CALCULUS Understanding Its Concepts and Methods

Archimedes of Syracuse (287--212 BC) --- Historical sketch

Much has been written about how calculus was invented in the 17th century by Isaac Newton and Gottfried Leibniz. However, a good argument can be made that calculus began with Archimedes some 1900 years before that. In his enjoyable little book The Great Mathematicians, Herbert Westren Turnbull states,

"Archimedes, one of the greatest of all mathematicians, was the practical man of common sense, the Newton of his day, who brought imaginative skill and insight to bear upon metrical geometry and mechanics, and even invented the integral calculus."

Perhaps you have heard the story that Archimedes ran naked through the streets of Syracuse (in Sicily, not New York) yelling Eureka , "I have found it," signalling his bathtub discovery of the principle of buoyancy, that the force of buoyancy equals the weight of the liquid displaced. That is why most wood will float in water, as will most humans, a brick of steel won't, but a ship with a steel hull will. Even though this discovery is in the realm of physics, Archimedes is regarded by many as the greatest mathematician of the ancient world, and one of the greatest mathematicians of all times. While he knew and used the works of Euclid, Archimedes ventured far beyond that.

It is generally accepted that Archimedes was 75 when he was killed by a Roman soldier in 212 B.C. If that is so, then he was born in Syracuse in 287 B.C., nearly 2300 years ago. One story is that he was the son of an astronomer named Phidias who was interested in the ratio of the diameters of the sun and the moon.

Archimedes computed the area and circumference of a circle and many other geometric figures. He used what he called his method which is the basis for the integral calculus. As for the area of a circle, Archimedes found how to compute the volume of a sphere by showing that it is two-thirds the volume of the smallest cylinder enclosing it (see Problem 1 below). He was so proud of this result that he wanted it engraved on his tombstone.

There are several books on Archimedes. One very accessible book is Archimedes: What Did He Do Besides Cry Eureka, by Sherman Stein, and a more mathematically comprehensive book is Archimedes, by E. J. Dijksterjuis, both (and others) available through bookstores.

1.

Deduce the formula for the volume of a sphere by using the statement that its volume is the volume of the smallest cylinder enclosing it.

2.

What is the surface area of a cube with all eight edges of equal length ? What is its volume? What is the volume of a sphere with the same surface area?

Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.