Is God a Mathematician_ - Mario Livio [29]
The basic idea behind the process of integration is quite simple (once it is pointed out!). Suppose that you need to determine the area of the segment of an ellipse. You could divide the area into many rectangles of equal width and sum up the areas of those rectangles (figure 14). Clearly, the more rectangles you use, the closer the sum will get to the actual area of the segment. In other words, the area of the segment is really equal to the limit that the sum of rectangles approaches as the number of rectangles increases to infinity. Finding this limit is called integration. Archimedes used some version of the method I have just described to find the volumes and surface areas of the sphere, the cone, and of ellipsoids and paraboloids (the solids you get when you revolve ellipses or parabolas about their axes).
In differential calculus, one of the main goals is to find the slope of a straight line that is tangent to a curve at a given point, that is, the line that touches the curve only at that point. Archimedes solved this problem for the special case of a spiral, thereby peeping into the future work of Newton and Leibniz. Today, the areas of differential and integral calculus and their daughter branches form the basis on which most mathematical models are built, be it in physics, engineering, economics, or population dynamics.
Archimedes changed the world of mathematics and its perceived relation to the cosmos in a profound way. By displaying an astounding combination of theoretical and practical interests, he provided the first empirical, rather than mythical, evidence for an apparent mathematical design of nature. The perception of mathematics being the language of the universe, and therefore the concept of God as a mathematician, was born in Archimedes’ work. Still, there was something that Archimedes did not do—he never discussed the limitations of his mathematical models when applied to actual physical circumstances. His theoretical discussions of levers, for instance, assumed that they were infinitely rigid and that rods had no weight. Consequently, he opened the door, to some extent, to the “saving the appearances” interpretation of mathematical models. This was the notion that mathematical models may only represent what is observed by humans, rather than describing the actual, true, physical reality. The Greek mathematician Geminus (ca. 10 BC–AD 60) was the first to discuss in some detail the difference between mathematical modeling and physical explanations in relation to the motion of celestial bodies. He distinguished between astronomers (or mathematicians), who, according to him, had only to suggest models that would reproduce the observed motions in the heavens, and physicists, who had to find explanations for the real motions. This particular distinction was going to come to a dramatic head at the time of Galileo, and I will return to it later in this chapter.
Figure 14
Somewhat surprisingly perhaps, Archimedes himself considered as one of his most cherished accomplishments the discovery that the volume of a sphere inscribed in a cylinder (figure 15) is always 2/3 of the volume of the cylinder. He was so pleased with this result that he requested it be engraved on his tombstone. Some 137 years after Archimedes’ death, the famous Roman orator Marcus Tullius Cicero (ca. 106–43 BC) discovered the great mathematician’s grave. Here is Cicero’s rather moving description of the event:
Figure 15
When I was a quaestor in Sicily I managed to track down his [Archimedes’] grave. The Syracusans knew nothing about it, and indeed denied that any such thing existed. But there it was, completely surrounded and hidden by bushes of brambles and thorns. I remembered having heard of some simple lines of verse which had been inscribed on his tomb, referring to a sphere and a cylinder modeled in stone on top of the grave. And so I took a good look around all the numerous tombs that stand beside the Agrigentine Gate. Finally I noted a little column just visible above the scrub: