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.