Let us suppose that we are walking along a street in a busy town; that we are familiar with it, and all the things that are usually to be seen in it, so that our attention is not likely to be arrested by anything unusual; and let us further suppose that we are thinking about something interesting but not intellectually difficult. In these circumstances all the sights of the town, and all the turmoil of the traffic fail to impress us, though we are, in a vague sort of way, conscious of it all. Electric trams approach and recede with a grinding noise; a taxicab passes and we hear the throb of the engine and the hooting of the horn, and smell the burnt oil; a hansom comes down the street and we hear the rhythmic tread of the horse’s feet and the jingle of the bells; we pass a florist’s shop and become aware of the colour of the flowers and of their odour; in a café a band is playing “ragtime.” There are policemen, hawkers, idlers, ladies with gaily coloured dresses and hats, newsboys, a crowd of people of many characteristics. It is all a flux of experience of which we are generally conscious without analysis or attention, and it is a flux which is never for a moment quite the same, for everything in it melts and flows into everything else. The noise of the tram-cars is incessant, but now and then it becomes louder; the music of the orchestra steals imperceptibly on our ears and as imperceptibly fades away; the smell of the flowers lingers after we pass the shop, and we do not notice just when we cease to be conscious of it; the rhythm of the ragtime continues to irritate after we have ceased to hear the band—all the sense-impressions that we receive melt and flow over into each other and constitute our stream of consciousness, and this changes from moment to moment without gap or discontinuity. It is not a condition of “pure sensation,” but it is as nearly such as we can experience in our adult intellectual life.

It is easy to discover that many things must have occurred in the street which did not affect our full consciousness. We may learn afterwards that we have passed several friends without recognising them; we may read in the newspapers about things that happened that we might have seen, but which we did not see; we may think we know the street fairly well, but we find that we have difficulty in recalling the names of three contiguous shops in it; if we happen to see a photograph which was taken at the time we passed through the street we are usually surprised to find that there were many things there that we did not see. Why is it, then, that so much that might have been perceived by us was not really perceived? We cannot doubt that everything that came into the visual fields of our eyes must have affected the terminations of the optic nerves in the retinas; the complex disturbances of the air in the street must have set our tympanic membranes in motion; and all the odoriferous particles inhaled into our nostrils must have stimulated the olfactory mucous membranes. In all these cases the stimulation of the receptor organs must have initiated nervous impulses, and these must have been propagated along the sensory nerves, and must have reached the brain, affecting masses of nerve cells there. Nothing in physiology seems to indicate that we can inhibit or repress the activity of the distance sense-receptors, visual, auditory, and olfactory, with their central connections in the brain; they must have functioned, and must have been physically affected by the events that took place outside ourselves, and yet we were unconscious, in the fullest sense of this term, of all this activity. Why is it, then, that our perception was so much less than the actual physical reception of external stimuli that we must postulate as having occurred? Sherlock Holmes would have said that we really saw and heard all these things although we did not observe them, but the full explanation involves a much more careful consideration of the phenomena of perception than this saying indicates.

There is, of course, no doubt that we did see and hear and smell all the things that occurred in the street during our aimless peregrination, that is, all the things which so happened that they were capable of affecting our organs of sense. This is true if we mean by seeing and hearing and smelling merely the stimulation of the nerve-endings of the visual, auditory, and olfactory organs, and the conduction into the brain of the nervous impulses so set up. But merely to be stimulated is only a part of the full activity of the brain; the stimulus transmitted from the receptor organs must result in some kind of bodily activity if it is to affect our stream of consciousness. Two main kinds of activity are induced by the stimulation of a receptor organ and a central ganglion, (1) those which we call reflex actions, and (2) those actions which we recognise as resulting from deliberation. We must now consider what are the processes that are involved in these kinds of neuro-muscular activity.

