Chapter VI: completes the preliminary course in the fundamentals of (4)

Now when no particular observer is specified, we must of course assume an observer connected with the train, or with whatever the body mentioned. To that observer it doesn't make the slightest difference what the train does; it may stand at rest with respect to some external system or it may move at any velocity whatsoever; its length remains always 1,000 feet. In order for this question to have the significance which its propounder means it to have, I must restate it as follows: A train is 1,000 feet long as measured by an observer travelling with it. If it passes a second observer at 60 miles per hour, what is its length as observed by him? The answer is now easy.]* [According to the formula the length of the moving train as seen from the ground will be

$$1,000 \times \sqrt{1 - (88)^2 / (186,000 \times 5,280)^2} =
999.999,999,999,996$$

feet, a change entirely too small for detection by the most delicate instruments. Examination of the expression K shows that in so far as terrestrial movements of material objects are concerned it is equal to 1]232 [within a far smaller margin than we can ever hope to make our observations. Even the diameter of the earth, as many of the essayists point out, will be shortened only 2 1/2 inches for an outside observer past whom it rushes with its orbital speed of 18.5 miles per second. But slight as the difference may be in these familiar cases, its scientific importance remains the same.]*

RELATIVITY AND REALITY

[A simple computation shows that this effect is exactly the amount suggested by Lorentz and Fitzgerald to explain the Michelson-Morley experiment.]188 [This ought not to surprise us, since both that explanation and the present one are got up with the same purpose. If they both achieve that purpose they must, numerically, come to the same thing in any numerical case. It is, however, most emphatically to be insisted that the present "shortening" of lengths]* [no longer appears as a "physical" shortening caused by absolute motion through the ether but is simply a result of our methods of measuring space and time.]188 [Where Fitzgerald and Lorentz had assumed that a body in motion has its dimensions shortened in the direction of its motion,]220 [this very form of statement ceases to possess significance under the relativity assumption.]* [For if we cannot tell which of two bodies is moving, which one is shortened? The answer is, both--for the other fellow. For each frame of reference there is a scale of length and a scale of time, and these scales for different frames are related in a manner involving both the length and the time.]220 [But we must not yield to the temptation to say that all this is not real; the confinement of a certain scale of length and of time to a single observation system does not in the least make it unreal.]* [The situation is real--as real as any other physical event.]165

[The word physical is used in two senses in the above paragraph. It is denied that the observed variability in lengths indicates any "physical" contraction or shrinkage; and on the heels of this it is asserted that this observed variability is of itself an actual "physical" event. It is difficult to express in words the distinction between the two senses in which the term physical is employed in these two statements, but I think this distinction ought to be clear once its existence is emphasized. There is no material contraction; it is not right to say that objects in motion contract or are shorter; they are not shorter to an observer in motion with them. The whole thing is a phenomenon of observation. The definitions which we are obliged to lay down and the assumptions which we are obliged to make in order, first, that we shall be able to measure at all, and second, that we shall be able to escape the inadmissible concept of absolute motion, are such that certain realities which we had supposed ought to be the same for all observers turn out not to be the same for observers who are in relative motion with respect to one another. We have found this out, and we have found out the numerical relation which holds between the reality of the one observer and that of the other. We have found that this relation depends upon nothing save the relative velocity of the two observers. As good a way of emphasizing this as any is to point out that two observers who have the same velocity with respect to the system under examination (and whose mutual relative velocity is therefore zero) will always get the same results when measuring lengths and times on that system. The object does not go through any process of contraction; it is simply shorter because it is observed from a station with respect to which it is moving. Similar remarks might be made about the time effect; but the time-interval is not so easily visualized as a concrete thing and hence does not offer such temptation for loose statement.

The purely relative aspect of the matter is further brought out if we consider a single example both backwards and forwards. Systems $S$ and $S'$ are in relative motion. An object in $S$ which to an observer in $S$ is $L$ units long, is shorter for an observer in $S'$--shorter by an amount indicated through the "correction factor" $K$. Now if we have, in the first instance, made the objectionable statement that objects are shorter in system $S'$ than they are in $S$, it will be quite natural for us to infer from this that objects in $S$ must be longer than those in $S'$; and from this to assert that when the observer in $S$ measures objects lying in $S'$, he gets for them greater lengths than does the home observer in $S'$. But if we have, in the first instance, avoided the objectionable statement referred to, we shall be much better able to realize that the whole business is quite reciprocal; that the phenomena are symmetric with respect to the two systems, to the extent that we can interchange the systems in any of our statements without modifying the statements in any other way.

Objects in $S$ appear shorter and times in $S$ appear longer to the external "moving" observer in $S'$ than they do to the domestic observer in $S$. Exactly in the same way, objects in $S'$ appear shorter to observers in the foreign system $S$ than to the home observer in $S'$, who remains at rest with respect to them. I think that when we get the right angle upon this situation, it loses the alleged startling character which has been imposed upon it by many writers. The "apparent size" of the astronomer is an analogy in point. Objects on the moon, by virtue of their great distance, look smaller to observers on the earth than to observers on the moon. Do objects on the earth, on this account, look larger to a moon observer than they do to us? They do not; any suggestion that they do we should receive with appropriate scorn. The variation in size introduced by distance is reciprocal, and this reciprocity does not in the least puzzle us. Why, then, should that introduced by relative motion puzzle us?

TIME AND SPACE IN A SINGLE PACKAGE

Our old, accustomed concepts of time and space, which have grown up through countless generations of our ancestors, and been handed down to us in the form in which we are familiar with them, leave no room for a condition where time intervals and space intervals are not universally fixed and invariant. They leave no room for us to say that]* [one cannot know the time until he knows where he is, nor where he is until he knows the time,]220 [nor either time or place until he knows something about velocity. But in this concise formulation of the difference between what we have always believed and what we have seen to be among the consequences of Einstein's postulates of the universal relativity of uniform motion, we may at once locate the assumption which, underlying all the old ideas, is the root of all the trouble. The fact is we have always supposed time and space to be absolutely distinct and independent entities.]*

[The concept of time has ever been one of the most absolute of all the categories. It is true that there is much of the mysterious about time; and philosophers have spent much effort trying to clear up the mystery--with unsatisfactory results. However, to most persons it has seemed possible to adopt an arbitrary measure or unit of duration and to say that this is absolute, independent of the state of the body or bodies on which it is used for practical purposes.]272 [Time has thus been regarded as something which of itself flows on regularly and continuously, regardless of physical events concerning matter.]150 [In other words, according to this view, time is not affected by conditions or motions in space.]272 [We have deliberately chosen to ignore the obvious fact that time can never appear to us, be measured by us, or have the least significance for us, save as a measure of something that is closely tied up with space and with material space-dimensions. Not merely have we supposed that time and space are separated in nature as in our easiest perceptions, but we have supposed that they are of such fundamentally distinct character that they can never be tied up together. In no way whatever, assumes the Euclidean and Newtonian intellect, may space ever depend upon time or time upon space. This is the assumption which we must remove in order to attain universal relativity; and while it may come hard, it will not come so hard as the alternative. For this alternative is nothing other than to abandon universal relativity. This course would leave us with logical contradictions and discrepancies that could not be resolved by any revision of fundamental concepts or by any cleaning out of the Augean stables of old assumptions; whereas the relativity doctrine as built up by Einstein requires only such a cleaning out in order to leave us with a strictly logical and consistent whole. The rôle of Hercules is a very difficult one for us to play. Einstein has played it for the race at large, but each of us must follow him in playing it for himself.

SOME FURTHER CONSEQUENCES

I need not trespass upon the subject matter of those essays which appear in full by going here into any details with regard to the manner in which time and space are finally found to depend upon one another and to form the parts of a single universal whole. But I may appropriately point out that if time and space are found to be relative, we may surely expect some of the less fundamental concepts that depend upon them to be relative also. In this expectation we are not disappointed. For one thing,]* [mass has always been assumed to be a constant, independent of any motion or energy which it might possess. Just as lengths and times depend upon relative motion, however, it is found that mass, which is the remaining factor in the expression for energy due to motion, also depends upon relative velocities. The dependence is such that if a body takes up an amount of energy $E$ with respect to a certain system, the body behaves, to measurements made from that system, as though its mass had been increased by an amount $E/C^2$, where $C$ is as usual the velocity of light.]194

[This should not startle us. The key to the situation lies in the italicized words above, which indicate that the answer to the query whether a body has taken up energy or not depends upon the seat of observation. If I take up my location on the system $S$, and you on the system $S'$, and if we find that we are in relative motion, we must make some assumption about the energy which was necessary, initially, to get us into this condition. Suppose we are on two passing trains.]* [The chances are that either of us will assume that he is at rest and that it is the other train which moves, although if sufficiently sophisticated one of us may assume that he is moving and that the other train is at rest.]272 [Whatever our assumption, whatever the system, the localization of the energy that is carried in latent form by our systems depends upon this assumption. Indeed, if our systems are of differing mass, our assumptions will even govern our ideas of the amount of energy which is represented by our relative motion; if your system be the more massive, more energy would have to be localized in it than in mine to produce our relative motion. If we did not have the universal principle of relativity to forbid, we might make an arbitrary assumption about our motions and hence about our respective latent energies; in the presence of this veto, the only chance of adjustment lies in our masses, which must differ according to whether you or I observe them.]*

