Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these things possible and in doing so finds a correspondence between real numbers and the real world is the subject of this dazzling book by a writer of extraordinary clarity and stylistic brio. Even as he initiates us into the mysteries of real numbers, functions, and limits, Berlinski explores the furthest implications of his subject, revealing how the calculus reconciles the precision of numbers with the fluidity of the changing universe.

"An odd and tantalizing book by a writer who takes immense pleasure in this great mathematical tool, and tries to create it in others."--New York Times Book Review

but inscrutable creations of the imagination, the silken thread that binds together the vagrant world’s far-flung concepts. Fabulous formulas bring anarchic speed panting to heel and make of its forward rush a function of time; the wayward area underneath a curve is in the end subordinated to the rule of number. Speed and area, the calculus reveals, are related, the revelation acting like lightning flashing between two distant mountain peaks, the tremendous flash of light showing in the moment

of Zeno, mad, bad, and dangerous to know. And there is not much else. But the analysis of change has been the mathematician’s stock in trade at least since the seventeenth century. It is change that is the concern of the calculus and the inscription of change that brings a coordinate system to vibrant life; and if the mathematician cannot define change he can sort out its characteristic forms, the ways in which it appears in this, our crowded world. We all of us live within hearing of the muted

in front of a pleased but stout and red-faced young girl: Gregory, you dance with Jessica here. The homely tableau succeeds in spite of itself. The sets A and B are represented by boys on the one hand, girls on the other, and the function itself by the Czar’s dancing mistress, mysteriously transposed to suburban Teaneck, New Jersey, and acting energetically to pick a boy and assign him to a girl. The functions of the calculus are mathematical functions, of course; they serve to bind real numbers

Leibnitz looks closely at his own well-made hand, the left one resting on the desk’s edge, a manly network of veins running from the top of his wrist forward to his fingers. Aber sicher. To be sure. And yet what if the distance between times were to become small? How small? Leibnitz asks himself. The clock chimes, and as it does, Leibnitz taps the table-top with his thumb. This is now, he says. With great delicacy, Leibnitz allows his forefinger to approach his thumb. Very small. He can sense

delicate distinctions begins. Unbounded meaning there is no largest natural number? Surely that is obviously true. Whatever candidate n might be proffered, n + 1 is larger yet. The requisite sense of unbounded lies elsewhere. Imagine numbers collected together in a set—K, say. K is bounded from above if there is a number x that is greater than or equal to any number in K. This is a definition. In symbols: x ≥ a for each and every a in K. Such an x is an upper bound for K. If K comprises the