# Popular Science Monthly/Volume 83/October 1913/The Fourth Dimension

(1913)
The Fourth Dimension
1580081Popular Science Monthly Volume 83 October 1913 — The Fourth Dimension1913Samuel M. Barton

 THE FOURTH DIMENSION

By Professor SAMUEL M. BARTON

UNIVERSITY OF THE SOUTH, SEWANEE, TENN.

I. Non-Euclidian Geometry

AS non-Euclidean geometry has not become popular enough to find a place in the ordinary college curriculum, and as its discovery preceded any serious consideration of space of more dimensions than three, it seems to me that, before taking up hyperspace proper, it would be well at least to mention the non-Euclidean geometries.

The mathematics of the college student is largely deductive, and he but faintly realizes the important part played by intuition, observation, induction and even imagination in the realms of higher mathematics. For instance, nothing surprises the layman—I use the word layman as including all who have not made some special study of higher mathematics—so much as to hear for the first time that the famous axiom of Euclid, namely: If two lines are cut by a third, and the sum of the interior angles on the same side of the cutting line is less than two right angles, the lines will meet on that side when sufficiently produced, is not necessarily true. And yet in very early times mathematicians began to doubt the truth of this axiom as it did not seem to be, like the rest, a simple elementary fact. The great geometer Legendre and other mathematicians attempted to give a proof of this so-called axiom, but without success. At last, to make a long story short, some mathematicians began to believe that this proposition was not only not self-evident, but was not capable of proof, and moreover that an equally consistent geometry could be built up on the supposition that it is not always true. Thus, out of various endeavors to prove Euclid's "parallel" axiom, arose non-Euclidean geometry, the beginning of which is sometimes attributed to Gauss; but as he did not publish anything on the subject, it is impossible to say what his ideas were. In the greatness of his heart, he generously gave full credit to Bolyai for his independent discoveries.

All honor, however, is due to two remarkable men, the Russian Lobatchevsky and the Hungarian Bolyai, who, about 1830, independently of each other, showed the denial of Euclid's parallel axiom led to a system of two-dimensional geometry as self-consistent as Euclid's. This new geometry is based on the assumption that through a given point a number of straight lines can be drawn parallel to a given straight line.

In 1854, the German Riemann[1] discovered another geometry. This geometry is based on the hypothesis that through a given point no straight line can be drawn parallel to a given straight line. Thus we have the three geometries: Euclid's (or the parabolic geometry), in which the "parallel axiom" holds, Lobatchevsky's (or the hyperbolic geometry), Riemann's (or the elliptic geometry). As is now well established, all three geometries are consistent with reality: Euclid's is true for a plane (a surface of zero curvature); Riemann's is true for a spherical surface (a two-dimensional space of constant positive curvature); Lobatchevsky's is true on the so-called pseudo-spherical surface of indefinite extent (a two-dimensional space of constant negative curvature). This pseudo-spherical surface is a saddle-shaped surface, like the inner surface of a solid ring.

It is to be noted that the straight line of one geometry is not the straight line of another, but in all three geometries it is the shortest distance between two points. Such straightest lines are "geodetic" lines. It will perhaps be evident now why in a sense the discovery of the non-Euclidean geometries was a stepping-stone to the consideration of hyperspace; though we should bear in mind that the two conceptions are entirely distinct, neither one being dependent upon the other. The logical conception of non-Euclidean geometry is far more difficult than the abstract notion of the fourth dimension. The study of the results arrived at by Lobatchevsky, Bolyai, Riemann, Beltrami and others forced men to think of "spaces," and it is hardly too much to say that the stimulus thus given to "high thinking" of this nature gave rise to the hypothetical acceptance of a fourth (or any higher) dimensional space.[2]

II. The Fourth Dimension

I come now to the consideration of hyperspace, which is space of any dimension above three, but for convenience and simplicity I shall confine myself mainly to fourth dimensional space.

