Alchemy is historically considered a proto-science and directly related to the birth of modern chemistry. Not to be confused with the anime, historical alchemy does not use transmutation circles. In fact, many of the laboratory practices and techniques used by chemist today were invented by alchemists. The major difference between the modern chemistry and alchemy is their affiliation with the supernatural. Alchemist claimed otherworldly powers, and often mixed religion into their practices. Incidentally, the main goal of many alchemists was to create the Philosopher's Stone - A fabled object said to hold the ability to create gold and silver from common metals, and create the elixir of life. Of course, none ever succeeded.
 In the Anime
 Equivalent Exchange
There is one golden rule of Alchemy, and that's the law of Equivalent Exchange. Based on the laws of conservation of energy and matter, it basically means the same thing: One cannot receive anything more from a system than what is initially put in. The whole of the Original Series is dedicated in finding the Philosopher's Stone, said to be able to break that rule.
Given this principle of equivalent exchange, one can theorize the ability to create humans - or resurrect the dead - from the material that they are made of. Unfortunately, this is impossible in the Fullmetal Alchemist universe, and in the attempt, not only will it not work, but a greater price has to be paid which may end the alchemist's life and give birth to a monster.
 Transmutation Circles
Alchemy in the anime is used primarily as a means to fight an opponent, but the alchemist is still bound to transmutation circle. Many of the Alchemists in the series circumvent this by having the circles drawn somewhere on their body. Roy Mustang is one such character as he wields special gloves made of ignition cloth with his transmutation circle sown onto the back. Once he snaps his fingers, the cloth provides a spark, and he activates the circle for the reaction.
Edward Elric and his run-in with The Gate early on in his life, learned to perform alchemy without a transmutation circle. He still has to clasp his hands together to have the energy flow which is more than indicative of what he's about to do. This, however, doesn't seem to give him any more of an advantage over your regular state alchemist.
In keeping with the law of equivalent exchange, all of he alchemy that is used in their world draws the energy necessary for the reactions from our world, or a world similar to it. The world where the Elric brother's are from is actually a parallel universe separated by The Gate.
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