How It Was Brought About that the Greeks Speak of Three Hypostases, the Latins of Three Persons. Scripture Nowhere Speaks of Three Persons in One God
7 For the sake, then, of speaking of things that cannot be uttered, that we may be able in some way to utter what we are able in no way to utter fully, our Greek friends have spoken of one essence, three substances; but the Latins of one essence or substance, three persons; because, as we have already said, essence usually means nothing else than substance in our language, that is, in Latin. And provided that what is said is understood only in a mystery, such a way of speaking was sufficient, in order that there might be something to say when it was asked what the three are, which the true faith pronounces to be three, when it both declares that the Father is not the Son, and that the Holy Spirit, which is the gift of God, is neither the Father nor the Son.
When, then, it is asked what the three are, or who the three are, we betake ourselves to the finding out of some special or general name under which we may embrace these three; and no such name occurs to the mind, because the super-eminence of the Godhead surpasses the power of customary speech. For God is more truly thought than He is altered, and exists more truly than He is thought. For when we say that Jacob was not the same as Abraham, but that Isaac was neither Abraham nor Jacob, certainly we confess that they are three, Abraham, Isaac, and Jacob.
But when it is asked what three, we reply three men, calling them in the plural by a specific name; but if we were to say three animals, then by a generic name; for man, as the ancients have defined him, is a rational, mortal animal: or again, as our Scriptures usually speak, three souls, since it is fitting to denominate the whole from the better part, that is, to denominate both body and soul, which is the whole man, from the soul; for so it is said that seventy-five souls went down into Egypt with Jacob, instead of saying so many men. Again, when we say that your horse is not mine, and that a third belonging to some one else is neither mine nor yours, then we confess that there are three; and if any one ask what three, we answer three horses by a specific name, but three animals by a generic one.
And yet again, when we say that an ox is not a horse, but that a dog is neither an ox nor a horse, we speak of a three; and if any one questions us what three, we do not speak now by a specific name of three horses, or three oxen, or three dogs, because the three are not contained under the same species, but by a generic name, three animals; or if under a higher genus, three substances, or three creatures, or three natures. But whatsoever things are expressed in the plural number specifically by one name, can also be expressed generically by one name.
But all things which are generically called by one name cannot also be called specifically by one name. For three horses, which is a specific name, we also call three animals; but, a horse, and an ox, and a dog, we call only three animals or substances, which are generic names, or anything else that can be spoken generically concerning them; but we cannot speak of them as three horses, or oxen, or dogs, which are specific names; for we express those things by one name, although in the plural number, which have that in common that is signified by the name.
For Abraham, and Isaac, and Jacob, have in common that which is man; therefore they are called three men: a horse also, and an ox, and a dog, have in common that which is animal; therefore they are called three animals. So three several laurels we also call three trees; but a laurel, and a myrtle, and an olive, we call only three trees, or three substances, or three natures: and so three stones we call also three bodies; but stone, and wood, and iron, we call only three bodies, or by any other higher generic name by which they can be called.
Of the Father, therefore, the Son, and the Holy Spirit, seeing that they are three, let us ask what three they are, and what they have in common. For the being the Father is not common to them, so that they should be interchangeably fathers to one another: as friends, since they are so called relatively to each other, can be called three friends, because they are so mutually to each other. But this is not the case in the Trinity, since the Father only is there father; and not Father of two, but of the Son only.
Neither are they three Sons, since the Father there is not the Son, nor is the Holy Spirit. Neither three Holy Spirits, because the Holy Spirit also, in that proper meaning by which He is also called the gift of God, is neither the Father nor the Son. What three therefore? For if three persons, then that which is meant by person is common to them; therefore this name is either specific or generic to them, according to the manner of speaking. But where there is no difference of nature, there things that are several in number are so expressed generically, that they can also be expressed specifically.
For the difference of nature causes, that a laurel, and a myrtle, and an olive, or a horse, and an ox, and a dog, are not called by the specific name, the former of three laurels, or the latter of three oxen, but by the generic name, the former of three trees, and the latter of three animals. But here, where there is no difference of essence, it is necessary that these three should have a specific name, which yet is not to be found. For person is a generic name, insomuch that man also can be so called, although there is so great a difference between man and God.
Source: On the Holy Trinity (New Advent)