To get any clear notion of the fourth dimension, one must make up his mind to exercise much patience, perhaps reading and re-reading many times articles by various authors. In this exposition of the subject, I would warn the reader against supposing that any attempt is here made to convince him of the possibility of the existence of fourth dimensional space. He is not even asked to believe in a material space other than our common, every day three-space. Fortunately a comparison with lower dimensional geometries furnishes so many analogies that the subject can be very fully explained in a non-mathematical way. Only let me say just here that the geometry of the fourth dimension is a perfectly logical system of theorems and proofs entirely independent of these analogies.

We, the dwellers in ${\displaystyle 3}$-space, can best realize the reasonableness of conceiving of a fourth or higher dimensional space by considering as best we may what would take place in lower-dimensional space did such exist.

Consider a pipe of indefinite length with a bore of diameter as small as you please, and suppose that there dwell within this pipe "worms" of such diameter that they just fill the pipe. We can not conceive of anything with no breadth or thickness, but let us consider for sake of the illustration that this one-dimensional animal (which for brevity I shall call a unodim) has only length. Of course these unodims may vary in length according to age or family traits, perhaps. Now it is evident that a unodim can never turn around. He may move forward or backward, but one unodim can never pass another. If he possesses an eye in front or behind he can see a neighboring unodim as a mere point. His world is a very limited one.

Again, we might imagine a two-dimensional animal, taking hold

Fig. 1

of a unodim, turning him around in his (two-dimensional) space and putting him back with his "tail" where his "head" was before. Evidently the unodim would be ignorant of the cause of his reversion, Fig. 2 for he has no knowledge of a two-dimensional space, and the two-dimensional animal is invisible to him. In other words, if ${\displaystyle AB}$ and ${\displaystyle A'B'}$ in the figure are equal in length but running in opposite directions, it is impossible to put ${\displaystyle A'B'}$ in the place of ${\displaystyle AB}$, that is, ${\displaystyle A'}$ where ${\displaystyle A}$ is and ${\displaystyle B'}$ where ${\displaystyle B}$ is. To accomplish this, it would be necessary to take ${\displaystyle A'B'}$ into ${\displaystyle 2}$-space and turn it around. While this would be an impossible feat for a unodim, a two-space animal could readily do it.

Now this one-dimensional space may not be "straight" (that is, of zero curvature); but it may be the space that we should get by bending the pipe around in the form of a circle, as in Fig. 2. In such a case, as his body would be constantly bent in the same direction and by the same degree, we may suppose that the unodim is totally unconscious of any curvature. It is well to note that in an exactly similar way our space may be curved without our being conscious of it. So he might feel just as certain that his space is "straight-line" space as we, the high and mighty ${\displaystyle 3}$-space beings, do that our space is Euclidean (or space of zero curvature).

Again, suppose the tube to be bent in an egg-shaped curve where the curvature is not constant. Here the unodim's world would still be one-dimensional, but as his body would be bent a little more in one part of his world than in another, it is possible that he may feel that there is some variety in his space. He may walk a little straighter at times and less straight at others. Whether his ${\displaystyle 1}$-space is straight or curved, and, if curved, whatever may be the variety of its convolutions, the unodim can not know of the existence of a world of ${\displaystyle 2}$-space or ${\displaystyle 3}$-space.

If a ${\displaystyle 2}$-space body, say a square, passes through his ${\displaystyle 1}$-space world, he sees only the ${\displaystyle 1}$-space section of the square.

In Fig. 3, ${\displaystyle xy}$, the ${\displaystyle 1}$-space world, is represented as being in the same

Fig. 3.

plane with the square ${\displaystyle mn}$. The square may cut ${\displaystyle xy}$ at right angles or obliquely. In any case the unodim sees at any moment only the part of the square common to his world and is not conscious that there is any more to the square.

Two-space

Next let us consider ${\displaystyle 2}$-space.

Assume a ${\displaystyle 2}$-space being, which we shall call a duodim, that is, a flat being (theoretically with no thickness) with length and breadth and confined to a surface having length and breadth but no thickness. Such a being could move to the right or left or forward or backward, we will say, but neither up nor down from the surface. In fact, he knows neither up nor down: the surface is his world.

