Mathematics and War

Mathematics and War
by Tomas Kubes

Introduction

Mathematics and war through ages


Stone Age

Grece and Romas

Renaissance

Industrial Revolution

Second World War
The future

List of Sources



What is it all about
Conflict, old as mankind itself, the clash of different opinions, and the will to put the conflict down, are twinning man from the very beginning of the times. There are many ways how to settle the dispute. Sometimes revealing the new facts may bring foes back together, or somebody may back up and agree on compromise, but when talking or diplomacy fails, the fight takes place. Fight, the violent way how to force your enemy to accept your will, your opinion, your way of life. As the civilizations developed, various differences, needs or just will to rule others led thousands of people to fight again each other, to fight a war, to conquer, to kill. The desire for victory, desire not to be the one who will have to step back, desire to not to be the one who loses, force each side to put everything into the war effort, to do their best, because there is nothing after defeat.
And as history proved many times, there is one thing determining the output of the war, knowledge. Knowing what opponent will do, knowing his weaknesses, knowing how to fight better. Some knowledge can be got in a short time, while some need to be researched for a long time. Ant that is what everything is about, knowing how to apply the psychical wealth of the nation to be stronger than the foe on the battlefield.
As the time progressed, the warfare become more and more sophisticated, from the sharpened poles, to cooper and bronze swords and long bows. The war also extended on the surface of the sea. And the weapons were gaining range. And immediately after you stop to shoot directly and start shooting by arc, you want to know how to set your "equipment" to hit the target. With long loading times, the guess of range and elevation was essential. But train a skilled shooter was hard, military leaders rather searched for other way how to shoot accurately. And that is the time where math comes and take the job. While in the early age, math was preliminary used during the construction of the war machines and war ships, with introduction of a cannons, the math moved more and more to the battlefield. Math was used to get the distance of the target; sophisticated devices were than used to count the elevation. Air warfare brought even more mathematicians to the drafting board to construct airplanes. The first long-range missiles did not have a guidance system, rather they were just fired in the chosen direction and than after exactly measured amount of the time, engine was turned off and rocket fell down. The precise counting was essential for hitting the target two hundreds miles away. Invention of the computer brought even more mathematics into the warring. Computers work with the logic, which is a division of mathematics. More and more computer systems are being implemented to the weaponry. The simple thrown stone evolved to the sophisticated self-propelled, self-guided device full of explosives able to deliver death over the five hundreds miles with accuracy of few feet.
Direct use of mathematics in the developing of artificial intelligence prophesy still larger and larger involvement of this general science in very specific field of fighting.

