Wednesday, March 28, 2018


Robert Hahn, The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles. SUNY series in ancient Greek philosophy. Albany: SUNY Press, 2017. Pp. 300. ISBN 9781438464893. $95.00.

Reviewed by Guy P.R. Métraux, York University (

Version at BMCR home site

Socrates, we are told, enjoyed visiting artists in their ateliers (Xen., Mem. 3.10.1-8): how artists moved from manual work with materials (marble, bronze) to considerations of the soul and metaphysical structures was a smooth intellectual trajectory. In addition, craftsmen/artists, especially architects, could supply philosophers with practical models for cosmic constructs. The interaction of material and intellectual realms is the topic of this study.

In past books and articles, Robert Hahn has started with the premise that our sparse and contradictory knowledge of the Ionian pre-Socratic philosophers—the monists Thales and Anaximander of Miletus and Anaxagoras of Clazomenae—can be given greater substance if seen in their local milieux. Greek intellectual thought and its origins in the archaic period were partly based on what such thinkers learned from the practice of architecture– the immense Ionic temples of the Greek east– and allied arts. Architecture is not confined to finished buildings but comprises the whole process, from quarrying, roughing out, and moving and lifting stone to the techniques for sculpting elements to ensure their stability, their proportional and spatial relationships, and even their decoration and roofing.1 The practical geometries of construction gave shape to the ordered geometries of metaphysical structure: in part, early Greek philosophers found answers to some cosmic questions from the building trade in their Ionian home-towns.2

In the elaborate Introduction (1-43), Hahn continues his theme of linking practical and metaphysical endeavors and, by adding Pythagoras of Samos, he proposes an Ionian comity of knowledge (from the late seventh into the early fifth centuries BCE) rather than a succession of intellectual discoveries. In addition, he posits that the philosophers shared an intention to "resolve a metaphysical problem" (5, 11-2, the author's emphasis) based in part on much earlier Mesopotamian and Egyptian methods of geometric mensuration. These methods had been illustrated with diagrams and figures, as evidenced by cuneiform tablets and mathematical papyri, but as such they were mainly used to train surveyors and storage officials in determining the areas of oddly configured land, the capacity of granaries, and other spatial and practical problems (7-25). There is also some evidence that Egyptian painters used reductions arrived at geometrically to define humans of lesser status in relation to those of higher rank (14-17)—using geometry to denote social significance. A tile incised with a geometrical sketch from the Ephesian Artemision (mid-seventh century BCE) was used to instruct painters on how to locate the centers and cast concentric circles of decorations on roofing-tiles (32-5)—using geometry to generate a small-scale template for monumental repetition.

The Introduction ends with an analysis of the digging of the mid-sixth century BCE Eupalinos tunnel on Samos, an aqueduct supplying the city with water (35-43). Hahn shows that the letters marked on the walls corresponded to a lettered diagram mapping the work of the two teams of diggers working toward each other from the north and south ends along a straight line, with a module of 20.6 m subdividing the estimated length of the tunnel under Mount Kastro. Unstable rock necessitated a detour, so Eupalinos calculated the simultaneous change of direction for both digging teams by "constructing"—underground—an isosceles triangle (marked with the same number of letters as the straight line but, in consequence, with a shorter module of 17.59 m). It was his knowledge of the properties of isosceles triangles that allowed Eupalinos to predict, within good tolerance, that the two teams of diggers would meet. Geometry is thus a universal device to solve problems of any kind.

The succeeding chapters refine the propositions of the Introduction by reviewing and updating the history of early geometry. Chapter 1 ("The Pythagorean Theorem", 45-89) analyzes the hypotenuse theorem attributed to Pythagoras, not, as Hahn points out (45) as a modern algebraic equation (a2 + b2 = c2) but as extensions of right triangles deriving from, and generating, squares of different sizes (as per Euc., Elements I.47; 46-66); the "enlargement" of the theorem broaches the relation of right triangles to other forms of triangles, rectangles, geometric shapes, and polygons (id. VI.31, with ratio values at VI.19-20; 66-81). The reducibility of all shapes to the ratios derived from right triangles "tells us how many small worlds fit into the big world . . . The whole cosmos is conceived aggregately out of small worlds" (71).

In Chapter 2 ("Thales and Geometry", 91-133), Hahn brings us to Thales' two famous feats: the measuring of the height of a pyramid and of the distance from shore of a ship at sea. The measurement of a pyramid on the day that the sun stands at a 45o angle to give a length of shadow to its height (97-106) is shown to be impractical: the days are few and the shadow can change within a few moments, resulting in measurements too varied to be convincing. (Some steep-sided pyramids do not throw shadows beyond their base at all.) Still, the principles of similarity and scalable proportions among triangles was understood, leading to Thales' deriving the height of a pyramid by simultaneously measuring its shadow and the shadow cast by a gnomon, then upscaling proportionally from smaller to larger (107-08, 113-15, later at 143). A similar method could determine the distance of ships at sea, either "on the flat" from the land or from elevated view-points (108-13). From these practical applications, the study of triangles, triangles within half-circles, and proportional relations among triangles would have led Thales to an understanding of the hypotenuse theorem as expressed in diagrams. The plenitude of the world was reducible to simple, single, but multipliable geometric figures, and gave rise to the search for a simple, single, universal guiding form. Apparent variety was reducible to a comprehensible One.

