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Greek geometer and writer on mechanical and physical subjects, probably flourished in the second half of the 1st century. This is the more modern view, in contrast to the earlier theory most generally accepted, according to which he flourished about loo B.C. The earlier theory started from the superscription of one of his works, "Hpwvos Kr77vc(51ov , eXoiroiirca, from which it was inferred that Hero was a pupil of Ctesibius. Martin, Hultsch and Cantor took this Ctesibius to be a barber of that name who lived in the reign of Ptolemy Euergetes II. (d. 117 B.C.) and is credited with having invented an improved water-organ. But this identification is far from certain, as a Ctesibius mechanicus is mentioned by Athenaeus as having lived under Ptolemy II. Philadelphus (285-247 B.C.). Nor can the relation of master and pupil be certainly inferred from the superscription quoted (observe the omission of any article), which really asserts no more than that Hero re-edited an earlier treatise by Ctesibius, and implies nothing about his being an immediate predecessor. Further, it is certain that Hero used physical and mathematical writings by Posidonius, the Stoic, of Apamea, Cicero's teacher, who lived until about the middle of the 1st century B.C. The positive arguments for the more modern view of Hero's date are (I) the use by him of Latinisms from which Diels concluded that the 1st century A.D. was the earliest possible date, (2) the description in Hero's Mechanics iii. of a small olive-press with one screw which is alluded to by Pliny ( Nat. Hist. viii.) as having been introduced since A.D. 55, (3) an allusion by Plutarch (who died A.D. 1 20) to the proposition that light is reflected from a surface at an angle equal to the angle of incidence, which Hero proved in his Catoptrica, the words used by Plutarch fitting well with the corresponding passage of that work (as to which see below). Thus we arrive at the latter half of the 1st century A.D. as the approximate date of Hero's activity.

The geometrical treatises which have survived (though not interpolated) in Greek are entitled respectively Definitiones, Geometria, Geodaesia, Stereometrica (i. and ii.), Mensurae, Liber Geoponicus, to which must now be added the Metrica recently discovered by R. Schdne in a MS. at Constantinople. These books, except the Definitiones, mostly consist of directions for obtaining, from given parts, the areas or volumes, and other parts, of plane or solid figures. A remarkable feature is the bare statement of a number of very close approximations to the square roots of numbers which are not complete squares. Others occur in the Metrica where also a method of finding such approximate square, and even approximate cube, roots is shown. Hero's expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios. Akin to the geometrical works is that On the Dioptra, a remarkable book on land-surveying, so called from the instrument described in it, which was used for the same purposes as the modern theodolite. It is in this book that Hero proves the expression for the area of a triangle in terms of its sides. The Pneumatica in two books is also extant in Greek as is also the Automatopoietica. In the former will be found such things as siphons," Hero's fountain," penny-in-theslot "machines, a fire-engine, a water-organ, and arrangements employing the force of steam. Pappus quotes from three books of Mechanics and from a work called Barulcus, both by Hero. The three books on Mechanics survive in an Arabic translation which, however, bears a title" On the lifting of heavy objects."This corresponds exactly to Barulcus, and it is probable that Barulcus and Mechanics were only alternative titles for one and the same work. It is indeed not credible that Hero wrote two separate treatises on the subject of the mechanical powers, which are fully discussed in the Mechanics, ii., iii. The Belopoiica (on engines of war) is extant in Greek, and both this and the Mechanics contain Hero's solution of the problem of the two mean proportionals. Hero also wrote Catoptrica (on reflecting surfaces), and it seems certain that we possess this in a Latin work, probably translated from the Greek by Wilhelm. van Moerbeek, which was long thought to be a fragment of Ptolemy's Optics, because it bore the title Ptolemaei de speculis in the MS. But the attribution to Ptolemy was shown to be wrong as soon as it was made clear (especially by Martin) that another translation by an Admiral Eugenius Siculus (12th century) of an optical work from the Arabic was Ptolemy's Optics. Of other treatises by Hero only fragments remain. One was four books on Water Clocks (IIEpi 5Spicov eopoarc07rEiwP), of which Proclus ( Hypotyp. astron., ed. Halma) has preserved a fragment, and to which Pappus also refers. Another work was a commentary on Euclid (referred to by the Arabs as" the book of the resolution of doubts in Euclid ") from which quotations have survived in an-Nairizi's commentary.

The Pneumatica, Automatopoietica, Belopoiica and Cheiroballistra of Hero were published in Greek and Latin in Thevenot's Veterum mathematicorum opera graece et latine pleraque nunc primum edita (Paris, 1693); the first important critical researches on Hero were G. B. Venturi's Commentari sopra la storia e la teoria dell'ottica (Bologna, 1814) and H. Martin's" Recherches sur la vie et les ouvrages d'Heron d'Alexandrie disciple de Ctesibius et sur tous les ouvrages mathematiques grecs conserves ou perdus,publies ou inedits, 9ui ont ete attribues a un auteur nomme Heron "( Mem. presentes a l Academie des Inscriptions et Belles-Lettres, i. serie, iv., 1854). The geometrical works (except of course the Metrica ) were edited (Greek only) by F. Hultsch (Heronis Alexandrini geometricorum et stereometricorum reliquiae, 1864), the Dioptra by Vincent ( Extraits des manuscrits relatifs a la geometrie pratique des Grecs, Notices et extraits des manuscrits de la Bibliotheque Imperiale, xix. 2, 1858), the treatises on Engines of War by C. Wescher ( Poliorcetique des Grecs, Paris, 1867). The Mechanics was first published by Carra de Vaux in the Journal asiatique (ix. serie, ii., 1893). In 1899 began the publication in Teubner's series of Heronis Alexandrini opera quae supersunt omnia. Vol. i. and Supplement (by W. Schmidt) contains the Pneumatica and Automata, the fragment on Water Clocks, the De ingeniis spiritualibus of Philon of Byzantium and extracts on Pneumatics by Vitruvius. Vol. ii. pt. i., by L. Nix and W. Schmidt, contains the Mechanics in Arabic, Greek fragments of the same, the Catoptrica in Latin with appendices of extracts from Olympiodorus, Vitruvius, Pliny, &c. Vol. iii. (by Hermann Schohe) contains the Metrica (in three books) and the Dioptra. A German translation is added throughout. The approximation to square roots in Hero has been the subject of papers too numerous to mention. But reference should be made to the exhaustive studies on Hero's arithmetic by Paul Tannery," L'Arithmetique des Grecs dans Heron d'Alexandrie "( Mem. de la Soc. des sciences phys. et math. de Bordeaux, ii. serie, iv., 1882)," La Stereometrie d'Heron d'Alexandrie "and" Etudes Heroniennes ( ibid. v., 1883), "Questions Heroniennes" ( Bulletin des sciences math., ii. serie, viii., 1884), "Un Fragment des Metriques d'Heron" ( Zeitschrift fir Math. and Physik, xxxix., 1894; Bulletin des sciences math., ii. serie, xviii., 1894). A good account of Hero's works will be found in M. Cantor's Geschichte der Mathematik, i. 2 (1894), chapters 18 and 19, and in G. Loria's studies, Le Scienze esatte nell' antica Grecia, especially libro iii. (Modena, 1900), pp. 103-128. (T. L. H.)