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  • Essays on the history of mechanics by Antonio Becchi; Massimo Corradi; Federico Foce;Orietta PedemonteReview by: MICHAEL DE VILLIERSThe Mathematical Gazette, Vol. 92, No. 523 (March 2008), pp. 180-181Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/27821755 .Accessed: 30/05/2014 17:05

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  • 180 THE MATHEMATICAL GAZETTE

    This is a translation of the book originally entitled Si Einstein m'?tait cont?

    published in 2005. It is another window on Einstein written in a very readable style. TONY CRILLY

    Middlesex Business School, The Burroughs, Hendon, London NW4 4BT e-mail: t.crilly@mdx.ac.uk

    Essays on the history of mechanics, (in memory of Clifford Truesdell and Edoardo Benvenuto), Antonio Becchi, Massimo Corradi, Federico Foce & Orietta Pedemonte,

    (edrs). Pp. 256. EUR 42.00 (hbk). 2003. ISBN 3 7643 1476 1 (Birkh?user Verlag). The history of mechanics applied to constructions in building and architecture is

    relatively recent. This book is dedicated in memory of Benvenuto (1940-1998) and Truesdell (1919-2000) who were two major figures in the development of this field in the past twenty to thirty years, and is based on a symposium held in their honour in Genoa in 2001. This book also forms part of an ongoing series by Birkh?user called 'Between mechanics and architecture'.

    In the first chapter Hey man (p. 10) writes that Truesdell as an applied mathematician was well aware of the contempt he was in danger of arousing when

    he first 'defected' from his 'proper' research work to study the history and

    development of mechanics. Sadly, it is often claimed that when inspiration for 'real'

    mathematical or scientific research runs dry, the scientist starts dwindling into

    becoming just 'a philosopher of science', implying that it carries a lower status.

    However, when one reads over the analyses of the historical works of major mathematicians and scientists such as Galileo, Bernoulli, Leibniz, Newton, and

    Euler, to name, but a few, one can only marvel at the major contributions they made

    to modern engineering, and architecture. For example, the tremendous challenge faced by Euler in 1744 with a hideous non-linear fourth order differential equation in

    modelling elasticity. Heyman (p. 18) points out that unfortunately in the past century there seems to

    have developed a divide between pure mathematics and applied mathematics such as

    engineering. For example, already in 1936 a Russian mathematician, Gvozdev,

    developed the fundamental theorems of the theory of plastic structures, whereas the

    engineering community carried on largely experimentally in their approach to plastic design. Not until the 1950s was a more rigorous structural theory for plastics

    developed, based on Gvozdev's work and the collaborative efforts of

    mathematicians, physicists and engineers.

    The book is spread over 10 more chapters discussing the works of a many number of scientists like Lagrange, Klein, Navier, Poisson, Beltrami, Michell, Mann,

    Hooke, Carnot, Laplace, Cauchy, Chasles, etc. It virtually reads like a Who's Who in science and mathematics. The main topics that are dealt with are deformation theory,

    mechanics of timbrel vaults, rose windows in churches, vectors, Newton's principles of motion, and impact theory.

    The chapter on timbrel vaults is nicely illustrated with photographs of timbrel domes under construction; and in other chapters original historical drawings by scientists or engineers are reproduced. Particularly interesting to

    me was the

    graphical methods used by many of the engineers, and also the reproduction of De la Hire's 1702 figure to show the parallelogram law of forces. Stevin in 1605 apparently was the first person to discover the parallelogram law of forces while studying the

    equilibrium of a thread loaded with some weights (also illustrated with a copy from the original). His further studies led to the analysis of the forces acting on a hanging chain or rope, i.e. the catenary. Notably, quite a lot of geometry was evident in much

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  • REVIEWS 181

    of this early work on mechanics extending to Daniel Bernoulli's and Huygen's further studies of the catenary, Newton's laws of motion, etc.

    The chapter by Caparrini dispels the misconception apparently portrayed in every book on the history of mathematics, that the birth of vector calculus was a

    direct consequence of the discovery of the geometrical interpretation of complex numbers by mathematicians such as Wessel, Argand, Gauss, and others in the period 1799 to 1831. Instead, the author argues and shows evidence that the roots of vector calculus can be traced further back to the sixteenth and seventeeth centuries (as alluded to by the work of Stevin and others), and even some of the ancient Greek authors. Moreover, he argues that the historical intepretation reflected in current

    books on the history of mathematics neglects to mention the fundamental influence

    that mechanics and geometry had on the development of the vector calculus.

    Basically, the scalar product arose from analytic geometry (the projection of line

    segments onto straight lines), while the vector product was the translation into the

    language of pure geometry the analytic expression for the moment of a force.

    Despite the common view that Newton discovered universal gravitation, and that

    since his time essentially no new principles to have been added Newtonian mechanics, this is also incorrect. In fact, it is argued in the chapter by Maltese (p. 203) Newton's Principia contains no equations for systems of more than two free

    mass-points, and gives no evidence of Newton having being able to set up differential

    equations of motion for more complex mechanical systems. Indeed, the common

    interpretation denies the major developments that Lagrange, Euler and Bernoulli, for example, made towards what is today known as 'Newtonian mechanics'.

    The last chapter is a philosophical reflection on the role and aim of studying the history of science. An obvious reason is that it enables deeper understanding of a

    scientist and his or her work. For example, Andr? Weil apparently once said in a

    private lecture: 'after having now penetrated into Euler's work on number theory, I

    think that I know him better than I know most of my best friends'.

    Perhaps the strongest case to be made is that it helps one to understand the origin of ideas and how theories developed, changed and matured over time. As the Dutch mathematician and educator Freudenthal was apt to say: it provides insight into how

    humans have learned over the ages, and by doing so, can also give us some useful

    insight into how children may learn the subject matter.

    However, an irritation of the book was that in several places quotes by mathematicians and scientists were given in the original language (e.g. French, Latin or Italian), but no English translation was provided alongside or as a footnote.

    The book will clearly be of general interest to mathematicians, engineers, and

    scientists with an interest in the history of science.

    MICHAEL DE VILLIERS

    University of KwaZulu-Natal, Edgewood Campus, Ashley 3605, South Africa

    Fibonacci numbers, by Nicolai N. Vorobiev. Pp. 176. EUR 28.97. 2002. ISBN 3 7643 6135 2. (Birkh?user Verlag).

    The Middle Ages really only produced one notable mathematical work, namely, Liber Abaci, by the Italian mathematician, Leonardo of Pisa, better known by his nickname Fibonacci. This famous book was first published in 1202 and again in

    1228, and contained almost all the arithmetical and algebraic knowledge of those

    times. Apart from also making Europeans aware of the value of the Hindu-Arabic

    number system, Liber Abaci contained a problem about breeding rabbits, that led to

    the introduction of the famous Fibonacci numbers.

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    Article Contentsp. 180p. 181

    Issue Table of ContentsThe Mathematical Gazette, Vol. 92, No. 523 (March 2008), pp. 1-192Front MatterNice polynomials with three roots [pp. 1-7]A geometric interpretation of equal sums of cubes [pp. 8-13]The Divine Proportion, matrices and Fibonacci numbers [pp. 14-21]The power of a point for some real algebraic curves [pp. 22-28]Twenty-one poi