The subject of four dimensional space hardly seems, at first glance, to have much connection with theosophical doctrines; except possibly that most persons would regard both as being vaguely mysterious, and many persons would consider both as arrant nonsense; and I am afraid that if I should claim that there was quite a definite relation between the study of the fourth dimension, and the fundamental principle of the T. S., universal brotherhood, even the members themselves might smile incredulously; but I hope I shall be able to show that such a relation is not preposterous, but quite natural.

We all know that from the very first records that we have of philosophy, especially of esoteric philosophy, there has been a peculiar and mystical virtue assigned to numbers. "All systems of religious mysticism are based on numerals." *Isis Unveiled* II, 407. Pythagoras said that the essence of things consisted in Number, and that the Cosmos was generated out of numbers.

And this mystical virtue has also been attributed, to various geometrical figures. The line, the circle, the triangle, the square, each has been used as a symbol of some great truth; as well as, on a lower plane, a potent instrument in magical ceremonies. Now geometry is the study of dimensions, surfaces and solids; and the study of four dimensional space, if such a thing there be, is merely a higher branch of geometry; and we may perhaps find in it teachings of an order beyond what we can get from a mathematics dealing with space of fewer dimensions.

Of course you will not expect that in the short time I shall use this evening, I can give you much idea of higher space and the laws which work in it; or perhaps a better expression would be, the forms under which, in higher space, are manifested those universal laws with whose working in our ordinary space we are familiar. To do this would be a long and not very easy task; I can only hope to show that there is something to study and to learn, something that will repay the study. And in what I shall say, I am almost entirely indebted to Mr. C. H. Hinton, whose books "Scientific Romances" and "A New Era of Thought" are most fascinating and instructive studies for any one who cares to look within the surface of things.

To begin at the beginning: — a point, mathematically speaking, has location, but no dimensions. When this point is moved in any direction, we have a line, extending from the point of starting to the point of stopping; and we may call this line space of one dimension, that of length. Suppose this whole line to be moved in a direction at right angles to itself, and we have a surface, a square; space of two dimensions, viz: — length and breadth. Let this square move at right angles to itself, that is, at right angles both to the original course of the point to form the line, and to the course of the line in forming the square; we have now a solid body, *i.e.,* space of three dimensions, length, breadth and thickness. Now just as the line moved at right angles to itself and formed the square; just as the square moved at right angles to itself to form the cube; so if the cube can be moved in a direction at right angles to itself, that is, at right angles to each and every line and surface in itself,
we shall get — what? We can't say now, but certainly something quite different from a line, a surface, or a solid.

Suppose the length of the line to be two inches; then the area of the square will be 2x2=4 inches, the contents of the cube 4x2 inches = 8 inches. If we represent the first (the line) by the algebraical expression a, the second (the square) will be a^{2}, the third (the cube) will be a^{3}; so that these three terms can be graphically represented to us. But in algebra the expression a^{4}, that is a^{3} multiplied by a, is perfectly proper; how shall we represent that graphically? We cannot; we have to stop at a^{3}.

The length of the line is 2 inches, the area of the square is 4 inches, the contents of the cube is 8 inches; but these inches are quite distinct from each other. No possible number of inches composing the line will make up one of the inches composing the square; no possible number of the inches composing the square can make up one of the inches composing the cube. So no conceivable number of the solid units composing the cube representing a^{3}, can make up that which we mean by a^{4}; the two are incommensurable. So as soon as we try to go beyond a^{3}, we come up against a wall, metaphorically speaking; and why should we not say that there *is* nothing beyond it?

The only way here is to apply a famous occult motto, what we might almost call a fundamental axiom of occult science; the words from the emerald tablet of Hermes: "As is that which is above, so is that which is below." We cannot directly perceive that which is above, but if we look at that which is below, we may learn from analogy.

Let us suppose beings existing in space of two dimensions; beings with senses and intelligence, like ourselves, but neither they nor the world in which they exist, having any dimensions but length and breadth; no such thing as thickness. Or, as such beings would be to us, and to our modes of thought, merely abstractions, let us suppose them to be exceedingly thin in the third dimension, say of no greater thickness than a single molecule of matter. In a work on astronomy or physics, when we wish to show how gravitation holds bodies on the earth, on whichever side of it they may be, we represent the latter by a section of it, a circle, along the circumference of which we place representations of the various bodies on it, which are held firmly to it by the force of gravity, drawing them towards the centre. Now to these beings of the plane world, this circle *is* their earth, not merely a section of it; they are free to move round it; by an effort opposing the force of gravity they can move from it,
as we can by an effort and by suitable appliances rise from the surface of our earth, as by a ladder or for an instant by jumping.

Looking at the corner of this room, we find three lines proceeding from it at right angles to each other; two horizontal and one vertical; now from that corner we can proceed to any point of space in the room by moving on those lines, or lines parallel to them. The plane being, supposing the plane he inhabits to be that of this floor, could reach any point in his space by one or both of the two directions, which proceed from the corner on his plane; to rise from that plane into what we call space, would be as inconceivable to him as for us to pass to some point not to be reached by either of our three lines or lines parallel to them.

Cut out of paper an equilateral triangle, each side say two inches long: cut this in two by a line from one angle to the middle of the opposite side: let the two parts lie on the table without changing their relative position from what it was before the original triangle was divided; you have now two triangles of the same dimensions, their angles and areas just the same: but as long as they lie in the same plane you may move them round and round as much as you like, and you can never make them coincide. But if you lift one of them from the surface on which it lies and turn it over, it will then coincide exactly with the other.

Draw a square on a piece of paper; put a coin on the paper inside the square; can you slide that coin on the paper in any way so as to bring it outside, without touching the square? No; but you can lift it up and set it down outside.

Now in these two apparently, and to us actually simple operations, we have accomplished what to the two-dimensional being is an impossibility, an actual absurdity; equally impossible and absurd as would be to me to turn round my left hand until it fitted my right glove just as my right hand does: or corresponding to the second example of two-dimensional impossibility, the moving of an object in and out of a closed room or box without any opening being made in the walls. To make the two triangles coincide, we lift one out of the plane in which it lies, and turning it over through the third dimension of space, lay it down on the other triangle, and thus accomplish what never could be done as long as we moved it about in the two dimensions. Now just as the triangle exists in space of two dimensions, so my hand exists in space of three dimensions; the two hands agree in every particular, dimensions the same in every respect, every curve and angle the same; but no possible way that I can move them will make them coincide. As the plane being would say as to the triangle problem, it is impossible. But as the two dimensional impossibilities are very simple things to us, so to a being existing in higher space, if such a being exist, our impossibilities must be equally simple matters.

(*Concluded in May.*)

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