What could primary substance be?
Composite substances (e.g. a human being, a chair)
Matter (e.g. the stuff from which the chair is made)
Form (e.g. the essence of a chair)
What are the criteria for being a primary substance?
1) Separable (i.e. capable of existing independently of anything else)
2) Primary in knowledge (i.e. x is primary in knowledge iff knowledge of it is not dependent on knowledge of anything else.
3) An individual (i.e. a 'this something', thereby excluding the imaginary since the imaginary cannot have attributes that are matters of fact)
Composite substances cannot be primary substances because their existence is quite obviously not seperable from that of their form and matter. Though one can can be consider it in terms of its component parts, one cannot separate it from them and be left with a 'this something'. Because of this, it is also not primary in knowledge; matter and/or form are.
Matter cannot be a primary substance either. Though it seems to pass both the separability test (one can break a tree down into logs of wood) and the individuality test (one can say 'this tree' and 'this log'), it cannot pass both at the same time, i.e. one cannot take 'this tree', separate the matter from its form (e.g. chop it into logs) and maintain that same individual, since it is no longer 'this tree' but now 'these logs', and you are now dealing with a new substance (has new form).
Indeed, by separating matter from its form, it ceases to be individual, since one can no longer refer to it as a 'this anything'; whenever one refers to matter, one implicitly refers to its form as well. It therefore isn't primary in knowledge either, as to know of matter one must also know of that matter's form.
Therefore form must be primary substance, and Aristotle begins to substitute form for essence, perhaps to disassociate himself from Plato, since his theory of forms doesn't hold them to be universal constants instantiated in physical things, but as the arrangement, or essence, of matter, that disappears with that matter. It might also be because when we talk of something's form, we can mean many different things, e.g. "Aristotle is a botanist", "Aristotle is a philosopher", however when we talk of essence we focus on that which is universal and central to an object, e.g. "Aristotle is a human being that is most fulfilled in a polis"
But then is primary substance the essence of an individual object, or of a class or species of objects, e.g. "humans", "cliffs", etc.? In terms of solving the problem of 'what is primary substance' this really doesn't matter, but it should matter to Aristotle. According to his criteria, it could be either, depending on how you define and deploy the term "essence". To make as safe a definition of essence as possible (by safe I mean one that makes essence most unobjectionably primary) one ought to lean towards a species interpretation, but then we can also assert a very individualistic sense of each person on earth having their own unique essence, and so lean towards an individual interpretation.
Objection: even though essence can persist whilst the underlying matter changes, the essence couldn't survive the total removal of that matter except in language, where we can talk of A without mentioning A's matter. But even in language we implicitly posit the existence of A's matter when discussing A's form, and we cannot conceive of form without matter. One could also say that essence is separable in time, in the 4th dimension, since it is in that dimension that essence doesn't depend upon any particular matter. But again, it still depends upon some matter to exist, unless we can think of essence as existing solely in the 4th dimension and not having the three spatial dimensions.
Aristotle seems to share this assumption that form cannot persist entirely without matter, as he rejects Plato's theory of universal forms and physical instantiations, suggesting he was caught in between a theory of materialism and a theory of forms. If one were to interpret Aristotle's criteria for primary substance strictly, essense doesn't seem to comply; in other words, it looks as though Aristotle has simply been liberal with his interpretation to fit his form/matter distinction with his theory of primary substances.
Objection: if I melt down a bronze statue, the matter doesn't change, and underlies the change between two forms. I can also, in a sense, refer to the bronze as a 'this something' in saying 'this bronze' on both occasions. That appears to be two honomonous uses of the term 'this bronze', suggesting that matter is both separable, individual and primary in knowledge, something we can accept if we interpret Aristotle's criteria liberally. So Aristotle's loose interpretation runs into trouble here again.
Objection: can anything be considred primary? If neither complex substances nor their form nor matter can be definitively described as primary according to a strict interpretation of Aristotle's criteria, we must conclude either that his criteria are flawed or that there is no such thing as primary substance. Given the fairly arbitrary nature of any other criteria for primary substance, it would seem that the very idea is simply false, a philosophical illusion that Aristotle conjured up to fit into his theory of coherent knowledge with foundations.