Robinson’s Arch

Now in the western quarters of the enclosure of the temple there were four gates; the first led to the king's palace, and went to a passage over the intermediate valley; two more led to the suburbs of the city; and the last led to the other city, where the road descended into the valley by a great number of steps, and thence up again by the ascent for the city lay over against the temple in the manner of a theater, and was encompassed with a deep valley along the entire south quarter.

— Josephus, Antiquities of the Jews 15:410.

It is not certain if Josephus listed these gates from right-to-left or left-to-right.  But if the “kings palace” refers to Fortress Antonia, then the first gate is describing Robinson’s Arch which spanned the Tyropoeon Valley on the left. The middle 2 gates included the Kiponus Gate, and the right-most gate leading to the “other city” with a “great number of steps” leads to the City of David.

Mishnah Three 

1)      There were five gates to the Temple Mount:

2)      The two Huldah gates on the south were used both for entrance and exit;

3)      The Kiponus gate on the west was used both for entrance and exit.

4)      The Taddi gate on the north was not used at all.   

5)      The Eastern gate over which was a representation of the palace of Shushan and through which the high priest who burned the red heifer and all who assisted with it would go out to the Mount of Olives. 

"But the other hill, which was called "Acra," and sustains the lower city, is of the shape of a moon when she is horned; over against this there was a third hill, but naturally lower than Acra, and parted formerly from the other by a broad valley. However, in those times when the Asamoneans reigned, they filled up that valley with earth, and had a mind to join the city to the temple. They then took off part of the height of Acra, and reduced it to be of less elevation than it was before, that the temple might be superior to it. Now the Valley of the Cheesemongers, as it was called, and was that which we told you before distinguished the hill of the upper city from that of the lower, extended as far as Siloam; for that is the name of a fountain which hath sweet water in it, and this in great plenty also. But on the outsides, these hills are surrounded by deep valleys, and by reason of the precipices to them belonging on both sides they are every where unpassable.Josephus Wars: Book 5 Ch:4: 1