[For most of the velocities with which we are familiar $E/C^2$ is, like the difference between $K$ and unity, such an extremely small quantity that the most delicate measurements fail to detect it. But the electrons in a highly evacuated tube and the particles shot out from radioactive materials attain in some cases velocities as high as eight-tenths that of light. When we measure the mass of such particles at different velocities we find that it actually increases with the velocity, and in accordance with the foregoing law.]194 [This observation, in fact, antedates Einstein's explanation, which is far more satisfactory than the earlier differentiation between "normal mass" and "electrical mass" which was called upon to account for the increase.]*

[But if the quantity $E/C^2$ is to be considered as an actual increase in mass, may it not be possible that all mass is energy? This would lead to the conclusion that the energy stored up in any mass is $mC^2$. The value is very great, since $C$ is so large; but it is in good agreement with the internal energy of the atom as calculated from other considerations. It is obvious that conservation of mass and of momentum cannot both hold good under a theory that translates the one into the other. Mass is then not considered by Einstein as conservative in the ordinary sense, but it is the total quantity of mass plus energy in any closed system that remains constant. Small amounts of energy may be transformed into mass, and vice versa.]194

[Other features of the theory which are often displayed as consequences are really more in the nature of assumptions. It will be recalled that when we had agreed upon the necessity of employing signals of some sort, we selected as the means of signalling the speediest messenger with which we happened to be acquainted. Our subsequent difficulties were largely due to the impossibility of making a proper allowance for this messenger's speed, even though we knew its numerical value; and as a consequence, this speed enters into our formulae. Now we have not said in so many words that $C$ is the greatest speed attainable, but we have tacitly assumed that it is. We need not, therefore, be surprised if our formulae give us absurd results for speeds higher than $C$, and indicate the impossibility of ever attaining these. Whatever we put into a problem the algebra is bound to give us back. If we look at our formula for $K$, we see that in the event of $v$ equalling $C$, lengths become zero and times infinite. The light messenger itself, then, has no dimension; and for it time stands still.

If we suppose $v$ to be greater than $C$, we get even more bizarre results, for then the factor $K$ is the square root of a negative number, or as the mathematician calls it an "imaginary" quantity; and with it, lengths and times become imaginary too.

The fact that time stops for it, and the fact that it is the limiting velocity, give to $C$ certain of the attributes of the mathematician's infinity. Certainly if it can never be exceeded, we must have a new formula for the composition of velocities. Otherwise when my system passes yours at a speed of 100,000 miles per second, while yours passes a third in the same direction at the same velocity, I shall be passing this third framework at the forbidden velocity of 200,000 miles per second--greater than $C$. In fact Einstein is able to show that an old formula, which had already been found to connect the speed of light in a material medium with the speed of that medium, will now serve universally for the composition of velocities. When we combine the velocities $v$ and $u$, instead of getting the resultant $v+u$ as we would have supposed, we get the resultant

$$(v + u) / (1 + (uv / C^2))\,,$$ or $$C^2(v + u) / (C^2 + uv)$$

This need not surprise us either, if we will but reflect that the second velocity effects a second revision of length and time measurements between the systems involved. And now, if we let either $v$, or $u$, or even both of them, take the value $C$, the resultant still is $C$. In another way we have found $C$ to behave like the mathematician's infinity, to which, in the words of the blind poet, if we add untold thousands, we effect no real increment.

ASSUMPTION AND CONSEQUENCE

A good many correspondents who have given the subject sufficient thought to realize that the limiting character of the velocity $C$ is really read into Einstein's system by assumption have written, in more or less perturbed inquiry, to know whether this does not invalidate the whole structure. The answer, of course, is yes--provided you can show this assumption to be invalid. The same answer may be made of any scientific doctrine whatever, and in reference to any one of the multitudinous assumptions underlying it. If we were to discover, tomorrow, a way of sending signals absolutely instantaneously, Einstein's whole structure would collapse as soon as we had agreed to use this new method. If we were to discover a signalling agent with finite velocity greater than that of light, relativity would persist with this velocity written in its formulae in the place of $C$.

It is a mistake to quote Einstein's theory in support of the statement that such a velocity can never be. An assumption proves its consequences, but never can prove itself; it must remain always an assumption. But in the presence of long human experience supporting Einstein's assumption that no velocity in excess of $C$ can be found, it is fair to demand that it be disputed not with argument but with demonstration. The one line of argument that would hold out a priori hope of reducing the assumption to an absurdity would be one based on the familiar idea of adding velocities; but Einstein has spiked this argument before it is started by replacing the direct addition of velocities with another method of combining them that fits in with his assumption and as well with the observed facts. The burden of proof is then on the prosecution; anyone who would contradict our assertion that $C$ is the greatest velocity attainable may do so only by showing us a greater one. Until this has been done, the admission that it may properly be attempted can in no way be construed as a confession of weakness on the part of Einstein.

