Galileo’s Two New Sciences is a scientific treatise published in 1638. The Third Day section through the Scholium of Theorem II is concerned with the constant speed of physical bodies and the uniform acceleration of falling bodies.
The section that focuses on the constant speed of physical bodies is written in a style similar to that employed by Euclid in the Elements. Galileo presents a definition of Uniform Motion: “one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal.” Then he derives axioms from this definition: “in the case of one and the same uniform motion, the distance traversed during a longer interval of time is greater than the distance traversed during a shorter interval of time.” Finally he forms theorems using the axioms: “If a moving particle, carried uniformly at a constant speed, traverses two distances the time-intervals required are to each other in the ratio of these distances.” The conclusions derived in this section are intuitive. For example, if body A is traveling at the same speed as body B, then body A will travel a greater distance than body B if body A travels at the same speed for a longer interval of time.
The second section, which treats of the uniform acceleration of falling bodies, is much more interesting and instructive. Galileo defines a motion to be uniformly accelerated “when starting from rest, it acquires, during equal time-intervals, equal increments of speed.” Then, instead of proceeding to provide axioms and forming theorems, Galileo writes a dramatic scene in which three characters argue over the merits of the definition, and whether the definition accurately describes the acceleration of falling bodies. The dramatic style is similar to the style of the Socratic dialogues. This type of writing increases the enjoyment and interest experienced by the reader, who might otherwise become bored and indifferent to the subject matter being discussed. Science is not the most stimulating topic for many people.
One of Galileo’s interlocutors states that it is intuitive that falling objects gain speed as they fall if one considers a simple experiment. A brick dropped from a height of 10 meters above a stick will drive the stick into the mud a certain length. A brick dropped from the height of 5 meters above the same stick will drive it into the mud a shorter length. A brick dropped from the height of a centimeter above the stick will not drive the stick into the mud at all. From this simple observation, it is logical to conclude that the brick attains a greater velocity, and consequently a greater force, when it falls for a longer interval of time.
So, having concluded that falling bodies accelerate, Galileo then attempts to prove that the acceleration is uniform. To determine this, Galileo performs a different experiment. He rolls a steel ball down an incline plane and measures the amount of time that passes during the ball’s descent from the top to the bottom, then from the top to ¾ of the incline, then ½, then ¼. After performing this experiment hundreds of times, he concludes that the distance traveled by the ball is proportional to the square of time passed during the descent. Using concrete numbers, if the ball travels one foot in one second, then it will travel four feet in two seconds, 9 feet in 3 seconds, etc.
Galileo is clearly one of the earliest proponents of the scientific method. He made observations, then hypothesized about certain aspects of nature, and finally performed experiments to determine whether his conjectures were correct. The next two authors on the ten year reading plan are Francis Bacon – who also helped to promote and develop the scientific method that is used by modern science – and Rene Descartes – who relied on intellectual principles to make deductions about the world rather than the empirical observations of Galileo and Bacon. I am eager to read the disparate opinions of these great thinkers.
“We have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties which have been, one after another, demonstrated by us.”