The Fourth Dimension

by Charles Howard Hinton

Our Representation of a Block of Tesseracts.

Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes 1, 2, 3, and place them in front of you, in any order, say running from left to right, placing 1 in the normal position, the red axis running up, the white to the right, and yellow away.

Now notice that in catalogue cube 2 the colours of each region are derived from those of the corresponding region of cube 1 by the addition of blue. Thus null + blue = blue, and the corners of number 2 are blue. Again, red + blue = purple, and the vertical lines of 2 are purple. Blue + yellow = green, and the line which runs away is coloured green.

By means of these observations you may be sure that catalogue cube 2 is rightly placed. Catalogue cube 3 is just like number 1.

Having these cubes in what we may call their normal position, proceed to build up the three sets of blocks.

This is easily done in accordance with the colour scheme on the catalogue cubes.

The first block we already know. Build up the second block, beginning with a blue corner cube, placing a purple on it, and so on.

Having these three blocks we have the means of representing the appearances of a group of eighty-one tesseracts.

Let us consider a moment what the analogy in the case of the plane being is.

He has his three sets of nine slabs each. We have our three sets of twenty-seven cubes each.

Our cubes are like his slabs. As his slabs are not the things which they represent to him, so our cubes are not the things they represent to us.

The plane being’s slabs are to him the faces of cubes.

Our cubes then are the faces of tesseracts, the cubes by which they are in contact with our space.

As each set of slabs in the case of the plane being might be considered as a sort of tray from which the solid contents of the cubes came out, so our three blocks of cubes may be considered as three-space trays, each of which is the beginning of an inch of the solid contents of the four-dimensional solids starting from them.

We want now to use the names null, red, white, etc., for tesseracts. The cubes we use are only tesseract faces. Let us denote that fact by calling the cube of null colour, null face; or, shortly, null f., meaning that it is the face of a tesseract.

To determine which face it is let us look at the catalogue cube 1 or the first of the views of the tesseract, which can be used instead of the models. It has three axes, red, white, yellow, in our space. Hence the cube determined by these axes is the face of the tesseract which we now have before us. It is the ochre face. It is enough, however, simply to say null f., red f. for the cubes which we use.

To impress this in your mind, imagine that tesseracts do actually run from each cube. Then, when you move the cubes about, you move the tesseracts about with them. You move the face but the tesseract follows with it, as the cube follows when its face is shifted in a plane.

The cube null in the normal position is the cube which has in it the red, yellow, white axes. It is the face having these, but wanting the blue. In this way you can define which face it is you are handling. I will write an “f.” after the name of each tesseract just as the plane being might call each of his slabs null slab, yellow slab, etc., to denote that they were representations.

We have then in the first block of twenty-seven cubes, the following—null f., red f., null f., going up; white f., null f., lying to the right, and so on. Starting from the null point and travelling up one inch we are in the null region, the same for the away and the right-hand directions. And if we were to travel in the fourth dimension for an inch we should still be in a null region. The tesseract stretches equally all four ways. Hence the appearance we have in this first block would do equally well if the tesseract block were to move across our space for a certain distance. For anything less than an inch of their transverse motion we should still have the same appearance. You must notice, however, that we should not have null face after the motion had begun.

When the tesseract, null for instance, had moved ever so little we should not have a face of null but a section of null in our space. Hence, when we think of the motion across our space we must call our cubes tesseract sections. Thus on null passing across we should see first null f., then null s., and then, finally, null f. again.

Imagine now the whole first block of twenty-seven tesseracts to have moved tranverse to our space a distance of one inch. Then the second set of tesseracts, which originally were an inch distant from our space, would be ready to come in.

Their colours are shown in the second block of twenty-seven cubes which you have before you. These represent the tesseract faces of the set of tesseracts that lay before an inch away from our space. They are ready now to come in, and we can observe their colours. In the place which null f. occupied before we have blue f., in place of red f. we have purple f., and so on. Each tesseract is coloured like the one whose place it takes in this motion with the addition of blue.

Now if the tesseract block goes on moving at the rate of an inch a minute, this next set of tesseracts will occupy a minute in passing across. We shall see, to take the null one for instance, first of all null face, then null section, then null face again.

At the end of the second minute the second set of tesseracts has gone through, and the third set comes in. This, as you see, is coloured just like the first. Altogether, these three sets extend three inches in the fourth dimension, making the tesseract block of equal magnitude in all dimensions.

We have now before us a complete catalogue of all the tesseracts in our group. We have seen them all, and we shall refer to this arrangement of the blocks as the “normal position.” We have seen as much of each tesseract at a time as could be done in a three-dimensional space. Each part of each tesseract has been in our space, and we could have touched it.

The fourth dimension appeared to us as the duration of the block.

If a bit of our matter were to be subjected to the same motion it would be instantly removed out of our space. Being thin in the fourth dimension it is at once taken out of our space by a motion in the fourth dimension.

But the tesseract block we represent having length in the fourth dimension remains steadily before our eyes for three minutes, when it is subjected to this transverse motion.

We have now to form representations of the other views of the same tesseract group which are possible in our space.

Let us then turn the block of tesseracts so that another face of it comes into contact with our space, and then by observing what we have, and what changes come when the block traverses our space, we shall have another view of it. The dimension which appeared as duration before will become extension in one of our known dimensions, and a dimension which coincided with one of our space dimensions will appear as duration.

Leaving catalogue cube 1 in the normal position, remove the other two, or suppose them removed. We have in space the red, the yellow, and the white axes. Let the white axis go out into the unknown, and occupy the position the blue axis holds. Then the blue axis, which runs in that direction now will come into space. But it will not come in pointing in the same way that the white axis does now. It will point in the opposite sense. It will come in running to the left instead of running to the right as the white axis does now.

When this turning takes place every part of the cube 1 will disappear except the left-hand face—the orange face.

And the new cube that appears in our space will run to the left from this orange face, having axes, red, yellow, blue.

Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views placed, with its orange face coincident with the orange face of 1, red line to red line, and yellow line to yellow line, with the blue line pointing to the left. Then remove cube 1 and we have the tesseract face which comes in when the white axis runs in the positive unknown, and the blue axis comes into our space.

Now place catalogue cube 5 in some position, it does not matter which, say to the left; and place it so that there is a correspondence of colour corresponding to the colour of the line that runs out of space. The line that runs out of space is white, hence, every part of this cube 5 should differ from the corresponding part of 4 by an alteration in the direction of white.

Thus we have white points in 5 corresponding to the null points in 4. We have a pink line corresponding to a red line, a light yellow line corresponding to a yellow line, an ochre face corresponding to an orange face. This cube section is completely named in Chapter XI. Finally cube 6 is a replica of 1.

These catalogue cubes will enable us to set up our models of the block of tesseracts.

First of all for the set of tesseracts, which beginning in our space reach out one inch in the unknown, we have the pattern of catalogue cube 4.

We see that we can build up a block of twenty-seven tesseract faces after the colour scheme of cube 4, by taking the left-hand wall of block 1, then the left-hand wall of block 2, and finally that of block 3. We take, that is, the three first walls of our previous arrangement to form the first cubic block of this new one.

This will represent the cubic faces by which the group of tesseracts in its new position touches our space. We have running up, null f., red f., null f. In the next vertical line, on the side remote from us, we have yellow f., orange f., yellow f., and then the first colours over again. Then the three following columns are, blue f., purple f., blue f.; green f., brown f., green f.; blue f., purple f., blue f. The last three columns are like the first.

These tesseracts touch our space, and none of them are by any part of them distant more than an inch from it. What lies beyond them in the unknown?

This can be told by looking at catalogue cube 5. According to its scheme of colour we see that the second wall of each of our old arrangements must be taken. Putting them together we have, as the corner, white f. above it, pink f. above it, white f. The column next to this remote from us is as follows:—light yellow f., ochre f., light yellow f., and beyond this a column like the first. Then for the middle of the block, light blue f., above it light purple, then light blue. The centre column has, at the bottom, light green f., light brown f. in the centre and at the top light green f. The last wall is like the first.

The third block is made by taking the third walls of our previous arrangement, which we called the normal one.

You may ask what faces and what sections our cubes represent. To answer this question look at what axes you have in our space. You have red, yellow, blue. Now these determine brown. The colours red, yellow, blue are supposed by us when mixed to produce a brown colour. And that cube which is determined by the red, yellow, blue axes we call the brown cube.

When the tesseract block in its new position begins to move across our space each tesseract in it gives a section in our space. This section is transverse to the white axis, which now runs in the unknown.

