Galileo Galilei on Motion and Inertia
Galileo's Two New Sciences, Third, Fourth and Additional Days
Together, the Third, Fourth and Additional Days of Galileo’s Two New Sciences can be understood as the culmination of more than 40 years of work by Galileo on motion, work begun even before he arrived in Padua in 1592. During those years, his understanding changed many times, ultimately arriving to a very succinct set of concepts, presented in this post, that laid the basis for modern mechanics.
This is the final post of the series examining Galileo’s Two New Sciences, his last work. Previous posts in this series are:
The Discorsi e dimostrazioni matematiche intorno a due nuove scienze, as originally published in 1638 by Lodewijk Elzevir in Leiden, included dialogues for each of four days. However, the version found in Galileo’s collected works1 includes an additional day, prepared from Galileo’s own notes, and which is included as part of Stillman Drake’s translation2 and Alessandro De Angelis’s modern adaptation.3 This additional day has as participants, as for the other days, both Sagredo and Salviati, but not Simplicio. Instead, appears Paolo Aproino, who had worked with Galileo in Padua during the years 1608-1610.
Galileo was very much aware of the importance of his work on motion. He clearly understood that he was making a clear initial step forward, from which future researchers would be able to make significant discoveries. Here is the introduction to the Third Day, entitled “On Local Motion”:
We bring forward [promovemus] a brand new science concerning a very old subject.
There is perhaps nothing in nature older than MOTION, about which volumes neither few nor small have been written by philosophers; yet I find many essentials [symptomata] of it that are worth knowing which have not even been remarked, let alone demonstrated. [p.147, emphasis mine]
He then summarizes the main result of the Third Day, namely that objects falling to earth are continually accelerated, and that this acceleration is constant:
Certain commonplaces have been noted, as for example that in natural motion, heavy falling things continually accelerate; but the proportion according to which this acceleration takes place has not yet been set forth. Indeed no one, so far as I know, has demonstrated that the spaces run through in equal times by a moveable descending from rest maintain among themselves the same rule [rationem] as do the odd numbers following upon unity. [p.147]
This result, of the distances covered through a sequence of successive instants corresponding to the sequence of odd numbers, was mentioned in the Dialogue, as I wrote in an earlier post Galileo on the Acceleration of Falling Bodies. However, since it was the Dialogue that got him in trouble with the Holy Inquisition, Galileo made it appear as if this was the first time it appeared in print.
Galileo then summarizes the main result of the Fourth Day, entitled “On The Motion of Projectiles”, namely that projectiles that are launched from the ground follow a trajectory in the shape of a parabola.
It has been observed that missiles or projectiles trace out a line somehow curved, but no one has brought out that this is a parabola. That it is, and other things neither few nor less worthy [than this] of being known, will be demonstrated by me, and (what is in my opinion more worthwhile) there will be opened a gateway and a road to a large and excellent science of which these labors of ours shall be the elements, [a science] into which minds more piercing than mine shall penetrate to recesses still deeper. [p.147, emphasis mine]
The dialogue of the Third Day can be summarized with two concepts:
“On Equable Motion”, i.e., on the motion of bodies with fixed velocities. The definition is as follows:
Equal or uniform motion I understand to be that of which the parts run through by the moveable in any equal times whatever are equal to one another. [p.148]
“On Naturally Accelerated Motion”, i.e., on the motion of bodies accelerating as they rise from and fall back to the earth, always with a fixed acceleration. The definition is as follows:
I say that that motion is equably or uniformly accelerated which, abandoning rest, adds on to itself equal momenta of swiftness in equal times. [p.154]
So the first definition focuses on motion with a constant velocity and the second definition on motion subject to a constant acceleration.
For uniformly accelerated motion, Galileo demonstrates that the distance travelled depends on the square of the time elapsed:
If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times; that is, are as the squares of those times. [p.166]
From the discussion in the Third Day, it becomes clear that Galileo has some understanding of inertia: motion in the horizontal does not change, should there be no outside forces.
It may also be noted that whatever degree of speed is found in the moveable, this is by its nature [suapte natura] indelibly impressed on it when external causes of acceleration or retardation are removed, which occurs only on the horizontal plane; for on declining planes there is cause of more [maioris] acceleration, and on rising planes, of retardation. From this it likewise follows that motion in the horizontal is also eternal, since if it is indeed equable it is not [even] weakened or remitted, much less removed. [p.197]
In the Fourth Day, entitled “On these two different kinds of motion are combined to demonstrate that a falling projectile follows the trajectory of a semiparabola.
