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 earlier. 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 your first knowledge of Archimedes was the (possibly apocryphal) story that he 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, while a ship with a steel hull will. Even though this discovery is in the realm of physics, Archimedes is today 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 those horizons.
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 287 B.C., nearly 2300 years ago, in Syracuse. 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 of what we now call limits of infinite sequences, and thus of integral calculus. Archimedes found the area of a circle by inscribing a circle in a regular n-gon and finding a formula for the area of the n-gon. He 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 2/3 the volume of the smallest cylinder enclosing it.
2. What is the surface area of a cube with all eight edges of equal length L? What is its volume? What is the volume of a sphere with the same surface area?