The book "Root Causes" appears on the right side bookcase in the Library after you talk to Katya. It is just a way to show you where the mandrake root is.
You previously exited the Library by breaking the chains with the crowbar. Now leave the front of the castle and go through the portal at left, which takes you back to the moat. To the far right is a tree and a tall wall, and at the top is the mandrake root. Use the crossbow to shoot some arrows into the wall, starting near the bottom, then climb up them before they fall out. (Some are in to stay.)
go outside the castle and go to the far right use the arrows to jump on top of the tree and the roots are above you
you use it to go up to the top of the rooom and get the garlic
You use it to make the potion for the serume
You use the water from the gaiser to fill up the bucket. :-)
Use one drop of the Mandrake and three drops each of the Wolfsbane and the Garlic.
3 garlic 3 wilted-wolfsbane 1 mandrake
find an eye , put it in the eyeless tigers hole, and a key falls out. use it to unlock the amory
Generally speaking, items found on one island can't be used on another island. The crossbow works the same way.
Once your in the library simply go near the fire. Now go to your pockets, click the use button on the log card in you pockets and use that to start a fire! (You also need the teen vampire book to start the fire.)
Once you have received the log and you're inside the castle, you put the log and a page of the vampire novel into the fireplace. Then hop on the thing next to it three times to light up a fire.
The missing girl is in the count's chambers, to the right of the laboratory. You can only open the cage when you have created an anti-vampirism serum (mixing the garlic, wolfsbane, and mandrake root). When you do, there is a silver cane at the right of the room that you can use to hook the key. (see related solution question)
Katya is in a cage in one of the castle towers. You cannot free her until you have formulated an anti-vampirism serum to use on Count Bram.