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What prevailed until the eighteenth century was the model of Nicolo Tartaglia of 1537. It was not until the arrival of a man named Benjamin Robins and his work New principles of gunnery (1742) that he completely revolutionized the study of ballistics. But let's go step by step in this story.

Nicolo Tartaglia is a scientist of the 16th century, whose fame has reached our days mainly due to the fight he had with Cardano over the algebraic resolution of cubic equations. However, Tartaglia also published La nova scientia in 1537, in which he described the trajectory of a projectile composed of an initial straight line, and a later fall:

Figure 1: Representation of mixture model and front page of Nova scientia [1]

Figure 2: Representation of a horizontal shot, according to Tartaglia [2]

Tartaglia himself ended up not believing his theory, as he acknowledged years later. But this Italian thinker laid the foundations to create a few abacus that tabulated his calculations, and to be able to calculate the length of the shot as a function of the inclination of the barrel. These abacuses were used until the 18th century.

Later, in 1638, Galileo Galilei proposed his parabolic firing model, which was a bit absurd, since Galileo, despite knowing that there was some kind of resistance to air, did not count it in his calculations. He obtained a symmetrical trajectory that was not realistic, and for that reason, the artillerymen continued using the model of Tartaglia. On certain occasions, it was even enough for them to use Daniel Santbech's triangular model, proposed in 1561.

Figure 3: Daniel Santbech shot model [1]

However, all this changed in the eighteenth century with our next character: Benjamin Robins. This English scientist was born in Bath in 1707, just the same year as Euler. He soon devoted himself to military engineering, and his greatest contribution to the study of ballistics resides in his suggestion that it was essential to take into account the strength of air resistance, and that it depended much more on the initial speed of the shot than on its range. To demonstrate this, Benjamin Robins designed a ballistic pendulum, thanks to which he could calculate the initial velocities of projectiles, taking into account Newtonian mechanics and Huygens pendulum theory. In fact, Robins is also considered the father of wind tunnels for this type of contributions.

Figure 4: Robins’ ballistic pendulum

The apparatus consisted in firing a bullet against the hanging mass of the pendulum and calculating its arc. The mass of the pendulum was much larger than that of the bullets of that time.

Benjamin Robins was already a member of the Royal Society since 1727, and fortunately, the greatness of this character was noticed right away. In 1742 he published his work New principles of gunnery. An Euler employer, Frederick the Great of Prussia, asked the mathematician what the best ballistics book was, and despite Robins' lack of attachment to his own book, Euler chose it. Not only that, but he translated it from English into German, and added a few comments about it. So the new manuscript went from 150 pages to 720. In 1783, it was translated into French, and according to Napoleon, it was one of the books that most influenced its scientists and engineers.

An artillery officer once wrote that Robins was to artillery as the immortal Newton was to philosophy. Before his influence, the success in the shot was a matter of chance subject to multiple variables and uncertainties. That same century, the president of the Royal Society went further and claimed that Robins had invented a whole new science.

REFERENCES

[1] Janet Heine Barnett (2009). Mathematics goes ballistic: Benjamin Robins, Leonhard Euler, and the mathematical education of military engineers, BSHM Bulletin: Journal of the British Society for the History of Mathematics, 24:2, 92-104, DOI: 10.1080/17498430902820887
[2] Ed Sandifer. How Euler did it. The cannonballs http://eulerarchive.maa.org/hedi/HEDI-2006-12.pdf
[3] The first wind tunnels.
https://www.centennialofflight.net/essay/Evolution_of_Technology/first_wind_tunnels/Tech34.htm
[4] McMurran, S., & Rickey, V. F. (2008, October). The impact of ballistics on mathematics. In Published in the Proceedings of the 16th ARL/USMA Technical Symposium https://www.revolvy.com/page/Benjamin-Robins
[5] Steele, B. D. (1994). Muskets and pendulums: Benjamin robins, Leonhard Euler, and the ballistics revolution. Technology and Culture, 35(2), 348-382