Popular Science Monthly/Volume 78/June 1911/Is Euclid's Geometry Merely a Theory?
|IS EUCLID'S GEOMETRY MERELY A THEORY?|
PROFESSOR OF PHILOSOPHY, WASHINGTON AND JEFFERSON COLLEGE
AS there are creeds in religion, so there are creeds in geometry. Most of us pin our faith upon Euclid, whose masterpiece of reasoning is still, after twenty-two centuries, the wonder of the world. The system that Euclid founded stands, as it were, four-square and solid; it meets every need in the only kind of space that we practically know. Over against this edifice, the modern geometries of hyperspace have been reared, from foundations which we Euclideans regard as fantastic. They are intangible structures, like the towers and battlements of a region of dreams.
The present writer holds no brief in favor of a fourth dimension of space. Hypothetical realms, wherein the dimensions of space are assumed to be greater in number than three, yield strange geometries, which are only card-castles, products of a sort of intellectual play, in the construction of which the laws of logic supply the rules of the game. The character of each system is determined by whatsoever assumptions its builder lays down at the start. The illustrious Euclid himself, whom none would rank as visionary, would probably set no great store by these hypergeometries. If he were to return to earth to-day, his interest in them would be that of a retired chess-champion who perceives that his old style of play has given rise to new varieties of the game. Nothing from out this fairyland of thinking could endanger Euclid's prestige; he might contemplate retirement on a professor's old-age pension.
Nevertheless, as soon as Euclid had viewed modern geometry throughout its entire range, the mere suggestion of a pension would in all likelihood ruffle his spirit. For by that time the master would know that geometers no longer blindly accept his teachings; that, moreover, our real space holds mysteries of which he never dreamed. When finally he should discover that experts have arisen who would undertake to instruct him at his own game, he would investigate the massive non-Euclidean systems—the Lobachevski-Bolyai or pseudospherical geometry, and another, the spherical, invented by Riemann—no mere card-castles, but valid in their application to every known space-condition of the universe. Like the rest of us, Euclid would ask himself: In which of these varieties of space does our actual universe belong?
Now, Euclid's geometry is, of course, something more than a game. Its rules—the twelve axioms and five postulates—taken as a group, Euclid might well be proud of. Most people believe that the whole body of his proof rests upon them as upon an eternally established foundation. Eternity is too long to contemplate, but we are certain that to the present time, in no instance, have these seventeen assumptions, when correctly used, ever led to a detectable error. His reasoning is always consistent in itself, always in perfect accord with the known laws of mechanics. One does not feel that the system is a Mahomet's coffin, hovering unsupported in mid-air; it seems to rest on the solid earth, and very firmly. Where, then, is any weakness in the foundation that he laid?
Possibly, in his sleep through the centuries, Euclid has turned over once or twice at doubts, first raised by Ptolemy in the second century a.d., who never became quite convinced that a certain momentous statement was perfectly self-evident, a statement which Euclid used without proving. Apparently it was an afterthought with Euclid in the first place, for not until he had reasoned himself well into the heart of his subject did the need for it, or for something like it, become imperative. Then he asserted quite dogmatically that: Through the same point there can not be two parallels to the same straight line. Ptolemy, hoping to strengthen Euclid's foundation, tried to prove this parallel postulate, but concerning the outcome, Poincaré, about eighteen centuries later, has recently said: "What vast effort has been wasted in this chimeric hope is truly unimaginable. Finally, in the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian, Bolyai and Lobachevski, established irrefutably that this demonstration is impossible: they have almost rid us of geometries 'sans postulatum'; since then the Académie des Sciences receives only about one or two new demonstrations a year." The parallel postulate, then, is a weak spot in the Euclidean system. The demonstration that beyond all doubt no proof of its correctness can be devised was an epoch-making discovery. Bolyai's share in this event took concrete form in a brief appendix to a work by his father, published in 1831. Halsted characterizes this document as "the most extraordinary two dozen pages in the history of human thought."
Our chief concern for the next few moments will be to comprehend why the truth of the Euclidean postulate can not be established by argument. After that, I shall try to show why it is not self-evident. These steps taken, I believe that the reader will agree with me that, since it can not be proved, one may freely choose some other postulate in its stead, and thus develop a different geometry every whit as trustworthy.
