August 1940. The Battle of Britain is reaching its peak. Britain is battered by Nazi Germany's air force: the Luftwaffe. Heavy bombings are causing immense suffering among military and civilians alike, but the British can count on a secret weapon: the radar.

Developed in the early 30s, by 1936 the first five Chain Home systems were operational. Chain home was the codename for the British system of coastal aircraft detection. By 1940 Chain Homes stretched across the entire UK including Northern Ireland. 

Beside the technological challenges associated with the development of radar systems, British intelligence faced another key challenge: only five radars were initially available; it was crucial to determine where to position them to maximise their effectiveness. 

Optimal positioning of radar bases along the UK coastline during the Battle of Britain represents one of the first attempts to apply mathematics to improve military operations; but is this really the case? 

"As soon as the first rank has fired together, they will march to the back. The second rank either marching forward or standing still, fire together then march to the back. After that, the third and following ranks will do the same. Thus, before the last ranks have fired, the first will have reloaded." 

The "counter march" - an innovation adopted by Maurice of Nassau Prince of orange - is only one of the new military practices introduced in the 16th century, but were military
practitioners of the 16th century simply following common sense, or did some of them follow a principled scientific approach akin to the one in use today? What would happen if we read their research works today? 

The 15th and 16th century saw impressive technological advances: painting made a quantum leap thanks to the development of perspective; improved maps, the use of compass and quadrant, as well as the new practice of double entry bookkeeping allowed merchants to manage ever-expanding trades; gunpowder and heavy artillery started to be systematically deployed on battlefields to revolution warfare.

Early attempts to apply algebra and numbers to the battlefield can be dated to the late 16th century. The equilibrium between attack and defense strategies that had endured for centuries had been destroyed by introduction of heavy artillery. 

The emergence of new theoretical knowledge should be intended as a consequence of an advanced and challenging technological context, but how did it all started? When was mathematics first applied to the battlefield? 

The decline and collapse of the Byzantine Empire heightened contact between its scholars and those of the West. Neoplatonism was the dominating school of the Byzantine Empire. The
collapse of the Empire brought an influx of Neoplatonic scholars into Western Europe. Georgius Gemistus, the leading scholar of neo platonic philosophy in the late Byzantine Empire, reintroduced Plato's thoughts to Western Europe during the Council of Florence. He founded a new platonic Academy that focused on the translation of Plato into Latin. 

The Renaissance revival of the rational tradition of Greek science led to a proliferation of studies and translations of Greek works including Euclid. Euclid's elements was translated into Arabic in the ninth century. Muslim mathematicians then combined geometry with Hindus arithmetic and algebra and developed new advances in the 12th century. The work was translated into Latin making it more accessible to European scholars. 

Neoplatonism influence was strong at the time of Leon Battista Alberti. Mathematics as a discipline tended to be framed in a neo platonic context. In his treatise on painting, Leon Battista Alberti wrote: "mathematicians measure the shapes and forms of things in the mind alone and divorced entirely from matter." Renaissance platonists had been interested in number mysticism rather than real mathematics. A key ingredient was missing: the connection between theoretical models and practical applications. 

The development of this connection is what motivated the studies discussed in Tartaglia's La Nova Scientia, written in 1537. In 1532, while he was living in Verona, a friend who was a bombardier asked Tartaglia at which angle the barrel of a cannon should be elevated to achieve the longest possible shot. Tartaglia did not have expertise in specialized areas connected to military activity. However, by using geometrical and algebraic arguments, he was able to establish the following result: "the maximum range would be achieved if the barrel of a cannon were raised at an angle of 45 degrees above the line of horizon." By answering this question Tartaglia consciously engaged in an inquiry that today we would have no problem in labeling as applied mathematics. 

In Tartaglia's time there had been a proliferation of mathematical treatises on topics such as algebra and combinatorics: both Ramon Llull and Gerolamo Cardano wrote an Ars Magna. What made Tartaglia's work new was the application of abstract mathematical models to achieve practical outcomes. Hence the title Nova Scientia. This was a revolutionary step that paved the way and inspired, as it is clear from the title of his treatise - "Discorso intorno a due nuove scienze" - Galileo's works. 

