A genie (as the story is told) lights a candle at a minute before midnight. After half of the minute has passed, the genie extinguishes the flame. Fifteen seconds later, she relights the candle, and again, halfway to midnight, she puts the flame out. This continues as midnight approaches, the time always divided in two, the flame soon leaping up and vanishing faster than we can see.

Now the genie asks you, “At midnight, will the flame be lit or out?”

Leaving aside the issue ofwhenthis question is asked, you are still left with some bewildering possibilities. The...

The rise of Rome at the expense of Greece marked a steep decline in the pursuit of pure mathematics in the western world. Although Roman culture borrowed freely from Greek religion, philosophy, and art, Roman mathematics largely confined itself to what was necessary for commerce and engineering. However, the economies and militaries of both Greece and Rome extended east as far as India, prompting trade not only of goods but of knowledge. Islamic versions of universities attracted thinkers and collected knowledge for the sake of science, acting as transfer points where thinkers carried mathematical ideas between cultures.

The scholar Abu...

Even if you sequester yourself in nature, away from the influences of humankind, the world moves in curious, patterned ways. Why does one falling leaf drift in a spiral while another twirls about its axis? What explains the eddy patterns in a creek? What forces act on a bird’s wing? Why do the stars travel a circle during the night? Thinkers from many cultures looked beneath the surface of questions like these; underlying all of the answers was mathematics.

Most thinkers who discover something about how the universe works want to share their discoveries, and some created notation intended to...

Every mathematical subject advances thanks to imaginative conjectures. One of the earliest examples of such risk-taking in calculus is due to Democritus (Greece, bornc. 460 bce), who lived about 200 years before Archimedes. He is credited with a claim such as the following:

If two solids are cut by a plane parallel to their bases and at equal distances to their bases, and the sections cut by the plane are equal, and if this is true for all such planes, then the two solids have equal volumes.

Although this claim does not directly state that solids are composed of...

Amid a flurry of discovery in the 1600s, European mathematicians began to recognize the underlying unity of their results. Quite often, a successful quadrature shed light on the mysteries inherent in series, or a clever use of geometry prompted advances in the theory of numbers. Each new connection fanned the intellectual fire. We see this effect in this chapter as we trace efforts to find the quadrature of the hyperbola.

One way to define a hyperbola is as the collection of points whose horizontal and vertical components are inversely proportional, as in themodern¹ notationxy= 1 ory=...

Evidence mounted in the late 1600s that efforts to understand quadrature and attempts to quantify instantaneous velocity could be unified in a single theory. This link was confirmed by Isaac Newton of England and Gottfried Leibniz of Germany. For this achievement, we honor them as the discoverers of calculus.

We may view an object’s velocity as therate of changeof its distance. Not all curves describe distance, but many curves allow for tangent lines. Thus, we speak of the rate of change of a curve from this point forward, unless the situation specifically describes motion.

We finally meet Isaac...

Flipping back through the pages of this book, you can see how important geometric figures were in the development of calculus. The figures become more sophisticated as the truths they reveal become deeper; Figure 6.5 of Leibniz, for example, goes to the heart of the connections within calculus, but falls just shy of being an impenetrable maze of lines. Leibniz, as much as anyone in his day, desired to push calculus past the point where its truths are a consequence of diagrams. The notation he invented allowed this, and we use many of his symbols today.

The notation of Leibniz...

“Where is that?” is a question as ancient as astronomy, often accompanied by, “Where is it going?” and, these days, “Is that thing going to hit us?” Because Greek thinkers of old believed that the earth was stationary and that celestial objects traveled in circular paths, the study of angles related to circles received careful attention. The word for this study,trigonometry, refers to the measure of triangles, which yield a multitude of curious and beautiful truths.

Greek astronomers were privy to many such truths, but it was Indian scholars in the years between 400 and 700 who began to...

This chapter highlights the story of how scholars engaged in a conversation with the aim to remove from calculus all ambiguity, especially with regard to the infinitely large and infinitely small. Despite the common goal, there was at first no agreement on the solution, and the conversation meandered as most great debates do. Some ideas, dropped for dozens of years, made surprising returns in pamphlets, private letters, books, and book reviews. Ultimately, the voices in the debate unified on a course of action that banished ambiguity while opening marvelous new possibilities.

Jean le Rond d’Alembert (France, born 1717) played a...

Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built. One might attempt to distinguish mathematics from other disciplines by arguing that, ultimately, weknowthat its results expresstruthin a way no other subject can boast; however, philosophical arguments of the early 1900s call even this claim into question.¹

While the story of calculus features plenty of intrigue and debate, readers should rest assured that controversy is...