We say that He both suffered, and rose again, not meaning that the Word of God suffered in His own nature .... but in so far as that which had become His own body suffered, then He Himself is said to suffer these things, for our sake, because the Impassible One was in the suffering body.1

(God) ever remains what He is and does not change or undergo alteration. Moreover all of us confess that the divine Word is impassible, even if in His all-wise economy of the mystery he is seen to attribute to Himself the sufferings that befall His own flesh (I Pet. 4.1.). He bears the suffering of His own flesh in an economic appropriation to Himself, as I have said, so that we may believe Him to be the Saviour of all.2

In these statements Saint Cyril explains to us that although God the Word is impassible (not susceptible to suffering), yet according to the economy (God’s plan for saving us), it can be said that God the Word Incarnate suffered, because he economically appropriated unto Himself the sufferings of the flesh. In the same way, we can say that God died for us (in the flesh) even though as God, He is Life.

The Word became flesh in the senses already exposed by us so often before. He has laid down His life for us, for since His death was to be the salvation of the world He “endured the cross, scorning the shame” (Heb.12.2)  even though, as God, He was Life by nature. How can Life be said to die? It is because Life suffered death in its very own body that it might be revealed as Life when it brought the body back to life again. 
    Come now, and let us carefully examine the manner of our own deaths. Is it not the case that men of good sense say that souls are not destroyed at the same time as are bodies that come from the earth? In my opinion this is something no one questions. However, what befalls us is still called the “death of a man”. This is how you should understand in the case of Emmanuel. For He was the Word in His own body born from a woman, and he gave it to death in due season, but He suffered nothing at all in His own nature for as such He is Life and Life-giver. Nonetheless he made the things of the flesh His own so that the suffering could be said to be His. The same is true in the rising up on behalf of all, having died for the sake of all to redeem all that is under heaven with His own blood, and to acquire for God the Father all that is on the face of the earth. ... For if He had not suffered for us as man He would not have achieved our salvation as God.3

The Word of God the Father is impassible and immortal. for the divine and ineffable nature is above all suffering, and this it is which gives life to all things and is greater than corruption or anything else that can normally cause us grief. Yet even though the Word of God the Father is so by his own being, He made His own the flesh which is capable of death so that by means of this which is accustomed to suffer He could assume sufferings for us and because of us, and so liberate us all from death and corruption by making His own body alive, as God, and by becoming the first fruits of those who have fallen asleep, and the first born from the dead (1 Cor. 15.20)4

In this sense Saint Cyril is following faithfully in the footsteps of his great master Saint Athanasius:

Whence it was that, when the flesh suffered, the Word was not external to it; and therefore is the passion said to be His.5

Let no one then stumble at what belongs to man, but rather let a man know that in nature the Word Himself is impassible, and yet because of that flesh which He put on, these things are ascribed to Him, since they are proper to the flesh, and the body itself is proper to the Saviour. And while He Himself, being impassible in nature, remains as He is, not harmed by these affections, but rather obliterating and destroying them, men, their passions as if changed and abolished in the Impassible.

For in the incorporeal, the properties of body had not been, unless He had taken a body corruptible and mortal; for mortal was Holy Mary, from whom was His body. Wherefore of necessity when He was in a body suffering, and weeping, and toiling, these things which are proper to the flesh, are ascribed to Him together with the body.7 

Whence, as I said before, the Word, since it was not possible for Him to die, as He was immortal, took to Himself a body such as could die, that He might offer it as His own in the stead of all, and as suffering, through His union  with it, on behalf of all, "Bring  to nought him that had the power of death, that is the devil; and might deliver them who through fear of death were all their lifetime subject to bondage." (Heb 2.14)8

But, to both Saint Athanasius and Saint Cyril, it was not only that the Word appropriated to Himself the things of the flesh, but He also did His divine works through the body. This is called "The Communication Of Properties".

