# How Round Is Your Circle? Where Engineering and Mathematics Meet

## Synopsis

How do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. How Round Is Your Circle? invites readers to explore many of the same fundamental questions that working engineers deal with every day--it's challenging, hands-on, and fun.

John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated, How Round Is Your Circle? reveals some of the hidden complexities in everyday things.

## Excerpt

I have been obliged to confide the greater portion of the
theoretical part of the present work to some mathematical
assistants, whose algebra has, I fear, sometimes risen to a
needless luxuriance, and in whose superfine speculations
the engineer may perhaps discern the hand of a tyro.

Bourne (1846)

There are many convincing ways to justify a result. A scientist gathers evidence by undertaking a systematic experiment. One can undertake mathematical experiments, such as a sequence of calculations. Another kind of experiment is to draw a picture, be it on paper or sketched in the sand with a stick. Few, if any, mathematicians would now accept a picture as a valid proof but sketches do provide us with the simplest and most direct form of mathematical experiment. When undertaking such an experiment we ask you to think of it as representing a whole class of similar ones. What can you change without removing the essence of what you are doing? What must stay the same? And then, of course, decide how you can justify this.

So that we might be definite in the difference between a mathematical proof and an illustration, let us begin with an example. This is a theorem from Euclid’s Elements, book III, part of proposition 31 (Euclid 1956, volume 2, p. 61), which is encountered early in school geometry connected with a circle.

Theorem 1.1. Take any circle, and any diameter (from A to B, say), and any other point P on the circle. Then the triangle APB is a right-angled triangle, with right angle at P. (This is illustrated in figure 1.1.) . . .

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