John Lightfoot on Matthew 4:5

[Upon the pinnacle of the Temple.] Whether he placed him upon the Temple itself, or upon some building within the holy circuit, it is in vain to seek, because it cannot be found. If it were upon the Temple itself, I should reflect upon the top of the porch of the Temple: if upon some other building, I should reflect upon the royal gallery. The priests were wont sometimes to go up to the top of the Temple, stairs being made for this purpose, and described in the Talmudic book entitled Middoth; and they are said to have ascended hither, "When fire was first put to the Temple, and to have thrown up the keys of the chambers of the Temple towards heaven, with these words; 'O thou eternal Lord, because we are not worthy to keep these keys, to thee they are delivered.' And there came, as it were, the form of a hand out of heaven, and took them from them: and they leaped down, and fell into the fire."

Above all other parts of the Temple the porch of the Temple, yea, the whole space before it, may not unfitly be called the wing of the Temple, because, like wings, it extended itself in breadth on each side, far beyond the breadth of the Temple: which we take notice of elsewhere.

If, therefore, the devil had placed Christ in the very precipice of this part of the Temple, he may well be said to have placed him upon the wing of the Temple, both because this part was like a wing to the Temple itself, and that that precipice was the wing of this part.

But if you suppose him placed upon the royal gallery, look upon it thus painted out by Josephus: "On the south part [of the court of the Gentiles] was the king's gallery, that deserves to be mentioned among the most magnificent things under the sun: for upon a huge depth of a valley, scarcely to be fathomed by the eye of him that stands above, Herod erected a gallery of a vast height; from the top of which if any looked down, he would grow dizzy, his eyes not being able to reach to so vast a depth."