) and gorge width also increase, obeying the trigonometric laws. In this manner, the proportions of the elements of the magistral line align with the size of the fortress base, in coordination with the expansion of the area it defends. However, the comprehensive geometric construction is actually absent; it is more about a scheme that provides instructions for the disposition of elements, or a sketch for trigonometric calculation. Recognizing that builders may lack a proficiency in geometry, he does not provide intricate geometric procedures; instead, he supplemented geometric calculus by providing tabular measurements for each polygon ranging from n = 4 to n = 12. Thus, even the construction according to Marolois system provided in (Fig. 4) relies primarily on the measurements from the table rather than on an intricate constructive procedure.

As for De Ville, bastions exhibit a shape that is minimally dependent on the form of the fortress polygon itself, given that the bastion angle (γ) is always right and the angles at the other two protruding vertices of the bastion change imperceptibly in dependence on the number (n) of the base polygon sides. Out of the five key elements of the magistral line that we highlighted, he proposes as many as four that have predefined measurements: the side length (t) of the n-sided internal polygon, the width of the curtain (c), flank (d), and the value of the bastion angle (γ). However, they arise from the construction itself, which can be performed synthetically, starting solely from the length of the side (t). Only the length of the bastion face (f) undergoes slight variations depending on (n), making its model the most rigid among the observed ones, concerning the shape of the bastion itself.

We note that he did not provide a comprehensive construction itself, a derivation of which can be inferred from his method. We present it in (Fig. 8), with a note that the rest of the magistral line is easily obtained by applying n-fold symmetry. Instead, he gives tabulated measures as a substitute for laborious calculations or geometric constructions, resulting in a more schematic representation of geometry.

Fig. 8

Geometric construction derived according to de Ville’s method.

No matter how elegant and implicitly synthetic this system was, it had numerous shortcomings in the result: weak variability of the surface of the bastion in relation to the surface of the base of the fortress itself, as well as impractical solutions for the bases n = 4 and n = 5. Hence, the pursuit of an optimal construction will persist.

Pagan provides a geometric construction of the magistral line where he also employs predefined measures rather than a procedure covering the general case. In fact, he furnishes more predefined measures than immediately apparent, as only two of them pertain to the magistral line. According to his model, four lengths from which we can derive the construction are constant: the lengths of the bastion face (f), the side of the external polygon (s), the segment CD, and the radius CM = CN. The only element that is variable and affected by changes in the number (n) of polygon sides is the bastion angle (γ). With an increase in (n), the bastion angle (γ) also increases, significantly influencing the bastion’s shape, having that the other elements of the magistral line are rigidly assembled and do not adapt to n. Similar to Marolois and de Ville, despite the provided construction details, Pagan offers builders a practical relief by providing simplified, and rounded measurements in description, sparing them from intricate trigonometric calculations. We can characterize his model as notably rigid, comprising a fully predefined segment of the magistral line that iterates n times in a radial arrangement.

As for the Vauban’s model, it can be considered an advanced version of Pagan’s model, starting also from the assigned side of the external polygon (s). Now, instead of a predefined value for the length of the bastion face (f), he defines it through the ratio f:s, as given in Eq. 3. Additionally, instead of a specified value of the radius CD, he again links this length to a ratio with respect to the polygon side (s), (Fig. 7). The step involving radius CM = CN is replaced by introducing another radius, r = AF = BE, which can be derived from the construction, knowing only a value of s. Practically, out of several pre-defined measures in Pagan’s model, he reduces them to derivations from one: the length of the polygon side s. Therefore, we can characterize Vauban’s approach as a step towards symbolic representation, unlike his predecessors who relied more on numerical representations. This signifies a higher level of development in scientific thought and theoretical foundations during that era, reflecting an expected evolutionary progression.

However, by comparing the values displayed in Tables (2–5) with the precisely calculated trigonometric ones, we concluded that the engineers we observed made some rounding adjustments to facilitate easier use in the field. As an illustration of this, Table 6 provides an example of the trigonometric precision of the magistral line elements according to Vauban’s system.

Table 6 The Trigometrically precise dimensions of the magistral line elements according to Voban’s system

Overall, this does not alter the fact that in Vauban’s case, as well as in Pagan’s, the focus is on defining a rigid segment of the magistral line, universally applicable to multiple bases and unaffected by changes in n, where the magistral line is assembled by a polar arrangement of such immutable segments. In (Table 7), we provide an overview of the input dimensions and conditionality of the constructions, based on the initially given parameters, in order to facilitate the observation of the evolution of geometric reasoning in the design of bastion fortresses. Figure 9 shows the disposition of the magistral line elements to which these dimensions apply, for easier reference to (Table 7).

Table 7 Comparative analysis of the geometric dependencies of the magistral line elements in the systems of four selected engineers

Fig. 9

The elements of the magistral line used in the geometric constructions of the four observed engineers.

For easier review and comparison of the values obtained by applying the construction process shown in Figs. 4–7, Table 8 presents the lengths of the magistral line elements in relation to the side length of the inner polygon (t) for fortresses bases from n = 5 to n = 12, along with the corresponding values of the bastion angles.

Table 8 The ratio of the lengths of the magistral line elements (f, d, c, s) to the length of the inner polygon side (t) and the values of the bastion (salient) angles (γ)

By observing and comparing the values of the measures of individual elements from Table 8, we conclude that they vary relatively little from one author to another. Therefore, it is the geometric construction that primarily defines the design and provides far more information about the author and the layout of the bastioned fortress than the numerical values alone.

Conclusions

After conducting the research, we conclude that the quest for the optimal configuration of the magistral line evolved from instructions for assembling its predefined components (Marlois, de Ville), through attempts to solve the general case with a single construction (de Ville), or efforts to formulate a general model for shaping any base with a universal segment of the magistral line (Pagan), to a model that transcends previous aspirations through a synthetic procedure—a construction achievable solely with a compass and a ruler (Vauban). All of them have in common a strict foundation on geometrical principles.

Finally, can we provide an answer to the question: do these principles differ across Europe, or do they converge to a universal style?

The evolution of the magistral line was intricately linked to the imperative of fortification survival. The knowledge that contributed to that objective accumulated spontaneously through the exploration of novel techniques and procedures, as well as the reciprocal exchange of insights encompassing both defensive and offensive tactics, weaponry, and theoretical acumen.

A shared attribute among the presented fortress designers is the endeavor to formulate most efficacious defense mechanisms, targeted at countering weaponry innovations of the epoch, as well as offensive ones, by means of geometric deliberations.

Geometry (more precisely, trigonometry grounded in geometric constructions) served as the scientific arsenal of that era, much akin to contemporary practices such as programming or utilization of computer software. Proficiency in theoretical knowledge gave individuals the ability to skilfully implement these principles in practical applications, thereby yielding tangible outcomes in both the battlefield and, consequentially, the geopolitical domain.

Concerning the realm of geometry, based on a comparative analysis of the provided examples, we infer that these examples essentially represent variations of a single model, initially formulated during the era of Italian authors. Consequently, in scholarly discourse, this design methodology is frequently and justifiably referred to as the “Italian model”.

Although we can identify distinctive characteristics of some key builders, they all operated within a common framework which incorporates minor or major innovations influenced by a multitude of factors: the time in which they were constructed, availability of data on contemporary war techniques, experiences in previous battles, available building materials and their characteristics, and finally—the skill of the author. In this regard, Vauban is certainly the most influential figure due to the strongest institutional support and skillful application and improvement of previously accumulated knowledge and experience. The fact remains that, geometrically speaking, he made a breakthrough that resolved the proportions within the magistral line of the fortress for any given length of the related polygon. This is a characteristic of an enhanced, synthetic approach, in contrast to predecessors who adhered to a more partial, numerical one. What was implicitly discerned in de Ville’s method and descriptively indicated in Pagan’s, Vauban synthesized into an almost general formulation of the magistral line construction.

The contributions of the described geometric procedures are reflected not only in the effectiveness of military strategies, but also in the scientific development of geometric procedures.

Why is the geometric precision of the magistral line significant, not only in the era of bastion fort construction but also in the present day?

In that era: Geometric deliberation served military purposes, involving the trajectory of projectiles, coverage of dead angles, and providing a better (if not the best) overview of the attack field. The accuracy of these calculations and precision in execution had a direct impact on the efficiency of both defense and offense. Subsequently, this efficiency influenced not only the preservation and survival of the fortress as an architectural entity but also the persistence of the population and even the authority itself. This change was driven by the demands of artillery warfare and the geopolitical pressures of the turbulent period during the 16th and 17th centuries, marked by the antagonism between the leading European powers of the time – the Holy Roman Empire, France, Spain, and the Netherlands. Thus, indirectly, geometry served the geopolitical relations of that time, which has had a long-term impact on the historical map of Europe up to the present day. Notably, during this period, the emergence of the Baroque style aligned with the intricate aesthetics of star fortresses, bringing together military function and artistic expression.

In modern times: The magistral line is the unassailable physical outline of a structure that exists undisturbed by time, reflecting its enduring strength and position as an inherited part of the landscape’s history and dictating the organization of space within and outside it. As an immobile inherited setting, it defines the directions of urban development through its morphology and influences the design of infrastructure, urban planning, and zoning, which is evident on the examples of preserved fortresses throughout Europe. Accordingly, the knowledge of its geometric principles, calculations, and dimensions, originally used for defense, can facilitate the entire process of reconstruction, conservation, restoration, and revitalization of fortresses. The use of these principles, in combination with modern analytical and imaging technologies constitutes a compatible approach to the preservation of culturally significant objects without damaging the original structures.

Therefore, this research, in addition to clarifying the geometric framework and underlying logic in the conceptualization of bastion fortresses from the observed period, facilitates identifying architectural influences in territories where these influences intertwined due to a turbulent history.