Galileo was not the first natural philosopher to have difficulty with Aristotle's physics. The explanation of projectile motion was also controversial among the Greeks. Aristotle held that violent or non-natural motion, that is, any motion of body other than going freely toward its "natural place", was the direct result of an external applied force. The motion continues as long as the force is applied and would cease when the force is removed. This seems to agree with our common-sense experience, say, of pushing a table across the floor. But it does not agree with the motion of projectiles, like an arrow, which continues awhile after the force that put them in motion stops. Aristotle's explanation is that an arrow continues to fly through the air because the air in front of the arrow was forced aside returns (nature abhors a vacuum) behind the arrow and pushes on the back of the arrow. Thus the violent motion of the arrow is caused first by the force of the bowstring and then continues because of the pushing of the air flowing in behind the arrow. Thus Aristotle believed that the external medium was necessary for continuous violent motion. This push of the air gradually diminishes and when it vanishes then the arrow begins to fall with its natural downward motion. By this mechanism Aristotle employed the medium both as motive force and resistance. Not only did he believe that the motive force had to be in constant physical contact with the body it moved, but also that the resistant medium was essential to slow and stop the motion, otherwise the motion would be instantaneous (or, as he put it, "beyond any ratio"), which is absurd. And since the resistance of the medium to motion increases as the density of medium increases and decreases as its density decrease, if the density of medium is decreased indefinitely, leaving a vacuum, the speed of the body would increase indefinitely and motion would be instantaneous. For this absurd consequence Aristotle rejected the existence of vacuum or empty space. He argued vehemently against the existence of vacuum in any form. He believed therefore that space had necessarily to be a plenum filled everywhere in the sublunary region and in the space beyond filled with the unchangeable divine aether.
This mechanism of violent motion did not seem plausible to John Philoponus. He rejected not only the necessity of the resistant medium for violent motion, but also that the external medium, especially air, is the motive force or agent of violent motion. He proposed that that some sort of incorporeal motive force is imparted to the projectile; it is used up in overcoming the resistance of the air during the flight of the arrow. Philoponus held that a body would fall through a vacuum at a finite speed and would require a greater time to fall through a resisting medium.
The Islamic scholars continued the controversy. Avempace (the Latinized form of Ibn Bajja), a Spanish Islamic philosopher (d. 1138 A.D.), agreed with Philoponus as to the speed of falling bodies. He denied Aristotle's claim that the time of fall is directly proportional to the density, and therefore to the resistance of the external medium through which it fell. Avempace argued that the time required to move from one point to another does not depend upon the resistive capacity of the intervening medium. Therefore the speed of the body will not be instantaneous in the absence of the intervening medium. To prove this important point, Avempace used Aristotle's own statements as counterargument. Aristotle had observed that the planets and stars, like all terrestrial bodies, do not move instantaneously from one point to another. And Aristotle had also insisted that the heavenly bodies move effortlessly through the material celestial aether, which offers no resistance. It was also obvious that the differing finite speeds of the planets did not depend upon the celestial medium. Avempace concluded that not only is the resistant medium unessential for the occurrence of motion, but that its sole function is to retard it. However, Averroes or Ibn-Rushd (1126-1198 A.D.), an Islamic philosopher born in Cordoba, Spain, who was famous for his commentaries on Aristotle and by his followers was called "the Commentator", as a mark of deep respect, just as they called Aristotle "the Philosopher", defended Aristotle's view of falling bodies. Avicenna or Ibn-Sina (980-1037 A.D.), a famous Persian Islamic philosopher, who combined Aristotelian and Neoplatonic themes in his philosophy, thought that the projectile received something which he called "mail" from the mover; it was something that made the body move and in absence of the medium (air) the motion would be indefinite in extent and duration, since there would be no reason for it come to rest. For this reason, among other, he rejected the existence of void space or vacuum.
In the 14th century, when the Scholastics began to look at the
problem, they also found difficulties with Aristotle's explanation.
(1) If the air continuously displaced from in front of the projectile
returns behind the arrow to push it, why is it that the projectile
eventually stops moving violently and falls to the ground?
(2) If two javelins are thrown, one with a pointed rear end and
the other with a blunt rear end, why does not the latter go farther,
since it has more surface area to push on?
These and other difficulties produced an open skepticism in 14th
century toward the Aristotelian explanation as to the role of
air in projectile motion. Jean Buridan (1295-1360) at the University
of Paris taught that there was there was no motion in nature that
is properly described by Aristotle. The expertise of the Paris
school was, not kinematics, but dynamics. Buridan, who was the
chief exponent of "impetus" dynamics, thought that something,
which he called "impetus", was put into a body by a
force. The impetus increased directly with the weight of the
body and with its velocity. This concept is similar to Newton's
"quantity of motion" or the modern concept of "momentum".
Unlike these concepts, Buridan thought of impetus as a kind of
force which kept the body moving with its violent motion until
the impetus was dissipated in overcoming the resistance to the
motion. If there is no resistance, the motion would continue
forever. Applying this to celestial motion, Buridan thought that
the celestial bodies, the stars, planets, the sun and moon, were
moving in empty outer space without resistance and hence moving
forever with the original impetus that God gave them at the beginning.
The teacher of John Buridan, Nicolas Oresme (c.1320-1382 A.D.), who taught at Paris and died as bishop of Lisieux in A.D. 1382, was an outstanding medieval physicist. He made several important discoveries in kinematics. He proved geometrically, for example, the famous mean-speed theorem, enunciated at Merton College of Oxford in the first half of the fourteenth century, that the distance moved by a body moving with uniform acceleration during a stated time interval was equal to the distance which the body would have moved with the mean speed. The mean speed is the instantaneous speed of the body at the middle of the time interval, during a given time interval. Oresme also anticipated Galileo's distance theorem, wherein a body uniformly accelerating from rest will, in equal time intervals, traverse distances that are related to each other as a sequence of odd numbers beginning with unity. More importantly, he suggested a way to represent the motion of bodies by mean of graphs, by drawing a horizontal straight base line representing time and constructing vertical straight lines whose lengths representing position of the body. When the end points of the vertical lines are connected together, a curve is obtain that represents the motion of the body. This geometrical device was an important step for science. It was not graphing by rectangular coordinates developed later by Descartes, but it did make possible the picturing motion. In A.D. 1337 Nicolas had written a Commentary on Aristotle's De Caleo, entitled Du ciel et du monde, in which he discusses the problem of the earth's motion. He maintains that all direct observation cannot afford a proof that the heavens rotates daily while the earth remains at rest. For the appearances would be precisely the same if it were the earth and not the heavens which rotates. He says, "I conclude that one could not show by any experience that the heavens was moved with a daily motion and the earth was not moved in this way." After considering arguments for and against the motion of the earth, he ends by rejecting the hypothesis that the earth rotates, "notwithstanding the reasons to the contrary, for they are conclusions which are not evidently conclusive." In other words he was not willing to abandon the commonly accepted view for a hypothesis which had not been conclusively proven.
There was also difficulties with Aristotle's description of terrestrial motion. Aristotle and his commentators spoke loosely about speed and motion. Four mathematical logicians at Merton College in Oxford in 1328 began the study of kinematics. They were Thomas Bradwardine (c.1290-1349), later the Archbishop of Canterbury, William Heytesbury, who became the chancellor of Oxford in 1371, Richard Swineshead, and John Dumbleton. These four men sharpened the terminology of Mechanics. Being careful logicians these Merton College men knew that space and time were completely different concepts. Thus the definition of uniform velocity involved the comparison of the space traversed and the elapsed time. Uniform velocity must then be defined as the motion of body in which equal space is traversed in equal times. They defined instantaneous velocity as the velocity at a point that would be proportional to the space which would have been traversed in an given interval time if the motion at the point had continued on as uniform. Accelerated motion is motion in which the instantaneous velocity varies with time. And if the instantaneous velocity is proportional to time, the accelerated motion is uniform. They stated and proved several theorems about uniform accelerated motion.
