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. Schöne 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-the-slot” 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 (Περὶ ὑδρίων ὡροσκοπείων), 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-Nairīzī’s commentary.
The Pneumatica, Automatopoietica, Belopoiica and Cheiroballistra of Hero were published in Greek and Latin in Thévenot’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’Héron d’Alexandrie disciple de Ctésibius et sur tous les ouvrages mathématiques grecs conservés ou perdus, publiés ou inédits, qui ont été attribués à un auteur nommé Héron” (Mém. presentés à l’Académie des Inscriptions et Belles-Lettres, i. série, 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 à la géométrie pratique des Grecs, Notices et extraits des manuscrits de la Bibliothèque Impériale, xix. 2, 1858), the treatises on Engines of War by C. Wescher (Poliorcétique des Grecs, Paris, 1867). The Mechanics was first published by Carra de Vaux in the Journal asiatique (ix. série, 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 Schöne) 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’Arithmétique des Grecs dans Héron d’Alexandrie” (Mém. de la Soc. des sciences phys. et math. de Bordeaux, ii. série, iv., 1882), “La Stéréométrie d’Héron d’Alexandrie” and “Études Héroniennes” (ibid. v., 1883), “Questions Héroniennes” (Bulletin des sciences math., ii. série, viii., 1884), “Un Fragment des Métriques d’Héron” (Zeitschrift für Math. und Physik, xxxix., 1894; Bulletin des sciences math., ii. série, 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.)
HERO, the Younger, the name given without any sufficient reason to a Byzantine land-surveyor who wrote (about A.D. 938) a treatise on land-surveying modelled on the works of Hero of Alexandria, especially the Dioptra.
See “Géodésie de Héron de Byzance,” published by Vincent in Notices et extraits des manuscrits de la Bibliothéque Impériale, xix. 2 (Paris, 1858), and T. H. Martin in Mémoires présentés à l’ Académie des Inscriptions, 1st series, iv. (Paris, 1854).
HEROD, the name borne by the princes of a dynasty which reigned in Judaea from 40 B.C.
Herod (surnamed the Great), the son of Antipater, who supported Hyrcanus II. against Aristobulus II. with the aid first of the Nabataean Arabs and then of Rome. The family seems to have been of Idumaean origin, so that its members were liable to the reproach of being half-Jews or even foreigners. Justin Martyr has a tradition that they were originally Philistines of Ascalon (Dial. c. 52), and on the other hand Nicolaus of Damascus (apud Jos. Ant. xiv. 1. 3) asserted that Herod, his royal patron, was descended from the Jews who first returned from the Babylonian Captivity. The tradition and the assertion are in all probability equally fictitious and proceed respectively from the foes and the friends of the Herodian dynasty.
Antipas (or Antipater), the father of Antipater, had been governor of Idumaea under Alexander Jannaeus. His son allied himself by marriage with the Arabian nobility and became the real ruler of Palestine under Hyrcanus II. When Rome intervened in Asia in the person of Pompey, the younger Antipater realized her inevitable predominance and secured the friendship of her representative. After the capture of Jerusalem in 63 B.C. Pompey installed Hyrcanus, who was little better than a figurehead, in the high-priesthood; and when in 55 B.C. the son of Aristobulus renewed the civil war in Palestine, the Roman governor of Syria in the exercise of his jurisdiction arranged a settlement “in accordance with the wishes of Antipater” (Jos. Ant. xiv. 6. 4). To this policy of dependence upon Rome Antipater adhered, and he succeeded in commending himself to Mark Antony and Caesar in turn. After the battle of Pharsalia Caesar made him procurator and a Roman citizen.