In order to determine the nature of the scientific method let us examine the analysis that Galileo Galilei (1564-1642) made of a problem he found in Aristotelian physics. Galilei's analysis not only solved this problem but also shows how modern science using the scientific method solves such problems.
The method which Galilei 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 Galilei and his contemporaries that projectiles do not move the way that Aristotle's physics describe it. This was Galilei's problem. Something was wrong with the Aristotelian theory of the motion of a projectile.
Galilei 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 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 a 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 itself 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 Galilei'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.
Galilei's analysis of this problem thus transformed his investegation 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 observeable phenomenon. He notes that there are three factors involved in the motion of a ball falling from his hand to the floor under the force of gravity:
These three observed factors suggested to Galilei three hypothees.
Galilei'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 empirical 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 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 Galilei with the other two hypotheses.
Galilei believed that he had 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 Galilei's proof was invalid. Nonetheless, the hypothesis can be shown to be false. Thus Galilei 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 Galilei 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 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 increase its velocity. In other words, it will move with a constant velocity that 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 Galilei'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 Galilei'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 Galilei'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 Galilei 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 Galilei's analysis and experiment to replace Aristotelian physics with the physics of Galilei, was required to replace the attendant Aristotelean 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 terrestial motion from the standpoint of Galilei'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 the terrestial were shown to be one rather than two realms.
Such are the consequences of Galilei's investigation of projectile motion.
The steps of his investigation are the steps in the new scientific method.
The following are the three main steps of this scientific method: