In the last chapter a short description has been given of the ideas of
the ancients as to the nature of the earth and heavens. Before we pass
to the changes introduced by modern science, it will be well to devote
a short space to an examination of ancient scientific ideas.
All science is really based upon a combination of two methods,
called respectively inductive and deductive reasoning. The first of
these consists in gathering together the results of observation and
experiment, and, having put them all together, in the formulation of
universal laws. Having, for example, long observed that all heavy
things tended to go towards the centre of the earth, we might conclude
that, since the stars remain up in the sky, they can have no weight.
The conclusion would be wrong in this case, not because the method
is wrong, but because it is wrongly applied. It is true that all
heavy things _tend_ to go to the centre of the earth, but if they are
being whirled round like a stone in a sling the centrifugal force
will counteract this tendency. The first part of the reasoning would
be inductive, the second deductive. All this reasoning consists,
therefore, in forming as complete an idea as possible respecting the
nature of a thing, and then concluding from that idea what the thing
will do or what its other properties will be. In fact, you form correct
ideas, or “concepts,” as they are called, and reason from them.
But the danger arises when you begin to reason before you are sure of
the nature of your concepts, and this has been the great source of
error, and it was this error that all men of science so commonly fell
into all through ancient and modern times up to the seventeenth century.
Of course, if it were possible by mere observation to derive a
complete knowledge of any objects, it would be the simplest method.
All that would be necessary to do would be to reason correctly from
this knowledge. Unfortunately, however, it is not possible to obtain
knowledge of this kind in any branch of science.
The ancient method resembled the action of one who should contend that
by observing and talking to a man you could acquire such a knowledge of
his character as would infallibly enable you to understand and predict
all his actions, and to take little trouble to see whether what he did
verified your predictions.
The only difference between the old methods and the new is that in
modern times men have learned to give far more care to the formation
of correct ideas to start with, are much more cautious in arguing
from them, and keep testing them again and again on every possible
opportunity.
The constant insistence on the formation of clear ideas and the
practice of, as Lord Bacon called it, “putting nature to the torture,”
is the main cause of the advance of physical science in modern times,
and the want of application of these principles explains why so little
progress is being made in the so-called “humanitarian” studies, such as
philosophy, ethics, and politics.
The works of Aristotle are full of the fallacious method of the old
system. In his work on the heavens he repeatedly argues that the
heavenly bodies must move in circles, because the circle is the most
perfect figure. He affects a perplexity as to how a circle can at the
same time be convex and also its opposite, concave, and repeatedly
entangles his readers in similar mere word confusion.
Regarded as a man of science, he must be placed, I think, in spite
of his great genius, below Archimedes, Hipparchus, and several other
ancient astronomers and physicists.
His errors lived after him and dominated the thought of the middle
ages, and for a long time delayed the progress of science.
The other great writer on astronomy of ancient times was Ptolemy of
Alexandria.
His work was called the “Great Collection,” and was what we should
now term a compendium of astronomy. Although based on a fundamental
error, it is a thoroughly scientific work. There is none of the false
philosophy in it that so much disfigures the work of Aristotle. The
reasons for believing that the earth is at rest are interesting.
Ptolemy argues that if the earth were moving round on its axis once in
twenty-four hours a bird that flew up from it would be left behind.
At first sight this argument seems very convincing, for it appears
impossible to conceive a body spinning at the rate at which the earth
is alleged to move, and yet not leaving behind any bodies that become
detached from it.
On the other hand, the system which taught that the sun and planets
moved round the earth, and which had been adopted largely on account
of its supposed simplicity, proved, on further examination, to be
exceedingly complicated. Each planet, instead of moving simply and
uniformly round the earth in a circle, had to be supposed to move
uniformly in a circle round another point that moved round the earth in
a circle. This secondary circle, in which the planet moved, was called
an epicycle. And even this more complicated view failed to explain the
facts.
A system which, like that of Aristotle and Ptolemy, was based on
deductions from concepts, and which consisted rather of drawing
conclusions than of examining premises, was very well adapted to
mediæval thought, and formed the foundation of astronomy and geography
as taught by the schoolmen.
[Illustration: FIG. 20.]
The poem of Dante accurately represents the best scientific knowledge
of his day. According to his views, the centre of the earth was a fixed
point, such that all things of a heavy nature tended towards it. Thus
the earth and water collected round it in the form of a ball. He had no
idea of the attraction of one particle of matter for another particle.
The only conception he had of gravity was of a force drawing all heavy
things to a certain point, which thus became the point round which the
world was formed. The habitable part of the earth was an island, with
Jerusalem in the middle of it _J_. Round this island was an ocean _O_.
Under the island, in the form of a hollow cone, was hell, with its
seven circles of torment, each circle becoming smaller and smaller,
till it got down into the centre _C_. Heaven was at the opposite side
_H_ of the earth to Jerusalem, and was beyond the circles of the
planets, in the _primum mobile_. When Lucifer was expelled from heaven
after his rebellion against God, having become of a nature to be
attracted to the centre of the earth, and no longer drawn heavenwards,
he fell from heaven, and impinged upon the earth just at the antipodes
of Jerusalem, with such violence that he plunged right through it to
the centre, throwing up behind him a hill. On the summit of this hill
was the Garden of Eden, where our first parents lived, and down the
sides of the hill was a spiral winding way which constituted purgatory.
