HUESTIS Relativity QC 6 I H83 m A New View of the Universe BY CHARLES HERBERT HUESTIS "Henceforth space by itself and time by itself shall sink to mere shadows, and only a union of the two shall preserve reality." Minkowski in 1908. 1922 151 FOREWORD THIS paper makes no particular claims of originality unless it be in the general form and method of pre- sentation. It was written first of all to clear the writer's mind upon a theme which has haunted him from boyhood. It was put into the present form so as. to in- troduce the subject to a number of ministerial friends who have expressed a desire to know more about Ein- stein. The material for the paper was gathered from various sources, some of which are indicated in the text and the bibliography at the end. It will be noticed that the writer shies off from the mathematical aspects of the question, for which he confesses a profound and unrepen- tant ignorance. The paper in practically its present form was printed in the Red Deer Advocate in 1921, with the exception of the quotations from Lord Haldane's book at the close. So many requests have come to the writer for copies of this article that he has had it put into shape for distribu- tion. Generous souls who would like to contribute to the expense of printing can do so by sending twenty-five cents to the writer. Red Deer, Alta., October, 1922. The Einstein Theories of Relativity It is a fact worthy of comment that the astonishing discoveries of Einstein have met with so little attention from the Church. When Galileo asserted that the earth moved around the sun, instead of being the centre of the universe about which the sun, moon, and all the galaxy of stars revolved, he was forced by the Church to recant his dangerous heresy, and to confess that he was only joking. Darwin was not burned at the stake as Bruno was they were not doing that sort of thing in the Victorian age but even today when the truths he announced as the result of his studies and investigations have become one of the sure bases of science, his middle name is Satan among ultra-orthodox people, and that of Huxley is anathema. But here a young Jew with a German name has thrown a monkey-wrench into the machinery of the Universe, and brought the scientific world up all standing; but there seems to have been not so much as a flutter in ecclesiastical dove cotes. Surely, in another sense than Galileo meant when he muttered after his recantation, Eppur si muove, the world does move. For the Einstein theories are quite as radical as those of Galileo, Copernicus, or Darwin. Moreover, they were announced in the usual sensational manner in the newspapers a considerable time, of course, after they had become known to scient- ists. "Euclid dethroned," "Newton a back number!" The Law of Gravitation scrapped*" "The Warping of Space !" Such were some of the headlines. One would have expected that the Congregation of the Vatican, which had just recently turned its guns upon the Y.M.C.A., would not have missed so shining a mark. But the Church remained silent. The special and general principles of Relativity which Einstein has announced are simple enough when understood, but exceedingly difficult to understand. More- over, the writers who have 'sought to give popular expositions of the theories have not all been particularly successful in lucidity of statement. To the reader, who like myself, is deficient in mathematics, but fairly familiar with philosophy, I would recommend as a starter Dr. H. Windon Carr's little book, published by Macmillan, "The General Principle of Relativity." The subject is there discussed in iia broader philosophical and historical bearings. Dr. Hugh Elliot, in the October, 1920 number of the Edinburgh Review, has a clear statement, though I must dissent from his conclusion that the theories mark the end of the anthropomorphisation of nature; for it seems to me they bring us nearer than ever to the Pythagorean conception of man as the measure of all things, as I shall hope to show in this paper. Dr. C. D. Broad, in the April, 1919 number of the Hibbert Journal, begins his article thus hopefully: "I propose to try in this present paper to put in simple terms which ehall neither make a layman feel dizzy nor a mathematician feel sick, the main points of Einstein's principle of relativity." I would voice a warning to the non- mathematical reader to turn a deaf ear to this invitation, and if he accepts it, to prepare himself for some pretty rough sledding. The state of mind in which I found myself when I finished reading the article in question can only be described by the phrase 'the morning after.' The Scientific American has published a number of the essays contributed in competition for the Eugene Higgin's prize of $5,000.00 offered for a popular exposition of Einstein, edited by J. M. Bird. These should be read, especially the winning essay by Mr. L. Bolton of London, England. The writer would express his indebtedness to all the above, and to the articles in the New Re- public by Dr. Morris R. Cowen. A further list of books will be found at the close of this paper. One's first introduction to Einstein is like an adventure with Alice through the Looking-glass. We have always supposed that a yard is at all times and every- where thirty-six inches long, that an hour must always be an hour, that a mass weighing a pound in one place will weigh the same in another place, and that when you have measured the length, breadth and thickness of an object, you were finished with the measurement, and could state the volume with confidence. But Einstein tells us that there are circumstances in which a yard may appear to be only a few inches in length, an hour may be contracted to a fraction of sixty minutes, and a mass which started weighing a few pounds may come to weigh a ton, and he is not speaking of snowballs rolling down hills either. All that is necessary to accomplish these miracles is to get the objects moving fast enough approaching the velocity of light. In such a world one thinks of the reason Pat preferred a train wreck to a shipwreck. In the former, he said, there you are, but in the latter, where are you ? And the worst of it is, Einstein proves that what he states is true. His world is not the conceptual world of mathematics, as we shall see farther along, but a real world. He followed the example of his forerunner, Galileo. It used to be thought that a heavy weight would fall faster than a light one. But one day Galileo went up into one of the top galleries of the leaning tower at Pisa, and let two objects of unequal weight fall together, and they both reached the ground at the same time. In like manner Einstein based his conclusions upon the observation of actual events and movements. Usually things do not travel fast enough for us to experience the transformations spoken of above. In the case of the planet Mercury, however, whose orbit is a comparatively small one, the pace was sufficiently rapid, and Ein- stein was able to explain from his theory an aberration in the perihelion of that planet which had never been understood upon the Newtonian hypothesis of gravita- tion ; an explanation which was found to be correct in the observations of the eclipse of the sun in 1919. The same observations, I may say in passing, proved the truth of Einstein's calculations of the gravitation of light rays passing near the sun's gravitative field. That light has weight is an odd idea, but the mind of Newton guessed it. In his Opticks he asks the question : Do not bodies act upon light at a distance, and by their action bend its rays, and in not this action (caeteris paribus) strongest at least distance? The thing about the theories of Relativity which forms their chief attraction is their paradoxical character. The haunting fear of paradox has always been the bane of science, but I have to confess that it has ever had a charm for me. I shall never forget how in my boyhood I found delight in the paradoxes of Zeno. Achilles can never overtake the tortoise, which has a start of him in the race, because he must first reach the point where the tortoise started, and -when he reaches it the tortoise will have moved on, therefore Achilles will always have a step to take. The arrow can never reach the target, because at every moment of its passage it is at rest at some point. A body can never move from one place to another, because in order to do so it must first cover half the distance, then half of that half and half of that half, and so on to infinity. To me, fed up at that time with