Let no one suppose, however, that the least dispute has ever arisen as to what parallel lines are. Euclid defined them as: Straight lines which are in the same plane, and which, being produced ever so far both ways, do not meet. All geometers accept this definition, just as at whist every one agrees on the meaning of certain terms—calls a spade a spade, and so forth. To the brilliant young Lobachevski no good reason presented itself why through a given point there should be only one such parallel to a given straight line. He accepted all Euclid's assumptions except this one, in place of which he substituted a contradictory statement of his own making; he hazarded the novel assertion that: Through a given point there can be two parallels to the same straight line. On this foundation he erected a new geometry, building proposition upon proposition until he had reared an edifice as coherent and in every respect as perfect as the geometry of Euclid. What conclusion may we draw? This: had Euclid's postulate been eternally true, then to deny it while holding to his other axioms would have led Lobachevski into endless inconsistencies. But the fact that its contrary
was substituted for it and a new geometry developed without encountering any logical obstacle shows that the postulate rests on nothing more fundamental than itself; shows that it swings, so to speak, in mid-air, unaffected by Euclid's other assertions. No statement can be proved by itself alone; consequently, this statement, having no logical connection with any other, can not be proved at all. Moreover, this achievement, broadly comprehended, set the entire Euclidean system aswing without support; its supposed connection with the solid earth was a fact only of the imagination.
I promised, in the next place, to show that Euclid's postulate lacks self-evidence. In Fig. 1 there is a point P, lying without a straight line CD. Another straight line AB passes through this point, and we shall imagine both AB and CD to be produced ever so far both ways.
Now AB will be parallel to CD, if they conform to Euclid's definition of what parallels are, namely, if both lines are straight, and in the same plane, and being produced indefinitely, do not meet. In that position the lines would be parallel, but let us start from the position shown in the figure, where the lines we are talking about do meet at the point M, and let us imagine further that this point of intersection M travels along the line CD. If then we keep turning the line AB slowly round the point P, eventually the point of intersection M must disappear at one end and reappear at the other end of CD, it matters not how far the two lines have been extended.
The assumption hidden in Euclid's assumption is that there can be one and only one position of the moving line AB at which it will be parallel to CD. Lobachevski contrariwise assumes that AB will have to be turned through a finite angle after parting from CD before it intersects with CD again. That angle to be passed through gives Lobachevski the opportunity of postulating not only two parallels to CD, but an aggregate of parallels, all passing through the point P. The same argument may be presented a little differently and more clearly perhaps, as follows: Imagine AB at first not merely parallel but at all points equidistant from CD. Will not AB have to dip through a certain distance before it can meet CD?
This problem, apparently so simple, is of such a nature that neither opponent can prove his assertion. It will be observed that when Euclid says only one parallel is possible, and when Lobachevski says an infinite number of them are possible, there is still room for a third champion who will say no parallels are possible, that the lines AB and CD if extended will always meet, which is precisely Riemann's position on the question. The three geometries are thus exactly upon a par; no one of them can establish itself against the other two; and the number of possibilities is complete, for among the assertions "one," "many" and "none," there is no position unoccupied in reference to the mystery of parallel lines; no chance left for any fourth geometry on this basis.
We are now on the threshold of non-Euclidean geometry, prepared, I trust, to enter a new variety of space where geometrical problems work out to results differing widely from those found in the books of Euclid. Compared with Lobachevski, Euclid was more sparing of parallels, and the effect of this parsimony upon Euclid's idea of space is very marked. I know of no better expression for the difference between their notions of space than to say that Lobachevski's space is roomier. In Lobachevski's space, if a man whose course was restricted to a perfectly straight line should wish to avoid crossing a perfectly straight road, he might set out in exactly the same direction as the road runs, or he might incline a very little toward it; in either case he would never meet it. This is equivalent to saying that Lobachevski's space is more expansive, more generously given, and if the reader will follow me a step further, I may say that this space becomes roomier increasingly with every step that the man takes forward.
It costs nothing to imagine ourselves entering this domain of Lobachevski; indeed, for aught we know, we may be actually in it now. And it will cost us no more to imagine our expedition equipped with instruments for measuring angles and for drawing straight lines, instruments more delicate and accurate than any that science has yet devised. A series of experiments may then be carried out to illustrate the properties of this hyperbolic region, I shall limit the narrative to some of the results we could obtain:
A. Parallels, really straight lines that never meet, have a point of nearest approach to each other, but if followed in either direction outward from this point, they will be found to diverge, spreading farther apart without limit (Fig. 2).
Fig. 2. The lines AB and CD are straight, as may be seen by viewing the figure from one side with the eye close to the paper. They are In the same plane and will never meet. Yet by an optical illusion we here obtain within a small compass the same appearance as would be furnished by two parallels to an eye located In Lobachevski's space and capable of surveying a tremendous stretch of the parallels from a very great distance. The diagram probably has no reference whatever to non-Euclidean geometry. It elucidates mental, not physical, phenomena.
B. If two perpendiculars are erected on a Lobachevski parallel, they will spread away from each other, becoming farther apart the farther we extend them outward from the base line.
