At an elementary level, it is easy to consider mathematics merely as a game of numbers, measurements, and figures without a valid practical application. However, the advent of mathematics was premised on seeking to solve practical problems faced by the early man (Kilpatrick, 2014). Solving practical problems is still the main premises for the relevance of mathematics in the contemporary world (Lewinter & Widulski, 2002). Many mathematical problems that are so easy to solve today when major formulas have been arrived at and availed were quite a feat to arrive at in the first instance (Kilpatrick, 2014). The arrival at of these formulas is indeed major mathematical happenings. Two major happenings took place between the 4 th and 3 rd centuries BCE regarding one of the most important areas of mathematics to wit establishing the areas and volumes of complex shapes (Kilpatrick, 2014). These processes are relatively easy now with the advent of calculus which was only developed in the 17 th century CE. The two major happenings during this era was the elucidation of the process of exhaustion by Eudoxus of Cnidus and the invention of Pi by Archimedes who advanced on the processes elucidated by Eudoxus.
Save for Archimedes, Eudoxus is defined as the greatest mathematician of the era within which the 4 th and 3 rd centuries BCE fall under (Kilpatrick, 2014). By the age of 23, Eudoxus has arguably received any possible education that a school could have taught and reverted to his own research mainly in areas of mathematics and astronomy (Kilpatrick, 2014). He was born around 390 BCE at Cnidus also called Knidos, currently within Turkey. He undertook most of his studies locally but also travelled to study in Tarentum and also the city of Athens, which was then a great learning center having inter alia the academy of Plato. History records that he was also a politician and a legislator in Cnidus (Kilpatrick, 2014).
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Among the areas of mathematics that he focused on was proportions and the process of exhaustion. Albeit the idea of exhaustion has been traced back to 5 th century by mathematician Antiphon, researchers seem to agree that Antiphon did not really understand the process (Lewinter & Widulski, 2002). It was Eudoxus who not only properly understood it but also applied it in the finding of areas and volumes of complex objects. The significance of this discovery lies in the fact that from a practical perspective, rarely with a regular shape like a square, triangle or rectangle be found or needed. Most natural and even artificial objects have extremely complicated shapes or circles. Their areas and volumes are almost always impossible to establish before Eudoxus (Lewinter & Widulski, 2002).
The process of exhaustion is premised on the proof by contradiction also referred to as reductio ad absurdum. In simple terms, it entails establishing a quantity that is impossible to calculate by proofing what it is not. If an amount is proofed not to be greater than a certain assumed amount and not lesser than the said assumed amount, then the assumed amount is the correct one (Lewinter & Widulski, 2002). The area of one region is first compared with the area of another region, assumed to be greater and proofed not to be greater than it. After it is also proofed not to be smaller than it, then the area has been established through exhaustion. Through this process, Eudoxus was able to establish the areas and volumes of many irregular objects. It is upon his work that the calculus is premised (Lewinter & Widulski, 2002).
With the mystery of establishing the area and volumes of irregular objects solved by Eudoxus, the mystery of doing the same about circular objects remained. This mystery was solved by Archimedes who expanded on the works of his predecessor Eudoxus. Archimedes was born in Syracuse, in what is today Sicily, around 287 BCE just over a century after Eudoxus (Lewinter & Widulski, 2002). Little is known about his private life save that he seemed to have been close to the royal family of Syracuse. It is also known that Archimedes died at the age of 75 after his hometown was overrun by the Romans (Saito, 2013). Apparently he was so engrossed in a geometrical problem that he ignored the summons of a Roman soldier. The soldier is said to have stubbed him to death. Another important historical fact about Archimedes was the practical application of mathematics in an era and region where theory was of main focus. This practical application resulted to the Archimedes principle, the Archimedes Screw and many other practical tools and weapons (Saito, 2013).
Many of the mathematical and engineering feats by Archimedes are legendary even by today’s standards. However, one of his mathematical feats was so important to him that he requested that it be engraved at his tomb. This feat entails a marvelous expansion of the process of exhaustion and using it to establish the volume of a sphere. It also relates to the establishment of one of the most common and important mathematical formula knows as Pi (Saito, 2013). His works in this arena were reduced into a book referred to as On the Sphere and the Cylinder published in two volumes around 225BCE (Saito, 2013). The book shows the first known instance when the area of a circle, the surface area of a sphere, spherical volume and dimensions of a cylinder were calculable.
To find the area of a circle, Archimedes used polygons slightly larger than the circle superimposed upon the circle (Lewinter & Widulski, 2002). Using exhaustion he established that the area of the polygon, divided by the square of the radius of the circle would always be congruent to 3+10/71 (Saito, 2013). This became the premises for Pi and a major breakthrough in understanding the dimensions of a circle. The invention that he was so proud of as indicated above was an expansion of the same and involved arriving at the relationship between a sphere and a cylinder in order to establish both the surface area of a the sphere and the volume of the ball within the sphere (Lewinter & Widulski, 2002).
He established that the volume of a cylinder whose height and diameter are equal is also equal to three halves of sphere which shares the same diameter. In retrospect, the volume of a sphere is two thirds of the cylinder whose base and diameter are equal to the diameter of the circle (Saito, 2013). From a practical perspective, when the diameter and height of a cylinder are equal to the diameter of a sphere, the sphere can be accurately inscribed within the cylinder. It is this picture of a sphere perfectly inscribed in a cylinder that was found by Marcus Tullius Cicero during the 2 nd century CE engraved in the tomb of Archimedes (Saito, 2013).
Due to the advancements in the field of mathematics made during the renaissance period around the 17 th century, calculations based on the discoveries of Eudoxus and Archimedes have been reduced to elementary mathematics. These developments, including calculus were however only made possible through the works of these two great men who lived over two millennia ago. Without the many advantages that are enjoyed today, Eudoxus and Archimedes sought to and managed to achieve the impossible by stretching the capacities of the process of exhaustion. Eudoxus arrived at the means of establishing the areas and volumes of irregular objects while Archimedes did the same for circular objects thus coming up with Pi. These are two exponentially great mathematical feats, achieved during the same era.
References
Kilpatrick, J. (2014). History of research in mathematics education. In S. Lerman Encyclopedia of mathematics education (pp. 267-272). Netherlands: Springer
Lewinter, M., & Widulski, W. (2002). The saga of mathematics: A brief history . United States: Prentice Hall.
Saito, K. (2013). Archimedes and double contradiction proof. Lettera Matematica , 1 (3), 97-104.