As the tesseract in its present position passes across our space, we should see first of all the first of the blocks of cubic faces we have put up—these would last for a minute, then would come the second block and then the third. At first we should have a cube of tesseract faces, each of which would be brown. Directly the movement began, we should have tesseract sections transverse to the white line.

There are two more analogous positions in which the block of tesseracts can be placed. To find the third position, restore the blocks to the normal arrangement.

Let us make the yellow axis go out into the positive unknown, and let the blue axis, consequently, come in running towards us. The yellow ran away, so the blue will come in running towards us.

Put catalogue cube 1 in its normal position. Take catalogue cube 7 and place it so that its pink face coincides with the pink face of cube 1, making also its red axis coincide with the red axis of 1 and its white with the white. Moreover, make cube 7 come towards us from cube 1. Looking at it we see in our space, red, white, and blue axes. The yellow runs out. Place catalogue cube 8 in the neighbourhood of 7—observe that every region in 8 has a change in the direction of yellow from the corresponding region in 7. This is because it represents what you come to now in going in the unknown, when the yellow axis runs out of our space. Finally catalogue cube 9, which is like number 7, shows the colours of the third set of tesseracts. Now evidently, starting from the normal position, to make up our three blocks of tesseract faces we have to take the near wall from the first block, the near wall from the second, and then the near wall from the third block. This gives us the cubic block formed by the faces of the twenty-seven tesseracts which are now immediately touching our space.

Following the colour scheme of catalogue cube 8, we make the next set of twenty-seven tesseract faces, representing the tesseracts, each of which begins one inch off from our space, by putting the second walls of our previous arrangement together, and the representation of the third set of tesseracts is the cubic block formed of the remaining three walls.

Since we have red, white, blue axes in our space to begin with, the cubes we see at first are light purple tesseract faces, and after the transverse motion begins we have cubic sections transverse to the yellow line.

Restore the blocks to the normal position, there remains the case in which the red axis turns out of space. In this case the blue axis will come in downwards, opposite to the sense in which the red axis ran.

In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1 and put 10 underneath it, imagining that it goes down from the previous position of 1.

We have to keep in space the white and the yellow axes, and let the red go out, the blue come in.

Now, you will find on cube 10 a light yellow face; this should coincide with the base of 1, and the white and yellow lines on the two cubes should coincide. Then the blue axis running down you have the catalogue cube correctly placed, and it forms a guide for putting up the first representative block.

Catalogue cube 11 will represent what lies in the fourth dimension—now the red line runs in the fourth dimension. Thus the change from 10 to 11 should be towards red, corresponding to a null point is a red point, to a white line is a pink line, to a yellow line an orange line, and so on.

Catalogue cube 12 is like 10. Hence we see that to build up our blocks of tesseract faces we must take the bottom layer of the first block, hold that up in the air, underneath it place the bottom layer of the second block, and finally underneath this last the bottom layer of the last of our normal blocks.

Similarly we make the second representative group by taking the middle courses of our three blocks. The last is made by taking the three topmost layers. The three axes in our space before the transverse motion begins are blue, white, yellow, so we have light green tesseract faces, and after the motion begins sections transverse to the red light.

These three blocks represent the appearances as the tesseract group in its new position passes across our space. The cubes of contact in this case are those determinal by the three axes in our space, namely, the white, the yellow, the blue. Hence they are light green.

It follows from this that light green is the interior cube of the first block of representative cubic faces.

Practice in the manipulations described, with a realization in each case of the face or section which is in our space, is one of the best means of a thorough comprehension of the subject.

We have to learn how to get any part of these four-dimensional figures into space, so that we can look at them. We must first learn to swing a tesseract, and a group of tesseracts about in any way.

When these operations have been repeated and the method of arrangement of the set of blocks has become familiar, it is a good plan to rotate the axes of the normal cube 1 about a diagonal, and then repeat the whole series of turnings.

Thus, in the normal position, red goes up, white to the right, yellow away. Make white go up, yellow to the right, and red away. Learn the cube in this position by putting up the set of blocks of the normal cube, over and over again till it becomes as familiar to you as in the normal position. Then when this is learned, and the corresponding changes in the arrangements of the tesseract groups are made, another change should be made: let, in the normal cube, yellow go up, red to the right, and white away.

Learn the normal block of cubes in this new position by arranging them and re-arranging them till you know without thought where each one goes. Then carry out all the tesseract arrangements and turnings.