C. With this base line and the two perpendiculars, we might think we had three sides of a rectangle, but no—for after making three of the corners right angles, the fourth must needs be an acute angle. A true rectangle is impossible (Fig. 3).
D. We can, however, draw a straight-sided triangle. In Euclidean apace the internal angles of such a triangle, added together, always equal two right angles, but here they fall short of two right angles, and the larger the triangle, the less the sum of its internal angles. Logically pursued, the largest triangle possible would have all its sides parallel and all its angles zero.
This last statement has touched the verge of infinity and that is no doubt treacherous territory. Coming back to our real universe—How can we prove that Euclid is right about it? The real universe is large. If, like the adversary in the Book of Job, we could go to and fro in the great world and walk up and down in it, then we might decide the controversy. But the limit of man's present astronomical measurements is only about 30 light-years—176 millions of millions of miles. Within this compass he has observed no drift or change in the direction of rays of light. If in our real space parallels are not exactly and everywhere equidistant, Euclid's geometry is incorrect. The slightest deviation in parallels would give the victory to Lobachevski or else to the third competitor, Riemann. The three justly claim equal consideration
in the light of present knowledge. Could such drift, if real, escape human observation? Yes; first, because our instruments are not absolutely accurate; and secondly, because eyesight is no infallible test.
If the human eye could survey a sufficiently tremendous expanse, then parallels running through it might present the appearance of the hyperbolic curves limiting the black and white areas in Fig. 4. These curves may represent, and in certain respects they do simulate, the parallels of Lobachevski. They are, to be sure, not parallels, for parallels are by definition straight; however, by placing the eye an inch and a half above the center of this figure, these lines can be made to look straight—a fact that confirms the statement that eyesight is not an infallible test of straightness.
Fig. 4 will do good service if it enables us to understand what people mean when they assert profoundly that "non-Euclidean space is curved." We ought to discard this misleading, though very prevalent, expression, for it is as confusing as to say that space is straight, or cold, or pink. It certainly sounds absurd to call a straight line curved under any circumstances, and so it is, so long as we confine our thinking to any one kind of space. But in carrying lines over from one space to another, there is this change of emphasis. For example, the parallels of Lobachevski, when transferred into Euclidean space, cease to be parallels and become, as shown in Fig. 4, hyperbolic curves.
Contrariwise, the parallels of Euclid, transferred into Lobachevski's space, retain merely their secondary property of equidistance, and pass under the name of equidistantials, since they are no longer true parallels, nor even straight, but rather they are very long curves.
We read in the Arabian Nights of the magical carpet of Tangu, which could be made to fly incredible distances by wishing it to do so. Imagination can furnish us with a similar carpet that will flit from one realm of space to another. Suppose then that our carpet is being woven at a non-Euclidean factory. It should be pliable, but it must not stretch, and it must possess truly princely size, having leagues upon leagues of surface. When spread out, it must lie perfectly flat and smooth in Lobachevski's space, but if transported into Euclidean space, it can only lie crinkly and fluted around its edges, for this environment, though boundless, is less roomy and the expanse will be too niggardly to accommodate the carpef s ample proportions. Suppose, in addition, that the carpet, while at the non-Euclidean factory, receives on its surface a checker-board pattern of black and white squares separated by criss-cross parallel lines, truly straight lines in the sense of being the shortest distances between points. The squares can not be perfect rectangles because, as already observed, such figures are not among the non-Euclidean possibilities. In Euclidean space, they would look like the black and white areas in Fig. 4. The figure is not, however, a perfect representation, because the carpet could not be made to lie flat in Euclidean surroundings without violent stretching, while to distend it would be to destroy the spatial relations of the lines of the pattern, after which, for geometrical purposes, it would no longer be the same carpet.
Spread out the carpet, nevertheless, as evenly as Euclidean space allows. Xo part will lie perfectly flat, of course; and only a limited portion can be made to lie smooth; the outlying portions will refuse to be spread out and must remain in folds. The smooth portion will then be slightly curved into the shape of a saddle, trending upward at front and back, and rolling off downward on either side, the whole forming a surface of constant negative curvature, called by mathematicians a pseudosphere, and being simply Lobachevski's plane surface after its transportation into the Euclidean environment. Upon such a surface we can draw diagrams suited to illustrate any problem in Lobachevski's plane geometry just as for Euclid's plane geometry we make use of the flat surface of a blackboard. Lines drawn on the pseudosphere can not be straight; they can only be the straightest lines that the surface will allow; but, limiting our discussion to lines lying wholly within the surface, these straightest lines will still be the shortest distances between points in the surface and would remain so, even if the surface were crushed into a wrinkled heap.