The term “reflex action” is one that denotes rather a scheme of sensori-motor activity than anything that actually happens in the animal body; it is a concept that is useful as a means of analysis of complex phenomena. In a reflex three things happen, (1) the stimulation of a receptor organ and of the nerve connecting this with the brain, (2) the reflection, or shunting, of the nervous impulse so initiated from the terminus ad quem of the afferent or sensory nerve, to the terminus a quo of the efferent or motor nerve, and (3) the stimulation of some effector organ, say a motor organ or muscle, by the nervous impulse so set up. The simplest case, perhaps, of a reflex is the rapid closure of the eyelids when something, say a few drops of water, is flicked into the face. Stated in the way we have stated it the simple reflex does not exist. In the first place, it is a concept based on the structural analysis of the complex animal where the body is differentiated to form tissues—receptor organs, nerves, muscles, glands, and so on. But a protozoan animal, a Paramœcium for instance, responds to an external stimulus by some kind of bodily activity, and yet it is a homogeneous, or nearly homogeneous, piece of protoplasm, and this simple protoplasm acts at the same time as receptor organ, conducting tissue or nerve, and effector organ. In the higher animal certain parts of the integument are differentiated so as to form visual organs, and the threshold of these for light stimuli is raised while it is lowered for other kinds of physical stimuli. Similarly other parts of the integument are modified for the reception of auditory stimuli, becoming more susceptible for these but less susceptible for other kinds of stimuli than the adjacent parts of the body. Within the body itself certain tracts of protoplasm are differentiated so that they can conduct molecular disturbances set up in the receptor organs in the integument better than can the general protoplasm; these are the nerves. Other parts are modified so that they can contract or secrete the more easily; these are the muscles and glands. The conception of a reflex action, as it is usually stated in books on physiology, therefore includes this idea of the differentiation of the tissues, but all the processes that are included in the typical reflex are processes which can be carried on by undifferentiated protoplasm.

It is also a schematic description that assumes a simplicity that does not really exist. As a rule a reflex is initiated by the stimulation of more than one receptor organ, and the impulses initiated may thus reach the central nervous system by more than one path. There is no simple shunting of the afferent impulse from the cell in which it terminates into another nerve, when it becomes an efferent impulse; but, instead of this, the impulse may “zigzag” through a maze of paths in the brain or spinal cord connecting together afferent and efferent nerves and ganglia. Further, the final part of the reflex, the muscular contraction, is far from being a simple thing, for usually a series of muscles are stimulated to contract, each of them at the right time and with the right amount of force, and every contraction of a muscle is accompanied by the relaxation of the antagonistic muscle. There are muscles which open the eyelids and others which close them, and the cerebral impulse which causes the levators to contract at the same time causes the depressors to relax.

It is quite necessary to remember that the simple reflex is really a process of much complexity and may involve many other parts and structures than those to which we immediately direct our attention. But leaving aside these qualifications we may usefully consider the general characters of the reflex, regarding it as a common, automatically performed, restricted bodily action, involving receptor organ, central nervous organ, and effector organ. There are certain kinds of external stimuli that continually affect our organs of sense, and there are certain kinds of muscular and glandular activity that occur “as a matter of course,” when these stimuli fall on our organs of sense. The emanation from onions or the vapour of ammonia causes our eyes to water; the smell of savoury food causes a flow of saliva; and anything that approaches the face very rapidly causes us to close the eyes. Reflexes are, in a way, commonly occurring, purposeful and useful actions, and their object is the maintenance of a normal condition of bodily functioning.

We dare hardly say that the simple reflex is an unconsciously performed action, although we are not conscious, in the fullest sense of the term, of the reflexes that habitually take place in ourselves. But even in the decapitated frog, which moves its limbs when a drop of acid is placed on its back, something, it has been said, akin to consciousness may flash out and light up the automatic activity of the spinal cord. We must not think of consciousness as that state of acute mentality which we experience in the performance of some difficult task, or in some keenly appreciated pleasure, or in some condition of mental or bodily distress; it is also that dimly felt condition of normality that accompanies the satisfactory functioning of the parts of the bodily organism. But this dim and obscure feeling of the awareness of our actions is easily inhibited whenever what we call intellectual activity proceeds.

Much of the stimulation of our receptor organs is of this generally occurring nature, and we are not aware of it although the stimuli received are such as to induce useful and purposeful bodily activity. In walking along the street we automatically avoid the people, and the other obstacles that we encounter, by means of regulated movements of the body and limbs, but this is activity that has become so habitual and easy that we are hardly aware of it, and not at all, perhaps, of the physical stimuli which induce it. But not only do we receive stimuli which are reflected into bodily actions without our being keenly aware of this reception, but we also receive stimuli which do not become reflected into bodily activity. It is, Bergson suggests, as if we were to look out into the street through a sheet of glass held perpendicularly to our line of sight; held in this way we see perfectly all that happens in front of us, but when we incline the glass at a certain angle it becomes a perfect reflector and throws back again the rays of light that it receives. This is, of course, a physical analogy, and no comparison of material things with psychical processes can go very far, but in a way it is more than an analogy. In our indolent absorbed state of mind we do not as a rule see the objects which we are not compelled to avoid, and which do not, in any way, influence our immediate condition of bodily activity. The optical images of all these things are thrown upon our retinas and are, in some way, thrown or projected upon the central ganglia, but there the series of events comes to an end, for the images are not reflected out towards the periphery of the body as muscular actions. We cannot doubt that this is why we do not perceive all the stimulation of our organs of sense that we are sure that take place. These stimuli pass through us, as it were, unless they are reflected out again as actions. In this reflection, or translation of neutral into muscular activity, perceptions arise.

