Now these advocates of the doctrine of the trinity, may God have mercy on thee, are divided into four sects, three of which are the older while the fourth appeared only recently. The first of these is of the opinion that the body, as well as the spirit of their Messiah, is derived from the Creator, exalted be He. The second holds the view that his body was created, his spirit alone having emanated from the Creator. The third, again, believes that both his body and his spirit were created, but that he also possessed another spirit that was derived from the Creator. As for the fourth group, it assigns to him the position of the prophets only, interpreting the sonship of which they make mention when they speak of him just as we interpret the Biblical expression: Israel is My first-born son (Exod. 4:22), which is merely an expression of esteem and high regard, or, as others interpret the meaning of the phrase: "Abraham, the friend of God."[Saadia Gaon, The Book of Beliefs and Opinions, Rosenblatt, tr. Yale University Press (New Haven: 1976) p. 109]
It's unclear whether and to what extent this is supposed to be a somewhat idealized classification -- the chronological remark in the first sentence suggests that it is intended to identify real groups, but the cleanness of the classification suggests that he might be partly just considering the logical possibilities. The second position seems to be Apollinarianism. I'm fairly sure that the third group is the Christology of the Church of the East -- it admits of both orthodox and Nestorian interpretation.
I don't know who would fall into the first and fourth groups, although the first position could be the kind of statement of Monophysitism that one might find in its critics. What is interesting about the fourth position is the comment that it is recent. Rosenblatt claims that Saadia means Muslims by the fourth group, which would account for the position. But Saadia clearly says he is talking about advocates of the doctrine of the Trinity (and reaffirms it at the end of the chapter), and it is impossible to imagine that Saadia, of all people, born in Egypt and writing in Baghdad in the tenth century, could possibly be ignorant of the Muslim rejection of the Trinity. Since Saadia is Jewish (albeit the greatest Jewish mind of the tenth century), one can allow for a bit of an outsider's perspective, so perhaps he is just approximating Christian positions he has only heard about. On the other hand, a look at the whole section in which Saadia criticizes Christian theology shows a clear familiarity with actual Christian arguments.
Saadia rejects the fourth position on the basis of arguments that Torah admits of no abrogation and that Christians have a false account of Messianic prophecy. (The latter is another reason to take him not to be discussing Muslims here.) Both of these arguments would apply to all the groups of Christians, of course. The first he argues against on the basis of the fact that a creature cannot be a portion or natural emanation of the Creator. With regard to the third, he argues that creatures cannot become God merely by association with the divine. And all of these arguments would apply against the second group.
One of the interesting things is the analogy he attributes later in the chapter to the third group (the fact that he is so much more precise about the third group is another reason to think that he explicitly has in mind the Church of the East): "They cite as an analogy the descent of the glory of God on Mount Sinai and its appearance in the Burning Bush and the Tent of Meeting." It's unlikely that the Assyrian Christians were actually arguing for the Incarnation on the basis of such an analogy (although Saadia does seem to take them to be doing so), but it's possible that it was brought up in arguments for clarification purposes, or may be the kind of imagery associated with the Incarnation in the liturgy and devotional life of Assyrian Christians in Saadia's day.
