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I would just like to add some nuance to Geremia's excellent answer.
We have to keep in mind that the intention of the Council Fathers at Nicaea (325) was not so much to defend the term ὁμοούσιος (h*m*ousios, consubstantial; the term is derived from οὐσία, which means "essence" or "substance") as to refute the heresy of Arius.
(For those who are not familiar, Arius, a priest originally from Libya but ministering at a prominent church in Alexandria in Egypt, essentially denied the full divinity of the Son, asserting instead that He is the first "creature" or emanation of the Father. We do not have any certain records of what he believed about the Holy Spirit, but we may deduce that he considered the Holy Spirit to be a creature or emanation inferior to the Son. He also denied the Son's—and we may presume the Spirit's—eternity. He is reported as stating, "There was a time when the Son was not.")
The consensus at the Council of Nicaea was that Arius was in error—in other words, that the Son is fully divine, just like the Father—but there was as yet no consensus as to the correct understanding of the relationship between Father and Son. There are essentially three possibilities:
Hardly anyone professing to be an orthodox Christian would have taken the second option (which smacks of polytheism), but there were a number who opted for the first: Marcellus of Ancyra, for example. This first option, the "fusion" of the Persons into one, is not very different from the heresy of Modalism, also known as Sabellianism (which professes that the Persons of the Trinity are merely "aspects" or "modes" of the one God).
And here is the problem: the term ὁμοούσιος was the very same one that Sabellius used to justify his heresy. Moreover, the very important distinction, advanced by St. Basil the Great, between οὐσία (ousia; i.e., essence or substance) and ὑπόστασις (hypostasis; i.e., "Person," in our terminology) had not yet been made: in fact, the Council of Nicaea condemns anyone who professes a distinction in "hypostasis" between Father and Son.
For this reason, many otherwise orthodox doctors and bishops viewed the term ὁμοούσιος with suspicion. Moreover, many who used the term ὁμοούσιος (Marcellus, for instance) used it in a way that, in retrospect, resembles Sabellianism.
This is the origin of the "h*m*eousian" party (note the extra "e"): they derive their name from their preference for the term ὁμοιούσιος (again, the extra iota is important). They professed that the Father and the Son were not the same substance (ὁμοούσιος) but similar in substance (ὁμοιούσιος).
Now, as St. Athanasius discovered after many discussions, it turns out that the Homoeousians were not necessarily heterodox at all, at least as regards the Father and the Son. When they used the term similar (ὁμοίος, h*m*eos), they understood a profound similarity: the way a human son is "similar" to his human father. They were, in fact, already on the right track: a human father is identical in nature with his human son, but not the same concrete substance (hypostasis). This idea was made much clearer later, when the Cappadocian Fathers (Basil the Great, Gregory of Nazianzus, and Gregory of Nyssa) introduced the distinction between οὐσία and ὑπόστασις, as I mentioned.
It would be a mistake, in my opinion, to characterize Homoeousianism as a "compromise" between "full" Arianism and orthodoxy. Rather, it was an honest, and, so far as it went, orthodox attempt to avoid both Sabellianism and Arianism. For this reason, I consider it misleading to apply the term "Semi-Arian" (which is a modern, anachronistic term anyhow) to this group.
The Homoeousians should be distinguished sharply from the Homoeans, who arose from other circumstances. Even after the Council of Nicaea, there continued to be a lot discussion and polemics regarding the οὐσία (Essence) of the Father and the Son: as we saw, the opinions and choice of terms were all over the map. It occurred to some of the Christian emperors (who wanted unity of religion in the Roman Empire at all costs) to suppress this discussion once and for all. The reasoning was, if they could simply make the bishops and doctors stop talking about οὐσία, then all could agree to a sort of "lowest common denominator": that the Father is similar (ὁμοίος) to the Son, not necessarily in Essence (οὐσία) but just "according to the Scriptures." In other words, the emperors wanted everyone simply to avoid the problem.
This is the origin of the "h*m*eian" party: those who agreed, basically, to tow the emperor's line and avoid discussion on the Divine Essence. Since this position was imposed practically by force, naturally there were many adherents; at one point, probably a large majority of bishops. It was Arianism in this form that would last for another three or four centures: as St. Jerome put it, "The world woke up and groaned to find itself Arian." If we are going to use the label "Semi-Arian" at all, then it should apply to this form of Arianism, which was a sort of compromise, in an attempt to please all parties.
