Chapter VI: completes the preliminary course in the fundamentals of (5)

With the postulates embodied in his definitions, those stated in his axioms, and those which he reads into his structure by his methods of proof, Euclid has a categorical set--enough to serve as foundation for a geometry. We may then climb into Euclid's shoes and take the next step with him. We follow him while he proves a number of things about intersecting lines and about triangles. To be sure, when he proves that two triangles are identically constituted by moving one of them over on top of the other, we may protest on the ground that the admission of motion, especially of motion thus imposed from without, into a geometry of things is not beyond dispute. If Euclid has caught our modern viewpoint, he will rejoin that if we have any doubts as to the admissibility of motion he will lay down a postulate admitting it, and we shall be silenced.

Having exhausted for the present the interest of intersecting lines, our guide now passes to a consideration of lines in the same plane that never meet. He defines such lines as parallel. If we object that he should show the existence of a derived concept like this before laying down a definition that calls for it to exist, he can show that two lines drawn perpendicular to the same line never meet. He will execute this proof by a special sort of superposition, which requires that the plane be folded over on itself, through the third dimension of surrounding space, rather than merely slid along upon itself.

We remain quiet while Euclid demonstrates that if two lines are cut by any transversal in such a way as to make corresponding angles at the two intersections equal, the lines are parallel. It is then in order to investigate the converse: if the lines are parallel to begin with, are the angles equal?

AXIOMS MADE TO ORDER

This sounds innocent enough; but in no way was Euclid able to devise a proof--or, for that matter, a disproof. So he took the only way out, and said that if the lines were parallel, obviously they extended in the same direction and made the angles equal. The thing was so obvious, he argued, that it was really an axiom and he didn't have to prove it; so he stated it as an axiom and proceeded. He didn't state it in precisely the form I have used; he apparently cast about for the form in which it would appear most obvious, and found a statement that suited him better than this one, and that comes to the same thing. This statement tells us that if the transversal makes two corresponding angles unequal, the lines that it cuts are not parallel and do meet if sufficiently prolonged. But wisely enough, he did not transplant this axiom, once he had arrived at it, to the beginning of the book where the other axioms were grouped; he left it right where it was, following the proposition that if the angles were equal the lines were parallel. This of course was so that it might appeal back, for its claim to obviousness, to its demonstrated converse of the proposition.

Euclid must have been dissatisfied with this cutting of the Gordian knot; his successors were acutely so. For twenty centuries the parallel axiom was regarded as the one blemish in an otherwise perfect work; every respectable mathematician had his shot at removing the defect by "proving" the objectionable axiom. The procedure was always the same: expunge the parallel axiom, in its place write another more or less "obvious" assumption, and from this derive the parallel statement more or less directly. Thus if we may assume that the sum of the angles of a triangle is always exactly 180 degrees, or that there can be drawn only one line through a given point parallel to a given line, we can prove Euclid's axiom. Sometimes the substitute assumption was openly made and stated, as in the two instances cited; as often it was admitted into the demonstration implicitly, as when it is quietly assumed that we can draw a triangle similar to any given triangle and with area as great as we please, or when parallels are "defined" as everywhere equidistant. But such "proofs" never satisfied anyone other than the man who made them; the search went merrily on for a valid "proof" that should not in substance assume the thing to be proved.

LOCATING THE DISCREPANCY

Saccheri, an Italian Jesuit, would have struck bottom if he had had a little more imagination. He gave an exhaustive reductio ad absurdum, on the basis of the angle-sum theorem. This sum must be (a) greater than or (b) equal to or (c) less than 180 degrees. Saccheri showed that if one of these alternatives occurs in a single triangle, it must occur in every triangle. The first case gave little trouble; admitting the possibility of superposing in the special manner mentioned above, which he did implicitly, he showed that this "obtuse-angled hypothesis" contradicted itself. He pursued the "acute-angled hypothesis" for a long time before he satisfied himself that he had caught it, too, in an inconsistency. This left only the "right-angled hypothesis," proving the Euclidean angle-sum theory and through it the parallel postulate. But Saccheri was wrong: he had found no actual contradiction in the acute-angled hypothesis--for none exists therein.

The full facts were probably first known to Gauss, who had a finger in every mathematical pie that had to do with the transition to modern times. They were first published by Lobatchewsky, the Russian, who anticipated the Hungarian John Bolyai by a narrow margin. All three worked independently of Saccheri, whose book, though theoretically available in Italian libraries, was actually lost to sight and had to be rediscovered in recent years.

Like Saccheri, Lobatchewsky investigated alternative possibilities. But he chose another point of attack: through a given point it must be possible to draw, in the same plane with a given line (a) no lines or (b) one line or (c) a plurality of lines, which shall not meet the given line. The word parallel is defined only in terms of the second of these hypotheses, so we avoid it here. These three cases correspond, respectively, to those of Saccheri.

