The Geometry of War

The mathematicians of the Renaissance applied their geometry to all manner of practical disciplines - from navigation and surveying to cartography, perspective and dialling. They aimed to demonstrate the usefulness of geometry as well as its ingenuity and certainty, and to associate it with action, achievement and progress. Many new instruments were designed in this context, as the collections of the Museum of the History of Science amply demonstrate.

Developments in the art of warfare in the late fifteenth and sixteenth centuries provided mathematicians with one such outlet for geometry. The ability to make a heavy gun in a single metal casting led to ordnance that was longer and capable of more accurate fire. Further, the addition of trunnions on which the barrel could turn in a vertical plane and the development of moveable carriages improved the setting of direction and elevation, and so encouraged the idea of predicting and calculating ranges. Practical mathematicians responded to these changes by devising techniques, designing instruments and writing books.

The speed with which the mathematicians responded to the novel set of problems thrown up by changes in the conduct of war is striking. It was in the sixteenth century that cannon came to be used in large numbers and their effect became critical to the outcome of military engagements; in the same period the geometry of war became a major branch of practical mathematics. To harness the capabilities of the new weaponry, gunners needed instruments to measure both the inclination of the barrel and the distance to the target, together with a means of relating these two measurements; the geometers offered a variety of solutions, as well as designs for fortifications to withstand attack from the new artillery. The frequent wars in sixteenth-century Europe added urgency to their work and helped justify claims for the importance of their discipline. As in other areas of practice, such as cartography or surveying, the mathematicians were quick to recognize an opportunity and they responded with enthusiasm - perhaps with over-enthusiasm, in view of the instruments and images on view in this exhibition.

The ingenuity and precision of the instruments on display, and the elegance, poise and delicacy of many of them, contrast with the harsh and uncompromising conditions of the battlefield. A review of such instruments is bound to raise the question of how useful they could really have been in practice. They were supposed to be employed in battle, but it is clear that their purported military value also had other functions - in justifying textbook geometrical problems, for example, and in attracting patronage.

The exhibition at the Museum of the History of Science seeks to draw attention to the fact that the advent, not of gunpowder directly, but of its more effective use in improved ordnance, precipitated a new variety of practical mathematics and a new disciplinary discourse that flourished through the sixteenth and seventeenth centuries. Though few historians of science or of art have given it much attention, this geometrical discipline engaged a great many practitioners in the period and was associated with a range of technological developments applied throughout Europe, both in military campaigns and in large works of military engineering. The exhibition itself has three main sections, dealing with gunnery, range-finding and surveying, and fortification. The mathematics of troop formations is the first of two additional topics, while the other is a reminder that the telescope - one of the most emblematic instruments of science - was originally introduced as an instrument of war.


The art of gunnery was complex and dangerous and the gunner's ability to fire reliably and accurately was frequently criticized. Although mathematicians could not remedy variations in powder or in the form of individual guns, they did seek to improve gunnery by devising instruments for the measurement of shot, the elevation of guns and mortars, and the calculation of the range of fire. Instruments for these operations include calipers, gauges, quadrants, sights, levels and specialized rules. Two or more of these elements were often combined in a single design, in a marketing initiative typical of mathematical instrument makers. Other mathematical instruments such as sundials were also incorporated in the more exotic hybrids, which typically date from the sixteenth or earlier seventeenth centuries. By the eighteenth century, as mathematical instrument makers became regular suppliers to professionalized ordnance departments, more standard patterns were emerging.

Early modern gunners had to cope with a bewildering variety of weaponry. Guns were named according to their size, weight and length, or by the type and weight of their shot. Each class of gun was different, and was primed with specific amounts of powder (often inserted with individualized ladles) before being loaded with its own shot. Printed sources frequently tabulated the appropriate information for each type of gun and some instruments likewise carry tables to serve as aides-mémoire. Instruments were also enlisted to match the correct size of shot to different guns. Calipers provided measurements of the diameters of gun bores and of spherical shot, and often directly indicated the weight of the shot. Gauging rods also accomplished the same linking of diameter and weight, and scales adapted for iron, lead and stone shot frequently appear on German instruments.