His position in his world is easily located by the Cartesian system of coordinates, that is, with reference to its distance from, say, two straight lines at right angles to each other. For illustration, define his world as the geometrical plane formed by the two lines ${\displaystyle xx',yy'}$ intersecting each other at right angles. Employing the usual notation, we consider distances measured perpendicular to the ${\displaystyle Y}$-axis as positive if measured to the right, and negative if to the left of the ${\displaystyle Y}$-axis. Such a distance is called the abscissa of the given point. Similarly, distances measured perpendicular to the ${\displaystyle X}$-axis are positive if measured above and negative if below the ${\displaystyle X}$-axis, such a distance being called the ordinate of the given point.

Thus it is evident that a point is fully determined in the plane if its abscissa and ordinate are given. Every school boy has met with this Fig. 4. principle in the location of a place on the earth's surface by latitude and longitude, where the axes of reference are great circles of the globe.

Now our duodim has a far more extended space than the unodim, and can do many things that the unodim is totally ignorant of. His space may not necessarily be one of zero curvature,—for it is perfectly consistent with our definition of ${\displaystyle 2}$-space for it to be the surface of a sphere, of an ellipsoid, of an egg-shaped figure, or what not. It is to be noticed that if the space has constant curvature (including no curvature), a body may be moved from any place to any other place on the surface without changing its shape.

If there are (Fig. 5) two triangles like ${\displaystyle E}$ and ${\displaystyle F}$ in which the sides

Fig. 5.

are equal each to each, but are arranged in reversed order, it is impossible in ${\displaystyle 2}$-space for ${\displaystyle F}$ to be made to take the position of ${\displaystyle E}$. Here ${\displaystyle E}$ is the reflection of ${\displaystyle F}$ in a mirror. An inhabitant of ${\displaystyle 3}$-space has no difficulty, however, in taking up ${\displaystyle F}$, turning it over and putting it on the position ${\displaystyle E}$.

A three-dimensional body as such is of course invisible to a ${\displaystyle 2}$-space being. If a ${\displaystyle 3}$-space body, say a cube, crosses a ${\displaystyle 2}$-space, the ${\displaystyle 2}$-space being is conscious only of its section with his world.

Fig. 6 represents the effect of a ${\displaystyle 3}$-space body, a cube, passing through the ${\displaystyle 2}$-space pictured by the plane "${\displaystyle mn.}$" The section ${\displaystyle ABCD}$ of the plane ${\displaystyle mn}$ with the cube ${\displaystyle G}$ is all that a duodim would be conscious of. As ${\displaystyle G}$ passes through ${\displaystyle mn}$, this section, certainly if ${\displaystyle G}$ is a homogeneous body, will appear the same until suddenly it vanishes as ${\displaystyle G}$ passes beyond ${\displaystyle mn}$.

Fig. 6.

Three-space

Let us next direct our attention to ${\displaystyle 3}$-space, an inhabitant of which we might call an animal, but which, to continue the nomenclature adopted, we shall sometimes in a general way speak of as a tridim. Here freedom of life is much more augmented, even more so than in Fig. 7. passing from ${\displaystyle 1}$-space to ${\displaystyle 2}$-space. For here we have added the up-and-down motion to the right-and-left and the forward-and-backward motions. Here any point is located by means of its distances from three mutually perpendicular planes, each plane being formed by two of the three lines that can be drawn mutually perpendicular to one another. In Fig. 7, ${\displaystyle Ox,Oy,Oz}$—representing directions to the right, hitherward and upward, respectively—are the axes of reference, each being perpendicular to the other two, forming the mutually perpendicular planes, ${\displaystyle xOy,yOz,zOx}$.

We saw that in ${\displaystyle 2}$-space the axes ${\displaystyle xx',yy'}$ divided the space into four equal parts of indefinite extent. A straight line in ${\displaystyle 2}$-space divides that space into two parts. In ${\displaystyle 3}$-space, it is evident that the coordinate planes divide space into eight equal parts of indefinite extent. Any point in ${\displaystyle 3}$-space is definitely determined when its distances from the three planes of reference is known. Distances perpendicular to the ${\displaystyle yz}$ plane, denoted by ${\displaystyle x}$, are positive if measured to the right, negative if measured to the left; distances perpendicular to the ${\displaystyle xz}$ plane, denoted by ${\displaystyle y}$, are positive if measured towards us, negative if measured away from us; distances perpendicular to the ${\displaystyle xy}$ plane are positive if measured above, negative if measured below. This notation enables us to locate any point in our space.