The first case Lobatchewsky ruled out just as did Saccheri, but accepting consciously the proviso attached to its elimination; the third he could not rule out. He developed the consequences of this hypothesis as far as Euclid develops those of the second one, sketching in a full outline for a system of geometry and trigonometry based on a plurality of "non-cutters." This geometry constitutes a coherent whole, without a logical flaw.

This made it plain what was the matter with Euclid's parallel axiom. Nobody could prove it from his other assumptions because it is not a consequence of these. True or false, it is independent of them. Trinity Church is in New York, Faneuil Hall is in Boston, but Faneuil Hall is not in Boston because Trinity is in New York; and we could not prove that Faneuil Hall was in Boston if we knew nothing about America save that Trinity is in New York. The mathematicians of 2,000 years had been pursuing, on a gigantic scale, a delusion of post hoc, ergo propter hoc.

WHAT THE POSTULATE REALLY DOES

Moreover, in the absence of an assumption covering the ground, we shall not know which of the alternatives (a), (b), (c) holds. But when one holds in a single case it holds permanently, as Saccheri and Lobatchewsky both showed. So we cannot proceed on this indefinite basis; we must know which one is to hold. Without the parallel postulate or a substitute therefor that shall tell us the same thing or tell us something different, we have not got a categorical set of assumptions--we cannot build a geometry at all. That is why Euclid had to have his parallel postulate before he could proceed. That is why his successors had to have an assumption equivalent to his.

The reason why it took so long for this to percolate into the understanding of the mathematicians was that they were thinking, not in terms of the modern geometry and about undefined elements; but in terms of the old geometry and about strictly defined and circumscribed elements. If we understand what is meant by Euclidean line and plane, of course the parallel postulate, to use the old geometer's word, is true--of course, to adopt the modern viewpoint, if we agree to employ an element to which that assumption applies, the assumption is realized. The very fact of accepting the "straight" line and the "flat" plane of Euclid constitutes acceptance of his parallel postulate--the only thing that can separate his geometry from other geometries. But of course we can't prove it; the prior postulates which we would have to use in such an attempt apply where it does not apply, and hence it cannot possibly be consequences of.

To all this the classical Euclidean rejoins that we seem to have in mind elements of some sort to which, with one reservation, his postulates apply. He wants to know what these elements look like. We can, and must, produce them--else our talk about generality is mere drivel. But we must take care that the Euclidean geometer does not try to apply to our elements the notions of straightness and flatness which inhere in the parallel postulate. We cannot satisfy and defy that postulate at the same time. If we do not insist on this point, we shall find that we are reading non-Euclidean properties into Euclidean geometry, and interpreting the elements of the latter as straight lines that are not straight, flat planes that are not flat. It is not the mission of non-Euclidean geometry thus to deny the possibility of Euclidean geometry; it merely demands a place of equal honor.

THE GEOMETRY OF SURFACES

Let us ask the Euclidean geometer whether he can recognize his plane after we have crumpled it up like a piece of paper en route to the waste basket. He will hesitate only long enough to recall that in the special case of superposition he has reserved for himself the privilege of deforming his own plane, and to realize that he can always iron his plane out smooth again after we are through with it. This emphasizes the true nature of the two-dimensionality which is the fundamental characteristic of the plane (and of other things, as we shall directly see). The plane is two-dimensional in points not because two sets of mutually perpendicular Euclidean straight lines can be drawn in it defining directions of north-south and east-west, but because a point in it can be located by means of two measures. The same statement may be made of anything whatever to which the term "surface" is applicable; anything, however crumpled or irregular it be, that possesses length and breadth without thickness. The surface of a sphere, of a cylinder, of an ellipsoid, of a cone, of a doughnut (mathematically known as a torus), of a gear wheel, of a French horn, all these possess two-dimensionality in points; on all of them we can draw lines and curves and derive a geometry of these figures. If we get away from the notion that geometry of two dimensions must deal with planes, and adopt in place of this idea the broader restriction that it shall deal with surfaces, we shall have the generalization which the Euclidean has demanded that we produce, and the one which in the hands of the modern geometer has shown results.

In this two-dimensional geometry of surfaces in general, that of the plane is merely one special case. Certain of the features met in that case are general. If we agree that we know what we mean by distance, we find that on every surface there is a shortest distance between two points, together with a series of lines or curves along which such distances are taken. These lines or curves we call geodesics. On the plane the geodesic is the straight line. On surfaces in general the geodesic, whatever its particular and peculiar shape, plays the same rôle that is played by the straight line in the plane; it is the secondary element of the geometry, the surface itself and all other surfaces of its type are the tertiary elements. And it is a fact that we can take all the possible spheres, or all the possible French-horn surfaces, and conceive of space as we know it being broken down by analysis into these surfaces instead of into planes. The only reason we habitually decompose space into planes is because it comes natural to us to think that way. But geometric points, lines and surfaces must be recognized as abstractions without actual existence, for all of them lack one or more of the three dimensions which such existence implies. These figures exist in our minds but not in the external world about us. So any decomposition of space into geometric elements is a phenomenon of the mind only; it has no parallel and no significance in the external world, and is made in one way or in another purely at our pleasure. There isn't a true, honest-to-goodness geometrical plane in existence any more than there is an honest-to-goodness spherical surface: so on intrinsic grounds one decomposition is as reasonable as another.

Certain of the most fundamental postulates are obeyed by all surfaces. As we attempt to discriminate between surfaces of different types, and get, for instance, a geometry that shall be valid for spheres and ellipsoids but not for conicoids in general, we must do so by bringing in additional postulates that embody the necessary restrictions. A characteristic shared by planes, spheres, and various other surfaces is that the geodesics can be freely slid along upon themselves and will coincide with themselves in all positions when thus slid; with a similar arrangement for the surface itself. But the plane stands almost unique among surfaces in that it does not force us to distinguish between its two sides; we can turn it over and still it will coincide with itself; and this property belongs also to the straight line. It does not belong to the sphere, or to the great circles which are the geodesics of spherical geometry; when we turn one of these over, through the three-dimensional space that surrounds it, we find that the curvature lies in the wrong way to make superposition possible. If we postulate that superposition be possible under such treatment, we throw out the sphere and spherical geometry; if we postulate that superposition be only by sliding the surface upon itself we admit that geometry--as Saccheri failed to see, as Lobatchewsky realized, and as Riemann showed at great length in rehabilitating the "obtuse-angled hypothesis." Lobatchewsky's acute-angled geometry is realized on a surface of the proper sort, which admits of unrestricted superposition; but it is not the sort of a surface that I care to discuss in an article of this scope.

Euclidean geometry is the natural and easy one, I suppose, because it makes it easy to stop with three dimensions. If we take a secondary element, a geodesic, which is "curved" in the Euclidean sense, we get a tertiary element, a surface, which is likewise curved. Then unless we are to make an altogether abrupt and unreasonable break, we shall find that just as the curved geodesic generated a curved surface, the curved surface must give rise to a "curved space"; and just as the curved geodesic needed a second dimension to curve into, and the curved surface a third, so the curved three-space requires a fourth. Once started on this sort of thing, there doesn't really seem to be any end.

EUCLIDEAN OR NON-EUCLIDEAN

Nevertheless, we must face the possibility that the space we live in, or any other manifold of any sort whatever with which we deal on geometric principles, may turn out to be non-Euclidean. How shall we finally determine this? By measures--the Euclidean measures the angles of an actual triangle and finds the sum to be exactly 180 degrees; or he draws parallel lines of indefinite extent and finds them to be everywhere equally distant; and from these data he concludes that our space is really Euclidean. But he is not necessarily right.

We ask him to level off a plot of ground by means of a plumb line. Since the line always points to the earth's center, the "level" plot is actually a very small piece of a spherical surface. Any test conducted on this plot will exhibit the numerical characteristics of the Euclidean geometry; yet we know the geometry of this surface is Riemannian. The angle-sum is really greater than 180 degrees; lines that are everywhere equidistant are not both geodesics.

The trouble, of course, is that on this plot we deal with so minute a fraction of the whole sphere that we cannot make measurements sufficiently refined to detect the departure from Euclidean standards. So it is altogether sensible for us to ask: "Is the universe of space about us really Euclidean in whatever of realized geometry it presents to us? Or is it really non-Euclidean, but so vast in size that we have never yet been able to extend our measures to a sufficiently large portion of it to make the divergence from the Euclidean standard discernible to us?"

This discussion is necessarily fragmentary, leaving out much that the writer would prefer to include. But it is hoped that it will nevertheless make it clear that when the contestants in the Einstein competition speak of a non-Euclidean universe as apparently having been revealed by Einstein, they mean simply that to Einstein has occurred a happy expedient for testing Euclideanism on a smaller scale than has heretofore been supposed possible. He has devised a new and ingenious sort of measure which, if his results be valid, enables us to operate in a smaller region while yet anticipating that any non-Euclidean characteristics of the manifold with which we deal will rise above the threshold of measurement. This does not mean that Euclidean lines and planes, as we picture them in our mind, are no longer non-Euclidean, but merely that these concepts do not quite so closely correspond with the external reality as we had supposed.

As to the precise character of the non-Euclideanism which is revealed, we may leave this to later chapters and to the competing essayists. We need only point out here that it will not necessarily be restricted to the matter of parallelism. The parallel postulate is of extreme interest to us for two reasons; first because historically it was the means by which the possibilities and the importance of non-Euclidean geometry were forced upon our attention; and second because it happens to be the immediate ground of distinction between Euclidean geometry and two of the most interesting alternatives. But Euclidean geometry is characterized, not by a single postulate, but by a considerable number of postulates. We may attempt to omit any one of these so that its ground is not specifically covered at all, or to replace any one of them by a direct alternative. We might conceivably do away with the superposition postulate entirely, and demand that figures be proved equivalent, if at all, by some more drastic test. We might do away with the postulate, first properly formulated by Hilbert, on which our ideas of the property represented in the word "between" depend. We might do away with any single one of the Euclidean postulates, or with any combination of two or more of them. In some cases this would lead to a lack of categoricity and we should get no geometry at all; in most cases, provided we brought a proper degree of astuteness to the formulation of alternatives for the rejected postulates, we should get a perfectly good system of non-Euclidean geometry: one realized, if at all, by other elements than the Euclidean point, line and plane, and one whose elements behave toward one another differently from the Euclidean point, line and plane.

Merely to add definiteness to this chapter, I annex here the statement that in the geometry which Einstein builds up as more nearly representing the true external world than does Euclid's, we shall dispense with Euclid's (implicit) assumption, underlying his (explicitly stated) superposition postulate, to the effect that the act of moving things about does not affect their lengths. We shall at the same time dispense with his parallel postulate. And we shall add a fourth dimension to his three--not, of course, anything in the nature of a fourth Euclidean straight line perpendicular, in Euclidean space, to three lines that are already perpendicular to each other, but something quite distinct from this, whose nature we shall see more exactly in the next chapter. If the present chapter has made it clear that it is proper for us to do this, and has prevented anyone from supposing that the results of doing it must be visualized in a Euclidean space of three dimensions or of any number of dimensions, it will have served its purpose.

VI

THE SPACE-TIME CONTINUUM

Minkowski's World of Events, and the Way It Fits Into Einstein's Structure

BY THE EDITOR, EXCEPT AS NOTED

Seeking a basis for the secure formulation of his results, and especially a means for expressing mathematically the facts of the dependence which he had found to exist between time and space, Einstein fell back upon the prior work of Minkowski. It may be stated right here that the idea of time as a fourth dimension is not particularly a new one. It has been a topic of abstract speculation for the best part of a century, even on the part of those whose notions of the fourth dimension were pretty closely tied down to the idea of a fourth dimension of Euclidean point-space, which would be marked by a fourth real line, perpendicular to the other three, and visible to us if we were only able to see it. Moreover, every mathematician, whether or not he be inclined to this sort of mental exercise, knows well that whenever time enters his equations at all, it does so on an absolutely equal footing with each of his space coordinates, so that as far as his algebra is concerned he could never distinguish between them. When the variables $x$, $y$, $z$, $t$ come to the mathematician in connection with some physical investigation, he knows before he starts that the first three represent the dimensions of Euclidean three-space and that the last stands for time. But if the algebraic expressions of such a problem were handed to him independently of all physical tie-up, he would never be able to tell, from them alone, whether one of the four variables represented time, or if so, which one to pick out for this distinction.

It was Minkowski who first formulated all this in a form susceptible of use in connection with the theory of relativity. His starting point lies in the distinction between the point and the event. Mr. Francis has brought this out rather well in his essay, being the only competitor to present the Euclidean geometry as a real predecessor of Newtonian science, rather than as a mere part of the Newtonian system. I think his point here is very well taken. As he says, Euclid looked into the world about him and saw it composed of points. Ignoring all dynamic considerations, he built up in his mind a static world of points, and constructed his geometry as a scientific machine for dealing with this world in which motion played no part. It could to be sure be introduced by the observer for his own purposes, but when so introduced it was specifically postulated to be a matter of no moment at all to the points or lines or figures that were moved. It was purely an observational device, intended for the observer's convenience, and in the bargain a mental device, calling for no physical action and the play of no force. So far as Euclid in his daily life was obliged to take cognizance of the fact that in the world of work-a-day realities motion existed, he must, as a true Greek, have looked upon this as a most unfortunate deviation of the reality from his beautiful world of intellectual abstraction, and as something to be deplored and ignored. Even in their statuary the Greeks clung to this idea. A group of marvelous action, like the Laocoon, they held to be distinctly a second rate production, a prostitution of the noble art; their ideal was a figure like the majestic Zeus--not necessarily a mere bust, be it understood, but always a figure in repose without action. Their statuary stood for things, not for action, just as their geometry stood for points, not for events.

Galileo and Newton took a different viewpoint. They were interested in the world as it is, not as it ought to be; and if motion appears to be a fundamental part of that world, they were bound to include it in their scheme. This made it necessary for them to pay much more attention to the concept of time and its place in the world than did the Greeks. In the superposition process, and even when he allowed a curve to be generated by a moving point, the sole interest which Euclid had in the motion was the effect which was to be observed upon his static figures after its completion. In this effect the rate of the motion did not enter. So all questions of velocity and time are completely ignored, and we have in fact the curious spectacle of motion without time.

To Galileo and Newton, on the other hand, the time which it took a body to pass from one point of its path to another was of paramount importance. The motion itself was the object of their study, and they recognized the part played by velocity. But Galileo and Newton were still sufficiently under the influence of Euclid to fit the observed phenomena of motion, so far as they could, upon Euclid's static world of points. This they effected by falling in with the age-old procedure of regarding time and space as something entirely disassociated and distinct. The motion of an object--in theory, of a point--was to be recorded by observing its successive positions. With each of these positions a time was to be associated, marking the instant at which the point attained that position. But in the face of this association, space and time were to be maintained as entirely separate entities.

