Part 5

“All particles of air are four-dimensional in magnitude when, in addition to their position in space, we also consider the variable densities which they assume, as they are expressed by the different heights of the barometer in the different parts of the atmosphere. Similarly all conceivable spheres in space are four-dimensional magnitudes, for their centres form a three-dimensional point-aggregate, and around each centre a one-dimensional totality of spheres, the radii of which can be expressed by every numerical magnitude from zero to infinity. Further, if we imagine a measuring-stick of invariable length to assume every conceivable position in space, the positions so obtained will constitute a five-dimensional aggregate. For in the first place one of the extremities of the measuring-stick may be conceived to assume a position at every point of space, and this determines for one extremity alone of the stick a three-dimensional totality of position, and, secondly, as we have seen above, there proceeds from every such position of this extremity a two-dimensional totality of directions, and by conceiving the measuring-stick to be placed lengthwise in every one of these directions, we shall obtain all the conceivable positions which the second extremity can assume, and consequently the dimensions must be 3 + 2 or 5 …” &c., &c.

Mathematicians have for long done problems in the seventh and eighth dimensions. They have told us that you cannot tie a knot in the second dimension, because there is no up or down, and the threads would not cross--nor in the fourth, because the knot would pull out in a new direction and would not hold. But it has only lately been realised that fourth and other dimensions may be actual fact in the world round us. Of course, from the point of view of a point there are only three dimensions to be known, but to a line in the same space there are five, to the surface probably six. Our intelligence at present does not go beyond the point; but if we could think of space from the point of view of a solid, worlds upon worlds would rise before our view.

Of the fourth dimension we can discover some facts by analogy. We can count the edges of its typical figure, and apply thought to determining some of its conditions. But a more interesting subject of research is the inquiry into the light thrown by the theory of four dimensions on the determination of certain atoms in chemistry, that are known to be distinct elements, but could only be determined actually in another dimension.

“In chemistry, the molecules of a compound body are said to consist of the atoms of the elements which are contained in the body, and these are supposed to be situated at certain distances from one another and to be held in their relative positions by certain forces. At first the centres of the atoms were conceived to lie in one and the same plane. But Wislicenus was led by researches in paralactic acid to explain the differences of isomeric molecules of the same structural formulæ by the different positions of the atoms in space. In fact, four points can always be so arranged in space that every two of them may have any distance from each other; and the change of one of the six distances does not necessarily involve the alteration of any other.

“But suppose our molecule consists of five atoms? Four of these may be so placed that the distance between any two of them can be made what we please. But it is no longer possible to give the fifth atom a position such that each of the four distances by which it is separated from the other atoms may be what we please. On the contrary, the fourth distance is dependent on the three remaining distances, for the space of experience has only three dimensions. If, therefore, I have a molecule which consists of five atoms, I cannot alter the distance between two of them without at least altering some second distance. But if we imagine the centres of the atoms placed in a four-dimensioned space, this can be done; all the ten distances which may be conceived to exist between the five points will then be independent of one another. To reach the same result in the case of six atoms we must assume a five-dimensional space, and so on.”

Here we see that if chemistry as a science is bound to take account of all its facts, the scientist is confronted with a problem of dimensions that is really a problem of Infinity applied not, as in the other cases quoted, to number, but to space.

And there is a reason which explains why the same problem tends to appear in these different ways. Both time and space can be most correctly thought of as series: the former known to us as possessing one direction, though possibly involving more, and the latter three, though possibly involving more. Time is not a thing nor a condition, but it is the way in which we are enabled to apprehend the relations of actions to one another. The assumption of the Pragmatist, that a different date in history is a new condition which might affect a chemical experiment, is meaningless, unless by that he intends to say that at the different date new conditions prevailed.

The general conclusion of recent thought is then to establish the Idealist position more strongly by an appeal to mathematical argument. This argument is strengthened by finding at the present time some support in scientific fact and experiment. The Idealist therefore appeals to fact, and his position rests ultimately on a truth which has its aspects of conformity with scientific experiment and with logical or mathematical proof.

FOOTNOTES:

This word is used here in the most general and inclusive sense as applying to all thinkers who accept the reality of relations as part of a higher Unity.

“A Pluralistic Universe,” p. 99.

Taylor, “Elements of Metaphysics,” p. 312.

Ibid., p. 350.

A succession of what is disconnected is not change. Change is a succession within an identity: if not within the identity, there is no change, only analysis and re-grouping. The closer our knowledge is of ourselves or anything else, the more we see that change is the expression in time of an identity.

Illingworth, “The Doctrine of the Trinity,” p. 6.

Ormond, “Foundations of Knowledge,” p. 19.

1908-10.

James, “A Pluralistic Universe,” p. 80.

“A Pluralistic Universe,” p. 230.

See A. T. Cameron, “Radio-Chemistry,” p. 17: “The curves illustrate two further points. They approach constant value towards the end of a month, but it is seen that they reach a final value only at infinite time. This property is common to all such curves; it illustrates the fact that the life of a radio-active element is infinite.” It is explained in the same book (p. 31) that “infinity is only a relative term; in this connection it only means a longer time than we can measure.”

His theology is not so good as his mathematics; he seems to think that in the Creed we assert our belief in the Incomprehensible, in the sense of that which is “not capable of being seized by the mind,” instead of in that which is “untrammelled by limitations.” The word is immensus, best translated infinite.

Hibbert Journal, 1909, pp. 626-28.

Hibbert Journal, 1909, p. 629.

Hibbert Journal, 1909, p. 632.

Schubert, “Mathematical Essays and Recreations,” pp. 70, 71.

See Van t’ Hoff, La Chimie dans l’espace, and Schubert, “Mathematical Essays and Recreations,” pp. 88-89.

Schubert, “Mathematical Essays and Recreations,” pp. 88-89. See also Mach, “Conservation of Energy” (trans. Open Court Publishing Co.).

Printed by BALLANTYNE, HANSON & CO. Edinburgh & London