But even then perception need not arise. It does not, as a rule, accompany the automatically performed reflex action, because the latter is the result of intra-cerebral activities that have become so habitual that they proceed without friction. There are innumerable paths in the brain along which impulses from the receptor organs may pass into the motor ganglia, but in the habitually performed reflex actions these paths have been worn smooth, so to speak. The images of objects which are perceived over and over again by the receptor organs glide easily through the brain and as easily translate themselves into muscular, or some other kind of activity. The things that matter in the life of an animal which lives “according to nature” are cyclically recurrent events in which, after a time, there is nothing new. Most of them proceed just as well in the animal deprived of its cerebral hemispheres by operation as in the intact cerebrate animal. In the performance of actions of this kind the organism becomes very much of an automaton.

Let something unusual happen in the street while we are walking through it—a runaway horse, or the fall of an overhead “live” wire, for instance, something that has seldom or never formed part of our experience, and something that may have an immediate effect on us as living organisms. Then perception arises at once because the stimulation of our organs of sense presents us with something which is unfamiliar, and yet not so unfamiliar that it does not recall from memory, or from derived experience, reminiscences of the images of somewhat similar things, and of the effects of these. The train of events that now proceeds in our central nervous system becomes radically different from that which proceeded in our former, rather aimless, series of actions. The stimuli no longer pass easily through the “lower” ganglia of the brain, but flash upwards into the cortical regions, where they become confronted with the possibility of innumerable alternative paths and connections with all the parts of the body. They waver, so to speak, before adopting one or other, or a combination of these paths; there is hesitation, deliberation, and finally choice of a path, with the result that a series of muscular organs become inervated and motor actions, of a type more or less competent to the situation in which we find ourselves, are set up. In this hesitation and deliberation perception arises. It is when the animal may act in a certain way as the result of a stimulus which is not a continually recurrent one, but at the same time may refrain from acting, or may act in one of several different ways, that perception of external things and their relations arises.

That is to say, we perceive and think because we act. We do not look out on the environment in which we are placed in a speculative kind of way, merely receiving the images of things, and classifying and remembering them, while all the time we are passive in so far as our bodily activities are concerned. If the results of modern physiology teach us anything in an unequivocal way they teach us this—that the organs of activity, muscles, glands, and so on, and the organs of sense and communication, are integrally one series of parts, and that apart from motor activity nervous activity is an aimless kind of thing. It is because we act that we think and disentangle the images of things presented to us by our organs of sense, and subject all that is in the stream of consciousness to conceptual analysis.1

That is to say, in thinking about the flux of consciousness we decompose it into what we regard as its constituent parts, and we confer upon these parts separate existence in space and time. But it is clear that none of the things which we thus regard as the elements of our consciousness has any real existence apart from the others. The smell of the flowers and that of the burnt oil interpenetrate in our consciousness of the stimulation of our olfactory organs just as do the jingle of the cab bells, the music of the orchestra, and the throb of the motor car in the impressions transmitted by our auditory organs. It is difficult to see that all these things, with the multitude of other things which we perceive, constitute a “multiplicity in unity,” that is an assemblage of things which are separate things, but which do not lie alongside each other in space and mutually exclude each other, but which are all jammed into each other, so to speak. It is easy to see that we are conscious of a heterogeneity, and whenever we think of this multitude of things it seems natural that we should separate them from each other. The stream of our consciousness is so complex that we cannot attend to it all at once, not even to the few things that we have picked out in our example. If we concentrate our attention on any part, or rather aspect of it, all the rest ceases to exist, or rather we agree to ignore it, and this very concentration of thought upon one part of our experience isolates it from all the rest. To a certain extent the analysis of the complex of sensation is the result of the work of different receptor organs; certain fields of energy, which we call light, radiation, etc., affect the nerve-endings in the retina; chemically active particles in the atmosphere affect the nerve-endings in the olfactory membranes; and rapidly repeated changes of pressure in the atmosphere (sound vibrations) affect the auditory organs in the internal ear, and so on. But this reception of different stimuli by different receptor organs exists only in the higher animal; there are no specialised sense organs in a Paramœcium, for instance, and the whole periphery of the animal must receive all these different kinds of external stimuli at once. The specialisation of its receptor organs in the higher animal is rather the means whereby the organism becomes more receptive of its environment, than the means whereby it analyses that environment. This analysis is the work of the consciousness of the animal.

