A Smoother Pebble: Mathematical Explorations

Isaac Newton once explained his own genius to Robert Hooke by writing, "If I have seen farther, it is by standing on the shoulders of giants." These are among the most widely quoted words in all of science. They justify, even require, that mathematicians and scientists build on the genius of those who came before. At least a dozen books now in print (and a brand of shoulder pads for hockey and lacrosse) have the words "shoulders of giants" in their titles.

Newton penned another poetic passage, this one on beauty and the limits of genius:

I seem to have been only like a child playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

This phrase "a smoother pebble" asks us to pause to examine the beautiful parts of mathematics, and, through them, try to imagine what lies unseen, still beyond our grasp.

It seems that there is only one book in print that uses Newton's lovely image of "a smoother pebble" it its title, namely Donald C. Benson's new book. Readers with sharp memories might recall Benson's 1999 book, The Moment of Proof. Benson seems to have aimed that book at a talented student, either in high school or the first couple of years at university. He tried to infect such students with the thrill of understanding that we experience when we discover a beautiful proof. He subtitled the book Mathematical Epiphanies.

A Smoother Pebble seems directed to the same student, a few years later. Now, our former student knows a bit of the history of mathematics, some geometry, a bit of group theory, and a good deal of calculus. Benson chooses a particularly sparkling topic from each of these various subjects. Often, the topic is already familiar, like border patterns or the Greek attitude towards incommensurability. Then Benson examines it, turns it over and examines it again from another point of view, and then asks questions that extend the topic beyond its usual limits.

For example, Chapter 1 is mostly about Egyptian arithmetic and unit fractions. Most treatments of Egyptian mathematics demonstrate their algorithms for multiplication and division, and point out their similarities to modern binary arithmetic. They also usually tell us that Egyptians used unit fractions exclusively. Benson examines it a little farther. He shows us how Egyptians added fractions and how to use those fascinating doubling tables found on the Rhind papyrus. Finally, he gives a (modern) proof that any positive rational number has a unit fraction representation.

We see the same pattern in Chapter 3, "The Music of the Ratios." Most readers know well that if you halve the length of a vibrating string, its pitch goes up an octave. Divide it by three and the pitch goes up an octave and a fifth. He continues to construct the familiar Pythagorean scale. He continues, dropping in details about how the fingering on a clarinet is different from the fingering on a flute, as well as how their acoustical spectra are different. He gives the best explanation I have ever seen of the Pythagorean comma, and finally, he works through the details of the differences between Pythagorean temperament and equal temperament. It is mostly familiar material, but richer in detail and more carefully examined.

Benson ends his book with three chapters, about 70 pages, on calculus. Chapter 11, 28 pages, is his "Six Minute Calculus." It took me a good deal more than ten minutes to read it, and I already know calculus. I don't think that someone who didn't already know calculus could have made much sense of it. Benson does give us a nice explanation of how any two of the three instruments in an automobile, a clock, an odometer and a speedometer, can be used to construct the function of the third. He really uses this chapter to set up the equations of motion for an object sliding down a frictionless curve, so that he can use them in his last chapter.

Chapter 12, "Roller-Coaster Science" is the highlight of the book, though the mathematics in it is an order of magnitude more difficult than what comes before. We move quickly through elementary max-min problems, Fermat's "three towns" problem (also known as the Steiner point problem) and Benford's law (when you use base-10 arithmetic, 30% of useful numbers start with a 1) to the problem of quickest descent, the so-called Brachistochrone problem. In a sense, Benson shows us several "pebbles" and chooses to examine the Brachistochrone more closely. He gives a good solution, but then he extends the problem. Suppose, for example, that the path must have a given minimum. Or suppose that we do (or do not) require that the function be monotone. These are all interesting extensions, worth the examination.

I have a few complaints about this book, though. Among my minor quibbles is the author's explanation of limit: y tends to b as x tends to a (page 188). I think that y can tend to b without ever actually approaching it, and I would prefer the more traditional term approaches. In Chapter 5, he uses an example (page 92) with data about the number of pages in the journal Nature that contain graphs. It clearly shows how graphs became more common and more popular, using data roughly every ten years from 1879 to 1957. One must ask why did he stop there? We must wonder about the last 45 years. And I don't think it is fair to call Galileo's intellectual tools "primitive" (page 85).

On the whole, Benson uses the "smoother pebble" metaphor very effectively. Like Newton's contemporary Robert Hooke, author of Micrographia, he shows us that there are beautiful things to be found by looking at things very closely. Most readers of these pages would enjoy Benson's A Smoother Pebble.

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Ed Sandifer (esandifer@earthlink.net) writes the column How Euler Did It for MAA Online. He is professor of mathematics at Western Connecticut State University and has run the Boston Marathon 32 times.