Military Subjects: Organization, Strategy & Tactics

Artillery Swings Like a Pendulum...

By Tom Holmberg

Note: This article is a summary and restatement of information and research from the two much-longer articles written by Dr. Brett D. Steele listed in the bibliography below. I recommend that readers refer to Dr. Steele's articles for further information on this topic.

A while back the question was asked as to the method used in the era of the Napoleonic wars to measure muzzle velocity. Prior to the 18th century the most widely used theory to explain ballistics was Galileo’s vacuum or parabolic trajectory theory. Unfortunately this theory did not consider the effect of air resistance on ballistics. Galileo’s theory was valid for heavy mortar shells fired at low speeds but he admitted that it was too inaccurate for high-velocity shots. After Galileo many of the great natural philosophers and mathematicians attempted and failed to solve the problem of a projectile moving through a resisting medium.

Benjamin Robins, an English mathematician born in Bath in 1707, eventually solved this fundamental problem of ballistics. In the 1730’s Robins, a member of the Royal Society, began a study of fortifications, hydraulics and ballistics. Blocked by political differences from a professorship at the recently established Royal Military Academy, Robins was spurred to publish his New Principles of Gunnery in 1742. Robins’ most important breakthrough was the invention of the ballistics pendulum, the first scientific instrument for measuring projectile velocity. Using this instrument allowed Robins to attack the fundamental theoretical questions of ballistics.

The ballistics pendulum allowed the researcher to measure both muzzle velocity and –by moving the pendulum progressively farther from the gun—the effects of air resistance on a projectile, the two basic parameters for understanding ballistics science. The ballistic pendulum was a simple pendulum comprising a flat plate connected to a bar that swung from a tripod. By measuring the length of the pendulum’s swing after the impact of a ball on the plate it was possible mathematically to determine the velocity of the projectile. For a diagram of how the ballistics pendulum works and the math involved see the following websites:

Conservation of Energy & Momemtum

The Ballistic Pendulum

The measurement of muzzle velocity is a function of interior ballistics. The science of ballistics is also concerned with exterior ballistics, or what happens to a projectile after it leaves the barrel of a gun. Using the ballistics pendulum allowed Robins to experimentally measure the enormous forces acting on high-speed projectiles. Robins discovered that the force of air resistance could be as high as 120 times the projectile’s weight. The being able to accurately measure these forces meant improved accuracy of artillery. Under Galileo’s vacuum trajectory theory as 24-pound ball from a cannon would travel 16 miles, whereas in practice, the maximum range was less than three miles.

Robins also investigated lateral deflection of high-speed projectiles. He set up a series of evenly spaced paper curtains that allowed him to measure the enormous deflections of a musket ball in flight. In one test a ball measured over a range of 760 yards deflected more that 100 yards to the left. Robins identified that it was the spin of the ball that caused this deflection. In a further experiment he bent a gun barrel a few degrees to the left. Although the bullet initially moved towards the left, eventually in reversed its later direction and crossed to the right of the barrel (this effect, which can be observed in baseball and tennis balls, is known as the Magnus effect).

Robins thus explained why it wasn't the tighter fit of a rifled bullet that caused it to be more accurate, but rather that the direction of the projectile’s forward motion coincides with its axis of rotation. Robins developed a series of ballistics tables that simplified the work of the gunnery in determining the range of a projectile. He demonstrated the accuracy of his tables by comparing his predicted trajectory with experimental data. For instance, a 24-pound cannon loaded with 10 pounds of gunpowder elevated to 5 degrees had actual ranges of 834, 872, 851, 845, 871, and 838 toises. Robins tables gave a range of 850 toises. Robins concluded that: “Our theory differs less from the experiments, than the experiments do from each other.”

The Swiss mathematician Leonhard Euler (1707-1783) and others expanded upon Robins’ work. Tables based on Robins’ work were still in use up to World War II for calculating low-velocity and high-angle mortar fire. As a young artillerist at Auxonne, Napoleon studied a French translation of Euler’s German translation and commentary on Robins’ work (Napoleon’s notes on the book survive). Napoleon’s scientific understanding of cannon and mortar fire was a significant factor in his successful strategy at Toulon. As R. Riehn has written, “Trained in the artillery sciences, [Napoleon] had a keen grasp of the principles of physics and the concepts of energy and force. No one understood better that he the relationships of mass, time and the distance that went into the creation of energy. This much emerges from his methods of conducting a campaign or battle…”


Steele, Brett D. "Muskets and Pendulums: Benjamin Robins, Leonhard Euler, and the Ballistics Revolution" Technology and Culture Vol. 35, No. 2. April 1994. pp. 348-382.

Steele, Brett D. "Napoleon and the Ballistics Revolution" Consortium on Revolutionary Europe Proceedings 1995. pp. 455-463.

Also a few websites:


Benjamin Robins

Leonhard Euler


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