Book Review János Bolyai, Non-Euclidean Geometry, and the Nature of Space Reviewed by Robert Osserman János Bolyai, Non-Euclidean Geometry, and the that is still very Nature of Space much alive today Jeremy J. Gray and whose contem- MIT Press, 2004 porary shape might Paperback, 256 pages, $20.00 profit from a deeper ISBN 0-262-57174-9 historical perspec- tive. This attractive little volume consists of two Gray starts by dis- major components, together with a number of cussing the way shorter sections. The major components are philosophers and labeled “Introduction” and “Appendix” respec- scientists, as well as tively, both designations grossly understating humankind in gen- their content. The “Appendix” consists of Bolyai’s eral, pictured the revolutionary tract, with the subtitle “THE SCIENCE physical space we in- ABSOLUTE OF SPACE Independent of the Truth or habit—the shape of Falsity of Euclid’s Axiom XI (which can never be the universe as a decided a priori)”. It appears here both as a whole—and the way facsimile in the original Latin and also in Halstead’s that such pictures 1896 English translation. It was indeed originally influenced and were influenced by the mathemat- published as an appendix to his father’s two- ical constructs that became known as “geometry”. volume treatise on mathematics, whence its name. In particular, the use made by Newton of the for- Jeremy Gray’s “Introduction”—or to give it its full malism of Euclidean geometry, together with the name, “Bolyai’s Appendix: An Introduction”—is in overwhelming success of his approach to physics fact a 122-page historical survey of Euclidean and astronomy based on the model of Euclidean and non-Euclidean geometry that focuses on space, enshrined Euclidean geometry and cloaked Bolyai and his appendix but goes far beyond that. it in a certainty and inevitability that made ques- Among other things, it includes a number of tioning it appear to be a sign of mental instability. fascinating historical remarks on the debates And indeed, that was the reaction by the Russian over the course of many centuries about how to mathematical establishment, the only ones to read teach Euclidean geometry in school—a debate Lobachevskii’s first articles on what was to be- come non-Euclidean geometry, published starting in 1829 in the obscure Russian journal the Kazan Robert Osserman is the special projects director of the Messenger. The news of this work apparently did Mathematical Sciences Research Institute in Berkeley. His email address is [email protected]. not reach the Bolyais until 1844, which is 1030 NOTICES OF THE AMS VOLUME 52, NUMBER 9 astonishing on many grounds, not the least of given his endorsement of the work publicly and which involves the shameful role played by Gauss brought it to wider attention, it would have changed in the sad story of János Bolyai’s brilliant beginnings the course of János’s life and career. But he kept and bitter end. his silence,1 and János not only got no public The story of Gauss and the Bolyais starts back support but was convinced that his father had before Gray picks it up, with the close friendship betrayed him and revealed what he had been doing between Gauss and János’s father, Farkas (also to Gauss, who was now claiming it as his own. known by his German name of Wolfgang), dating When Gauss later became aware of Lobachevskii’s back to their student days together in Göttingen. work, he made no attempt to inform his old friend, The family obsession with Euclid’s parallel postu- Farkas. late apparently dates to that time, and indeed, Recognition of Bolyai’s achievement did not father Bolyai published his first work on the sub- come until too late. Gauss died in 1855, Farkas and ject, The Theory of Parallels, in 1804. He was only Lobachevskii the following year, and János in 1860. one of many who tried to prove that the parallel As Gauss’s correspondence became public, his postulate follows from Euclid’s other axioms. views on non-Euclidean geometry finally became Gray describes in some detail the history of such known. Bolyai’s appendix was translated into Ital- attempts throughout the eighteenth and nineteenth ian in 1868, the year and the place that were to be centuries, even well after their futility had been fully decisive for the future of the new geometry that he demonstrated. Among the prominent mathemati- had invented. cians who fell into the trap were Legendre, who per- Between 1832, when it was originally published, sisted in publishing false proofs over many years, and 1868, when it became more widely known, the and none other than Lagrange, who did not go so critical events for the fate of Bolyai’s