Now we know of ${\displaystyle 2}$-space only as a section of ${\displaystyle 3}$-space, and a duodim is purely an imaginary being to us; and we know of ${\displaystyle 1}$-space only as a section of ${\displaystyle 2}$-space (and therefore of ${\displaystyle 3}$-space), and the unodim is imaginary. We have seen that a duodim might interfere with life in ${\displaystyle 1}$-space, but the unodim would not know at all what had caused the

Fig. 8.

interference. We have also seen that a tridim might in a similar way interfere with life in ${\displaystyle 2}$-space. The important point to observe is that in either case the inhabitant of the lower space would not understand what had caused the change.

A duodim could lock up his treasure in circular or polygonal vaults, such as "${\displaystyle a}$" or "${\displaystyle b}$," safe from ${\displaystyle 2}$-space intruders, but a tridim could help himself to anything he pleased without breaking the sides of the vault. By analogy, a ${\displaystyle 4}$-space being could do many things in ${\displaystyle 3}$-space impossible to man and entirely inexplicable to him. No ${\displaystyle 3}$-space safe or vault would be secure from a ${\displaystyle 4}$-space burglar. He could get a ball out of a hollow shell without breaking the surface, he could get out the

Fig. 9.

contents of an egg without cracking the shell and enjoy the kernel of a nut without the use of a nut-cracker.

A geometrical illustration similar to those already given is found in Fig. 9. Here "${\displaystyle a}$" and "${\displaystyle b}$" are symmetrical tetrahedrons,[3] in length ${\displaystyle AB=A'B',AC=A'C',BC=B'C',AD=A'D',BD=B'D',}$ ${\displaystyle CD=CD'}$. It is evidently impossible in ${\displaystyle 3}$-space to put "${\displaystyle b}$" in the position "${\displaystyle a}$" or vice versa. It would be possible to make "${\displaystyle b}$" coincide with the image of "${\displaystyle a}$" in a mirror. In fact it is obvious that "${\displaystyle b}$" is the image of "${\displaystyle a}$" as seen in a mirror.

Fig. 10. Readers of that classic nonsense book by Lewis Carroll (Rev. C. L. Dodgson), "Alice Behind the Looking-Glass," will be interested in the fact that Mr. Dodgson, himself a mathematician of no mean note, is poking fun at the fourth-dimension students.

Now, while it is impossible for a tridim to make "${\displaystyle b}$" take the position "${\displaystyle a}$," there would be no difficulty in a fourth-dimensional animal interchanging "${\displaystyle a}$" and "${\displaystyle b}$." In other words, to make "${\displaystyle a}$" and "${\displaystyle b}$" coincide, one must be taken up into ${\displaystyle 4}$-space, turned over and put down on the other.

It is' easy now to see that, while there is no proof of the material existence of ${\displaystyle 4}$-space or space of any dimension higher than three, and while we can not even say that there is any likelihood that such exists, yet the conception of hyperspace is a perfectly real and logical conception; moreover, it is by no means an idle question or a useless idea. Assuming hyperspace, mathematicians have built up a perfectly consistent geometry which throws much light upon problems of ${\displaystyle 3}$-space.

We have seen that by many analogies it is a simple matter to conceive of hyperspace. Let us next observe how algebra invites us to consider the possible existence of higher space.

The solution of two simultaneous equations in two variables ${\displaystyle x,y}$, gives us a point in a plane. The solution of three simultaneous equations in three variables ${\displaystyle x,y,z}$, gives us a point in ${\displaystyle 3}$-space. The solution of four simultaneous equations in four variables, ${\displaystyle x,y,z,w}$, which is easily performed, gives what? Is there a geometrical equivalent here? Can the values of ${\displaystyle x,y,z,w}$ be represented graphically? The answer to both questions is No, at least not in our space. Four-space is necessary if we are to give a geometrical representation to the solution of four simultaneous equations, such as:

${\displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}w=e_{1},}$
${\displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}w=e_{2},}$
${\displaystyle a_{3}x+b_{3}y+c_{3}z+d_{3}w=e_{3},}$
${\displaystyle a_{4}x+b_{4}y+c_{4}z+d_{4}w=e_{4},}$

Again,

${\displaystyle x=a}$

represent a point in ${\displaystyle 1}$-space. [Incidentally it would also denote a line in ${\displaystyle 2}$-space and a plane in ${\displaystyle 3}$-space.]