Although heavy guns were often fired for maximum destructive effect at point blank range (with the barrel horizontal), greater range could be achieved by elevating the gun. Quadrants, sights and levels enabled the gunner to set the barrel at specified elevations. These instruments were constructed in an enormous variety of forms and styles, with their scales diversely graduated in degrees, inches or 'gunner's points'. In his text of 1537 inaugurating the 'new science' of artillery, Niccolò Tartaglia described a quadrant which was inserted into the muzzle of the gun. Subsequent authors noted that this exposed the gunner to enemy fire and they offered alternative instruments which could be set up at the gun's breech. The many surviving styles of levels and sights fall into this safer category of instrument. Levels measure the inclination of the barrel using a plumb line or a rigid plummet and a graduated arc. Sights have a moveable pinhole or a series of fixed holes through which the gunner observed, making an alignment with the top of the muzzle.

To determine appropriate elevations for his cannons and mortars, a gunner had to know the distance of the target and also be able to relate this range to the elevation of the piece. The question of determining the correct elevation necessary to fire a shot a given distance (and its inverse, the prediction of range at a given elevation) was the most taxing problem of gunnery as a mathematical art. Gunners had their own rules of thumb, and instruments were used to embody rules relating range and elevation. But questions of ballistics also engaged the most prominent mathematicians. Tartaglia set the terms of the debate by seeking to portray the geometry of a projectile's trajectory based on the opposed natural and violent motions of Aristotelian physics. His work provided the basis for many subsequent accounts in textbooks and manuals. Galileo offered a new foundation in his Discorsi of 1638, demonstrating the parabolic path of projectiles and reinforcing the military relevance of his work with a complete table of ranges. Through the seventeenth and eighteenth centuries mathematicians of the highest stature, such as Newton, sought more accurate depictions of projectile trajectories by attempting to take into account disturbing factors such as air resistance. Their sophisticated efforts may have eluded the grasp of the average gunner, but do demonstrate that the study of projectile motion was a point of intersection for the art of war, the mathematical sciences and contemporary natural philosophy.

Rangefinding and Surveying

How was the gunner to determine the distance of his target? The traditional method of linear measurement in land surveying was simply to lay ropes or poles between the two stations concerned - hardly an option when the distant station was a hostile position. However, sixteenth-century geometers were seeking to introduce the technique of triangulation, and range-finding was part of their case for a new geometry of surveying. Distant stations could be located by sighting from either end of a measured baseline; their distances were found by measuring the angles formed with the baseline and by subsequent calculation, or by a more straightforward graphical method. In ordinary surveying, the possibility of triangulating as many features as were required from a single linear measurement was presented as an innovation that would greatly improve efficiency and convenience. Range-finding offered a particularly appropriate application, since access to the target feature was not simply inconvenient, but impossible. Further it gave the designers of new surveying instruments an immediate and telling example of the value of the novel method, so that relevant instruments - particularly the universal instruments, with all manner of claimed uses - were frequently illustrated in dramatic and urgent action.

Two types of surveying instrument were applied to triangulation and illustrated in use in warfare. In the first group, comprising what are called 'triangulation instruments', parts of the instrument are arranged to form a triangle similar to that on the ground, for example by aligning rules with the lines of sight from the baseline. Scaled measurements can then be taken directly from graduations on these arms. The other category of instrument employs scales typically found on the backs of astrolabes - the circular degree scale and the shadow square or geometrical quadrant. This class includes, of course, the astrolabe itself, but extends to the theodolite, circumferentor and graphometer.

Such instruments were made relevant to military life not only through the range-finding of the gunners. It was claimed that they were generally helpful to the military surveyor, either when laying out a new fortification, or when measuring and representing an existing one. Accounts of the instruments show them being used in such contexts and point to the particular advantage of being able to survey from a safe distance.


A further relevant aspect of military surveying is the design of fortifications. If guns precipitated new branches of the mathematical sciences dealing with projectile motion and with range-finding, they also created the conditions for a new military architecture. Just as contemporary civil architecture was founded on geometry, expressed through the classical style, so too the new genre of fortification rested on a geometrical formalism.

The high walls of the medieval fortress were good for repelling attack from beneath, but were vulnerable to heavy guns: they presented large targets without providing suitable platforms for defensive fire. Yet if walls were to be low and stout, so as to withstand artillery bombardment, how were they to be defended against direct infantry assault? The new style of fortification emerged as a response to this problem. The solution was to create squat, thick walls that were defended by sidelong or flanking fire aimed from projecting gun emplacements or bastions. These bastions had to offer protection to the adjoining walls on either side and the whole fort had to be enclosed, resulting in polygonal outline plans. The angled shape of the bastions was progressively refined so that each bastion offered covering fire to its neighbours and no 'dead space' remained hidden from defensive fire. Beyond the primary wall and bastions, further positions and defensive structures could be extended outwards. These limits and rules set the conditions for the development of elaborate geometrical designs, not always constrained by the features of actual sites and the limited budgets of genuine commissions.

