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What did Mercator expect to write about in his book on ratios? The list of contents looks rather curious to us: the book was to contain sections on devising calendars, astronomical predictions, the proper division of the musical scale, and military constructions. Mercator believed that an ingenious application of the theory of ratios could help with the solution of all these kinds of problems. I am particularly interested in the musical material.

In Mercator's day there was a long tradition of considering ratios as things quite different from numbers, and quite different from fractions. They had special properties and combined in an unusual way. For example, the most natural way to combine two ratios was to multiply their terms: a:b and c:d made ac:bd. To take one ratio away from another, you divided the terms: c:d taken from a:b gave (a/c) : (b/d). In fact, these two operations were often called 'adding' and 'subtracting' ratios until the late seventeenth century, terms which can seem quite confusing to us.

If these are 'adding' and 'subtracting', an interesting problem arises: what are 'multiplying' and 'dividing'? It isn't easy to see how to 'multiply' a ratio by another ratio. What you can do, though, is to multiply a ratio by a number, so that for example two times a:b is the same as a:b 'plus' a:b – that is, a^{2}:b^{2}. And so on. So we can in fact 'multiply' a given ratio by any whole number, simply by repeatedly 'adding' it to itself.

But what would it mean to 'divide' a ratio by another? This isn't at all easy, and from perhaps the fourteenth century onwards certain mathematicians considered that finding a 'ratio of ratios', or how to 'divide' one ratio by another, was an unsolved problem. The difficulty can be put another way by asking, how many times does the ratio a:b go into the ratio c:d? That is, what number must we 'multiply' a:b by – how many times must we 'add' a:b to itself – in order to get c:d?

You might be able to see how to find the answer. Mercator did too. But first, what do ratios have to do with music?

Benjamin Wardhaugh, "A Plague of Ratios - The Uses of Ratios," *Convergence* (July 2010)