THE FOUR-DIMENSIONAL WORLD OF EVENTS

This severe separation of time and space Minkowski has now questioned, with the statement that the elements of which the external world is composed, and which we observe, are not points at all, but are events. This calls for a revision of our whole habit of thought. It means that the perceptual world is four-dimensioned, not three-dimensioned as we have always supposed; and it means, at the very least, that the distinction between time and space is not so fundamental as we had supposed.

[This should not impress us as strange or incomprehensible. What do we mean when we say that a plane is two-dimensional? Simply that two coordinates, two numbers, must be given to specify the position of any point of the plane. Similarly for a point in the space of our accustomed concepts we must give three numbers to fix the position--as by giving the latitude and longitude of a point on the earth and its height above sea-level. So we say this space is three-dimensional. But a material body is not merely somewhere; it is somewhere now,]182 or was somewhere yesterday, or will be somewhere tomorrow. The statement of position for a material object is meaningless unless we at the same time specify the time at which it held that position. [If I am considering the life-history of an object on a moving train, I must give three space-coordinates and one time-coordinate to fix each of its positions.]182 And each of its positions, with the time pertaining to that position, constitutes an event. The dynamic, ever-changing world about us, that shows the same aspect at no two different moments, is a world of events; and since four measures or coordinates are required to fix an event, we say this world of events is four-dimensional. If we wish to test out the soundness of this viewpoint, we may well do so by asking whether the naming of values for the four coordinates fixes the event uniquely, as the naming of three under the old system fixes the point uniquely.

Suppose we take some particular event as the one from which to measure, and agree upon the directions to be taken by our space axes, and make any convention about our time-axis which subsequent investigation may show to be necessary. Certainly then the act of measuring so many miles north, and so many west, and so many down, and so many seconds backward, brings us to a definite time and place--which is to say, to a definite event. Perhaps nothing "happened" there, in the sense in which we usually employ the word; but that is no more serious than if we were to locate a point with reference to our familiar space coordinate system, and find it to lie in the empty void of interstellar space, with no material body occupying it. In this second case we still have a point, which requires, to insure its existence and location, three coordinates and nothing more; in the first case we still have an event, which requires for its existence and definition four coordinates and nothing more. It is not an event about which the headline writers are likely to get greatly excited; but what of that? It is there, ready and waiting to define any physical happening that falls upon it, just as the geometer's point is ready and waiting to define any physical body that chances to fall upon it.

A CONTINUUM OF POINTS

It is now in order to introduce a word, which I shall have to confess the great majority of the essayists introduce, somewhat improperly, without explanation. But when I attempt to explain it, I realize quite well why they did this. They had to have it; and they didn't have space in their three thousand words to talk adequately about it and about anything else besides. The mathematician knows very well indeed what he means by a continuum; but it is far from easy to explain it in ordinary language. I think I may do best by talking first at some length about a straight line, and the points on it.

If the line contains only the points corresponding to the integral distances 1, 2, 3, etc., from the starting point, it is obviously not continuous--there are gaps in it vastly more inclusive than the few (comparatively speaking) points that are present. If we extend the limitations so that the line includes all points corresponding to ordinary proper and improper fractions like 1/4 and 17/29 and 1633/7--what the mathematician calls the rational numbers--we shall apparently fill in these gaps; and I think the layman's first impulse would be to say that the line is now continuous. Certainly we cannot stand now at one point on the line and name the "next" point, as we could a moment ago. There is no "next" rational number to 116/125, for instance; 115/124 comes before it and 117/126 comes after it, but between it and either, or between it and any other rational number we might name, lie many others of the same sort. Yet in spite of the fact that the line containing all these rational points is now "dense" (the technical term for the property I have just indicated), it is still not continuous; for I can easily define numbers that are not contained in it--irrational numbers in infinite variety like $\sqrt{2}$; or, even worse, the number pi = 3.141592 ... which defines the ratio of the circumference of a circle to the diameter, and many other numbers of similar sort.

If the line is to be continuous, there may be no holes in it at all; it must have a point corresponding to every number I can possibly name. Similarly for the plane, and for our three-space; if they are to be continuous, the one must contain a point for every possible pair of numbers $x$ and $y$, and the other for every possible set of three numbers $x$, $y$ and $z$, that I can name. There may be no holes in them at all.

A line is a continuum of points. A plane is a continuum of points. A three-space is a continuum of points. These three cases differ only in their dimensionality; it requires but one number to determine a point of the first continuum, two and three respectively in the second and third cases. But the essential feature is not that a continuum shall consist of points, or that we shall be able to visualize a pseudo-real existence for it of just the sort that we can visualize in the case of line, plane and point. The essential thing is merely that it shall be an aggregate of elements numerically determined in such a way as to leave no holes, but to be just as continuous as the real number system itself. Examples, however, aside from the three which I have used, are difficult to construct of such sort that the layman shall grasp them readily; so perhaps, fortified with the background of example already presented, I may venture first upon a general statement.

