Abstract

Most historians and mathematicians consider Archimedes of Syracuse as one of the greatest mathematicians of all time. His achievements in mathematics were outstanding. He contributed many theorems in geometry that have applications in other fields such as engineering, physics, and astronomy. One of which he is most famous for is his theorem that measures the weight of a body immersed in a liquid (called the Archimedes’ principle). Archimedes is great both in theory and application as he did not only gain reputation for being a mathematical genius but also as an inventor.

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Archimedes’ Geometry and Inventions

Introduction

Archimedes was a native of Syracuse, Sicily. His date of birth was dated 287 BC based from John Tzetzes’ assertion that Arhimedes lived for 75 years. Little is known about his family and his early life but he had written in his The Sandreckoner that his father was named Phidias and that he was an astronomer. Other information about his life was written by Heracleides, which was Archimedes’ friend, but sadly, the work was lost.

We can not tell as it is unknown, whether he had a wife or children. Some of his writings showed that he probably spent part of his life under Euclid’s contemporaries in Alexandria, Egypt. He had communicated with mathematicians in Alexandria, especially Conon of Samos and Eratosthenes of Cyrene. He sent them mathematical theorems and problems to his own delight (and revenge). One of the problems which he sent to them was the Cattle Problem presented in a forty-four-line poem that took a number of years before it was completely solved. It was in 1880 that A. Amthor finally found the general solution.

Other than being a mathematician, Archimedes also invented many mechanical devices that were used mostly in the defense of his native city upon the invasion of Roman soldiers. One of which is the Archimedes’ screw which is still used in some parts of the world today. This is a pump made of a revolving screw inside a cylinder that is turned by hand and is able to transfer water from a lower position to a higher one. He also designed block and tackle pulley system, as describe by Plutarch, which enabled people to lift heavy objects with lesser amount of force.

Perhaps Archimedes’ most popular discovery is the Archimedes’ principle allowing people to compute the volume of an irregularly-shaped object. This principle states that “a body immersed in a fluid is pushed upward by a buoyant force that is equal to the weight of the fluid displaced.” This is useful in hydrostatics and is known to be the Archimedes’ principle. According to Vitruvius, Archimedes made this discovery upon the request of his friend, King Hieron II, that the new gold crown in the shape of a laurel wreath be tested if it were made of pure gold or not. However, he must do it without changing the crown’s original form. If it were only a cube, he would have easily measured its volume and thus the density. It was when he was taking a bath when he observed that water is displaced whenever a body is immersed on it. To his excitement, he jumped off the tub and ran across the streets naked shouting “Eureka (I have found it)!”

Archimedes died in the hands of a Roman soldier who killed him despite the announcement of General Marcus Claudius Marcellus that he should not be harmed. This was the time when Syracuse was captured in the two year long siege during the Second Punic War. There were three accounts on his death. The most popular was that he was working on a mathematical theorem when the soldier rushed in. The soldier obliged him to come but he refused. The impatient soldier drew his sword and struck Archimedes to death. Marcellus, upon knowing this event, was mad and had the soldier executed. Archimedes was buried according to his will. He wanted that his tomb be decorated with his favorite mathematical diagram: a sphere enclosed in a cylinder of the same height and same width. This was what Cicero, a Roman orator who served as quaestor (Roman government official) in Sicily, also found in Archimedes’ neglected tomb.

Education

Archimedes probably went to Alexandria, Egypt to study mathematics under the disciples of Euclid. As we all know, Euclid is called the Father of Geometry. His principles of geometry were accepted as the backbone of mathematics. Euclid did not only discover and proved mathematical theorems by logical deduction but he also inspired mathematicians to follow his path as shown by Archimedes. Euclid’s influence to Archimedes, though indirect, can be observed from the latter’s extraordinary love in solving and proving theorems, especially in geometry. He enjoyed geometry very much as Plutarch described this saying,

“Oftimes Archimedes’ servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. And while they were anointing of him with oils and sweet savours, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.”

If only we have that enthusiasm, we would have been able to discover things beyond our imagination just like Archimedes. But we must accept the fact that not all of us are gifted with talents in mathematics. There are those who enjoy mathematics very much while others are finding it hard to solve even the simplest operation in polynomials.

Works in Mathematics

We all know Archimedes’ enthusiasm in mathematics, particularly in geometry. He had extensive and intensive studies whose findings still find its applications in modern mathematical sense. He had written several books containing results of his experiments with the geometrical shapes as he tried to solve some problems. His works that have survived until today are the following: On plane equilibriums (two books), Quadrature of the parabola, On the sphere and cylinder (two books), On spirals, On conoids and spheroids, On floating bodies (two books), Measurement of a circle, and The Sand Reckoner. The latest addition to these is The Method discovered by a classical philology professor named J.L. Heiberg from University of Copenhagen. The ordering of these works were followed from the one suggested by Heath (1921) in his book.

