HERO OF ALEXANDRIA

(fl. Alexandria, maybe first century CE),

mathematics, optics, pneumatics, mechanics. For the original article on Hero see DSB, vol. 6.

The main points in the present postscript are the problem of Hero’s dating, returned to scholarly attention; an updated account of the treatise ascribed to him and known as De speculis; the extent, contents and aims of his commentary on Euclid’s Elements; and his peculiar approach to the procedure of analysis and synthesis. Scholarship is beginning to assess Hero’s mathematical works in a way unbiased by his traditional renown as the mechanician. It appears that he was a skilled and original author who played a key role in the development of Greek mathematics, not only as a “vital link in a continuous tradition of practical mathematics” (DSB, p. 314), but also in mastering and adapting purely theoretical tools.

Hero’s Dating . A lunar eclipse was described in Dioptra 35 as observable in Alexandria at the fifth hour (and in Rome at the third hour) of the night ten days before the vernal equinox. The eclipse was identified as the one that occurred on 13 March 62 CE, although with some latitude, because the vernal equinox fell on 20 March 62 CE. This provides a terminus post quem. The argument was completed by observing that the eclipse must have really been observed by Hero because its proximity to the equinox makes it particularly ill-suited to the graphical solution he is proposing for the analemma-problem solved in Dioptra 35. The identification has been challenged by two kinds of arguments. The first argument is a statistical one. References to Hero in Pappus of Alexandria and to Archimedes in Hero leave a span of approximately five hundred years as the one in which Hero must have lived. Within such a time span there is a probability of about three over four that at least one lunar eclipse actually fits the date, and even more if some margin for error is allowed. The probability reduces to about 10 percent if one inserts the time datum, but remains nonnegligible. Therefore, the fact that one eclipse fitting Hero’s data has been found does not authenticate his description as a reference to a historical eclipse rather than a textbook example. The second argument concerns the identification of the eclipse itself. Two other lunar eclipses fit the temporal data as well as the one of 62, and the date actually better than that. In any of the three cases, however, considerable latitude must be allowed for the determination both of the date of the equinox and of the hours in which the eclipse took place in Alexandria and in Rome: one might well wonder how much can be tolerated for Hero’s description to keep the status of an accurate observational report. The argument that the eclipse is ill-suited to a graphical solution does not stand, because it can be shown that the same problem can be solved exactly, and of course also the other two eclipses are ill-suited as well.

To these arguments it can be added that Hero’s reference to the eclipse was worded in a very peculiar way. He asserted that if simultaneous observations of a lunar eclipse cannot be found in the almanacs, then the reader can provide one, because they occur at intervals of five or six months. After that, the mere existence of the eclipse and its temporal data are set out, by using imperatives, in the typical format of geometrical propositions. This puts the assertions under the hypothetical mode of expression and strengthens the suspicion that the data of the eclipse in Dioptra 35 have been invented by Hero. Finally, care should be taken in handling astronomical phenomena for dating purposes. They are easily transformed from termini post quem into absolute chronological determinations by a very questionable application of a principle of economy of hypotheses. The same holds for chronological determinations that have been proposed on the basis of other passages in Hero’s works. What is more, all of these concur to a year near 62 and are in fact intended to support the dating by means of the eclipse. The search for textual clues is thus biased by the assumption that this is correct and hence they are of no independent value.

The De speculis . A new edition led to an improved assessment of this short monograph, preserved only in the Latin translation of William of Moerbeke. The ascription to Ptolemy in the title was long ago recognized as a copyist’s mistake: for the late optical author Damianus ascribed to Hero a proof that straight lines reflected at equal angles are minimal among those that inflect on the same (straight) line. This is precisely the proof found in the De speculis, even if Damianus’s testimony does not guarantee that the short treatise is originally Heronian. It is now clear that the Greek text translated by Moerbeke was a late compilation. A first clue comes from the phrase in rymis sive in plateis (in the streets or lanes) in the enunciation of section 22: it transliterates a Greek expression in Luke 14:12 and otherwise found only in Christian writers. This presupposes in the author an acquaintance with Christian literature that suggests a late dating. The very initial portion of the introduction refers to Plato and develops considerations on the harmony of the spheres that are at odds with the typically direct argumentations of Hero’s introductions.