Dante, having descended into hell, and passed the centre, found his
head immediately turned round so as to point the other way up, and,
having ascended a tortuous path, came out upon the hill of Purgatory.
Having seen this, he was conducted to the various spheres of the
planets, and in each sphere he became put into spiritual communion with
the spirits of the blessed who were of the character represented by
that sphere, and he supposes that he was thus allowed to proceed from
sphere to sphere until he was permitted to come into the presence of
the Almighty, who in the _primum mobile_ presided over the celestial
hosts.
The astronomical descriptions given by Dante of the rising and setting
of the sun and moon and planets are quite accurate, according to the
system of the world as conceived by him, and show not only that he was
a competent astronomer, but that he probably possessed an astrolabe and
some tables of the motions of the heavenly bodies.
Our own poet Chaucer may also be credited with accurate knowledge of
the astronomy of his day. His poems often mention the constellations,
and one of them is devoted to a description of the astrolabe, an
instrument somewhat like the celestial globe which used to be employed
in schools.
But with the revival of learning in Europe and the rise of freedom of
thought, the old theories were questioned in more than one quarter.
It occurred to Copernicus, an ecclesiastic who lived in the sixteenth
century, to re-examine the theory that had been started in ancient
times, and to consider what explanation of the appearance of the
heavenly bodies could be given on the hypothesis put forward by
Pythagoras, that the earth moved round on its own axis, and also round
the sun.
It may appear rather curious that two theories so different, one that
the sun goes round the earth and the other that the earth goes round
the sun, should each be capable of explaining the observed appearances
of those bodies. But it must be remembered that motion is relative. If
in a waltz the gentleman goes round the lady, the lady also goes round
the gentleman. If you take away the room in which they are turning,
and consider them as spinning round like two insects in space, who is
to say which of them is at rest and which in motion? For motion is
relative. I can consider motion in a train from London to York. As I
leave London I get nearer to York, and I move with respect to London
and York. But if both London and York were annihilated how should I
know that I was in motion at all? Or, again, if, while I was at rest
in the train at a station on the way, instead of the train moving the
whole earth began to move in a southward direction, and the train in
some way were left stationary, then, though the earth was moving, and
the train was at rest, yet, so far as I was concerned, the train would
appear to have started again on its journey to York, at which place it
would appear to arrive in due time. The trees and hedges would fly by
at the proper rate, and who was to say whether the train was in motion
or the earth?
The theory of Copernicus, however, remained but a theory. It was
opposed to the evidence of the senses, which certainly leads us to
think that the earth is at rest, and it was opposed also to the ideas
of some among the theologians who thought that the Bible taught us that
the earth was so fast that it could not be moved. Therefore the theory
found but little favour. It was in fact necessary before the question
could be properly considered on its merits that more should be known
about the laws of motion, and this was the principal work of Galileo.
The merit of Galileo is not only to have placed on a firm basis the
study of mechanics, but to have set himself definitely and consciously
to reverse the ancient methods of learning.
He discarded authority, basing all knowledge upon reason, and protested
against the theory that the study of words could be any substitute for
the study of things.
Alluding to the mathematicians of his day, “This sort of men,” says
Galileo in a letter to the astronomer Kepler, “fancied that philosophy
was to be studied like the ‘Æneid’ or ‘Odyssey,’ and that the true
reading of nature was to be detected by the collating of texts.” And
most of his life was spent in fighting against preconceived ideas. It
was maintained that there could only be seven planets, because God
had ordered all things in nature by sevens (“Dianoia Astronomica,”
1610); and even the discoveries of the spots on the sun and the
mountains in the moon were discredited on the ground that celestial
bodies could have no blemishes. “How great and common an error,”
writes Galileo, “appears to me the mistake of those who persist in
making their knowledge and apprehension the measure of the knowledge
and apprehension of God, as if that alone were perfect which they
understand to be so. But ... nature has other scales of perfection,
which we, being unable to comprehend, class among imperfections.
“If one of our most celebrated architects had had to distribute the
vast multitude of fixed stars over the great vault of heaven, I believe
he would have disposed them with beautiful arrangements of squares,
hexagons, and octagons; he would have dispersed the larger ones among
the middle-sized or lesser, so as to correspond exactly with each
other; and then he would think he had contrived admirable proportions;
but God, on the contrary, has shaken them out from His hand as if by
chance, and we, forsooth, must think that He has scattered them up
yonder without any regularity, symmetry, or elegance.”
In one of Galileo’s “Dialogues” Simplicio says, “That the cause that
the parts of the earth move downwards is notorious, and everyone knows
that it is gravity.” Salviati replies, “You are out, Master Simplicio:
you should say that everyone knows that _it is called_ gravity; I do
not ask you for the name, but for the nature, of the thing of which
nature neither you nor I know anything.”
Too often are we still inclined to put the name for the thing, and to
think when we use big words such as art, empire, liberty, and the
rights of man, that we explain matters instead of obscuring them. Not
one man in a thousand who uses them knows what he means; no two men
agree as to their signification.
