In the century preceding the French Revolution advanced mathematics began to play a role in ordinary affairs. If you wanted to find the position of a ship at sea, design fortifications or price annuities, then you needed mathematics. In a process accelerated by the Revolution, first the French and then the German states set up or adapted existing institutions to train engineers and school teachers in mathematics.

The teachers of these new students taught two subjects. One was the classical geometry of the Greeks, less useful than it might appear, but universally held up as a model of reasoning in which rigorous argument lead from clearly stated first principles to final conclusions. The other was modern mathematics, in particular calculus, which lacked any such clear structure.

Certainly modern mathematics dealt with numbers but numbers seemed to be drawn from a rag bag of different objects.

Certainly modern mathematics dealt with numbers but numbers seemed to be drawn from a rag bag of different objects. There were numbers like 1 and 2 which everybody understood, and fractions like 1/3 or 3/9 which were the same number and 2/3 and 1/4 which were different numbers. Then there was 0 which was a number like every number except that you were not allowed to divide by it. To these were adjoined the negative numbers which were every bit like positive numbers, provided that you remembered that the product of two negative numbers was a positive number. Negative numbers had no square root unless you allowed mysterious entities called complex numbers which were also exactly like other numbers except when they were not (for example complex numbers usually had three complex cube roots). Finally there were objects like π which were not fractions, or like Euler’s ϒ which may or may not be a fraction, but which everybody agreed were bona fide numbers to be treated exactly the same as other numbers.

It is doubtful if this worried many students then who, like most students now, wished just to pass their exams, get a good job and enjoy themselves. However it did worry some of their professors and during the course of the 19th century, with many fits and starts, they completed the difficult task of rigourising calculus and the linked task of providing a coherent account of the numbers used in calculus.

In 1930, Landau published a little book *Foundations of Analysis* setting out this account at undergraduate level. Landau’s much loved text is still in print but, as Landau says, is written `in merciless telegraph style . . . as befits such easy material’.

There is, I think, room for a more relaxed account which gives some idea of where the ideas come from and why they are used in the way they are used. My book is an attempt at such an account.

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