HERO OF ALEXANDRIA

hero of alexandria, 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 100 b.c. the earlier theory started from the superscription of one of his works, [greek: heronos ktesibiou belopoiika], 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 (1) 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. 120) 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. schone 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_ ([greek: peri hydrion horoskopeion]), 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, qui 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 schone) 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 fur math. und 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.)