To illustrate the part played by mathematics in the construction of scientific theories, we consider the development of optics.

Euclid's Optics

We begin with Euclid (c. 300 BC). Not unnaturally for a geometer, he wished, as doubtless had many geometers before him, to apply geometry to optics. Unlike the others he was successful. Conceiving light as propagated in straight lines enabled him to apply geometry to optics. On second thought this statement cannot stand. Until Euclid had applied geometry to optics there was, to use the Irish idiom, no such subject as optics. Nowadays, when diagrams are used as an ingredient of educated common sense, of course it is obvious that light is propagated in straight lines. If light rays could not be represented by lines, optical phenomena could not be illustrated by diagrams. We, with the arrogance of hindsight, cannot begin to understand Euclid's foresight in making his basic assertion that light is rectilinearly propagated. When the needle in the haystack has been pointed out to us, we are prone to suppose that finding it was no problem at all.

Physical objects that more or less crudely approximate straight lines readily come to mind, for example, a taut wire. But surely a shaft of sunlight piercing the shutters of a darkened room is singularly apt. Isn't this the perfect example? Euclid must have been well pleased with his observation. Yet note that his basic assertion embraces metaphysical speculation as well as physical observation.