To start the quest "Lost to the Ages", you will need to read a book called "The Aetherium Wars". This side quest is available to BOTH factions.

The Aetherium Wars Location

The easiest way to find the book depends on the faction you chose:

This book is in Fort Dawnguard, near a crafting station on some shelves.

Vampire Lords: This book is in Castle Volkihar (Volkihar Keep), on top of a small round table in one of the balconies overlooking the main dining hall

Once you read this book, you receive an objective:
Investigate the ruins of Arkngthamz

Note: Some readers have reported a bug where the quest does not start. If that happens, try reading another copy of the book. There are more copies in Bard's College, The Castle Volkihar Undercroft,  Hall of the Vigilant, Ilinalta's Deep, Rannveig's Fast, Redwater Den and Ruunvald Temple. If none of the books trigger the quest, try going straight to the ruins of Arkngthamz as described below.

This objective can be found under the Miscellaneous tab in your quest log. Your map is updated with the location of the ruins. Exit the fort/castle and fast travel there. It is located in The Reach, to the south/southeast of Markarth and to the south of Dushinkh Yal. See map below:

Lost to the Ages map Arkngthamz location

After you get to the destination marked on your map, enter Arkngthamz. Go north down the hall, and then west. You will hear a ghostly voice telling you "Turn back... please turn back, before it's too late". You arrive at a chamber with an abyss going through the center. Three beams lead down to a platform in the middle of the abyss. Climb down to the platform on one of those ramps.

While you are exploring these ruins, a ghost called Katria appears. She asks you what you are still doing here. She guesses you are here for the treasure, just like all the others.

Ask her who she is. She tells you her name is Katria. She was an adventurer. Raided ruins like this for nigh on twenty years. She was on the trail of something big. It led her here, and she didn't make it.

If you tell her that her name sounds familiar, she says it is because you have read the book, The Aetherium Wars (which is "dedicated" to Katria, by Taron Dreth). That was her theory, her research and life's work. All of it was lost, stolen by her own apprentice. That is how she ended up here and she cannot rest until she finds the forge and can prove that it was all her discovery. But she thinks it is hopeless. She died here just like all the others, and suggests you turn back before you become the next victim.

Your options are:
I can handle myself (PERSUADE)
I won't give up that easily
Get out of my way

Try the persuasion option first, and if that doesn't work than tell her you won't give up that easily. Eventually, Katria is convinced. She offers her help, says you will need it if you want to reach the summit. Tell her "Sure".

This starts the quest:

Lost to the Ages
In Arkngthamz, I met Katria, the ghost of an adventurer who perished there. She offered to join me in my fight to the summit, and I accepted her help.

Reach the Summit of Arkngthamz

Before proceeding, make sure to search Katria's body (on the platform) for Katria's Journal.

The Lost to the Ages achievement guide continues here: Getting the Zephyr Bow