In his On plane equilibriums, he worked on finding center of gravity of plane figures (book 1) such as parallelograms, triangles, and trapezium and also finding the center of gravity of a line segment of a parabola. The Quadrature of the parabola is dedicated in finding the area of a segment of a parabola. In the On the sphere and cylinder, he was able to show that the sphere has four times the surface of a circle; he also proved in the book that the surface and volume of a sphere is two-thirds that of the surface and volume of a circumscribed cylinder. On spirals contains Archimedes’ definition of a spiral and its fundamental properties that connects the length of the radius vector to the angles where it revolved. In On conoids and spheroids, he recorded his works on paraboloids and hyperboloids of revolution and spheroids formed from the rotation of an ellipse through its axes. On floating bodies contains the Archimedes’ principle. Also included in this is his study on the stability of objects with different shapes and specific gravities. His computation on the exact value of π is written in his Measurement of the Circle. In his The Sandreckoner, he proposed a number system that could express large numbers up to 8 x 1063 in our modern notation. He also cited in this work the achievements of Aristarchus, Eudoxus and Phidias (his father).

Perhaps his most popular work is On floating bodies where Archimedes introduced his famous Archimedes’ principle, the principle that has many applications particularly in hydrostatics. As we can observe, most of his works were on areas and volumes of shapes, that is, the precursor of modern calculus. He was able to use infinitesimals in order to compute for the area of curves and planes. He was also able to compute the volumes of solids, especially the paraboloids and hyperboloids of revolution. However, these bodies have regular shapes and so the challenge was to take the volume (and surface area) of an irregular body. Thanks to his friend, King Hieron II, that he was able to think for an answer and succeeded upon developing the solution. He did this when he found out that when an object is place in a liquid, it displaces from its initial height. What he did was to find a way to compute for the specific gravity and finally, the volume of the object by utilizing the excess liquid. Thus, Archimedes’ principle is very useful when we want to know the volume of objects that is of irregular shape. In Physics, we are also able to compute for the density of the matter by getting the ratio of mass into volume.

Inventions and Discoveries

Other than being a mathematician, Archimedes is also known for his inventions and discoveries. He gained a reputation not just for being an “expert” in mathematical ideas but also as an inventor of machines used to fulfill the needs of his countrymen, especially in the defense of his hometown during the Roman invasion. They were just very effective in battling the incoming troops lead by General Marcellus. As Plutarch puts:

… 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 beak 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. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.

Some of Archimedes’ inventions include the Archimedes’ screw and the Claw of Archimedes and he also designed a block and tackle pulley and the Syracusia, a huge ship that was designed for luxury travel, carrying supplies, as well as a warship. He was also credited to have improved the catapult’s power and accuracy and also the actual invention of the odometer that was used to measure distance traveled.

Conclusion

The reason why Archimedes is my choice is that he was a very brilliant mathematician. He made extensive and intensive research on many subjects, especially in Geometry. Considering his time, he already had the idea of exponents (as shown in his Cattle Problem). He devoted his life in research and problem solving which benefited later generations of mathematicians and eventually, ordinary people. Today, we may not remember him but we should note that his contributions to physics, engineering, astronomy, among others, mathematics were indispensable and without them, our modern concept of mathematics might not be that “advanced” and computers might come later than we thought. Aside from this, Archimedes is my choice because he was able to use his knowledge in mathematics in everyday life. He was able to build machines that aided his fellowmen from generation to generation. Some of it is still used today in some parts of the world. Although he did not succeed in defending Syracuse from its invaders, he did enjoy a lot playing with his inventions and seeing them working. His knowledge in theoretical and practical living is enormous. We owe Archimedes a lot for sharing what he knew.

References

Brodie, S.E. (1980). Archimedes’ Axioms for Arc-Length and Area. Mathematics Magazine, 53(1), 36-39.

This journal article describes Archimedes proofs on measuring the length of a segment in parabola as well as computing for the area using infinitesimals.

Gow, M. (2005). Archimedes: Mathematical genius of the ancient world. NJ: Enslow Publishers, Inc.

This book contains information about Archimedes’ education by examining the places in Alexandria and Syracuse, where he came from. It also describes his works in geometric and other fields.

Heath, T.L. (1921). A history of Greek Mathematics, II. Oxford: Clarendon Press

This book, part of a two-volume work, is considered by some as the best English language source on Greek Mathematics. It is an indispensable reference for those who are interested in Greek contribution to Mathematics. It discusses Euclid’s Elements and also suggested an ordering of Archimedes’ geometry.

Knorr, W.R. (1978, September) Archimedes and the ‘Elements’ : proposal for a revised chronological ordering of the Archimedean corpus. Archive for History of Exact Sciences, 19 (3), 211-290.

This article reexamine the existing orderings of Archimedes’ works by using several indicators which showed how weak the foundations of the accepted orderings were and how Knorr’s re-ordering differ from the accepted one.

Osborne, C. (1983). Archimedes on the dimensions of the cosmos. Isis, 74(2), 234-242.

This article provides concise information on Archimedes’ mathematical astronomy. Included in this text is an explanation of Archimedes’ findings in The Sandreckoner.

Rorres, C. (2004). Completing Book II of Archimedes’s On Floating Bodies. The Mathematical Intelligencer, 26(3), 32-42.

This article contains the Archimedean principle on floating bodies, especially his principle in hydrostatics. It provides a comprehensive study on different objects with different shapes, sizes, specific gravity, and other variables.