The equation

${\displaystyle ax+by=c}$

represents a line in ${\displaystyle 2}$-space, but has no meaning in ${\displaystyle 1}$-space.

The equation

${\displaystyle ax+by+cz=d}$

represents a plane in ${\displaystyle 3}$-space, but has no meaning in ${\displaystyle 2}$-space or ${\displaystyle 1}$-space.

So, by analogy, the equation

${\displaystyle ax+by+cz+dw=e}$

would have a meaning in ${\displaystyle 4}$-space,—say a ${\displaystyle 3}$-space section of ${\displaystyle 4}$-space—but has no meaning in ${\displaystyle 3}$-space.

In general, an algebraic equation of ${\displaystyle k}$ variables has no meaning in a space of lower dimension than ${\displaystyle k}$, but has a meaning in ${\displaystyle n}$-space, where ${\displaystyle n>=k}$.

Discarding experience and reasoning wholly from analogy, we introduce some properties of the fourth dimension as follows.

Four-dimensional measure depends upon length, breadth, height and a fourth dimension all multiplied together. In the graphical representation of ${\displaystyle 3}$-space, points are referred to three mutually perpendicular planes formed by three lines mutually at right angles. In a similar way, to represent ${\displaystyle 4}$-space we must assume another axis at right angles to each of the other three. In the present development of human thought, this is purely subjective, a mere mental conception, and it is upon this conception that the theory of hyperspace is built.

The position of a point in a plane may be determined, as we have seen, by its distance from each of two perpendicular right lines; in ${\displaystyle 3}$-space, by its distance from each of three mutually perpendicular planes; and in ${\displaystyle 4}$-space, by its distance from each of four mutually perpendicular ${\displaystyle 3}$-spaces, for there are four arrangements of the four axes taken three at a time, and each independent set of three perpendicular axes define a ${\displaystyle 3}$-space, for example, ${\displaystyle wxy,wxz,wyz,xyz}$. Just as in our space it requires at least three points to determine a plane (${\displaystyle 2}$-space), so in ${\displaystyle 4}$-space four points are necessary to determine a ${\displaystyle 3}$-space.

As portions of our space are bounded by surfaces, plane or curved, so portions of ${\displaystyle 4}$-space are bounded by hyperspace (three-dimensional).

In our space, a point moving in an unchanging direction generates a straight line.

This straight line (say of ${\displaystyle a}$ units in length), moving perpendicular to its initial position through the distance a, generates a square.

This square, moving perpendicular to its initial position through the distance ${\displaystyle a}$, generates a cube.

This cube, we will suppose, moving perpendicular to our space for a distance equal to one of its sides (that is, equal to ${\displaystyle a}$), will generate a hypercube.

Now the line contains ${\displaystyle a}$ units, the square ${\displaystyle a^{2}}$ units, the cube ${\displaystyle a^{3}}$ units, the hypercube ${\displaystyle a^{4}}$ units.

Again, to repeat in words slightly different from the foregoing (Fig. 11) considering the ${\displaystyle a}$ units as ${\displaystyle a}$ points (an indefinite number), the square ${\displaystyle ABCD}$ is derived from the line ${\displaystyle AB}$, which for convenience suppose to be one foot in length, by letting ${\displaystyle AB}$ with its a points move through a distance of one foot in a direction perpendicular to itself, that is, perpendicular to the one dimension of ${\displaystyle AB}$, every point of ${\displaystyle AB}$ describes a line, and ${\displaystyle ABCD}$ contains therefore ${\displaystyle a}$ lines and ${\displaystyle a^{2}}$ points.

Fig. 11.

