David Berlinski’s A Tour of the Calculus provides a thorough yet witty account of what the calculus is and how it came about. He begins at the foundations of mathematics, discussing numbers and how they came to be, and works his way steadily, both historically and conceptually, through to the heart of Calculus as we know it. Taking the perspective of contemporaries of the greatest mathematicians known, Berlinski conveys each idea, from the conception of integers, negatives, fractions to derivatives, limits, and the mean value theorem, with the same excitement and enthusiasm as these mathematicians must have felt upon first discovering such remarkable mathematical concepts. He leaves nothing to be taken for granted—each area of thought is explained in its entirety, leaving nothing to simply be accepted by the reader. Berlinski shows the reader the fundamental theorems of the calculus, including continuity, instantaneous speed, areas under curves, and the like—he simply knows his stuff.
The book as a whole was surprising. Berlinski adds a certain “readability” to his discussion of the calculus that text books often lack. In this sense, A Tour of the Calculus is more of an exploration than an explanation, as the author skillfully uses humor and conversational prose to make the reader feel welcome; facts and theorems are not simply highlighted and listed, but rather are examined in a personal sense. Still, such a comfort level can be deceiving, since it is often possible to overlook a critically important mathematical concept as it is woven into Berlinski’s dialog and conveyed train of thought. But still, Berlinski does a stunning job of making mathematics less distant to the average reader.
Although Berlinski discusses of variety of challenging and nearly incomprehensible topics, I particularly found his explanations of the development of the real number system most interesting, as its existence, or possible lack thereof, is often taken for granted. Yo, Berlinski says, and yo is right—there are actual deduced reasons why these various types of numbers exist or don’t exist. The square root of two, for example, Berlinski shows through a colorful scene between mathematician and well-versed taxi cab driver, not to correspond to any measurable distance. Yet he also shows how such a number was created for use in our mathematics, and ultimately the calculus. Berlinski sets the scene and shows us why. Therein lies his talent. Therein lies A Tour of the Calculus.