The relativity of motion mentioned above was very elegantly illustrated
by Galileo. He called attention to the fact that if an artist were
making a drawing with a pen while in a ship that was in rapid passage
through the water, the true line drawn by the pen with regard to the
surface of the earth would be a long straight line with some small
dents or variations in it. Yet the very same line traced by the pen
upon a paper carried along in the ship made up a drawing. Whether you
saw a long uneven line or a drawing in the path that the pen had traced
depended altogether on the point of view with which you regarded its
motion.
[Illustration: FIG. 21.]
But the first great step in science which Galileo made when quite a
young professor at Pisa was the refutation of Aristotle’s opinion that
heavy bodies fell to the earth faster than light ones. In the presence
of a number of professors he dropped two balls, a large and a small
one, from the parapet of the leaning tower of Pisa. They fell to the
ground almost exactly in the same time. This experiment is quite an
easy one to try. One of the simplest ways is as follows: Into any beam
(the lintel of a door will do), and about four inches apart, drive
three smooth pins so as to project each about a quarter of an inch;
they must not have any heads. Take two unequal weights, say of 1 lb.
and 3 lbs. Anything will do, say a boot for one and pocket-knife for
the other; fasten loops of fine string to them, put the loops over the
centre peg of the three, and pass the strings one over each of the side
pegs. Now of course if you hitch the loops off the centre peg _P_ the
objects will be released together. This can be done by making a loop
at the end of another piece of string, _A_, and putting it on to the
centre peg behind the other loops. If the string be pulled of course
the loop on it pulls the other two loops off the central peg, and
allows the boot and the knife to drop. The boot and the knife should be
hung so as to be at the same height. They will then fall to the ground
together. The same experiment can be tried by dropping two objects from
an upper window, holding one in each hand, and taking care to let them
go together.
[Illustration: FIG. 22.]
This result is very puzzling; one does not understand it. It appears as
though two unequal forces produced the same effect. It is as though a
strong horse could run no faster than a weaker one.
The professors were so irritated at the result of this experiment, and
indeed at the general character of young Professor Galileo’s attacks on
the time-honoured ideas of Aristotle, that they never rested till they
worried him out of his very poorly paid chair at Pisa. He then took a
professorship at Padua.
Let us now examine this result and see why it is that the ideas we
should at first naturally form are wrong, and that the heavy body will
fall in exactly the same time as the light one.
We may reason the matter in this way. The heavy body has more force
pulling on it; that is true, but then, on the other hand there is more
matter which has got to be moved. If a crowd of persons are rushing out
of a building, the total force of the crowd will be greater than the
force of one man, but the speed at which they can get out will not be
greater than the speed of one man; in fact, each man in the crowd has
only force enough to move his own mass. And so it is with the weights:
each part of the body is occupied in moving itself. If you add more to
the body you only add another part which has itself to move. A hundred
men by taking hands cannot run faster than one man.
But, you will say, cannot a man run faster than a child? Yes, because
his impelling power is greater in proportion to his weight than that of
a child.
If it were the fact that the attraction of gravity due to the earth
acted on some bodies with forces greater in proportion to their
masses than the forces that acted on other bodies, then it is true
that those different bodies would fall in unequal time. But it is
an experimental fact that the attractive force of gravity is always
exactly proportional to the mass of a body, and the resistance to
motion is also proportional to mass, hence the force with which a
body is moved by the earth’s attraction is always proportional to the
difficulty of moving the body. This would not be the case with other
methods of setting a body in motion. If I kick a small ball with all
my might, I shall send it further than a kick of equal strength would
send a heavier ball. Why? Because the impulse is the same in each case,
but the masses are different. But if those balls are pulled by gravity,
then, by the very nature of the earth’s attraction (the reason of which
we cannot explain), the small ball receives a little pull, and the big
ball receives a big pull, the earth exactly apportioning its pull in
each case to the mass of the body on which it has to act. It is to this
fact, that the earth pulls bodies with a strength always in each case
exactly proportional to their masses, that is due the result that they
fall in equal times, each body having a pull given to it proportional
to its needs.
The error of the view of Aristotle was not only demonstrated by
Galileo by experiment, but was also demonstrated by argument. In this
argument Galileo imitated the abstract methods of the Aristotelians,
and turned those methods against themselves. For he said, “You” (the
Aristotelians) “say that a lighter body will fall more slowly than a
heavy one. Well, then, if you bind a light body on to a heavy one by
means of a string, and let them fall together, the light body ought
to hang behind, and impede the heavy body, and thus the two bodies
together ought to fall more slowly than the heavy body alone; this
follows from your view: but see the contradiction. For the two bodies
tied together constitute a heavier body than the heavy body alone, and
thus, on your own theory, ought to fall more quickly than the heavy
body alone. Your theory, therefore, contradicts itself.”
The truth is that each body is occupied in moving itself without
troubling about moving its neighbour, so that if you put any number of
marbles into a bag and let them drop they all go down individually, as
it were, and all in the time which a single marble would take to fall.
