Book Review: János Bolyai, Non-Euclidean Geometry, and the Nature of Space, Volume 52, Number 9

This attractive little volume consists of two major components, together with a number of shorter sections. The major components are labeled " Introduction " and " Appendix " respectively , both designations grossly understating their content. The " Appendix " consists of Bolyai's revolutionary tract, with the subtitle " THE SCIENCE ABSOLUTE OF SPACE Independent of the Truth or Falsity of Euclid's Axiom XI (which can never be decided a priori) ". It appears here both as a facsimile in the original Latin and also in Halstead's 1896 English translation. It was indeed originally published as an appendix to his father's two-volume treatise on mathematics, whence its name. Jeremy Gray's " Introduction " —or to give it its full name, " Bolyai's Appendix: An Introduction " —is in fact a 122-page historical survey of Euclidean and non-Euclidean geometry that focuses on Bolyai and his appendix but goes far beyond that. Among other things, it includes a number of fascinating historical remarks on the debates over the course of many centuries about how to teach Euclidean geometry in school—a debate that is still very much alive today and whose contemporary shape might profit from a deeper historical perspective. Gray starts by discussing the way philosophers and scientists, as well as humankind in general , pictured the physical space we in-habit—the shape of the universe as a whole—and the way that such pictures influenced and were influenced by the mathematical constructs that became known as " geometry ". In particular, the use made by Newton of the formalism of Euclidean geometry, together with the overwhelming success of his approach to physics and astronomy based on the model of Euclidean space, enshrined Euclidean geometry and cloaked it in a certainty and inevitability that made questioning it appear to be a sign of mental instability. And indeed, that was the reaction by the Russian mathematical establishment, the only ones to read Lobachevskii's first articles on what was to become non-Euclidean geometry, published starting in 1829 in the obscure Russian journal the Kazan Messenger. The news of this work apparently did not reach the Bolyais until 1844, which is Robert Osserman is the special projects director of the Mathematical Sciences Research Institute in Berkeley. His email address is [email protected].