But the author also reveals (rather worryingly) that he belongs among those willing to read significance into the dimensions of pyramids. The Egyptians apparently chose 1.273, rather than 1.272 as ⁴/_n the value of a constant embodied in their monumental tombs, but which is not otherwise known to anything like this accuracy in written Egyptian sources.

Chinese mathematics follows, after which the diet is Greek. Wilson deals with the symmetries of the regular solids, sliding from Platoā€™s Timaeus and Haeckelā€™s Radiolaria to Zeus and God; about Aristotle he explains, among other things, the classical syllogisms and how they got their names. A further two chapters are anachronistic dialogues about axioms and proofs in geometry (with a nod to David Hilbertā€™s approach around 1900).

I enjoyed the chapters about Archimedes, Apollonius and, a welcome addition, Hero, which are more closely tied to the relevant surviving works, and so more successful. Only the account of Apolloniusā€™s work seems difficult, but Apollonius is notoriously forbidding.

Then comes India under the ominous title ā€œThe dark subcontinent of Indiaā€, which raises even darker suspicions, although some of the explanation of Indian number theory is attractive. A final chapter on Islam covers spherical trigonometry quite well.

Wilsonā€™s book shows some of the temptations inherent in the genre. The extensive passages of political history are not always up to the standards of a Stephen Jay Gould: for example, the defeat of the Arab army at Poitiers in 732 was not decisive in preserving Christian Europe, but the European victory in the East was, as Princeton historian Bernard Lewis has shown. There are more substantial problems: the author of the Rhind mathematical papyrus (one of the main sources of our knowledge of Egyptian mathematics) fails to work out correctly the area of an isosceles triangle. The numerical error is 2 per cent here, so why does Wilson fail to discuss why we should then take seriously a difference of less than 0.1 per cent in the height of a pyramid?

The mathematics is not always well understood. For instance, to claim that the Egyptians evaluated volumes of pyramids by considering infinitesimal slices is to presume too much of them from the evidence we have of their mathematical knowledge. But if the author disagrees, why did he finish the proof in a clumsy way? And in the piece devoted to Brahmaguptaā€™s formula for the area of a quadrilateral as a function of the lengths of its sides, Wilson fails even to hint that there can be no such formula: the side lengths determine neither the shape nor the area.

The Infinite in the Finite does start well, but it loses its way. Both its good qualities and its faults, in part, go with the genre of popular history of science. I would have liked footnotes so that I could check his sources. And an uneven tone does not help, with oddities such as ā€œthe only real numbers are the primesā€.

George Green was not much appreciated in his lifetime. No portrait of him
survives (if one was ever made), and we know little about his life. Indeed,
until the indefatigable Mary Cannell began work, and despite the best
efforts of other, active, friends of Green such as Lawrie Challis, Green was
nearly an invisible man. This attractive and well-illustrated book
commemorates the life and work of Green, one of the most important British
mathematicians of the 19th century, whose bicentenary falls this year (see
ā€˜There was a jolly miller . . .ā€™ 10 July). It is a significant addition to
our knowledge of him.

Although we can still only conjecture who fostered Greenā€™s talent for
mathematics and guided his early reading ā€“ Cannell makes some highly
plausible suggestions ā€“ we can now see clearly for the first time the milieu
in which Green worked. He earned a prosperous living as a miller in
Nottingham, and had access to the best new French mathematics. He wrote his
remarkable Essay in 1828, in which Greenā€™s theorem and Greenā€™s functions
first appeared. This work brought him to the attention of Sir Edward ffrench
Bromhead, a local dignitary and Cambridge graduate, and eventually to the
University of Cambridge where Green became a fellow for a brief time. During
this period, he wrote his few other papers, which deal with magnetism,
electricity and hydrodynamics.

Greenā€™s work was saved for posterity by William Thomson, later Lord Kelvin,
who tracked down the original Essay and saw that it was republished in an
influential German journal. From then on, as this book shows so well,
Greenā€™s theorem and Greenā€™s functions have steadily enjoyed the attentions
of mathematicians and physicists. Not only particle physics, but fields as
diverse as soil science and pure mathematics, continue to benefit from these
ideas. A useful account of the Essay is given here in an Appendix by M. C.
Thornley.

Cannell also describes Greenā€™s family. His steadfast common law wife, Jane
Smith, bore him seven children, all illegitimate, but if she survived what
surely was a local stigma, some of the children seem to have found it harder
to bear. Their lives, and the precarious path of knowledge about Green down
to the present day, make fascinating reading. Indeed, the book is eloquent
testimony to the difficulties and pleasures of the historianā€™s task. It
provides a lucid insight into the times, and as much as is known of the
life, of one of Britainā€™s most creative mathematicians.

Jeremy Gray is an historian of mathematics.