For any other result would be a contradiction. If you cut a piece of
bread in two, and put the two halves together, and tie them together
with a thread, will the mere fact that they are two pieces make each of
them fall more slowly than if they were one? Yet that is what you would
be bound to assert on the Aristotelian theory. Hold an egg in your
open hand and jump down from a chair. The egg is not left behind; it
falls with you. Yet you are the heavier of the two, and on Aristotelian
principles you ought to leave the egg behind you. It is true that when
you jump down a bank your straw hat will often come off, but that is
because the air offers more resistance to it than the air offers to
your body. It is the downward rush through the air that causes your hat
to be left behind, just as wind will blow your hat off without blowing
you away. For since motion is relative, it is all one whether you jump
down through the air, or the air rushes past you, as in a wind. If
there were no air, the hat would fall as fast as your body.
This is easy to see if we have an airpump and are thus enabled to
pump out almost all the air from a glass vessel. In that vessel so
exhausted, a feather and a coin will fall in equal times. If we have
not an airpump, we can try the experiment in a more simple way. For
let us put a feather into a metal egg-cup and drop them together. The
cup will keep the air from the feather, and the feather will not come
out of the cup. Both will fall to the ground together. But if the
lighter body fall more slowly, the feather ought to be left behind. If,
however, you tie some strings across a napkin ring so as to make a sort
of rough sieve, and put a feather in it, and then drop the ring, then
as the ring falls the air can get through the bottom of the ring and
act on the feather, which will be left floating as the ring falls.
Let us now go on to examine the second fallacy that was derived from
the Aristotelians, and that so long impeded the advance of science,
namely, that the earth must be at rest.
The principal reason given for this was that if bodies were thrown
up from the earth they ought, if the earth were in motion, to remain
behind. Now, if this were so, then it would follow that if a person
in a train which was moving rapidly threw a ball vertically, that is
perpendicularly, up into the air, the ball, instead of coming back into
his hand, ought to hit the side of the carriage behind him. The next
time any of my readers travel by train he can easily satisfy himself
that this is not so. But there are other ways of proving it. For
instance, if a little waggon running on rails has a spring gun fixed in
it in a perpendicular position, so arranged that when the waggon comes
to a particular point on the rails a catch releases the trigger and
shoots a ball perpendicularly upwards, it will be found that the ball,
instead of going upwards in a vertical line, is carried along over the
waggon, and the ball as it ascends and descends keeps always above the
waggon, just as a hawk might hover over a running mouse, and finally
falls not behind the waggon, but into it.
So, again, if an article is dropped out of the window of a train, it
will not simply be left behind as it falls, but while it falls it will
also partake of the motion of the train, and touch the ground, not
behind the point from which it was dropped, but just underneath it.
The reason is, that when the ball is dropped or thrown it acquires
not only the motion given to it by the throw, or by gravity, but it
takes also the motion of the train from which it is thrown. If a ball
is thrown from the hand, it derives its motion from the motion of the
hand, and if at the time of throwing the person who does so is moving
rapidly along in a train, his hand has not only the outward motion
of the throw, but also the onward motion of the train, and the ball
therefore acquires both motions simultaneously. Hence then it is not
correct reasoning to say, because a ball thrown up vertically falls
vertically back to the spot from which it was thrown, that therefore
the earth must be at rest; the same result will happen whether the
earth is at rest or in motion. You can no more tell whether the earth
is at rest or in motion from the behaviour of falling bodies than you
can tell whether a ship on the ocean is at rest or in motion from the
behaviour of bodies on it.
But you will say. Then why do we feel sea-sick on a ship? The answer
is, that that is because the motion of the ship is not uniform. If the
earth, instead of turning round uniformly, were to rock to and fro,
everything on it would be flung about in the wildest fashion. For as
soon as the earth had communicated its motion to a body which then
moved with the earth, if the earth’s motion were reversed, the body
would go on like a passenger in a train on which the break is quickly
applied, and he would be shot up against the side of the room. Nay,
more, the houses would be shaken off their foundations. Changes of
motion are perceptible _so long as the change is going on_. We are
therefore justified in inferring from the behaviour of bodies on the
earth, not that the earth is at rest, but that it is either at rest, or
else, if it is in motion, that its motion is uniform and not in jerks
or variable.
[Illustration: FIG. 23.]
For if it were not so, consider what would be happening around us. The
earth is about 8,000 miles in diameter, and a parallel of latitude
through London is therefore about 19,000 miles long, and this space
is travelled in twenty-four hours. So that London is spinning through
space at the rate of over 1,000 feet a second, due to the earth’s
rotary motion alone, not to speak of the motion due to the earth’s
path round the sun. If a boy jumped up two and a half feet into the
air, he would take about half a second to go up and come down, but if
in jumping he did not partake of the earth’s motion, he would land
more than 500 feet to the westward of the point from which he jumped
up, and if he did it in a room, he would be dashed against the wall
with a force greater than he would experience from a drop down from
the top of Mont Blanc. He would be not only killed, but dashed into an
indistinguishable mass. If the earth suddenly stood still, everything
on it would be shaken to pieces. It is bad enough to have the
concussion of a train going thirty miles an hour when dashed against
some obstacle. But the concussion due to the earth’s stoppage would
be as of a train going about 800 miles an hour, which would smash up
everything and everybody.
