The beginning of the modern period, confronted by new discoveries and interrogations regarding scientific knowledge, marks the emergence of new languages. The question of images and scientific illustrations as copies of reality, of representation of instruments as complements of the written discourse, had been gaining strength since the sixteenth century. The alert given by the Jesuits in relation to the deceit created by the senses appears within this context, appealing therefore to the use of mathematical concepts and scientific instruments (such as the telescope) and wagering in the practical dimension of these same instruments (beyond the symbolic dimension they already had). It was the Jesuits that gave Mathematics the responsibility to explain/demonstrate the physical world, countering the Aristotelic primacy of Natural Philosophy; for them, the principle of all sciences should be as evident and universal as the Euclidian postulates.
One of the strongest examples of the application of mathematical knowledge is situated at the level of military engineering, which reveals, in its engineers, excellent mathematicians, some of them with a Jesuit education. Starting from the book Disciplinae Mathematicae traditae anno institutae societatis Iesu secularie (Louvain, 1639-1640) and from the representation of the mathematical instruments included in this work by Jan Ciermans (1602-1648), a Jesuit, we shall try to appraise the influence of this work in the interventions of its author as chief engineer and superintendent of the fortresses in the South of Portugal, contextualizing both Author and book in the scientific production of their time. In this work, which illustrates the diversity of mathematical disciplines (mixed and pure) we shall highlight the chapters dedicated to Fortification and to the Machines of War, as well as the whole ensemble of illustrations, and we shall look for both continuity and innovation in terms of military engineering treatises (which call upon mathematical instrumentation) in Portugal in the seventeenth century.