How to Study Mathamatics for College Students and Higher Secondary School students

Mathematics and war have been directly linked since the Babylonian times around 1800 B.C. and will continue to develop together well into the distant future.In 1939, the British crystallographer and science historian John Desmond Bernal wrote: "Science and warfare have always been most closely linked; in fact, except for a certain portion of the nineteenth century, it may be fairly claimed that the majority of significant technical and scientific advances owe their origin directly to military or naval requirements."

This project intends to look at the role of maths and mathematicians throughout the history of warfare, looking specifically at Archimedes and the siege of Syracuse, fortifications and gunnery.

The siege of Syracuse was fought from 214 B.C. to 212 B.C. between the rebellious city of Syracuse, and a Roman army under command of Marcus Claudius Marcellus, sent to put down the city's rebellion. Marcellus attacked the coastal walls of Syracuse with sixty quinqueremes (battleships with five man oar banks) while his co-commander, Appius Claudius Pulcher, attacked the inland walls with ground troops.

"The Romans' wicker screens, missiles and other siege apparatus had been made ready beforehand, and they felt confident that with the number of men at their disposal they could within five days bring their preparations to a point which would give them the advantage over the enemy. But here they failed to reckon with the talents of Archimedes or to foresee that in some cases the genius of one man is far more effective than superiority in numbers."

Archimedes, the Greek mathematician had been King Hiero's military advisor for many years and had well prepared Syracuse for any attack. Archimedes had built ingenious defences including advanced catapults, scorpions and trebuchets, Polybius describes some of these defences

"Archimedes had constructed artillery which could cover a whole variety of ranges, so that while the attacking ships were still at a distance he scored so many hits with his catapults and stone-throwers that he was able to cause them severe damage and harass their approach. Then, as the distance decreased and these weapons began to carry over the enemy's heads, he resorted to smaller and smaller machines, and so demoralized the Romans that their advance was brought to a standstill."

Archimedes also devised the 'Archimedes Claw' and the 'Archimedes' Death Ray'.
The 'Archimedes claw' was in essence a large crane, at the outer walls of the city, equipped with a grappling hook that could lift attacking ships partly out of the water, and then either cause the ship to capsize or suddenly drop it. Plutarch depicts the devastating effects of the 'claw'

"Other were seized at the bows by iron claws or by beaks like those of cranes, hauled into the air by means of counterweights until they stood upright upon their sterns, and then allowed to plunge to the bottom, or else they were spun round by means of windlasses situated inside the city and dashed against the steep cliffs and rocks which jutted out under the walls, with great loss of life to the crews. Often there would be seen the terrifying spectacle of a ship being lifted clean out of the water into the air and whirled about as it hung there, until every man had been shaken out of the hull and thrown in different direction, after which it would be dashed down empty upon the walls."

Lucian wrote that during the Siege of Syracuse, Archimedes repelled an attack by Roman forces with a burning glass. Archimedes "constructed a kind of hexagonal mirror, and at an interval proportionate to the size of the mirror, he set similar small mirrors with four edges, moving by links and by a kind of hinge, and made the glass the centre of the suns beams...So after that, when the beams were reflected into this, a terrible kindling of flame arose upon the ships, and he reduced them to ashes. Thus by his contrivances did [Archimedes] vanquish Marcellus."

Archimedes magnificent inventions were so effective that "at a council of war the decision was reached to abandon the assault, as all attempts were baffled, and to confine operations to a blockade by sea and land."

However in 212 B.C. while the inhabitants were participating in a festival to their goddess Artemis, the Romans managed to get over the walls and the onslaught began.
The city of Syracuse fell and "was turned over to the troops to pillage as they pleased."
It was at this time that Archimedes was killed while "carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him," much to the distress of Marcellus who "pleased with the man's exceptional skill, he gave out that his life was to be spared, putting almost as much glory in saving Archimedes as in crushing Syracuse."

The fact that the besieging army's commander is distraught at the death of the man who masterminded the destruction of his forces shows the incredible impact that this great mathematician and his ingenious inventions had.

Fortifications are military constructions and buildings designed for defence in warfare. The Renaissance period was the golden age of fortification. During these 400 years, fortification achieved the stature of art and science. Fortification's most striking achievement was the construction of many impressive fortresses found all over the world.

During the 15th century, a revolution in the development of arms, in the form of the canon made it necessary for fortifications and fortresses to be made stronger and harder to be breached. The original medieval castle walls were high and constructed to prevent the scaling of the castle wall, by means of ladders.

However with the new developments in artillery, the high walls were easy targets and simply shattered under the accuracy and strength of the cannon. This necessitated a change in the design of fortifications. In the early 1500's, a simple square with bastions was the first, most basic design. However, the small flanks and sharp angle characteristics of this design produced cramped interiors and hence limited the troops and cannon that could be garrisoned there. The square bastion design was quickly replaced by polygonal shaped fortifications. These polygonal walls offered more sides and were clearly easier to defend. It also allowed for expansion to achieve even greater interior space - this was carried out by increasing the number of bastions and the length of the enclosing walls. Although most theories for a bastioned fortress were based on geometrical designs, nature often called for readjustments in the original design.

Many fortifications had to accommodate terrain with mountains, swamps and rivers and were hence constructed as irregular polygons.By the end of the 16th Century, the system of fortification was quite well developed and new elements were added to the bastion design. Defences located the near the castle walls, but behind the enclosing ditch were developed, known as Outworks. A ravelling, a free-standing triangular outwork equidistant between the bastions, was situated almost as an island in the moat in front of the castle wall. The ravelling was designed to produce crossfire over the ground in front of the neighbouring bastions. If an attacker captured the ravelling, he would find himself isolated in the middle of the ditch, and in the midst of vicious flanking fire. The defensive fortification structured in this way facilitated transportation of cannon and ammunitions from one defensive point to another during period of siege. The final shape of the new defensive structures resembled a star, and for this reason they were known as star forts.
Gunnery became a subject for practical mathematics in the 16th century. Printed books and new mathematical instruments dealt with the measurement of shot, the elevation of guns and mortars, and the calculation of the range of fire. The prediction of range in relation to the elevation of a gun was considered the pinnacle of artillery as a mathematical science.
Tartaglia, who had experimented with almost every type of cannon in existence in Europe, had a great deal of data on cannons and so was able to develop the first ballistic firing tables; these tables were instrumental in educating gunners and developing artillery as a precise military tool.
The next major contribution to ballistics came from Galileo who showed that the acceleration due to gravity is the same for all objects and air drag was the factor that changed their descent velocities. He was able to determine that ballistic trajectories are parabolic.

He theorized that the velocity of a projectile was related to the drag acting upon the projectile.
Sir Isaac Newton made the most important contributions to ballistics and the study of aerodynamic drag. In Principia, he derived formulas and explained the mechanics of ballistics. He concluded that the retarding force (drag) that acts on a projectile through air is proportional to the density of air, the cross sectional area of the projectile and approximately the square of its velocity.

In conclusion it can be seen that maths and mathematicians play a key role in the art of warfare, and the two disciplines directly affect the development and style of one another. Also maths and war have continued to work together to advance civilisation and provide protection for the free world, seen through the developments of the computer and other types of machinery constructed for military application by mathematicians. Truly the power, influence and wealth of the armed forces offered and continues to offer fantastic opportunities for mathematical advancements.