Archimedes and the Method of Exhaustion |
Calculus has its roots in the work of the mathematicians and philosophers of the golden age of Greece. Thought was most important to the early Greeks, who longed for an understanding of how the world around them was put together.
Today the most well known of these great men of the mind was a philosopher named Archimedes of Susa, who lived around 225 B.C. It is Archimedes who laid the foundations for what we know today as Integral Calculus, in his development of the ‘Method of Exhaustion’.
Archimedes approximated the areas of otherwise-unmeasurable shapes by approximating them using shapes whose area could be calculated. The most common example of this was Archimedes’ calculation of the area of a circle using inscribed polygons. Archimedes found that by increasing the number of sides of an inscribed polygon, the area of the polygon became closer to that of the circle. By continually increasing the number of sides, 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, and defined the constant of proportionality – what we today know as ‘
’.
Another prominent example of Archimedes work was his method for calculating the area of a parabolic section, using triangles. Using this method Archimedes calculated quite a large number of properties still in use today – including the surface area of a sphere, the volumes of cylinders and cones, and many more. As with the majority of mathematics of the Greek time, Archimedes work was based upon geometric working ratios and area, rather than the algebraic definitions we are familiar with. However the principles that Archimedes established in this method were used many centuries later in developing calculus as we know it today.