If you want to understand the subject, but do not see your way clearly, if it does not seem natural and easy to you, practise these turnings. Practise, first of all, the turning of a block of cubes round, so that you know it in every position as well as in the normal one. Practise by gradually putting up the set of cubes in their new arrangements. Then put up the tesseract blocks in their arrangements. This will give you a working conception of higher space, you will gain the feeling of it, whether you take up the mathematical treatment of it or not.

APPENDIX II
A LANGUAGE OF SPACE

The mere naming the parts of the figures we consider involves a certain amount of time and attention. This time and attention leads to no result, for with each new figure the nomenclature applied is completely changed, every letter or symbol is used in a different significance.

Surely it must be possible in some way to utilise the labour thus at present wasted!

Why should we not make a language for space itself, so that every position we want to refer to would have its own name? Then every time we named a figure in order to demonstrate its properties we should be exercising ourselves in the vocabulary of place.

If we use a definite system of names, and always refer to the same space position by the same name, we create as it were a multitude of little hands, each prepared to grasp a special point, position, or element, and hold it for us in its proper relations.

We make, to use another analogy, a kind of mental paper, which has somewhat of the properties of a sensitive plate, in that it will register, without effort, complex, visual, or tactual impressions.

But of far more importance than the applications of a space language to the plane and to solid space is the facilitation it brings with it to the study of four-dimensional shapes.

I have delayed introducing a space language because all the systems I made turned out, after giving them a fair trial, to be intolerable. I have now come upon one which seems to present features of permanence, and I will here give an outline of it, so that it can be applied to the subject of the text, and in order that it may be subjected to criticism.

The principle on which the language is constructed is to sacrifice every other consideration for brevity.

It is indeed curious that we are able to talk and converse on every subject of thought except the fundamental one of space. The only way of speaking about the spatial configurations that underlie every subject of discursive thought is a co-ordinate system of numbers. This is so awkward and incommodious that it is never used. In thinking also, in realising shapes, we do not use it; we confine ourselves to a direct visualisation.

Now, the use of words corresponds to the storing up of our experience in a definite brain structure. A child, in the endless tactual, visual, mental manipulations it makes for itself, is best left to itself, but in the course of instruction the introduction of space names would make the teachers work more cumulative, and the child’s knowledge more social.

Their full use can only be appreciated, if they are introduced early in the course of education; but in a minor degree any one can convince himself of their utility, especially in our immediate subject of handling four-dimensional shapes. The sum total of the results obtained in the preceding pages can be compendiously and accurately expressed in nine words of the Space Language.

In one of Plato’s dialogues Socrates makes an experiment on a slave boy standing by. He makes certain perceptions of space awake in the mind of Meno’s slave by directing his close attention on some simple facts of geometry.

By means of a few words and some simple forms we can repeat Plato’s experiment on new ground.

Do we by directing our close attention on the facts of four dimensions awaken a latent faculty in ourselves? The old experiment of Plato’s, it seems to me, has come down to us as novel as on the day he incepted it, and its significance not better understood through all the discussion of which it has been the subject.

Imagine a voiceless people living in a region where everything had a velvety surface, and who were thus deprived of all opportunity of experiencing what sound is. They could observe the slow pulsations of the air caused by their movements, and arguing from analogy, they would no doubt infer that more rapid vibrations were possible. From the theoretical side they could determine all about these more rapid vibrations. They merely differ, they would say, from slower ones, by the number that occur in a given time; there is a merely formal difference.

But suppose they were to take the trouble, go to the pains of producing these more rapid vibrations, then a totally new sensation would fall on their rudimentary ears. Probably at first they would only be dimly conscious of Sound, but even from the first they would become aware that a merely formal difference, a mere difference in point of number in this particular respect, made a great difference practically, as related to them. And to us the difference between three and four dimensions is merely formal, numerical. We can tell formally all about four dimensions, calculate the relations that would exist. But that the difference is merely formal does not prove that it is a futile and empty task, to present to ourselves as closely as we can the phenomena of four dimensions. In our formal knowledge of it, the whole question of its actual relation to us, as we are, is left in abeyance.

