Fall 2005 Edition

**Aaron Timpson**

Expository 1010 1st place

To be classified as a genius of mathematics, a person would have to discover and add a new concept to the world of mathematics, greatly simplify a previous concept, or have his or her ideas/proofs accepted and never proven wrong. Examples of such genius mathematicians are Pythagoras, Euclid, and Newton. In good company with these brilliant mathematicians, Archimedes discovered so many principles in the world of mathematics that his geometrical discoveries (circles, spheres, and cones) and inventions (pulleys and levers) laid the foundations for modern day mathematics and physics. Because of his accomplishments, Archimedes can be considered to be a genius of mathematics.

Today, it would be almost impossible for anyone to qualify as a genius of mathematics, but in the time of Archimedes discovering a new concept identified a mathematician. In his famous theorem, for example, Pythagoras asserted the relation of squares on the sides of a right triangle to one another. Euclid found the way area and lengths of sides related to any sided polygon. Discoveries can also come from the simplification of a previous concept or theorem. For example, Pythagoras derived his theorem from the knowledge he gained at Plato’s Academy, which Euclid used to prove his polygonal geometry. Archimedes used Euclid’s works to derive π, by the “method of exhaustion.” As one commentator notes, “It was Archimedes who laid the foundations for what we know today as *integral calculus*, in his development of the ‘method of exhaustion’” (Harding 2). Newton, recognizing the genius behind the “method of exhaustion,” built upon the idea, and created calculus. Through calculus, Newton showed that Pythagoras, Euclid, and Archimedes were correct in their discoveries and have not yet been proven wrong.

Archimedes also used this method to calculate the area of a circle by using the area of polygons. “By continually increasing the number of sides [a polygon has as the polygon changes from a triangle, to a square, to a pentagon, etc.], he ‘exhausted’ the circle by reducing the area of the circle not covered by the polygon. He established that the area of the circle was exactly proportional to the square of its radius,” a commentator explains, pointing out that Archimedes thus defined this constant of proportionality π (Harding 3). As soon as Archimedes had derived this constant, he began working on the geometry of three dimensional figures. He was “the first, in fact, to resolve the complicated geometry of circles, spheres, cones, conoids, spheroids, and spirals” (Frye 53).

Not only did Archimedes derive his constant of proportionality, but he used it to prove the volume calculations of some three dimensional figures (spheres, cones, etc.), which opened up an entirely new mathematical field, because his geometry was not confined to the two dimensional plain. Archimedes greatly expanded upon the knowledge he gained from Pythagoras’s and Euclid’s work, by opening up mathematics to a three dimensional plane. Even though he did not greatly simplify a previous concept of mathematics, he moved mathematics into the three dimensional world.

By developing and expanding his knowledge to a three dimensional plane, Archimedes has been titled as a physicist as well as a mathematician because of the many things he invented and the principles he discovered. Among these principles is that of the pulley, which is a circular device with a rope around it that is used to change the direction of force acting on an object; this change of direction allows the force being exerted on the rope to be focused on an object to pull it in a specific direction. Another principle is that of the lever, which is a bar or piece of wood with one of its ends inserted under a heavy object with a force pushing down on the other end of the lever, against a pivot point, to move the object. Archimedes became the “first physicist to describe the principle of the lever” (Frye 54). He demonstrated these principles (pulley and lever) to the common people by attaching a boat to a pulley and lever system, and he pulled the boat onto the shore. It is from demonstrations like this that King Heiro II, of Syracuse, Sicily, noticed the discoveries Archimedes was making and called on him to construct a defense system for the city of Syracuse (Archimedes’ home town) against the Romans. Archimedes used the principles of pulleys and levers to engineer and invent his machinery. One of these inventions is known to posterity as Achimedes’ Crow,” [which] sank not only ships but Roman morale as well. Known for their “iron hands,” Archimedes’ most fearsome weapons also utilized beams that could be swung out over the water. These arms, however, dropped hooks rather than stones. Controlled by an operator behind the walls, the grapple would clutch at the ship until it grasped the prow. Then, driven by heavy weights that forced down the other end of the beam, the claw would spring back up and yank the ship by the prow, practically standing it on its stern before letting loose and dropping it. Some ships fell on their sides. Many took on water. Others capsized. (Frye 54)

These weapons made it very difficult for Rome to attack the city from the water, which was their greatest battle tactic of the time. The Romans were forced to attack by land, but Archimedes invented other contraptions like the catapult, a device that is used to hurl big boulders across a distance, to keep the Roman army outside the city walls. Rome attempted to conquer the city many times, but Archimedes continued to show that he could withstand the Roman army by utilizing his inventions. From all of the mathematics used in his inventions, Archimedes completely intertwined mathematics and physics.

Even though many mathematicians today achieve their PhD in mathematics, only rarely can any of them can come up with their own undiscovered theorems as Archimedes did. If a modern-day mathematician were to discover something remarkably true, he or she would be classified as a genius. The main reason that men and women are not achieving such world changing discoveries in mathematics is because so many concepts have already been unveiled; in ancient days, there were so many concepts that were yet to be disclosed that mathematicians were painting on a clean canvas. Of course, such opportunity came with a challenge. Archimedes not only had to develop the geometry he did, he had to do so on the foundation of the work of only two other mathematicians-- Pythagoras and Euclid.

The world of mathematics has been greatly improved since the ancient days because of all the discoveries that Archimedes made throughout his life time. Not only did Archimedes discover so many true concepts and prove that they were true, but all of his work has led to furthering the over-all knowledge of mathematics that we have today. This is why Archimedes is known not only as a mathematician, but as a genius of mathematics.

**Works Cited:**

Frye, David. “Archimedes’ Engines of War.” *Military History* Oct 2004: 50-56. *SIRS Researcher*. SUU Sherratt Lib., Cedar City, UT. 19 Oct 2005 <http://web8.epnet.com/ >.

Harding, Simon, Paul Scott. “The History of the Calculus.” *Australian Mathematics Teacher* Jun 2005: 2-5. *SIRS Researcher*. SUU Sherratt Lib., Cedar City, UT. 19 Oct 2005 <http:// web8.epnet.com/>.