Mathematics and war through ages
At the dawn of ages the fight was more about personal strength, training of warriors and little bit about the science. Using of just sharpened poles and stones did not provide space to employ knowledge. Just raw brutal strength decided. Even the first bowmen had no idea about math, maybe except of counting remaining arrows. Some science was probably used while designing swords and chariots. But all was still more about marksmanship.
The great change came with Greek civilization. Greeks were great scientists. They made achievements mostly in philosophy but also mathematics. Moreover they introduced the war machines like catapult or ballista. Construction of these devices required direct involvement of mathematics and mathematicians. Vessels also started to cruise the seas and navigation become example of applied mathematics. The achievements of Pythagoras made more new things possible.
Than came the Romans. Their vast empire required superior military power. Fortunately with their conquer, they also spread the knowledge. Education was widely supported and direct applications for it were found. Mathematics was directly used in the building of military and civilian structures, including defense walls, barracks, forts, coloseums, bridges, aqueducts and roads. Position of empire also supported sea trade and great advancements in navigation and astronomy were made. Their fighting style also switched from the heroic actions of a single man to a coordinated fight of a group. Weapons were carefully designed and many new war machines were constructed. Battering rams, ballistae and catapults become normal equipment of the army. Heavy use of the math to construct those was evident.
We must not forget that the Greek civilization did not diminish. It produced one of the greatest mathematicians of old age. Archimedes was not only very intelligent scientist; he also knew how to use his knowledge.
Archimedes was a native of Syracuse, Sicily. When he was a young man, he studied the in the Alexandria. Certainly he was completely familiar with the Euclidian mathematics developed there.
He had gained a reputation in his own, which few other mathematicians of this period achieved. The reason for this was not a widespread interest in new mathematical ideas but rather invention of many machines, which were used as engines of war. He had been persuaded by his friend and relation King Hieron to build these war instruments, which were particularly effective in the defense of Syracuse when the Romans under the command of Marcellus attacked it.
... when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's 1beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them.
The achievements of Archimedes are quite outstanding. He is considered by most historians of mathematics as one of the greatest mathematicians of all time. He perfected methods of integration, which allowed him to find areas, volumes and surface areas of many bodies. His most famous theorem gives the weight of a body immersed in a liquid, is called Archimedes' principle.
Archimedes was killed in 212 BC during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed. It is perhaps surprising that the mathematical works of Archimedes were relatively little known immediately after his death.
Than most of his wisdoms become forgotten and they were not used in action, especially with fall of Roman Empire (between 300-500 AD), the interest in science died out. Europeans were once more ruled on basis of religion. The fighting stile of typical Dark Age knight did not allowed much space for tactics or science. Mathematics was sometimes used while castle building, but most of the science banished from the live.
Then came the Renaissance, it brought the interest in sciences again. Typical renaissance citizen should be physically strong, have knowledge in various areas from Greek poetry to math and other sciences. New discoveries were made, mostly in geometry. And there was other thing, which changed the warfare, invention of gunpowder. Now, heavy plating was not useful anymore. Instead the focus of blacksmiths moved to designing better guns. With enhancing of methods of iron casting new longer, bigger, and heavier guns could be made. That brought problems to gunners, it was no longer possible to shoot accurately just by practice. At first, guns were hard to use and they also did not have very long lifespan, therefore new ways how to aim needed to be discovered. Mathematicians quickly reacted to the new needs. They introduced new tools and techniques for measuring. Sights, levels, calipers and gauges were used to measure distances, differences between elevation of the two spots, weight of projectile, amount of gun powder and the elevation of the gun. That provided gunners with lots of valuable data, however it also required high math skills to be able to employ this new measurements in action. The distance was especially difficult task, but the mathematicians come with various new solutions how to accomplish this task. Variety of triangular techniques was used to measure the distance without moving from the spot. New books were printed giving instructions how to calculate required information from the given data, or the tables directly showing the range with given elevation and the amount of gunpowder. Famous scientists like Newton were also interested in this job.
Usage of heavy guns also made old type of castle obsolete. Now the builders of new fortifications had to use math to find the best shape and dimensions for their walls. Walls still had to be able to stand the infantry assault, however they had to be able to survive hits by very heavy artillery shells and also provide the place where to put the defensive guns. Soon new building techniques were developed; fortifications were constructed in the regular polygon shape, with bastions on each of the corners. Mathematicians again had to come up with methods how to find out the sizes and angles in order to construct regular polygon. Many mathematicians employed to improve the warfare made contributions to the mathematics as a science itself.
Since Renaissance mathematics was widely accepted science, and as the technology progressed it become more and more common. Together with physics they presented the engines of the industrial revolution in eighteenth century. Math stood by the invention of steam engine and more or less hidden behind most of inventions of that age. This credit is often being taken away from mathematics, because as the science itself, it does not bring that much direct use in the life, but it is essential for other sciences like physics. During that time several mathematicians made discoveries in the field contributing to the warfare.
Adrien-Marie Legendre Born on 18 Sept 1752 in Toulouse (France), but the family moved to Paris when he was very young. He came from a wealthy family and he was given a top quality education in mathematics and physics at the College Mazarin in Paris.
From 1775 to 1780 he taught with at Ecole Militaire. He then decided to enter for the 1782 prize on projectiles offered by the Berlin Academy. The actual task was stated as follows:-
Determine the curve described by cannonballs and bombs, taking into consideration the resistance of the air; give rules for obtaining the ranges corresponding to different initial velocities and to different angles of projection.
His essay Recherches sur la trajectoire des projectiles dans les milieux résistants won the prize and launched his research career. In 1782 Lagrange become Director of Mathematics at the Academy in Berlin.
He moved back to Paris to work in the Académie des Sciences. However after few years the Académie des Sciences was closed due to the Revolution in 1793 and Legendre had special difficulties since he lost the capital, which provided him with a comfortable income. He began a major task of producing logarithmic and trigonometric tables, the Cadastre. Than the academy was reopen and his career was again continuing very well, he published lot of his work.
However in 1824 Legendre refused to vote for the government's candidate for the Institut National. As a result of refusal to vote for the government's candidate, his pension was stopped and he died in poverty on 10 Jan 1833.
As the time passed, war machinery become more and more sophisticated. Newly invented steel vessels with steam engine, required lot of hidden mathematics work. Now we starting to loose track of mathematics, because it is becoming normal, that great deal of routine and hard numerical work and computing is done by the drawing board, without anybody letting know. But is it the mathematics behind, what is making things work. Mathematics and physics stand together behind the invention of new guns.
The many wars happened in the world during this time period. Every time they were followed with new discoveries in fields related to the war. Education become more common between normal people and mathematics stood behind everything. People just stopped to build things from scratch; they relied on usage of mathematics and physics to see if the things will be able to work, before constructing it. Now we can see and praise mathematics for everything and nothing. It just become normal, that with need of solving new problems mathematicians come with new solutions.
The First World War came. New trench warfare forced both sides to try new fighting techniques. Even more heavy artillery pieces could be made and math helped them to hit the targets. Designing of plane also required hard mathematics work, but no mathematician was directly honored for contributions to war. They just did the work, which allowed others to find their way to fame. Introduction of new ways of counting made lot of things easier and more complicated designs possible. Yet no mathematician won the war medal.
Great leap in technology and science in twenties, and thirties made newly coming war even more violent. Researchers were spending long and long hours with pen and paper making new machines of war.
The Second World War come and brought death to millions, but brought also the golden age of science. Both Allies and Axis powers spent endless effort (and money) to find the ultimate weapon. New heavy bombers, new faster fighter planes, new deadlier explosives, new heavier tanks, new deadlier guns and new bigger ships, all that would be impossible to invent and construct without hours of hard mathematics work. Nuclear physics also brought new requirements to mathematical brains. But this work was done deep behind the front, far away from the battlefield, and it was considered obvious for it to be done. But it could not be done without the hard work of mathematicians discovering the new ways how to make calculations faster. The most in combat use of mathematics were probably connected with new German secret weapon. New rocket was able to deliver explosives on never before seen distance of 200 miles, without any possibility of being shot down. There was only problem, no self-guidance system was yet invented and rocket was simply shot on blind. The wind speed in various heights was measured, missile was fired under the specified angle and than they predicted the path by mathematical calculation, and when they though that rocket is above the targeted city, they sent radio signal and shut down the engine, letting the missile fall on the target. Although German rocket did not change the course of the war, it did great contributions to the upcoming space race.
War ended but even more scientist sit down to the drafting tables to design more horrible and more powerful atomic weapons of mass destruction. The physics got the credit for the atomic device, however without precise calculations no inventions in that field were possible.
Invention of the computer was revolution in the math. It opened the doors to the even more difficult calculations and predictions. Things what took weeks to calculate could be done in the seconds or minutes. And the military did not over see that advantage. The data from the sensors could be processed in real time, allowing humans to concentrate on critical thinking rather than on routine work, so the computers conquered military much sooner, than the normal world. They provided the possibility that machine could partially control itself, making human presence in the dangerous situations unnecessary.
Yet more hidden mathematics fights for us. For example we can look on American missile Tomahawk. Carrying more than ton of conventional payload, missile is given the exact location of the target and than it is fired. It than navigates itself, using the GPS satellites. The path could be controlled by the human operator, however missile is designed to be able to fly on its own and still hit the target hundreds miles away with pinpoint accuracy. But no mathematicians get direct credit for that, it is just the hidden teamwork. We just consider normal that computer works how it works, but there is lots and lots of mathematics and hard work hidden in it.

