You obtain the quest "Hitting the Books" from the Arch-Mage, after completing the Under Saarthal quest.

Hitting the Books
Since Tolfdir is not available, I've been asked by Mirabelle Ervine to look into information on the object we found in Saarthal by talking to Urag gro-Shub in the Acanaeum

Objectives
Speak with Urag gro-Shub

You can enter the Arcanaeum through the Hall of the Elements. Speak to Urag behind the desk. Tell him you want to learn about something you found in Saarthal. He will tell you he can't help since his books got stolen by Orthorn when he went to the summoners in Fellglow Keep. One of the volumes Orthorn stole has the information you need.

If you look at the quest journal, it has been updated:
I've been asked to find more information on the object found in Saarthal. Urag suggested that useful books may have been stolen from the Arcanaeum by Othorn, a former apprentice at the colllege. His last known location was Fellglow Keep.

Find the Stolen Books (0/3)

Ancano will ask you some questions as you leave. Answer his questions and leave the college. Fast Travel to Whiterun. Exit Whiterun through the main gate, and travel East to Fellglow Keep, past the Honningbrew Meadery and White River Watch.

The main door is locked and requires a key. Enter through the stairs in the ruined tower near the water instead.

Work your way through the flooded caverns. Go east, then south, defeat a mage and 2 spiders, then head NE and arrive at a room with some cells. Activate the levers to release the imprisoned vampires. Keep going NE to another chamber with jail cells. The mage there will release wolves at you. Defeat all the enemies. Talk to Orthon through the bars. Tell him you are here for the books.

He will tell you that if you free him, he will help you find the books. Free Orthorn by activating the middle lever. Open the wooden door and descend down the stairs. Fight a group of  mages, then climb up the stairs. Go to the end of the corridor and turn left. Open the door to Fellglow Keep. Head Northeast towards the Fellglow Keep Ritual Chamber. Work your way to a circular room to the SE. Then go south, annd west. Enter a room and kill the mage there to get the Fellglow Keep Key. Go through the wooden door to the North. Climb up the stairs and open the ritual chamber door.

Find the Caller inside. Tell her you are not leaving without the books. Fight and kill her. Get the three stolen books. Loot the caller for the Ritual chamber key and enter a small room nearby with lots of treasures. Then exit down through the grate on the floor. Raise the bar and open the wooden door. Use another door to exit to Skyrim. Fast travel to the College of Winterhold and go the the Arcanaeum to return the books to Urag.

This complete the quest "Hitting the Books" and unlocks the next quest in the College of Winterhold series, "Good Intentions"