the conceptual world of mathematics, it seemed that Zeno's position was impregnable, nor do 1 think it has ever been reduced, from that standpoint, nor can be except upon the basis of the philosophy of Bergson, that movement is the soul of reality, and is not, like a static line in space, infinitely divisible, but is in its nature indivisible. Later on I was awakened one day from my dogmatic slumbers when my little boy brought to me a stick of wood and asked, Show me the inside, Daddy. I promptly took my pocket knife and cut the stick in two. But my lad's mind was too acute for me. But that's the outside now, he cried in glee; show me the inside. It was then that I first realized dimly that the mind cannot penetrate the fraction of an inch into the inner heart of things, but must be contented with surfaces only. That is why nature is so full of paradox. And Einstein's word to us is just this, to wit, that the data of experience must be accepted no matter how paradoxical they may seem. How different the real everyday world we experience is from the world of science and mathematics. Take the ideas of time and space with which relativity is chiefly concerned. We move to and fro about the room, or let our eyes rove, and we get an idea of space. We put our finger on our pulse and count its beats, we remember that a moment ago we heard the clock strike, and that in a half hour we have an engagement; and the idea of time is given. Then in the interests of science and mathematics, which must be exact, we invent standards and instruments for the measurement of time and space : clocks whose faces are divided into sections of twelves for the hours and sixties for the minutes (which, it is to be noted, are really space and not time measurements) ; and measuring sticks and rules which are di- vided into inches and feet and yards and rods. This is public time and space with which we are dealing, and very necessary when we wish to communicate with one another, or to write of our experiences and observations. But it is the height of absurdity to say that an hour spent in agreeable company is the same length as an hour waiting alone at an isolated station for a late train; or that ten miles in a motor car is the same distance as ten miles in an ox cart. How slow ye move, ye weary hours As ye were wae and weary; It was not sae ye glinted by When I was wi' my dearie. In our boyhood we wondered mildly why it was that Euclid, who was so clever in proving that if a line cuts two parallel straight lines, it makes the interior and opposite angles equal to one another, did not seem to be able to prove the con- verse, namely, that if the angles are equal the lines must be parallel; but had to make a sort of pseudo axiom out of this, which, however, he had not seen his way to include in the axioms at the beginning of hia book. He told us, too, that the two sides of a triangle are together greater than the third side, though no one has ever really proved this proposition, and innocent school boys have been told it can be done and they seem to be able to put it over their examiners. That the three angles of a triangle are together equal to two right angles, is another proposition of Euclid, but suppose your triangle is drawn upon the floor of the Parthenon at Athens, which conforms to the convexity of the earth, the statement is not true. That one and one make two is true in arithmetic, but it is not true of two drops of mercury which happen to come together, nor of two fishes in a pond. Life laughs at mathematics, as indeed does the new mathematics laugh at the old. We have too long been under the sway of the ancient neo-Platonic delusion that to a mind which approaches divine insight the book of nature is written in simple geometric lines. All the great founders of modern science, Copernicus, Kep- ler, Galileo, Descartes, Newton, shared this faith. By means of it they built up a noble structure of knowledge which is approximately correct but only approximately. This faith has been undermined on its experimental side by the progressive improve- ment of instruments of measurement, which fail to give us the results exactly as the laws predict, so that again and again physicists have had to abandon a simple for a more complex law; and on the mathematical side by the discovery of non-Euclidean geometry and the conception of a four-dimensional continuum of space-time. That great imaginative mathematician, Riemann, questioned the assumption of geometry that only one straight line could be drawn through any two points. If a straight line is the shortest distance between the two points, any captain who sails the seaa will be of the opinion of Riemann, for he knows that through the poles of the earth any number of straight lines can be drawn. More and more science and philosophy reveal the fact that we impose our human qualities upon external nature to such an extent that it may be asked with seriousness whether the outside world has any real or absolute existence outside our minds. Kant, as you know, asked this question and gave a negative reply. We have sought comfort in the above faith in the pos- session of axioms and laws which were untainted by human sense-perceptions. In- tellects upon other worlds would, he believed, find them as immutable as we. This assumption Einstein has challenged There is no need that I should point out that the conceptual universe of science, while in some mysterious way it parallels the real universe, is totally unlike it. Matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space, and time do not exist actually and really as we sense them, nor do they actu- ally and really have the qualities with which we endow them. We know nothing at all about the real world, and these things of which we talk so learnedly are merely devices and symbols which enable us to talk about reality. Euclid's world of points and lines and planes is clearly a conceptual world; you cannot make it concrete. As soon as you attempt to do so, as when you put a 'point' upon the blackboard, it vanishes, for your point has magnitude. We make use of a Euclidean terminology and a mechanistic mode of reasoning only because we have found from experience that they facilitate our reasoning. They are the tools with which we do our work. They are fitted to our minds, as Bergson has pointed out, because the mind is built geometrically and is suited to deal with the outside surfaces of things only. Time and space as we leel them are entirely different from time and space as they are measured by clocks and foot rules, as we saw above. The result is that we are called upon from time to time to change our ways of thinking. In the last century it was the study of biology which played havoc with men's orthodox ideas, and Spencer, Darwin, and Huxley received the maledictions of those who desired to walk in the old paths. Today we read the literature of this contro- versy with astonishment and wonder why the Church could ever have been so stupid as to combat the doctrines of Evolution and of the Higher Criticism which helped to sweep away the cobwebs of Verbal Inspiration and to disclose the truth which people have always suspected, to wit, that the Bible is written chiefly in the language of poetry because it is that medium to which we resort when we desire to express our deepest thoughts. .Today it is physics which is pointing out new ways. Mass, as we noted above, has been found not to be unalterable, but to vary with velocity; matter seems to vanish altogether, being resolved into tiny specks of energy, with no bodily support, scattered at relatively wide intervals, and darting to and fro with incredible velocity. Heat, light, and even motion are found to have weight quite apart from matter. Our philosophic preconceptions were clearly too confined to hold these facts. Something had to "bust." It did bust, and our unquestioned convictions were reduced to smithereens by the new principles of Relativity. I have neither the space nor the competence to give in this brief paper an ader quate account of the theories which Einstein has announced, much less his proof which involves mathematical gymnastics which Einstein is reported to have said only some twelve men in the world are able to perform. All I can do is to whet the appetite of the reader for further knowledge of scientific doctrines which give us a new world-view of the nature of the continuity and infinity of the universe. The idea of the relativity of knowledge is by no means a new one. It simply means that knowledge is not absolute