AFTER Galileo had felt the strong hand of the
Inquisition, in 1632, he was careful to confine
his researches, or at least his publications, to topics
that seemed free from theological implications. In
doing so he reverted to the field of his earliest studies
—namely, the field of mechanics; and the Dialoghi delle Nuove Scienze,
which he finished in 1636, and
which was printed two years later, attained a celebrity
no less than that of the heretical dialogue that had
preceded it. The later work was free from all apparent
heresies, yet perhaps it did more towards the
establishment of the Copernican doctrine, through
the teaching of correct mechanical principles, than
the other work had accomplished by a more direct
method.
Galileo's astronomical discoveries were, as we have
seen, in a sense accidental; at least, they received their
inception through the inventive genius of another.
His mechanical discoveries, on the other hand, were
the natural output of his own creative genius. At
the very beginning of his career, while yet a very
young man, though a professor of mathematics at
Pisa, he had begun that onslaught upon the old
Aristotelian ideas which he was to continue throughout
his life. At the famous leaning tower in Pisa, the
young iconoclast performed, in the year 1590, one of
the most theatrical demonstrations in the history of
science. Assembling a multitude of champions of the
old ideas, he proposed to demonstrate the falsity
of the Aristotelian doctrine that the velocity of falling
bodies is proportionate to their weight. There is
perhaps no fact more strongly illustrative of the temper
of the Middle Ages than the fact that this doctrine,
as taught by the Aristotelian philosopher, should so
long have gone unchallenged. Now, however, it was
put to the test; Galileo released a half-pound weight
and a hundred-pound cannon-ball from near the top
of the tower, and, needless to say, they reached the
ground together. Of course, the spectators were but
little pleased with what they saw. They could not
doubt the evidence of their own senses as to the
particular experiment in question; they could suggest,
however, that the experiment involved a violation of
the laws of nature through the practice of magic.
To controvert so firmly established an idea savored of
heresy. The young man guilty of such iconoclasm
was naturally looked at askance by the scholarship of
his time. Instead of being applauded, he was hissed,
and he found it expedient presently to retire from
Pisa.
Fortunately, however, the new spirit of progress had
made itself felt more effectively in some other portions
of Italy, and so Galileo found a refuge and a following
in Padua, and afterwards in Florence; and while, as
we have seen, he was obliged to curb his enthusiasm
regarding the subject that was perhaps nearest his
heart—the promulgation of the Copernican theory—yet
he was permitted in the main to carry on his
experimental observations unrestrained. These experiments
gave him a place of unquestioned authority
among his contemporaries, and they have transmitted
his name to posterity as that of one of the greatest
of experimenters and the virtual founder of modern
mechanical science. The experiments in question
range over a wide field; but for the most part they
have to do with moving bodies and with questions
of force, or, as we should now say, of energy. The
experiment at the leaning tower showed that the
velocity of falling bodies is independent of the weight
of the bodies, provided the weight is sufficient to
overcome the resistance of the atmosphere. Later
experiments with falling bodies led to the discovery
of laws regarding the accelerated velocity of fall.
Such velocities were found to bear a simple relation to
the period of time from the beginning of the fall.
Other experiments, in which balls were allowed to
roll down inclined planes, corroborated the observation
that the pull of gravitation gave a velocity proportionate
to the length of fall, whether such fall were
direct or in a slanting direction.
These studies were associated with observations on
projectiles, regarding which Galileo was the first to
entertain correct notions. According to the current
idea, a projectile fired, for example, from a cannon,
moved in a straight horizontal line until the propulsive
force was exhausted, and then fell to the
ground in a perpendicular line. Galileo taught that
the projectile begins to fall at once on leaving the
mouth of the cannon and traverses a parabolic course.
According to his idea, which is now familiar to every
one, a cannon-ball dropped from the level of the
cannon's muzzle will strike the ground simultaneously
with a ball fired horizontally from the cannon. As to
the paraboloid course pursued by the projectile, the
resistance of the air is a factor which Galileo could
not accurately compute, and which interferes with
the practical realization of his theory. But this is a
minor consideration. The great importance of his
idea consists in the recognition that such a force as
that of gravitation acts in precisely the same way
upon all unsupported bodies, whether or not such
bodies be at the same time acted upon by a force of
translation.
