A History of Mathematical Tables: From Sumer to Spreadsheets (review)

In 1823 the English-born machinist and instrument maker Timothy Claxton sailed from Russia to the United States. Claxton spent much of the nine weeks of his voyage doing calculations on a slide rule. He later published the results in a small book titled Concise Decimal Tables for Facilitating Arithmetical Calculations . . .(1830). Few twenty-first-century travelers spend odd moments doing calculations by hand, and even fewer publish the results in printed tables. As the volume under review demonstrates, however, the humble Mr. Claxton was part of a rich tradition of table making that stretches back to ancient Sumer. Moreover, though table making is now done largely by electronic computers, the tabular presentation of data lives on in software such as spreadsheets.

In September 2001, the British Society for the History of Mathematics brought together a remarkable group of historians for a conference on the history of mathematical tables. The volume under review is an outgrowth of the papers presented at that conference, providing readers with an excellent introduction to a growing body of literature in the history of mathematics, astronomy, and computing. The opening chapter describes the oldest surviving tables, which are from ancient Mesopotamia. These range from a simple table for land measurement from Sumer (circa 2600 B.C.E.) to a tabular account of sheep and goats from the archives of Ur (circa 2028 B.C.E.) to more standardized administrative tables of the eighteenth century B.C.E.Later, Babylonian astronomers regularly produced ephemerides listing the times of astronomical events; the last of these appeared after the birth of Christ.

From this wide-ranging opening the book moves on to innovations in printed tables, particularly the early logarithmic and actuarial tables prepared in seventeenth-century Britain. Subsequent chapters discuss groups of people organized to produce tables, from the unemployed hairdressers of postrevolutionary France to the table-making enthusiasts of the late-nineteenth- and early-twentieth-century British Association for the Advancement of Science to those employed by the Mathematical Tables Project of the Works Projects Administration in the United States.

While some organized people to compute tables, others sought to automate the process. Two chapters are devoted to machines known as difference engines, which were designed in the nineteenth and early twentieth centuries both to compute tables and to print the results of calculation. Entries in a table, however they are calculated, depend on underlying assumptions about the interpretation and classification of data. Detailed accounts of the theory underlying the preparation of planetary tables, of the selection of categories used to present census data, and of table making at Britain's Nautical Almanac Office (NAO) suggest some of the complexity that underlies the ordered columns of tables. Many of the tables of the NAO are now presented exclusively in electronic form. A concluding chapter examines another electronic version of the mathematical table, the spreadsheet, and focuses in particular on the history of Visicalc, Lotus 1-2-3, and Microsoft Excel.

One characteristic of ardent table makers that emerges from this book is a meticulous—some might say obsessive—attention to detail. Happily, this spirit seems to have worn off on the authors, editor, and publisher. Each section includes not only notes and suggestions for further reading but also well-chosen illustrations of tables, people, and places. Relevant historical, biographical, and technical details are presented in separate text boxes. Such careful design, combined with thorough planning, a thoughtful introduction, cross-referencing, and an index, does much to unify the disparate contributions.

A book that surveys more than four thousand years of accomplishment is, perforce, selective. This reviewer would have been happy to learn more about the hundreds of inexpensive books of tables like Claxton's, as these played a much larger role in nineteenth-century engineering and commerce than the handful of difference engines discussed at length. The book also says very little about tables published in Germany, important...