The topic of geometry is specifically mentioned and discussed on p.55-92 (Chapter: “Principes et pratiques de geometrie, necessaires pour travailler aux fortifications”). In this part, Jean Du Breuil touches upon 18 practices to describe various methods of constructing fortifications, on the basis of geometrical principles, forms and simple tools, which should be used to achieve the desired result. His descriptive and highly directed practices, are further categorised into three main methods of geometric principles. These geometric principles and methods are key to understand the basis of fortification and the ability to construct effective and competitive defensive structures. De Breuil creates a guide for the construction of fortifications, which he illustrates by drawings and the principles of geometry. His descriptions are reminiscent of an instruction manual. The drawings are simple and marked by numbers or letters to guide the reader through the process.
The first Method describes geometric practices necessary to trace fortifications on paper and on the ground. De Breuil elaborates on the concept of fortifications outside the geometric figures. This is done using a try-square (l’equaire/l’équerre), which can be a useful tool to draw fortifications. De Breuil also mentions the importance of the geometric principle of angles, that is used to divide circles into equal parts. He then specifies the use of certain angles to create regular polygonal structures. This concept is then further elaborated to highlight the importance of circular figures or polygons which are the basis of regular fortifications. The first method also mentions a system to divide and split lines, in order to take measurements of fortifications. This is done by using fathoms (unit of length), feet and a scale. De Breuil then elaborates on the possibilities of forming any structure with desired angles. He then uses this first method to illustrate the ability to trace regular French fortifications with a ruler and a compass, a drawing tool which is used by architects. (le compas). This method is then used to further display that his method works by demonstrating the fortification of a pentagonal structure or hexagonal structure. The method also works on octagonal (eight-angle) and decagonal (ten-angle) structures. It’s important to notice that he defines these fortifications as French, as there are distinguishable differences according to each country.
The second method is distinguished by teaching us about the fortification on the inside of the geometric figure instead of the outside, thus the sides are defined by the distance between the points of the bastions. De Breuille then describes the fortification of a square with the second method with only a ruler and a compass. He then expands on the second method by demonstrating the same principles on the fortification of a pentagonal and further polygonal structures.
On the basis of the third method it is possible to fortify all polygons with a ruler and a compass. De Breuil states that this method makes the fortification of any place whatsoever very easy, provided that one has an angle which directs the structure and allows to measure everything on-site. This method can be summarised as the director angle. Since everything depends on this angle, De Breuil goes on by illustrating the rather simple fortification of hexagonal structures and other polygonal structures. This serves as an exemplification that by the means of this third method, we can fortify irregular structures. The third method is also used by De Breuil specifically to plot Spanish fortifications or to strengthen all regular figures according to the Italian order. He further touches upon distinct foreign fortifications, which have no specific rules or maxims.
In conclusion one could argue that the theory behind it is of geometrical or mathematical nature. A practical use for the principles is illustrated by De Breuil, by guiding the reader through the practices of constructing fortifications. He elaborates the importance of precision and the beauty that stems from it. Geometry forms the basis of fortification and in its principles lies the structure of it. This can be accessed by putting the ideas into practice, using simple tools to evolve an idea into a structure, built on principles. It shows the importance of the practical side of science, where geometry was only able to flourish due to the engineers and professors who were willing to not only rely on the theory but also the practice of geometry in the construction of fortifications.
“If everyone was of the opinion of these best, who are satisfied to know the theory of the arts, without wanting to take the trouble to put them into practice: for example, to take a ruler and a compass, when this science requires it, like that of mathematics, we would not now have such excellent master professors, nor such admirable engineers in France, which by the grace of God are better supplied with them than all the rest of the world. I say this to invite the youth and especially the nobility, not to follow the first of whom science is only for them, but to do boldly and without shame like the last, since it is the science of kings and princes, who take pleasure in putting down on paper the beautiful thoughts they have for the plan of a city, the design of a castle, as it is necessary to besiege and attack a place: also to esteem and singularly love those who know how to work there like them, since it is one of the beautiful marks of wit a good man can have.” (p.56: L’Art universel des fortifications, Jean du Breuil)