The original Heronian work appears, as far as can be discerned from a critical assessment of the extant De speculis, to have aimed at providing tighter foundations for the theory of reflection than the ones found in the Catoptrics ascribed to Euclid. To this end, the equal-angle rule for reflection is based on a less contrived assumption than the one in the Catoptrics. The argument leading to this basic assumption is a combination of mathematical and physical arguments that is typically Heronian. Visual rays move with infinite speed, and what moves with the greatest speed moves along a straight line, for, because of the speed, it tends to move along the shortest path and straight lines realize the shortest path between two fixed points. In case of reflection, direct vision being excluded, the assumption naturally reduces to the requirement of realizing the shortest path through broken straight lines. The compiler completed the original theoretical part, that possibly contained also a proof of the contrived assumption of the Catoptrics, with a rather arbitrary choice from an initial segment of the Catoptrics itself. The subsequent description of the several mirror devices was shortened to an extent that cannot be determined, as the introduction mentions arrangements that are not found in the extant text. The bad status of many of the excerpted descriptions must more likely be ascribed to accidents in the transmission of the text than to ineptitude of the compiler. An exception is section 23, where a defective trivialization of Catoptrics 14 is expounded. Marks of original Heronian conception surface here and there, however: for instance, the proof in section 22 is shaped in the format of analysis and synthesis and a dioptra is employed to perform a part of the construction.

The Commentary to the Elements . Extensive and literal quotations from Hero’s commentary to Euclid’s Elements are found in Proclus and in a number of Arabic commentators, most notably in an-Nayrizi. The title of the commentary as reported in the Fihrist and in other Arabic sources is Book of the Solution of the Difficulties in Euclid. It appears that Hero was one of the first authors of technical commentaries. He possibly originated the genre by fixing some canonical critical attitudes toward the commented text:

addition of missing cases, of lemmas and corollaries or of mathematical complements;

proposal of alternative proofs suited to strengthen the deductive structure or to simplify it;

replacement of proofs by reduction to impossible with direct proofs or, less frequently, the opposite;

structural adjustments, such as inversion of the order of certain propositions, suppression or addition of definitions and axioms;

even rewriting entire segments of the text, such as propositions 2–10 of Book 2.

As is clear, such a commentary was less exegetical and more aggressive than the ones compiled in late antiquity. It speaks for an author concerned with issues of logical coherence and deductive structure, adapting demonstrative techniques from other fields, as in the case of the rewritten proofs of 2.2–10. If a comparison can be made, it is more to the Alexandrian commentaries of the literary genre, where difficult passages of the edited author were explained but on the same grounds substantial modifications to the text were proposed. However, it is unlikely that Hero procured a new text of the Elements. Proposals of modifications rather than direct changes are found there and the Arabic title at least suggests that the work might have had a format and a circulation independent from the Euclidean treatise. Minor changes in the text cannot be detected with the available critical methods.

What is unquestionable is that Hero’s commentary, in the hands of later editors, interfered with the transmission of the Elements, producing modifications that affected both the received Greek and Arabic versions, often in independent ways. For instance, the alternative proof to 3.10 proposed by Hero was is now attested as such in the Greek manuscripts of the Elements, whereas the case added to 3.11 directly became 3.12. Additional cases of 3.20, 25, and 30 found their way into the text, whereas the proposed displacement of 3.25 affected a part of the Arabic tradition only. It is not clear whether Hero’s commentary aimed at didactical purposes, as is usually assumed for any such work. The fact that it belonged to a well-established literary genre appears to provide enough reasons for its existence, and its aims are more properly described as scholarly than as pedagogical.

Analysis and Synthesis . Two heterodox variants of the technique of analysis and synthesis are found in the Heronian corpus. The first occurs in the Metrica, the second in the alternative proofs to Elements 2.2–10. The proofs in the Metrica concern problems that are set out assigning numerical values to the relevant quantities, such as, for instance, the sides of a triangle whose area is sought. Typically, the sought quantity is obtained at the end of an analysis, consisting of a series of steps framed in the language of givens. The analysis is followed by a calculation. The latter is termed synthesis by Hero, who overcame in this way any concern about constructions. The synthesis–calculation repeats step by step the chain of the givens. The calculation is performed with specific numbers, but they must be taken as paradigmatic, as always in Hero, and hence the calculation must be read as a description of an algorithm. Proofs similar to those in the Metrica can be found in other treatises of applied mathematics, for instance in Ptolemy’s Almagest, where the format of the chain of givens is employed to show how to single out one magnitude from an expression in which it is involved. The unifying feature of such proofs is that the calculation is legitimated once the analysis has shown that the magnitude in question is uniquely determined by the givens of the problem.

