By Pfeffer, Riemannian
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Extra info for Approach To Integration
It’s fairly obvious that, in general, Heron has a more physical and environmental treatment of geometrical objects anyway. Or rather, none of the extant works is a work of geometry. 21 I am somewhat sceptical, since, I think, it at most reflects Heron’s own views about applied geometry. First, the use of ‘void (κενόν)’ looks like the ordinary use, indeed, as does the use of ‘hollow/concave (κοιλῆς)’, for Heron does not say that the region is empty of everything. One might be led to think that something unusual is going on by the fact that the inner ﬁgure in II 12–13 and 20 Heron’s division of methodologies in the Metrica is somewhat subtle.
In any case, it is very optimistic to read ontological commitments about geometry from a treatment of applied mathematics, as is reading the end of Metrica II as providing the level of abstraction for the entire work. Nonetheless, we shall see in § 6 that Heron takes into account practical considerations in apparently purely geometrical constructions. 23 Here is how Euclid deﬁnes ‘given in position’ in the Data (Deﬁnition 4): Points and lines and angles that always keep the same place are said to be given in position.
41 A body, as Apollodorus says in the Physics, is what’s extended in three ways, in length, in width, in depth. This is also called ‘solid body’. A surface is a limit of a body or what has length and width alone and not depth. Posidonius in the ﬁfth [book] on Things Seen Above [meteorology] admits this both in attentive-thought and in substance. And line is a limit of a surface or widthless length or what has length alone. 68 First, the deﬁnitions would all seem to be from Apollodorus, but endorsed by Posidonius.