Mathematics and War

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 [ships] 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 ravelin, 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 ravelin was designed to produce crossfire over the ground in front of the neighbouring bastions. If an attacker captured the ravelin, 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.
The author graduated with a BSc Hons in Mathematics from the University Of St. Andrews before moving to Heriot Watt University in Edinburgh where he attained an MSc in Applied Mathematics. The author is also an associate member of the Institute of Mathematics and its Applications. During his study and since leaving university the author has held a number of customer services and marketing roles within major retail organisations.For a free marketing course visit http://www.scottemcclelland.info

Scott E McClelland MSc Applied Mathematics AMIMA.

Authors Homepage: http://www.whoisscottemcclelland.info/

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Calculus eBook Makes Learning Calculus Easier

For most of us learning calculus isn't easy. How many times have you looked up the answer to a problem in the back of the textbook and smacked yourself on the head saying "how did they get that"?

Well those days are over. A new calculus ebook called "Calculus without Limits" approaches the subject in a different way-showing you in detailed step-by-step fashion how to do calculus problems. Discussions of theorems and proofs are avoided in favor of a plain-English approach based on problem solving.

The book is organized to cover most first year college calculus courses. Introductory chapters include limits, derivatives, and the applications of the derivative. Although the book is called "Calculus without Limits", that's just a pun, as calculating limits is covered in detail. Solved examples include basic limit computation, one-sided limits, infinite limits and limits at infinity. Limits of trig functions, exponentials, and logarithms is also included.

Two chapters on differentiation are extensive. The first one lays out the basics of computing derivatives like the power rule and derivative of a constant, then its on to the basic rules everyone struggles with, the product rule, the quotient rule, and the chain rule. The next chapter, applications of the derivative, covers standard topics like max-min problems, Rolle's theorem, and implicit differentiation, along with computing derivatives of the inverse trig and hyperbolic functions.

The second part of the book is primarily devoted to integratin, and this part of the book is really helpful. Each topic is accompanied by a solutions where dozens of integrals are worked out in complete step-by-step detail. The first chapter goes over the basics, teaching you about Reimann sums, basic integration, and integration of trig, exponential, and log functions. Then a really extensive and information-packed chapter shows how to do problems involving u-substitution, integration by parts, trig substitution, partial fractions, and rational functions. There are also chapters on improper integration and integrating powers of trig functions. Each chapter has mutliple examples with detailed problem solutions.

The last chapter covers sequences and infinite series. This includes a clear English explanation of what sequences and infinte series are, along with examples showing how to take limits of sequences. Then all the standard tests for convergence are covered and applied to examples: the root test, the integral test and so on. Power series and Taylor series are also covered.

This calculus ebook is also accompanied by two bonus ebooks. The first one is L'Hopitals rule, a favorite subject of calculus haters. After studying this chapter, they won't believe how easy calculus really is. Finally there is a very helpful ebok that introduces ordinary differential equations and their solutions.

If you've thought calculus is hard or you can't pass it, you might want to give this calculus ebook a try.
David McMahon is a freelance author. He invites you to visit http://calculus-without-limits.com/ to learn more.

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Make Mathematics Fun on Interactive Whiteboards

Interactive whiteboards allow students to actually interact with the subject matter that is being presented and you'll find that there are many great touch screen dynamics available that make them helpful for teaching math. Whiteboards are used with a computer and projector and put the images from a desktop onto the wall where they can be manipulated by touch or by using a pointer. When used along with free teaching resources whiteboards can make a huge difference in a math class.

When using these resources along with an interactive whiteboard, suddenly number is a subject that is more hands-on for students. Instead of being a concept that they can't see, they are now able to see and touch the concept. Everyday ways of using math can be displayed with the whiteboard so they have both an auditory and a visual example of what is going on and why the information is important. The more senses engaged during learning, the more likely students are to retain the information that is being taught.

There is a large selection of free teaching resources available for math teachers. On the web there are many mathematical specific options that can be implemented into the classroom. Along with these resources, PowerPoint is a huge help as well. With this program you are able to create lessons, graphs to help illustrate lessons. Graphs that are in 3D are perfect for teaching math. With interactive whiteboards you can make simple 3D graphs come to life, changing them when things occur to illustrate what is happening.

You'll also find that interactive whiteboards are perfect when you want to play games in the classroom. There are many great math games that help to illustrate tough concepts. Pupils are able to better understand the concepts with games and activities that make them use the material that they have learned. Not only can you use games that you find online, but you can easily create games of your own to play or to stretch advanced students they can create their own mathematical games, a great way to get the most out of pupil's abilities and free teaching resources!
If you are looking for a wide range of Free Teacher Resources, please check out the Promethean Planet website. Teaching resources, teaching software and Interactive Whiteboard Resources are available.

The Promethean Planet site also contains a huge range of Free Teaching resources.

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