THE CONTINUUM IN GENERAL

Suppose we have a set of "elements" of some sort--any sort. Suppose that these elements possess one or more fundamental identifying characteristics, analogous to the coordinates of a point, and which, like these coordinates, are capable of being given numerical values. Suppose we find that no two elements of the set possess identically the same set of defining values. Suppose finally--and this is the critical test--that the elements of the set are such that, no matter what numerical values we may specify, it we do specify the proper number of defining magnitudes we define by these an actual element of the set, that corresponds to this particular collection of values. Our elements then share with the real number system the property of leaving no holes, of constituting a continuous succession in every dimension which they possess. We have then a continuum. Whatever its elements, whatever the character of their numerical identifiers, whatever the number n of these which stands for its dimension, there may be no holes or we have no continuum. There must be an element for every possible combination of n numbers we can name, and no two of these combinations may give the same element. Granted this condition, our elements constitute a continuum.

As I have remarked, it is not easy to cite examples of continua which shall mean anything to the person unaccustomed to the term. The totality of carbon-oxygen-nitrogen-hydrogen compounds suggested by one essayist as an example is not a continuum at all, for the set contains elements corresponding only to integer values of the numbers which tell us how many atoms of each substance occur in the molecule. We cannot have a compound containing $\sqrt2$ carbon atoms, or $3\pi$ oxygen atoms. Perhaps the most satisfactory of the continua, outside the three Euclidean space-continua already cited, [is the manifold of music notes. This is four-dimensional; each note has four distinctions--length, pitch, intensity, timbre--to distinguish it perfectly, to tell how long, how high, how loud, how rich.]263 We might have a little difficulty in reducing the characteristic of richness to numerical expression, but presumably it could be done; and we should then be satisfied that every possible combination of four values, $l$, $p$, $i$, $t$ for these four identifying characteristics would give us a musical effect, and one to be confused with no other.

There is in the physical world a vast quantity of continua of one sort or another. The music-note continuum brings attention to the fact that not all of these are such that their elements make their appeal to the visual sense. This remark is a pertinent one; for we are by every right of heritage an eye-minded race, and it is frequently necessary for us to be reminded that so far as the external world is concerned, the verdict of every other sense is entirely on a par with that of sight. The things which we really see, like matter, and the things which we abstract from these visual impressions, like space, are by no means all there is to the world.

EUCLIDEAN AND NON-EUCLIDEAN CONTINUA

If we are dealing with a continuum of any sort whatever having one or two or three dimensions, we are able to represent it graphically by means of the line, the plane, or the three-space. The same set of numbers that defines an element of the given continuum likewise defines an element of the Euclidean continuum of the same dimensionality; so the one continuum corresponds to the other, element for element, and either may stand for the other. But if we have a continuum of four or more dimensions, this representation breaks down in the absence of a real, four-dimensional Euclidean point-space to serve as a picture. This does not in the least detract from the reality of the continuum which we are thus prevented from representing graphically in the accustomed fashion.

The Euclidean representation, in fact, may in some cases be unfortunate--it may be so entirely without significance as to be actually misleading. For in the Euclidean continuum of points, be it line, plane or three-space, there are certain things which we ordinarily regard as secondary derived properties, but which possess a great deal of significance none the less.

In particular, in the Euclidean plane and in Euclidean three-space, there is the distance between two points. I have indicated, in the chapter on non-Euclidean geometry, that the parallel postulate of Euclid, which distinguishes his geometry from others, could be replaced by any one of numerous other postulates. Grant Euclid's postulate and you can prove any of these substitutes; grant any of the substitutes and you can prove Euclid's postulate. Now it happens that there is one of these substitutes to which modern analysis has given a position of considerable importance. It is merely our good old friend the Pythagorean theorem, that the square on the hypotenuse equals the sum of the squares on the sides; but it is dressed in new clothes for the present occasion.

Mr. Francis' discussion of this part of the subject, and especially his figure, ought to make it clear that this theorem can be considered as dealing with the distance between any two points. When we so consider it, and take it as the fundamental, defining postulate of Euclidean geometry which distinguishes this geometry from others, we have a statement of considerable content. We have, first, that the characteristic property of Euclidean space is that the distance between two points is given by the square root of the sum of the squares of the coordinate-differences for these points--by the expression

$$D = \sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2}\,,$$

where the large letters represent the coordinates of the one point and the small ones those of the other. We have more than this, however; we have that this distance is the same for all observers, no matter how different their values for the individual coordinates of the individual points. And we have, finally, as a direct result of looking upon the thing from this viewpoint, that the expression for $D$ is an "invariant"; which simply means that every observer may use the same expression in calculating the value of $D$ in terms of his own values for the coordinates involved. The distance between two points in our space is given numerically by the square root of the sum of the squares of my coordinate-differences for the two points involved; it is given equally by the square root of the sum of the squares of your coordinate-differences, or those of any other observer whatsoever. We have then a natural law--the fundamental natural law characterizing Euclidean space. If we wish to apply it to the Euclidean two-space (the plane) we have only to drop out the superfluous coordinate-difference; if we wish to see by analogy what would be the fundamental natural law for a four-dimensional Euclidean space, we have only to introduce under the radical a fourth coordinate-difference for the fourth dimension.

