Magic Squares: Their History and Construction from Ancient Times to AD 1600

Historians and mathematicians come across magic squares more as a side curiosity than as a central part of historical or mathematical narratives. Creating a magic square — that is to say a square grid filled with positive whole numbers, all different, arranged so that the sum of the numbers in horizontal rows, in vertical columns, and along the main diagonals of the grid is the same — is an intriguing challenge. It rarely, however, at the center of mathematical instruction or research. Indeed, Sesiano reports that western mathematicians inherited from the medieval era only two sets of seven magic squares. These were associated with the seven planets and thought to have differing magical properties. Hence the term “magic square.” Similarly, going through a rich collection of objects associated with centuries of American mathematics education, this reviewer has run across only one example of magic squares — a set of simple drawings from a high school teacher in Worcester, Massachusetts.

As this volume amply demonstrates, however, Arabic and Indian texts on the construction of magic squares reveal a rich tradition of ideas and cultural interchange. Sesiano has spent years uncovering these interactions in wide-ranging sources, many of them highly dispersed manuscripts. He not only describes in exacting detail procedures used but reports on recently rediscovered manuscript sources linking Arabic texts to ancient Greek mathematics. Despite the time restrictions stated in the title, he also links these early sources to later work on magic squares.

This volume is not a quick and easy read. It offers a rich source for anyone interested in magic squares or more generally in the complexity of change in the history of mathematics and in cultural interactions. It would provide a rich resource for undergraduates undertaking a senior research project.

Peggy Aldrich Kidwell is Curator of Mathematics at the Smithsonian’s National Museum of American History. With Amy Ackerberg-Hastings and David Lindsay Roberts, she is coauthor of Tools of American Mathematics Teaching: 2800-2000 (Johns Hopkins University Press).