Out of these studies of moving bodies was gradually
developed a correct notion of several important
general laws of mechanics—laws a knowledge
of which was absolutely essential to the progress of
physical science. The belief in the rotation of the
earth made necessary a clear conception that all bodies
at the surface of the earth partake of that motion
quite independently of their various observed motions
in relation to one another. This idea was hard to
grasp, as an oft-repeated argument shows. It was
asserted again and again that, if the earth rotates, a
stone dropped from the top of a tower could not fall
at the foot of the tower, since the earth's motion
would sweep the tower far away from its original
position while the stone is in transit.
This was one of the stock arguments against the
earth's motion, yet it was one that could be refuted
with the greatest ease by reasoning from strictly
analogous experiments. It might readily be observed,
for example, that a stone dropped from a moving cart
does not strike the ground directly below the point
from which it is dropped, but partakes of the forward
motion of the cart. If any one doubt this he has but
to jump from a moving cart to be given a practical
demonstration of the fact that his entire body was
in some way influenced by the motion of translation.
Similarly, the simple experiment of tossing a ball from
the deck of a moving ship will convince any one that
the ball partakes of the motion of the ship, so that it
can be manipulated precisely as if the manipulator
were standing on the earth. In short, every-day experience
gives us illustrations of what might be called
compound motion, which makes it seem altogether
plausible that, if the earth is in motion, objects at its
surface will partake of that motion in a way that does
not interfere with any other movements to which
they may be subjected. As the Copernican doctrine
made its way, this idea of compound motion naturally
received more and more attention, and such experiments
as those of Galileo prepared the way for a new
interpretation of the mechanical principles involved.
The great difficulty was that the subject of moving
bodies had all along been contemplated from a wrong
point of view. Since force must be applied to an object
to put it in motion, it was perhaps not unnaturally
assumed that similar force must continue to be applied
to keep the object in motion. When, for example, a
stone is thrown from the hand, the direct force applied
necessarily ceases as soon as the projectile leaves the
hand. The stone, nevertheless, flies on for a certain
distance and then falls to the ground. How is this
flight of the stone to be explained? The ancient
philosophers puzzled more than a little over this
problem, and the Aristotelians reached the conclusion
that the motion of the hand had imparted a propulsive
motion to the air, and that this propulsive motion was
transmitted to the stone, pushing it on. Just how
the air took on this propulsive property was not explained,
and the vagueness of thought that characterized
the time did not demand an explanation.
Possibly the dying away of ripples in water may have
furnished, by analogy, an explanation of the gradual
dying out of the impulse which propels the stone.
All of this was, of course, an unfortunate maladjustment
of the point of view. As every one nowadays
knows, the air retards the progress of the stone,
enabling the pull of gravitation to drag it to the
earth earlier than it otherwise could. Were the
resistance of the air and the pull of gravitation removed,
the stone as projected from the hand would
fly on in a straight line, at an unchanged velocity,
forever. But this fact, which is expressed in what we
now term the first law of motion, was extremely difficult
to grasp. The first important step towards it
was perhaps implied in Galileo's study of falling
bodies. These studies, as we have seen, demonstrated
that a half-pound weight and a hundred-pound weight
fall with the same velocity. It is, however, matter
of common experience that certain bodies, as, for example,
feathers, do not fall at the same rate of speed
with these heavier bodies. This anomaly demands an
explanation, and the explanation is found in the
resistance offered the relatively light object by the
air. Once the idea that the air may thus act as an
impeding force was grasped, the investigator of mechanical
principles had entered on a new and promising
course.
