Archimedes: The Last Great Mathematician of the Ancient World
Jan 20 '05
The Bottom Line A high school senior historical essay on the life, death, and achievements of Archimedes of Syracuse. Internet use permitted by author.
Ancient Greece emerged as the first civilization in the world to fully explore and understand the depths of the human mind. Great philosophers, writers, scientists, and mathematicians found their success and livelihood in the Ancient Greek world. However, as its days of glory slowly faded away with the end of the Ancient World, so did these fantastic and unprecedented men. As the Greek city-states began to topple, and these great men begin to disappear, one man still remained to take his place in the history of Ancient Greece. His ways of life beginning to dissipate, Archimedes strode forward against the changing times. He made incredible contributions to both his city-state and to the development of human knowledge. He proved to be master of mathematics and never missed the opportunity to broaden and expand the horizon. Caught in a changing world, he showed loyalty, ingenuity, and courage on the battlefield. His works and achievements gained him fame and honor in his land. Rightfully so, he has been dubbed as the last great mathematician of the Ancient World. Archimedes’ countless discoveries and creations continue to serve importance in our world today.
Archimedes was born in 287 BC in the seaport city-state of Syracuse, Sicily. The only record of his family is his father, Phidias, an astronomer. As Archimedes grew, so did his genius at mathematics. Always looking for any opportunity to build upon his knowledge, he spent most of his time contemplating new problems to solve often forgetting to eat and sleep. Except for his studies in Alexandria under the mathematician Euclid, Archimedes spent his entire life in Syracuse. In time, due to his grand contributions and incredible mathematical skills, he was called “The Master”, “The Wise One”, or “The Great Geometer”. He excelled in few known mathematical concepts of the time, but showed his true worth in the discovery and development of new, more useful ideas.
Archimedes was the first person in history to introduce the concept of a center of gravity. He proposed that the center of gravity is the average location of an object’s weight. To prove his point, he showed that the torque exerted on a lever by weights resting at various spots on the lever would be the same as if all the weights were moved to a single point on the lever, the center of gravity. For example, three uniform weights applied two inches from the lever’s axis point would be the same as one weight applied six inches from the axis point. The torque is the force applied to the lever and the distance from the axis point this force is applied. Archimedes, further proving the validity of his claim, developed techniques to find the center of gravity of objects of uniform density. The objects he found the most success in determining the center of gravity in were triangles, hemispheres, and circular paraboloids. Today, the center of gravity serves a supreme significance in aeronautics. In an aircraft, the amount of mass forward or behind the center of gravity needs to be moved in order to pitch the plane up or down. Often, the center of gravity is located parallel to where the wings get their lift. With these two in the same place, the net torque that each exerts will cancel each other out to keep a stable and flat plane. Remember that torque is force plus mass. If moved too far apart, the torque will become uncontrollable and the results can be disastrous.
The fantastic geometrical capabilities of Archimedes led him to incredible discoveries with pi, parabolas, and triangles. First, Archimedes found in his many trials and experiments, that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. Upon doing so, Archimedes formulated the oldest known example of a geometric series. The series has a ratio of 1/4 and when the geometric sum is calculated, it finally becomes 4/3, the equation he just proved above. In the world of pi, Archimedes, employing his unequaled geometric ability, gave the closest approximation of pi of the time. He calculated pi to be somewhere between 3.14 and 3.15. Today, the first seven terms of pi have been found to be roughly 3.1415926. This approximation was very close indeed considering the limited technology and innovation of the time. Some people even credit him as the founder of pi though, older historical documents make mention of it. Still, his calculation was closer than anyone had come before him and would assist later mathematicians into finding its exact value. In his other investigations of circles, Archimedes found that the ratio of the perimeter to the diameter is equal to the ratio of the area to the radius squared.
One of Archimedes’ most refined and well-known skills was the finding and developing of geometric proofs. He spent his twilight hours with these proofs and by using them, he soon formulated his own method of integration which allowed him to find the area, volume, and surface area of numerous bodies. This method of integration was based quite similarly to the already well established method of exhaustion of that time. The method of exhaustion was used to find the area of an object by using the areas of a sequence of polygons that are smaller than the object but, together occupy the same amount of space. Archimedes’ method of integration was found by his work with proofs of infinitesimals. Infinitesimals are infinitely small numbers that are greater in absolute value than zero yet, smaller than any positive real number. Archimedes was the first person in history to use infinitesimals to find the volume or area of a figure by breaking the figure into infinitely small parts. Specific examples of his work with infinitesimals can be found in his manuscript Archimedes’ Palimpsest. The manuscript combines all his ideas on infinitesimals, torque and the center of gravity. He calls this his mechanical method. Today, the problems addressed in his Palimpsest would be solved by integral calculus. That is why many mathematicians believe that Archimedes took the first step into the discovery and development of integral calculus. However, this topic has been debated. Archimedes himself disbelieved the existence of infinitesimals and could not guarantee to what extent his results were correct. Nevertheless, his findings at the very least suggest a preliminary notion to what is now integral calculus.
Never seeming to become bored or distracted from of his love for geometry, Archimedes pushed forward into discoveries related to spheres and cylinders. Through trial and error, he soon found that the surface of a sphere is 2/3 the surface of a circumscribed cylinder. (The sphere is inscribed in the cylinder.) He then found the area of all segments of the sphere. However, what he would soon come to call his greatest achievement was his discovery that the volume of a sphere is 2/3 the volume of a circumscribed cylinder. It is unclear why Archimedes believed this to be his greatest achievement but so proud of it was he, that he made it his epitaph.
One of the most famous and popular discoveries Archimedes is credited for is his discovery of buoyancy. Buoyancy, or the Archimedes Principle, states that any object completely or partially submerged in a fluid is acted upon by an upward force which is equal to the weight of the fluid displaced by the object. The upward force on the object enables it to float or at least appear lighter. In detail, Archimedes found that if the weight of the object is less than that of the fluid that it would displace, then the object is less dense than the fluid and will float. If the object weighs more than the fluid it will displace, then the object will sink. Today, this discovery is a highly important topic to address in the construction of ships and balloons. The Archimedes Principle has been developed and contemplated over the years in order to find a way for heavy objects, such as a metal ship, to float. The object can float if it retains a suitable shape that keeps a sort of air pocket under the surface of the fluid. This will add density to the liquid and perhaps then the ship will be light enough to float. As buoyancy is Archimedes’ most famous discovery, it has thus been surrounded with two incredibly popular and somewhat comical legends. The first, and probably the most well known tells of how Archimedes’ discovered this concept. One day, as Archimedes lowered himself into the bathtub, he noticed that the amount of water that spilled out of the tub seemed incredibly similar and close to his bodyweight. As he sat in the tub, it finally became all clear to him. He had found buoyancy. Stricken with so much excitement, he leaped from the tub and ran naked through the streets of Syracuse yelling “Eureka!”, “Eureka!” (I found it.) The second legend begins with King Hieron of Syracuse employing a welder to create him a crown of solid gold. When he received the crown, it was the proper weight yet he thought it might be partially silver. Turning to his friend, Archimedes, he asked him to see if his crown was truly solid gold. First, Archimedes took equal weights of silver and gold, immersed them in water, and took observations. Next, he took the king’s crown, an identical pure silver crown, and immersed them in water. The results proved the crown was not solid gold and the fraudulent welder was thrown into prison.
Besides his many contributions to the mathematical world, Archimedes, full of endless strength and desire for knowledge, continued to advance society with his various inventions. Many, of which, have been developed or are still employed today.