If we were not able to attach any concrete meaning to the expression for $D$ the value of all this would be materially lessened. Consider, for instance, the continuum of music notes. There is no distance between different notes. There is of course significance in talking about the difference in pitch, in intensity, in duration, in timbre, between two notes; but there is none in a mode of speech that implies a composite expression indicating how far one note escapes being identical with another in all four respects at once. The trouble, of course, is that the four dimensions of the music-note continuum are not measurable in terms of a common unit. If they were, we should expect to measure their combination more or less absolutely in terms of this same unit. We can make measurements in all three dimensions of Euclidean space with the same unit, with the same measuring rod in fact. [This presents a peculiarity of our three-space which is not possessed by all three-dimensional manifolds. Riemann has given another illustration in the system of all possible colors, composed of arbitrary proportions of the three primaries, red, green and violet. This system forms a three-dimensional continuum; but we cannot measure the "distance" or difference between two colors in terms of the difference between two others.]130

Accordingly, in spite of the fact that the Euclidean three-space gives us a formal representation of the color continuum, and in spite of the fact that the hypothetical four-dimensional Euclidean space would perform a like office for the music-note continuum, this representation would be without significance. We should not say that the geometry of these two manifolds is Euclidean. We should realize that any set of numerical elements can be plotted in a Euclidean space of the appropriate dimensionality; and that accordingly, before allowing such a plot to influence us to classify the geometry of the given manifold as Euclidean, we must pause long enough to ask whether the rest of the Euclidean system fits into the picture. If the square root of the sum of the squares of the coordinate-differences between two elements possesses significance in the given continuum, and if it is invariant between observers of that continuum who employ different bases of reference, then and only then may we allege the Euclidean character of the given continuum.

If under this test the given continuum fails of Euclideanism, it is in order to ask what type of geometry it does present. If it is of such character that the "distance" between two elements possesses significance, we should answer this question by investigating that distance in the hope of discovering a non-Euclidean expression for it which will be invariant. If it is not of such character, we should seek some other characteristic of single elements or groups of elements, of real physical significance and of such sort that the numerical expression for it would be invariant.

If the continuum with which we have to do is one in which the "distance" between two elements possesses significance, and if it turns out that the invariant expression for this distance is not the Pythagorean one, but one indicating the non-Euclideanism of our continuum, we say that this continuum has a "curvature." This means that, if we interpret the elements of our continuum as points in space (which of course we may properly do) and if we then try to superpose this point-continuum upon a Euclidean continuum, it will not "go"; we shall be caught in some such absurdity as trying to force a sphere into coincidence with a plane. And of course if it won't go, the only possible reason is that it is curved or distorted, like the sphere, in such a way as to prevent its going. It is unfortunate that the visualizing of such curvature requires the visualizing of an additional dimension for the curved continuum to curve into; so that while we can picture a curved surface easily enough, we can't picture a curved three-space or four-space. But that is a barrier to visualization alone, and in no sense to understanding.

OUR WORLD OF FOUR DIMENSIONS

It will be observed that we have now a much broader definition of non-Euclideanism than the one which served us for the investigation of Euclid's parallel postulate. If we may at pleasure accept this postulate or replace it by another and different one, we may presumably do the same for any other or any others of Euclid's postulates. The very statement that the distance between elements of the continuum shall possess significance, and shall be measurable by considering a path in the continuum which involves other elements, is an assumption. If we discard it altogether, or replace it by one postulating that some other joint property of the elements than their distance be the center of interest, we get a non-Euclidean geometry. So for any other of Euclid's postulates; they are all necessary for a Euclidean system, and in the absence of any one of them we get a non-Euclidean system.

Now the four-dimensional time-space continuum of Minkowski is plainly of a sort which ought to make susceptible of measurement the separation between two of its events. We can pass from one element to another in this continuum--from one event to another--by traversing a path involving "successive" events. Our very lives consist in doing just this: we pass from the initial event of our career to the final event by traversing a path leading us from event to event, changing our time and space coordinates continuously and simultaneously in the process. And while we have not been in the habit of measuring anything except the space interval between two events and the time interval between two events, separately, I think it is clear enough that, considered as events, as elements in the world of four dimensions, there is a less separation between two events that occur in my office on the same day than between two which occur in my office a year apart; or between two events occurring 10 minutes apart when both take place in my office than when one takes place there and one in London or on Betelgeuse.

