# Tons of Sums

## Michael Mason

- Charles Babbage: Pioneer of the Computer by Anthony Hyman

Oxford, 287 pp, £12.50, July 1982, ISBN 0 19 858170 X

Most people know that Charles Babbage was a pioneer of the computer. This absorbing, though hagiographical, new life makes very clear how many other things he was as well: pure mathematician, economist, inventor, reformer of scientific institutions, craftsman, even salon host. But Anthony Hyman does not seek to displace the computers from centre-stage in Babbage’s life, and this seems correct. They provide the most sustained theme in his variegated career, and may be seen as a point of convergence for most of the tendencies of his mind. To consider the computers is to consider Babbage. Indeed his story falls under a triple rubric that belongs to computers from Babbage onwards: software, hardware, and applications.

The young Babbage was a good mathematician by international standards, at a time (the 1810s) when English maths was generally backward. He conceived his first calculating-machine in 1821, when he was 30. Thereafter he did no interesting work in mathematics, not even in connection with the mathematics of computing. There is something shocking in the spectacle of this fertile mind (which could raise the strategy of noughts-and-crosses to a sophisticated level of mathematical description) making nothing of the theory of computation by machine: especially since modern computing has spawned so much abstractness about itself.

There is admittedly a danger of unhistorical judgment here. Babbage was working quite on his own, as far as the broad design of the machines was concerned, and he had to do what he could, with the technical means at his command. He designed two computers, which were themselves families of machines. The first, the Difference Engine, was entirely arithmetical, being intended to compute and print out long series of values for various arithmetical operations. Its workings depended on a solitary arithmetical principle, which Babbage brilliantly saw could be given a mechanical representation. It is a fact that many functions, applied to the series of natural numbers, will yield results that are separated by a constant amount. For multiplication by two, for example, this constant amount is two and separates the results themselves(2,4,6,8 ... ). But it may be that the ‘constant difference’ is a second-order one: separating, not the results proper, but the differences between the results. Squaring successive numbers, for example, gives results that are separated by a series of increasing amounts, and it is these which are separated by a constant interval of 2 (2^{2}-1^{2}=3, 3^{2}-2^{2}=5, 4^{2}-3^{2}=7, etc). There can be constant third-order differences, and so on.

The upshot is that many operations in arithmetic can be expressed as additions, and a machine that adds quantities, as a gas-meter does, can perform them. The vertical axles of the Difference Engine with their cogs (part of one that Babbage constructed may be seen in the Science Museum – Babbage’s own brain is preserved in the Royal College of Surgeons) are simply columns of differences. The end column retains whatever constant value you set it at; the others will successively add the preceding column’s value to their own, and hold the result. You transfer the constant value time and again into the machine, the other columns change by non-constant amounts, and, depending on their initial settings, the results of some arithmetical operation or other will emerge, in order of magnitude, at the other end.

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Vol. 4 No. 17 · 16 September 1982 » Michael Mason » Tons of Sums

pages 7-8 | 2605 words