These ideas spread throughout Europe, but did not replace the dominate views of Aristotle, especially in Roman Catholic countries where the massive synthesis of Christian theology and of Aristotelianism by Thomas Aquinas was the official view of the Roman Catholic Church. Even in Protestant countries where the Protestants rejected the theology of this Thomistic synthesis, the educational system and intellectual life was based on Aristotle's philosophy. Thus at the beginning of the 17th century, Aristotelianism dominated western thinking, both Protestant and Roman Catholic, with only a few dissident voices here and there. These dissidents anticipated Galileo's work; he was not the first to raise these difficulties.
The difficulties with Aristotle's physics can be traced to three
deficiencies:
(1) Aristotle believed that mathematics was of little value in
describing change and motion; mathematics to him was geometry
and this is inadequate in dealing with motion. The rise of modern
mechanics was accompanied with the development of algebraic techniques
in mathematics.
(2) Aristotle put emphasis upon qualitative observations as the
basis for all theorizing, almost to the complete exclusion of
quantitative observations; being a biologist, he would emphasize
qualitative observation, since it is characteristic of the method
of biology. This neglect of quantitative observation led also
to the neglect of measurement and experimentation, in favor of
classification.
(3) Aristotle believed that experiments distorted nature.
According to Aristotle the inductive method does not involve experimentation.
Aristotle like Plato, who mocked the Pythagoreans for the use of torturing
instruments in order to obtain knowledge, disparged experimentation.
The motion of a body involves, not only its speed, but also its direction. In the foregoing discussion Galileo was concerned only with vertical motion, but in the discussion on later days he will consider the direction as well as speed of the motion. When both speed and direction are to be considered the term "velocity" is used. When a velocity is given, the direction as well as the speed must be given. When the direction as well as the numerical magnitude of quantity must be given or specified, the quantity is called a vector; velocity is an example of a vector quantity. A quantity that has magnitude only is called a scalar. In Physics, time and speed are scalars. Speed is the magnitude of velocity. Another important vector quantity is displacement. When completely describing the motion of a body, not only must the distance that the body moves be given, but the direction of the motion must also be given. The quantity with both the distance and the direction of the motion specified, is called displacement. Displacement is a vector quantity and the magnitude of displacement is the distance the body travels; distance is the magnitude of displacement. Thus the velocity of a body may be defined as the time rate of displacement of that body.
When vector quantities are added, the directions of the quantities
must be used to determine the arithmetic operations performed
to obtain the magnitude and direction of the resultant. That
is, if two vector quantities like velocities are in the same direction
then numerical magnitudes are simply added and direction of the
resultant velocity is the common direction.
For example, if an airplane is flying north at 100 mph and has
a tail wind of 30 mph north, the resultant velocity (ground speed) is
100 + 30 = 130 mph north.
But if they are in opposite directions, the smaller
magnitude is subtracted from the larger magnitude and the direction
of the resultant is the direction of the larger magnitude. For
example, if the airplane in previous example encounters instead a head
wind of 30 mph south, the resultant velocity (ground speed) is
100 - 30 = 70 mph north.
If the direction of one vector quantity is at right angles
to the other, then the Pythagorean theorem is used to determine
the magnitude, and the tangent trigonometric function is used
to determine the direction.
For example if the airplane flying at 100 mph north has
a cross wind of 30 mph west,
the resultant speed is square root of (100 square + 30 square) = 104.4 mph
and its direction is
the angle whose tangent is 30/100 = 16.7 degrees west of north.