It may be well to point out that in no event may analogy be drawn with sound, as many have tried to do. In the first place sound requires a material medium and its velocity with regard to this rather than relative to the observer we know to be fixed; in the second place, requiring a material medium, sound is not a universal signalling agent; in the third place, we know definitely that its velocity can be exceeded, and are therefore barred from making the assumption necessary to establish the analogy. The very extraordinary behavior of light in presenting a velocity that is the same for all observers, and in refusing to betray the least material evidence of any medium for its transmission, rather fortifies us in believing that Einstein's assumption regarding the ultimate character of this velocity is in accord with the nature of things.

RELATIVITY AND THE LAYMAN

A great deal can be said in the direction of general comment making the Special Theory and its surprising accompaniments easier of acceptance, and we shall conclude the present discussion by saying some of these things.]* [It has been objected that the various effects catalogued above are only apparent, due to the finite velocity of light--that the real shape and size of a body or the real time of an event cannot be affected by the point of view or the motion of an observer. This argument would be perfectly valid, if there were real times and distances; but there are not. These are earth-bound notions, due to our experience on an apparently motionless platform, with slow-moving bodies. Under these circumstances different observations of the same thing or of the same event agree. But when we no longer have the solid earth to stand on, and are dealing with velocities so high that the relativity effects become appreciable, there is no standard by which to resolve the disagreements. No one of the observations can claim to be nearer reality than any other. To demand the real size of a thing is to demand a stationary observer or an instantaneous means of information. Both are impossible.

When relativity asks us to give up our earth-bound notions of absolute space and absolute time the sensation, at first, is that we have nothing left to stand on. So must the contemporaries of Columbus have felt when told that the earth rested on--nothing. The remedy too is similar. Just as they had to be taught that falling is a local affair, that the earth is self-contained, and needs no external support--so we must be taught that space and time standards are local affairs. Each moving body carries its own space and time standards with it; it is self-contained. It does not need to reach out for eternal support, for an absolute space and time that can never quite be attained. All we ever need to know is the relation of the other fellow's space and time standards to our own. This is the first thing relativity teaches us.]141

[The consequences of Einstein's assumptions have led many to reject the theory of relativity, on the ground that its conclusions are contrary to common sense--as they undoubtedly are. But to the contemporaries of Copernicus and Galileo the theory that the earth rotates on its axis and revolves around the sun was contrary to common sense; yet this theory prevailed. There is nothing sacred about common sense; in the last analysis its judgments are based on the accumulated experience of the human race. From the beginning of the world up to the present generation, no bodies were known whose velocities were not extremely small compared with that of light. The development of modern physics has led to discovery of very much larger velocities, some as high as 165,000 miles per second. It is not to be wondered at that such an enlargement of our experience requires a corresponding enlargement or generalization of the concepts of space and time. Just as the presupposition of primitive man that the earth was flat had to be given up in the light of advancing knowledge, so we are now called upon to give up our presupposition that space and time are absolute and independent in their nature.

The reader must not expect to understand the theory of relativity in the sense of making it fit in with his previous ideas. If the theory be right these ideas are wrong and must be modified, a process apt to be painful.]223 [All the reader can do is to become familiar with the new concepts, just as a child gets used to the simple relations and quantities he meets until he "understands" them.]221 [Mr. Francis has said something of the utmost significance when he points out that "understanding" really means nothing in the world except familiarity and accustomedness.]* [The one thing about the relativity doctrine that we can hope thus to understand at once and without pain is the logical process used in arriving at our results.]221 [Particularly is it hard to give a satisfactory explanation of the theory in popular language, because the language itself is based on the old concepts; the only language which is really adequate is that of mathematics.]223 [Unless we have, in addition to the terms of our ordinary knowledge, a set of definitions that comes with a wide knowledge of mathematics and a lively sense of the reality of mathematical constructions, we are likely to view the theory of relativity through a fog of familiar terms suddenly become self-contradictory and deceptive. Not that we are unfamiliar with the idea that some of our habitual notions may be wrong; but knowledge of their illusory nature arises and becomes convincing only with time. We may now be ready to grant that the earth, seemingly so solid, is really a whirling globe rushing through space; but we are no more ready immediately to accept the bald assertion that this space is not what it seems than our ancestors were to accept the idea that the earth was round or that it moved.]156 [What we must have, if we are to comprehend relativity with any degree of thoroughness, is the mathematician's attitude toward his assumptions, and his complete readiness to swap one set of assumptions for another as a mere part of the day's work, the spirit of which I have endeavored to convey in the chapter on non-Euclidean geometry.]*