Suppose that we draw a curve AB freehand with a single undivided sweep of the pencil. By making a certain assumption—that the curve which we drew was one that might be regarded as cyclical, that is, might be repeated over and over again—we can subject it to harmonic analysis. We can decompose it into a number of other curves (CD, EF, etc.), each of which is a separate “wave” rising above and falling below the axis OX in a symmetrical manner. If we draw any vertical line MN cutting these curves, we shall find that the distance between the axis OX and the main curve AB is always equal to the algebraic sum of the distances between the axis and the other curves. These latter we call the harmonic constituents of the curve AB, supposing them to “add up” so as to form it. But AB was something quite simple and elemental and its constituents cannot be said to have existed in it when we drew it freehand; it was only by an artifice of practical utility in mathematical computations that we constructed them. It may be, of course, that the harmonic constituents of a curve had actual existence apart from the curve itself, but, in the case that we take, they certainly had not. Now we must think of our stream of consciousness in much the same way. It is something immediately experienced and elementary; it is the concomitant, if we choose so to regard it, of the external processes that go on outside our bodies. We can investigate it by thinking about it, and attending to one aspect of it after another, thus arbitrarily detaching one “part” of it from all the rest, but immediately we do this we rise above the flux of experience into the region of intellectual concepts. We have converted a multiplicity of states of consciousness, all of which co-exist along with each other, and in each other, and which have no spatial existence, into a multiplicity of states, visual, auditory, olfactory, etc., which have become separated from each other and have therefore acquired extension. This dissociation of the flux of experience is the process of conceptual analysis carried out by thought.

If we dissociate the stream of consciousness in this way, breaking it up into states which we choose to regard as separate from each other, we shall see that of the elements which we thus isolate many are like each other and can be associated. Obviously there is a greater resemblance between different smells than between smells and sounds. Different musical sounds are more like each other than are sounds, and feelings of heat and cold. There is a greater likeness between the states of consciousness which arise from the stimulation of the same receptor organ, than between those that arise from the stimulation of different receptors. Those differences of sensation accompanying the stimulation of different sense organs we regard as different in kind; there is absolutely no resemblance between a colour and a sound, we say, however much the modern annotator of concert programmes may suggest the analogy. But we say that there may be different degrees of stimulation of the same sense organ, and that the sensations that we thus receive are of the same kind though they differ in intensity. The whistle of a railway engine becomes louder as the train approaches, that is to say, more intense, and if we study the physical conditions that are concomitant with the stimulation of our tympanic membranes we shall see that waves of alternate rarefaction and compression are set up in the atmosphere outside our ears. All the time that the train approaches the frequency of these waves remains the same, that is, just as many occur in a second when the train is distant as when it is near. But the amplitude of the waves has been increasing, and the velocity with which the molecules of air strike against the tympanic membranes becomes greater the nearer is the source of sound. Fig. 2. We can represent this by means of a diagram which shows that the amplitude of the waves—which represents the loudness of the sound—increases while the frequency—which represents the pitch—remains the same. The amplitude is represented by the straight vertical lines, 11, 22, 33, etc., which are of increasing magnitude. Thus we represent the physical cause of the increasing loudness of the sound by space-magnitudes, and then we transfer these magnitudes to the states of consciousness concomitant with the vibrating molecules of air. Suppose that we knew nothing at all about the cause of the differences of pitch of musical sounds and that we listen to the notes of the octave, C, D, E,——C, sounded by an organ; all that we should experience would be that the sounds were different. If we were to sing the notes we might attain the intuition that the notes G, A, B were “higher” than the notes C, D, E, because a greater effort was required in order to produce these sounds, but obviously this is a different thing from saying that the notes themselves were “higher” or “lower.” But let us match the notes by striking tuning-forks, and then having selected forks which give the notes of the octave let us fix them so that they will make a tracing, while still vibrating, on a revolving strip of paper. We shall then find that the fork emitting the note C makes (say) 256 vibrations per second, the fork D  9/8 256 vibrations, the fork E  5/4 256 vibrations, and so on. Thus we associate the notes of the octave together and we say that their quality was the same but that their pitch differed, and since the pitch depends on the frequency of vibration of the fork, or of the air in its vicinity, we say that pitch differences are quantitative ones, and that the states of consciousness which accompany these physical events are also quantitatively different.

