When you enter the Furnace, you'll be attacked by Cyclope. There are also hatcheries here. Try to destroy them after dealing with Cyclope. On right side of arena is a place almost without them.
When you defeat the Cyclope, break hatcheries and defeat woken up enemies to open a small room full of coal.
Nearby you find a chamber closed by a wooden wall which you have to destroy. On left you find a gearwheel thanks to which you pull a platform and open a shaft in the room on right.
Use a gearwheel and jump onto a platform before it drives away. After getting on the other side of the abyss, go left until you reach an upper level of the Furnace.
Here you find Archimedes Note, his corpses, red and white chests and a lever.
Nearby the red chest you find also Archimedes' Treatise.
Then push a lever to open the Furnace's chimney and block it using your clone. Jump inside to get onto an Archimedes' Furnace.
Here you find two white chests.
Moving towards the screen, you find a gearwheel. Rotate it to open the gate and make sparks in a small niche on right.
Go through the gate, grab a platform with coal and move it to the niche where the sparks were . Rotate the nearby gearwheel what burns the coal.
Place this coal back onto its place and use Amulet. Regenerate the platform to light a fire in the tower over it.
Grab the platform with coal again and push it to the niche, closed before with the wooden wall. Now use the gearwheel to move platform, to fill the platform with the hot fluid.
Grab the coal platform for the last time and push it through Archimedes' Furnace. Then use an Amulet on it and regenerate it so the flames appear.