Thus, then, the first effect of the new ideas formulated by Galileo was
to show that the Copernican theory that the earth moved round on its
axis, and round the sun, was in agreement with the laws of motion. In
fact, he introduced quite new ideas of force, and these ideas I must
now endeavour to explain.
Let us consider what is meant by the word “force.” If I press my
hand against the table, I exert force. The harder I press, the more
force there is. If I put a weight on a stand, the weight presses the
stand down with a force. If I squeeze a spring, the spring tries to
recover itself and exerts a certain force. In all these cases force
is considered as a pressure. And I can measure the force by seeing
how much it will press things. If I take a spring, and press it in an
inch, it takes perhaps a force of 1 lb. It will take a force of 2 lbs.
to press it in another inch. Or again, if I pull it out an inch, it
takes a force of 1 lb. If I pull it out another inch, it takes a force
of 2 lbs. We thus always get into the habit of conceiving forces as
producing pressures and being measured by pressures.
[Illustration: FIG. 24.]
This is a perfectly legitimate way of looking at the matter, just as
the cook’s method of employing a spring balance to weigh masses of meat
is a perfectly legitimate way of estimating the forces acting upon
bodies at rest. But when you come to consider the laws of the pendulums
of clocks, to which all that I am saying is a preparation, then you
have to deal with bodies in motion. And for this purpose a new idea of
force altogether is requisite. We shall no longer speak of forces as
producing _pressures_. We shall treat them quite independently of their
pressing power. The sun exerts a force of attraction on the earth, but
it does not press upon it. It exerts its force at a distance. Hence
then we want a new idea of “force.” This idea is to be the following.
We will consider that when a force acts upon a body it endeavours to
cause it to move; in fact, it tries to impart motion to the body. We
may treat this motion as a sort of thing or property. The longer the
force acts on the body, the more motion it imparts to it, provided the
body is free to receive that motion. So that we may say that the test
of the strength of the force is how much motion it can give to a body
of a given mass in a given time. It does not matter how the force acts.
It may act by means of a string and pull it; it may act by means of a
stick and push it; it may act by attraction and draw it; it may act
by repulsion and repel it; it may act as a sort of little spirit and
fly away with it. In all these cases it _acts_. The more it acts, the
more effect it has. In double the time it produces double the motion.
If the mass is big, it takes more force to make the mass move; if the
mass of the body is small, it is moved more easily. Therefore when we
want to measure a force in this way we do not press it against springs
to see how much it will press them in. What we do is to cause it to act
on bodies that are free to move and see what motions it will produce
in them. Of course we can only do this with things that are free to
move. You cannot treat force in this way if you have only a pair of
scales; in that case you would have to be content with simply measuring
pressures. It is important clearly to grasp this idea. If a body has
a certain mass, then the force acting on it is measured by the amount
of motion that will in a given time be imparted to that mass, provided
that the mass is free to move. This is to be our definition of force.
Therefore, by the action of an attraction or any other force on a body
free to move; motion is continually being imparted to the body. Motion
is, as it were, poured into it, and therefore the body continually
moves faster and faster.
Here is a ball flying through the air. Let us suppose that forces are
acting on it. How can we measure them? We cannot feel what pressures
are being exerted on it. The only thing we can do is to watch its
motions, and see how it flies. If it goes more and more quickly, we
say, “There is propelling force acting on it”; if it begins to stop,
we say again, “There is retarding force acting on it.” So long as it
does not change its speed or direction, we say, “There is no force
acting on it.” By this method, therefore, we tell whether a body is
being acted on by force, simply by observing its speed or its change of
speed. Merely to say a body is _moving_ does not tell us that force is
acting on it. All we know is that, if it is moving, force _has_ acted
on it. It is only when we see it changing its speed or direction, that
is changing its motion, that we say _force_ is acting. Every change of
motion, either in direction or speed, must be the result of force, and
must be proportional to that force. This is what we mean when we say
motion is the test and measure of force.
This most interesting way of looking at the matter lies at the root
of the whole theory of mechanics. It is the foundation of the system
which the stupendous genius of Newton conceived in order to explain the
motion of the sun, moon, and stars.
Forces were treated by him as proportional to the motions, and the
motions proportional to the forces, and with this idea he solved a part
of the riddle of the universe. Galileo had partly seen the same thing,
but he never saw it so clearly as Newton. Great discoveries are only
made by seeing things clearly. What required the force of a genius in
one age to see in the next may be understood by a child.
Hence then we say a force is that which in a given time produces a
given motion in a given mass which is free to move.
You must have time for a force to act in; for however great the force,
in no time there can be no motion. You must have mass for a force to
act on; no mass, no effect. You must have free space for the mass to
move in; no freedom to move, no movement.