PHYSICS VS. METAPHYSICS

[The ideas of relativity may seem, at first sight, to be giving us a new and metaphysical theory of time and space. New, doubtless; but certainly the theory was meant by its author to be quite the opposite of metaphysical. Our actual perception of space is by measurement, real and imagined, of distances between objects, just as our actual perception of time is by measurement. Is it not less metaphysical to accept space and time as our measurements present them to us, than to invent hypotheses to force our perceptual space into an absolute space that is forever hidden from us?]182 [In order not to be metaphysical, we must eliminate our preconceived notions of space and time and motion, and focus our attention upon the indications of our instruments of observation, as affording the only objective manifestations of these qualities and therefore the only attributes which we can consider as functions of observed phenomena.]47 [Einstein has consistently followed the teachings of experience, and completely freed himself from metaphysics.]114 [That this is not always easy to do is clear, I think, if we will recall the highly metaphysical character often taken by the objections to action-at-a-distance theories and concepts; and if we will remind ourselves that it was on purely metaphysical grounds that Newton refused to countenance Huyghens' wave theory of light. Whether, as in the one case, it leads us to valid conclusions, or, as in the other, to false ones, metaphysical reasoning is something to avoid. Einstein, I think, has avoided it about as thoroughly as anyone ever did.]*

V

THAT PARALLEL POSTULATE

Modern Geometric Methods; the Dividing Line Between Euclidean and Non-Euclidean; and the Significance of the Latter

BY THE EDITOR

The science of geometry has undergone a revolution of which the outsider is not informed, but which it is necessary to understand if we are to attain any comprehension of the geometric formulation of Einstein's results; and especially if we are to appreciate why it is proper and desirable to formulate these results geometrically at all. The classical geometer regarded his science from a narrow viewpoint, as the study of a certain set of observed phenomena--those of the space about us, considered as an entity in itself and divorced from everything in it. It is clear that some things about that space are not as they appear (optical illusions), and that other things about it are true but by no means apparent (the sum-of-squares property of a right triangle, the formulæ for surface and volume of a sphere, etc.). While many things about space are "obvious," these need in the one case disproof and in the other discovery and proof. With all their love of mental processes for their own sake, it is then not surprising that the Greeks should have set themselves the task of proving by logical process the properties of space, which a less thoughtful folk would have regarded as a subject only for observational and experimental determination.

But, abstract or concrete, the logical structure must have a starting point; and it is fair to demand that this consist in a statement of the terms we are going to use and the meanings we are going to attach to them. In other words, the first thing on the program will be a definition, or more probably, several definitions.

Now the modern scientist has a somewhat iconoclastic viewpoint toward definitions, and especially toward the definition of his very most fundamental ideas.

We do not speak here in terms of dictionary definitions. These have for object the eminent necessity of explaining the meaning and use of a word to some one who has just met it for the first time. It is easy enough to do this, if the doer possesses a good command of the language. It is not even a matter of grave concern that the words used in the definition be themselves known to the reader; if they are not, he must make their acquaintance too. Dr. Johnson's celebrated definition of a needle stands as perpetual evidence that when he cannot define a simple thing in terms of things still simpler, the lexicographer is forced to define it in terms of things more complex. Or we might demonstrate this by noting that the best dictionaries are driven to define such words as "and" and "but" by using such complex notions as are embodied in "connective," "continuative," "adversative," and "particle."

It is otherwise with the scientist who undertakes to lay down a definition as the basis of further procedure in building up the tissue of his science. Here a degree of rigorous logic is called for which would be as superfluous in the dictionary as the effort there to attain it would be out of place. The scientist, in building up a logical structure that will withstand every assault, must define everything, not in terms of something which he is more or less warranted in supposing his audience to know about, but actually in terms of things that have already been defined. This really means that he must explain what he is talking about in terms of simpler ideas and simpler things, which is precisely what the lexicographer does not have to worry about. This is why it is quite trivial to quote a dictionary definition of time or space or matter or force or motion in settlement of a controversy of scientific or semi-scientific nature.

TERMS WE CANNOT DEFINE

But the scientist who attempts to carry out this ideal system of defining everything in terms of what precedes meets one obstacle which he cannot surmount directly. Even a layman can construct a passable definition of a complex thing like a parallelopiped, in terms of simpler concepts like point, line, plane and parallel. But who shall define point in terms of something simpler and something which precedes point in the formulation of geometry? The scientist is embarrassed, not in handling the complicated later parts of his work, but in the very beginning, in dealing with the simplest concepts with which he has to deal.

Suppose a dictionary were to be compiled with the definitions arranged in logical rather than alphabetical order: every word to be defined by the use only of words that have already been defined. The further back toward the beginning we push this project, the harder it gets. Obviously we can never define the first word, or the second, save as synonymous with the first. In fact we should need a dozen words, more or less, to start with--God-given words which we cannot define and shall not try to define, but of which we must agree that we know the significance. Then we have tools for further procedure; we can start with, say, the thirteenth word and define all the rest of the words in the language, in strictly logical fashion.

