These terms arose in the early centuries of the Christian church as theologians sought to understand and to define more clearly how they understood the person and work of Christ.
First, in contrast to orthodox Christian doctrine that the incarnate Christ was fully divine and fully man, monophysites hold that Christ had only a single—i.e., the divine—nature in the Person of the Incarnate Christ.
Monophysitism first became a distinct doctrine following the pronouncement of Christ's dual nature by the Council of Chalcedon in 451 and became a sigificant matter of debate during the 5th through to the 6th Cent.
Monophysitism came in different flavours. Eutychianism (after Eutyche, the leader of a monastary in Constantinople, early 5th Cent) held that the human nature of Christ was so diluted by the Divine—dissolved like a drop of honey in the sea—to the point where it is essentially obliterated. Apollinarianism (after Apollonarius, bishop of Laodicea, early 4th Cent.)holds that Christ had a human body, even a human "living principle" but his nous (think of its as analogous to the mind) has been so overtaken by the Divine Logos as to have lost its human nature.
Monophysitism remains the official stand of the Coptic, Abyssinian, the Syrian Jacobite and the Armenian churches.
Miaphysitism (from Greek mia, meaning 'one') is today held mainly by the Eastern Orthodox churches and teaches that the incarnate Christ has only one nature but one that is simultaneously fully divine and fully human.
Dyophytisitism (from Greek duo, for two) teaches, on the other hand that the incarnate Christ has two natures, that he was fully human and fully divine.
Resources:
Louis Berkhof, History of Christian Doctrines
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