Objection 1. It would seem not right to say that the three persons are of one essence. For Hilary says (De Synod.) that the Father, Son and Holy Ghost "are indeed three by substance, but one in harmony." But the substance of God is His essence. Therefore the three persons are not of one essence.
Objection 2. Further, nothing is to be affirmed of God except what can be confirmed by the authority of Holy Writ, as appears from Dionysius (Div. Nom. i). Now Holy Writ never says that the Father, Son and Holy Ghost are of one essence. Therefore this should not be asserted.
Objection 3. Further, the divine nature is the same as the divine essence. It suffices therefore to say that the three persons are of one nature.
Objection 4. Further, it is not usual to say that the person is of the essence; but rather that the essence is of the person. Therefore it does not seem fitting to say that the three persons are of one essence.
Objection 5. Further, Augustine says (De Trin. vii, 6) that we do not say that the three persons are "from one essence [ex una essentia]," lest we should seem to indicate a distinction between the essence and the persons in God. But prepositions which imply transition, denote the oblique case. Therefore it is equally wrong to say that the three persons are "of one essence [unius essentiae]."
Objection 6. Further, nothing should be said of God which can be occasion of error. Now, to say that the three persons are of one essence or substance, furnishes occasion of error. For, as Hilary says (De Synod.): "One substance predicated of the Father and the Son signifies either one subsistent, with two denominations; or one substance divided into two imperfect substances; or a third prior substance taken and assumed by the other two." Therefore it must not be said that the three persons are of one substance.
On the contrary, Augustine says (Contra Maxim. iii) that the word homoousion, which the Council of Nicaea adopted against the Arians, means that the three persons are of one essence.
I answer that, As above explained (13, 1,2), divine things are named by our intellect, not as they really are in themselves, for in that way it knows them not; but in a way that belongs to things created. And as in the objects of the senses, whence the intellect derives its knowledge, the nature of the species is made individual by the matter, and thus the nature is as the form, and the individual is the "suppositum" of the form; so also in God the essence is taken as the form of the three persons, according to our mode of signification. Now in creatures we say that every form belongs to that whereof it is the form; as the health and beauty of a man belongs to the man. But we do not say of that which has a form, that it belongs to the form, unless some adjective qualifies the form; as when we say: "That woman is of a handsome figure," or: "This man is of perfect virtue." In like manner, as in God the persons are multiplied, and the essence is not multiplied, we speak of one essence of the three persons, and three persons of the one essence, provided that these genitives be understood as designating the form.
Reply to Objection 1. Substance is here taken for the "hypostasis," and not for the essence.
Reply to Objection 2. Although we may not find it declared in Holy Writ in so many words that the three persons are of one essence, nevertheless we find it so stated as regards the meaning; for instance, "I and the Father are one (John 10:30)," and "I am in the Father, and the Father in Me (John 10:38)"; and there are many other texts of the same import.
Reply to Objection 3. Because "nature" designates the principle of action while "essence" comes from being [essendo], things may be said to be of one nature which agree in some action, as all things which give heat; but only those things can be said to be of "one essence" which have one being. So the divine unity is better described by saying that the three persons are "of one essence," than by saying they are "of one nature."
Reply to Objection 4. Form, in the absolute sense, is wont to be designated as belonging to that of which it is the form, as we say "the virtue of Peter." On the other hand, the thing having form is not wont to be designated as belonging to the form except when we wish to qualify or designate the form. In which case two genitives are required, one signifying the form, and the other signifying the determination of the form, as, for instance, when we say, "Peter is of great virtue [magnae virtutis]," or else one genitive must have the force of two, as, for instance, "he is a man of blood"--that is, he is a man who sheds much blood [multi sanguinis]. So, because the divine essence signifies a form as regards the person, it may properly be said that the essence is of the person; but we cannot say the converse, unless we add some term to designate the essence; as, for instance, the Father is a person of the "divine essence"; or, the three persons are "of one essence."
Reply to Objection 5. The preposition "from" or "out of" does not designate the habitude of a formal cause, but rather the habitude of an efficient or material cause; which causes are in all cases distinguished from those things of which they are the causes. For nothing can be its own matter, nor its own active principle. Yet a thing may be its own form, as appears in all immaterial things. So, when we say, "three persons of one essence," taking essence as having the habitude of form, we do not mean that essence is different from person, which we should mean if we said, "three persons from the same essence."
Reply to Objection 6. As Hilary says (De Synod.): "It would be prejudicial to holy things, if we had to do away with them, just because some do not think them holy. So if some misunderstand homoousion, what is that to me, if I understand it rightly?... The oneness of nature does not result from division, or from union or from community of possession, but from one nature being proper to both Father and Son."
Source: Summa Theologica (New Advent)