To start this quest, you must first complete "Murder and Mayhem" and then head to the location known as Long Bay, situated on the northwestern coast of Jamaica. Roberts will already be waiting for you on his ship near the shore.
Edward and Bartholomew Roberts fight their way through treacherous jungles to reach the entrance of the Observatory.
The first part is relatively simple. First, follow Adéwalé, and after a moment, you'll have to lead the way.
Generally, keep heading toward the next markers, staying up high whenever possible - moving along tree branches, platforms, and the like. This way, you'll avoid encountering wild animals lurking below - jaguars or crocodiles. At some point, you'll reach the shore of a lake with a few huts nearby.
Talk to your companion. Adéwalé will decide to leave you, and from then on, you'll continue alone.
Enter the tunnel directly ahead of you, then take the uphill path leading to a large cave. Inside, you'll run into some natives. Head to the right and slowly make your way to the other side of the cave - staying up high and knocking out any natives standing in your path.
If the two natives aren't moving, you can hit the one on the far side with a sleep dart and when the one on your side goes to check on him, quickly slip past behind his back.
On the other side of the cave, continue along the winding path and take down another native standing at the end of it with his back turned to you. Jump across the tree branches to the other side of the clearing with the strange stone, near another native sitting by a campfire with his back to you as well. Knock him out and move on.
Head straight, and when you see the waterfall, cross to its left side along the tree branches.
A bit further ahead, you'll reach a clearing and spot the Observatory.
As you enter the clearing, Roberts and his men will arrive right after you and head toward the Observatory. Roberts will open the entrance, then dispose of his companions, leaving just the two of you.
Pick up the box of blood vials lying on the ground and follow Roberts deeper into the observatory. After a short while, you'll reach a strange device standing in one of the rooms.
Put the box down next to the device as Roberts asks. Roberts will demonstrate how the observatory works. A cutscene will then begin, in which Roberts breaks your deal, sending you into the pool of water below.
First, climb back up toward the device by scaling the cubic rock blocks.
The door you entered through will close, so you'll need to find another way out.
Climb up through the breach on the north wall and move onto the wall on the right.
Performing a back eject to jump over to the wall on the left.
Climbing further to the left, you will reach the tunnel entrance.
The tunnel will lead you outside where, in a short cutscene, Edward slides down a steep slope, gets badly battered, and falls into a pond by the beach.
Edward is severely injured and struggling to move. Even so, get out of the pond and head toward Roberts and his men standing on the beach.
Roberts' men will head toward you. Simply let them deplete your health, and a cutscene will trigger in which Roberts informs you that he intends to hand you over to the authorities to collect the bounty on your head.
This will complete this quest and automatically start the next one - To Suffer Without Dying.