ERRATUM On page 143 the title of the book reviewed should read: Joan L. Richards,Mathematical Visions: The Pursuit of Geometry in Victorian England, Academic P ress,...... BOOK REVIEWS

Joan L. Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England, San Diego etc; 1988, 13+266 pages, ISBN 0-12- 587445-6.

In Victorian England there were few eminent creative mathematicians, but mathematics was widely regarded as a very important part of ed­ ucation. Mathematics was esteemed, not so much for its own sake but as a training for the mind, with moral and religious principles enhanced by the knowledge of the certain truths taught in mathematics. Above all, the geometry of Euclid was the paradigm of certain knowledge. As John F. W. Herschel wrote in 1840 (page 25),

The truths of geometry exist and are verified in every part of space, as the statue in the marble. They may depend upon the thinking mind for their conception and discovery, but they cannot be contradictory to that which forms their subject-matter, and in which they are realized, in every place and at every instant of time. But, during the 1850s, some mathematicians became aware of the pub­ lications by Nikolai Lobachevskii (from 1829) and Janos Bolyai (1832) of geometries which differed from the classical geometry of Euclid. In 1865 Arthur Cayley published a puzzled “Note on Lobatchewsky’s Imaginary Geometry” , and for several decades thereafter the interpretation of non- Euclidean geometries remained a central concern for British mathemati­ cians. Bernhard Riemann delivered his inaugural lecture at Gottingen “On the hypotheses which lie at the bases of geometry” in 1854, and it was published in 1866. The German physiologist Hermann von Helmholtz promptly presented more comprehensible accounts of the very general view of geometry advocated by Riemann. He combined ideas from physiology, philosophy and mathematics in a manner which impressed many Englishmen. At University College London, the eminent mathe­ matician William Kingdon Clifford (1845-1879) studied the writings of Lobachevskii, Bolyai, Gauss, Riemann and Helmholtz, and he became a leading exponent of the new geometries. He translated Riemann’s lec­ ture into English ( Nature 8 (1873), 14-17 and 36-37), and in his famous lecture on “The postulates of the science of space” (1873) he proclaimed Lobachevskii as one of the most brilliantly original of scientific thinkers:

143 What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatchewsky to Euclid .... there is a real parallel between the work of Copernicus and his suc­ cessors on the one hand, and the work of Lobatchewsky and his successors on the other. In both of these the knowl­ edge of Immensity and Eternity is replaced by the knowl­ edge of Here and Now.

Clifford emphasized the need to re-think the foundations of physics, since space could not be assumed to have any particular geometry. In par­ ticular, he suggested that the curvature of space might not be constant, and that atoms might consist of local singularities in the curvature of space. The logician and economist William Stanley Jevons disputed Helmholtz’s concept of the applicability of mathematics, and insisted that, whatever mathematical interest various geometries may possess, Euclidean ge­ ometry remained fundamental to the understanding of physical space. Other British philosophers, including Samuel Roberts and J.P.N. Land, insisted upon the primacy of Euclidean geometry. In 1869 the Meta­ physical Society was founded by several eminent scientists, theologians and writers with the aim of reducing the tensions between science and religion, with non-Euclidean geometries and Darwinian evolution as the central concerns. Projective geometry had been developed largely in France during the first half of the 19th century. In England, projective geometry was first considered seriously by Cayley, in 1859, when he emphasized the im­ portance of cross-ratio and used that to develop a distance function which was invariant under projection and section. J.J. Sylvester ex­ citedly extended Cayley’s ideas about Projective geometry being more fundamental than Euclidean geometry, and in 1871 Felix Klein presented a projective interpretation of non-Euclidean geometry, based on Cayley’s definition of distance. However, in 1879 R.S. Ball pointed out a circu­ larity in Cayley’s argument, since cross-ratio itself was defined in terms of distances. In 1889 Cayley re-considered the entire problem, without coming to any definite conclusion. He did not accept fully the approach of von Staudt, who had (as we now realise) succeeded in treating cross- ratio without assuming a distance function beforehand. AH of this metaphysical disputation had little effect upon the teaching of mathematics at English schools, where Euclid was canonised. Despite pleas by Sylvester and others to abandon the use of Euclid’s Elements

144 as a textbook, Augustus De Morgan, Charles Lutwidge Dodgson and others insisted upon the benefits to be gained by schoolboys in study­ ing the text of Euclid. (And indeed, geometry continued to be taught from diluted adaptations of Euclid’s Elements until the 1960s in New Zealand, as in the U.K.) By 1910, Bertrand Russell had convinced most English mathemati­ cians that the old certainties in geometry had vanished, and G. H. Hardy was leading mathematics in England into new directions which went far outside the range of discussion in the 19th century. Some minor misprints should be corrected in later printings, includ­ ing Charles Biddel Airy for George Biddell Airy (page 13), Grattan- Guinness, Ivar (for Ivor) on page 251, the date 1978 for Thomas L. Han­ kins’s book on Sir W illiam Rowan Ham ilton (1980), and the date 1934 for T.T. Thompson’s curious work G eom etry W ithout Axiom s (1834). This book is a stimulating and thought-provoking historical study of the interactions between the philosophy of mathematics and social thought. G. J. Tee

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