or positive. Any bit of knowledge we may possess is related to and conditioned by some other bit or bits of knowledge. More- over, all knowledge is conditioned by the character of the mind that knows, and the channels through which the knowledge flows. A change in viewpoint or in experience may cause a revision of all previous knowledge. Einstein has simply introduced subjectivity into positive science. He points out that there is no such thing as ab- solute motion, absolute time; that space and time are not independent, but inter- dependent (this is important), and that everything is relative to something else. Mark Twain said that the street in Damascus was called Straight because while it was not as straight as a rainbow it was straighter than a cork-screw. That expresses the basic idea of relativity, namely, comparison. Things are big or little, long or short, fast or slow, only by comparison. An atom may be as large compared with an electron as a cathedral compared with a fly. The Relativity theory of Einstein emphasises two cases of relative knowledge, to wit, our knowledge of time and space and our knowledge of motion. Only Ein- stein carries the idea of relativity to its logical and empirical conclusion. If relativity is to be admtted at all, it must be admitted in toto; no matter what else it contra- dicts we have no appeal from its conclusions unless it contradicts itself. As has been well said, the essence of Einstein's generalisation is its final disentanglement of that part of a physical event which is contributed by the observer from that which is inherent in the nature of things. Kant attempted to do this by means of his dia- lectic, but Einstein has done so by means of a mathematical method introduced by Riemann, the calculus of terosorS; which like a modern harvesting machine threshes out the laws of nature and separates the subjective and objective elements. I will now state briefly the two principles of Relativity. The Special principle of Relativity is that the velocity of light is constant for all observers and independent of their system of reference, and that space and time are variable, dependent on the relative translation of the systems. The General theory goes farther. It extends the principle to all the laws of nature, and results in the successful application of the principle to the formulation of a new law of gravitation. These two principles revo- lutionise the current conceptions of time and space and gravitation. Let me try to illustrate what is meant, with Dr. H. W. Carr's assistance. Suppose you are walking up and down the deck of a steamer at night, carry- ing a light, and suppose the steamer is proceeding at the rate of some four miles an hour; the space that you cover and the interval of time that you occupy are the same to you whether you are moving forward with the steamer's motion or moving aft against that motion. Now suppose someone on the shore is observing the light while not able to see the steamer. Your movement, or rather that of the light, will appear very different to him from what it does to you ; for while you are walking forward it will seem to him that you are going twice as fast as you really are (for to your movement must be added that of the steamer), and while you are walking aft it will seem to him that you are not moving at all. The time measurements would also seem different to the observer on the shore, for while to you each movement would be measured by an equal space covered, to him at one moment you would be moving rapidly, a little later at rest. All this is simple and easy to understand, for it is in accordance with expedience. But now suppose both you and the shore observer are watching a thunder- storm coming from the direction toward which the ship is moving. Experience would say that the sound and the light, if we were measuring their velocity, would be faster to you than to him ; on the other hand, if the ship were moving with the sound and the light, their propulsion, we should say, would be slower to you than to him. Of course the immense velocity of light, 186,000 miles in a second, would make the difference in a movement of four miles an hour so infinitsimal as to be quite inap- preciable; but it would not be nothing, and you would be quite sure that if it were measured the infinitesimal quantity would appear as the result. But, now, suppose having measured it with perfect exactness, you made the discovery that there was no difference at all, what would you be be obliged to think? Certainly, if the ve- locity of the light waves was the same to an observer moving toward the source of the light as to one moving away from the same, you would be forced to the con- clusion that in some mysterious way space and time must be different to each. Now, strange to say, this measurement has been made over and over again with this sur- prising result. It was due to an experiment made by two physicists, Messrs. Michel- son and Morley, to determine the movement of the earth through the ether of space. As Einstein's principles are based upon the result of this experiment, let me tell you about it. As you know, the ether hypothesis was adopted to explain the passage of light. The above-mentioned experiment is not difficult to understand. The earth is moving through this hypothetical ether relative to the sun about 18.5 miles a second. If we think of the earth as at rest we may imagine the ether as a stream rushing past our earth at this rate of speed, just as the landscape seems to rush past the railway train in which we are sitting at rest. Now, in the experiment light waves were sent to mirrors up the ether stream and down the ether stream and across the ether stream. In every case the return was instantaneous! This seemed to prove that the ether hypothesis was not necessary, and it introduced a new absolute into the world of science, to wit, the velocity of light. It was upon this basis that Einstein built his system. He assumed boldly that the universe is so constructed that uniform straightforward motion of an observer and his apparatus will not produce any dif- ference whatever in the result of any physical process or experiment of any kind. A corollary to this is that by no experiment conducted on his own system can an observer detect the unaccelerated motion of that system. Everyone who has sat in a slowly moving train passing another train on an adjoining track, so that the land- scape was blotted out from observation, has experienced this fact. He has to wait for accelerations, or bumps, or a glance between the passing cars, to detect his or the other train's motion. Motion, therefor, depends upon matter. We are conscious of the motion of the earth only at sunset and sunrise, and it took ages for man to realise that it was really the earth and not the sun that was moving. The true meaning of Relativity in this instance may be appreciated if we imagine the universe as containing nothing but a single, undifferentated globe of matter. Suppose we desired to discover whether this globe was moving, could we do it ? We could not, because we would have nothing to measure the movement by. The globe is not getting nearer to or farther away from anything, for there is nothing to get near to or far from. In such a universe space does not exist, because our conception of space is something through which bodies can move. If we lose the idea of motion, we lose at the same time the idea of space. Space is therefor not an entity, but simply an idea based upon our conception of matter. While we are speaking of space, perhaps the -most astounding conclusion drawn by Einstein is that the universe is not infinite, and yet it has no boundaries. It is thought that it may be curved in form, maybe like a globe, but Einstein sug- gests a cylinder. And the magnitude of the universe depends upon its density. If it were of the density of water it would measure not more than 350 million miles in diameter. But we know that there are stars so distant that the light which comes to us today and makes them visible started hundreds of thousands and maybe mil- lions of years ago; so the universe must be much larger than that. Some have cal- culated the diameter as 400 trillion miles, but we need not bother about that. Let us try to get an idea of a universe finite but yet unbounded. Suppose we liken it to a sphere upon which a flat sort of insect is crawling. It might crawl to all eter- nity but it would never reach the end of the sphere. We spoke a moment ago of the light of the distant stars travelling with