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the thought of whether he could devise a mechanical aid for the sense of hearing which would prove both certain and ingenious. Such an aid would be similar to the compasses, rules and optical instruments designed for the sense of sight. Likewise the sense of touch had scales and the concepts of weights and measures. By some divine stroke of luck he happened to walk past the forge of a blacksmith and listened to the hammers pounding iron and producing a variegated harmony of reverberations between them, except for one combination of sounds.
    According to Iamblichus, Pythagoras immediately ran into the forge to investigate the harmony of the hammers. He noticed that most of the hammers could be struck simultaneously to generate a harmonious sound, whereas any combination containing one particular hammer always generated an unpleasant noise. He analysed the hammers and realised that those which were harmonious with each other had a simple mathematical relationship – their masses were simple ratios or fractions of each other. That is to say that hammers half, two-thirds or three-quarters the weight of a particular hammer would all generate harmonious sounds. On the other hand, the hammer which was generating disharmony when struck along with any of the other hammers had a weight which bore no simple relationship to the other weights.
    Pythagoras had discovered that simple numerical ratios were responsible for harmony in music. Scientists have cast some doubt on Iamblichus’ account of this story, but what is more certain is how Pythagoras applied his new theory of musical ratios to the lyre by examining the properties of a single string. Simply plucking the string generates a standard note or tone which is produced by the entire length of the vibrating string. By fixing the string at particular points along its length, it is possible to generate other vibrations and tones. Crucially, harmonious tones only occur at very specific points. For example, by fixing the string at a point exactly half-way along it, plucking generates a tone which is one octave higher and in harmony with the original tone. Similarly, by fixing the string at points which are exactly a third, a quarter or a fifth of the way along it, other harmonious notes are produced. However, by fixing the string at a point which is not a simple fraction along the length of the whole string, a tone is generated which is not in harmony with the other tones.
    Pythagoras had uncovered for the first time the mathematical rule which governs a physical phenomenon and demonstrated that there was a fundamental relationship between mathematics and science. Ever since this discovery scientists have searched for the mathematical rules which appear to govern every single physical process and have found that numbers crop up in all manner of natural phenomena. For example, one particular number appears to guide the lengths of meandering rivers. Professor Hans-Henrik Stølum, an earth scientist at Cambridge University, has calculated the ratio between the actual length of rivers from source to mouth and their direct length as the crow flies. Although the ratio varies from river to river, the average value is slightly greater than 3, that is to say that the actual length is roughly three times greater than the direct distance. In fact the ratio is approximately 3.14, which is close to the value of the number π, the ratio between the circumference of a circle and its diameter.
    The number π was originally derived from the geometry of circles and yet it reappears over and over again in a variety of scientific circumstances. In the case of the river ratio, the appearance of π is the result of a battle between order and chaos. Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to fastercurrents on the outer side, which will in turn result in more erosion and a sharper bend. The sharper the