Such celebrated practical geometers of the Renaissance as Francesco di Giorgio, Filippo Brunelleschi, Leonardo da Vinci, Albrecht Dürer and Simon Stevin were concerned with the solutions to these problems, as were artists who might not be immediately associated with a programme of this kind, such as Bramante and Michelangelo. A species of practical geometry that began in Italy in the later fifteenth century, spread to other parts of Europe in the sixteenth, supported by the design of instruments and the publication of books. The general recommendation was for arrow-headed bastions projecting from the corners of a walled polygon, but beyond this there was plenty of scope for individual styles and schools. In practice the basic polygon often had to accommodate the natural conditions of the site or its existing structures. The French school took the lead in the development of such systems in the second half of the seventeenth century, with the practice and theory of Sébastien le Prestre de Vauban dominant into the eighteenth. The outcome was a vast programme of work and a thriving specialist geometrical discipline.

Geometers had to consider not only the plan of an ideal fortification: the section offered them further possibilities. Fortress defenders wanted both to impede the progress of any attackers and to oblige them to present an advantageous target for their guns. A large ditch in front of the rampart was fundamental to defence, and the earth from the ditch provided an embankment on the outer border that both obscured the walls and provided a sloping approach, angled so as to be swept by defensive fire. Other features could be incorporated, such as the covered way on the outer side of the ditch, for patrols shielded from fire. As with the plan or 'trace', the result was an elaborate structure subject to revision and adjustment by different practitioners.

A Practical or a Polite Science?

As a new branch of mathematical science, the geometry of war was ambiguously positioned in a number of respects and these ambiguities deserve attention from historians. Was this a practical science or a polite one? Did it belong at war or at court? Was it driven by practice or by theory? Was it characterized by action or by rhetoric? Was it part of mathematics or of natural philosophy? Answers to each of these questions fall somewhere between the two respective alternatives, and their positions on each of these ranges of possibilities change over time. It is within these ambiguities and changes that some of the most interesting aspects of the subject are to be found.

The tension between the practical and the polite is perhaps most obvious in the contemporary literature. Instruments are illustrated in use where conflict is imminent or already begun, and the coolness of the practitioners who apply their geometry in the heat of battle can seem improbable. Such examples of purported use initially served to stress the potential importance of the practitioners' geometrical techniques. With the acceptance that aspects of warfare were amenable to geometry, the association could be used to justify a group of mathematical problems that in practice would remain confined, for the great majority of readers, to textbook instruction.

The same can be observed in the progress of popular interest in fortification. Both gunnery and fortification become established in general encyclopaedic works aimed at the gentle market, and polygonal scales are included in a conventional way on a variety of instruments whose links to professional military surveying seem implausible. By the end of the period covered by the exhibition, popular lecturers in eighteenth-century London, such as Benjamin Martin and Erasmus King, were offering fortification as mathematical entertainment. Another lecturer, W. Griffiss, tackled the subject in a tavern in the evening, 'illustrated by various Plans, and a large Model of a fortified Town', and repeated the performance the following morning 'for the better Accommodation of the Ladies' (Morton & Wess, p. 73).

Military mathematics likewise featured in academic as well as coffee house teaching. To take a local example, David Gregory proposed a new mathematics course for Oxford in 1700 which included 'a lecture on fortification, so far as 'tis necessary for understanding it without actually serving in an army or forttifying a town or camp' (quoted in Bryden). The museum building itself witnessed instruction in fortification in the very room where the present exhibition has been mounted, when John Whiteside introduced the subject into his course on mathematics in 1723. At the same time as these popular and academic activities, there was at the practical level an active programme of fortress construction across Europe, and this lent credibility and status to the polite interest.