But what is this “mass”? We do not know; it is a mystery. We call it
“quantity of matter.” In uniform substances it varies with size. Double
the volume, double the mass. Cut a cake in half, each half has the
same “mass.” But then is mass “weight”? No, it is not. _Weight_ is the
action of the earth’s attraction on matter. No earth to attract, and
you would have no weight, but you would still have “mass.” What then
is matter? Of that we have no idea. The greatest minds are now at work
upon it. But _mass_ is quantity of matter. Knock a brick against your
head, and you will know what mass is. It is not the weight of the brick
that gives you a bump; it is the mass. Try to throw a ball of lead, and
you will know what mass is. Try to push a heavy waggon, and you will
know what mass is. _Weights_, that is earth attractions on masses, are
proportional to the masses at the same place. This, as we have seen, is
known by experiment.
Therefore, when a force acts for a certain time on a mass that is free
to move, however small the force and however small the time, that body
will move. When a baby in a temper stamps upon the earth it makes the
earth move—not much, it is true, but still it moves; nay, more, in
theory, not a fly can jump into the air without moving the earth and
the whole solar system. Only, as you may imagine they do not show it
appreciably. Still, in theory the motion is there.
Hence then there are two different ways of considering and estimating
forces, one suitable for observations on bodies at rest, the other
suitable for observations of bodies that are free to move. The force
of course always tends to produce motion. If, however, motion is
impossible, then it develops pressures which we can measure, and
calculate, and observe. If the body is free to move, then the force
produces motions which we can also measure, calculate, and observe.
And we can compare these two sets of effects. We can say, “A force
which, acting on a ball of a mass of one pound, would produce such and
such motions, would if it acted on a certain spring produce so much
compression.”
The attraction of the earth on masses of matter that are not free to
move gives rise to forces which are called weights. Thus the attraction
of gravitation on a mass of one pound produces a pressure equal to a
weight of one pound. Unfortunately the same word “pound” is used to
express both the mass and the weight, and has come down to us from days
when the nature of mass was not very well appreciated. But great care
must be taken not to confuse these two meanings.
But the earth’s attractions and all other forces acting upon matter
which is free to move give rise to changes of motion. The word used for
a change of motion is “acceleration” or a quickening. “He accelerated
his pace,” we say. That is, he quickened it; he added to his motion. So
that _force_, acting on _mass_ during a _time_, produces acceleration.
From this, then, it follows that if a _force_ continues to act on a
body the body keeps moving quicker and quicker. When the force stops
acting, the motion already acquired goes on, but the acceleration
stops. That is to say, the body goes on moving in a straight line
uniformly at the pace it had when the force stopped.
If, then, a body is exposed to the action of a force, and held tight,
what will happen? It will, of course, remain fixed. Now let it go—it
will then, being a free body, begin to move. As long as the force
acts, the force keeps putting more and more motion into the body, like
pouring water into a jug, the longer you pour the faster the motion
becomes. The body keeps all the motion it had, and keeps adding all the
motion it gains. It is like a boy saving up his weekly pocket-money:
he has what he had, and he keeps adding to that. So if in one second
a motion is imparted of one foot a second, then in another second a
motion of one foot a second more will be added, making together a
motion of two feet a second; in another second of force action the
motion will have been increased or “accelerated” by another foot per
second, and so on. The speed will thus be always proportional to
the force and the time. If we write the letter V to represent the
motion, or speed, or velocity; F to represent the acceleration or gain
of motion; and T to represent the time, then V = FT. Here V is the
velocity the body will have acquired at the end of the time T, if free
to move and submitted to a force capable of producing an acceleration
of F feet per second in a unit of time.
V is the final velocity. The average velocity will be 1/2 V, for it
began with no velocity and increased uniformly. How far will the body
have fallen in the interval? Manifestly we get that by multiplying the
time by the average velocity, that is S = 1/2 VT, where V, as I said,
is the final velocity, but we found that V = FT. Hence by substitution
S = 1/2 FT × T = 1/2 FT².
It is to be carefully borne in mind that these letters V, S, and T
do not represent velocities, spaces, and times, but merely represent
arithmetical numbers of units of velocities, spaces, and times. Thus
V represents V feet per second, S represents S feet, and T represents
T seconds. And when we use the equation V = FT we do not mean that
by multiplying a force by a time you can produce a velocity. If, for
instance, it be true that you can obtain the number of inhabitants (H)
in London by multiplying the average number of persons (P) who live
in a house by the number of houses (N), this may be expressed by the
equation H = PN. But this does not mean that by multiplying people into
houses you can produce inhabitants. H, P, and N are numbers of units,
and they are _numbers only_.
Therefore when a body is being acted on by an accelerating force
it tends to go faster and faster as it proceeds, and therefore its
velocity increases with the time. But the space passed through
increases faster still, for as the time runs on not only does the
space passed through increase, but the rate of passing also gets
bigger. It goes on increasing at an increasing rate. It is like a man
who has an increasing income and always goes on saving it. His total
mounts up not merely in proportion to the time, but the very rate of
increase also increases with the time, so that the total increase is
in proportion to the time multiplied into the time, in other words to
the square of the time. So then, if I let a body drop from rest under
the action of any force capable of producing an acceleration, the space
passed through will be as the square of the time.
Now let us see what the speed will be if the force is gravity, that is
the attraction of the earth.
Turning back to what was said about Galileo, it will be remembered that
he showed that all bodies, big and small, light and heavy, fell to the
earth at the same speeds. What is that speed? Let us denominate by G
the number of feet per second of increase of motion produced in a body
by the earth’s action during one second. Then the velocity at the end
of that second will be V = GT. The space fallen through will be S = 1/2
GT².
What I want to know then is this: how far will a body under the action
of gravity fall in a second of time?
This, of course, is a matter for measurement. If we can get a machine
to measure seconds, we shall be able to do it; but inasmuch as falling
bodies begin by falling sixteen feet in the first second and afterwards
go on falling quicker and quicker, the measurements are difficult.
Galileo wanted to see if he could make it easier to observe. He said
to himself, “If I can only water down the force of gravity and make
it weaker, so that the body will move very slowly under its action,
then the time of falling will be easier to observe.” But how to do it?
This is one of those things the discovery of which at once marks the
inventor.
[Illustration: FIG. 25.]
The idea of Galileo was, instead of letting the body drop vertically,
to make it roll slowly down an incline, for a body put upon an incline
is not urged down the incline with the same force which tends to make
it fall vertically.
Can any law be discovered tending to show what the force is with which
gravity tends to drag a mass down an incline?
There is a simple one, and before Galileo’s time it had been
discovered by Stevinus, an engineer. Stevinus’ solution was as follows.
Suppose that _A B C_ is a wedge-shaped block of wood. Let a loop of
heavy chain be hung over it, and suppose that there is a little pulley
at _C_ and no friction anywhere. Then the chain will hang at rest. But
the lower part, from _A_ to _B_, is symmetrical; that is to say, it
is even in shape on both sides. Hence, so far as any pull it exerts
is concerned, the half from _A_ to _D_ will balance the other half
from _B_ to _D_. Therefore, like weights in a scale, you may remove
both, and then the force of gravity acting down the plane on the part
_A C_ will balance the force of gravity acting vertically on the part
_C B_. Now the weight of any part of the chain, since it is uniform,
is proportional to its length. Hence, then, the gravitational force
down the plane of a piece whose weight equals _C A_ is equal to the
gravitational force vertically of a piece whose weight equals _C B_. In
other words, the force of gravity acting down a plane is diminished in
the ratio of _C B_ to _C A_.
But when a body falls vertically, then, as we have seen, S = 1/2 GT²,
where S is the space it will fall through, G the number of feet per
second of velocity that gravity, acting vertically on a body, will
produce in it in a second, and T the number of seconds of time. If
then, instead of falling vertically, the body is to fall obliquely down
a plane, instead of G we must put as the accelerating force
G × (vertical height of the end of the plane)/(length of the plane).
To try the experiment, he took a beam of wood thirty-six feet long with
a groove in it. He inclined it so that one end was one foot higher than
the other. Hence the acceleration down the plane was 1/36 G, where G is
the vertical acceleration due to gravity which he wanted to discover.
Then he measured the time a brass ball took to run down the plane
thirty-six feet long, and found it to be nine seconds. Whence from
the equation given above 36 feet = 1/2 acceleration of gravity down
the plane × (9 seconds)². Whence it follows that the acceleration of
gravity down the plane is (36 × 2)/(9)² feet per second.
But the slope of the plane is one thirty-sixth to the vertical.
Therefore the vertical acceleration of gravity, _i.e._, the velocity
which gravity would induce in a vertical direction in a second, is
equal to thirty-six times that which it exercises down the plane,
_i.e._,
36 × (36 × 2)/(9)²; and this equals 32 feet per second.
Though this method is ingenious, it possesses two defects. One is the
error produced by friction, the other from failure to observe that
the force of gravity on the ball is not only exerted in getting it
down the plane, but also in rotating it, and for this no allowance has
been made. The allowance to be made for rotation is complicated, and
involves more knowledge than Galileo possessed. Still the result is
approximately true.
[Illustration: FIG. 26.]
The next attempt to measure G, that is the velocity that gravity will
produce on a body in a second of time, was made by Attwood, a Cambridge
professor. His idea was to weaken the force of gravity and thus make
the action slow, not by making it act obliquely, but by allowing it to
act, not on the whole, but only on a portion of the mass to be moved.
For this purpose he hung two equal weights over a very delicately
constructed pulley. Gravity, of course, could not act on these, for
any effect it produced on one would be negatived by its effect on
the other. The weights would therefore remain at rest. If, however,
a small weight _W_, equal say to a hundredth of the combined weight
of the weights _A_ and _B_ and _W_, were suddenly put on _A_, then it
would descend under an accelerating force equal to a hundredth part of
ordinary gravity. We should then have
S (the space moved through by the weights) = 1/2 × G/100 × t².
With such a system, he found that in 7½ seconds the weights moved
through 9 feet. Whence he got
9 = 1/2 G/100 × (7½)².
From which
G = (2 × 9 × 100)/(7½)² = 32 feet per second nearly.
Thus by letting gravity only act on a hundredth part of the total
weight moved, namely _A_, _B_, and _W_, he weakened its action 100
times, and thus made the time of falling and the space fallen through
sufficiently large to be capable of measurement. To sum up, when a body
free to move is acted upon by the force of gravity, its speed will
increase in proportion to the time it has been acted upon, and the
space it will pass through from rest is proportional to the square of
the time during which the accelerating force has acted on it.