What we have said about definitions applies equally to statements of fact, of the sort which are going to constitute the body of our science. In the absence of simpler facts to cite as authority, we shall never be able to prove anything, however simple this may itself be; and in fact the simpler it be, the harder it is to find something simpler to underlie it. If we are to have a logical structure of any sort, we must begin by laying down certain terms which we shall not attempt to define, and certain statements which we shall not try to prove. Mathematics, physics, chemistry--in the large and in all their many minor fields--all these must start somewhere. Instead of deceiving ourselves as to the circumstances surrounding their start, we prefer to be quite frank in recognizing that they start where we decide to start them. If we don't like one set of undefined terms as the foundation, by all means let us try another. But always we must have such a set.

The classical geometer sensed the difficulty of defining his first terms. But he supposed that he had met it when he defined these in words free of technical significance. "A point is that which has position without size" seemed to him an adequate definition, because "position" and "size" are words of the ordinary language with which we may all be assumed familiar. But today we feel that "position" and "size" represent ideas that are not necessarily more fundamental than those of "line" and "point," and that such a definition begs the question. We get nowhere by replacing the undefined terms "point" and "line" and "plane," which really everybody understands, by other undefined terms which nobody understands any better.

In handling the facts that it was inconvenient to prove, the classical geometer came closer to modern practice. He laid down at the beginning a few statements which he called "axioms," and which he considered to be so self-evident that demonstration was superfluous. That the term "self-evident" left room for a vast amount of ambiguity appears to have escaped him altogether. His axioms were axioms solely because they were obviously true.

LAYING THE FOUNDATION

The modern geometer falls in with Euclid when he writes an elementary text, satisfying the beginner's demand for apparent rigor by defining point and line in some fashion. But when he addresses to his peers an effort to clarify the foundations of geometry to a further degree of rigor and lucidity than has ever before been attained, he meets these difficulties from another quarter. In the first place he is always in search of the utmost possible generality, for he has found this to be his most effective tool, enabling him as it does to make a single general statement take the place and do the work of many particular statements. The classical geometer attained generality of a sort, for all his statements were of any point or line or plane. But the modern geometer, confronted with a relation that holds among points or between points and lines, at once goes to speculating whether there are not other elements among or between which it holds. The classical geometer isn't interested in this question at all, because he is seeking the absolute truth about the points and lines and planes which he sees as the elements of space; to him it is actually an object so to circumscribe his statements that they may by no possibility refer to anything other than these elements. Whereas the modern geometer feels that his primary concern is with the fabric of logical propositions that he is building up, and not at all with the elements about which those propositions revolve.

It is of obvious value if the mathematician can lay down a proposition true of points, lines and planes. But he would much rather lay down a proposition true at once of these and of numerous other things; for such a proposition will group more phenomena under a single principle. He feels that on pure scientific grounds there is quite as much interest in any one set of elements to which his proposition applies as there is in any other; that if any person is to confine his attention to the set that stands for the physicist's space, that person ought to be the physicist, not the geometer. If he has produced a tool which the physicist can use, the physicist is welcome to use it; but the geometer cannot understand why, on that ground, he should be asked to confine his attention to the materials on which the physicist employs that tool.

It will be alleged that points and lines and planes lie in the mathematician's domain, and that the other things to which his propositions may apply may not so lie--and especially that if he will not name them in advance he cannot expect that they will so lie. But the mathematician will not admit this. If mathematics is defined on narrow grounds as the science of number, even the point and line and plane may be excluded from its field. If any wider definition be sought--and of course one must be--there is just one definition that the mathematician will accept: Dr. Keyser's statement that "mathematics is the art or science of rigorous thinking."

The immediate concern of this science is the means of rigorous thinking--undefined terms and definitions, axioms and propositions. Its collateral concern is the things to which these may apply, the things which may be thought about rigorously--everything. But now the mathematician's domain is so vastly extended that it becomes more than ever important for him to attain the utmost generality in all his pronouncements.

One barrier to such generalization is the very name "geometry," with the restricted significance which its derivation and long usage carry. The geometer therefore must have it distinctly understood that for him "geometry" means simply the process of deducing a set of propositions from a set of undefined primitive terms and axioms; and that when he speaks of "a geometry" he means some particular set of propositions so deduced, together with the axioms, etc., on which they are based. If you take a new set of axioms you get a new geometry.

The geometer will, if you insist, go on calling his undefined terms by the familiar names "point," "line," "plane." But you must distinctly understand that this is a concession to usage, and that you are not for a moment to restrict the application of his statements in any way. He would much prefer, however, to be allowed new names for his elements, to say "We start with three elements of different sorts, which we assume to exist, and to which we attach the names A, B and C--or if you prefer, primary, secondary and tertiary elements--or yet again, names possessing no intrinsic significance at all, such as ching, chang and chung." He will then lay down whatever statements he requires to serve the purposes of the ancient axioms, all of these referring to some one or more of his elements. Then he is ready for the serious business of proving that, all his hypotheses being granted, his elements A, B and C, or I, II and III, or ching, chang and chung, are subject to this and that and the other propositions.