Returning to the original question, then, regarding the Homoeousians (who do not, in my opinion, merit the epithet of "Semi-Arian") we can answer as follows:
(For an interesting insight on the efforts of St. Athanasius to reconcile with members of the Homoeousian party, see his Tomus ad Antiochenos.)
Regarding the Homoeans (who might be called "Semi-Arians"):
There was not a unity of opinion regarding the eternity of the Son, just as there was not unity of opinion regarding the fullness of His divinity. The tendency would have been towards a "moderate" Arianism: the Son was not exactly a creature, nor was there a time when He did not exist, but He was subordinate to the Father (not fully divine).
I think I have answered this question above.
Their position regarding the Holy Spirit was varied, for the same reasons as in the case of the Homoeousians. The tendency, however, was to subordinate the Holy Spirit to the Son. (So, basically, if the Son were not "fully" divine, then the Spirit was even "less" divine than the Son.)
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The Rise and Fall of the Homoiousianism
After Nicaea, the 'Arian' Controversy raged for another 55 years. During that period, 'Arianism' dominated the church. But 'Arianism' consisted of several strands. This article explains the theology of the Homoiousians, which was one of those strands.
This article series is largely based on two books:
RH = Bishop RPC Hanson The Search for the Christian Doctrine of God - The Arian Controversy 318-381, 1987
LA = Lewis Ayres Nicaea and its legacy, 2004 Ayres is a Professor of Catholic and Historical Theology
It is often claimed that Homo-i-ousianism (meaning, similar substance) arose as an attempt to reconcile two opposing teachings, namely, Homo-ousianism and Homo-ianism:
Homo-ousios is a keyword in the Nicene Creed of the year 325 and means "same substance." Homo-ousianism was a continuation of that concept and taught that the Son is of the same (ὁμός, homós, "same") substance as the Father. Consequently, the Son is co-equal and co-eternal with the Father.
Homo-ianism, on the other hand, refused to use the term οὐσία (ousía, "essence"). Homo-ians believed that the Son is "like" or "similar" (ὅμοιος, hómoios) to the Father but subordinate to Him.
It is then proposed that Homo-i-ousian (similar substance) theology was an attempt to reconcile the Homo-ousian (same substance) theology with the Homoian notion of similarity. For example, “Gwatkin described the group as a 'Semi-Arian position modified by an Athanasian influence.” (RH, 349) Athanasius was the great defender of the Nicene Creed.
However, Homo-i-ousianism was “most prominently associated with … Basil of Ancyra” (RH, 349) and "the term h*m*iousios plays no role in Basil's surviving texts" (LA, 150). This implies that such a compromise was not the purpose. More recently, Lewis Ayres proposed that Homo-i-ousianism was not merely a compromise but "a significant and persistent strand in earlier eastern theology." (LA, 150)
There are indications that this theology was a restatement or development of the theology of Eusebius of Caesarea, as stated in the letter he wrote to his home church after the Nicene Council, to explain why he accepted that Creed:
Ritter described Homoiousianism "as the right wing of the Eusebian party.” (RH, 349)
“Basil … prefers the term 'image of the ousia' to define the Son's relationship to the Father; it is worth noting that this term was favoured by Eusebius of Caesarea … and also is found in the Second ('Dedication') Creed of Antloch 341.” (RH, 353)
Eusebius was “universally acknowledged to be the most scholarly bishop of his day.” (RH, 46) Eusebius was the most influential theologian present at the Council of Nicaea in AD 325.
Both Lewis Ayres and R.P.C. Hanson stated that the formulation of Homoiousian theology in 358 by a council of bishops called by Basil of Ancyra was a response to, what Ayres calls, "the emerging shape of Heterousian theology" in the form of the creed of “Sirmium 357,” which was based on the teachings of Aetius. Hanson refers to this as "Neo-Arianism" and as “a new and radical theology” that appears for the first time in the extant ancient records in the form of the “Second Creed of Sirmium of 357,” afterward approved by a larger synod at Antioch (probably in AD 358). 'Neo-Arianism' may be an appropriate name because it was “a development" of Arius' theology. (RH, 348; LA, 149-150)
“Basil's council sent a delegation to the Emperor Constantius … and this embassy met with success.” The Emperor condemned “Aetius and his teaching” and exiled Aetius and his supporters. This supports the view that this formulation of h*m*-i-ousianism was particularly intended to oppose the Neo-Arians. (LA, 152-153)
“The statement which emerged from this council … marks the emergence of a new and coherent theological point of view. This is the theology of those whom Epiphanius, quite undeservedly, calls 'Semi-Arians', but who are usually today thought of as Homoiousians, a designation which is more accurate.” (RH, 348-9)
This statement was written by Basil of Ancyra himself (LA, 150) and “is of the highest importance for an understanding of Homoiousian theology.” (RH, 350) It includes “nineteen anathemas which reveal more clearly the position which Basil is attacking." (RH, 355)
Homoianism was a dominant Christology during the mid-fourth century. For example, the creeds of the councils of Sirmium in 358, Ariminum in 359, and the key council at Constantinople in 359 / 360 were h*m*ian. It refused to use ousia (substance) language in the formulation of any statement of faith. Against them, Basil insisted that substance language is necessary to reflect the closeness of the Father and Son expressed by the concepts “Father/Son” and “begotten.” He wrote:
“God must be both Father and creator” (of His Son) (RH, 353). “If we remove this resemblance of ousia,” the Son is merely a created being; “not a Son.” (RH, 353, 354)
Since human sons are like their fathers, the Son of God is like His Father (RH, 352). “The salient irreducible element” in a father/son relationship is “the begetting of a living being that is like in ousia.” (RH, 352-3)
“If the Father gives the Son to have life in himself (John 5:26) … then the Son must have the same life and thus have ‘everything according to essence and absolutely as does the Father’.” (LA, 152)“
It is often claimed that the term h*m*-ousios in the Nicene Creed means "one substance," namely, that the substance of the Son is one and the same as the Father's substance. It is on this basis alone that we can argue that the Son is co-eternal and co-equal with the Father. However:
Hanson concluded that “we can … be pretty sure that h*m*ousios was not intended to express the numerical identity of the Father and the Son.” (RH, 202)
Philip Schaff stated: "The term h*m*ousion ... differs from monoousion. ... and signifies not numerical identity, but equality of essence or community of nature among several beings. It is clearly used thus in the Chalcedonian symbol, where it is said that Christ is “h*m*ousios with the Father as touching the Godhead, and h*m*ousios with us [and yet individually distinct from us] as touching the manhood.” [Philip Schaff, History of the Church volume 3 (Grand Rapids, MI: Eerdmans Publishing Co., 1981 edition) pp.672-673.]
The idea of "one substance," therefore, developed later. In the Homo-ousianism of the Nicene Creed, the Son's substance is identical with the Father's, meaning two substances that are identical.
This is based on a material interpretation of the terms “Father,” Son,” and “only-begotten,” as if God has a body and bodily gave existence to the Son, comparable to how human sons are brought forth. Consequently, the Son’s substance is identical to the Father’s. For example:
“Anathema 13 links the error of thinking of the Father/Son relationship in corporeal terms with that of making the Son identical with the Father.” (RH, 356)
Homo-i-ousianism did not accept this notion. (RH, 352-3) In Homo-i-ousianism, the Son is subordinate to the Father. (RH, 355) If this is true, then the Son's substance cannot be identical to the Father's. Basil explained:
“The Son is like the Father in ousia but not identical with him.” (RH, 352-3).
“As He … was in the likeness of men (John 1:14) … yet not a man in all respects;” “not identical with human nature,” for example. He was not born through natural conception, “so the Son … is God in that he is Son of God,” was “in the form of God," and is "equal to God (Phil 2:6, 7),” “but not identical with the God and Father.” (RH, 354)
Anathema 13 “damns him who declares ... that the Son is identical with the Father … This is manifestly directed against N (the Nicene Creed).” (RH, 355)
In Sabellianism, the Son is not a distinct Person. Rather, the Father and Son are parts of one Person. Basil responded:
“This argument that God must be both Father and creator and that the likeness in ousia is necessary … as a safeguard against Sabellianism: that which is like can never be the same as that to which it is like'.” (RH, 353)
The anathemas also attack the apparent Sabellianism of Marcellus of Ancyra. (RH, 355)
In Neo-Arianism, which was “a new and radical” (RH, 348) adaptation of Arius’ theology, the terms “Father,” Son,” and “only-begotten” symbolize that the Son is the very image of the Father, but not in a corporeal (material) sense. For that reason, in this view, "the Son is 'unlike' (anh*m*ios) in ousia to the Father” Ayres refers to this as "Heterousian (different substance) theology." (LA, 149) For example, “Anathema 12 strikes him who declares that the Son's likeness to the Father consists in power but not in ousia.” (RH, 355)
Homoiousianism was somewhere between the Homoousian (identical substance) view and the Neo-Arian (different substance) view.
