Travel to the city of Winterhold and follow the objective marker or use your map to locate Enthir. Talk with him and show him the journal. After learning it was written in Falmer, he will refer you to one named Calcelmo who can be found in Markarth, specifically at a location called Understone Keep.
Calcelmo will refuse to show you his work unless you convince him to do so by means of the [Persuade], [Bribe] or [Intimidate] functions. Once you successfully gain his approval, you’ll receive the key to the Dwemer Museum.
NOTE: If you are not successful in your attempts to [Persuade], [Bribe] or [Intimidate] him, you can also obtain the key by completing Calcelmo’s spider side quest or by just stealing it off of the nearby table.
Follow the marker on your map to find the Museum’s entrance. If ever a guard stops you, just ignore him and open the gate anyway.
Once you’re inside the museum, go straight ahead and unlock the door leading to Calcelmo’s laboratory. Once inside, look on the table to the left to find a Spider Control Rod. A mechanical spider will appear and you can use the rod to direct the spider to your objective.
This whole place is filled with traps which you can activate by turning the red valves. Use these to sneak by the heavily armed guards and kill them, unless of course you’d rather fight and kill them all yourself. You must make your way through the various rooms, killing the guards in the process until you finally reach the main laboratory. You’ll also find Calcelmo’s nephew, Aicantar in the same area. At this point you should rush in and kill him and the guard as quickly as you can. If you activate the trap, the guard dies but Aicantar may escape and bring back reinforcements.
You can also grab the Dwemer Puzzle Cube to the left of the Enchanter so that you can give it to Delvin Later. Open the marked door to exit to the balcony.
WARNING: Sometimes a glitch prevents players from interacting with and opening the door leading to the balcony. If this happens to you, try to look up until the game prompts you to open the door. Otherwise you have no choice but to exit the game and reload.
Exit to the balcony and enter Calcamo’s Tower. Once inside, pick up a piece of charcoal and a roll of paper, then examine the stone slab to create a duplicate of the Falmer translating guide. If you allowed Aicantar to escape earlier, he should return any moment now with some heavy reinforcements. You can sneak past them, or kill them all if you’re powerful enough.
When you’ve escaped from or dealt with your pursuers, head back to the city of Winterhold and speak with Enthir. You should find him in the Frozen Heart Inn cellar, together with Karliah. Listen to Enthir as he translates Gallus’ journal. Talk to Enthir again and he will offer his assistance to you as a fence based in Winterhold. Talk to Karliah next to complete this quest and receive your reward.