HlSTORlA MATHEMATICA 17 t 1990). 76-80

REVIEWS

Editedby KAREN HUNGER PARSHALL

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Die Geschichte der Geometrischen Mechanik im 19. Jahrhundert. By Renatus Ziegler. Franz Steiner Vet-lag Wiesbaden GmbH Stuttgart. Boethius XIII. 1985. vii + 260 pp. DM 48.

Reviewed by Jeremy Gray

Faculty of Mathematics, The Open University, Milton Keynes MK7 6AA, England

This book discusses an important and neglected branch of the history of mathe- matics, and so the author is able to reap the rewards of being the first to enter a new area. He brings us a singularly informative and clearly written treatment of the following connected topics: the study of forces acting on a rigid body from a geometric point of view; line geometry; and a non-Euclidean, specifically elliptic geometry of space. How do these subjects come together, and what gives this book its coherence?

Ziegler starts his story with the attempts by Poinsot, Chasies, Mobius, and Plucker to rederive the theory of rigid body motion under forces from a geometric rather than an analytic or algebraic point of view. Since forces in space act along lines, the problem of dealing geometrically with the set of all lines in space arises, and so the mechanical questions led naturally to the invention of coordinate descriptions of a line in space. A system of forces can be represented, it was soon discovered, by the composition of a force and a couple; the ensuing motion can be regarded as a translation of the center of gravity of the body combined with a rotation of the body around its center of gravity. This hint of symmetry intrigued the early writers, and even, as Ziegler shows, sometimes led them briefly astray.

76 0315-0860190 $3.00 Copyright 0 1990 by Academic Press. Inc. All rights of reproduction in any form reserved.

HM 17 REVIEWS 77

In fact, there is no simple relationship between a force and a translation, a couple and a rotation. It was noted that the composition of rotations is dependent on the order of the rotations, unlike the composition of translations, which do commute. However, infinitesimal rotations commute, and this provided Mobius with what Ziegler considers his most fundamental insight into geometrical mechanics-the analogy between infinitesimal rotations and forces.

In a satisfying and long overdue study of Plucker, Ziegler shows how this fascinating geometer was led to his theory of line geometry, in which the basic element is a line in three-space (and the space of all lines is four-dimensional). He shows just what Plucker accomplished, and in this way sets the scene for Klein, who did so much to bring line geometry and mechanics together. Klein’s work in this area forms his first original contribution to mathematics and, as Ziegler points out, these early researches are important to understanding his later work, but most authors are content to jump over them to get to papers that are better known. Ziegler shows us what we have missed.

He also shows us what riches are to be found in the work of Klein’s student Lindemann, today chiefly remembered as the man who showed that 7~ is transcen- dental. This brings us to elliptic non-Euclidean geometry, for it was Lindemann’s fortune to realize that the way to unify geometrical mechanics was to exploit Klein’s newly created non-Euclidean metrics. To see why this is, observe that a duality which pairs translations and rotations is required. This must be metrical, because these quantities can be measured, and the metric must impose a maxi- mum length on a translation. Such a duality is familiar to us from spherical geometry, in which lengths along great circles are essentially measured by the angles subtended at the center of the sphere. Indeed, the formulae of spherical trigonometry do display a duality between lengths and angles. Klein had shown how to introduce metrics into projective geometry via a choice of absolute conic. Lindemann showed how the introduction of a suitable hyperboloid of one sheet made projective three-space into an elliptic space with the required duality. In- deed, as Ziegler ably discusses, Lindemann showed how to embed all of Klein’s ideas about line geometry and mechanics in a similarly rich duality.

There is much else to like in this book, including informative sections on English and Italian developments. It should be read by anyone interested in the history of applied mathematics, line geometry, and elliptic non-Euclidean geometry. One looks forward to the many interesting studies this book will help to make possible.