The objection will be urged that the mathematician who does all this usurps the place of the logician. A little reflection will show this not to be the case. The logician in fact occupies the same position with reference to the geometer that the geometer occupies with reference to the physicist, the chemist, the arithmetician, the engineer, or anybody else whose primary interest lies with some particular set of elements to which the geometer's system applies. The mathematician is the tool-maker of all science, but he does not make his own tools--these the logician supplies. The logician in turn never descends to the actual practice of rigorous thinking, save as he must necessarily do this in laying down the general procedures which govern rigorous thinking. He is interested in processes, not in their application. He tells us that if a proposition is true its converse may be true or false or ambiguous, but its contrapositive is always true, while its negative is always false. But he never, from a particular proposition "If A is B then C is D," draws the particular contrapositive inference "If C is not D then A is not B." That is the mathematician's business.

THE RÔLE OF GEOMETRY

The mathematician is the quantity-production man of science. In his absence, the worker in each narrower field where the elements under discussion take particular concrete forms could work out, for himself, the propositions of the logical structure that applies to those elements. But it would then be found that the engineer had duplicated the work of the physicist, and so for many other cases; for the whole trend of modern science is toward showing that the same background of principles lies at the root of all things. So the mathematician develops the fabric of propositions that follows from this, that and the other group of assumptions, and does this without in the least concerning himself as to the nature of the elements of which these propositions may be true. He knows only that they are true for any elements of which his assumptions are true, and that is all he needs to know. Whenever the worker in some particular field finds that a certain group of the geometer's assumptions is true for his elements, the geometry of those elements is ready at hand for him to use.

Now it is all right purposely to avoid knowing what it is that we are talking about, so that the names of these things shall constitute mere blank forms which may be filled in, when and if we wish, by the names of any things in the universe of which our "axioms" turn out to be true. But what about these axioms themselves? When we lay them down in ignorance of the identity of the elements to which they may eventually apply, they cannot by any possibility be "self-evident." We may, at pleasure, accept as self-evident a statement about points and lines and planes; or one about electrons, centimeters and seconds; or one about integers, fractions, and irrational numbers; or one about any other concrete thing or things whatever. But we cannot accept as self-evident a statement about chings, changs and chungs. So we must base our "axioms" on some other ground than this; and our modern geometer has his ground ready and waiting. He accepts his axioms on the ground that it pleases him to do so. To avoid all suggestion that they are supposed to be self-evident, or even necessarily true, he drops the term "axiom" and substitutes for it the more color-less word "postulate." A postulate is merely something that we agreed to accept, for the time being, as a basis of further argument. If it turns out to be true, or if we can find circumstances under which and elements to which it applies, any conclusions which we deduce from it by trustworthy processes are valid within the same limitations. And the propositions which tell us that, if our postulates are true, such and such conclusions are true--they, too, are valid, but without any reservation at all!

Perhaps an illustration of just what this means will not be out of place. Let it be admitted, as a postulate, that $7+20$ is greater, by 1, than $7+19$. Let us then consider the statement: "If $7 + 19 = 65$, then $7 + 20 = 66$." We know--at least we are quite certain--that $7 + 19$ is not equal to 65, if by "7" and "19" and "65" we mean what you think we mean. We are equally sure, on the same grounds, that $7 + 20$ is not equal to 66. But, under the one assumption that we have permitted ourselves, it is unquestionable that if $7 + 19$ were equal to 65, then $7 + 20$ certainly would be equal to 66. So, while the conclusion of the proposition which I have put in quotation marks is altogether false, the proposition itself, under our assumption, is entirely true. I have taken an illustration designed to be striking rather than to possess scientific interest; I could just as easily have shown a true proposition leading to a false conclusion, but of such sort that it would be of decided scientific interest as telling us one of the consequences of a certain assumption.

WHAT MAY WE TAKE FOR GRANTED?

This is all very fine; but how does the geometer know what postulates to lay down? One is tempted to say that he is at liberty to postulate anything he pleases, and investigate the results; and that whether or not his postulate ever be realized, the propositions that he deduces from it, being true, are of scientific interest. Actually, however, it is not quite as simple as all that. If it were sufficient to make a single postulate it would be as simple as all that; but it turns out that this is not sufficient any more than it is sufficient to have a single undefined term. We must have several postulates; and they must be such, as a whole, that a geometry flows out of them. The requirements are three.

