And then Newton said, now let's shr

And then Newton said, now let's shrink the triangle, which is formed by the curve and the coordinate differences, by moving the two points on the curve closer and closer together. As we do so, the straight line between the two points will come closer and closer to the curve, and the error in calculating the speed between the two points will be smaller and smaller. Finally, when we reachthe limit ofinfinitely small difef rences-this is the crucial step!-thetwo points on the curve rrterge into one, and we get the exactspeed at that point. Geometrically the straight line will then be a tangent to the curve.To shrink this triangle to zero mathematically and calculate the ratio between two infinitely small differences is far from trivial. The precise definition of the limit of the infinitely small is the crux of the entire calculus. Technically an infinitely small difference is called a "differential," and the calculus invented by Newton and Leibniz is therefore known as differential calculus. Equations in­ volving differentials are called differential equations.For science, the invention of the differential calculus was a giant step. For the first time in human history the concept of the infinite, which had intrigued philosophers and poets from time immemo­ rial, was given a precise mathematical definition, which opened countless new possibilities for the analysis of natural phenomena. The power of this new analytical tool can be illustrated with the celebrated paradox of Zeno from the early Eleatic school of Greek philosophy. According to Zeno, the great athlete Achilles can never catch up with a tortoise in a race in which the tortoise is granted an initial lead. For when Achilles has completed the distance corre­ sponding to that lead, the tortoise will have covered a farther distance; while Achilles covers that, the tortoise will have advanced again; and so on to infinity. Although the athlete's lag keeps de­ creasing, it will never disappear. At any given moment the tortoise will always be ahead. Therefore, Zeno concluded, Achilles, thefastest runner of antiquity, can never catch up with the tortoise.Greek philosophers and their successors argued about this para­ dox for centuries, but they could never resolve it because the exact definition of the infinitely small eluded them. The flaw in Zeno'sargument lies in the fact that even though it will take Achilles an infinite number of steps to reach the tortoise, this does not take an infinite time. With the tools of Newton's calculus it is easy to show that a moving body will run through an infinite number of infi­ nitely small intervals in a finite time.In the seventeenth century Isaac Newton used his calculus to describe all possible motions of solid bodies in terms of a set of differential equations, which have been known as "Newton's equations of motion" ever since. This feat was hailed by Einstein as "perhaps the greatest advance in thought that a single individ­ ual was ever privileged to make."2