In their historical narratives, philosophy and mathematics can be framed as progressive, with one breakthrough following upon another linearly. Other narratives exist which frame philosophy and mathematics as explorations of intellectual phenomena within a shared mental topography. In Chapter 3 ("Pythagoras and the Famous Theorems", 135-212), Hahn combines the two narratives by back-dating the hypotenuse theorem (or a grasp of it) to the later seventh / early-sixth century BCE rather than to Pythagoras and his followers in the late sixth / early fifth century. His complex argument takes in the "squaring of the circle" (quadrature of lunes; 1137-40) by Hippocrates of Chios (fl. 440 BCE) and the study of the incommensurability of the hypotenuse (and other odd-even numerical relations; 144-8) by Hippasius of Metapontum (mid-5th century BCE). "Squaring the circle" and irrational numbers both presuppose a long knowledge of the hypotenuse theorem that had been clinched long before by Thales' measurement of the height of a pyramid (143). The application of geometry to musical intervals (octave, fifth, fourth) and the "harmony of the spheres" is analyzed in detail. Hahn then returns to the geometric strategies by which Eupalinos laid out the Samos tunnels; levelling from sea-level, determining (and changing) the direction of the digging so the tunneling-teams could meet, and the geometric means—"microcosmic-macrocosmic reasoning" and "analogia– by means of ratio and proportions"—by which the problem was solved (158-65). Thales (for the pyramid) and Eupalinus (for the tunnel) "had to supply a number" to show the authorities that their calculations were practical, but the number was arrived at by a well-established tradition of geometric speculation (148).

The remainder of Chapter 3 concerns Pythagoras and his associates as geometers (168-212). As with the hypotenuse theorem, Hahn (with other scholars) pushes back the date of some of the so-called Platonic solids—tetrahedron, cube, octahedron, dodecahedron, icosahedron— more than a century, to Pythagoras and his associates (201-12, esp. 202, 210) rather than to the mid-fourth century and Theaetetus, because these shapes (surfaces of the same size and angles meeting at the same number of vertices) were all derived or derivable from right angles.

A final Chapter 4 (Epilogue, 213-39) is a reprise and a useful recapitulation, for the non-specialist, of the many arguments in the history of Greek geometry and mathematics, in terms that cover the intellectual meaning, cosmic structure, and practical application of the early philosophers' work. The arguments are graphically accompanied by some repetitions of illustrations to keep the necessarily complex arguments clear and accessible. Hahn ends with the story of Pythagoras, who in some accounts became a vegetarian in old age and a believer in animal reincarnation, celebrating his youthful discoveries with a splendid sacrifice of oxen.

It is a mark of Hahn's generosity that he includes, in every chapter, clear analysis of the geometric problems together with a thorough history of both ancient sources and modern mathematical theory and interpretations before proposing his own extensions; this allows the non-specialist reader to perceive, if not participate in, the often quite divergent current debates about early Greek thought. The handsome production-values and clarity of organization of this book by SUNY Press are exemplary, as are its general index and index of Greek terms.3

The only regret the reader may have is that Hahn does not address, even in an endnote, the relation of early Greek geometry with the architectural diagrams of later Greek architects. The evidence of architectural drawings on the floor surfaces at the Didymaion (temple of Apollo at Didyma) is well known and could provide further material for Hahn's methodology.4


1.   R. Hahn, Anaximander and the Architects. The Contributions of Egyptian and Greek Architectural Technologies to the Origins of Greek Philosophy (Albany, NY: SUNY Press, 2001); "Proportions and Numbers in Anaximander and Early Greek Thought" in D. L. Couprie, R. Hahn, and G. Naddaf, eds. Anaximander in Context. New Studies in the Origins of Greek Philosophy (Albany, NY: SUNY Press, 2003), 71-163; Archaeology and the Origins of Philosophy (Albany, NY: SUNY Press, 2010); and many articles.
2.   The method is not without its critics: see, e.g. D. L. Couprie in Aestimatio 8 (2011), 78-96.
3.   The only editorial mistake I noticed is the non-indentation of the paragraph of the English translation of a lengthy Greek quotation from the Timaeus (196).
4.   See L. Haselberger, "Die Bauzeichnungen des Apollontempels von Didyma," Architectura 13 (1983), 13-26.

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