It is not at all unreasonable, a priori, then, to seek a numerical measure for the separation, in space-time of four-dimensions, of two events. If we find it, we shall doubtless be asked just what its subjective significance to us is. This must be answered with some circumspection. It will presumably be something which we cannot observe with the visual sense alone, or it would have forced itself upon our attention thousands of years ago. It ought, I should think, to be something that we would sense by employing at the same time the visual sense and the sense of time-passage. In fact, I might very plausibly insist that, by my very remarks about it in the above paragraph, I have sensed it.

Minkowski, however, was not worried about this phase of the matter. He had only to identify the invariant expression for distance; sensing it could wait. He found, of course, that this expression was not the Euclidean expression for a four-dimensional interval. He had discarded several of the Euclidean assumptions and could not expect that the postulate governing the metric properties of Euclid's space would persist. Especially had he violated the Euclidean canons in discarding, with Einstein, the notion that nothing which may happen to a measuring rod in the way of uniform translation at high velocity can affect its measures. So he had to be prepared to find that his geometry was non-Euclidean; yet it is surprising to learn how slightly it deviates from that of Euclid. Without any extended discussion to support the statement, we may say that he found that when two observers measure the time- and the space-coordinates of two events, using the assumptions and therefore the methods of Einstein and hence subjecting themselves to the condition that their measures of the pure time-interval and of the pure space-interval between these events will not necessarily be the same, they will discover that they both get the same value for the expression

$$S = \sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2 - (CT - Ct)^2}\,.$$

If our acceptance of this as the numerical measure of the separation in space-time between the two events should lead to contradiction we could not so accept it. No contradiction arises however and we may therefore accept it. And at once the mathematician is ready with some interpretative remarks.

THE CURVATURE OF SPACE-TIME

The invariant expression for separation, it will be seen, is in the same form as that of the Euclidean four-dimensional invariant save for the minus sign before the time-difference (the appearance of the constant $C$ in connection with the time coordinate $t$ is merely an adjustment of units; see page 153). This tells us that not alone is the geometry of the time-space continuum non-Euclidean in its methods of measurement, but also in its results, to the extent that it possesses a curvature. It compares with the Euclidean four-dimensional continuum in much the same way that a spherical surface compares with a plane. As a matter of fact, a more illuminating analogy here would be that between the cylindrical surface and the plane, though neither is quite exact. To make this clear requires a little discussion of an elementary notion which we have not yet had to consider.

Our three-dimensional existence often reduces, for all practical purposes, to a two-dimensional one. The objects and the events of a certain room may quite satisfactorily be defined by thinking of them, not as located in space, but as lying in the floor of the room. Mathematically the justification for this viewpoint is got by saying that we have elected to consider a slice of our three-dimensional world of the sort which we know as a plane. When we consider this plane and the points in it, we find that we have taken a cross-section of the three-dimensional world. A line in that world is now reduced, for us, to a single point--the point where it cuts our plane; a plane is reduced to a line--the line where it cuts our plane; the three-dimensional world itself is reduced to our plane itself. Everything three-dimensional falls down into its shadow in our plane, losing in the process that one of the three dimensions which is not present in our plane.

For simplicity's sake it is usual to take a cross-section of space parallel to one of our coordinate axes. We think of our three dimensions as extending in the directions of those axes; and it is easier to take a horizontal or vertical section which shall simply wipe out one of these dimensions than to take an oblique section which shall wipe out a dimension that consists partly of our original length, and partly of our original width, and partly of our original height.

If we have a four-dimensional manifold to begin with, we may equally shake out one of the four dimensions, one of the four coordinates, and consider the three-dimensional result of this process as a cross-section of the original four-dimensional continuum. And where, in cross-sectioning a three-dimensioned world, we have but three choices of a coordinate to eliminate, in cross-sectioning a world of four dimensions we have four choices. By dropping out either the $x$, or the $y$, or the $z$, or the $t$, we get a three-dimensioned cross-section.

Now our accustomed three-dimensional space is strictly Euclidean. When we cross-section it, we get a Euclidean plane no matter what the direction in which we make the cut. Likewise the Euclidean plane is wholly Euclidean, because when we cross-section it in any direction whatever we get a Euclidean line. A cylindrical surface, on the other hand, is neither wholly Euclidean nor wholly non-Euclidean in this matter of cross-sectioning. If we take a section in one direction we get a Euclidean line and if we take a section in the other direction we get a circle (if the cylindrical surface be a circular one). And of course if we take an oblique section of any sort, it is neither line nor circle, but a compromise between the two--the significant thing being that it is still not a Euclidean line.

The space-time continuum presents an analogous situation. When we cross-section it by dropping out any one of the three space dimensions, we get a three-dimensional complex in which the distance formula is still non-Euclidean, retaining the minus sign before the time-difference and therefore retaining the geometric character of its parent. But if we take our cross-section in such a way as to eliminate the time coordinate, this peculiarity disappears. The signs in the invariant expression are then all plus, and the cross-section is in fact our familiar Euclidean three-space.