80 SCIENCE IN GRECO-ROMAN ANTIQUITY 



the determination of the areas of fields having a given 

 shape ; 



(2) Propositions on the method of calculating the 

 volumes of certain solids, with applications to buildings 

 used as theatres, baths, banqueting-halls, etc. 



(3) A rule for finding the height of inaccessible 

 objects. 



(4) A table of weights and measures. 



Amongst his writings on mathematics, must be 

 mentioned, besides the Definitions and a Commentary 

 on the Elements of Euclid, a recently discovered work 

 on Measurements, in which the rules and formulae 

 for estimating the most important volumes and 

 surfaces are enunciated together with theoretical 

 proofs. The main part is borrowed from Euclid 

 and Archimedes ; even the formula which gives the 

 surface of a triangle in terms of its three sides a, b, c — i.e. 



S = Vp(p — a)(p — b)(p — c) (where p = the semi- 

 perimeter) — is not an original invention, for it was 

 probably used by the Egyptian land-surveyors, and 

 it is only the demonstration which can be attributed 

 to Hero. He also attempted to perfect the levelling 

 instrument hitherto used in surveying. These improve- 

 ments are carefully described and theoretically correct, 

 but they reveal the great practical ignorance of their 

 author. The work entitled The Construction of Vaults 

 was also probably written with a practical aim in view, 

 and at any rate had the honour of being studied and 

 commented upon by one of the architects of St. Sophia, 

 Isidore of Miletus. Inspired by previous works, Hero 

 has been able to give very exact information on The 

 Construction of Catapults ; on the other hand, some of 

 his writings, which are similar in conception to those of 

 Archimedes and Philo, display great defects, especially 

 the Pneumatics, in which the theory of the pressure of 

 the air is applied to various apparatus. These are for 

 the most part borrowed from Philo, and their descrip- 



