The basis of the Philosopher's Stone is the transmutation of the four elements, the method of Elemental Dignities. Cornelius Agrippa and the great alchemist Paracelsus were both students of Abbot Trithemius, whose most famous work Steganographia concerned ciphers and angelic magic.
The importance of Cornelius Agrippa and his analysis of the four elements is indicated by this quotation from Israel Regardie's Golden Dawn book. Agrippa is the inspiration for Mathers when the elemental subtleties of Tarot were introduced.
Henry Cornelius Agrippa (1468-1535) defined the nature of the Four Elements thus:
It is necessary that we should know and understand the nature and quality of the Four Elements, in order to our being perfect in the principles and groundwork of our studies in the Talismanic or Magical Art.
Therefore, there are four Elements, the original grounds of corporeal things, viz: Earth, Air, Fire, and Water, of which elements all inferior bodies are compounded, not by way of being heaped up together, but by transmutation and union; and when they are destroyed, they are resolved into elements.
But there are none of the sensible elements that are pure. They are more or less mixed, and apt to be changed into one into the other, even as earth being moistened and dissolved becomes water, but the same being made thick and hard becomes earth again, and being evaporated through heat it passes into air, and that being kindled into fire; and this being extinguished into air again; but being cooled after burning becomes water again, or else stone or sulphur, and this is clearly demonstrated by lightning.
Now every one of these Elements has two specific qualities: the former whereof it retains as proper to itself; in the other as a mean, it agrees with that which comes directly after it. For Fire is hot and dry; Water cold and moist; and Air hot and moist; and so in this manner, the Elements, according to two contrary qualities are opposite one to the other, as Fire to Water, and Earth to Air.
Likewise the Elements are contrary one to the other on another account. Two are heavy, as Earth and Water; and the others are light as Fire and Air. Therefore the Stoics called the former `passives,' but the latter `actives.'
And Plato distinguishes them after another manner, and assigns to each of them three qualities, viz.: To the Fire brightness, thinness and motion. To the Earth, darkness, thickness, and quietness. And according to these qualities the Elements of Fire and Earth are contrary. Now the other Elements borrow their qualities from these, so that the Air receives two qualities from the Fire, thinness and motion, and from the Earth one, darkness. In like manner, Water receives two qualities from the Earth, darkness and thickness; and from the Fire one, motion. But Fire is twice as thin as Air, thrice more moveable, and four times lighter. The Air is twice more bright, thrice more thin, and four times more moveable than Water. Therefore, as Fire is to Air, so is Air to Water, and Water to Earth. And again, as the Earth is to the Water, so is Water to Air, and Air to Fire. And this is the root and foundation of all bodies, natures, and wonderful works. And he who can know and thoroughly understand these qualities of the Elements and their mixtures shall bring to pass wonderful and astonishing things in Magic.
Therefore, as Fire is to Air, so is Air to Water, Water to Earth. And again, as the Earth is to the Water, so is Water to Air, and Air to Fire.
First Order
Now each of these Elements has a threefold consideration, so that the number of four may make up the number of twelve; and by passing by the number of seven into ten, there may be a progress to the Supreme Unity upon which all virtue and wonderful things do depend. Of the first Order, are the pure Elements, which are neither compounded, changed, nor mixed, but are incorruptible and not of which but through which the virtues of all natural things are brought forth to act. No man is fully able to declare their Virtues, because they can do all things upon all things. He who remains ignorant of these, shall never be able to bring to pass any wonderful matter.
Second Order
Of the second order are Elements that are compounded, changeable and impure, yet such as may, by art, be reduced to their pure simplicity, whose virtue, when they are thus reduced, doth above all things perfect all occult and common operations of Nature; and these are the foundations of the whole of natural Magic.
Third Order
Of the third Order are those elements which originally and of themselves are not elements, but are twice compounded, various, and changeable unto another. These are the infallible medium, and are called the Middle Nature, or Soul of the Middle Nature; very few there are that understand the deep mysteries thereof. In them is, by means of certain numbers, degrees and orders, the perfection of every effect in what thing soever, whether natural, celestial, or super-celestial. They are full of wonders and mysteries, and are operative in Magic, natural or divine. For, from these, through them, proceeds the binding, loosing, and transmutation of all things - the knowledge and foretelling of things to came, also the expelling of evil and the gaining of Good Spirits. Let no one, therefore, without these three sorts of Elements, and the true knowledge thereof, be confident that he can work anything in the occult science of Magic and Nature.
Transforming the Elements
But whosoever shall know how to reduce those of one order into another, impure into pure, compounded into simple, and shall understand distinctly the nature, virtue, and power of them, into number, degrees and order, without dividing the substance, he shall easily attain to the knowledge and perfect operation of all natural things, and celestial secrets likewise; and this is the perfection of the Qabalah, which teaches all these before mentioned; and by a perfect knowledge thereof, we perform many rare and wonderful experiments. In the original and exemplary world all things are all in all. So also in this corporeal world. And the elements are not only in these inferior things; but are in the Heavens, in stars, in devils, in angels, and likewise in God Himself, the maker and original example of all things.
Now it must be understood that in these inferior bodies the elements are gross and corruptible, but in the heavens they are, with their natures and virtues, after a celestial and more excellent manner than in sublunary things. For the firmness of the celestial earth is there without the grossness of water, and the agility of Air without exceeding its bounds. The heat of fire without burning, only shining, giving light and life to all things by its celestial heat.
The qualities and combinations of the elements can be summarised thus: