I answer that, It was most fitting that Christ should suffer the death of the cross.
First of all, as an example of virtue. For Augustine thus writes (Questions. lxxxiii, qu. 25): "God's Wisdom became man to give us an example in righteousness of living. But it is part of righteous living not to stand in fear of things which ought not to be feared. Now there are some men who, although they do not fear death in itself, are yet troubled over the manner of their death. In order, then, that no kind of death should trouble an upright man, the cross of this Man had to be set before him, because, among all kinds of death, none was more execrable, more fear-inspiring, than this."
Secondly, because this kind of death was especially suitable in order to atone for the sin of our first parent, which was the plucking of the apple from the forbidden tree against God's command. And so, to atone for that sin, it was fitting that Christ should suffer by being fastened to a tree, as if restoring what Adam had purloined; according to Ps. 68:5: "Then did I pay that which I took not away." Hence Augustine says in a sermon on the Passion [*Cf. Serm. ci De Tempore]: "Adam despised the command, plucking the apple from the tree: but all that Adam lost, Christ found upon the cross."
The third reason is because, as Chrysostom says in a sermon on the Passion (De Cruce et Latrone i, ii): "He suffered upon a high rood and not under a roof, in order that the nature of the air might be purified: and the earth felt a like benefit, for it was cleansed by the flowing of the blood from His side." And on Jn. 3:14: "The Son of man must be lifted up," Theophylact says: "When you hear that He was lifted up, understand His hanging on high, that He might sanctify the air who had sanctified the earth by walking upon it."
The fourth reason is, because, by dying on it, He prepares for us an ascent into heaven, as Chrysostom [*Athanasius, vide A, III, ad 2] says. Hence it is that He says (Jn. 12:32): "If I be lifted up from the earth, I will draw all things to Myself."
The fifth reason is because it is befitting the universal salvation of the entire world. Hence Gregory of Nyssa observes (In Christ. Resurr., Orat. i) that "the shape of the cross extending out into four extremes from their central point of contact denotes the power and the providence diffused everywhere of Him who hung upon it." Chrysostom [*Athanasius, vide A. III, ad 2] also says that upon the cross "He dies with outstretched hands in order to draw with one hand the people of old, and with the other those who spring from the Gentiles."
The sixth reason is because of the various virtues denoted by this class of death. Hence Augustine in his book on the grace of the Old and New Testament (Ep. cxl) says: "Not without purpose did He choose this class of death, that He might be a teacher of that breadth, and height, and length, and depth," of which the Apostle speaks (Eph. 3:18): "For breadth is in the beam, which is fixed transversely above; this appertains to good works, since the hands are stretched out upon it. Length is the tree's extent from the beam to the ground; and there it is planted---that is, it stands and abides---which is the note of longanimity. Height is in that portion of the tree which remains over from the transverse beam upwards to the top, and this is at the head of the Crucified, because He is the supreme desire of souls of good hope. But that part of the tree which is hidden from view to hold it fixed, and from which the entire rood springs, denotes the depth of gratuitous grace." And, as Augustine says (Tract. cxix in Joan.): "The tree upon which were fixed the members of Him dying was even the chair of the Master teaching."
The seventh reason is because this kind of death responds to very many figures. For, as Augustine says in a sermon on the Passion (Serm. ci De Tempore), an ark of wood preserved the human race from the waters of the Deluge; at the exodus of God's people from Egypt, Moses with a rod divided the sea, overthrew Pharaoh and saved the people of God. the same Moses dipped his rod into the water, changing it from bitter to sweet; at the touch of a wooden rod a salutary spring gushed forth from a spiritual rock; likewise, in order to overcome Amalec, Moses stretched forth his arms with rod in hand; lastly, God's law is entrusted to the wooden Ark of the Covenant; all of which are like steps by which we mount to the wood of the cross.