Objection 1. It would seem that "person" is the same as "hypostasis," "subsistence," and "essence." For Boethius says (De Duab. Nat.) that "the Greeks called the individual substance of the rational nature by the name hypostasis." But this with us signifies "person." Therefore "person" is altogether the same as "hypostasis."
Objection 2. Further, as we say there are three persons in God, so we say there are three subsistences in God; which implies that "person" and "subsistence" have the same meaning. Therefore "person" and "subsistence" mean the same.
Objection 3. Further, Boethius says (Com. Praed.) that the Greek ousia, which means essence, signifies a being composed of matter and form. Now that which is composed of matter and form is the individual substance called "hypostasis" and "person." Therefore all the aforesaid names seem to have the same meaning.
Objection 4.On the contrary, Boethius says (De Duab. Nat.) that genera and species only subsist; whereas individuals are not only subsistent, but also substand. But subsistences are so called from subsisting, as substance or hypostasis is so called from substanding. Therefore, since genera and species are not hypostases or persons, these are not the same as subsistences.
Objection 5. Further, Boethius says (Com. Praed.) that matter is called hypostasis, and form is called ousiosis--that is, subsistence. But neither form nor matter can be called person. Therefore person differs from the others.
I answer that, According to the Philosopher (Metaph. v), substance is twofold. In one sense it means the quiddity of a thing, signified by its definition, and thus we say that the definition means the substance of a thing; in which sense substance is called by the Greeks ousia, what we may call "essence." In another sense substance means a subject or "suppositum," which subsists in the genus of substance. To this, taken in a general sense, can be applied a name expressive of an intention; and thus it is called "suppositum." It is also called by three names signifying a reality--that is, "a thing of nature," "subsistence," and "hypostasis," according to a threefold consideration of the substance thus named. For, as it exists in itself and not in another, it is called "subsistence"; as we say that those things subsist which exist in themselves, and not in another. As it underlies some common nature, it is called "a thing of nature"; as, for instance, this particular man is a human natural thing. As it underlies the accidents, it is called "hypostasis," or "substance." What these three names signify in common to the whole genus of substances, this name "person" signifies in the genus of rational substances.
Reply to Objection 1. Among the Greeks the term "hypostasis," taken in the strict interpretation of the word, signifies any individual of the genus substance; but in the usual way of speaking, it means the individual of the rational nature, by reason of the excellence of that nature.
Reply to Objection 2. As we say "three persons" plurally in God, and "three subsistences," so the Greeks say "three hypostases." But because the word "substance," which, properly speaking, corresponds in meaning to "hypostasis," is used among us in an equivocal sense, since it sometimes means essence, and sometimes means hypostasis, in order to avoid any occasion of error, it was thought preferable to use "subsistence" for hypostasis, rather than "substance."
Reply to Objection 3. Strictly speaking, the essence is what is expressed by the definition. Now, the definition comprises the principles of the species, but not the individual principles. Hence in things composed of matter and form, the essence signifies not only the form, nor only the matter, but what is composed of matter and the common form, as the principles of the species. But what is composed of this matter and this form has the nature of hypostasis and person. For soul, flesh, and bone belong to the nature of man; whereas this soul, this flesh and this bone belong to the nature of this man. Therefore hypostasis and person add the individual principles to the idea of essence; nor are these identified with the essence in things composed of matter and form, as we said above when treating of divine simplicity (3, 3).
Reply to Objection 4. Boethius says that genera and species subsist, inasmuch as it belongs to some individual things to subsist, from the fact that they belong to genera and species comprised in the predicament of substance, but not because the species and genera themselves subsist; except in the opinion of Plato, who asserted that the species of things subsisted separately from singular things. To substand, however, belongs to the same individual things in relation to the accidents, which are outside the essence of genera and species.
Reply to Objection 5. The individual composed of matter and form substands in relation to accident from the very nature of matter. Hence Boethius says (De Trin.): "A simple form cannot be a subject." Its self-subsistence is derived from the nature of its form, which does not supervene to the things subsisting, but gives actual existence to the matter and makes it subsist as an individual. On this account, therefore, he ascribes hypostasis to matter, and ousiosis, or subsistence, to the form, because the matter is the principle of substanding, and form is the principle of subsisting.
Source: Summa Theologica (New Advent)