Galileo could not demonstrate the retarding influence
of air in the way which became familiar a
generation or two later; he could not put a feather and
a coin in a vacuum tube and prove that the two would
there fall with equal velocity, because, in his day,
the air-pump had not yet been invented. The experiment
was made only a generation after the time
of Galileo, as we shall see; but, meantime, the great
Italian had fully grasped the idea that atmospheric
resistance plays a most important part in regard to
the motion of falling and projected bodies. Thanks
largely to his own experiments, but partly also to the
efforts of others, he had come, before the end of his
life, pretty definitely to realize that the motion of a
projectile, for example, must be thought of as inherent
in the projectile itself, and that the retardation
or ultimate cessation of that motion is due to the
action of antagonistic forces. In other words, he had
come to grasp the meaning of the first law of motion.
It remained, however, for the great Frenchman Descartes to
give precise expression to this law two years after Galileo's
death. As Descartes expressed it in his Principia Philosophiæ,
published in 1644, any body once in motion tends to go on
in a straight line, at a uniform rate of speed, forever.
Contrariwise, a stationary body will remain forever at
rest unless acted on by some disturbing force.
This all-important law, which lies at the very foundation
of all true conceptions of mechanics, was thus
worked out during the first half of the seventeenth
century, as the outcome of numberless experiments
for which Galileo's experiments with failing bodies
furnished the foundation. So numerous and so gradual
were the steps by which the reversal of view regarding
moving bodies was effected that it is impossible to
trace them in detail. We must be content to reflect
that at the beginning of the Galilean epoch utterly
false notions regarding the subject were entertained
by the very greatest philosophers—by Galileo himself,
for example, and by Kepler—whereas at the close of
that epoch the correct and highly illuminative view
had been attained.
We must now consider some other experiments of
Galileo which led to scarcely less-important results.
The experiments in question had to do with the movements
of bodies passing down an inclined plane, and
with the allied subject of the motion of a pendulum.
The elaborate experiments of Galileo regarding the
former subject were made by measuring the velocity
of a ball rolling down a plane inclined at various angles.
He found that the velocity acquired by a ball was
proportional to the height from which the ball descended
regardless of the steepness of the incline.
Experiments were made also with a ball rolling down
a curved gutter, the curve representing the are of a
circle. These experiments led to the study of the curvilinear
motions of a weight suspended by a cord; in
other words, of the pendulum.
Regarding the motion of the pendulum, some very
curious facts were soon ascertained. Galileo found, for example,
that a pendulum of a given length performs its oscillations
with the same frequency though the arc described by the
pendulum be varied greatly.
[6]
He found, also, that the rate of oscillation for pendulums
of different lengths varies according to a simple
law. In order that one pendulum shall oscillate one-half
as fast as another, the length of the pendulums
must be as four to one. Similarly, by lengthening the
pendulums nine times, the oscillation is reduced to
one-third, In other words, the rate of oscillation of
pendulums varies inversely as the square of their
length. Here, then, is a simple relation between the
motions of swinging bodies which suggests the relation
which Kepler bad discovered between the relative
motions of the planets. Every such discovery coming
in this age of the rejuvenation of experimental science
had a peculiar force in teaching men the all-important
lesson that simple laws lie back of most of the diverse
phenomena of nature, if only these laws can be discovered.
Galileo further observed that his pendulum might
be constructed of any weight sufficiently heavy readily
to overcome the atmospheric resistance, and that,
with this qualification, neither the weight nor the material
had any influence upon the time of oscillation,
this being solely determined by the length of the cord.
Naturally, the practical utility of these discoveries
was not overlooked by Galileo. Since a pendulum
of a given length oscillates with unvarying rapidity,
here is an obvious means of measuring time. Galileo,
however, appears not to have met with any great
measure of success in putting this idea into practice.
It remained for the mechanical ingenuity of Huyghens
to construct a satisfactory pendulum clock.
As a theoretical result of the studies of rolling and
oscillating bodies, there was developed what is usually
spoken of as the third law of motion—namely, the law
that a given force operates upon a moving body with
an effect proportionate to its effect upon the same
body when at rest. Or, as Whewell states the law:
"The dynamical effect of force is as the statical effect;
that is, the velocity which any force generates in a
given time, when it puts the body in motion, is proportional
to the pressure which this same force produces in a body
at rest.''[7] According to
the second law of motion, each one of the different forces,
operating at the same time upon a moving body, produces
the same effect as if it operated upon the body while at rest.