So also with colour. If we had no such apparatus as prisms or diffraction gratings, which enable us to find what is the wave length of light, should we have any idea of the spectral hues, red, yellow, orange, green, etc., as differing from each other quantitatively? It is certain that we should not. But observation and experiment have shown that the nerve-endings of the optic nerve in the retina are stimulated by vibrations of something which we agree to call the ether of space, and that the frequency of vibration of light which we call red is less than that which we call orange, while the frequency of vibration of orange light is less again than that of blue light, and so on. To our consciousness red, orange, yellow, and blue light are absolutely different, but we disregard this intuition and we say that our perceptions of light are similar in kind but differ, in some of them are more intense than are some others. Again, have we any intuitive knowledge of increasing temperature? If we dip our hands into ice-cold water the sensation is one of pain, if the water has a temperature of 5° C. it feels cold, if it is at 15° C. we have no particular appreciation of temperature, if at 25° C. it feels very warm, if it is at 60° it is very hot, and if it is at 90° we are probably scalded and the feeling is again one of pain. If we place a thermometer in the water we notice that each sensation in turn is associated with a progressive lengthening of the mercury thread, and if we investigate the physical condition of the water we find that at each stage the velocity of movement of the molecules was greater than that at the preceding stage. We say, then, that our different perceptions were those of heat of different degrees of intensity, so transferring to the perceptions themselves the notions of space-magnitudes acquired by a study of the expansion of the mercury in the thermometer, or by the adoption of the physical theory of the kinetic structure of the water. Yet it is quite certain that what we experienced were quite different things or conditions, cold, warmth, heat, and pain, and indeed, in this series of perceptions different receptor organs are involved.

Thus we decompose our stream of consciousness into a series of quantitatively different and qualitatively different things, upon each of which we confer independent existence. We attribute to these different aspects of our consciousness extension, but the extension is due only to our analysis; for the qualities of pitch, loudness, colour, odour, etc., which we disentangle from each other, did not exist apart from each other, any more than do the sine and cosine curves into which we decompose an arbitrarily drawn curved line. The multiplicity of our consciousness is intensive, like the multiplicity that we see to exist in the abstract number ten. This number stands for a group of things, but its multiplicity is intensive and only exists because we are able to subdivide anything in thought to an indefinite extent. Now, so far we have only separated what we agree to regard as the elemental parts of our general perception of the environment, but it is to be noted that we have not given to these elements anything like spatial extension.

We may, if we like, regard our intuition of space as that of an indefinitely large, homogeneous, empty medium which surrounds us and in which we may, in imagination, place things. So regarded it is difficult to see in what way our notion of space differs from our idea of “nothing,” a pseudo-idea incapable of analysis, except into the idea of something which might be somewhere else. The more we think about it the more we shall become convinced that space, that is the “form” of space, represents our actual or potential modes of motion, that is, our powers of exertional activity. Space, we say, has three dimensions; in all our analysis of the universe, and of the activities that we can perceive in it, this idea of movement in three dimensions, forward and backward, up and down, and right and left, occurs; and we have to recognise that in it there is something fundamental, as fundamental as the intuitive knowledge that we possess of the direction of right and left. It is because we can move in such a way that any of our motions, no matter how complex, can be resolved into the components of backward and forward, right and left, and up and down, these directions all being at right angles to each other, that we speak of our movements as three-dimensional ones. Our geometry is founded, therefore, on concepts derived from our modes of activity; and there is nothing in the universe, apart from our own activity, that makes this the only geometry possible to us. Euclidean geometry does not depend on the constitution of the external universe, but on the nature of the organism itself.

There is a little Infusorian which lives, in its adult phase, on the surface of the spherical ova of fishes. These ova float freely in sea water, and the Infusorian crawls on their surfaces, moving about by means of ciliary appendages. It does not swim about in the water, but adheres closely to the surface of the ovum on which it lives. Let us suppose that it is an intelligent animal and that it is able to construct a geometry of its own; if so, this geometry would be very different from our own.