The future
The future? We can expect that even higher improvements in the speed of computers will make the artificial intelligence possible. Than almost anything can happen, computer piloted device will fight for us and even work for us. There are no limits where technology can go. Until there will be rivalry between the nations, millions will be spent on arms, making sure that every new technology will be tested for possible military use.
But we must not forget that nothing from this could be possible, if the humans in the stone age did not noticed that 1+1 is not 3 but 2, if Pythagoras (and maybe Egyptians before him) did not discovered his famous c2=a2+b2, and if any from almost unknown mathematicians did not discover his/hers small part...


Sources
·http://www.mhs.ox.ac.uk/g eometry/summary.htm; The Geometry of War essay
·http://www.acm.org/pubs/; Application of heuristic search techniques to the problem of flight path generation in a military hostile environment.
·http://www-history.mcs.st -andrews.ac.uk/history/Ma thematicians/Legendre.htm l;
·http://www-history.mcs.st -andrews.ac.uk/history/Ma thematicians/Archimedes.h tml; University of St. Andrews, Scotland
·http://forum.swarthmore.e du/epigone/math-history/; Ubitan D Ambrosio; Military mathematics
·Archimedes's Revenge; P.Hoffman; Fawcett Crest, NY
·Biographical Encyclopedia of Mathematicians; D. R. Franceschetti; Marshall Cavendish, NY


Copyright: (c) Tomas Kubes 2000
Author is giving the permission to use this material for any non-comercial purpose, but allways with displaying the name of the source.