It is probably the surviving instruments that most eloquently address the tension between the courtly context and the battlefield. Elegant and ingenious instruments, perhaps in gilt brass and accompanied by tooled leather cases, were far beyond the material and probably the educational resources of the ordinary gunner. They could, however, enhance the image of an active officer abreast of the finer points of contemporary warfare. The maker offered a rare accessory to enhance a courtly posture and in exchange the patron offered a valuable and perhaps prominent commission. Instrument design might itself accommodate a rhetorical or gestural aspect, as when mathematical instruments are made to resemble, or even to act as, weapons. Yet not all surviving instruments fall into this class: they range from the improbably ornate to the crude and humble, and we must remember that the former are much more likely to have found their way into museum collections. We know, for example, that the British Board of Ordnance ordered gunner's quadrants and perpendiculars in quantity. On the other hand again, John Muller, Professor of Fortification and Artillery at the Royal Military Academy, Woolwich, says that gunners should learn to judge by eye and dismisses the use of instruments in his Treatise of Artillery of 1757.

The geometry of war sits ambiguously too between practice and theory. It is an ambiguity characteristic of the mathematical sciences and one from which they derived their potential for influential development in the Renaissance. Leonard Digges points clearly to this in his Pantometria of 1571, where he asserts that the perfection of gunnery is beyond both the mathematical novice, 'though hee turmoile in powder and shot all the dayes of his life', and equally the geometer, 'leaning onely to discourse of reason', who 'shall fall into manifolde errors, or inextricable Laberinthes.'The tension between deeds and words, between action and rhetoric, is particularly evident in contributions from the practitioners themselves, or at least from writers who present themselves as practitioners. Robert Norton, who describes himself as 'one of his Maiesties Gunners and Enginiers' on the title-page of The Gunner of 1628, asserts firmly that '… lead on by Experience the Mistris of all Arts, Action being the best Tutor … I … haue endeuoured herein more to respect a few experimented truthes, then many Rhethoricall imbellishments of words.' An 'experiment' at the time has the sense of a deed or venture, as distinct from a discussion or projection, without the more specific reference it came to have in later science. But it is clear that the sentiment of the Royal Society's motto, 'nullius in verba', had an earlier currency in this empirical tradition.

Finally there is the changing position of gunnery between mathematical science and natural philosophy. The former of these separate branches of Renaissance learning dealt with mathematics and its practical applications, while the latter was concerned with causal explanations of natural phenomena. Norton again was bullish and provocative: gunnery, he said, was a profound study, 'euen able to spose the knowne parts of Naturall Philosophy, Arithmetick, Geometry, and Perspectiue, each of which her handmayd is'. In reality the relationship was much more ambiguous, but the fact that this aspect of practical geometry raised questions pertinent to natural philosophy may be relevant to the emergence of a new relationship between the two - one where mathematics, experiment and instruments are all recruited to a reformed explanatory programme for the natural world.

The exhibition concludes with one such connection. Galileo began his career as a mathematical scientist, a teacher of practical mathematics, including fortification, and a designer of mathematical instruments. His ambitions, however, were towards the higher science of natural philosophy and one instrument he was involved with - the telescope - though it began as an instrument of warfare, became in his hands a reforming tool of natural philosophy. Reform was foreshadowed in two ways - through the telescopic observations Galileo marshalled in support of Copernicus, and through his use of an instrument as a means of investigating the natural world.

The telescope was a new departure in the methodology of natural philosophy, but it was through projectile motion that the discipline was most directly implicated in the geometry of war. Early writers on ballistics - Tartaglia and his successors - couch their discussions in terms of the natural and violent motions of Aristotelian physics. The form of the ballistic trajectory, traced by an initially rising projectile which then begins to falter and to descend, was presented as the combined outcome of the two opposed motions and the transition between them. The superiority of Galileo's technical achievement in the Discorsi should not blind us to the significance of a century of problem setting and the significance too of setting the terms of a solution - that is, as a geometrical trajectory with universal reference yielding predictable consequences. The treatment from Tartaglia onwards is thus characteristic of the mathematical sciences and particularly of their most sophisticated branch, astronomy.

Although Newton's success in Principia is often presented as a transcendent achievement of the geometry of reason, part of the prior development that made 'the mathematical principles of natural philosophy' a viable conjunction was contained in the geometry of war. Almost two hundred years after the Renaissance discourse of ballistics was opened by Tartaglia, Newton, in his System of the World (published posthumously in 1728), explained the dynamics of planetary motion in terms of a body projected horizontally from a mountain top with successively greater initial velocity. The problem is the old one of point-blank range but, not only has Newton escaped from the language of gunnery into that of abstract mechanics, the range of his projectile extends until, instead of descending to earth, it falls forever in an orbital trajectory.

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