Here we get the short-form version of Augustine's exposition of the Trinity. (The long form is Augustine's work De Trinitate.) It is for all intents an purposes both a rephrasing of the Creed and a articulation of the question that would eventually and finally divide the Eastern church from the Western church. Augustine in general is content to articulate truth while leaving appropriate mystery:
When we ask concerning each Person individually, the answer must be that each one is God and each is Almighty; and when we inquire concerning the three together, the reply must be that there are not three Gods or three Almighties, but a single God Almighty. Such is the indivisible unity in the Three and such is the way it should be stated.To paraphrase one of the Cappacadocian Fathers (I can never remember which), we cannot think of the One True God without being caught up into the mystery of the Trinity, and we cannot think of the Trinity without being back to the unity of the One True God.
Beyond that, however, Augustine hesitates to step into fray that would divide the church in later centuries. Namely, "I am not prepared to offer any premature opinion as to whether or not the Holy Spirit of the good Father and of the good Son should be called the goodness of both since He is common to both."
Though Augustine inclines to what would become the Western view (that the Spirit proceeds from both the Father and the Son), he instead looks at the practical implications of the nature of the Trinity for creation. Because of the economy of the Trinity, we see that the stamp of the image of God on the world is a stamp of goodness, holiness, reason, life, wisdom, joy, and community. All of these exist in God Himself between the Father, Son, and Holy Spirit, and flow from Him into creation.
We see a picture of the Trinity even in worldly philosophy, where the three categories of Hellenistic thought (physics, logic, and ethics) line up with ways the Trinity is reflected in the world. That is, through existence, wisdom, and purpose. This is not of course to say that the worldly philosophers understand or believe in the Trinity, just that the stamp of God is so firmly on the world that the exercise of reason itself teaches us something about His character. In fact, these worldly philosophers were pursuing only natural ends, which would be appropriate "if we were the cause of our own nature." But since we are not, since "in fact, God is the Author of the existence of our nature... therefore, He must be our Teacher if we are ever to be wise, and He must be the Source of our inmost consolation if we are ever to be happy." Worldly philosophy can uncover our desires and teach us that what we need cannot be found within it, but it can never satisfy the longings of the heart or be a ground for the thought of the mind.
When we reflect on ourselves, we "recognize... an image of God, in the sense of an image in the Trinity." Augustine is careful to note that "it is merely an image, and, in fact, a very remote one," lest we be tempted to begin to identify ourselves too much with our Creator. Even at that, it is still a clearer image than is found anywhere else in creation. Specifically, we see in ourselves that "I am certain that I am, that I know that I am, and that I love to be and to know." This reflects the Trinitarian economy in which God the Father exists as the ground of the Godhead, God the Son is the Wisdom and the Word (the Logos) of the Father, and God the Holy Spirit is the Love of the Father for the Son and of the Son for the Father. This same idea is expressed by Edwards:
And this I suppose to be that blessed Trinity that we read of in the Holy Scriptures. The Father is the Deity subsisting in the prime, un-originated and most absolute manner, or the Deity in its direct existence. The Son is the Deity generated by God's understanding, or having an idea of Himself and subsisting in that idea. The Holy Ghost is the Deity subsisting in act, or the Divine essence flowing out and breathed forth in God's Infinite love to and delight in Himself. And I believe the whole Divine essence does truly and distinctly subsist both in the Divine idea and Divine love, and that each of them are properly distinct Persons.When we examine ourselves, we find similar interactions and inclinations within us. This is not to make the argument that each human being is actually three persons, just that we are made in God's image and so even reflect this truth about Him.
Augustine notes, in passing, that this provides a response to the theories of the Skeptics, who would question everything--even whether or not we exist. Anticipating Descartes, Augustine notes that even if we're wrong, we can conclude that we exist, as there must be someone there to be wrong. And if I can know that I exist, then I must not be wrong about my knowledge that I exist. And since such knowledge is a good thing--a pleasing thing to be loved, since it is knowledge--then I can know 11) that I exist; 2) that I have knowledge; 3) that there is something to love. Which gets us right back to our picture of the Trinity.