Mathematics of firearms

Summary: Mathematicians have long studied and analyzed firearms and projectile motion to create more accurate weapons.

The successful construction and use of many types of offensive weaponry rely on mathematical principles. Ancient people typically used body-powered projectiles, like spears and stones thrown from slings, which required judgments of force and angles to achieve the correct parabolic motion to hit the target. Archimedes of Syracuse designed weapons like mechanical catapults to defend Syracuse from attack by the Romans. The development of gunpowder-propelled field artillery, the successor to mechanical missile weapons like catapults and ballistae, created a demand for sophisticated mathematics. Mathematicians studied and solved problems of ballistic velocities and trajectories to increase accuracy and impact. Handheld firearms of all types rely on similar principles. There are a number of interesting mathematical properties related to firearms, including weapon caliber, rate of fire, rifling, muzzle velocity, and propulsion, as well as telescopic sights and other characteristics. Mathematics training or degrees are suggested for firearms identification and bullet matching, which are increasingly used to match weapons to crimes, and mathematics skills are one of the requirements cited for careers in firearms repair.

Brief History of Firearms

As artillery and projectiles began to play a much larger role in warfare, kings, generals, and powerful concerns in society began looking for more powerful and more accurate weapons. They called upon scientists and mathematicians to address the problem. Niccolo Tartaglia, Galileo Galilei, Evangelista Torricelli, Rene Descartes, Isaac Newton, and Johann Bernoulli are some of the people who worked on the problem of projectile trajectories. Two of the foremost mathematicians to work in this area were Benjamin Robins and Leonhard Euler.

Tartaglia published an important work on cannon trajectory in the sixteenth century. Using the science and mathematics of the time (Aristotelian dynamics, named for Aristotle, and Euclidean geometry, named for Euclid of Alexandria), he thought of the flight of a cannonball as moving from a line with slope determined by the angle of the cannon, the final trajectory by a vertical line and a circular segment on which the apex of the trajectory occurs joining these two lines. In his 1537 text La nova scientia and his 1546 text Questi et inventioni diverse, he indicated that this was only an approximation to the actual trajectory. However, it was such a good approximation—and so comparatively easy—that it was used by artillery groups well into the eighteenth century. His model took into account the practical knowledge gained through working with gunners and their experience in the field.

Galileo stated that if there is no air resistance, the trajectory of a projectile is a parabola. This conjecture appears first in the work of his student Bonaventura Cavalieri in 1632 and, later, in Galileo’s 1638 work Discorsi e dimostrazioni matematiche: intorno a due nuoue scienze. Torricelli also worked with Galileo. His book De motu contained a geometric method for computing the range of a projectile. Galileo asserted that the path of the trajectory and the shape of a hanging curve (the catenary) are the same, leading him to work with the idea that the trajectory curve is symmetric. This idea results in erroneous computations for range.

In the next era, the important work of Christiaan Huygens, Bernoulli, and Newton on air resistance set the stage for great strides forward in understanding projectile trajectories . The first to explicitly consider air resistance was Benjamin Robins, an English mathematician who was a student of Henry Pemberton and a protégé of Newton. Robins became interested in military engineering in the 1730s from his work on Newton’s fluxions and their utility in describing objects in motion. In 1736, he wrote a detailed critique of Euler’s Treatise on Motion and his extensive use of algebra versus geometry. He was subsequently barred from an appointment as mathematics professor at the new Royal Military Academy in Woolrich in 1741 because of a political dispute. In order to bolster his application for this position, he returned to his work on ballistics and in 1742 published New Principles of Gunnery. In 1747, the Royal Society awarded him its prestigious Copley Medal for his work in ballistics. A major contribution of Robins was in determining that the important consideration for ballistics was the initial velocity of the projectile and the effect of air resistance, not the range, which was a function of initial velocity. Experiments showed that the assumption of Huygens and Newton that air resistance was proportional to the square of the velocity was true only at low velocities. Also, Robins hypothesized that lateral deviations were caused by random spinning of the projectile. He advocated the use of rifled barrels with ovoid (rather that spherical) bullets to control this effect.

Robins’s work ultimately had a broad impact. In a time of very poor English-Continental mathematical relations, Euler himself found Robins’ work from 1742 so important that he translated it into German in 1745 and made extensive additions. He contributed and acclaimed the work of Robins. Napoleon Bonaparte, an avid student of mathematics who is widely considered to have revolutionized the use of field artillery, had Euler’s translation translated into French for his study. Euler is credited with bringing the study of trajectory motion into the modern mathematics realm. In his 1753 work, he described motion in terms of second-order differential equations, allowing him to make appropriate changes in assumptions about air resistance and to give better approximate solutions that matched experimental results. Work was undertaken to create tables of trajectories for army artillery units. As technology advanced, mathematics had to evolve to keep pace. There were groups of mathematicians who worked for the war departments in both World War I and World War II. For example, British mathematician John Littlewood improved and simplified calculation formulas for range, flight time, and angle of descent of projectiles and updated ballistics tables. The Applied Mathematics Panel in the United States in World War II looked at various trajectory issues, including aerial dogfights and projectile trajectory. The U.S. Navy maintained the Aberdeen Proving Grounds after the war and had panels of mathematicians there to help model projectile motion and explosions.

There are a number of other interesting mathematical connections related to artillery and firearms, such as caliber and barrel rifling. The caliber of a firearm is the approximate diameter of the barrel and the projectile used in it, usually measured in inches or millimeters. Rifling is traditionally the process of making helical grooves down the entire length of a firearm’s barrel to impart a spin to the projectile. Polygonal rifling is another method that shapes the interior of the barrel like a polygon with rounded edges to achieve a similar effect, most commonly with hexagons but sometimes with octagons or decagons. Overall, rifling gives the projectile gyroscopic stability and improves its trajectory. Since a rifled barrel is noncircular, as opposed to a smoothbore (nonrifled) weapon, there are different ways of measuring caliber. In the case of helical rifling, measurements may be taken of the bore diameter, which is the diameter across the lands or high points in the rifling, or the groove diameter, which is the diameter across the grooves or low points. Rifling grooves create striations on the bullet, which, together with caliber, are used in forensics to identify the firearm that shot a bullet. Twist rate for rifling is the distance the projectile must travel down the barrel to complete one full revolution about its own axis, which is often given in units of turns per inches or centimeters. A shorter distance indicates a higher turning rate and a faster spin.

Bibliography

Barnett, Janet Heine. “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, no. 2 (2009): http://dx.doi.org/10.1080/17498430902820887.

McCleary, J., and D. E. Rowe. “Airborne Weapons Accuracy: Topologists and the Applied Mathematics Panel.” The Mathematical Intelligencer 28, no. 4 (2006).

McMurran, Shawn, and V. Frederick Rickey. “The Impact of Ballistics on Mathematics.” http://www.math.usma.edu/people/rickey/talks/08-10-25-Ballistics-ARL/08-10-23-BallisticsARL-pulished.pdf.

2009 Product Engineering Processes. “Archimedes Death Ray: Idea Feasibility Testing.” October 2005. http://web.mit.edu/2.009/www/experiments/deathray/10‗ArchimedesResult.html.