Gravity is, of course, not the only accelerating force with which
we are acquainted. If a spring be suddenly allowed to act on a body
and pull it, the body begins to move, and its action is gradually
accelerated, just as though it were attracted, and the acceleration
of its motion will be proportional to the time during which the
accelerating force acts. Similarly, if gunpowder be exploded in a
gun-barrel, and the force thus produced be allowed to act on a bullet,
the motion of the bullet is accelerated so long as it is in the barrel.
When the bullet leaves the barrel it goes on with a uniform pace in a
straight line, which, however, by the earth’s attraction is at once
deflected into a curve, and altered by the resistance of the air.
[Illustration: FIG. 27.]
It has been already stated that motions may be considered independently
one of another, so that if a body be exposed to two different forces
the action of these forces can be considered and calculated each
independently of the other. Let us take an example of this law. We have
seen if a body is propelled forwards, and then the force acting on it
ceases, that it proceeds on with uniform unchanging velocity, and if
nothing impeded it, or influenced it, it would go on in a straight line
at a uniform speed.
We have also seen that if a body is exposed to the action of an
accelerating force such as gravity it constantly keeps being
accelerated, it constantly keeps gaining motion, and its speed becomes
quicker and quicker.
[Illustration: FIG. 28.]
Let us suppose a body exposed to both of these forces at the same time.
Shoot it out of a cannon, and let an accelerating force act on it, not
in the direction it is going, but in some other direction, say at right
angles. What will happen? In the direction in which it is going, its
speed will remain uniform. In the direction in which the accelerating
force is acting, it will move faster and faster. Thus along _A B_ it
will proceed uniformly. If it proceeded uniformly also along _A C_ (as
it would do if a simple force acted on it and then ceased to act), then
as a result it would go in the oblique line _A D_, the obliquity being
determined by the relative magnitude of the forces acting on it. But
how if it went uniformly along _A B_, but at an accelerated pace along
_A C_? Then while in equal times the distances along _A B_ would be
uniform the distances in the same times along _A C_ would be getting
bigger and bigger. It _would not describe a straight line; it would go
in a curve_. This is very interesting. Let us take an example of it.
Suppose we give a ball a blow horizontally; as soon as it quits the bat
it would of course go on horizontally in a straight line at a uniform
speed; but now if I at the same instant expose it to the accelerating
force of gravity, then, of course, while its horizontal movement will
go on uniformly, its downward drop will keep increasing at a speed
varying as the time. And while the total distances horizontally will
be uniform in equal times, the total downward drop from _A B_ will
be as the squares of the times. Here, then, you have a point moving
uniformly in a horizontal direction, but as the squares of the times in
a vertical direction. It describes a curve. What curve? Why, one whose
distances go uniformly one way, but increase as the squares the other
way.
[Illustration: FIG. 29.]
This interesting curve is called a parabola. With a ball simply hit by
a bat, the motion is so very fast that we cannot see it well. Cannot
we make it go slowly? Let us remember what Galileo did. He used an
inclined plane to water down his force of gravity. Let us do the same.
Let us take an inclined plane and throw on it a ball horizontally.
It will go in a curve. Its speed is uniform horizontally, but is
accelerated downwards. If we desire to trace the curve it is easy to
do. We coat the ball with cloth and then dip it in the inkpot. It will
then describe a visible parabola. If I tilt up the plane and make the
force of gravity big, the parabola is long and thin; if I weaken down
the force of gravity by making the plane nearly horizontal, then it is
wide and flat.
One can also show this by a stream of peas or shot. The little bullets
go each with a uniform velocity horizontally, and an accelerated force
downwards.
Instead of peas we can use water. A stream of it rushing horizontally
out of an orifice will soon bend down into a parabola.
Thus then I have tried to show what force is and how it is measured. I
repeat again, when a body is free to move, then, if no further force
acts on it, it will go on in a straight line at a uniform speed, but
if a force continues to act on it in any direction, then that force
produces in each unit of time a unit of acceleration in the direction
in which the force acts, and the result is that the body goes on moving
towards the direction of acceleration at a constantly increasing speed,
and hence passing over spaces that are greater and greater as the speed
increases. This is the notion of a “force.” In all that has been said
above it has been assumed that the attraction of gravity on a body
does not increase as that body gets nearer to the earth. This is not
strictly true; in reality the attractive force of gravity increases as
the earth’s centre is approached. But small distances through which
the weights in Attwood’s machine fall make no appreciable difference,
being as nothing compared to the radius of earth. For practical
purposes, therefore, the force may be considered uniform on bodies that
are being moved within a few feet of the earth’s surface. It is only
when we have to consider the motions of the planets that considerations
of the change of attractive force due to distance have to be considered.
I am glad to say that the most tiresome, or rather the most difficult,
part of our inquiry is now over. With the help of the notions already
acquired, we are now ready to get to the pendulum, and to show how it
came about that a boy who once in church amused himself by watching the
swinging of the great lamps instead of attending to the service laid
the foundation of our modern methods of measuring time.