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80 Books harmony manifest themselves, and where in the vast literature of mathematics such qualities may be found. These difficulties are considerably eased by reading Huntley’s personal interpretations of the subject. In a simple, non-technical way the reader is introduced to the curious properties of Pascal’s triangle, the logarithmic spiral, the number ratios underlying the tempered musical scale, magic squares and the conic sections. Throughout these discussions, for example, of plane and solid geometry , number theory and probability one finds the persistent recurrence of the golden section. The author’s reliance on the golden ratios is not, however , an arbitrary whim, indeed a comparable book might have been written using group theory or topology as its basis. Huntley’s argument, developing as it does from the historical foundations of mathematics, stressing its relatedness to art, music and philosophy, makes the presence of number series virtually inescapable. Moreover, those artists to whom the use of numerical ratios in art appears to be overly academic and hackneyed would do well to read the section (pp. 62-65) that describes the researches into aesthetic preferences undertaken by Fechner and Lalo during the last century. In this connexion, it is worth mentioning the paper ‘Duality and Synthesis in the Music of Bela Bartok’ by Ern0 Lendvai (in Module, Symmetrv, Proportion, Rhythm, Ed. G. Kepes, Brazillier, New York, 1966), which describes the appearance of proportions identical to the golden ratios in the music of Bartok. The fact that the composer was not consciously aware of the existence of such proportions in his work may be seen as lending weight to Huntley’s thesis that aesthetic satisfaction lies in the perception of metrical and geometrical patterns. Working from the view that the discernment of such elementary structures and relationships forms the core of the human aesthetic sense, the author then proceeds to show that the creation and/orappreciation of art and music, or ofa mathematical theorem, are far from being distinct processes determined by different sets of criteria. Rather, in so far as they may all be considered beautiful or harmonious, they all satisfy the same primordial psychological drive. The author wisely omits references to specific works of art, since the subject of the rigorous application of golden (or any other) ratios to traditional painting and architecture is disputed by historians. Although the late Greek architect Hermogenes advocated an architecture based on fixed arithmetical ratios, there is little archaelogical or analytical evidence to show that any Hellenic building was systematically determined by mathematical formulae. There is a similar lack of indisputable evidence to support the theory that mathematical procedures were, in general, used by the painters of the Italian Renaissance. Nevertheless, it is a pity that Huntley makes no mention of either Vitruvius or Palladio, especially the latter, who did employ a strict proportional system in architectural design. In addition, the painters Piero della Francesca and Albrecht Durer might have been included. Both were actively concerned with mathematical researches, although, again, we cannot be sure of the extent to which they appliPd mathematical principles to painting. Huntley’s literary style is consistently economical and lively, all the mathematical topics are simply explained and he manages to maintain an easily readable balance between wordage and mathematical symbolism, a rare achievement in books of this kind. The book covers a very broad area of mathematical history and theory, taking the reader through two thousand years of creative endeavour, easily encompassing widely divergent realms of human achievement and incorporating them into a unified whole, whilst never failing to communicate the writer’s love for his subject. Considering the scope and quality of this compact volume, I must add that my earlier criticisms should be regarded as very minor ones. The Discovery of Talent. The Walter Van Dyke Lectures on the Development of Exceptional Abilities and Capabilities. Ed., Dael Wolfle. Harvard University Press, Cambridge, Mass., 1969. 316 pp. $9.50. Reviewed by: Harold K. Hughes* The editor of this collection of eleven invited lectures, presented between 1954 and 1965, signed his introduction in October 1968 before ‘the urgent social problem of preventing potential talent from being killed or stunted’ dramatically shifted to the urgent social problem of...

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