Possibly a new apprehension of nature may come to us through the practical, as distinguished from the mathematical and formal, study of four dimensions. As a child handles and examines the objects with which he comes in contact, so we can mentally handle and examine four-dimensional objects. The point to be determined is this. Do we find something cognate and natural to our faculties, or are we merely building up an artificial presentation of a scheme only formally possible, conceivable, but which has no real connection with any existing or possible experience?

This, it seems to me, is a question which can only be settled by actually trying. This practical attempt is the logical and direct continuation of the experiment Plato devised in the “Meno.”

Why do we think true? Why, by our processes of thought, can we predict what will happen, and correctly conjecture the constitution of the things around us? This is a problem which every modern philosopher has considered, and of which Descartes, Leibnitz, Kant, to name a few, have given memorable solutions. Plato was the first to suggest it. And as he had the unique position of being the first devisor of the problem, so his solution is the most unique. Later philosophers have talked about consciousness and its laws, sensations, categories. But Plato never used such words. Consciousness apart from a conscious being meant nothing to him. His was always an objective search. He made man’s intuitions the basis of a new kind of natural history.

In a few simple words Plato puts us in an attitude with regard to psychic phenomena—the mind—the ego—“what we are,” which is analogous to the attitude scientific men of the present day have with regard to the phenomena of outward nature. Behind this first apprehension of ours of nature, there is an infinite depth to be learned and known. Plato said that behind the phenomena of mind that Meno’s slave boy exhibited, there was a vast, an infinite perspective. And his singularity, his originality, comes out most strongly marked in this, that the perspective, the complex phenomena beyond were, according to him, phenomena of personal experience. A footprint in the sand means a man to a being that has the conception of a man. But to a creature that has no such conception, it means a curious mark, somehow resulting from the concatenation of ordinary occurrences. Such a being would attempt merely to explain how causes known to him could so coincide as to produce such a result; he would not recognise its significance.

Plato introduced the conception which made a new kind of natural history possible. He said that Meno’s slave boy thought true about things he had never learned, because his “soul” had experience. I know this will sound absurd to some people, and it flies straight in the face of the maxim, that explanation consists in showing how an effect depends on simple causes. But what a mistaken maxim that is! Can any single instance be shown of a simple cause? Take the behaviour of spheres for instance; say those ivory spheres, billiard balls, for example. We can explain their behaviour by supposing they are homogeneous elastic solids. We can give formulæ which will account for their movements in every variety. But are they homogeneous elastic solids? No, certainly not. They are complex in physical and molecular structure, and atoms and ions beyond open an endless vista. Our simple explanation is false, false as it can be. The balls act as if they were homogeneous elastic spheres. There is a statistical simplicity in the resultant of very complex conditions, which makes that artificial conception useful. But its usefulness must not blind us to the fact that it is artificial. If we really look deep into nature, we find a much greater complexity than we at first suspect. And so behind this simple “I,” this myself, is there not a parallel complexity? Plato’s “soul” would be quite acceptable to a large class of thinkers, if by “soul” and the complexity he attributes to it, he meant the product of a long course of evolutionary changes, whereby simple forms of living matter endowed with rudimentary sensation had gradually developed into fully conscious beings.

But Plato does not mean by “soul” a being of such a kind. His soul is a being whose faculties are clogged by its bodily environment, or at least hampered by the difficulty of directing its bodily frame—a being which is essentially higher than the account it gives of itself through its organs. At the same time Plato’s soul is not incorporeal. It is a real being with a real experience. The question of whether Plato had the conception of non-spatial existence has been much discussed. The verdict is, I believe, that even his “ideas” were conceived by him as beings in space, or, as we should say, real. Plato’s attitude is that of Science, inasmuch as he thinks of a world in Space. But, granting this, it cannot be denied that there is a fundamental divergence between Plato’s conception and the evolutionary theory, and also an absolute divergence between his conception and the genetic account of the origin of the human faculties. The functions and capacities of Plato’s “soul” are not derived by the interaction of the body and its environment.