Being God, He had His own body, and using this as an instrument, He became man for our sakes. And on account of this, the properties of the flesh are said to be His, since He was in it, such as to hunger, to thirst, to suffer, to weary, and the like, of which the flesh is capable; while on the other hand the works proper to the Word Himself, such as to raise the dead, to restore sight to the blind, and to cure the woman with an issue of blood, He did through His own body. And the Word bore the infirmities of the flesh, as His own, for His was the flesh; and the flesh ministered to the works of the Godhead, because the Godhead was in it, for the body was God's.9

Whence it was that, when the flesh suffered, the Word was not external to it; and therefore is the passion said to be His: and when He did divinely His Father's works, the flesh was not external to Him, but in the body itself did the Lord do them. Hence, when made man, He said, “If I do not the works of the Father, believe Me not; but if I do, though ye believe not Me, believe the works, that ye may know that the Father is in Me and I in Him.” (John 10.37, 38) And thus when there was need to raise Peter's wife's mother, who was sick of a fever, He stretched forth His hand humanly, but He stopped the illness divinely. And in the case of the man blind from the birth, human was the spittle which He gave forth from the flesh, but divinely did He open the eyes through the clay. And in the case of Lazarus, He gave forth a human voice as man; but divinely, as God, did He raise Lazarus from the dead. These things were so done, were so manifested, because He had a body, not in appearance, but in truth; and it became the Lord, in putting on human flesh, to put it on whole with the affections proper to it; that, as we say that the body was His own, so also we may say that the affections of the body were proper to Him alone, though they did not touch Him according to His Godhead. If then the body had been another's, to him too had been the affections attributed; but if the flesh is the  Word's (for “the Word became flesh”), of necessity then the affections also of the flesh are ascribed to Him, whose the flesh is. And to whom the affections are ascribed, such namely as to be condemned, to be scourged, to thirst, and the cross, and death, and the other infirmities of the body, of Him too is the triumph and the grace. For this cause then, consistently and fittingly such affections are ascribed not to another, but to the Lord; that the grace also may be from Him, and that we may become, not worshippers of any other, but truly devout towards God, because we invoke no originate thing, no ordinary man, but the natural and true Son from God, who has become man, yet is not the less Lord and God and Saviour.10  

More than a century earlier, Saint Hippolytus, bishop of Rome, wrote almost the same:

But the pious confession of the believer is that, with a view to our salvation, and in order to connect the universe with unchangeableness, the Creator of all things incorporated with Himself a rational soul and a sensible body from the all-holy Mary, ever-virgin, by an undefiled conception, without conversion, and was made man in nature, but separate from 
wickedness: the same was perfect God, and the same was perfect man; the same was in nature at once perfect God and man. In His deity He wrought divine things through His all-holy flesh,-such things, namely, as did not pertain to the flesh by nature; and in His humanity He suffered human things,-such things, namely, as did not pertain to deity by nature, by the upbearing of the deity. He wrought nothing divine without the body; nor did the same do anything human without the participation of deity.11

1. Saint Cyril Of Alexandria: Second Letter To Nestorius in:  John McGuckin: Saint Cyril of Alexandria and the Christological Controversy p 203
2. Saint Cyril Of Alexandria: Letter to John of Antioch in John McGuckin: Saint Cyril of Alexandria and the Christological Controversy p 203
3. Saint Cyril Of Alexandria: Letter to the Monks of Egypt in: John McGuckin: Saint Cyril of Alexandria and the Christological Controversy pp 259, 260
4. Saint Cyril Of Alexandria: Explanation of the Twelve Anathemas in John McGuckin: Saint Cyril of Alexandria and the Christological Controversy p 292
5. Saint Athanasius, Four Discourses Against The Arians, Discourse III, 32,  Nicene and Post-Nicene Fathers Series II, Volume IV
6. Saint Athanasius, Four Discourses Against The Arians, Discourse III, 34, Nicene and Post-Nicene Fathers Series II, Volume IV
7. Saint Athanasius, Four Discourses Against The Arians, Discourse III 56, Nicene and Post-Nicene Fathers Series II, Volume IV
8. Saint Athanasius, On The Incarnation  Of The Word of God 20.6, Nicene and Post-Nicene FathersSeries II, Volume IV
9. Saint Athanasius, Four Discourses Against The Arians, Discourse III 31, Nicene and Post-Nicene Fathers Series II, Volume IV
10. Saint Athanasius, Four Discourses Against The Arians, Discourse III 34, Nicene and Post-Nicene Fathers Series II, Volume IV, read online
11. Saint Hippolytus, Extant works and fragments, Part II. E. Fragment VIII, Ante-Nicene Fathers Volume V, read online

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