It would be a two-dimensional geometry, for the animal can move backward and forward, and right and left, but not up and down; it is a stereotropic organism, as Jacques Loeb would say, that is, it is compelled by its organisation to apply its body closely to the surface on which it lives. But its two-dimensional geometry would, on this account, be different from ours. Our straight lines are really the directions in which we move from one point to another point in such a way as to involve the least exertion; they are the shortest distances between two points, and if we deviate from them we exert a greater degree of activity than if we had moved along them. For us there is only one straight line that can be drawn between two points, but this is not necessarily true for our Infusorian, and its straight line need not be the shortest distance between two points. It might be either the longest or the shortest distance between the points, for the latter can always be placed on a great circle passing through the two points and the poles of the egg, and in moving from a point on which it is placed the animal could reach the other point by moving in two directions, just as we could go round the earth along the equator by moving to the east or to the west. Therefore the straight line of the Infusorian would be not only a scalar quantity but a vector quantity, that is, it would represent, not only a quantity of energy, but a quantity of energy that has direction. For us only one straight line can be drawn between two given points, but this limitation would not exist in the two-dimensional geometry of a curved surface. Suppose that the two points are situated on a great circle and that they are exactly 180° apart; then the Infusorian could move from one pole to another pole along an infinite number of straight lines or meridians all of which had a different direction, but all of which were of the same length; that is to say, in this geometry an infinite number of straight lines can be drawn between the same two points. Again, its triangles might be different from ours; our triangles are figures formed by drawing straight lines between three points, and on a plane surface the sum of the angles of the triangle are together equal to two right angles, though on a curved surface they may be greater or less than two right angles. But our Infusorian could not imagine a triangle in which the sum of the angles was not greater than two right angles, for all its figures would be drawn on a convex surface.

Our three-dimensional geometry depends, therefore, on our modes of activity and the concepts with which it operates; points, straight lines, etc. are conceptual limits to those modes of activity. We can imagine a straight line only as a direction along which we can move without deviating to the right or the left, or up or down. But even if we draw such a line on paper with a fine pencil the trace would still have some width, and we can imagine ourselves small enough to be able to deviate to the right or the left within the width of the line drawn on the paper. We might make a very small mark on the paper, but no matter how small this mark is it would still have some magnitude; otherwise we should be unable to see it. If the straight line had no width and the point no magnitude they would have no perceptual existence. Our perceptual triangles are not figures, the angles of which are necessarily equal to two right angles. If we drive three walking sticks into a field and then measure the angles between them by means of a sextant we shall find that the sum is nearly 180°, but in general not that amount. If we stick a darning needle into the heads of each of the walking sticks and then remeasure the angles by means of a theodolite we shall obtain values which are nearer to that of two right angles, but we should not, except by “accident,” obtain exactly this value. We do not, therefore, get the “theoretical” result, and we say this is because of the errors of our methods of observation; but why do we suppose that there is such a theoretical result from which our observations deviate, if our observations themselves do not in general give this ideal result? We might accumulate a great series of measurements of the angles of our triangle, and we should then find that these results would tend to group themselves symmetrically round a certain value which would be 180°. Some of the results would be considerably less than the ideal, and some of them would be considerably more; but these relatively great deviations would be small in number and most of the results would be a very little less than 180° or a very little more, and there would be as many which would be a little less as those that were a little more. We should have formed a “frequency distribution”2 with its “mode” at 180°.

But by “reasoning” about the “properties” of these lines and triangles in plane two-dimensional space, we should arrive at the conclusion that the angles of a triangle were equal to 180°, and neither more nor less. We should then think of a straight line as still a path along which we move in imagination, and a path which still has some width. But we imagine the width of the path to become less and less, so that, even if we imagine ourselves to become thinner and thinner, we should be unable to deviate either to the right or left in moving along the path, because the thinner we make ourselves the thinner becomes also the path. We imagine our intuition of a deviation to the right or left becoming keener and keener, so that, no matter how small the deviation we should still be able to appreciate it by the extra exertion which it would involve. We think of a point as a little spot, and we think of ourselves as being very small indeed, so that we can move about on this spot. But we can reduce the area of the spot more and more, until it becomes “infinitesimally” small; and at the same time we think of ourselves as becoming smaller and smaller, so that we can still move about on the spot. But we think of the area of the spot as becoming so small that no matter how small we make ourselves we are unable to move on it.

This means that we substitute conceptual lines and points and triangles for the perceptual ones of our experience, and then we operate in imagination with these concepts. That is to say, we carry our modes of exertional activity to their limits,3 in the way which we have tried to indicate above—a process of thought which is the foundation of the reasoning of the infinitesimal calculus.

What we call space, therefore, depends on our intuition of bodily exertion. This intuition includes the knowledge that a certain change has occurred as the consequence of the expenditure of a certain amount of bodily energy, and that, as the result of this change, the relation of the rest of the universe to our body has become different. We think of our body as the origin, or centre, of a system of co-ordinates:—

We imagine three lines at right angles to each other to extend indefinitely out into space, and we think of ourselves as being situated at the point of intersection of these three straight lines. If anything moves in the universe outside ourselves we can resolve this motion into three components, each of which is to be measured along one of the axes of our system of co-ordinates. But any motion whatever in the universe outside ourselves can be represented equally well by supposing that the origin of the system of co-ordinates has been changed; that is, by supposing that we have changed our position relative to the rest of the universe. Therefore motion outside ourselves is not to be distinguished from a contrary motion of our own body—a statement of the “principle of relativity”—except that any change outside ourselves may be distinguished from that compensatory change in the position of our body which appears to be the same thing, by the absence of the intuition that we have expended a certain quantity of energy in producing the change. Conscious motion of our own body is something absolute; all other motion is relative.

So far we have been speaking of our crude bodily motion, but a very little consideration will show that our knowledge of space attained by scientific measurements depends just as much on our intuition of our bodily activity, and its direction; the measurement of a stellar parallax, or that of the meridian altitude of the sun, for instance, by astronomical instruments, involves bodily exertion, though of a refined kind. Three-dimensional space, that is our space, therefore represents the manner of our activity, just as convex two-dimensional space represents the manner of the activity of the Infusorian, and one-dimensional space would represent the manner of activity of an animal which was compelled to live in a tube, the sides of which it fitted closely, so that it could move only in one direction—up and down. A parasite, living attached to some fixed object, and the movements of which were represented only by the growth of its tissues, could not form any idea of space; and the “higher” forms of geometry, that is, space of four or more dimensions, present no clear notion to our minds, even although we regard the operations included in mathematics of this kind as pure symbolism, because we cannot relate this imaginary space to any form of bodily exertion. Geometry, then, represents the manner in which our bodily exertion cuts up the homogeneous medium in which we live.

Motion, whether it be that of our own body in controlled muscular activity, or that imaginary motion of the environment which we call giddiness, or a sensibly perceived motion of some part of the environment, that is, a motion which we can compensate by some actual or imaginary change in the position of our own body produced by our own exertion, is an intuitively felt change, and is incapable of intellectual representation. It is not clearly conceived either in ancient or in modern geometry. Euclidean geometry is, as we have seen, based directly on our intuition of bodily exertion, but it is essentially static in treatment. Let it be admitted that we can draw a straight line of any length and in any direction, and so on; then we regard these straight lines, etc., as motionless, abstract things, and we proceed to discuss their relationships. Cartesian geometry, and the methods of the infinitesimal calculus, do not treat of real motion, and the concept, if it is introduced at all, is introduced illegitimately and surreptitiously. Consider what we do when we “plot a curve.” Let the latter be a parabola having the equation y =  1/2 x. Now a parabola is defined as “the locus of a point which moves, so that its distance from a fixed point is in a constant relation to its distance from a fixed straight line.” How do we construct such a curve?

We proceed to fix the positions of a series of points in this way: there are two straight lines, OX and OY, at right angles to each other, and we measure off certain steps along the line OX; these steps are OX_0·5, OX₁, OX_1·5, OX₂, and so on, the small numerals indicating the distance of each point (OX_0·5, etc.) from the origin O. We then draw lines perpendicular to the X-axis through these points. We have now to calculate one-half of the square of each of these lengths OX_0·5, OX₁, etc., and then we mark off these calculated lengths along the perpendicular lines. The point A, for instance, is  1/2(0·5)² from the point X_0·5, B is  1/2(1)² from X₁, and so on. In this way we obtain a series of points, A, B, C, D, E, etc., and these are points on the locus of the “moving” point.