The line described by a heavy moveable, when it descends with a motion compounded from equable horizontal and natural falling [motion] is a semiparabola. [p.221]
It should be understood that the following assumptions are being made.
The semiparabola trajectory is only an approximation of reality, as it does not take into account air resistance.
Once a projectile is in the air, its movement is the simple combination of a fixed-velocity horizontal motion, and of a fixed-acceleration vertical motion downwards.
There is nothing pushing along the projectile, i.e., the Aristotelian assumption of the air pushing the projectile is just plain false.
The counterpart, of a semiparabola trajectory for the launching of a projectile from the ground, is the subject of a number of calculations to determine the altitude of the highest point in the trajectory, as well as the amplitude of the trajectory. One of the key conclusions is that a launch angle of 45º will produce the trajectory with greatest amplitude.
The Additional Day, entitled “On the Force of Percussion”, is a first step in the study of collisions, which would become a major subject of research of the rest of the seventeenth century.
In the dialogue of that day, Galileo presents a new machine of his invention, consisting of a large pulley, with a chord hanging over the pulley, with two identical heavy weights attached to the ends of the chord. A number of experiments could be undertaken with this machine, and it is likely that Galileo envisaged more experiments than for which we have records.
One of the simplest experiments consists of pushing down one of the weights, forcing the other weight to move upwards.
And in this same way the two equal weights, hanging from the ends of the rope, will be at rest when placed in balance, and if impetus downward shall be given to one, it will always conserve this equably. Here it is to be noted that all these things would follow if there were removed all external and accidental impediments, as of roughness and heaviness of rope or pulleys, of friction in the turning of these about the axle, and whatever others there may be of these. […]
Now it is evident that this degree of speed will not go on increasing when its cause of increase is taken away, this being the weight of the descending body itself; for its weight no longer acts when its propensity to descend is taken away by the repugnance to rising of its companion of equal weight. Hence the maximum degree of speed will be conserved, and the motion will be converted from one of acceleration to uniform motion. [p.297]
This might be a simple experiment, but its implications were huge, for here was the first proof of the principle of inertia being applicable not just in the horizontal plane, but also the vertical plane, despite the uniform acceleration to which falling bodies are subject to. Here is what Roberto Vergara Caffarelli,4 who redid many of Galileo’s experiments with reconstructed machines, wrote:
In this experiment the principle of inertia is verified in a more general form than with the horizontal plane: on each of the two masses hung from the cable the resultant of the forces is zero, so each of them remains still or, if put into motion, moves with constant velocity. [Vergara Caffarelli, p.233]
Alessandro De Angelis, commenting on the same passage, supported Newton’s claim that the principle of inertia should be attributed to Galileo:
The formulation of the principle of inertia in this additional day removes all doubts, supporting the interpretation of the most authoritative of critics: Isaac Newton, who explicitly in the Principia attributed the principle of inertia to Galilei. [De Angelis, p.150, n.a]
Galileo Galilei’s Discorsi e dimostrazioni matematiche intorno a due nuove scienze, or Two New Sciences, as it is commonly referred to in English, is without a doubt one of the most important texts in the history of science. The study of motion therein made giant steps, at both the experimental and theoretical levels. As he correctly predicted, “there will be opened a gateway and a road to a large and excellent science of which these labors of ours shall be the elements, [a science] into which minds more piercing than mine shall penetrate to recesses still deeper.”
Galileo Galilei. Le Opere, VIII:12-346. Firenze: Tipografia di G. Barbara, 1898.
Galileo Galilei. Two New Sciences, Including Centers of Gravity & Force of Percussion. Translated by Stillman Drake. University of Wisconsin Press, 1974.
Alessandro De Angelis. Galileo Galilei’s “Two New Sciences” for Modern Readers. Springer, 2021.
Roberto Vergara Caffarelli. Galileo Galilei and Motion: A Reconstruction of 50 Years of Experiments and Discoveries. Bologna: Società Italiana di Fisica, and Berlin, Heidelberg, New York: Springer, 2009.
Thanks for your blog! When Newton said he stood on the shoulders of giants, Galileo must have been one of the giants he meant.
I was initially confused by the word "amplitude" which is defined as "the extent of a vibratory movement (as of a pendulum) measured from the mean position to an extreme". That made me think the way to get the largest amplitude is to fire a projectile straight up. As used here the word means the distance travelled by the projectile, or the width of the trajectory, but not the height or altitude of the trajectory. Does Galileo himself use the word "amplitude" ?