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# Militarizing Mathematics

## Overview

Humanity has an uncanny knack of finding a military application for almost any discovery made by science. Mathematics is no exception and, in fact, because of its utility in describing many physical phenomena, has been extraordinarily useful to the military. This process began during the Renaissance, continues today, and is likely to be ever more important as we develop heavily computerized weapons and armies for the future.

## Background

Early warfare was a rather hit-or-miss affair. Armies were generally composed of a large number of poor people on foot with sharp weapons, and a small number of wealthy people on horses with sharp weapons. They would rush into battle, shooting arrows and hacking at each other until one side realized it could not win and either withdrew or was slaughtered. Sometimes, one army would ensconce itself inside a castle or some other fortification, where they would be relatively immune from the ravages of the other side for as long as their food and water held out.

Sometime around the fourteenth century, with the introduction of gunpowder from China, all this began to change. Almost simultaneously, firearms and artillery pieces gave foot soldiers a weapon to use at longer range and against those in armor, and gave besieging armies a weapon that could bring down castle walls. Because of this, gunpowder is possibly the most significant advance in the history of warfare, and one of the most important inventions in human history.

With the advent of gunpowder and artillery, battlefield strategies changed. Enemy troops could now be attacked from a greater distance, giving the side with superior artillery a tactical advantage. Since city walls were no longer impenetrable, they began to be replaced by designs that were more effective against cannons, while making best use of the cities' own cannons. In addition, the fledgling science of ballistics, based almost entirely on mathematics, helped make the flight of shot from a cannon more predictable; another significant advance.

Much of the these new weapons' effectiveness was due to the use of mathematical analysis to perfect both their design and use. Of course, battles remained anything but predictable, but the characteristics of an army's main weapons were better known, making their use in battle more certain.

At sea, mathematics found increasing use, too. As navies and merchant fleets ranged further from shore, becoming true blue water fleets, their dependence on accurate navigation grew. This, in turn, was dependent on the accurate use of trigonometry to determine a ship's location from the position of the sun or various stars. Some sea captains, especially those with more scientific knowledge, began to try to balance the forces acting on their ships to maximize their performance. Finally, waging naval warfare on an intercontinental scale put a heavy emphasis on logistics to make sure that a ship could be supplied with food, water, gunpowder, shot, and other supplies, travel all over the world and still fight effectively. This, too, benefited from the increased use of mathematics rather than operating solely by rule of thumb.

## Impact

The increased use of mathematics in warfare directly affected military technology and strategy, making warfare more efficient and more predictable. These are significant developments, and are in some ways intimately related because the process of making weapons more predictable allows them to be used to their maximum advantage. This, in turn, makes them more efficient in battle. The reason for this is not obvious, but is relatively simple, as the following example will demonstrate.

## Land warfare

Take two cannons, both of which are the same size and weight and hold the same amount of gunpowder. One cannon was cast and bored by eye, based on the judgment of the person overseeing the process while the other was cast and bored according to a master set of plans drawn up by an engineer. The first is loaded with a batch of gunpowder mixed according to an old formula that has been handed down over the decades, and poured into the barrel to a level that "looks" right. The other receives a standardized amount of a standardized batch of powder. They are loaded with cannonballs, aimed and fired.

The second gun will fire more consistently every time. Because the barrel is a standard size, it can fire a shot that has been designed to fit as closely as possible without jamming, so the shot will travel farther. Because the powder is uniform and the same amount is used every time, the crew will know exactly how far the shot will travel for a given cannon elevation. In addition, the new science of ballistics will have provided the gun crew with a table of different trajectories and their effects. In short, the performance of the second gun is more predictable than that of the first. Because of this, a general will know exactly where to place his artillery pieces to get the best use out of them. Of course, this example is a great simplification, but the fact remains that the mathematical control of ammunition and weapons use could give a great advantage to the gun crews using those weapons.

Naturally, both attackers and defenders used artillery, and defenders gained the same advantages in their use of guns. In addition, defenders soon learned that elevating their weapons gave even greater range, while replacing their high walls with lower, sloping embankments helped mitigate the worst effects of cannonballs. In this, too, mathematical analysis of the variety of angles from which attacks could come helped prepare for the most likely scenarios.

## Naval warfare

Naval gunnery benefited from mathematics in the same way that land-based artillery did, although firing a gun from a rolling ship at a moving target was much more difficult than hitting a stationary target with stationary weaponry. Mathematics was also useful in such seemingly mundane tasks as determining the optimum amount of food and water with which to provision a ship. (Even these simple tasks can help a ship to fight more efficiently: carrying too much food left less room for powder and shot, while scrimping on supplies adversely affected a crew's efficiency and effectiveness over long periods of time.) In addition, improved navigational techniques facilitated coordinating with other ships, and a better understanding of the physical stresses on naval vessels helped captains to sail them better and more quickly. All of these improvements, in turn, helped navies defend their national interests and project national power to newly discovered colonies and trading partners.

## Effects on society

The same technology that produced cannon barrels of precisely the same dimensions helped make the first internal combustion engines efficient enough for use in automobiles and aircraft. It also helped create piston cylinders on early steam engines, an invention that powered both the Industrial Revolution and our current automotive age. Making gunpowder as consistent and effective as possible helped produce what is now the science of chemistry and all that it has given humanity. The navigational improvements that made navies so much more efficient also made commercial travel more efficient, opening up the world for exploration, exploitation, and colonization by the European powers.

These spin-offs weren't always positive. The military changes that accompanied the use of gunpowder made noncombatants more vulnerable during wartime because they had no place to retreat for protection. The growing efficiency and complexity of warfare also helped spur the formation of professional armies, specialists in warfare who had mastered the weapons and techniques of their trade. The growing effectiveness of weapons on killing noncombatants, helped lead to rules of warfare, which were designed to protect civilian populations while allowing the wholesale slaughter of soldiers. So warfare changed, from the standpoint of the military and civilians both.

This is not to suggest that mathematics alone is to blame for these trends. Most of these "improvements" would have happened regardless of the use of mathematics. Just about everything mentioned above would have been determined empirically at some point. The use of mathematics simply made the process more efficient, a trend that has continued to this day.

P. ANDREW KARAM