The orthodox theologians consistently, therefore, offer arguments for the unique personality or personhood of the divine Son and his eternal generation from the Father—whether against the classical heresies and their more recent representatives or, in the high and late orthodox eras, specifically against Le Clerc, the Socinians, various other antitrinitarians, and those who followed their arguments.
—Richard A. Muller, Post-Reformation Reformed Dogmatics: The Rise and Development of Reformed Orthodoxy; Volume 4: The Triunity of God (Grand Rapids, MI: Baker Academic, 2003), 275.
In the first general council of the church which, consisting of three hundred and eighteen bishops, was called together by the Emperor Constantine at Nice, in Bithynia, AD. 325, there were found to be three great partiesrepresenting different opinions concerning the Trinity.
1st. The orthodox party, who maintained the opinion now held by all Christians, that the Lord Jesus is, as to his divine nature, of the same identical substance with the Father. These insisted upon applying to him the definite term oJ moou > sion (homoousion), compounded of oJ mo > v, sameand ouj si > a, substance, to teach the great truth that the three persons of the Godhead are one God, because they are of the same numerical essence.
2nd. The Arians, who maintained that the Son of God is the greatest of all creatures, more like God than any other, the only–begotten Son of God, created before all worlds, through whom God created all other things, and in that sense only divine. They held that the Son was eJ terou > sion of different or generically unlike essence from the Father.
3rd. The middle party, styled Semiarians, who confessed that the Son was not a creature, but denied that he was in the same sense God as the Father is. They held that the Father is the only absolute self–existent God; yet that from eternity he, by his own free will, caused to proceed from himself a divine person of like nature and properties. They denied, therefore, that the Son was of the same substance (homoousion) with the Father, but admitted that he was of an essence truly similar, and derived from the Father (homoiousion, oJ moio > usoin, from, o[ uiov, like, and oj usi > a, substance), generically though not numerically one.
The opinions of the first, or orthodox party, prevailed at that council, and have ever since been represented by the technical phrase, homoousian. For the creed promulgated by that council, see Chapter 7. 10.
What are the several propositions essentially involved in the doctrine of the Trinity?
1st. There is but one God, and this God is one, i.e. , indivisible.
2nd. That the one indivisible divine essence, as a whole, exists eternally as Father, and as Son, and as Holy Ghost; that each person possesses the whole essence, and is constituted a distinct person by certain incommunicable properties, not common to him with the others.
3rd. The distinction between these three is a personal distinction, in the sense that it occasions (l) the use of the personal pronouns, I, thou, he, (2) a concurrence in counsel and a mutual love, (3) a distinct order of operation.
4th Since there is but one divine essence, and since all attributes or active properties are inherent in and inseparable from the essence to which they pertain, it follows that all the divine attributes must be identically common to each of the three persons who subsist in common of the one essence. Among all creatures every distinct person is a distinct numerical substance, and possesses a distinct intelligence, a distinct will etc. In the Godhead, however, there is but one substance, and one intelligence, one will, etc., and yet three persons eternally co–exist of that one essence, and exercise that one intelligence and one will, etc. In Christ on the contrary, there are two spirits, two intelligences, two wills, and yet all the while one indivisible person.
5th. These divine persons being one God, all the divine attributes being common to each in the same sense, nevertheless they are revealed in the Scriptures in a certain order of subsistence and of operation. (1) Of subsistence inasmuch as the Father is neither begotten nor proceedeth, while the Son is eternally begotten by the Father, and the Spirit eternally proceedeth from the father and the Son; (2) of operation, inasmuch that the first person sends and operates through the second, and the first and second send and operate through the third. Hence the Father is always set forth as first, the Son as second, the Spirit as third.
6th. While all the divine attributes are common equally to the three persons, and all divine works wrought ad extra such as creation, providence, or redemption, are predicated alike of the one being––the one God considered absolutely––and of the Father, and of the Son, and of the Holy Ghost severally; nevertheless the Scriptures attribute some divine works wrought ad intra, exclusively to each divine person respectively, e. g., generation to the Father, filiation to the Son, procession to the Holy Ghost; and thereare likewise some divine works wrought ad extra which are attributed pre–eminently to each person respectively, e.g., creation to the Father, redemption to the Son, and sanctification to the Holy Ghost.
In order, therefore, to establish this doctrine in all its parts by the testimony of Scripture, it will be necessary for us to prove the following propositions in their order:
1st. That God is one.
2nd. That Jesus of Nazareth, as to his divine nature, was truly God, yet a distinct person from the Father.
3rd. That the Holy Spirit is truly God, yet a distinct person.
4th. That the Scriptures directly teach a trinity of persons in one Godhead.
5th. It will remain to gather what the Scriptures reveal as to the eternal and necessary relations which these three divine persons sustain to each other. These are distributed under the following heads: (1) The relation which the second person sustains to the first, or the eternal generation of the Son; (2) the relation which the third person sustains to the first and second, or the eternal procession of the Holy Ghost; and, (3) their personal properties and order of operation, ad extra.
Then he goes on to prove these. I highly recommend Hodge. It’s $2 on kindle.
A.A. Hodge Outlines in Theology