Plato was engaged on a variety of problems, and his religious and ethical thoughts were so keen and fertile that the experimental investigation of his soul appears involved with many other motives. In one passage Plato will combine matter of thought of all kinds and from all sources, overlapping, interrunning. And in no case is he more involved and rich than in this question of the soul. In fact, I wish there were two words, one denoting that being, corporeal and real, but with higher faculties than we manifest in our bodily actions, which is to be taken as the subject of experimental investigation; and the other word denoting “soul” in the sense in which it is made the recipient and the promise of so much that men desire. It is the soul in the former sense that I wish to investigate, and in a limited sphere only. I wish to find out, in continuation of the experiment in the Meno, what the “soul” in us thinks about extension, experimenting on the grounds laid down by Plato. He made, to state the matter briefly, the hypothesis with regard to the thinking power of a being in us, a “soul.” This soul is not accessible to observation by sight or touch, but it can be observed by its functions; it is the object of a new kind of natural history, the materials for constructing which lie in what it is natural to us to think. With Plato “thought” was a very wide-reaching term, but still I would claim in his general plan of procedure a place for the particular question of extension.

The problem comes to be, “What is it natural to us to think about matter qua extended?”

First of all, I find that the ordinary intuition of any simple object is extremely imperfect. Take a block of differently marked cubes, for instance, and become acquainted with them in their positions. You may think you know them quite well, but when you turn them round—rotate the block round a diagonal, for instance—you will find that you have lost track of the individuals in their new positions. You can mentally construct the block in its new position, by a rule, by taking the remembered sequences, but you don’t know it intuitively. By observation of a block of cubes in various positions, and very expeditiously by a use of Space names applied to the cubes in their different presentations, it is possible to get an intuitive knowledge of the block of cubes, which is not disturbed by any displacement. Now, with regard to this intuition, we moderns would say that I had formed it by my tactual visual experiences (aided by hereditary pre-disposition). Plato would say that the soul had been stimulated to recognise an instance of shape which it knew. Plato would consider the operation of learning merely as a stimulus; we as completely accounting for the result. The latter is the more common-sense view. But, on the other hand, it presupposes the generation of experience from physical changes. The world of sentient experience, according to the modern view, is closed and limited; only the physical world is ample and large and of ever-to-be-discovered complexity. Plato’s world of soul, on the other hand, is at least as large and ample as the world of things.

Let us now try a crucial experiment. Can I form an intuition of a four-dimensional object? Such an object is not given in the physical range of my sense contacts. All I can do is to present to myself the sequences of solids, which would mean the presentation to me under my conditions of a four-dimensional object. All I can do is to visualise and tactualise different series of solids which are alternative sets of sectional views of a four-dimensional shape.

If now, on presenting these sequences, I find a power in me of intuitively passing from one of these sets of sequences to another, of, being given one, intuitively constructing another, not using a rule, but directly apprehending it, then I have found a new fact about my soul, that it has a four-dimensional experience; I have observed it by a function it has.

I do not like to speak positively, for I might occasion a loss of time on the part of others, if, as may very well be, I am mistaken. But for my own part, I think there are indications of such an intuition; from the results of my experiments, I adopt the hypothesis that that which thinks in us has an ample experience, of which the intuitions we use in dealing with the world of real objects are a part; of which experience, the intuition of four-dimensional forms and motions is also a part. The process we are engaged in intellectually is the reading the obscure signals of our nerves into a world of reality, by means of intuitions derived from the inner experience.

The image I form is as follows. Imagine the captain of a modern battle-ship directing its course. He has his charts before him; he is in communication with his associates and subordinates; can convey his messages and commands to every part of the ship, and receive information from the conning-tower and the engine-room. Now suppose the captain immersed in the problem of the navigation of his ship over the ocean, to have so absorbed himself in the problem of the direction of his craft over the plane surface of the sea that he forgets himself. All that occupies his attention is the kind of movement that his ship makes. The operations by which that movement is produced have sunk below the threshold of his consciousness, his own actions, by which he pushes the buttons, gives the orders, are so familiar as to be automatic, his mind is on the motion of the ship as a whole. In such a case we can imagine that he identifies himself with his ship; all that enters his conscious thought is the direction of its movement over the plane surface of the ocean.