The cube ${\displaystyle ABCD-G}$ is derived from the square ${\displaystyle ABCD}$ which moves one foot in a direction perpendicular to its two dimensions, its ${\displaystyle a}$ lines and ${\displaystyle a^{2}}$ points describing ${\displaystyle a}$ squares and ${\displaystyle a^{2}}$ lines respectively. The cube ${\displaystyle ABCD-G}$ therefore contains ${\displaystyle a}$ squares, ${\displaystyle a^{2}}$ lines and ${\displaystyle a^{3}}$ points.

Similarly, the four-dimensional unit is derived from the cube, ${\displaystyle ABCD-G}$, by letting that cube move one foot in a direction perpendicular to each of its three dimensions, that is, in the direction of the fourth dimension; its ${\displaystyle a}$ squares, ${\displaystyle a^{2}}$ lines, and ${\displaystyle a^{3}}$ points describing respectively ${\displaystyle a}$ cubes, ${\displaystyle a^{2}}$ squares, ${\displaystyle a^{3}}$ lines. The hypercube, therefore, contains ${\displaystyle a}$ cubes, ${\displaystyle a^{2}}$ squares, ${\displaystyle a^{3}}$ lines and ${\displaystyle a^{4}}$ points.

Boundaries

Now, as to the boundaries of the units, ${\displaystyle AB}$ has two bounding points, ${\displaystyle ABCD}$ has four, two each from the initial and the final position of the moving line, ${\displaystyle ABCD-G}$ has eight,—four each from the initial and the final position of the moving square,—and the hypercube[4] has sixteen,—eight each from the initial and the final position of the moving cube.

Bounding Lines.—Of bounding lines, ${\displaystyle AB}$ has one (or is itself one), ${\displaystyle ABCD}$ has 4, one each from the initial and the final position of the moving line and 2 generated by the 2 bounding points of that line; ${\displaystyle ABCD-G}$ has 12,—4 each from the initial and the final position of the moving square and 4 generated by the 4 bounding points of that square; and the hypercube has 32,—12 each from the initial and the final position of the moving cube and 8 generated by the 8 bounding points of that cube.

Bounding Squares.—Of bounding squares, ${\displaystyle ABCD}$ has one (itself); ${\displaystyle ABCD-G}$ has 6,—one each from the initial and the final position of ${\displaystyle ABCD}$, and 4 described by the bounding lines of the moving square; and the hypercube has 24,—6 each from the initial and the final position of the moving cube, and 12 described by the bounding lines of the moving cube.

Bounding Cubes.—Finally, of bounding cubes, ${\displaystyle ABCD-G}$ has one (itself); and the hypercube has 8,—one each from the initial and the final position of the moving cube, and 6 described by the bounding squares of the moving cube.

The results obtained for the boundaries may be conveniently exhibited by the following table:

Boundaries

 Points Lines Squares Cubes One-dimensional unit 2 1 0 0 Two-dimensional unit 4 4 1 0 Three-dimensional unit 8 12 6 1 Four-dimensional unit 16 32 24 8

Freedom of movement is greater in hyperspace than in our space. The degrees of freedom of a rigid body in our space are 6, namely, 3 translations along and 3 rotations about 3 axes, while the fixing of 3 of its points, not in a straight line, prevents all movement. In hyperspace, however, with 3 of its points fixed, it could still rotate about the plane of those 3 points. A rigid body has 10 possible different movements in hyperspace, namely, 4 translations along 4 axes, and 6 rotations about 6 planes, while at least 4 of its points must be fixed to prevent all movement.

In hyperspace, a sphere of flexible material could without stretching or tearing be turned inside out. Two links of a chain could be separated without breaking them. Our knots would be useless. In hyperspace, as we have seen, it would be entirely possible to pass in and out of a sphere

Fig. 12.

(or other enclosed space). A right glove turned over through space of four dimensions becomes a left glove, but notice that when the glove is turned over, it is not turned inside out.[5] This may be made clear by analogy. Suppose we have in a plane (Fig. 12) a nearly closed polygon ${\displaystyle ABCDEF}$. This can be turned into its symmetrical form ${\displaystyle A'BCD'E'F'}$, the lower half of (a), by opening it out straight and bending it over the other way so that it is turned inside out. This process takes place entirely in the plane and can be performed by a two-dimensional being. The polygon may also be changed into its symmetrical form (b), Fig. 12, by being turned over, in ${\displaystyle 3}$-space, but in this process it is not turned inside out at all. On the other hand, if it is sufficiently flexible, it may be turned inside out by twisting each part upon itself through 180 degrees, and in this process it is not changed into its symmetrical form.