We do not know upon what kind of a surface Euclid drew his diagrams, perhaps upon sand, but it is reasonable to presume that it was approximately flat. Had he used a pseudospherical surface, he might have developed a different conception of space. Had he, on the contrary, chosen a sphere, he might have arrived at the geometry of Riemann, for the plane surface of Eiemannian space becomes simply a sphere under Euclidean conditions. The opportimity is so favorable just now that I may be permitted briefly to set down some of the results derived from this third type of geometry. Obviously, the straightest line that one could draw upon a sphere, as, for instance, by stretching a string between two points on the surface, would, if extended, go completely round and form a great circle. Certain conclusions follow:
A. No parallels are possible; all great circles (straightest lines) must somewhere meet.
B. Euclid's axiom that: Through any two points only one straight can pass, is, in most cases, correct; but when the points chosen are diametrically opposed, as are the poles of the earth, then an infinity of great circles cross at such points, and any two of these lines enclose a space.
C. Concerning triangles, the internal angles are together always equal to more than two right angles, the excess increasing with the size of the triangle. Rectangles are impossible. Two plane figures (except circles) can not be similar in shape unless equal in size.
D. The Riemannian space is not infinite in extent, but returns into itself. It is, however, boundless; one could never come to the end of it. With eyes adapted to enormous distances a creature looking in any direction might see the back of its own head.
E. On the hypothesis that our own universe is of this nature, "a finite number of our common building bricks," as Halsted says, "might be written down which might be more than our universe could contain." And if our earth should increase in bulk, at last the lower surface would advance upon us from above, and, reaching us, would fill the whole universe.
It is commonly supposed that these peculiarities of Riemannian space are easier to conceive than are the results at which Lobachevski arrives, but this is probably not the case. Long ago, Beltrami discovered that the whole space of Lobachevski, notwithstanding that it is infinite in extent in all directions, can be conceived as packed within a hollow globe of finite radius. Imagine, therefore, a great sphere of a hundred yards' radius, with a door leading into it. Looking in, let us suppose that we can discover a railroad track on a trestle extending from the doorway diametrically across to the other side, and a small man—an inch high—standing between the rails and at the exact center of the sphere. Nothing about this view suggests anything but ordinary space to us; it is only for the little being at the center that this enclosure constitutes Lobachevski's universe.
Another assumption is now to be granted: let the man dwindle in size whenever he moves out of the center toward the shell of the sphere. Growing less and less, he would have no size at all upon reaching the shell, but he could never reach it, for the length of his stride would lessen in proportion to his lessening stature. To him, therefore, the sphere is infinite in extent. Likewise all other objects—the boards on the footpath, the foot-rule and the keys in his pocket, and the pocket itself—^have their sizes determined by their location within the sphere, let only the rails of the track be continuously parallel both according to his and our own notion of things, and also according to Euclid's definition.
Going to and fro along the track, the little man would judge that the rails are not equidistant at all. Applying his shrinkable foot-rule, he would decide that the space between the rails varied in width at different places. He would come to the same conclusion that we have set down, namely, that parallels may at first approach but, following them further, they diverge more and more. All this he would discover and never suspect that his own variable dimensions were the cause of.a deception, for would not his surroundings shrink always in proportion to himself? Beltrami's illustration thus attains its purpose by making solid objects expand and contract in place of allowing space itself to grow any more roomy than our Euclidean notions permit.
If Euclid were to return to earth to-day, he would find many geometries, but the three here described would probably interest him above all others. A fitting task for Euclid would be to coordinate this trilogy of systems. With the three volumes spread open before him, he could write a dictionary by the aid of which a student could translate any proposition stated in one volume into the corresponding proposition given in either of the other two. It is at bottom a matter of words, or at least the facts lend themselves to that interpretation. With intense satisfaction Euclid could still contemplate his own geometry. "Where long and involved phrases are necessary to convey the idea presented in the other systems, his own ideas are always lucid and tersely expressible. His system is consequently by far the most convenient, so much more convenient, that if we should ever discover any discrepancy between it and the facts of the physical universe, we would probably prefer to change our laws of physics or mechanics rather than to adjust ourselves to a less convenient system of geometry.
- Poincaré, "Science and Hypothesis," transl. by G. B. Halsted, p. 30; where a clear account of these geometries will be found.
- Nothing in the definition, as established by Euclid himself, compels one to believe that two parallel lines must be equidistant. The requirements are that they be straight, that they lie in the same plane, and that they do not meet. Euclid discovers that his parallels are at all corresponding points equidistant from one another; but his parallels are peculiar in this respect, and it should be borne in mind that they owe their existence to the postulate which no one can validate.