Plato inherited the four-element idea from Empedocles. About 360 B.C. he wrote Timaeus where the foundations of alchemy are set forth. These underpinnings take the form of two concepts: Being & Becoming, and Transmutation. Note that Plato was not an alchemist. He was a philosopher, but so pervasive are his ideas that later alchemists almost always style themselves as philosophers also. Sorry for the long quotations below.

Being & Becoming. Plato, and his pupil Aristotle, are categorizers. Always organizing ideas and object in various categories. Foundationally are the two categories of Being and Becoming. Things that are being are perfect, have no change, are always right. Things which are becoming are imperfect, in the process of becoming more like God (as is the whole Earth), and thus are always changing and are not yet true. Into the Being category he places reason. Into the Becoming category he places opinion and observation. 

First then, in my judgment, we must make a distinction and ask, What is that which always is and has no becoming, and what is that which is always becoming and never is? That which is apprehended by intelligence and reason is always in the same state, but that which is conceived by opinion with the help of sensation and without reason is always in a process of becoming and perishing and never really is. Now everything that becomes or is created must of necessity be created by some cause, for without a cause nothing can be created. The work of the creator, whenever he looks to the unchangeable and fashions the form and nature of his work after an unchangeable pattern, must necessarily be made fair and perfect, but when he looks to the created only and uses a created pattern, it is not fair or perfect. Was the heaven then or the world, whether called by this or by any other more appropriate name – assuming the name, I am asking a question which has to be asked at the beginning of an inquiry about anything – was the world, I say, always in existence and without beginning, or created, and had it a beginning? Created, I reply, being visible and tangible and having a body, and therefore sensible, and all sensible things are apprehended by opinion and sense, and are in a process of creation and created. Now that which is created must, as we affirm, of necessity be created by a cause. But the father and maker of all this universe is past finding out, and even if we found him, to tell of him to all men would be impossible. This question, however, we must ask about the world. Which of the patterns had the artificer in view when he made it – the pattern of the unchangeable or of that which is created? If the world be indeed fair and the artificer good, it is manifest that he must have looked to that which is eternal, but if what cannot be said without blasphemy is true, then to the created pattern. Everyone will see that he must have looked to the eternal, for the world is the fairest of creations and he is the best of causes. And having been created in this way, the world has been framed in the likeness of that which is apprehended by reason and mind and is unchangeable, and must therefore of necessity, if this is admitted, be a copy of something . . .

From the translation of the Timaeus by Benjamin Jowett, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Bollingen Series LXXI (New York: Pantheon Books, 1961);

Here Plato is using reason and observation in the opposite sense we do today; for us observation is undeniable fact, and reason can take any fancy it likes. But for the alchemist who reads the Timaeus dialog, reason is paramount and observation is suspect. Aristotle will take this to new levels.

Plato expounds on the four-element theory of Empedocles by doing several things. First, he established via reason that the creator is good, perfect, and wants everything to be like He is. The world is not perfect, but it is changing to become so. Thus, the world is alive, like plants and animals are alive; he called it anima mundi, the "aliveness of the world".

TIMAEUS: Let me tell you then why the creator made this world of generation. He was good, and the good can never have any jealousy of anything. And being free from jealousy, he desired that all things should be as like himself as they could be. This is in the truest sense the origin of creation and of the world, as we shall do well in believing on the testimony of wise men. God desired that all things should be good and nothing bad, so far as this was attainable. Wherefore also finding the whole visible sphere not at rest, but moving in an irregular and disorderly fashion, out of disorder he brought order, considering that this was in every way better than the other. Now the deeds of the best could never be or have been other than the fairest, and the creator, reflecting on the things which are by nature visible, found that no unintelligent creature taken as a whole could ever be fairer than the intelligent taken as a whole, and again that intelligence could not be present in anything which was devoid of soul. For which reason, when he was framing the universe, he put intelligence in soul, and soul in body, that he might be the creator of a work which was by nature fairest and best. On this wise, using the language of probability, we may say that the world came into being – a living creature truly endowed with soul and intelligence by the providence of God.

This being supposed, let us proceed to the next stage. In the likeness of what animal did the creator make the world? It would be an unworthy thing to liken it to any nature which exists as a part only, for nothing can be beautiful which is like any imperfect thing. But let us suppose the world to be the very image of that whole of which all other animals both individually and in their tribes are portions. For the original of the universe contains in itself all intelligible beings, just as this world comprehends us and all other visible creatures. For the deity, intending to make this world like the fairest and most perfect of intelligible beings, framed one visible animal comprehending within itself all other animals of a kindred nature. Are we right in saying that there is one world, or that they are many and infinite? There must be one only if the created copy is to accord with the original. For that which includes all other intelligible creatures cannot have a second or companion; in that case there would be need of another living being which would include both, and of which they would be parts, and the likeness would be more truly said to resemble not them, but that other which included them. In order then that the world might be solitary, like the perfect animal, the creator made not two worlds or an infinite number of them, but there is and ever will be one only-begotten and created heaven. [27c–31b]

From the translation of the Timaeus by Benjamin Jowett, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Bollingen Series LXXI (New York: Pantheon Books, 1961)

Secondly, Plato establishes that in the Creation, the world contains all the fire, air, water and earth that could exist, and that each of these contains differing properties that we observe. For example, earth always moves down toward the center of the spherical earth, and fire always moves up. Air and water behave similarly, but to a lesser extent.

Now the creation took up the whole of each of the four elements, for the creator compounded the world out of all the fire and all the water and all the air and all the earth, leaving no part of any of them nor any power of them outside. His intention was, in the first place, that the animal should be as far as possible a perfect whole and of perfect parts, secondly, that it should be one, leaving no remnants out of which another such world might be created, and also that it should be free from old age and unaffected by disease. Considering that if heat and cold and other powerful forces surround composite bodies and attack them from without, they decompose them before their time, and by bringing diseases and old age upon them make them waste away – for this cause and on these grounds he made the world one whole, having every part entire, and being therefore perfect and not liable to old age and disease. And he gave to the world the figure which was suitable and also natural. Now to the animal which was to comprehend all animals, that figure would be suitable which comprehends within itself all other figures. Wherefore he made the world in the form of a globe, round as from a lathe, having its extremes in every direction equidistant from the center, the most perfect and the most like itself of all figures, for he considered that the like is infinitely fairer than the unlike . . . Of design he was created thus – his own waste providing his own food, and all that he did or suffered taking place in and by himself. For the creator conceived that a being which was self-sufficient would be far more excellent than one which lacked anything, and, as he had no need to take anything or defend himself against anyone, the creator did not think it necessary to bestow upon him hands, nor had he any need of feet, nor of the whole apparatus of walking. But the movement suited to his spherical form was assigned to him, being of all the seven that which is most appropriate to mind and intelligence, and he was made to move in the same manner and on the same spot, within his own limits revolving in a circle. All the other six motions were taken away from him, and he was made not to partake of their deviations. And as this circular movement required no feet, the universe was created without legs and without feet.

From the translation of the Timaeus by Benjamin Jowett, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Bollingen Series LXXI (New York: Pantheon Books, 1961)

Plato continues then to the transmutation of the elements by introducing the idea of prima materia, primeval matter, matter which has no properties. Yet. To explain the act of matter taking on properties, he first needs matter with a soul, matter which is in some sense alive.

Such was the whole plan of the eternal God about the god that was to be; he made it smooth and even, having a surface in every direction equidistant from the center, a body entire and perfect, and formed out of perfect bodies. And in the center he put the soul, which he diffused throughout the body, making it also to be the exterior environment of it, and he made the universe a circle moving in a circle, one and solitary, yet by reason of its excellence able to converse with itself, and needing no other friendship or acquaintance. Having these purposes in view he created the world a blessed god. Now God did not make the soul after the body, although we are speaking of them in this order, for when he put them together he would never have allowed that the elder should be ruled by the younger, but this is a random manner of speaking which we have, because somehow we ourselves too are very much under the dominion of chance. Whereas he made the soul in origin and excellence prior to and older than the body, to be the ruler and mistress, of whom the body was to be the subject. [32c–34c] Now when the creator had framed the soul according to his will, he formed within her the corporeal universe, and brought the two together and united them center to center. The soul, interfused everywhere from the center to the circumference of heaven, of which also she is the external envelopment, herself turning in herself, began a divine beginning of never-ceasing and rational life enduring throughout all things. [36d–e]

From the translation of the Timaeus by Benjamin Jowett, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Bollingen Series LXXI (New York: Pantheon Books, 1961)

Now he is ready to introduce transmutation, the shifting of the properties of matter, by the matter itself, to move it toward perfection.

Thus far in what we have been saying, with small exceptions, the works of intelligence have been set forth, and now we must place by the side of them in our discourse the things which come into being through necessity – for the creation of this world is the combined work of necessity and mind. Mind, the ruling power, persuaded necessity to bring the greater part of created things to perfection, and thus and after this manner in the beginning, through necessity made subject to reason, this universe was created. But if a person will truly tell of the way in which the work was accomplished, he must include the variable cause as well, and explain its influence. Wherefore, we must return again and find another suitable beginning – as about the former matters, so also about these. To which end we must consider the nature of fire and water and air and earth, such as they were prior to the creation of the heaven, and what was happening to them in this previous state, for no one has as yet explained the manner of their generation, but we speak of fire and the rest of them, as though men knew their natures, and we maintain them to be the first principles and letters or elements of the whole, when they cannot reasonably be compared by a man of any sense even to syllables or first compounds . . .

In the first place, we see that what we just now called water, by condensation, I suppose, becomes stone and earth, and this same element, when melted and dispersed, passes into vapor and air. Air, again, when inflamed, becomes fire, and, again, fire, when condensed and extinguished, passes once more into the form of air, and once more, air, when collected and condensed, produces cloud and mist – and from these, when still more compressed, comes flowing water, and from water comes earth and stones once more – and thus generation appears to be transmitted from one to the other in a circle. Thus, then, as the several elements never present themselves in the same form, how can anyone have the assurance to assert positively that any of them, whatever it may be, is one thing rather than another? No one can . . .

From the translation of the Timaeus by Benjamin Jowett, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Bollingen Series LXXI (New York: Pantheon Books, 1961)

And there it is. Matter, having a soul, can direct the properties it has to become more perfect. It can transform by altering those properties. Plato gives us examples, centered around gold.

Let me make another attempt to explain my meaning more clearly. Suppose a person to make all kinds of figures of gold and to be always remodeling each form into all the rest; somebody points to one of them and asks what it is. By far the safest and truest answer is, ‘That is gold,’ and not to call the triangle or any other figures which are formed in the gold ‘these,’ as though they had existence, since they are in process of change while he is making the assertion, but if the questioner be willing to take the safe and indefinite expression, ‘such,’ we should be satisfied. And the same argument applies to the universal nature which receives all bodies – that must be always called the same, for, inasmuch as she always receives all things, she never departs at all from her own nature and never, in any way or at any time, assumes a form like that of any of the things which enter into her; she is the natural recipient of all impressions, and is stirred and informed by them, and appears different from time to time by reason of them . . . Wherefore the mother and receptacle of all created and visible and in any way sensible things is not to be termed earth or air or fire or water, or any of their compounds, or any of the elements from which these are derived, but is an invisible and formless being which receives all things and in some mysterious way partakes of the intelligible, and is most incomprehensible. In saying this we shall not be far wrong; as far, however, as we can attain to a knowledge of her from the previous considerations, we may truly say that fire is that part of her nature which from time to time is inflamed, and water that which is moistened, and that the mother substance becomes earth and air, in so far as she receives the impressions of them. [47e–51b]

Of all the kinds termed fusile [by which he means metals], that which is the densest and is formed out of the finest and most uniform parts is that most precious possession called gold, which is hardened by filtration through rock; this is unique in kind, and has both a glittering and a yellow color. A shoot of gold, which is so dense as to be very hard, and takes a black color, is termed adamant. There is also another kind which has parts nearly like gold, and of which there are several species; it is denser than gold, and it contains a small and fine portion of earth and is therefore harder, yet also lighter because of the great interstices which it has within itself, and this substance, which is one of the bright and denser kinds of water, when solidified is called copper. [59b–c]

From the translation of the Timaeus by Benjamin Jowett, in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Bollingen Series LXXI (New York: Pantheon Books, 1961)

Aristotle will take these ideas, and run with them.