The Method of Archimedes: A Mechanical Approach for Calculating Areas and Volumes
Archimedes of Syracuse (c. 287-212 BCE) is often referred to as the greatest mathematician of antiquity. Aside from his wide array of mathematical achievements, his reputation was also formed through his mechanical inventions, such as those of war machines. This paper will examine two of Archimedes' brilliant contributions to mathematics, one of which was famously engraved in his tombstone. In The Method, we will see an often-neglected side to mathematics. It is in this text where he revealed how he discovered the area of a parabolic segment and the volume of a sphere. Through the application of the law of the lever, the science of weights, centers of gravity, and equilibria, Archimedes elevated the method of exhaustion into an art form. His use of mechanical balancing and the summation of indivisibles, techniques which foreshadow methods used in calculus by 2,000 years - provide more than the standard formulas to memorize when working with area and volume. Before examining the inspiration behind these results, we will demonstrate his geometric proof on finding the area of a segment of a parabola. Lastly, we will provide solutions to these problems using calculus. This paper should serve as a nice supplement to anyone interested in teaching or learning about areas and volumes, as it will provide an illustrative and refreshing method of understanding both.
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