Just as velocity of a body can be defined as the time rate of
change of the displacement of the body, so the acceleration
of a body can be defined as the time rate of change of velocity
of the body. That is, acceleration is the change of velocity
divided by the time in which change takes place. The change
of velocity is defined as the difference between the final
instantaneous velocity at the end of the change and initial instantaneous
velocity at the beginning of the change; that is,
change of velocity = final velocity - initial velocity:
Δv = v2 - v1,
where Δv stands for the change of the velocity,
where the Δ means "the change of", and
v2 is the finial instantaneous velocity and
v1 is the initial instantaneous velocity.
Thus, acceleration = change of velocity / time of the change,
a = Δv / Δt, or,
a = (v2 - v1) /
(t2 - t1),
where Δt is the time of the change or the elapsed time
of the change between the initial time t1 and
the final time t2. That is,
acceleration = (final velocity - initial velocity) / time of the change.
This defines a constant acceleration if the velocity is changing
at a constant rate, that is, the velocity has changed the same
amount in each unit of time. Acceleration is measured in units
that are units of velocity divided by time; for example, if velocity
is measured in feet per second, then the unit of acceleration
would be in feet per second, per second, or ft/sec².
For a falling body its acceleration is the acceleration due to
gravity g, that is, a = g. Thus the distance
that a body freely falls can be calculated, if the time of fall
since it was dropped is known. This is called the Law of Falling
Bodies: if a body is dropped in vacuum, the distance it falls
is directly proportional to the square of the time that it has
fallen that distance, where the constant of proportionality is
one-half the acceleration due to the force of gravity, or ½g.
That is,
s = ½gt².
Galileo discovered this law and experimentally verified it on
the Third Day of his book The Two New Sciences. Galileo's
verification of Free Fall Law is the famous inclined plane experiment.
In opposition to Aristotle's hypothesis that the speed of the falling
body is constant, Galileo proposes two hypotheses in which the
speed of falling bodies increase with time. Galileo and some
of his precursors were sure that the falling bodies gains speed
as they fall. Galileo's first hypothesis was that a body gains
speed as it falls so that its average speed, over the time of
fall, is proportional to the distance it has fallen. Galileo
showed that this hypothesis is logical absurd; it implied that
the time interval is constant, which is a variable, increasing
as the body falls. Galileo's second hypothesis was that body
gains speed as it falls so that average speed over any time interval
is proportional to that time. This hypothesis implies that the
distance that it falls is directly proportional to the square
of the time interval of the fall. Thus there are two hypotheses
to be tested:
(1) Aristotle's hypothesis that the distance is proportional
the time interval, and
(2) Galileo's hypothesis that the distance is proportional to
the square of the time interval of fall.
Since bodies fall at such a high speed and for such a short time,
that it is difficult to test; Galileo had no instruments to measure
such speeds and times. So he proposed to use a body, like a rolling
ball, moving down a very long, smooth inclined plane, so that
he could easily measure the distances accurately. Thus the distances
traversed were in time intervals sufficiently long so that they
could be measured by his own pulse rates. When that was not accurate
enough, he used a water clock. When he performed the experiment,
he found that his hypothesis, not Aristotle's, was the correct
hypothesis. The body accelerated as it rolled down the inclined
plane, so that the distance it traveled is proportional to the
square of the time intervals. And as the angle of the inclined
plane was increased, the body would move faster, but it would
still accelerate according his hypothesis. As the angle approached
a right angle, so that the plane would be nearer to the vertical,
the ball would roll so fast that the distances and times could
not be accurately measured; Galileo reasoned that if they could
be measured, they would fit his hypothesis. Thus a freely falling
body would accelerate uniformly, that is, equal amounts of velocity
would be added in equal times, so that the distance it falls is
proportional to the square of the time interval. Thus Galileo
had proved that Aristotle was wrong and he had discovered the
Law of Falling Bodies.