appendix far as publishing any but did have the embarrass- were, as already mentioned, the publication ment of presenting one at the prestigious Institute between 1860 and 1865 of the correspondence of France. As for Farkas Bolyai, it was his friend between Gauss and Schumaker, including a letter Gauss who pointed out the error in his argument, Gauss wrote in 1846 praising Lobachevskii’s work but he persisted for at least ten more years before on non-Euclidean geometry and saying that he had giving up in despair. No wonder then, when his son, shared the same views for fifty-four years (since János, who had turned out to be something of a 1792, when Gauss was fifteen), and the presenta- mathematical prodigy, appeared as a teenager to tion of Riemann’s Habilitationsvortrag, entitled have already been bitten by the parallel-postulate “On the hypotheses that lie at the foundation bug, Farkas wrote him the often-quoted feverish ad- of geometry” in 1854. Not that Riemann refers monishment to profit from his own example and directly to Bolyai or even to non-Euclidean geom- guard against this will-o-the-wisp: “I have traversed etry as it is usually understood. Rather, Riemann this bottomless night, which extinguished all light proposes a radical rethinking of the entire subject and joy in my life. I entreat you, leave the science of geometry, not based on axioms in the fashion of parallels alone.” of Euclid, Bolyai, and Lobachevskii, but on the As it turned out, Bolyai Sr. was right, but not for perhaps shakier but far more flexible foundations the reasons he thought. János soon concluded that of measurement, the calculus, and the whole field proving the parallel postulate was hopeless, and he of differential geometry as developed by Euler, gradually became convinced that one could con- Gauss, and the French school. struct a perfectly consistent geometry in which it Gray’s “Introduction” devotes about twenty-five was not true. By 1825, at the age of twenty-three, pages to Bolyai’s “Appendix” itself and provides an János was able to show his achievement to his excellent overview of Bolyai’s approach, with its father but not to convince him that there were no emphasis on an “absolute” geometry of space, hidden flaws. It was not until 1832 that Farkas with parallel (if one may use the word) results in agreed to publish his son’s work in the form of an appendix to his own book. When Gauss received a 1This was not the first time that Gauss’s silence on the sub- copy of the appendix, he wrote back a letter that ject of non-Euclidean geometry had a devastating effect. effectively ended János’s brilliant career, as well Another of Gauss’s correspondents, named Taurinus, pub- as, Gray tells us, the relationship between father lished a brochure in 1826 in which he derived a number and son, who did not speak to each other for years of the same trigonometric formulae as Bolyai. In the pref- afterward. In the letter Gauss gives an infuriat- ace to the brochure he asked Gauss to state his views on the subject, after which Gauss terminated the correspon- ingly mixed message, saying that he had himself dence. In the book by B. A. Rosenfeld, A History of Non- carried out much the same program but had Euclidean Geometry: Evolution of the Concept of a written very little of it down, and how pleased he Geometric Space, English translation Springer-Verlag, was that it should turn out to be the son of his 1988, p. 219), the author tells us, “Gauss’s reaction reduced old friend who had written it down, thereby spar- Taurinus to despair, and he burned all copies of the ing him the trouble of doing it himself. Had Gauss brochure in his possession.” OCTOBER 2005 NOTICES OF THE AMS 1031 both Euclidean and non-Euclidean geometry, but point that start out along a given two-dimensional also special results, such as the possibility of section. That surface will have a curvature whose “squaring the circle” in the non-Euclidean case, a value is given explicitly by a formula due to Gauss. construction taking up the last ten sections of The Gauss curvature of the surface at the given Bolyai’s appendix and described in detail by Gray. point is by definition the Riemannian (sectional) cur- It is in the description of the contributions by vature of the manifold at the point in the given Riemann and his successors Beltrami and Poin- two-dimensional direction. As an illustration, he caré that I find Gray’s version of the history of the gives explicit formulae for Riemannian metrics of subject to be somewhat lacking. After noting that constant positive, negative, and zero curvature in Riemann chose not to limit his differential geom- any number of dimensions.