If Water, Air, and the like are a matter of which the natural bodies consist, as some thinkers in fact believe, these elements must be either one, or two, or more. Now they cannot all of them be one-they cannot, e.g. all be Air or Water or Fire or Earth-because Change is into contraries. For if they all were Air, then (assuming Air to persist) there will be alteration instead of coming-to-be. Besides, nobody supposes a single element to persist, as the basis of all, in such a way that it is Water as well as Air (or any other element) at the same time. So there will be a certain contrariety, i.e. a differentiating quality: and the other member of this contrariety, e.g. heat, will belong to some other element, e.g. to Fire. But Fire will certainly not be hot Air. For a change of that kind (a) is alteration, and (b) is not what is observed. Moreover (c) if Air is again to result out of the Fire, it will do so by the conversion of the hot into its contrary: this contrary, therefore, will belong to Air, and Air will be a cold something: hence it is impossible for Fire to be hot Air, since in that case the same thing will be simultaneously hot and cold. Both Fire and Air, therefore, will be something else which is the same; i.e. there will be some matter, other than either, common to both.
The same argument applies to all the elements, proving that there is no single one of them out of which they all originate. But neither is there, beside these four, some other body from which they originate-a something intermediate, e.g. between Air and Water (coarser than Air, but finer than Water), or between Air and Fire (coarser than Fire, but finer than Air). For the supposed intermediate will be Air and Fire when a pair of contrasted qualities is added to it: but, since one of every two contrary qualities is a privation, the intermediate never can exist-as some thinkers assert the Boundless or the Environing exists-in isolation. It is, therefore, equally and indifferently any one of the elements, or else it is nothing.
Since, then, there is nothing-at least, nothing perceptible-prior to these, they must be all. That being so, either they must always persist and not be transformable into one another: or they must undergo transformation-either all of them, or some only (as Plato wrote in the Timaeus). Now it has been proved before that they must undergo reciprocal transformation. It has also been proved that the speed with which they come-to-be, one out of another, is not uniform-since the process of reciprocal transformation is relatively quick between the elements with a complementary factor, but relatively slow between those which possess no such factor. Assuming, then, that the contrariety, in respect to which they are transformed, is one, the elements will inevitably be two: for it is matter that is the mean between the two contraries, and matter is imperceptible and inseparable from them. Since, however, the elements are seen to be more than two, the contrarieties must at the least be two. But the contrarieties being two, the elements must be four (as they evidently are) and cannot be three: for the couplings are four, since, though six are possible, the two in which the qualities are contrary to one another cannot occur.
These subjects have been discussed before: but the following arguments will make it clear that, since the elements are transformed into one another, it is impossible for any one of them-whether it be at the end or in the middle-to be an originative source of the rest. There can be no such originative element at the ends: for all of them would then be Fire or Earth, and this theory amounts to the assertion that all things are made of Fire or Earth. Nor can a middle-element be such an originative source-as some thinkers suppose that Air is transformed both into Fire and into Water, and Water both into Air and into Earth, while the end-elements are not further transformed into one another. For the process must come to a stop, and cannot continue ad infinitum in a straight line in either direction, since otherwise an infinite number of contrarieties would attach to the single element. Let E stand for Earth, W for Water, A for Air, and F for Fire. Then (i) since A is transformed into F and W, there will be a contrariety belonging to A F. Let these contraries be whiteness and blackness. Again (ii) since A is transformed into W, there will be another contrariety: for W is not the same as F. Let this second contrariety be dryness and moistness, D being dryness and M moistness. Now if, when A is transformed into W, the white persists, Water will be moist and white: but if it does not persist, Water will be black since change is into contraries. Water, therefore, must be either white or black. Let it then be the first. On similar grounds, therefore, D (dryness) will also belong to F. Consequently F (Fire) as well as Air will be able to be transformed into Water: for it has qualities contrary to those of Water, since Fire was first taken to be black and then to be dry, while Water was moist and then showed itself white. Thus it is evident that all the elements will be able to be transformed out of one another; and that, in the instances we have taken, E (Earth) also will contain the remaining two complementary factors, viz. the black and the moist (for these have not yet been coupled).
We have dealt with this last topic before the thesis we set out to prove. That thesis-viz. that the process cannot continue ad infinitum-will be clear from the following considerations. If Fire (which is represented by F) is not to revert, but is to be transformed in turn into some other element (e.g. into Q), a new contrariety, other than those mentioned, will belong to Fire and Q: for it has been assumed that Q is not the same as any of the four, E W A and F. Let K, then, belong to F and Y to Q. Then K will belong to all four, E W A and F: for they are transformed into one another. This last point, however, we may admit, has not yet been proved: but at any rate it is clear that if Q is to be transformed in turn into yet another element, yet another contrariety will belong not only to Q but also to F (Fire). And, similarly, every addition of a new element will carry with it the attachment of a new contrariety to the preceding elements. Consequently, if the elements are infinitely many, there will also belong to the single element an infinite number of contrarieties. But if that be so, it will be impossible to define any element: impossible also for any to come-to-be. For if one is to result from another, it will have to pass through such a vast number of contrarieties-and indeed even more than any determinate number. Consequently (i) into some elements transformation will never be effected--viz. if the intermediates are infinite in number, as they must be if the elements are infinitely many: further (ii) there will not even be a transformation of Air into Fire, if the contrarieties are infinitely many: moreover (iii) all the elements become one. For all the contrarieties of the elements above F must belong to those below F, and vice versa: hence they will all be one.