THE FIRST CONCORD.
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 system 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.
[Illustration: 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 initial 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 then
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.
[Illustration: 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
originally 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 planets
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; we 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°.
[Illustration: Fig. 47.—A RIGHT ANGLE DIVIDED INTO TEN PARTS.]
But 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 other 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
gallant 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 that 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 arrangement, 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 primæval 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 particular 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.
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