Such is the relation, as I imagine it, of the soul to the body. A relation which we can imagine as existing momentarily in the case of the captain is the normal one in the case of the soul with its craft. As the captain is capable of a kind of movement, an amplitude of motion, which does not enter into his thoughts with regard to the directing the ship over the plane surface of the ocean, so the soul is capable of a kind of movement, has an amplitude of motion, which is not used in its task of directing the body in the three-dimensional region in which the body’s activity lies. If for any reason it became necessary for the captain to consider three-dimensional motions with regard to his ship, it would not be difficult for him to gain the materials for thinking about such motions; all he has to do is to call his own intimate experience into play. As far as the navigation of the ship, however, is concerned, he is not obliged to call on such experience. The ship as a whole simply moves on a surface. The problem of three-dimensional movement does not ordinarily concern its steering. And thus with regard to ourselves all those movements and activities which characterise our bodily organs are three-dimensional; we never need to consider the ampler movements. But we do more than use the movements of our body to effect our aims by direct means; we have now come to the pass when we act indirectly on nature, when we call processes into play which lie beyond the reach of any explanation we can give by the kind of thought which has been sufficient for the steering of our craft as a whole. When we come to the problem of what goes on in the minute, and apply ourselves to the mechanism of the minute, we find our habitual conceptions inadequate.

The captain in us must wake up to his own intimate nature, realise those functions of movement which are his own, and in virtue of his knowledge of them apprehend how to deal with the problems he has come to.

Think of the history of man. When has there been a time, in which his thoughts of form and movement were not exclusively of such varieties as were adapted for his bodily performance? We have never had a demand to conceive what our own most intimate powers are. But, just as little as by immersing himself in the steering of his ship over the plane surface of the ocean, a captain can lose the faculty of thinking about what he actually does, so little can the soul lose its own nature. It can be roused to an intuition that is not derived from the experience which the senses give. All that is necessary is to present some few of those appearances which, while inconsistent with three-dimensional matter, are yet consistent with our formal knowledge of four-dimensional matter, in order for the soul to wake up and not begin to learn, but of its own intimate feeling fill up the gaps in the presentiment, grasp the full orb of possibilities from the isolated points presented to it. In relation to this question of our perceptions, let me suggest another illustration, not taking it too seriously, only propounding it to exhibit the possibilities in a broad and general way.

In the heavens, amongst the multitude of stars, there are some which, when the telescope is directed on them, seem not to be single stars, but to be split up into two. Regarding these twin stars through a spectroscope, an astronomer sees in each a spectrum of bands of colour and black lines. Comparing these spectrums with one another, he finds that there is a slight relative shifting of the dark lines, and from that shifting he knows that the stars are rotating round one another, and can tell their relative velocity with regard to the earth. By means of his terrestrial physics he reads this signal of the skies. This shifting of lines, the mere slight variation of a black line in a spectrum, is very unlike that which the astronomer knows it means. But it is probably much more like what it means than the signals which the nerves deliver are like the phenomena of the outer world.

No picture of an object is conveyed through the nerves. No picture of motion, in the sense in which we postulate its existence, is conveyed through the nerves. The actual deliverances of which our consciousness takes account are probably identical for eye and ear, sight and touch.

If for a moment I take the whole earth together and regard it as a sentient being, I find that the problem of its apprehension is a very complex one, and involves a long series of personal and physical events. Similarly the problem of our apprehension is a very complex one. I only use this illustration to exhibit my meaning. It has this especial merit, that, as the process of conscious apprehension takes place in our case in the minute, so, with regard to this earth being, the corresponding process takes place in what is relatively to it very minute.

Now, Plato’s view of a soul leads us to the hypothesis that that which we designate as an act of apprehension may be a very complex event, both physically and personally. He does not seek to explain what an intuition is; he makes it a basis from whence he sets out on a voyage of discovery. Knowledge means knowledge; he puts conscious being to account for conscious being. He makes an hypothesis of the kind that is so fertile in physical science—an hypothesis making no claim to finality, which marks out a vista of possible determination behind determination, like the hypothesis of space itself, the type of serviceable hypotheses.

And, above all, Plato’s hypothesis is conducive to experiment. He gives the perspective in which real objects can be determined; and, in our present enquiry, we are making the simplest of all possible experiments—we are enquiring what it is natural to the soul to think of matter as extended.

Aristotle says we always use a “phantasm” in thinking, a phantasm of our corporeal senses a visualisation or a tactualisation. But we can so modify that visualisation or tactualisation that it represents something not known by the senses. Do we by that representation wake up an intuition of the soul? Can we by the presentation of these hypothetical forms, that are the subject of our present discussion, wake ourselves up to higher intuitions? And can we explain the world around by a motion that we only know by our souls?

Apart from all speculation, however, it seems to me that the interest of these four-dimensional shapes and motions is sufficient reason for studying them, and that they are the way by which we can grow into a fuller apprehension of the world as a concrete whole.