“In AD 359 Constantius decided to emulate his father's action in calling Nicaea and summon a general council. … A small group of bishops met at Sirmium to draw up a draft creed for discussion. Those present included not only Basil, but also some who were far more suspicious of ousia language. The creed on which they finally agreed … asserts that all ousia language should be avoided. … … Thus, although Basil of Ancyra was influential with the imperial authorities at one point during 358–9, it was not for long, and he never seems fully to have overcome long-standing Homoian influence at court. (LA, 157-8)
Constantius was becoming somewhat hostile to the influence of all of the new movements which had sprung up after the Nicene council. The result was that the Homoiousians disappeared from the stage of history and the struggle to define Church dogma became a two-sided battle between the Homo-ousians and the Homo-ians.
The 55 years of Controversy after the Nicene Creed of 325 revolved specifically around the word h*m*ousios. Since, in the Nicene Creed, this term was an interpretation of the term "begotten," the differences between the various Christological views are essentially different interpretations of the terms “Father,” "Son,” and “only-begotten.” These interpretations result in different views with respect to the substance of the Son, on the basis of which the five views may be summarized:
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Fr. Hardon's Modern Catholic Dictionary gives this definition of Semi-Arianism:
The teaching of certain theologians who, after the Council of Nicaea (A.D. 325), sought a compromise between Arianism and the doctrine of Christ's consubstantiality with the Father. They were led by Basil, Bishop of Ancyra, and their sympathies were toward orthodoxy, although they substituted h*m*iousios (similar to) the Father. St. Athanasius treated them kindly and their influence was felt in the reaffirmation of the Nicene Creed at the ecumenical Council of Constantinople in the year 381.
From Fr. Réginald Garrigou-Lagrange, O.P.'s, The Trinity & God the Creator, a commentary on St. Thomas Aquinas's articles on the Trinity in his Summa Theologica, the Treatise on the Trinity (qq. 27-43):
According to Arius, God the Father alone is eternal; the Father created the Son, not of His own substance but out of nothing, and then God made use of the Son as an instrument to create the universe and redeem men.
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After Arianism was thus condemned by the Church as a heresy, the Arians tried to dissimulate their error and said that the Son was similar in nature to the Father, h*m*iousion or h*m*ion, but they refused to say that He was consubstantial or h*m*ousion. Such was the teaching of Basil of Ancyra and Auxentius of Milan, who are called Semi-Arians. Arianism lasted into the sixth century, when it completely disappeared.
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Following the principles that misled Arius, Eunomius concluded that the Holy Ghost was not God but a creature made by the Son of God, inferior to Him and similar to the angels. At about the same time, the Macedonians like the Semi-Arians denied the divinity and consubstantiality of the Holy Ghost. Eunomius was refuted by St. Gregory of Nyssa, St. Basil of Caesarea, and St. Ambrose. Macedonianism [and thus also Semi-Arianism] was condemned by [Pope] St. Damasus [I] in the fourth Council of Rome (380) and in the following year by the second ecumenical Council of Constantinople. The most important definition of the Council is: "If anyone shall say that the Holy Ghost is not truly and properly of the Father, like the Son, of the divine substance, and true God, let him be anathema." Thus in the fourth century, opposing these heresies, the Church explicitly taught a Trinity of distinct persons, upheld their divinity and consubstantiality, and so preserved the unity of essence together with the distinction of persons. In the earliest centuries, therefore, the Church explicitly condemned that Unitarianism which the liberal Protestants have recently revived.
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In 325 the Council of Nicaea defended the true tradition against Arius, who taught that the Father alone was truly God, that the Word was the most excellent of creatures, created in time out of nothing, and that the Holy Ghost was also a creature, inferior to the Son. After long discussion it was defined that the Word was consubstantial with the Father, h*m*usion: "We believe in one God the Father almighty, maker of all things, visible and invisible. And in one Lord, Jesus Christ, the Son of God, the only-begotten Son of the Father, that is, of the substance of the Father, God of God, light of light, true God of true God, begotten, not made, of one substance with the Father, as the Greeks say, h*m*usion, by whom all things were made. And in the Holy Ghost."
After this condemnation the heretics tried to cover up their error by teaching that the Son was not properly h*m*usion or consubstantial with the Father, that is, of the same essence, but that He was similar in nature, or h*m*iousion. Such was the teaching of the Semi-Arians; the Acacians said the Son was h*m*ion, that is, similar with regard to form and accidents. These teachings were refuted by St. Alexander, the bishop of Alexandria, and by St. Athanasius.
So, to answer your questions: