certain remarkable coincidences—the plane of movement of a planet—consideration of planes of several planetary orbits—a characteristic of the actual planetary motions not to be explained by chance—the first concord—the planes not at random—a division of the right angle—statement of the coincidences—an illustration by parable—the cause of the coincidences—the argument strengthened by the asteroids—an explanation by the nebular theory.
in the present chapter, and in the two chapters which are to follow, i propose to give an outline of those arguments in favour of the nebular theory which are presented by certain remarkable coincidences observed in the movements of the bodies of our solar system. there are, indeed, certain features in the movements of the planets which would seem so inexplicable if the arrangement of the system had taken place by chance, that it is impossible not to seek for some physical explanation. we have already had occasion to refer in previous chapters to the movements of the bodies of our system. it will be our object at present to show that it is hardly conceivable that the movements could have acquired the peculiar characteristics they possess unless the solar 295system has itself had an origin such as that which the nebular theory assigns.
the argument on which we are to enter is, it must be confessed, somewhat subtle, but its cogency is irresistible. for this argument we are indebted to one of the great founders of the nebular theory. it was given by kant himself in his famous essay.
we will commence with a preliminary point which relates to elementary mechanics. it may, however, help to clear up a difficult point in our argument if i now state some well-known principles in a manner specially adapted for our present purpose.
let us think of two bodies, a and s, and, for the sake of clearness, we may suppose that each of these bodies is a perfect sphere. we might think of them as billiard balls, or balls of stone, or balls of iron. we shall, however, suppose them to be formed of material which is perfectly rigid. they may be of any size whatever, large or small, equal or unequal. one of them may be no greater than a grain of mustard-seed, and the other may be as large as the moon or the earth or the sun. let us further suppose that there is no other body in the universe by which the mutual attraction of the two bodies we are considering can be interfered with. if these two bodies are abandoned to their mutual attraction, let us now see what the laws of mechanics assure us must necessarily happen.
fig. 45.—a spiral presented edgewise (n.g.c. 4631; in coma berenices).
(photographed by dr. isaac roberts, f.r.s.)
let a and s be simply released from initial positions of absolute rest. in these circumstances, the two points will start off towards each other. the time that must elapse before the two bodies collide will depend upon circumstances. the greater the 296initial distance between the two balls, their sizes being the same, the longer must be the interval before they come together. the relation between the distance separating the bodies and the time that must elapse before they meet may be illustrated in this way. suppose that two balls, both starting from rest at a certain distance, should take a year to come together by their mutual attraction, then we know that if the distance of the two balls had been four times as great eight years would have to elapse before the two balls collided. if the distances were nine times as great 297then twenty-seven years would elapse before the balls collided, and generally the squares of the times would increase as the cubes of the distances. in such statements we are supposing that the radii of the balls are inconsiderable in comparison with the distances apart from which they are started. the time occupied in the journey must also generally depend on the masses of the two bodies, or, to speak more precisely, on the sum of the masses of the two bodies. if the two balls each weighed five hundred tons, then they would take precisely the same time to rush together as would two balls of one ton and nine hundred and ninety-nine tons respectively, provided the distances between the centres of the two balls had been the same in each case. if the united masses of the two bodies amounted to four thousand tons, then they would meet in half the time that would have been required if their united masses were one thousand tons, it being understood that in each case they started with the same initial distance between the centres.
instead of simply releasing the two bodies a and s so that neither of them shall have any impulse tending to make it swerve from the line directly joining them, let us now suppose that we give one of the bodies. a, a slight push sideways. the question will be somewhat simpler if we think of s as very massive, while a is relatively small. if, for instance, s be as heavy as a cannon-ball, while a is no heavier than a grain of shot, then we may consider that s remains practically at rest during the movement. the small pull which a is able to give will produce no more than an inappreciable effect on s. if the two bodies come together, a will practically do all the moving.
298
fig. 46.—the plane of a planet’s
orbit.
we represent the movement in the adjoining figure. if a is started off with an initial velocity in the direction a t, the attraction of s will, however, make itself felt, even though a cannot move directly towards s. the body will not be allowed to travel along a t; it will be forced to swerve by the attraction of s; it will move from p to q, gradually getting nearer to s. to enter into the details of the movement would require rather more calculation than it would be convenient to give here. even though s is much more massive than a, we may suppose that the path which a follows is so great that the diameter of the globe s is quite insignificant in comparison with the diameter of the orbit which the smaller body describes. we shall thus regard both a and s as particles, and kepler’s well-known law, to which we so often refer, tells us that a will revolve around s in that beautiful figure which the mathematician calls an ellipse. for our present purpose we are particularly to observe that the movement is restricted to a plane. the plane in which a moves depends entirely on the direction in which it was first started. the body will always continue to move in the same plane as that in which its motion originally commenced. this plane is determined by the point s and the straight line in which a was 299originally projected. it is essential for our argument to note that a will never swerve from its plane so long as there are not other forces in action beside those arising from the mutual attractions of a and s. the ordinary perturbations of one body by the action of others need not here concern us.
the case we have supposed will, of course, include that of the movement of a planet round the sun. the planet is small and represented by the body a, which revolves round the great body s, which stands for the sun. however the motion of the planet may actually have originated, it moves just as if it had received a certain initial impulse, in consequence of which it started into motion, and thus defined a certain plane, to which for all time its motion would be restricted.
so far we have spoken of only a single planet; let us now suppose that a second planet, b, is also to move in revolution about the same sun. this planet may be as great as a, or bigger, or smaller, but we shall still assume that both planets are inconsiderable in comparison with s. we may assume that b revolves at the same distance as a, or it may be nearer, or further. the orbit of b might also have been in the same plane as a, or—and here is the important point—it might have been in a plane inclined at any angle whatever to the orbit of a. the two planes might, indeed, have been perpendicular. no matter how varied may be the circumstances of the two planets, the sun would accept the control of each of them; each would be guided in its own orbit, whether that orbit be a circle, or whether it be an ellipse of any eccentricity whatever. so far as the attraction of the sun is concerned, each of these 300planets would remain for ever in the same plane as that in which it originally started. let us now suppose a third planet to be added. here again we may assume every variety in the conditions of mass and distance. we may also assume that the plane which contains the orbit of this third planet is inclined at any angle whatever to the planes of the preceding planets. in the same way we may add a fourth planet, and a fifth; and in order to parallel the actual circumstance of our solar system, so far as its more important members are concerned, we may add a sixth, and a seventh, and an eighth. the planes of these orbits are subjected to a single condition only. each one of them passes through the centre of the sun. if this requirement is fulfilled, the planes may be in other respects as different as possible.
in the actual solar system the circumstances are, however, very different from what we have represented in this imaginary solar system. it is the most obvious characteristic of the tracks of jupiter and venus, and the other planets belonging to the sun, that the planes in which they respectively move coincide very nearly with the plane in which the earth revolves. we must suppose all the orbits of our imaginary system to be flattened down, nearly into a plane, before we can transform the imaginary system of planets i have described into the semblance of an actual solar system.
if the orbits of the planets had been arranged in planes which were placed at random, we may presume they would have been inclined at very varied angles. as they are not so disposed, we may conclude that the planes have not been put down at random; 301we must conclude that there has been some cause in action which, if we may so describe it, has superintended the planes of these orbits and ordained that they should be placed in a very particular manner.
two planets’ orbits might conceivably coincide or be perpendicular, or they might contain any intermediate angle. the plane of the second planet might be inclined to the first at an angle containing any number of degrees. to make some numerical estimate of the matter, we proceed as follows: if we divide the right angle into ten parts of nine degrees each (fig. 47), then the inclination of the two planes might, for example, lie between o° and 9°, or between 18° and 27°, or between 45° and 54°, or between 81° and 90°, or in any one of the ten divisions. let us think of the orbit of jupiter. then the inclination of the plane in which it moves to the plane in which the earth moves must fall into one of the ten divisions. as a matter of fact, it does fall into the angle between 0° and 9°.
fig. 47.—a right angle divided into ten parts.
302but now let us consider a second planet, for example venus. if the orbit of venus were to be placed at random, its inclination might with equal probability lie in any one of the ten divisions, each of nine degrees, into which we have divided the right angle. it would be just as likely to lie between forty-five and fifty-four, or between seventy-two and eighty-one, as in any other division. but we find another curious coincidence. it was already remarkable that the plane of jupiter’s orbit should have been included in the first angle of nine degrees from the orbit of the earth. it is therefore specially noteworthy to find that the planet venus follows the same law, though each one of the ten angular divisions was equally available.
the coincidences we have mentioned, remarkable as they are, represent only the first of the series. what has been said with respect to the positions of the orbits of jupiter and venus may be repeated with regard to the orbits of mercury and mars, saturn, uranus, and neptune. if the tracks of these planets had been placed merely at random, their inclinations would have been equally likely to fall into any of the ten divisions. as a matter of fact, they all agree in choosing that one particular division which is adjacent to the track of the earth. if the orbits of the planets had indeed been arranged fortuitously, it is almost inconceivable that such coincidences could have occurred. let me illustrate the matter by the following little parable.
there were seven classes in a school, and there were ten boys in each class. there was one boy named smith in the first class, but only one. there was also one smith, but only one, in each of the 303other classes. the others were named brown, jones, robinson, etc. an old boy, named captain smith, who had gone out to australia many years before, came back to visit his old school. he had succeeded well in the world, and he wanted to do something generous for the boys at the place of which he had such kindly recollections. he determined to give a plum-cake to one boy in each class; and the fortunate boy was to be chosen by lot. the ten boys in each class were to draw, and each successful boy was to be sent in to captain smith to receive his cake.
the captain sat at a table, and the seven winners were shown in to receive their prizes. “what is your name?” he said to the boy in the first class, as he shook hands with him. “smith,” replied the boy. “dear me,” said the captain, “how odd that our names should be the same. never mind, it’s a good name. here’s your cake. good-bye, smith.” then up came the boy from the second class. “what is your name?” said the captain. “smith, sir,” was the reply. “dear me,” said the visitor. “this is very singular. it is indeed a very curious coincidence that two smiths should have succeeded. were you really chosen by drawing lots?” “yes, sir,” said the boy. “then are all the boys in your class named smith?” “no, sir; i’m the only one of that name in the ten.” “well,” said the captain, “it really is most curious. i never heard anything so extraordinary as that two namesakes of my own should happen to be the winners. now then for the boy from class three.” a cheerful youth advanced with a smile. “well, at all events,” said the good-natured old boy, “your name is not smith?” “oh, but it is,” said the youth. the 304gallant captain jumped up, and declared that there must have been some tremendous imposition. either the whole school consisted of smiths, or they called themselves smiths, or they had picked out the smiths. the four remaining boys, still expecting their cakes, here burst out laughing. “what are your names?” shouted the donor. “smith!” “smith!!” “smith!!!” “smith!!!!” were the astounding replies. the good man could stand this no longer. he sent for the schoolmaster, and said, “i particularly requested that you would choose a boy drawn by lot from each of your seven classes, but you have not done so. you have merely picked out my namesakes and sent them up for the cakes.” but the master replied, “no, i assure you, they have been honestly chosen by lot. nine black beans and one white bean were placed in a bag; each class of ten then drew in succession, and in each class it happened that the boy named smith drew the white bean.”
“but,” said the visitor, “this is not credible. only once in ten million times would all the seven smiths have drawn the white beans if left solely to chance. and do you mean to tell me that what can happen only once out of ten million times did actually happen on this occasion—the only occasion in my life on which i have attempted such a thing? i don’t believe the drawing was made fairly by lot. there must have been some interference with the operation of chance. i insist on having the lots drawn again under my own inspection.” “yes, yes,” shouted all the other boys. but all the successful smiths roared out, “no.” they did not feel at all desirous of another trial. they knew enough of the theory of probabilities to be aware 305that they might wait till another ten million fortunate old boys came back to the school before they would have such luck again. the situation came to a deadlock. the captain protested that some fraud had been perpetrated, and in spite of their assurances he would not believe them. the seven smiths declared they had won their cakes honestly, and that they would not surrender them. the captain was getting furious, the boys were on the point of rebellion, when the schoolmaster’s wife, alarmed by the tumult, came on the scene. she asked what was the cause of the disturbance. it was explained to her, and then captain smith added that by mathematical probabilities it was almost inconceivable that the only seven smiths in the school should have been chosen. the gracious lady replied that she knew nothing, and cared as little, about the theory of probabilities, but she did care greatly that the school should not be thrown into tumult. “there is only one solution of this difficulty,” she added. “it is that you forthwith provide cakes, not only for the seven smiths, but for every one of the boys in the school.” this resolute pronouncement was received with shouts of approval. the captain, with a somewhat rueful countenance, acknowledged that it only remained for him to comply. he returned, shortly afterwards, to his gold-diggings in australia, there to meditate during his leisure on this remarkable illustration of the theory of probabilities.
this parable illustrates the improbability of such arrangements as we find in the planets having originated by chance. the chances against their having thus occurred are 10,000,000 to 1. hence we find it reasonable to come to the conclusion that the 306arrangement, by which the planets move round the sun in planes which are nearly coincident, cannot have originated by chance. there must have been some cause which produced this special disposition. we have, therefore, to search for some common cause which must have operated on all the planets. as the planets are at present absolutely separated from each other, it is impossible for us to conceive a common cause acting upon them in their present condition. the cause must have operated at some prim?val time, before the planets assumed the separate individual existence that they now have.
we have spoken so far of the great planets only, and we have seen how the probability stands. we should also remark that there are also nearly 500 small planets, or asteroids, as they are more generally called. among them are, no doubt, a few whose orbits have inclinations to the ecliptic larger than those of the great planets. the great majority of the asteroids revolve, however, very close to that remarkable plane with which the orbits of the great planets so nearly coincide. every one of these asteroids increases the improbability that the planes of the orbits could have been arranged as we find them, without some special disposing cause. it is not possible or necessary to write down the exact figures. the probability is absolutely overwhelming against such an arrangement being found if the orbits of the planets had been decided by chance, and chance alone.
we may feel confident that there must have been some particular circumstances accompanying the formation of the solar system which rendered it absolutely necessary for the orbits of the planets to possess this 307particular characteristic. we have pointed out in chapter xii. that the nebular theory offers such an explanation, and we do not know of any other natural explanation which would be worthy of serious attention. indeed, we may say that no other such explanation has ever been offered.
