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Postby mingshey » Fri Oct 28, 2011 5:49 pm

The Physics textbook I'm using in my class referred to Archimedes, "On the Equilibrium of Planes", so I looked it up, and in Book I, Postulate 7, it said,

T. L. Heath's translation reads,
In any figure whose perimeter is concave in (one and) the same direction the centre of gravity must be within the figure.

Here the term "concave", in any English dictionary available to me, refers to a "concave figure", whose center of gravity could be outside of it. So I thought the postulate sounded too wrong for a great Greek geometer, although "to be concave in one and the same direction" was not very clear to me at the first time. And the whole book talks about nothing like a concave figure, and makes no reference to this postulate ever after. So I looked up the Greek text, and it was
Παντὸς σχήματος, οὗ κα ἁ περίμετρος ἐπὶ τὰ
αὐτὰ κοῖλα ᾖ, τὸ κέντρον τοῦ βάρεος ἐντὸς εἶμεν δεῖ
τοῦ σχήματος.

"Of any figure, if whose perimeter is "κοῖλα" to the same (direction or whatever), the center of gravity must be inside the figure"

κοῖλος, in LSJ, as to Geometry, usually means "concave". But it also means hollow, arched, and even "curved". (A line curved in one way would crudely look like the cutaway view of a hollow vessel.) And since the text is referring to the perimeter, not the figure itself, finally I thought it has to mean "curved"; The perimeter is curved in a same direction, so that the figure itself is "convex", which is the only interpretation that makes the postulate correct.

Either my understanding of the term "concave" was incomplete, or Heath was using "concave" in a loose way like its Latin origin "concavus" that includes the sense of "curved" as well.

Please tell me if I have erred in getting to the conclusion.

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