6 Epistemology of the Quadrant for Bombardiers


The extent to which Tartaglia’s ballistic theory was open to and influenced by the practical knowledge from which and for which it emerged has been clearly demonstrated. It has been shown that ballistics found itself on the borderline between the activity of the bombardier and the natural philosophy typical of the time, which was essentially Aristotelian. Tartaglia drew upon both as sources of knowledge and was thus able to construct his deductive-theoretical framework. 

Looking at this from another perspective, i.e. starting from Tartaglia’s ballistics, the first question is, what were the points of contact with these sources of knowledge? As far as the Aristotelian doctrine was concerned, it was without doubt the definitions of natural and violent motion which, according to Tartaglia, failed to provide a sufficient answer to the fundamental question concerning the curvilinear segment of a trajectory. This question later led to the formulation of an idea of “mixed motion,” and thus formed the basis for a concept of compositions of motions. Leaving aside this aspect, however, as it would require a digression leading at least as far as the works which Galileo published more than one hundred years later,1 the question remains as to how this ballistic theory was applied to the activity of the bombardier. The latter was, of course, not interested in natural philosophical discourse or in questions regarding the nature of motion. What he found most relevant was knowing how to hit a target with precision from as far away as possible. 

What linked this demand to Tartaglia’s theory was the above-mentioned mathematical instrument, commonly known as the “bombardier’s quadrant.” Tartaglia describes this instrument, its design and its use in the dedication letter that opens the book. In particular, the quadrant allowed the angle of elevation to be calculated while the bombardier remained sheltered behind the piece of artillery. Nevertheless, Tartaglia’s firing table would have been even more relevant for the bombardier, and, above all, the method for calculating his own table based on the data he would have obtained from a single shot. Having a table of this kind meant that the bombardier would no longer have to resort to several adjustment shots each time the target changed. If he had at his disposal an exact survey of the fortress which he was to attack, or information on how the attackers were positioned, the efficiency of the artillery battery would have been considerably improved. Tartaglia promises this table and these guidelines in the incipit, but the books which should have contained them were never written. Still, the idea was the right one. Within a few decades, such tables began to appear in a wide range of publications, although there were great discrepancies in the values they showed throughout the sixteenth century.2 The quadrant had become the physical instrument that was able to incorporate the new science and apply it materially, and remained so for a very long time. 

Yet, as was mentioned in Section 2.1, the quadrant was not an invention of the sixteenth century. In Tartaglia’s day, it had already been in use for 120 years, at least in some regions. As a measuring instrument, the quadrant was generally an application of the geometrical properties of a triangle, relatively simple to understand and covered in the sixth book of Euclid’s Elements. As such, it was thus indeed an ancient instrument. 

What the bombardier’s quadrant was used for prior to the era of ballistics is not clear from the known literature, but there is evidence of its existence and there are even graphic representations. Nonetheless, it is natural to assume that the quadrant was used as an instrument of registration, the angle of elevation being measured and noted before each shot. Were the shot successful, an annotation would be used to realign the position of the piece of artillery, which would have been lost through recoil. In relatively recent times, some authors have hypothesized that firing tables already existed during the fifteenth century, compiled as a kind of record.3 Even though the available sources do not appear to corroborate this hypothesis, it is clear that a good series of annotations of various angles of elevation, relating to one specific piece of artillery, at use at specific intervals of time, firing similar projectiles and maintaining the same quality and quantity of gunpowder, effectively amount to a firing table. Considering the growing diffusion and the increasingly intense use of heavy artillery during the fifteenth century, the inevitable conclusion is that a considerable amount of empirical data was accumulated owing to the use of the quadrant. An anachronistic evaluation of the data collected by the bombardier of the fifteenth century could lead to the formulation of a hypothesis that the accumulation of this data formed the empirical basis from which the theory of ballistics emerged. Under scrutiny, however, this hypothesis remains improbable. The accumulation of data, if it did in fact occur, would have been of a very particular nature, with reference to specific pieces of artillery, projectiles and gunpowder, moreover, it would have remained quite local, since no institutional structure capable of collecting and preserving such data existed in the fifteenth century. Even if this information had been gathered, there would not have been any way to use it to formulate general, abstract rules using a more or less inductive method. The absence of any information regarding the context of the annotations made of each shot, for example, a formal and normative description of the characteristics of the piece of artillery, the projectiles and the gunpowder, as well as the influence of particular atmospheric conditions present at the time the annotation was made, would have rendered such a collection of annotations incomprehensible and pointless. 

By way of conclusion, there is only one way to interpret the relationship between 120 years of accumulated experience in the use of the bombardier’s quadrant and the birth of ballistic theory with the consequent scientific use of the same instrument. The annotations made during the fifteenth century represent the beginning of a codified written recording of the experience of the bombardier and his practical knowledge in general. In principle, these annotations could be of use only to the very bombardier who had written them, since only he was able, on the basis of his memory and accumulated experience, to give them a practical significance. The angle of elevation was only one of many aspects the bombardier had to take into consideration when exercising his profession, but it was probably the first of these aspects which found a way towards establishing written rules of recording in the form of measurements. From this perspective, the annotation regarding the angle of elevation of the piece of artillery is a first step in a process of abstraction, and therefore in theoretical reflection on the bombardier’s own actions. The diffusion of the quadrant and of its use thus led to an increasing formalization of the descriptions of certain specific aspects of the bombardier’s activity. As was mentioned at the beginning of this paper, Tartaglia states that the motivation that led him to dedicate himself to ballistics was a specific question from a bombardier regarding the relationship between the angle of elevation and the maximum range of a shot. Had there not been deep-rooted experience in the use of the quadrant in the period preceding Tartaglia, nobody would have been able to come up with such a question formulated in such a specific way, let alone understand it. 

To conclude, the quadrant is the epistemological instrument that initiated a process of theoretical abstraction, which ended with the formulation of the bombardier’s question. Thanks to the instrument, the bombardier is able to describe his activity in a comprehensible way to somebody who is not familiar with his work but possesses the necessary mathematical understanding, such as that of Euclidean geometry, or the necessary physical understanding, such as that of Aristotelian dynamics. The quadrant is thus not only the link between theory and practice in the period following Tartaglia’s Nova scientia, but also the means by which the transition was made from experience alone to the birth of a new theoretical subject. 



There are many studies dedicated to the relationship between the science of ballistics of the sixteenth century and the formulation of the law of free fall. Among the most significant, see Renn et.al. 2011. For a more general overview of the entire development of pre-classical mechanics, see also Damerow et.al. 2004. Finally, as an introduction see Büttner et.al. 2003.


Thomas Harriot (1560–1621), like Galileo, arrived at a conception of the parabolic trajectory of projectiles. As Matthias Schemmel demonstrates, Harriot considered different firing tables produced by different authors and in different locations in order to put the theory itself to the test. There were great discrepancies between the data which he collected, however, so that Harriot was forced to formulate mathematical procedures to extract more credible data. For this interesting study, see Schemmel 2008, particularly the eighth chapter of the first volume.


Schmidtchen 1977, 150–161.