A History of the Circle: Mathematical Reasoning and the Physical Universe. – Review – book reviews
Michael M. Abrams
A History of the Circle: Mathematical Reasoning and the Physical Universe Ernest Zebrowski Jr. RUTGERS UNIVERSITY PRESS, 1999, $28.
THE CIRCLE AND ZERO HAVE A COUPLE of things in common beyond roundness: Both are crucial for mathematicians and scientists, and both are abstractions.
Although circular wheels, orbits, pennies, and pizzas may be mere approximations, the circle is so important to math that Ernest Zebrowski Jr. has built a 200-page history of the subject around it. In A History of the Circle, he begins with a chapter devoted to pi. Zebrowski points out that dividing the circumference of a circle by its diameter yields a number that never terminates or repeats for the same reason that a true circle doesn’t exist: because measurement can never be exact. Every digit added to pi takes us a step further toward unachievable accuracy But wait a minute. If no measurement of the circumference or diameter of a circle can be exact, how was pi calculated? In the third century B.C., Zebrowski tells us, Archimedes approximated a circle with various polygons, whose sides and perimeter he could
easily calculate. The more sides a polygon has, the closer it is to a circle. “[U]ltimately he decided that a 96-sided polygon … would be a reasonable approximation to the circumference of a circle. And, in fact, if you build a 96-sided polygon and roll it on a flat surface, it will indeed roll along reasonably well.”
That observation leads easily into the next chapter, on wheels and rollers. Long before the advent of the wheel, people rolled their heavy burdens on logs. But because rollers need to be uniform in size and shape to work well, Zebrowski suggests the Egyptians opted for a different technique to schlepp stone blocks up the pyramids — by adding “circular segments” to each side, creating rounded-edged rectangular cylinders that could be rolled up the pyramid. The celebrated wheel, he writes, wasn’t much use until ball bearings were invented. The Romans, for instance, used chariots only for short distances and special occasions, because the crude bearings they used on axles quickly wore out.
Later chapters stray further from the circle. From the difficulties of inventing the clock and the development of our understanding of the universe, conics, and sine waves, to the discovery of neutrinos, Zebrowski explains the major advances of physics and math simply and understandably eventually leaving the circle behind. The same book with a different name, minus the first chapter or two, could easily be mistaken for a review of the math you should have learned in high school. Although the concept of the circle was very real for thinkers throughout history the idea of zero took much longer to come into its own, explains Robert Kaplan in The Nothing That Is. There was a time when arithmetic was conducted without it, albeit painstakingly (imagine long division with Roman numerals).
Nowadays most people take the number for granted. It has some special rules — everyone knows you can’t divide by zero — but on the whole, it seems to function just like any other digit. Nonetheless, it makes our entire system of numerals possible, allowing us to tell the difference among, say 32, 302, and 30,200,000. It also serves as the boundary between positive and negative integers.
There’s not much historical evidence telling us when zero first appeared. Did the Babylonians invent it first, when they used a pair of diagonal wedges to show an empty column in their addition? Perhaps the Greeks beat them to it, using a bar over a circle to indicate nothing. However it started, by the Middle Ages, the use of the zero was widespread in Europe, though viewed with some distrust; centuries passed before the new number was fully accepted. “[T]echnical difficulties, combined with the slow spread of knowledge before books were printed and writing in the vernacular was common, added to the reputation that the Arabic numerals already had for being dangerous Saracen magic … So in Florence the city council passed an ordinance in 1299 making it illegal to use numbers when entering amounts of money in account books: Sums had to be written out in words.”
The story of the slow acceptance of zero is the most interesting part of Kaplan’s book. But as the text continues, his use of a literary quote here and there to sweeten the prose degrades into paragraphs that groan under passages from the likes of Henry James, Samuel Beckett, and Fyodor Dostoyevski, making this tome at times seem more like a version of Bartlett’s Familiar Qoutations than a history of math.
Both books provoke amazement at our ability to understand the world through abstractions. But both also acknowledge the probability of limits to that understanding. Near the end of A History of the Circle, Zebrowski comments: “[W]hat is not detectable cannot be considered to be real. At some level — and perhaps this is it — science may be destined to merge once again with philosophy the discipline it split off from many centuries ago.” It’s a reunion both authors seem to anticipate.
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