After completing Speaking with Silence head on over to Winterhold to locate Enthir in the Frozen Hearth. After a very informative conversation with the high elf, you will learn to seek out Calcelmo, court wizard of Markarth, to gain his much needed research on Ancient Falmer, the long forgotten language of the Snow Elves.
Head to Markarth, enter Understone Keep at the Western edge of town. Once inside, head South to locate Calcelmo; now if you have already completed the quests Neutralizing Nimhe or The book of Love , then you already have access to his museum. If you have not completed either of them, you can at this time or you can pickpocket the key from Calcelmo or the guard at the museum door. Alternatively, if your speech is high enough, you can persuade him to give you the key; intimidation and bribery will not work. Note: persuading Calcelmo to provide the keymakes the next part of this quest exponentially easier as the guards inside will not attack, but if you want a challenge, you may just pick the lock of the door, or steal the key for the museum.
Inside the museum, there are three city guards; if Calcelmo give you the key and permission to enter the museum, the guards have been informed and will not bother you. Note: This room is the best place in all of Skyrim to level up lockpicking by the way. Walk, stealth, or kill your way through the museum but make sure you scour the area as there are some valuable items in the room to each side here. Once the museum holds no more interest for you, head towards the West wall and enter the door to Calcelmo’s Laboratory.
Enter the lab and listen to the exchange between Aicantar and the guard to learn that these guards will attack anyone on sight. Luckily for you, these guards are not city guards so killing them won’t cause you to earn a bounty. Note: stealth is nigh impossible in this section so don’t bother with it unless you’re a masochist. When you enter, the room to your immediate left contains a Spider Control Rod and Aicantar’s Lab Journal detailing the use of the rod.
The door in the second room is locked but can be entered one of three ways, it can be picked, you can simply loot a key off one of the guards that you killed on the way in, or you can find a key in Aicantar’s bedroom on the North side of this room. Once through the doors, you will overhear some guards talking; there are two options here, fight your way through the guards and their backup (fast but may be tough depending on your level), or you can sneak past the guards travel through the toxic gas (step on the pressure plates for a break from the fumes or use the spider for this task). You will emerge on the other side of the hallway with the guards where you can use the red valve to activate the traps and kill them easily.
Open the door near the valve, travel up the stairs and around the corner to reach a room with the two Dwarven Sphere statues. Just beyond the door between them is another red valve that activates all the traps in the room below. This kills everyone giving you easy access to the door leading to Markarth Wizards’ Balcony. Note: Just before you go through the aforementioned door, look to the South to spot the next Larceny Target for Delvin. Once on the balcony, there is only one place left to go: Calcelmo’s Tower. This is where you will find the Calcelmo’s research which you need for your quest but there is a problem, you have nothing to copy it down with. Search the room for a Roll of Paper and Charcoal, then head back to Calcelmo’s Stone and make a rubbing of the stone.
As soon as you do, guards flood the room and fight or stealth are your only means of escape. Once you are outside, you can either backtrack through the museum, or you can take a quicker means of escape. At the bottom of the stairs that lead up to the Tower door, there is a small path to the right, follow it and jump off at the end.
You should now be safe enough to fast travel back to Winterhold where Karliah and Enthir will be waiting for you in the Frozen Hearth. Speak with Enthir and follow along as even more secrets about the Guild are revealed. After all conversation options have been explored with Enthir, speak with Karliah and exhaust her conversation options to complete this quest and begin The Pursuit.