When mathematicians began to talk of higher space, the spiritualists seized upon the idea as affording a habitation for their spirits. These men, naturally wanting a home for their spirits, were rather too eager to believe in the actual existence of the fourth dimension. It is astonishing with what avidity the advocates of spirit rappings and occult demonstrations appropriated the fourth dimension for the abiding place of their unearthly beings. This was, of course, unwarranted as are perhaps most of the claims of such people. While somewhat interesting, it is too trivial to claim our serious attention.

In conclusion, we have no material evidence of a fourth dimension. Our knowledge of the phenomena of ${\displaystyle 3}$-space is empirical. Our experience tells us nothing of ${\displaystyle 4}$-space, if it exists. But the conception, not being dependent upon experience or experiment, is not unreasonable. As a working hypothesis it is not without decided value, as it throws light upon many propositions of our (${\displaystyle 3}$-space) geometry.

The existence of ${\displaystyle 4}$-space might explain certain phenomena in physics and chemistry; for instance, rotation in hyperspace would explain the changes of a body producing a right-handed polarization of light into one giving a left-handed.

A few months ago an article appeared in the Scientific American by E. L. DuPuy setting forth the use of four dimensions in representing certain chemical compounds graphically. He took as an example a "special steel" consisting of iron, carbon, silicon-manganese and nickel-vanadium.

In this short sketch of what is meant by the fourth dimension, it must be borne in mind that the mathematical investigation of the geometry of the fourth dimension has been omitted altogether. It is hardly necessary to add that all arguments for the existence of a fourth dimension apply equally well for the existence of 5, 6, or ${\displaystyle n}$ dimensional space. The geometry of ${\displaystyle n}$-space, where ${\displaystyle n}$ is any number, is just as logical as that of ${\displaystyle 4}$-space.

[In this paper the author claims no originality, except to some extent in the mode of presentation and in the manner of introducing the illustrations; but he has not knowingly made use of any ideas that have not now become public property. The following are some of the works that he has consulted: "Non-Euclidean Geometry," by Henry P. Manning; "The Elements of Non-Euclidean Geometry," by L. L. Coolidge; "The Fourth Dimension Simply Explained—a collection of Popular Essays with an Introduction and Editorial Notes by the Editor," Henry P. Manning, editor (published by the Scientific American): "The Fourth Dimension," by C. H. Hinton; "The Fourth Dimension," in "Mathematical Essays and Recreations," by Hermann Schubert, translated by T. J. McCormack.]

1. Riemann, "Ueber die Hypothesen seiche der Geometrie zu Grunde liegen," first read in 1854.
2. Lobatschevsky's "The Theory of Parallels" and Bolyai's "The Science Absolute of Space" were translated into English by George Bruce Halsted and first appeared in Scientiæ Baccalaureus, a journal published for a short time by the Missouri School of Mines. By this and other publications Professor Halsted did much to popularize non-Euclidean geometry. Perhaps the most available short treatise on the subject in America is Professor Henry P. Manning's "Non-Euclidean Geometry."
3. A model of "${\displaystyle a}$" and "${\displaystyle b}$" can be readily constructed as follows:

Cut out the figure (Fig. 10) from a piece of cardboard, perforated along the lines ${\displaystyle AB,BC,CA}$, and having ${\displaystyle AF=AE,CE=CD}$ and ${\displaystyle BD=BF}$. Fold over the triangle ${\displaystyle ABF,ACE,CBD}$ till the points ${\displaystyle F,E}$ and ${\displaystyle D}$ meet in a point, thus making one tetrahedron: fold the triangles in the opposite direction and the symmetrical tetrahedron will be formed. The one corresponds to the image of the other in a mirror.

4. This four-dimension unit is often called the "tesseract."
5. A right glove turned inside out in our space becomes a left glove.