A ship in a canal could be located at any given time by a knowledge of its distance from some town, since its motion from that town has been restricted to one direction. When space is of such a nature that a point in it may be located by one measurement from some fixed or standard point, that space is said to be linear or one-dimensional.

   The same boat on the ocean, however, could not be located unless two measurements were given -- its latitude and longitude. The nature of such space is defined by the words "surface" or "two-dimensional area."

   If, now, our vessel were converted into an airship or a submarine, we should be obliged to add to our other data its distance above or below the sea level in order to place it accurately. With three basic elements (in our illustration; the equator, the prime meridian, and the sea level) and with three known distances from these elements, we can locate any point that comes within our consciousness, whether above, on, or below the surface of the earth. Any additional measurements would be either superfluous or misleading. Hence we say that our space is three-dimensional.

   In this discussion it will be necessary for us to use graphic representations of changes in one-dimensional, two-dimensional, and three-dimensional space, and for

   The rising and falling of the mercury is a one-dimensional movement. If we wish to keep an automatic record of the temperature during a given period, it would be an easy matter to pass a strip of photographic paper behind a thermometer, and allow the sun or some artificial light to darken the part above the mercury. If this paper were kept stationery, the only record we could obtain would be that of the minimum height of the mercury. Therefore, some movement of the strip is necessary. If this motion were to be in the length direction of the thermometer, every part of the paper would be exposed to the action of the light, and no record at all would be obtained. We could obviate this trouble, however, by covering the strip while it moved through a distance equal to the length of the thermometer, then exposing it for a short time, and then again moving it. Thus, without involving a second dimension, we would get a permanent record of various successive heights of the mercury. These pictures would be intermittent, and we would miss the changes that took place while the picture film was moving. In order to get a complete and continuous chart of the changes, we must move the paper in a direction other than that of the length of the thermometer. In short, we are forced to introduce a second dimension. The strip may be moved by clock work, and then we would have a two-dimensional chart, from which we could determine the temperature at any required time, the horizontal measurement

   It is not difficult to imagine a being whose percepts are confined to a linear representation of objects; for instance, a man whose sense of touch is paralyzed and whose eye is covered by a cataract in which a vertical slit has been successfully cut. Better yet, we may conceive of one whose retina itself is merely a line instead of a spherical surface. He could not imagine such a thing as an angle, and it would be as hard to explain parallel lines to him as to describe color to a man born blind. He could see the changes in the height of the mercury just as well as we, but a triangle passed before his line of vision would present the same sort of picture, viz., a line increasing in length; and there would be no way of convincing him of the simultaneous existence of all its parallel elements, which to us is a very simple concept. He could, however, picture from his memory, and re-produce, two or more lines which represent the height of the mercury at different times, but they would all lie in his one-dimensional consciousness as separate pictures.

   His knowledge of a growing tree would be confined to a line with various colored parts which change, both as the tree grows and as he moves his line of vision, but the most complex of these changes could be reproduced by a picture on a plane surface, slowly passed before his eye. In brief, such a being could

   When we come to consider changes in two dimensions, such, for instance, as are caused by the motion of the arms of a semaphore, how are we to represent them. A series of photographs might be taken in rapid succession, and if these were placed behind each other, a solid would be formed of which we might say each picture was a cross section. A book made up of these; pictures in their order is such a solid, and the little pocket mutoscope exactly satisfies this description. If its pages are rapidly turned, the successive sections are presented to our sight, and we apparently see the arms-of the semaphore changing their position. The kinetoscope with its two-dimensional strip and its shutter does the same thing more steadily, and presents the illusion of motion in a two-dimensional area even better than the little hand mutoscope. The pictures taken by the mutograph are really always two-dimensional; it is only our experience in shadow and perspective which gives us the illusion of motion in three dimensions when the ordinary "moving picture" is thrown on the screen. If we left the camera film unmoved while the semaphore was moving, only a picture of the stationary parts would be taken, the rest would be a blur. Hence we must move our picture film.

   If we move it continuously, no record of any position of the semaphore will be taken. Here again we must obviate the difficulty by shutting out the light while the film moves over a distance equal to the size of the picture it is to take, then exposing it, and then covering it again. But no matter how quickly the camera shutter is snapped, the representations of the

   If an imaginary being with a two-dimensional sense, an "Inhabitant of Flat-Land," were to have this solid passed through his plane, he would see reproduced the continuous motion of the semaphore arms. Like our slit-eyed friend, the "Line-lander," and for analogous reasons, he could not conceive the simultaneous existence of all these cross sections. But by using his memory, he could reproduce some of them as separate pictures in his two-dimensional world -- such pictures, perhaps, as we have in our kinetoscope film.

