coefficients 9pq of the quadratic form can be interpreted, as in Einstein's theory, as the potentials of gravitation, while the four coefficients op of the linear form can be interpreted as the scalar-potential and the three components of the vector-potential in Maxwell's electromagnetic theory. Thus Weyl succeeded in exhibiting both gravitation and electricity as effects of the metric of the world. The enlargement of geometrical ideas thus achieved was soon followed by still wider extensions of the same character, due to Eddington, Schouten, Wirtinger and others. From the point of view of the geometer, they constituted striking and valuable advances in his subject, and they seemed to offer an attractive prospect to the physicist of combining the whole of our knowledge of the material universe into a single unified theory. The working out of the various possible alternative schemes for identifying these more general geometries with physics has been the chief occupation of relativists during the last nine years. Many ingenious proposals and adaptations have been published, and more than one author has triumphantly announced that at last the problem has been solved. But I do not think that any of the theories can be regarded as satisfactory, and within the last year or two a note of doubt has been perceptible; were we after all on the right track? At last Einstein himself1 has made up his mind and renounced the whole movement. The present position, then, is that the years 1918-1926 have been spent chiefly in researches which, while they have contributed greatly to the progress of geometry, have been on altogether wrong lines so far as physics is concerned, and we have now to go back to the pre-1918 position and make a fresh start, with the definite conviction that the geometry of space-time is Riemannian. Granting then this fundamental understanding, we have now to inquire into the axiomatics of the theory. This part of the subject has received less attention in 4 Math. Ann. 97 (1926), p. 99. our country than elsewhere, perhaps because of the is more or less accidental circumstance that the most prominent and distinguished exponents of relativity in England happened to be men whose work lay in the i field of physics and astronomy rather than in mathe-de matics, and who were not specially interested in questions of logic and rigor. It is, however, evidently of t the highest importance that we should know exactly what assumptions must be made in order to deduce our equations, especially since the subject is still in a de rather fluid condition, and there is a possibility of th effecting some substantial improvement in it by a partial reconstruction of the foundations. What we want to do, then, is to set forth the axiomatics of general relativity in the same form as we have been accustomed to give to the axiomatics of any other kind of geometry-that is, to enunciate the primitive or undefined concepts, then the definitions, the axioms, and the existence-theorems, and lastly the deductions. In the course of the work we must prove that the axioms are compatible with each other, and that no one of them is superfluous. The usual way of introducing relativity is to talk about measuring-rods and clocks. This is, I think, a very natural and proper way of introducing the doctrine known as "special relativity," which grew out of FitzGerald's hypothesis of the contraction of moving bodies, and was first clearly stated by Poincaré in 1904, and further developed by Einstein in 1905. But general relativity, which came ten years later, is a very different theory. In general relativity there are no such things as rigid bodies that is, bodies for which the mutual distance of every pair of particles remains unaltered when the body moves in the gravitational field. That being so, it seems desirable to avoid everything akin to a rigid body-such, for example, as measuring-rods or clocks—when we are laying down the axioms of the subject. The axioms should obviously deal only with the simplest constituents of the universe. Now if one of my clocks or watches goes wrong, I don't venture to try and mend it myself, but take it to a professional clockmaker, and even he is not always wholly successful, which seems to me to indicate that a clock is not one of the simplest constituents of the universe. Some of the expounders of relativity have recognized the existence of this difficulty, and have tried to turn it by giving up the ordinary material clock with its elaborate mechanism, and putting forward in its place what they call an atomic clock; by which they mean a single atom in a gas, emitting light of definite frequency. Unfortunately the atom is apparently quite as complicated in its working as a material clock, perhaps more so, and is less understood; and the statement that the frequency is the same under all conditions, whatever is happening to the atom, is (whether true or not) a highly complex assumption which could Er scarcely be used in an axiomatic treatment of the subject until it has been dissected into a considerable i: number of elementary axioms, some of them perhaps of a disputable character. 让 It seems to me that we should abandon measuringrods and accurate clocks altogether, and begin with te something more primitive. Let us then take any system of reference for events-a network of points to each of which three numbers are assigned-which can serve as spatial coordinates, and a number indicating the succession of events at each point to serve as a temporal coordinate. Let us now refer to this coordinate system, the paths which are traced by infinitesimal particles moving freely in the gravitational field. Then it is one of the fundamental assumptions of the theory that these paths are the geodesics belonging to a certain quadratic differential form The truth or falsity of this assumption may, in theory at any rate, be tested by observation, since if the paths It are geodesics they must satisfy certain purely geogmetrical conditions, and whether they do or not is a question to be settled by experience. Granting for the present that the paths do satisfy these conditions, let us inquire if a knowledge of the paths or geodesics is sufficient to enable us to determine the