The first of his many inventions is the hydraulic screw or Archimedes Screw. The hydraulic screw is a circular pipe enclosing a helix. The lower end of the enclosed helix was dipped into a body of water and then rotated to raise the water up through the pipe. This contraption was incredibly useful to raise water to a more appropriate location. It was commonly used to empty the holds of large ships and for transporting water for irrigation. Today, it has been developed and still used for these purposes as well as for loading grain. The Ballard Locks are a direct example of the ingenious and efficiency of the Archimedes Screw
In relation to his studies in geometry, Archimedes found two ways to lift extremely heavy weights. Weights that would otherwise seem impossible to lift. The compound pulley, the first innovation, was created by tying a rope to the object needing to be hoisted. The other end of the rope was then threaded through a rotatable wheel supported by a wood or metal frame. This end was then pulled to lift the object. The amount of weight that could now be lifted was increased dramatically by this invention. According to legend, Archimedes told King Hieron he could lift any known weight to man, including the world. Unimpressed, the king asked him to pull a fully stocked warship into the harbor. Using his compound pulley, Archimedes smoothly and effortlessly pulled the ship into the dock. Today, adaptations of the compound pulley are used by climbers, mine workers, and sailors. Archimedes’ other innovation was the lever. The lever, like the pulley, was devised to lift heavy weights except by using something small like a plank. As the lever is related to Archimedes’ study of the center of gravity, he gave mathematical proofs to explain its physics. So proud of these two accomplishments and somewhat arrogant, Archimedes once said, “Give me a place to stand and I will move the earth.”
The diversity of Archimedes’ explorations continued as he also made observations in astronomy. Using the stars, planets, and sun, he estimated the total length of the year to within a few minutes. He then properly estimated the length to the sun, moon, and planets. Apparently, he created a device that mapped the sky on a sphere and the motions of the moon, sun, and planets. However, it was uncovered after his death and it is uncertain if he was really the one who created it. None of his works make mention of it. During the First Punic war, he may have invented the odometer which measures the distance traveled, but there is no solid proof of this claim.
As war broke out between Rome and Carthage in the First Punic War, Syracuse found itself stuck in the growing violence. Soon, Syracuse was attacked by the Roman army. General Marcus Claudius Marcellus led the Roman fleet that laid siege to Syracuse. His writings and memoirs would later provide much of the information of Syracuse’s city life as well as a biography of Archimedes. During this bloody and violent time, Archimedes put forth his endless knowledge into the battlefield. His popularity began to soar and patriotism erupted as he formulated the first machines of war.
Archimedes’ first war machine used in the defense of Syracuse was the catapult. The catapult was a missile-hurling construction that inflicted severe damage to buildings and ships. From jagged boulders to rotted animal carcasses, just about anything could be launched from the catapult. So useful was this invention that when Rome finally sacked Syracuse, they took one of Archimedes’ impressive catapults back to Rome. Engineers then examined the catapult and further developed it. The catapult and adaptations of were used all the way through the middle ages up to the French trebuchet.
In attempt to further stem the Roman invasion, Archimedes created the burning mirror. The burning mirror was a large lens that reflected light from the sun at Roman sails and masts. If done correctly, it supposedly set fire to the ship. Surrounded in mystery and exaggeration, this invention seems implausible. It is nearly impossible to focus enough light to set fire to an entire warship. The Claw of Archimedes was the last innovation of war developed by Archimedes. The Claw was used to defend the seaward portion of Syracuse’s city wall from amphibious assault. The exact nature of The Claw is unclear but historical accounts describe it as a being similar to a crane. It was then equipped with a grappling hook that would lift ships out of the water and then either suddenly drop or capsize the ship. Modern day industrial cranes undergone much evolution compared with these archaic constructions.
After two long wars of holding back the Roman invasion, Archimedes’ efforts and machines of war finally failed in 212 BC. As the walls were breached and enemy soldiers stormed the city, Archimedes was killed during the battle, fighting as a true patriot. Syracuse had been the place of his birth, exploration, life, and death. On his grave lies an inscription of pi and the sphere and circumscribed cylinder, his greatest achievement. Today, most of our knowledge of his discoveries and inventions come from his many books and manuscripts that have been recovered over the years. However, biographical information of Archimedes is gained from ancient historians and the account of General Marcellus. In the middle ages, Archimedes’ works were essentially unimportant. The few that could understand his work had no means or opportunity of applying it. Additionally, the majority of his works, including his mechanical method, were lost when the library at Alexandria was destroyed. It was not until 1900 that his mechanical method and Palimpsest were recovered. One can only wonder what effect this early form of calculus may have had on the development of the 16th and 17th centuries. Archimedes helped dig deeper into capability and potential of human minds. His devotion to the betterment of humanity and the will to expand our horizons is most honorable. Without his accomplishments and discoveries, much of what we depend upon today would not exist. By his strength, will, and determination, our world has developed into a better, more informed society.
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