(Article I.8.) Concerning the Position of ‘‘Heron’s Formula’’ in the Metrica (With a Platonic Note)

The article compares the way ‘‘Herons’s formula’’ for the area of a triangle is presented in Metrica I.7 – 8, with context Metrica I.4–6, with the way it appears in Dioptra 30, and also with the way circular segments are dealt with in Metrica I.27–32. Both sequences in the Metrica present obvious stylistic incongruities, seemingly indicating that both ‘‘Heron’s formula’’ and the Archimedean calculation in I.32 are interpolations – the former borrowed from Dioptra or a close source. At closer inspection, however, both insertions can be argued to have been made by Heron himself, within a treatise which is otherwise a re-elaboration of a technical manual. Only an initial numerical example in I.8 may have been inserted by a later editorial hand, together with the renowned ‘‘Heronian’’ method for approximating the square roots of non-square numbers A final note suggests a possible link between, on one hand, Heron’s replacement of μο $$ \hat{\upiota } $$ ραι(‘‘lots’’ or ‘‘shares’’) by μονάδες(units) when he transfers the ‘‘formula’’ from the Dioptra to the Metrica; and, on the other, Plato’s rejection of the splitting of unity into πολλὰ μóρια(‘‘many parts’’) in vulgar computation.