Any ciphers with your supper, sir?

The human hard drives who crunched numbers and tracked comets fascinate Tony Hoare

Intensive numerical calculations have always played an essential role in the progress of science. Calculations are needed to confirm the validity of newly discovered scientific laws by comparing their predictions with observation. They are needed again to apply the discoveries of science to the advancement of all branches of technology, commerce and even politics.

Until the middle of the 20th century, all such calculations were carried out by individual human labour, or by teams of men and women (or sometimes boys), in professionally managed organisations known as computing laboratories. At the time, these human calculators were called computers. That explains the title of this fascinating book. It is extraordinarily well written, full of stories that will elicit and satisfy the interest of students and amateurs of the history of science and technology.

The story starts with Edmond Halley, who wished to confirm Newton's theory of gravitation by calculating the orbits of four recently observed comets, including the one he had taken measurements of in 1682.

Three comets confirmed Newton's hypothesis that their orbits were parabolas, which meant they would never return to the neighbourhood of the Sun. But Halley noticed that his own comet followed a path similar to that of two previous sightings, one in 1531 and one in 1607. He brilliantly conjectured that they were the same comet, whose path would therefore have to be an ellipse. His calculations confirmed this conjecture, and led him to predict its return in 1758.

Or perhaps, he admitted, it would actually be 1759. His calculations were only approximate because there is no exact formula to predict the path of three or more gravitationally interacting bodies. In addition to the Sun and the comet itself, it was necessary to include also the effects of the heavy outer planets. Halley had other duties as Astronomer Royal, and he had to leave the issue "to the care of posterity".

The challenge was taken up in 1757 by an ambitious French nobleman, Alexis-Claude Clairant. He devised a scheme of calculation that enabled him to share the work with two friends, Joseph Lalande and Nicole-Reine Lapaute, the wife of the King's clockmaker. They worked with quills and linen paper, sharing a large table at which they also took their meals brought in by servants. They tested the scheme of calculation by applying it to the data from 1531, 1607 and 1682. In November 1757, Clairant announced that the perihelion of the comet (its nearest approach to the Sun) would occur on April 15, 1758, plus or minus one month. In the event, it was March 13: his prediction was ten times more accurate than Halley's.

For the 1835 return of Halley's comet, the most complete calculations were made over five years by another Frenchman, Philippe Gustave le Doulcet. He devised a more laborious and accurate numerical method, and by a series of step-by-step approximations he got within four days of the perihelion on November 16.

For the 1910 return, the calculations were carried out by a team at the Royal Greenwich Observatory, led by an Englishman, Andrew Claude Cromelin.

They used le Doulcet's method, but had the benefit of the Brunsviga and other mechanical calculators. Nevertheless, the estimate for the perihelion was nearly three days in error, and it remained so even when the calculations were repeated after the event. Among the conjectured causes of the discrepancy was the possibility that the comet's tail was slowing it down.

The most recent return was in 1985. This time the calculations were done on a Univac 1107 computer, capable of calculating a million times faster than any human being. The program was written by Donald Yeomans at the US Jet Propulsion Laboratory. The physical parameters of the system, including the "outgassing" of the comet's tail, were recomputed by re-running an analysis of all previous known returns. The final estimate was just five hours wrong.

Perhaps this result will be improved in 2061, when the available computing resources will be yet another million times more plentiful than they were in 1985. Or perhaps the error will persist. There may be more random fluctuations in the outer reaches of the solar system than we can ever know about. Let us leave the issue to the care of posterity.

These highlights of the story of Halley's comet are only punctuation marks in the long history of calculation by teams of human computers. The calculation methods devised for the pure science of astronomy were soon adapted to more practical concerns such as marine navigation. Determination of the longitude of a ship at sea required an accurate estimate of the local time in comparison with the time at Greenwich. This could be obtained by observation of the position of the moon against the stars, and comparison with standard tables (ephemerides), which were computed by teams at the Royal Greenwich Observatory, and published from 1767 onwards in the Nautical Almanac . The computation followed tables of instructions (now known as programs), with blanks to be filled by answers; they were devised originally by Nevil Maskelyne, the fifth Astronomer Royal.

David Alan Grier has collected in this book many equally fascinating stories of the original contributions made by human computers to the development and application of all branches of applied science. Their services were exploited in surveying, mapmaking, agriculture, insurance, statistics, ballistics, economics, artillery and many other aspects of warfare.

In each story, the reader learns something of the leading figures and the mathematical challenges and political opposition that had to be overcome to finance, recruit, train and manage the teams of computers. Even after an enjoyable browse through its stories, readers will find that the book, which I highly recommend, will continue to serve as a scholarly work of reference, with a clear index, an extensive bibliography, 40 pages of notes and a summary of key personalities and institutions.

Sir Tony Hoare is senior researcher, Microsoft Research. He learnt numerical calculation with Brunsviga manual calculators at the Oxford University Unit of Biometry in 1959 and programmed his first computer in 1960.