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

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

## Synopsis

## 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.)* . . .