Now consider the experiment in which the projectile is fired, not horizontal, but at an angle above the horizontal. An analysis of this experiment show that projectile's motion is also composed two components: one horizontal and the other vertical. The horizontal components of the projectile motion is uniform motion (x = vt); the vertical components of the projectile motion is accelerated motion, but decelerating upward until the velocity upward becomes zero, and having reached the top it stops and reverses its direction and the projectile begins accelerating downward. The path of followed by the projectile is called a trajectory and has the shape of a parabola. The horizontal distance from the point of projecting to point where the projectile crosses the horizontal is called the range of the projectile. The maximum range is obtained when the angle of projection is 45 degrees.
The method which Galileo used began with a problem left by Aristotelian physics. The problem arose in the motion of a projectile such as a shell shot from a cannon. It became more and more evident to Galileo and to his contemporaries that projectiles do not move the way that Aristotle's physics describe it. This was Galileo's problem. Something was wrong with the Aristotelian theory of the motion of a projectile.
Galileo began his scientific investigation of this problem by analyzing the roots of this problem. He accomplished this by stating clearly what the problem was and the traditional assumptions which generated it.
When this was done, it became clear that this problem centered not on the projectile but on the Aristotelian definition of force, a definition that applied not only to projectiles but to any motion whatever. It became clear to him that it was not necessary to pay attention any more just to projectiles and how they moved. For the difficulty concerning projectiles arose not from the motion of projectiles but from how Aristotelian physics viewed that motion. According to Aristotelian physics force is that which shows itself as the velocity of the object upon which it acts. In other words, force is that which produces velocity. From this it followed that when a force ceases to act upon a body, the body ceases to move.
In many cases this definition of force is apparently confirmed. When one pushes a table, the table moves, and when the force is removed, the table ceases to move. But in the motion of the projectile this is not so. The force ceases to act the instant that the shell leaves the cannon, but the shell continues to move over great distances of space and over a considerable interval of time. Thus the analysis of the assumptions of the problem, which disclosed themselves in the motion of the projectile, located the difficulty in the basic concept of force as it relates itself to any kind of motion.
As a result of this analysis, Galileo's problem took on a more fundamental and general form. The difficulty centered not solely on the motion of the projectile but on the Aristotelian concept of force and motion in general. Clearly, a new concept of force and motion was required. This alone would be sufficient to solve the problem. It would have consequences far beyond the motion of projectiles and would require the alteration of the basic concepts of physics. It would lead to the complete discarding of Aristotelian physics.
Galileo's analysis of this problem thus transformed his investigation of motion into the finding of a new and correct concept of force to understand the motion of any kind of object. This allowed him to choose the simplest case of force acting on a moving object that he could find, that is, a body falling freely under the force of gravity. This is much simpler case of a moving object than that of a projectile in which the freely falling vertical motion is compounded with a horizontal motion.
Having now restricted his problem to the motion of a ball which he can let drop from his hand to the floor, he proceeds to observe the factors involved in this directly observable phenomenon. He notes that are three factors involved in the motion of a ball falling from his hand to the floor under the force of gravity: (1) the weight of the ball, (2) the distance through which the ball falls, and (3) the time during which it falls.
These three observed factors suggested to Galileo three hypotheses.
(1) The force is simply proportional to the weight of the body
upon which the force acts.
(2) The force is simply proportional to the distance through which
the body moves as the force acts. And
(3) the force is simply proportional to the time during which
the force acts.
Galileo's analysis having thus led him to these relevant hypotheses, his next task is to determine which one, or whether any one them, is correct. This he does by deducing from each hypothesis what follows if it were true and then attempts to put this deduced consequence to an experimental test.
If the force is proportional to the weight of the body upon which it acts, it follows that bodies of different weights dropped at the same instant and acted upon the same gravitational force, should arrive at the ground at different times, the heavier bodies arriving first. The famous Tower of Pisa experiment, which apparently was apocryphal as an historical fact, could have led to the rejection of this hypothesis, had it been performed with objects that would not be affected by air resistant. This would leave Galileo with the other two hypotheses.