In the first place, the system of postulates must be "categorical" or complete--there must be enough of them, and they must cover enough ground, for the support of a complete system of geometry. In practice the test for this is direct. If we got to a point in the building up of a geometry where we could not prove whether a certain thing was one way always, or always the other way, or sometimes one way and sometimes the other, we should conclude that we needed an additional postulate covering this ground directly or indirectly. And we should make that postulate--because it is precisely the things that we can't prove which, in practical work, we agree to assume. Even Euclid had to adopt this philosophy.

In the second place, the system of postulates must be consistent--no one or more of them may lead, individually or collectively, to consequences that contradict the results or any other or others. If in the course of building up a geometry we find we have proved two propositions that deny one another, we search out the implied contradiction in our postulates and remedy it.

Finally, the postulates ought to be independent. It should not be possible to prove any one of them as a consequence of the others. If this property fails, the geometry does not fail with it; but it is seriously disfigured by the superfluity of assumptions, and one of them should be eliminated. If we are to assume anything unnecessarily, we may as well assume the whole geometry and be done with it.

The geometer's business then is to draw up a set of postulates. This he may do on any basis whatever. They may be suggested to him by the behavior of points, lines and planes, or by some other concrete phenomena; they may with equal propriety be the product of an inventive imagination. On proceeding to deduce their consequences, he will discover and remedy any lack of categoricity or consistence or independence which his original system of postulates may have lacked. In the end he will have so large a body of propositions without contradiction or failure that he will conclude the propriety of his postulates to have been established, and the geometry based on them to be a valid one.

AND WHAT IS IT ALL ABOUT?

Is this geometry ever realized? Strictly it is not the geometer's business to ask or answer this question. But research develops two viewpoints. There is always the man who indulges in the pursuit of facts for their sake alone, and equally the man who wants to see his new facts lead to something else. One great mathematician is quoted as enunciating a new theory of surpassing mathematical beauty with the climacteric remark "And, thank God, no one will ever be able to find any use for it." An equally distinguished contemporary, on being interrogated concerning possible applications for one of his most abstruse theorems, replied that he knew no present use for it; but that long experience had made him confident that the mathematician would never develop any tool, however remote from immediate utility, for which the delvers in other fields would not presently find some use.

If we wish, however, we may inquire with perfect propriety, from the side lines, whether a given geometry is ever realized. We may learn that so far as has yet been discovered there are no elements for which all its postulates are verified, and that there is therefore no realization known. On the other hand, we may more likely find that many different sets of elements are such that the postulates can be interpreted as applying to them, and that we therefore have numerous realizations of the geometry. As a human being the geometer may be interested in all this, but as a geometer it really makes little difference to him.

When we look at space about us, we see it, for some reason grounded in the psychological history of the human race, as made up in the small of points, which go to make up lines, which in turn constitute planes. Or we can start at the other end and break space down first into planes, then into lines, finally into points. Our perceptions and conceptions of these points, lines and planes are very definite indeed; it seems indeed, as the Greeks thought, that certain things about them are self-evident. If we wish to take these self-evident properties of point, line and plane, and combine with them enough additional hair-splitting specifications to assure the modern geometer that we have really a categorical system of assumptions, we shall have the basis of a perfectly good system of geometry. This will be what we unavoidably think of as the absolute truth with regard to the space about us; but you mustn't say so in the presence of the geometer. It will also be what we call the Euclidean geometry. It has been satisfactory in the last degree, because not only space, but pretty much every other system of two or three elements bearing any relations to one another can be made, by employing as a means of interpretation the Cartesian scheme of plotting, to fit into the framework of Euclidean geometry. But it is not the only thing in the world of conceptual possibilities, and it begins to appear that it may not even be the only thing in the world of cold hard fact that surrounds us. To see just how this is so we must return to Euclid, and survey the historical development of geometry from his day to the present time.

EUCLID'S GEOMETRY

Point, line and plane Euclid attempts to define. Modern objection to these efforts was made clear above. Against Euclid's specific performance we urge the further specific fault that his "definitions" are really assumptions bestowing certain properties upon points, lines and planes. These assumptions Euclid supplements in his axioms; and in the process of proving propositions he unconsciously supplements them still further. This is to be expected from one whose justification for laying down an axiom was the alleged obvious character of the statement made. If some things are too obvious to require demonstration, others may be admitted as too obvious to demand explicit statement at all.

Thus, if Euclid has two points A and B in a plane, on opposite sides of a line M, he will draw the line AB and without further formality speak of the point C in which it intersects M. That it does so intercept M, rather than in some way dodges it, is really an assumption as to the nature of lines and planes. Or again, Euclid will speak of a point D on the line AB, between or outside the points A and B, without making the formal assumption necessary to insure that the line is "full" of points so that such a point as D must exist. That such assumptions as these are necessary follows from our previous remarks. If we think of our geometry as dealing with "chings," "changs," and "chungs," or with elements I, II and III, it is no longer in the least degree obvious that the simplest property in the world applies to these elements. If we wish any property to prevail we must state it explicitly.