Myst

From the starting position go back to the platform with red umbrella. Put lever at the most right position and press button. This will drain the water in the lighthouse which is opposite of this platform. Get there, down the ladder and press the lever mechanism of the barrel to drain it. Press again the lever so the barrel will be closed and stay empty.

Go back to the platform, put lever on the middle position and press again the button on the left. This will drain the rooms in the middle of the age and flood the water in the lighthouse.

Go to the lighthouse. Now the barrel with the chest is in place so click on the hanging key, unlock the chest, take the new key and use it to the keyhole on the wall. This will make the lowest part of the ladder on the wall to emerge. Use the ladder and use the machine there to fully charge the lighthouse. This gives for a limited time power to the rooms but also to the lighthouse itself.

Go back to the ship, take the stairs to the next part of the ship and up to the path that go around the rock. At the top there is a spyglass/telescope. Looking through the device the only point of interest you will see is the lighthouse blink, @ 135° in the classic playthrough, random in the randomized playthrough.

Go back to the lighthouse recharge the machine there and go through the left path that leads to the rooms, the path left of the stairs that connect the two parts of the ship. Halfway down there is an open door to the left,it is open only when the lights are on and will lead to the compass room.

Save the game!
The north in the compass is the yellow arrow with the strange corner. So you have to find the button in the same degrees as the lighthouse on spyglass and press it. In classic playthrough it is @135°. It is the switch that the southeast red arrow of the compass points. In randomized playthrough you have to find the according button. User UnusedKael in comments provide a way to find the correct button

If you have done it right, the lights will stay on, more lights will open and the achievement will pop.

After raising the ship, go into the cabin to find the entrance to the Stoneship Age. Restore power to the underground passages and you'll find a room with a giant compass, and a bunch of buttons. For those who don't know, the fancy arrow on the compass is due north. Knowing that, find due east, count four buttons toward south and hit that one. Achievement unlocked!