The alternative proofs to Elements 2.2–10 are reported by an-Nayrizi. They form a strict deductive sequence. Each proposition is considered as proved insofar as it is reduced to previous results. Such a feature makes these proofs similar to the kind of analysis of complex deductive chains developed within the framework of Stoic logic. Both analysis and synthesis are framed as reductive sequences, in which two expressions, combining squares and rectangles formed from suitable lines, are transformed until it is manifest that they are equal. As a consequence, the analysis already ends with the proof of the sought equality. No chain of givens is displayed. Analysis and synthesis really become two independent and full-fledged ways of proof, as Hero himself is reported, in the short account preceding the alternative proofs, to have already stressed in his definition of the method. Such an approach can be found also in treatises dealing with number theory, such as Diophantus’s De polygonis numeris. To the Heronian proofs of 2.2–10 the alternative ones to 13.1–5 attested in the manuscript tradition of the Elements are usually but not properly compared.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY HERO OF ALEXANDRIA

Jones, Alexander. “Pseudo-Ptolemy De Speculis.” SCIAMVS 2(2001): 145–186. This is the new edition of the De Speculis, based on a fresh reading of the autograph of William of Moerbeke’s translation.

OTHER SOURCES

an-Nayrizi. Anaritius’ Commentary on Euclid. The Latin Translation I–IV. Edited by P. M. J. E. Tummers. Nijmegen, Netherlands: Ingenium Publishers, 1994. The Latin version of an-Nayrîzî’s commentary is now read in this critical edition, superseding, as far as the first four books are concerned, Ernst Curtze’s edition published as a Supplementum to Johan l. Heiberg and Heinrich Menge’s Euclidis Opera Omnia.

Argoud, Gilbert, ed. Science et vie intellectuelle à Alexandrie (I^er–III e siècle après J.-C.). Saint-Étienne, France: Publications de l’Université de Saint-Étienne, 1994. Much scholarly attention is being paid to the mechanical works of Hero. Besides the entire volume containing Souffrin’s note cited below, several contributions specifically dealing with Heronian matters may be found here.

Cambiano, Giuseppe. “Automaton.” Studi Storici 35 (1994): 613–633. This and the following article investigate the relationships between Hero’s mechanics and the philosophical background.

Guillaumin, Jean-Yves. “L’Éloge de la géométrie dans la préface du livre 3 des Metrika d’Héron d’Alexandrie.” Revue des etudes anciennes 99 (1997): 91–99.

Høyrup, Jens. “Hero, Ps.-Hero, and Near Eastern Practical Geometry. An Investigation of Metrica, Geometrica, and other Treatises.” In Antike Naturwissenschaft und ihre Rezeption, edited by Klaus Döring, Bernard Herzhoff, and Georg Wöhrle, 67–93 Band 7. Trier, Germany: Wissenschaftlicher Verlag Trier, 1997. This text focuses on Hero as a link between Babylonian and post-Greek mathematics.

Keyser, Paul. “Suetonius Nero 41.2 and the Date of Heron Mechanicus of Alexandria.” Classical Philology 83 (1988): 218–220. A textual finding that supports (with the reservations expressed in the text) Neugebauer’s proposal is found here.

———. “A New Look at Heron’s ‘Steam Engine.’” Archive for History of Exact Sciences 44 (1992): 107–124. The author proposes to interpret a device described by Hero in the Pneumatics as an experimental refutation of one Aristotelian thesis on self-motion.

Knorr, Wilbur R. “Arithmêtikê Stoicheiôsis: On Diophantus and Hero of Alexandria.” Historia Mathematica 20 (1993): 180–192. Knorr makes an attempt to challenge the traditional ascription to Hero of the core of the Definitiones.

Sidoli, Nathan. “Heron’s Dioptra 35 and Analemma Methods: An Astronomical Determination of the Distance between Two Cities.” Centaurus 47 (2005): 236–258. Neugebauer’s identification of the eclipse is criticized in this article.

Souffrin, Pierre. “Remarques sur la datation de la Dioptre d’Héron par l’éclipse de lune de 62.” In Autour de la Dioptre d’Héron d’Alexandrie, edited by Gilbert Argoud and Jean-Yves Guillaumin, 13–17 Saint-Étienne, France: Publications de l’Université de Saint-Étienne, 2000. This short note contains the statistical argument concerning Hero’s dates.

Tybjerg, Karin. “Doing Philosophy with Machines: Hero of Alexandria’s Rhetoric of Mechanics in Relation to the Contemporary Philosophy.” PhD diss., University of Cambridge, 2000. A rich and updated bibliography may be found here.

Vitrac, Bernard. “A Propos des démonstrations alternatives et autres substitutions de preuves dans les Éléments d’Euclide.” Archive for History of Exact Sciences 59 (2004): 1–44. A preliminary assessment of aims and extent of Hero’s commentary on the Elements is given here.

———. “Peut-on parler d’algèbre dans les mathématiques grecques anciennes?” Ayene-ye Miras n.s. 3 (2005): 1–44. The algorithmic character of some proofs in the Heronian corpus is analyzed in this article.

Fabio Acerbi