incredible velocity through the universe. If the universe is finite and curved it will some time come back to where it started. Then Prof. Eddington may be right with his suggestion that the spiral nebulae are just ghosts of our own and other solar systems in their beginnings returning to their haunts of millions of years ago! , I know I am getting ahead of my subject, but this theme is so fascinating and so odd that we may dwell upon it a little longer. In a recent lecture Prof. Einstein gave an exposition of this conception of a finite and yet unbounded space. He chose as his illustration a number of circular beetles living on a spherical surface with no consciousness of any space outside. Only a certain number of such beetles of definite size could inhabit the sphere. He pictured a source of light at one point on this sphere casting a shadow of the beetles on an infinite plane touching the sphere at the opposite end of the same diameter. As the beetles approached this source of light, their shadows on the plane would recede to a great distance and increase in size; but if they carried imaginary measuring rods shadows of the real ones carried by the beetles these would increase in size at the same rate, and thus the shadows would not themselves be aware of the change of size. The geometry of the infinite space inhabited by these shadows (interpreting the word geometry in its physical sense as the set of laws governing the possible positions of the shadows on the plane), would be essentially identical with the geometry of a set of circles on the sphere. Owing to the increase of the size of one of the shadow objects as it moved off into infinity, only a finite number of such shadows could occupy the plane; and thus, said Einstein, the picture was afforded of a space which in one sense was fi- nite, and which was not bounded. You will get something of this idea if you study your reflection in a polished globe as you move toward or away from the same. But I am straying far away from a clear pathway. Let me now by means of a few illustrations try to make more clear the principles of Relativity. The pilot of a moving airplane drops an object overboard. To him it seems to fall straight to the ground. But to an observer who is watching the airplane it seems to fall in a curve. Which of the two is right ? Einstein says they are both right, that there is no way of deciding between them ; that each observer measures the event from the standpoint of his own system at rest. If the object falling were seen from the North Pole it would seem to describe a circle, to an inhabitant in Mars it would seem to dart spirally about the sun, and to an observer in the stars it would gyrate through the sky. All motions are relative. This is the principle of Relativity ap- plied to space. Now let us imagine two pilots on rapidly moving airplanes and able to watch the clocks marking time on the other's machine. It wUl seem to each that his neigh- bor's clock goes more slowly than his own. Who is right ? Both, says Einstein. That is the principle of Relativity applied to time. All time is relative. Or suppose an observer standing upon a whirling disc (like a Brobdignagian disc-record, say), and that he is not able to observe anything but his whirling disc, which, indeed, constitutes his universe. He will notice as he walks about the sur- face of the disc that he is affected by a force directed from some point on the disc and which increases with his distance from that point. (I may say in passing, and this is exceedingly important to observe, that this force is exactly like that of gravi- tation, only working in an opposite way.) Now, if he stands at the point on the disc at which he feels no motion, the centre, and watches someone engaged in meas- uring the circumference with a measuring rod, the length of which he knows, it will seem to him that it takes more applications of the rod than it ought to take on the basis of the relation of circumference to diameter; but if the measurements are taken from circumference to centre it will seem to him that they are correct. The only reason which can be alleged for this phenomenon is that all objects are shortened by motion, and as the rod is in motion it will need more applications than seems to the observer necessary. But to the person engaged in the measurements this will not be apparent because the motion which contracts the rod contracts the edge of the moving disc also. We remarked this phenomenon in the above illustration of the beetles and their world. Moreover, if two clocks are placed, one at the centre and one at the circumference of the disc, the outer clock will seem to the observer at the cent/re to go more slowly than his own. All this seems like dream stuff, but it has been found to be true by the most exact * measurements and observations. Perhaps I can bring the matter closer to experience. Let us suppose a clock standing still and an observer moving away from it. Suppose as he moves he is able to count the ticks of the clock, he will find that they are slower and slower, and if he should move as rapidly as sound is propagated 1090 feet in a second he would never hear a second tick. Precisely the same is true if you substitute light waves for those of sound. If with a telescope you were able to watch the hands of the clock as you moved away from it they would seem to go more and more slowly, and if your motion was as swift as light, you would see the minute hand standing ever at the same place. But when you returned to the clock you would find that it had kept the same time as your own watch. This is the principle of Relativity, namely, that the laws of physical phenomena are the same when referred to one system as when referred to another system, moving uniformly with respect to the first ; i.e. Einstein asserts the dependence of natural law upon the movement of the observer. This explains why Michelson was not able to detect the motion of the earth through ether of space, as Einstein pointed out in 1905. I must caution the reader here against an error into which a number of writers on this subject have fallen, namely, that in this shortening of rods and slowing of clocks in motion, there is any actual physical change. They are phenomena of ob- servation only. Since two moving systems passing one another in empty space can- not tell which system is moving, the answer to the question, on which system are things shortened ? is : On both for the other fellow. The geometry of Euclid will not fit into Einstein's world for it is built upon the idea of a motionless, three-dimensional space. Einstein tells us that we cannot understand or explain natural phenomena upon the basis of a world of only three dimensions; another dimension, namely, time must be added; that there is no such thing as time independent of space or of space independent of time, but only time- space, which are interdependent. According to Newton when once a body is started in motion, it will move in a straight line unless acted upon by some other force. Einstein pointed out that as a matter of fact bodies do not move in space at all but in space-time, and not in straight lines but in 'geodesies', i.e., paths which conform to the nature of the space through which they travel. Space in the neighborhood of a field of gravity is held to be warped or twisted, and a body encountering such a field follows the curve of the same. Hence gravitation is accounted for without the need of a mysterious force acting instantaneously at a distance, as we shall see farther on. It was Minkowski who first gave expression to this idea of space-time by speak- ing not of points but of events. Things not only are but they happen. Euclid's world was made up, of things which are. He begins with points which generate lines, -which in turn generate solids. In Euclid's world motion had no place, and some of us remember the shook we felt when in the fourth proposition of his first book he proposed to make his proof by lifting one triangle and placing it on the other. It seemed to us a simple, but very improper thing to do. Euclid was a true Greek and admired repose more than he admired motion. But when Galileo and Newton contemplated nature they saw that account had to be taken of motion and also of time; and yet they found it necessary to cling to Euclid's static world. This they were able to do by regarding time and space as entirely dissociate. Then Min- kowski questioned this independence of time and space, and pointed out that the ex- ternal world does not consist of points but of events. Things happen. A pitcher in a ball game throws the ball toward the batter. It describes a curve until it comes in contact with the bat, when it takes another direction, curv- ing outward and downward till its motion ends on the ground or in some fielder's mitt. Now, as is well known it is possible to chart out the successive positions of the ball by means of a frame of reference consisting of the east and north sides, say, of the field, and the surface of the same. But this does not give a true descrip- tion of the ball's adventure, for time has elapsed, and time is an important element in ball games. So of all events, says Einstein ; they happen in space-time. The world of events is four-dimensional, -which is nothing more terrible than that you have to make four measurements to locate an event. This theme is so interesting it is worth our while to stay with it a little longer, and moreover we shall be helped to under- stand better what Einstein has to say of gravitation and the curvature of space. When the fielder out on the grounds catches the ball he lessens its force by acceler- ating his 'frame of reference', i.e., letting his hand go with the ball, producing a twist in the ball's 'geodesic path', and thus foists the characteristics of his own mo- tion on the path of the ball. All this is the ball's 'world-line' in its adventure. Now let us think not of one, but of a number of moving objects, say the move- ments of those hypothetical ultimate atoms of matter which we call electrons. Let us imagine the world lines of all the electrons in the universe threading space-time like lines running through a jelly. The intersections of these threads would be a history of all physical events. Now suppose this jelly to be shaken, at once the dis- tances between the events will be shortened or lengthened, but the natural order of the intersections will remain unchanged. The shaking of the jelly with its con- sequent distortion corresponds to the choice of a certain framework of reference by the observer, but he cannot by any choice of framework change the natural sequence of events, which is independent of all observers. The complete history of any event, such as the pitching, striking and catching of a ball, is summarized in a set of equa- tions giving the positions of the ball at every instant. A mathematician named Gauss has worked out a geometry of space-time by means of a system of charting known as Gaussian coordinates; but from my experi- ence I would advise the non-mathematical reader to give it a wide berth. Fortun- ately for all practical purposes Euclid will serve as well as Gauss. But I may give a hint of this geometry from the illustration above. Suppose we were able to indi- cate by mathematical symbols all these intersections, as we indicate the position of houses in a city by numbers on them, we could by means of the same locate all space-time events which these intersections stand for. But to come back to our illustration, there are two possibilities: Either these equations are the same in form for all space time reference frameworks, or they subsist only when some special framework is made use of, altering their form as they are referred to different frameworks. If the second possibility holds, the equations do not describe anything real in the universe, but simply some observer's charac- teristic way of observing nature. But if the first holds then the equations describe some real relations of nature, independent of the observer. Now Einstein's general theory is that those relations and those alone which persist unchanged in form for all possible space-time reference frameworks are the inherent laws of nature. I shall speak later on of some of the philosophical and theological implications of Einstein's theories, but I may say here in passing the accomplishment of nearly all great spiritual leaders has been of like character as this conclusion. They have sought and revealed to the world some moral equation by which the problems of men in association with one another and with the Infinite might be solved. Jesus did this when he summed up the teachings of "Moses and the Prophets" in one word love. Paul in his poem upon love in the thirteenth chapter of first Corinth- ians does the same. He refers to the "reference frameworks" of his day: the He- brew, Prophecy; the Greek, Knowledge, and the Christian, Charity or benevolence. All such, he says, are partial and unsatisfactory systems of life due to characteristic ways of reaction. We know in part and we prophesy in part, but when that which is perfect is come, that which is in part shall be done away. If we would be one with God and one with one another, so that all discord shall cease, all differences be dissolved, LOVE must be our reference framework. I am tempted to pursue this tempting theme, but I must pass on. All this seems to have to do with a topsy-turvy world like that of Alice's adventures behind the looking-glass. But this is not the case, for this bewildering profusion of different times and places, time rates and velocities, are readily har- monized by the proper mathematical formulae and operations; just as we are able to calculate French kilometers and English pounds into our miles and dollars and cents. It will be seen, however, that the old 'conception of an absolute space in which all things are contained, and an 'absolute, true and mathematical time' as something 'which in itself and from its own nature flows equally' and with no lia- bility to change, must be given up. Time is nothing else than a certain intercom- munication of events which are said to occur in time. But the doctrine goes deeper still. Einstein laid down the principle that the conception of time is achieved only by following the life history of some recognisable entity in the universe, i.e., our earth. His space is measured space and his time measured time. Each entity has its own private time, and the inhabitants of the earth only achieve public time for 10 the universe by judgments founded upon their system of communication with the rest of the universe. Public time for us will differ from that of other beings on other systems and under different circumstances. The reason we find it difficult to understand Einstein's space-time world is because we cannot visualize it. The same difficulty must have been experienced by people when for the first time the rotundity and revolution of the earth was taught. People said this was absurd because if the world turned over we should all fall off. We are still largely earth-bound in our conceptions when we talk of up and down, back and forth, right and left, as though these were absolute directions in space. The pre-millennialists today look for the Lord's coming in the clouds, so that every eye shall see him, from the standpoint of a flat world. The only difficulty we shall have with Einstein is to get rid of these earth-bound conceptions and habits of thought. They are not the only habits of thought of which we must rid ourselves if we are to make use of the rich flood of knowledge which has come to us of recent times, as James Harvey Robinson has been telling us in his The Mind in the Making. If the physical world cannot be known from the standpoint of the old conceptions of time and space, neither can the spiritual and social world be known from the stand- point of our savage and mediaeval ideas. Once more let us return to this idea of a four-dimensional space-time, which seems to be the most difficult part of the Relativity theory for laymen to understand. Professor A. S. Eddington in a recent book emphasises the importance of Minkowski's discovery, but protests against the delusion that the fourth dimension is something wholly beyond the conception of the ordinary man, and I think he is right. To every one it is evident that the world of solid and permanent objects has three di- mensions and no more. It is no less obvious to everyone that the world of events is of four dimensions; that events are arranged in the four-fold order of right-and- left, backwards-and-forwards, up-and-down, sooner-and-later. This world of events is external nature, common to all observers, but represented by them differently in their private frames of space and time. Even Euclid could not start upon his geometrical work without laying down certain postulates which involved the fourth dimension. The proposition that a line can be drawn from any point to any other point, or that a line can be produced to any length in a straight line involves at once the time element, for you cannot have motion of any kind, even in a one-dimensional universe, without time. Nor is It possible to imagine rest if the possibility of imagining motion is removed, for a thing changes or does not change; there is no intermediate state. And the time dimension is as necessary in the one case as in the other. The conception of dimension carries with it the idea of time since it involves distance. The Hebrew poet who spoke of a universe in which time should be no more was dealing with a spiritual and not a physical universe. I said a little while ago that the conception of four-dimensional space-time was nothing more terrible than that you had to make four measurements to locate an object. I see lying on the table before me the two volumes of the Standard Dic- tionary. With a foot rule I have found that the length, breadth and thickness are respectively 12%, 9, and 5% inches. Now suppose someone else gives the measure- ments when the books are standing on end, the respective measurements will differ, but the volume as calculated by him will be the same as calculated by me. But now suppose the books standing on the edge of our whirling disc-world of which I spoke above, the books would be contracted, or seem to be, by the motion, and it 'would seem that a different estimate of their volume would be obtained, and Einstein's law of relativity negatived. But Einstein says No. There will be compensation, and that takes place in the fourth dimension, namely, time. The slowing down of events on the moving disc-world exactly compensates for the shortening or contrac- tion. As this dimension never varies for our world we have not developed a sense of the same. This leads us to another interesting conception of Einstein, which was hinted at in our jelly-world illustration above. As the measurement of the volume of a body is constant, so in space-time there is a certain quantity unaffected by any variations in methods of measurements. It is called the interval between events. The observer on & rapidly moving system would differ from us as to measurements of objects and standards of time, but he would always agree with us as to the interval between two events measured through space-time. This interval is not intelligible like one 11 in geometry or music, but is simply a mathematical formula. Maybe this illustra- tion will assist us in appreciating it. Suppose I am travelling on the prairie and wish to reach a place ten miles away, as the crow flies, to the north-west. I may do this by driving eight miles west and six miles north. But supposing, owing to the valley of a river it is not possible to drive at right angles, but is necessary, as I found it the other day, to take a trail which runs roughly south-west and north- west, the same point will be reached by driving something over seven miles south- west and then something over seven miles north-west. The directions and the dis- tances differ but the interval is the same. An illustration of Einstein's proposition was given by Dr. J. H. Jeans at a meeting of the Royal Society in 1920. The interval between a man's birth and his death may be to one observer 1,000 miles and 75 years, and to another observer 1,000,000 miles and 76 years. The quantity remaining constant for both is the square of the distance travelled by the man between birth and death minus the square of the distance travelled by light during that period. This is not very clear, at least to the writer. The following by the same scientist will make the matter clearer. He says, "The instant of time and the point of space at which an event occurs can be fixed by a single point in the continuum, so that the interval between two events will be represented by a finite line. The event* and the interval are absolute, but the interval will be split into time and space in different ways by different observers. The interval between any two events, such as the Great Fire of London and the outburst of the star Nova Persei, may be measured by one set of observers as so many years and so many millions of miles, but another set of observers may divide the interval quite differently. For instance a terrestrial astronomer may reckon that the outburst on Nova Persei occurred a century before the Great Fire of London, but an astronomer on the Nova may reckon with equal accuracy that the great fire occurred a century before the outburst on the Nova. A third astronomer might in- sist that the events were simultaneous. All will be equally right, although none will be right in an absolute sense." As I mentioned above, the Newtonian theory of gravitation is overthrown by the Einstein hypothesis. Newton was never able to reveal the machinery by which the apple was drawn from the tree to the ground, and as the force was sup- posed to act at a distance and instantaneously, Einstein asked boldly whether New- ton's failure might not have been due to the fact that there was no such machinery and no such force. Perhaps gravitation was just a fundamental property of the world in which it occurred, and not the ugly duckling child of nature it had seemed to be. We saw a while ago that the observer on the whirling disc felt a force- exactly like that of gravitation, Einstein asked whether an observer on another sys- tem than that of Newton might not have seen the earth moving upward toward the apple rather than otherwise. Such a suggestion is very difficult for us to accept (though the idea of action at a distance is just as difficult, had we not accustomed our minds to it), but Ein- stein tries to make it clear by means of an illustration which we can understand. Let us imagine a sort of room, like a large box, and situated so far from all objects that no gravitative action is felt. Everything in the room would have at first to be tied in place, and the observer can imagine himself standing on the 'floor' of the room. Now let us suppose the room attached, at what we may call the top outside, to a rope by means of a hook, and a pull exerted 'upward' upon the rope. Immediately the inclination of all objects in the room will be in the other direction, or to the 'floor' of the room, that is, to the observer they will become 'heavy'. But to the observer outside the room the explanation will simply be the upward pull on the rope. Here again the principle of relativity is asserting itself, for things are judged from the observer's standpoint. But gravity appears not as a force but rather as a property of a space or field. The reader can make the above experiment clear to himself if he will imagine himself in an elevator going down with uniform accelera- tion like that of a freely falling body. In such an elevator no free object will fall to the ground and an object flipped across the elevator, which ordinarily would de- scribe a curve will be seen to describe a straight line. On the other hand an object moving horizontally outside the elevator will seem to the observer within to be mov- ing in a curve. As the outcome of these observations we are enabled to formulate an Equiva- lence Hypothesis, which Prof. Eddington states as follows: A gravitational field of force is precisely equivalent to an artificial field of force, so that in any small region it is impossible, by any conceivable experiment, to distinguish between them. Ein- 12 stein thus extends his equivalence hypothesis to cover all manifestations of force, and states that by no kind of experiment can we distinguish between the two de- , scriptions. This is the heart of his general theory of Relativity. I should add here that Einstein's law of gravitation, though essentially dif- ferent from Newton's, corresponds so nearly with it in the deductions derivable from it, that very few criteria can be found by which to test the accuracy of the one or the other law. One of these deductions was that the major axis of the ecliptic orbit of a planet round the sun must slowly turn round the sun in the plane of mo- tion of the planet in the same direction. Such a shifting of the axis of each plane- tary orbit is already produced in the solar system by the mutual disturbance of the various planets, but Einstein's law adds a slight additional motion after the former has been allowed for. But only in the case of the planet Mercury is this additional motion large enough, as we saw above, to have been observed as yet, and in this case a discrepancy had long been recognized between the actual motion and that deducible by Newton's law. Einstein's law precisely accounts for this discrepancy. To sum up, if we assume that space-time in the neighborhood of matter is non-Euclidean, and that this curvature or distortion increases as the mass and de- creases as the distance, and that every particle of matter not interefered with passes through space-time in the most direct path possible in that continuum J then the ob- served facts of gravitation are accounted for as a geometrical property of this space- time world. Gravitation is no longer a mysterious force acting at a distance, but a fundamental property of things. The reason why we accept Einstein's conclusions so readily is not because he has been able to prove them to us to most of us non- mathematical students of his work this proof cannot come but because of that passion for unity which the human mind possesses. What philosophy has ever tried to do in the past Einstein has done for science. He has for the first time brought mechanical, electro-magnetic, and gravitational phenomena into one structure. We are open-minded to the theory because we want it to be true. And moreover it gives a simple answer to the mystery of gravitaton, to wit, Because the world is so constructed. Einstein's theories have carried us to a height of knowledge which surpasses all elevations hitherto reached in the past history of the race. From this lofty peak we find ourselves contemplating nature with an insight such as no man has ever had before. We already know that matter is made up of electrons, and that radiant energy is electro-magnetic, and before Einstein it was regarded as certain that all physical phenomena, except gravitation, were manifestations of the electro-magnetic field. This belief is now confirmed by the Einstein theories. As we have seen, it is found that the gravitational and electro-magnetic conditions of the universe are completely defined if to each point is space-time a gravitational and electrical po- tential are ascribed. There is thus no need of bringing in some outside force to ex- plain them; they are the stuff of which the world is built. And this stuff is not matter but energy. I trust I have not been unsuccessful in giving you some conception of the subject under discussion. Let me now give one more illustration, for which I am indebted to Dr. Morris R. Cohen, which may help to link up the General theory with common experience. Let us suppose a group of scientists confined since birth in a moving railway car, and unable to learn anything of the world outside the car except by means of the light which comes through the windows of the same. (It may be well to remind the reader that our world is just such a car.) If the scien- tific observers in the car attempt to formulate the Laws of nature, they will natur- ally do so on the supposition that their car is at rest and all other things in motion in divers ways. If their car did not move with uniform motion, their system of laws would be very much more complicated than ours in a uniformly moving world. Now suppose one of these scientists, a very great and gifted one like New- ton, saying to himself, Why not suppose the world outside the car windows is at rest and our car in motion ? If he did this he would be able to simplify very greatly his system of natural laws. The sudden lurch forward or backward of any loose objects in the car would not have to be explained in some mysterious way, but as due to the inertia of things in motion. Well, suppose he has managed to prove to his fellows his new conception based on the idea of a world outside at rest and the car in motion, when suddenly one of them who has not been bothered much with past academic knowledge, but with the insight of Einstein, cries out : Wait a moment ; I have an idea. You have no doubt 13 been able to discover a new and better system of laws and equations to explain the course of nature, but what right have you to suppose that it is more true than that we have had in the past. Nature may fulfil herself in many ways and is not care- ful to adapt herself to the scientific laws enunciated by men. I think I can show you a system of equations by which you can pass from every proposition in the old account of things to a corresponding proposition in the new. And this is just Einstein's achievement. If there is no such thing as a unique and absolute space, it is just as true to say the earth moves with reference to the car as that the car moves with reference to the earth ; and the laws of nature are the same whether we suppose the observer to be at rest or in some kind of motion. Relegating space and time to their proper source, Einstein bids us contemplate the residuum, This is the true world of space-time. It is true we cannot visualise it, nor can we, for that matter visualise the world of geology which deals with things of the past, nor the worlds of astronomy and physics which deal with matters and figures far beyond our powers of conception,' nor even the world of Euclid made up of points and lines and planes, which no one ever saw or can see, if the definitions are correct. Einstein's world is one which we might see if we had two eyes, one of them stationary, and the other darting to and fro with the rapidity of light. It is a mathematical world which only mathematics can describe. Einstein reinforces by his theories the great philosophic tradition of viewing things in their eternal aspect and brings to our minds the intuition of the Hebrew poet to whose God a thousand years were as one day. What effect will Einstein's discoveries have upon philosophic thought ? Cer- tainly a profound one, corroborating many of the intuitions of the poets and philo- sophers of the past. The world we study is not independent of the mind which con- templates it. Space and time are not absolute, public to every one and eternal in the heavens, but .each observer is himself the absolute, and not as we have hitherto supposed, the relative, centre of the universe. We carry with us, inseparably at- tached to us, the form by which we judge the universe of natural laws. Philosophy has long held this view, "but the scientists were inclined to scoff at the philosophers. Absolute science must give way to that of humanism for which James and Schiller and Bergson have been pleading. Man is the measure of all things, and he may accept with confidence the deliverance of his experience. But to creatures of other worlds than ours experience will differ from ours. Our clocks cannot set the time for the Universe; our little system is but a broken light of the Absolute. Think for a moment of experience measured in terms of dimension. Let us be- gin with a creature inhabiting a one-dimensional world a world having length only. It could never know whether its world was a straight line or a curve, for curvature in any direction involves another dimension. So of a flat creature living in a flat world of but two dimensions, length and breadth. His experience will be larger than his neighbor's, but he will not know whether he inhabits a plane or a spherical surface. For this larger knowledge another dimension of height or depth must be added. You go for a walk with your dog as companion. How different your respec- tive interests: his for bones hidden in secret places, and the entrancing smells of corners; yours for the clouds and the trees and the far-flung landscape. Your dog's world is largely a two-dimensional one; he lives chiefly upon the surfaces of things. It must have been a tremendous moment in the history of our remote ancestors when they learned at last to stand securely upon their hind legs and could use their fore ones to explore heights! To your dog you are as a god, as was Setebcs to Caliban in Browning's poem. Thus we are led to a conception of deity, but of a philosophical deity who can be described in metaphysical terms only-^a God of a many-dimensional universe. Such a God can never be known until we have translated him into terms of our own four-dimensional world. Materialists from earlier times have criticised the hu- man conceptions of deity because God is made in the image of man. But only such a God a God who lives and loves could be known by man. And so the greatest religious Master speaks of God as Father: there must be a human nature in God if we are to know Him. Alice in Wonderland carries with her into that topsy-turvy world the memory of her former experiences and her former standards of measurement and judgment. She does not feel her own strangeness, but the strangeness of the world about her. When, in those fairy-tales which were the delight of our childhood, and which the wooden pedagogics of this day would eliminate from its culture material, the prin- cess was turned into a fawn, the fawn's consciousness was still that of the princess. 14 ' The continuity of conciousness was not broken. The continuity ol the universe is the continuity ol consciousness. Is not this the basis of the Christian doctrine of the Incarnation ? Jesus brought with him into this world the life and conscious- ness of the Father. The wise man of old who said that God had put eternity into the heart of man was giving expression to the same idea. Not in entire forgetfulness, and not in utter nakedness, But trailing clouds of glory do we come from God who is our home. And that ineradicable sense of personal immortality, whence comes it ? The wish that of the living whole No life may fail beyond the grave, Derives it not from what we have The likest God within the soul? It's a far cry from Quoheleth and Wordsworth and Tennyson to Einstein, but the poets are liberating gods, and have in their symbolic way anticipated much that science has later on confirmed by observation and experiment. To quote Dr. H. W. Carr: Ultimately, in spite of its claim for independence, all that an object or event is, in substance or in form, it derives from the activity of the mind for which alone it possesses the meaning which makes it an object or event If we adopt in mathematics and in physics the principles of Relativity (and have we any choice?), the obstinate, resistant form of objectivity of the physical world dis- solves into thin air and disappears. Space and time, its rigid framework, sink to shadows. Translated into the language of Christian mysticism may we not say with Paul, We look not at things seen, but at things unseen; for things seen are tem- poral, but things unseen are eternal ? POSTSCRIPT Lord Haldane's recent book on Relativity came into my hands after I had finished this paper. Some of his conclusions are of interest. Einstein is concerned, as we have seen, with the principle of Relativity in physics; Ha-ldane follows the principle outside of physics adopting the philosophic method. He is led thereby: (a) to recognise a higher order in which the distinction between thinking and what is thought is "in the end ideally superseded"; (b) to a tentative acceptance of Time as a genuine form of reality and the consequent exclusion of a timeless Absolute; (c) to the conception of God as an entity within whom all distinctions and resulting re- lations must fall : whence He is not a subject as differentiated from object, and is no entity separate from ourselves; (d) to the recognition of the search after God as after something which is immanent within us; (e) to the acknowledgment of the im- plications of the personal self as more than merely finite; (f) to the admission that death "at its own level", is an actual event, but that at a level of a different order, it does not touch the subject-self, which is no transitory physical organism; (g) to the acknowledgment of the self as that in which the Universe centres. In this larger sense of the term Haldane asks at the end of his book, What guidance does it offer for the conduct of our individual lives ? and answers, I do not think the question is a difficult one to answer. I quote : The real lesson which the principle of the relativity of knowledge teaches us is always to remember that there are different orders in which our knowledge and the reality which it seeks have differing forms. These orders we must be careful to distinguish and not to confuse. We must keep ourselves aware that truth in terms of one order may not necessarily be a sufficient guide in the search for truth in another one. We have, in other words, to be critical of our categories. As an aid to our practice, the principle points us in a direction in which we may possess our souls with tranquility and courage. We stand warned against "other worldliness" in a multitude of concealed forms. We are protected, too, if the doctrine be well- founded, against certain spectres which obtrude themselves in the pilgrim's path Materialism, scepticism, and obscurantism alike vanish. The real is there, but it is akin in its nature to our own minds, and it is not terrifying. Death loses much of 15 * its sting and the grave of its victory Such things cease to be of the old importance when they lose the appearance of final reality. There may come to us, too, contentment of spirit, and a peace which passes our everyday understanding. We grow in tolerance, for we see that it is in expres- sion rather than in intention that our fellow-men are narrow. We realise that we are all of us more, even in moments of deep depression, than we appear to ourselves to be, and that humanity extends beyond the limits that are assigned even by itself to itself. Our disposition to be gentle to those who may seem to misinterpret us be- cause of dissent from our outlook on life grows with the recognition that, as Spinoza wrote two hundred and fifty years ago, in his answer to the letter offering him refuge in a chair at Heidelburg from his theological persecutors, "religious dissensions arise not so much from the ardor or men's zeal for religion itself, as from their various dispositions and love of contradiction, which leads them into a habit of decrying and condemning everything, however justly it be said." .' (Spinoza's) life and his attitude of soul remain a lesson of high value for those who seek to believe, as he did, Est Deus in nobis It is this which seems to have been in sub- stance the creed, varying in expression but ever indicative of a common faith, pro- claimed by some of the greatest guides of mankind in ancient and modern times. It is a creed that if it be true helps those who can make it their own to dispel obscuri- ties, and to lighten for themselves the burden and the apparent mystery of human life. It is a creed that stimulates the practice of unselfishness in social and religious life, interpreted as fully harmonising with the dictates of philosophic thought. "If any man shall do His will, he shall know the doctrine." BIBLIOGRAPHY The following are the titles of a few books which should prove helpful to any one who desires to pursue the study of Relativity : Relativity : the Special and General Theory; a Popular Exposition, by Albert Einstein. New York, Henry Holt & Co. As Bergson is the best expositor of his doctrine of Creative Evolution, so is Einstein the best and clearest expositor of the Relativity theories. This book is a fine achievement in clear exposition, and ought not to be difficult to the general reader, except in a few spots. The Einstein Theory of Relativity; by H. A. Lorentz. New York, Brentanos. This was originally a newspaper article, and is by a great physicist, the author of the electro-magnetic theory of relativity. Worth reading. Easy Lessons in Einstein ; by E. E. Slosson. New York, Harcourt, Brace & Co. Full of pleasant and helpful illustrations, but not the best guide to the essential theories. From Newton to Einstein; by B. Harrow. New York, VanNostrand Co. Per- haps the easiest book yet published for a beginner, and a good introduction to the general meaning of relativity. Space, Time, and Gravitation; by A. S. Edington. New York, Cambridge University Press. A sympathetic treatise by the leading English expositor of Ein- stein. If you can manage the mathematical parts, no book is better, so I am told. The General Principle of Relativity in its Philosophic and Historical Aspect; by H. W. Carr. New York, The Macmillan Co. I have referred to this, book at the beginning of my paper. I found it helpful. It is by a philosopher rather than a physicist, and presents that viewpoint. Einstein's Relativity and Gravitation ; edited by J. M. Bird. New York, Scien- tific American Publishing Co. This book contains the papers contributed in compe- tition for the Higgins prize of $5000.00, and is in many ways the most useful book in the above list, because it contains a number of attempts to give the A. B.C. of Einstein. An Enquiry Concerning the Principles of Natural Knowledge, and The Concept of Nature; both by A. N. Whitehead. New York, Cambridge University Press. These two books by one of the greatest living mathematicians are not to be recom- mended to the general reader, but are remarkable expositions of the laws of nature which are destined, I believe, to have a profound influence not only upon our con- cepts of the physical world, but also upon our understanding of the spiritual nature of man. 16 THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW.