The New International Encyclopædia/Hero of Alexandria

HERO, or HERON, OF ALEXANDRIA. The leading Greek mathematician and physicist of his time. Not only are the dates of his birth and death unknown, but there is great uncertainty as to the century in which he lived. The most recent investigation of the evidence, by Schmidt (1899, work cited below, Bd. 1, p. ix.), leads to the conclusion that he may have lived in the first century A.D., but other writers, who, it must be said, have not considered the question so fully, have usually placed him in the first or even the second century B.C. There is much confusion concerning the works of Hero of Alexandria, there having been no less than eighteen Greek writers of this name. It is, however, fairly certain that he wrote at least thirteen books on mathematics and physics. He seems to have been an Egyptian, and it is certain that his style is not that of a Greek. He contributed little to pure mathematics, his chief work on this subject being the extension of the ancient mathematics so as to allow the consideration of the fourth power of lines. Thus, in his geodesy, γεωδαισία, contained in his Μετρικά, upon which subject he was the only Greek writer, he gives the well-known formula for finding the area of a triangle with sides, $a,b,c$, and semiperimeter $s$, ${\sqrt {s(s-a)(s-b)(s-c)}}$, a formula known by his name. (The proof is given, possibly an interpolation, in his Ηερὶ διόπτρας.) He seems also to have had some idea of trigonometry, and in his geometry is to be found the first definite use of a trigonometric formula. He asserts in substance, using modern symbols, that if $A$ represents the area of a regular $n$-gon of side $s$, and if $c$ be the numerical coefficient by which $s^{2}$ must be multiplied to produce $A$, i.e. so that $A=cs^{2}$ then must $c={\frac {n}{4}}\cot {\frac {180\deg }{n}}$. He proceeds to compute $c$ for the values $n=3,4,...12$, with considerable accuracy, but his method is unknown. He could also solve the complete quadratic equation $ax^{2}+bx=c$, where $a,b,c$, are positive, but not the general form. Hero is credited with a number of mechanical inventions, including a contrivance for utilizing the force of steam and a fountain which bears his name. Consult: Martin, “Recherches sur la vie et les ouvrages d'Héron d'Alexandrie,” in vol. iv. of Mémoires présentées par divers savants à l'Académie d'Inscriptions (Série I., Paris, 1854); Hultsch, Heronis Alexandrini Geometricorum et Stereometricorum Reliquiæ (Berlin, 1864); Schmidt, Heronis Alexandrini Opera quæ Supersunt Omnia (Leipzig, 1899—).