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From this site, one of Parmenides' fragments is translated:

Since, then, it has a furthest limit, it is complete on every side, like the mass of a rounded sphere, equally poised from the centre in every direction; for it cannot be greater or smaller in one place than in another. For there is no nothing that could keep it from reaching out equally, nor can aught that is be more here and less there than what is, since it is all inviolable. For the point from which it is equal in every direction tends equally to the limits.

I think Parmenides is arguing that the object on enquiry (i.e. what exists) has to necessarily be uniform and perfect, but I don't see how he reaches this conclusion. It seems like he's using his previous argument that the object of enquiry must be bounded (by bonds and limits), but I can't see the reasoning behind this argument?

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The argument for the uniformity (or homogeneity, or self-sameness) doesn't follow from the boundedness of what is. Parmenides introduces both uniformity and the argument for uniformity in the passage you're citing. The argument is:

[...] it cannot be greater or smaller in one place than in another. For there is no nothing that could keep it from reaching out equally, nor can aught that is be more here and less there than what is, since it is all inviolable.

The Stanford Encyclopedia of Philosophy presents a paraphrasis of this argument when touching the modal interpretation of Parmenides' On Nature, but it's still general enough that it doesn't really lean towards any particular interpretation. Maybe it's a little bit clearer than Burnet's version:

What must be must be free from any internal variation. Such variation would involve its being something or having a certain character in some place(s) while being something else or having another character in others, which is incompatible with the necessity of its (all) being what it is.

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