Hero of Alexandria: Mathematics.

The historical evaluation of Hero’s mathematics, like that of his mechanics, reflects the recent development of the history of science itself. Compared at first with figures like Archimedes and Apollonius, Hero appeared to embody the “decline” of Greek mathematics after the third century b.c. His practically oriented mensurational treatises then seemed to be the work of a mere “technician,” ignorant or neglectful of the theoretical sophistication of his predecessors. As Neugebauer and others have pointed out, however, recovery of the mathematics of the Babylonians and greater appreciation of the uses to which mathematics was put in antiquity have necessitated a reevaluation of Hero’s achievement.¹ In the light of recent scholarship, he now appears as a well-educated and often ingenious applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe.

The breadth and depth of Hero’s mathematics are revealed most clearly in his Metrica, a mensurational treatise in three books that first came to the attention of modern scholars when a unique manuscript copy was found in Constantinople in 1896.² The prologue to the work gives a definition of geometry as being, both etymologically and historically, the science of measuring land. It goes on to state that out of practical need the results for plane surfaces have been extended to solid figures and to cite recent work by Eudoxus and Archimedes as greatly extending its effectiveness. Hero meant to set out the “state of the art,” and the thrust of the Metrica is thus always toward practical mensuration, with a resulting ambiguity toward the rigor and theoretical fine points of classical Greek geometry. For example, Hero notes in regard to circular areas:

Archimedes shows in the Measurement of the Circle that eleven squares on the diameter of the circle are very closely equal [ΐσα γίγνεται ώςέγγιστα] to fourteen circles.... The same Archimedes shows in his On Plinthides and Cylinders that the ratio of the circumference of any circle to its diameter is greater than 211875 to 67441 and less than 197888 to 62351, but since these numbers are not easily handled, they are reduced to least numbers as 22 to 7 [Metrica I, 25, Bruins ed., p. 54].³

That is, Hero’s use of approximating values for irrational quantities arose not out of ignorance of their irrationality or of theoretically more precise values, but out of the need for values that can be handled efficiently. In the case of (n non-square) he set out an iterative technique for ever closer approximation, although he himself usually stopped at the first.⁴

Book I of the Metrica deals with plane figures and the surfaces of common solids. It proceeds in each case by numerical example (with no specified units of measure), presuming a knowledge of elementary geometry and supplying formal geometrical demonstrations where they might be unfamiliar. Beginning with rectangles and triangles, Hero gave, in proposition 1.8, the famous “Heronic formula” for determining the area of a triangle from its three sides. (the proposition is actually derived from Archimedes). He then proceeded to treat general quadrilaterals by dividing them into rectangles and triangles. Metrica I. 17–25 treats the regular polygons of from three to twelve sides, directly deriving the relation of side to radius in all cases except 9 and 11, where Hero appeals to the “Table of Chords” (τά περί τών έν κύκλω εύυεΐων). For n = 5, 6, and 7, the relations derived are the same as those found in the Babylonian texts at Susa.⁵ After discussing the circle and annulus, Hero dealt extensively with the segment (but not the sector) of a circle, offering three approximating formulas, two “ancient” (which he criticized) and his own, Which treats the segment as closely approximating a segment of a parabola (area = 4/3 inscribed triangle); Archimedes’ Method is the explicit source for the latter. For the ellipse and parabola, and for the surfaces of a cone, sphere, and spherical segment, Hero did no more than cite Archimedes’ results.

Book II moves on to solid figures. Beginning with the cone and the cylinder, Hero then dealt with prisms on various rectilinear bases and with regular and irregular frustra (including the famous βωμίσκος).⁶ For the sphere he turned again to Archimedes; for the torus, to Dionysodorus. He concluded with the five “Platonic” solids (regular polyhedra).

In book III the treatment of the problem of dividing plane and solid figures into segments bearing fixed ratios to one another brought Hero’s work more closely in line with the pure mathematical tradition. Very similar in style and content to Euclid’s On Divisions, the subject matter forced Hero after proposition III.9 to give up numerical calculation in favor of geometrical construction of the lines and planes sought. Nonetheless, the problem of dividing the pyramid, cone, and conical frustrum required an approximating formula for the cube root of a number.⁷

Hero’s concern in the Metrica—to extract from the works of such mathematicians as Archimedes only the results conducive to efficient mensuration—takes full effect in the other works that have come down to us bearing his name. Geometrica is essentially book I of the Metrica; Stereometrica is essentially book II. In both cases, numerical examples are used to eliminate geometrical derivations, concrete rather than general units of measure are employed, and the Greek mode of expressing fractions yields to the then more common and familiar Egyptian mode of unit fractions. Geodaesia and De mensuris contain nothing more than excerpts from the Geometrica. In all these texts, it is difficult to locate precisely Hero’s original contribution, for they, rather than the Metrica, are the texts that circulated widely, were edited frequently, and were used for instruction. That their fate conformed at least in part to Hero’s intention is indicated by his Definitiones and Commentary on Euclid’s Elements, both of which show clear pedagogical concerns. As Heath notes,⁸ the Definitiones, which contains 133 definitions of geometrical terms, is a valuable source of knowledge about alternative notions of geometry in antiquity and about what was taught in the classroom; it, like the others, shows the effect of many editors.

Hero’s works enjoyed a wide audience. This is clear not only from what has been said above, but also in that fragments of his works can be found in the writings of several Arab mathematicians, including al-NayrĪzĪ and al-Khwārtzmī.

NOTES

1. Otto Neugebauer, Exact Sciences in Antiquity, ed, 2 (New York, 1962), p. 146.

2. See Bibliography in section above for various modern editions.

3. Tannery has suggested correcting the numerators to read 211872 and 195882, respectively; respecectively; cf; T. L. Heath. History of Greek Mathematics, 1 (Oxford, 1921), 232–233.

4.Metrica I, 8 (Bruins ed., p. 41): let N = a² ± r; is a first approximation for is a second, more accurate one, and so on. On the history of this method, see Heath, II, 324, note 2.

5. Neugebauer, p. 47.

6. For a discussion, see Heath, II, 332–333.

7. Heath, II, 341–342. ero’s method must be reconstructed from a single numerical example. The best conjecture (Wertheim’s) seems to be that if a³ < N < (a + 1)³, d₁ = N-a³, and d₂ = (a + 1)³-N, then

8. II, 314; pp. 314–316 present a summary of the contents of the Definitiones.

BIBLIOGRAPHY

T. L. Heath, History of Greek Mathematics, vol. II (Oxford, 1921), ch. 18, remains the most complete secondary account of Hero’s mathematics and is the source of many of the details given above.

Michael S. Mahoney