In the frontispiece of his work, Tartaglia provides a motto. The motto states that the mathematical disciplines are seen as the only method to understand the reasons of things and that is open to everyone. This sets a precedent to Galileo's book of nature and represents a clear cut from the hermetic and neo-platonic traditions, in which knowledge is esoteric. 

However, the connection between theoretical models and applications requires a further ingredient not yet discussed: measurement. Influenced by Euclid's work, Leon Battista Alberti in his work "Ex ludis rerum matematicarum" discussed applications of trigonometry to surveying. The methods discussed relied on instruments called equilibra. Equilibra were a simple extension of the plumb line, and could be used to measure angles in everyday activities. These applications of mathematical tools were different from those devised by Tartaglia: measurement records properties of the world; Tartaglia applications to
ballistics aimed at influencing the world, not merely observing it.

The first main contribution of Tartaglia consisted in perfecting the quadrant and developing a systematic methodology for its use in ballistics. Tartaglia did not invent the quadrant. A quadrant can be seen as an enhanced equilibra, and equilibria had been in use for a long time.

In Tartaglia's work the quadrant was described in two versions: one developed for aiming Canon with a longer leg to be inserted into the gun mouth and a plumb line; another developed to measure the height and distance of a target, in the latter case the quadrant had legs of the same length and a plumb line. 

Prior to the era of ballistics the quadrant was chiefly used as an instrument of recording. Before each shot, the angle of elevation would be measured and noted. If the shot was successful, the annotation would be used to realign the position of the piece of artillery, which would have been lost through recoil. 

Tartaglia's work is not about me recording and repositioning. He illustrates techniques for estimating distances of target and aiming canons accordingly. These techniques are based on Euclidean geometry. They are illustrated in the form of propositions similar to the ones found in Euclid's elements. Results presented are supported by geometrical reasoning. 

Tartaglia's explicit aim was to create a science that was strictly mathematical and of an axiomatic deductive nature. He begins with definitions postulates and axioms - the influence of Euclid is apparent. In fact, Tartaglia also wrote the first translation of Euclid in vernacular Italian. Finally, propositions and corollary follow through a process of deductive reasoning. There is hardly any difference from this structure in that of a modern research article in applied mathematics. Further, some illustrations in Tartaglia's work are clearly reminiscent of Galileo's later studies on the fall of bodies. 

Tartaglia had to operate within the conceptual framework of Aristotelian physics, as no other viable framework existed. In line with Aristotelian physics, he described the trajectory of projectiles as a sequence of a violent motion followed by a natural motion. These were connected by a circular phase. Tartaglia had limited formal tools to model trajectories mathematically, so he did what every researcher would do: he forced the trajectory to take an approximate form that he could analyze with the formal tools he had. 

Tartaglia knew that the real trajectory was not made up of two straight lines joined up by an arc of circumference. He therefore states:

"nevertheless that part [of the transit]
that is not perceived as being curved is
assumed to be straight, and that part
that is evidently curved is assumed to be 
part of the circumference of a circle,
as this [assumption] does not influence
the argument." 

On the one hand, Tartaglia strived to achieve the greatest possible abstraction from the bombardier problem. His aim was to construct an exact science based on the Euclidian model. On the other, Tartaglia was aware of the applied nature of the Nova Scientia he was introducing. Tartaglia's argument cannot be explained on a purely geometrical basis. They require observation and experience in order to be considered valid. 

Tartaglia's thought was incredibly close to today's practice. In his book "Principles of Statistical Inference" David Cox, an eminent British statistician, states: "Some previous knowledge of statistics is assumed, and preferably some understanding of the role of statistical methods in applications. The latter understanding is important because many
of the considerations involved are essentially conceptual, rather than mathematical, and relevant experience is necessary to appreciate what is involved."

Finally it is worth spending some words on Tartaglia's use of the quadrant as an epistemic instrument. 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 quadrant links theory and practice and enables the transition from bombardier's experience to ballistics as a new theoretical subject.
It is this process of abstraction that enables the application of the methods of science to problems arising in practical settings.