Notched book

Notched book, also known as the "Book of the Elemental Gloves", is obtained from the bookshelf on the ground floor of the Sorcerer's Tower, after completing the Elemental Workshop III quest. When obtained for the first time, you will be given the start menu for the Elemental Workshop IV quest. The book features the symbol of a Cosmic rune on the front. It is reobtainable from the Sorcerer's Tower bookcase on the ground floor or from the Bookcase in your POH.

Transcript
Although the building of the body machine was time consuming and, ultimately, did not meet with the success he desired, Vitruvius was still resolutely determined to find a way to reanimate his bumbling apprentice. He had a clear idea of what to try next; just like he always tried to drum into his apprentice:''' Your ideas must be sharp and keen like a blade. Put your nose to the grindstone, and the edges of that idea will begin to form.'''

Indeed, it wasn't too long after the body machine proved unsuccessful that Vitruvius set about creating his next device one that generated energy akin to that of cosmic magic. He hoped that the state-changing effects cosmic magic caused in other magically receptive objects could be used to change the state of an inanimate object based upon its form.

Vitruvius set about constructing a cosmic energy matrix. The matrix consisted of pillars that each contained a number of precious gemstones, which had been ground into focussing lenses on the workshop's newly installed whetstone. Through these pillars, the four base elements were passed in such a way as to generate vast amounts of cosmic energy. It was an expensive proposition that required Vitruvius to sell off much of his property, but he was so stoically determined to return his apprentice that money was no object.

After the previous failed experiment with his apprentice's body, Vitruvius decided this time to first create a dedicated test subject: a prototype shell build from primed elemental metal. Unfortunately, his own expertise lay in metallurgy, not human anatomy, and so his prototype didn't turn out exactly human-shaped...but it worked! The cosmic energy turned the lifeless lump of metal into a walking, talking being, though one without a soul. Vitruvius named it 'Da-vi', inspired by something that an old artist friend of his had once drawn. With the prototype working, he set about building a second elemental shell, but this time using the elemental metal in which his old apprentice resided. He figured that the boy's consciousness, which was held within the metal, would bring back his old apprentice, albeit in a different form...

Trivia

 * According to the quest log, "you use the books spine as a guide when creating the Keyblade". This seems to mean that you use the book's notches as a guide to fashion the Keyblade.
 * You do not need the book in your inventory when smithing elemental gloves (of any type) after completing the quest.