Seeking the Philosophers' Stone: Alchemy as an early science

Alchemy may seem a pseudoscience today, but it was one of the first experimental sciences in the Middle Ages and influenced the development of chemistry. One of the major goals of alchemy was to transmute base metals such as lead into gold with the help of the Philosophers’ Stone. But what was medieval and early modern alchemy really like? What was the logic behind transmutation, and why was alchemy either forbidden or sponsored by monarchs?