quadratic form. The answer to this is in the negative, as may easily be seen if we consider for a moment the non-Euclidean geometry defined by a Cayley-Klein metric in three-dimensional space. In the Cayley-Klein geometry the geodesics are the straight lines of the space; but a knowledge of this fact is not sufficient to determine the metric, since the absolute may be any arbitrary quadric surface. In order to determine the quadratic form in general relativity we must then be furnished with some information besides the knowledge of the paths of material particles. It is sufficient, as Levi-Civita has remarked, that we should be given the null geodesics, i.e., the geodesics along which the quadratic form vanishes. In the Cayley-Klein geometry these are the tangents to the absolute; in general relativity they are simply the tracks of rays of light. So from our knowledge of the paths of material particles and the tracks of rays of light we can construct the quadratic form with respect to all point-transformations of coordinates." The theory is now fairly launched and I need not describe its axiomatic development further. The point I wish specially to make is that in the above treatment there has been no mention either of length or of time: neither measuring-rod nor clock has been introduced in any way. We have left open the question whether the quadratic form does or does not represent anything which can be given directly by measuring-rods and clocks. For my own part I incline to think that the notions of length of material bodies, and time of clocks, are really rather complex notions which do not normally occur in the early chapters of axiomatic physics. The results of the ether-drift experiments of D. C. Miller at Mount Wilson in 1925, if confirmed, would seem to indicate that the geometry which is based on rigid measuring-rods is actually different from the geometry which is based on geodesics and light-rays. The actual laws of nature are most naturally derived, it seems to me, from the Minimum Principle enunciated in 1915 by Hilbert, that "all physical happenings (gravitational, electrical, etc.) in the Universe are determined by a scalar world-function being, in fact, such as to annul the variation of the integral SSSSdxdxdxdx ̧.” This principle is the grand culmination of the movement begun 2,000 years ago by Hero of Alexandria with his discovery that reflected light meets the mirror at a point such that the total path between the source of light and the eye is the shortest possible. In the seventeenth century Hero's theorem was generalized by Fermat into his "Principle of Least Time" that "Nature always acts by the shortest course," which suffices for the solution of all problems in geometrical optics. A hundred years later this was further extended by Maupertuis, Euler and Lagrange into a general principle of "Least Action" of dynamical systems, and in 1834 Hamilton formulated his famous principle which was found to be capable of reducing all the known laws of nature-gravitational, dynamical and electrical-to a representation as minimumproblems. Hilbert's minimum principle in general relativity is a direct application of Hamilton's principle, in which the contribution made by gravitation is the integral of the Riemann scalar curvature. Thus gravitation acts so as to make the total amount of the curvature of space-time a minimum: or as we may say, gravitation simply represents a continual effort of the universe to straighten itself out. This is general relativity in a single sentence. I have already explained that the curvature of space-time at any point at any instant depends on the physical events that are taking place there: in statical systems, where we can consider space of three dimensions separately from time, the mean curvature (ie., the sum of the three principal curvatures) of the space at any point is proportional to the energydensity at the point. Since, then, the curvature of space is wholly governed by physical phenomena, the suggestion presents itself that the metric of spacetime may be determined wholly by the masses and energy present in the universe, so that space-time can not exist at all except in so far as it is due to the existence of matter. This doctrine, which is substantially due to Mach, was adopted in 1917 by Einstein, and has led to some interesting developments. The point at issue may be illustrated by the following concrete problem: if all matter were annihilated except one particle which is to be used as a test-body, would this particle have inertia or not? The view of Mach and Einstein is that it would not; and in support of this view it may be urged that, according to the deductions of general relativity, the inertia of a body is increased when it is in the neighborhood of other large masses; it seems needless, therefore, to postulate other sources of inertia, and simplest to suppose that all inertia is due to the presence of other masses. When we confront this hypothesis with the facts of observation, however, it seems clear that the masses of whose existence we know the solar systems, stars, and nebulæ are insufficient to confer on terrestrial bodies the inertia which they actually possess; and therefore if Mach's principle were adopted, it would be necessary to postulate the existence of enormous quantities of matter in the universe which have not been detected by astronomical observation, and which are called into being simply in order to account for inertia in other bodies. This is, after all, no better than regarding some part of inertia as intrinsic. Under the influence of Mach's doctrine, Einstein made an important modification of the field-equations of gravitation. He now objected to his original equations of 1915 on the ground that they possessed a solution even when the universe was supposed void of matter, and he added a term-the "cosmological term" as it is called-with the idea of making such a solution impossible. After a time it was found that the new term did not do what it had been intended to do, for the modified field-equations still possessed a solution-the celebrated "De Sitter World"-even when no matter was present; but the De Sitter world was found to be so excellent an addition to the theory that it was adopted permanently, and with it of course the cosmological term in the field-equations; so that this term has been retained for exactly the opposite reason to that for which it was originally introduced. The "De Sitter World" is simply the universe as it would be if all minor irregularities were smoothed out just as when we say that the earth is a spheroid, we mean that the earth would be a spheroid if all mountains were leveled and valleys filled up. In the case of the De Sitter universe the leveling is a more formidable operation, since we have to smooth out the earth, the sun, and all the heavenly bodies, and reduce the world to a complete uniformity. But after all, only a very small fraction of the cosmos is occupied by material bodies; and it is interesting to inquire what space-time as a whole is like when we simply ignore them. The answer is, as we should expect, that it is a manifold of constant curvature. This means that it is isotropic (i.e., the Riemann curvature is the same for all orientations at the same point), and is also homogeneous. As a matter of fact, there is a wellknown theorem that any manifold which is isotropic in this sense is necessarily also homogeneous, so that the two properties are connected. A manifold of constant curvature is a projective manifold, i.e., ordinary projective geometry is valid in it when we regard geodesics as straight lines; and it is possible to move about in it any system of points, discrete or continuous, rigidly, i.e., so that the mutual distances are unaltered. The simplest example of a manifold of constant curvature is the surface of a sphere in ordinary threedimensional Euclidean space; and the easiest way of constructing a model of the De Sitter world is to take a pseudo-Euclidean manifold of five dimensions in which the line-element is specified by the equation − ds2 = dx2 + dy2 + dz2 – du2 + dv2 where u is time. Hyperplanes which do not intersect the absolute are spatial, so spatial measurements are elliptic, i.e., the three-dimensional world of space has the same kind of geometry as the surface of a sphere, differing from it only in being three-dimensional instead of two-dimensional. In such a geometry there is a natural unit of length, namely, the length of the complete straight line, just as on the surface of a sphere there is a natural unit of length, namely, the length of a complete great circle. The We are thus brought to the question of the dimensions of the universe: what is the length of the complete straight line, the circuit of all space? answer must be furnished by astrophysical observations, interpreted by a proposition which belongs to the theory of De Sitter's world, namely, that the lines of the spectrum of a very distant star should be systematically displaced; the amount of displacement is proportional to the ratio of the distance of the star from the observer to the constant radius of curvature R of the universe. In attempting to obtain the value of R from this formula we meet with many difficulties: the effect is entangled with the ordinary Doppler effect due to the radial velocity of the star; it could in any case only be of appreciable magnitude with the most distant objects; and there is the most serious difference of opinion among astronomers as to what the distance of these objects really is. Within the last twelve months the distance of the spiral nebula M 33 Trianguli has been estimated by Dr. Hubble, of the Mount Wilson Observatory, at 857,000 light-years, and by Dr. Perrine, the director of the Cordoba Observatory, at only 30,000 light-years; and there is a similar uncertainty of many thousands per cent. in regard to all other very remote objects. Under these circumstances we hesitate to assign a definite length for the radius of curvature of the universe; but it is millions of light-years, though probably not greater than about a hundred millions. The curvature of space at any particular place due to the general curvature of the universe is therefore quite small compared to the curvature which may be imposed on it locally by the presence of energy. By a strong magnetic field we can produce a curvature with a radius of only 100 light-years, and of course in the presence of matter the curvature is far stronger still. So the universe is like the earth, on which the local curvature of hills and valleys is far greater than the general curvature of the terrestrial globe. In concluding these remarks I ought perhaps to apologize for having said nothing about the relation of general relativity to the new wave-mechanics. My excuse must be that, at the request of the secretary of the British Association, this address was sent to the printer many weeks before the meeting; and the wave-mechanics is developing so rapidly that, as one eminent worker has declared, anything printed is ipso facto out of date. E. T. WHITTAKER CHARLES FULLER BAKER—A SKETCH CHARLES FULLER BAKER, scientist, collector and pioneer, is dead-conquered on the very eve of the release which his indomitable will had long promised a harassed body. The doctors scarcely said whether it 70 was malignant malaria or amoebic dysentery or tuberculosis to which he succumbed at last. Five or six years ago, when I knew him as well as most men ever came to know him, Baker was living in a bamboo "bahai" on the outskirts of the dank little village of Los Baños, forty miles south of Manila. There, in his two rooms among the tops of palm trees, with the stench of his neighbors' pigs and carabaos floating up through the cracks in his floor, he made additions to his superb collections of insects and fungi, and "thanked the Lord daily" for the ships which brought him letters from scores of unseen, unknown friends who had come to know and revere his solitary work as a scientist. Though he was then only a little over fifty years, fever and a hundred tropic diseases had wasted his body and parched his skin, so that he looked more than seventy-very white of hair and intense of eye. Baker lived apart from the faculty of the College of Agriculture of which he was dean. Between him and most of us was an intangible though not unfriendly something which kept him from knowing the men intimately. Perhaps he found some compensation in the pioneer conditions, which, under earlier Wisconsin skies, had stirred the blood of his father, living there among the natives, cared for only by a Japanese servant and his wife, cooling his water in a swinging earthen jar and writing his innumerable letters. At any rate, few persons knew when intense pain made agony of his nights, or whether despair ever killed the stoic courage in his eyes. Once, when I learned he was suffering from one of his recurrent attacks, I climbed the ladder-stairs of his shack and entered the gloom of his large single room. He was lying on a narrow rattan couch, very wizened, very pale, and yet very fierce in the still, dark heat. "Buenos Dios, senor," he greeted me gaily, without moving. I urged him to let us care for him, but it was obvious that that day at least he could not be moved. The next noon he sent a note: "You are placing before me a fine temptation to be sick. You probably don't know that you are also tempting me to go back on one of my most cherished principles, not to give up, or to resign myself to conditions until the Angel Gabriel blows his horn." After a week he was up once more, riding behind the gray nag along the blazing three miles of road to the college, and greeting natives and Americans alike with his sweeping, faintly mocking friendliness. Baker virtually built the Philippine College of Agriculture. He fought for the appropriations which kept it going; he sought eagerly for a faculty fired by a kindred zeal to his own, for using the tropics as a great laboratory in which to enrich human knowledge. A work fraught ab initio with disappointment! A quest implicit with futility! Baker found few men so free of ambition for personal glory, so urged by passionate scientific curiosity, that they would suffer his exile unmindful of loneliness, disease, perilous trips, neither seeking nor expecting gratitude, wealth or even academic recognition. Next to the college he organized, to which come native lads from every part of the islands (Baker could capture their imaginations and stir their hopes as no other member of the faculty could, or bothered to do), Baker was interested chiefly in his entomological and mycological collections. He had a surpassing knowledge of insects and fungi and he showered the laboratories of collectors in the Orient and Europe with his specimens. His own collections he gave in part to the College of Agriculture of the University of the Philippines, to the University of Hawaii and to the Smithsonian Institute in Washington, D. C. Impressive monuments though they are to his intrepid, tireless spirit, the generations whose knowledge and whose living will be richer because of them, can scarcely glean from them a sense of the heroism of this rare and daring personality. Yet Baker was not coldly impersonal. In strange contradiction to his own stoicism, he was generous and sympathetic with people whose difficulties were not a fraction so severe as his own. Once he gave up a long-cherished plan for a trip to another more remote part of the islands, because a native boy who was dying of tuberculosis had neither money nor friends to care for him. Baker took the money he had put aside for the trip and sent the lad to the mountains. For his own part, he stayed in his shack and classified his treasured insects. In his death, science has lost a worker whose invaluable contributions were all too obscured by his indifference to public recognition, and a host of scattered admirers must be reminded of his countless kindnesses. COLIN G. WELLES The sessions of the Electrotechnical Commission, the Bureau announced orally, were held at Bellagio on Lake Como from September 5 to September 13, with a special trip to Como on September 11, when the centenary of the death of Volta was commemorated in conjunction with the International Congress of Physics. After the technical sessions at Bellagio, the delegates to the meeting made a 10-day tour to various power plants and industrial establishments in Italy, ending at Rome, where a final meeting was held for formal acceptance of the results of the Commission meetings. According to the preliminary program which has just been issued for the meeting, the Commission dealt with the standardization of electrical machinery and related problems, such as prime movers (steam engines and water turbines). During the meetings at Bellagio, consideration was given to specifications for such prime movers for switches, measuring instruments, insulating oils, lamp bases and holders, traction motors and radio electron tubes. There was also a discussion of the methods of rating the power of electrical machinery, of rating rivers in connection with water-power development, and of an international technical vocabulary covering the field of work of the Commission. In addition to the government representatives mentioned, the sessions were attended by prominent engineers and executives, including representatives of the General Electric Company, the Westinghouse Electric & Manufacturing Company, the Edison Electric Illuminating of Boston, the New York Edison Company, the Electrical Testing Laboratories and a number of universities. The International Commission on Illumination, which met at Bellagio from August 31 to September 3, included national committees in Great Britain, France, Italy, Germany, Belgium, Switzerland and Japan, in addition to the United States. Its work included the unification of practice in making photometric tests, the establishment of standard technical vocabularies, and in general the furtherance of good practice in lighting in the several countries. The Bellagio meeting considered several technical problems a primary standard of light, standard methods of comparing lights of different colors and the investigation of glare. Other matters dealing more directly with practice include proposed specifications for electric lamps, for street lighting and for the regulation of automobile headlights. There was also some general discussion of the teaching of the science and art of illumination and of the activities of lamp manufacturers in Europe and in America looking toward the improvement of illumination. In accordance with the action taken, at the last 15 |