.. How do I cross these ledges?

Jump to the first ledge and look at the wall painting. The circle with the heads rotates, and if you put the correct head on top and push the body in the middle, the next ledge will extend. If not, all the extended ledges retract a bit. Make three mistakes, and you're part of the exhibit.

Luckily, you have two pieces of information to help you figure out which head is the right one --

-- but unluckily, neither one is really that much help. In fact, this whole puzzle is rather messed up, so I won't tell anyone if you just go to the end of this hint and read the answer.

The two pieces of information are the riddle Jamila just gave you (which says "Half the key to the order's hidden exit") and the very last page of the journal (in which the writer says she has the first half of the key).

The riddle has four verses, and there are four ledges, so you'd think they'd apply one for one ... but the verses are complicated and have several images apiece.

The first one says, "... the ancient reptile leading the way." That sounds like we want the crocodile, and we do, so turn the dial to the croc and press the middle.

Hey presto, the ledge extends, and we think we've got the idea. But the second verse says, "Underneath a blazing sun, the jackal joins the mix of beasts that stand as one betwixt the sky and the Underworld." So ... that sounds like we put the falcon with the sun on his head on top and the jackal on the bottom. Right?

Wrong! Trial and error reveals that the correct head here is the falcon with nothing on his head.

The next one is even worse; it says, "Dressed in red, she is joined by the child. As the evening dew appears in the cat's footprints, they learn of knowledge." There's no one in red; there are no children; there are three cats; and it's not the one with the ibis head (who is Thoth, governing knowledge).

Finally, we have, "Drawn onward by the steadfast cow, a throne accompanied by hieroglyphs appears before the balance of the Mother." There is such a confusion of symbols here it's impossible to pick out a single one.

The info on the last page of the journal (the first page you found in the upper room) says the first is last, the second is fourth, the third is third, and the fourth is first. That should apply here, and it does, but not clearly.

On the first ledge, for example, the crocodile is the last head if you turn the dial clockwise. Great!

But on the second one, the bare falcon is only fourth if you turn the dial counterclockwise. Oy.

The third one is also counterclockwise, and the last one is clockwise again. So it's really not that much help. What good is a riddle if you have to guess?

In the end, until we can figure out the logic here, the best way to solve this problem is trial and error. That is, save before you try a head, then keep trying different heads until you get the one that doesn't kill you. Or just read on and I'll tell you the answer.

The heads that work are:
First ledge -- crocodile
Second ledge -- falcon with nothing on its head
Third ledge -- green cat with nothing on its head
Fourth ledge -- woman with sun on her head