Galileo believed that he demonstrated mathematically that the hypothesis that the force is proportional to the distance through which the body falls leads to a contradiction. Ernst Mach has shown that Galileo's proof was invalid. Nonetheless, the hypothesis can be shown to be false. Thus Galileo did not err in rejecting it. This left him with the hypothesis that the force is simply proportional to the time during its fall.
Now his problem has become that of putting this hypothesis to an empirical test. He began first by deducing from it the consequence that the distance moved must be simply proportional to the square of time. This meant that if a body moves one unit of distance in one unit of time, then it must move four units of distance in two units of time, nine units of distance in three units of time, and so on. To put this consequence of his third hypothesis to an experimental test Galileo proposed his famous experiment in which a ball is allowed to roll down an inclined plane. The purpose of the inclined plane was to slow down the fall of the ball, so that it would be possible to measure the distance moved in different units of time and thereby determine whether the relation between distance and time is as prescribed by the hypothesis. The confirmation of this hypothesis is well known.
The result was a new concept of force. Force is that which produces, not motion or velocity as Aristotle supposed, but a change of velocity or acceleration. This new concept of force is the foundation of modern mechanics and physics. According to this conception of force, when a force ceases to act upon a body it will not cease to move; it will merely cease to change its velocity.
Once this new conception of force was found, the difficulty with projectile motion was removed. During the brief interval of time during which the powder in the cannon is exploding, it follows from this concept of force that the velocity of the projectile will be continuously changing. In other words, the projectile will undergo a continuous acceleration from zero velocity when the explosion begins, to the finite velocity which it reaches when the explosion ends. As the projectile leaves the cannon, when the force ceases to act, the body will cease further to change its velocity. In other words, it will move with a constant velocity it attained when the force ceased to act. Thus the fact that the projectile goes on moving when the force has ceased to act is accounted for. Thus was Galileo's initial problem was solved.
But this new conception of force not only explained the initial motion of the projectile, but also the subsequent motion of the projectile. In addition to the force of the explosion in the cannon, the force of gravity acts on the projectile in a vertical direction after it leaves the cannon. This force of gravity, according to Galileo's new conception of force, will accelerate the projectile uniformly towards the earth. Thus the motion of the projectile will be composed of two components, a horizontal component of uniform velocity and a vertical component of uniform acceleration toward the earth. Mathematically it can be shown that the path of the projectile with these two components of motion is a parabola.
Besides explaining projectile motion this new conception of force and motion had implications beyond that for physics. The earliest part of Galileo's analysis anticipated this result. For the analysis showed that the difficulty with respect to the motion of projectiles centered on the Aristotelian conception of force applied to any form of motion. To solve this problem Galileo had to formulate a new conception of force and motion. This implied a rejection of the whole of Aristotelian physics. Since there is no major concept in Aristotle's metaphysics that does not appear in his physics, this change has the additional consequence of the rejection of Aristotelian philosophy and of the Thomistic theology built on it. The modern world, once forced by Galileo's analysis and experiment to replace Aristotelian physics with the physics of Galileo, was required to replace the attendant Aristotelian philosophy with a new philosophy built on a new foundation. This was attempted first by Descartes in France and later by Locke in England.
In addition, when Newton began to look at celestial as well as terrestrial motion from the standpoint of Galileo's new concept of force and motion, the modern science of mechanics, as formulated in Newton's Principia, was founded, and Kepler's previously verified three laws of planetary motion came out as the logical consequence. Thereby the previously separated realms of the celestial and terrestrial were shown to be one rather than two realms.
Such are the consequences of Galileo's investigation of projectile
motion. The steps of his investigation are the steps in the new
scientific method. The following is a summary of the steps of this
scientific method:
(1) The determination and analysis of a problem which initiates
the procedure and whose solution is its goal.
(2) The formulation of an hypothesis which is a proposed solution
of the problem.
(3) The verification of the hypothesis which confirms it and solves
the original problem. This step may involve experimentation or
observation. If the hypothesis fails to be confirmed, then
return to step (1) and repeat the steps until an hypothesis is found
that is confirmed by the verification step (3).