However correct the nominalistic conception of number may be in such a case as that of numbering the five apples, it is wholly incorrect to restrict the concept of number itself to one valid for this kind of occurrence. We shall see this immediately if we take one of the apples and cut it across. There we find the number five confronting us in the well-known star-like figure, represented by the fivefold pericarp in the centre of the apple. What man, restricted as he was to the mode of understanding, has completely overlooked is this: although the act of counting, by which we establish the number five, is the same in both cases, the quality of the number five is totally different. For in the case of the five pericarps this number is a quality immanent in the apple, which it shares with the whole species of Rosaceae. The apple itself is just as much 'five' as it is 'round', 'sweet', etc. In the supersensible type which creates in the plant its own organ of manifestation, the creation of a number - in the apple the number five - is part of the form-creating activities characteristic of the type. The numerical relationships which appear between natural phenomena depend upon the way in which the chemical ether participates. This is true equally of those discovered by chemistry in the sphere of inorganic matter and used to-day with such great success.

Let us be quite clear that the relationship of unity to plurality in the case of the five apples is totally different from what it is in the fivefold pericarp. In the first case unity is the smallest quantity represented by each of the five apples. There, the step from one to two is made by joining together two units from outside. The path from one to many is by way of continuous addition. In the second case the unity is represented by the pericarp - i.e. by the one comprising the many, the latter appearing as parts of the whole. In such a case two is part of one and so are three, four, five, etc. Plurality arises from a continuous process of division of unity.

The ancient world knew the idea of number only in the last-mentioned form. There unity appeared as an all-embracing magnitude, revealed through the Universe. The world's manifoldness was felt to be not a juxtaposition of single things, externally connected, but the content of this unity, and therefore derived from it. This was expressed by the pre-Socratic Greek philosophers in the formula έν και Ïαν (the One and the All).

With the appearance of the Arabs on the scene of history, human thought turned to the additive concept of number, and the original distributive concept receded gradually into oblivion. The acceptance of the new concept made it possible for the first time to conceive the zero. It is clear that by a continuous division of unity one is carried to a constantly growing number of constantly diminishing parts, but without ever reaching the nothing represented by the number zero. To-day we should say that in this way we can reach zero only by an infinite series of steps. Yet the idea of the infinite did not exist in this form for ancient man. On the other hand, in the arabic conception of number the steps necessary to reach zero are finite. For just as by the external addition of unities we can step forward from one number to the next, so we can also step back on the same path by repeated subtractions of unities. Having thus reached One, nothing can stop us from going beyond it by one more such step. The arabic numeral system, therefore, is the only one to possess its own symbol for zero.

It has been correctly noted that the penetration into European thought of this additive concept of number was responsible for developing the idea of the machine; for it accustomed human beings to think calmly of zero as a quantity existing side by side with the others. In ancient man the idea of nothingness, the absolute void, created fear; he judged nature's relation to the void accordingly, as the phrase 'natura abhorret vacuum' indicates. His capacity to think fearlessly of this vacuum and to handle it thus had to be developed in order to bring about the Machine Age, and particularly the development of efficient steam engines. Consider also the decisive part played by the vacuum in Crookes's researches, through which the path to the sub-physical realm of nature was laid open.

Yet nature makes use of number as a regulating factor in quite a different way from its appearance in the purely electrical and gravitational connexions of inorganic matter, namely where sound-ether from the upper boundary of nature so regulates nature's dynamic that the manifold sense-qualities appear in their time-and-space order. When we interpret the arrangement of numbers found there on a nominalistic basis, as is done when the axis- and angle-relationships of crystals are reduced to a mere propinquity of the atoms distributed like a grid in space, or when the difference in angle of the position of the various colours in the spectrum is reduced to mere differences in frequency of the electromagnetic oscillations in a hypothetical ether - then we bar the way to the comprehension not only of number itself, as a quality among qualities, but also of all other qualities in nature.

(e) LIFE

As already mentioned, the three kinds of ether, warmth, light and sound, are not sufficient in themselves to bring into existence what in its proper sense we call 'life' in nature, i.e. the formation of single living organisms. This requires the action of a fourth kind of ether, the life-ether, ranged above the other three. We can best comprehend the life-ether's contribution to the total activity of the ether in nature by considering the interaction of the four kinds of ether with the four physical elements.

We have seen that the warmth-ether has the double function of being at once the lowest ether and the highest physical element, thus acting as a sphere of reflexion for the other kinds of ether and the elements respectively. Each stage in the etheric has its reflexion in the physical, as the above table shows. Thus to the physical air the etheric light is related. (The affinity of light and air is best seen in the plant and its leaf-formation.) To bring about real changes in the material composition of the physical world requires the stronger powers of the chemical ether. Therefore it is also the first ether of which we had to speak as 'magical' ether. Its effects reach into the watery element which is already bound up with gravity, but by its own strength it cannot penetrate beyond that. The causation of material changes in the liquid sphere would in fact be all that these three kinds of ether could achieve together.