   If a small quantity of yeast were allowed to ferment between the slide and cover glass of a microscope, we should have under our observation the growth of an object in practically two dimensions. Now, its phases at very small intervals could be photographed, but the same conditions that met us in the case of the semaphore, face us again. The only way to represent all the changes that take place would involve the tracing of each point from one position to another. This would produce a line; and since two dimensions are required to present all the points in their relative positions at any given time, this line, in order not to he obscured, must extend beyond the two-dimensional space in which the growth takes place. We must, therefore, create a solid, whose successive sections would be recognized by the two-dimensional mind as the growth of the object which was passing through the plane of their consciousness.

   In our previous illustrations we were able by the use of two-dimensional space to fix permanently variations of position and magnitude of a one-dimensional object, and in three-dimensional space we were able to fix permanently the changes of an object moving or growing in two dimensions.

   Coming now to the phenomena of our every-day world, we know that changes in position and growth take place continuously in our three-dimensional space, and that the time element is necessary to determine exactly the conditions of any variable or movable thing. Thus the description of a tree would give an entirely false impression, if only its dimensions were given without adding the particular time when these were taken; and the position of a planet would be incompletely given, unless the time of observation were

   If we could only picture to ourselves that a three-dimensional object is merely the cross section of a permanent four dimensional thing, that what we are cognizant of is merely a phase of a thing which exists in its entirety, and of whose other phases we are ignorant, till they are brought to our own consciousness or till our consciousness reaches them, then we could conceive the physical nature of a four-dimensional object. Considering, for instance, our own material bodies, we are conscious of a gradual change of shape and position of all the parts, and yet, at the same time, we are conscious of a continuing identity throughout all these changes. Our past experiences are as real as the experiences we are now undergoing. Those past experiences, or phases of our existence, are as much a part of us as the present ones, and yet owing to the limitations of our three-dimensional consciousness we can reproduce past conditions only in memory. Nevertheless our lives in their completeness are made up of the sum of all our experiences and if our whole lives are considered as units, and each period of which we are conscious requires a three-dimensional space, then each individual may be considered as a four-dimensional solid.

   Let us, however, take a more simple illustration. A biologist wishes to present to his class a concrete

   Now, although every particle of the living jellyfish is constantly changing, either in size, or position, or in its relation to neighboring particles, we say it is the same jellyfish; there is a something that persists through all the changes; an individuality which differentiates this animal from all others, although to-day it is as different from what it was previously as any two models.

   These models may be considered copies of mere phases of the jellyfish, just as photographs may be said to represent phases of the fermenting yeast, and two separate lines may be said to represent corresponding phases of the mercury length in the thermometer.

   But no matter how small the interval which elapses between the making of two successive models, if there be any change at all, that change must have involved many, nay an infinite number, of smaller changes, and these changes in the case of each atom of the living organism must have been continuous; that is, they must be represented by a line, and not by a succession of separated points, if we would preserve the individuality of the animal in question.

   Now this line cannot be represented in our three-dimensional space without interfering with other atoms which surround it in three directions. We are compelled, therefore, as in the previous illustrations, to go outside the space in which the change takes place, in order to represent completely the continuous change in anything which preserves its individuality while

   Mind you, I do not say that a growing jellyfish is necessarily a fixed four-dimensional object, passing through three-dimensional space, but I do say it could be so represented; and that then a four-dimensional mentality could see any or all of its three-dimensional phases simultaneously, just as we can in a two-dimensional chart perceive simultaneously all the lengths of a varying line. To get a vague conception of such a four-dimensional figure, it is necessary for us to group all our three-dimensional memories of some changing object between two definite times, and imagine them merged into a something of such a nature that no part of one memory picture overlaps a different part of another, and yet that each of these concepts is itself complete. This is, of course, impossible to most of us, but so are many other mathematical and physical concepts.

   More scientific but somewhat similar considerations than those quoted above, have forced all the great mathematicians and many great physicists to accept the fourth dimension as a solution of many difficulties. Its use is recognized, almost unconsciously, even by the elementary student, when he computes the area of a triangle, for here he multiplies four dimensions and extracts their square root to obtain a two-dimensional result, namely,
s(s-a) (s-b) (s-c). Furthermore, this theory lends itself to the simplification of many physical and metaphysical problems. Therefore, its adherents

   The transparency of the jellyfish was the exceptional feature which permitted its use to illustrate a three-dimensional object whose changes could be studied without dissecting it.

   Our three-dimensional concepts generally are mere inferences from our two-dimensional knowledge, and we are easily deluded by our senses in forming them. When our knowledge of solids becomes as nearly perfect as our present knowledge of surfaces, then the vague four-dimensional figure may assume a more concrete form. Will this ever happen? Who can tell? Many more revolutionary theories have found concrete expression, and then obtained a firm foothold against stronger opposition and with less necessity for their existence.