In the last episode of this series, I started to examine Galileo’s last book his Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) published in 1638.
I noted there that although first published at the end of his active life as a scientist, the work it contained was all carried out much earlier, most of it during his years as a professor for mathematics in Padua.
I also noted that it is obviously two volumes published together, as the first two days are written in Italian, whereas the third and fourth days are in Latin. In the previous episode I dealt with Days I & II, which deal mostly with the strength of materials, a branch of statics. Today we turn to Days III &IV, which deal with motion. Day III with naturally accelerated motion and Day IV with projectile motion.
Day III contains Galileo’s famous experiments with balls and inclined planes with which he proved the laws of fall. He is often falsely credited with having singlehandedly, so to speak, refuted and replaced the Aristotelian laws of fall, However, as we have seen in this series, in the sixth century, John Philoponus (c. 490–c. 570) was the first to question and reject Aristotle in this area. We also saw that the Oxford Calculatores in the fourteenth century made great strides toward the correct laws of fall deriving the mean speed theorem that is at the core of those laws. Also, in the fourteenth century the so-called Paris Physicists provided a geometric proof of the mean speed theorem. Moving forward in the sixteenth century Giambattista Benedetti (1530–1590), a pupil of Tartaglia (1499–1557), formulated much of the theory of falling bodies attributed to Galileo fifty years before Galileo. The question is then, how much did Galileo actually contribute and how?
The answer is that Galileo contributed something very important. Although, both Philoponus and Simon Stevin (c. 1548–1620) both dropped balls off high towers to empirically prove Aristotle wrong, the work of all of Galileo’s predecessors on the topic was purely theoretical. Galileo is a series of carefully planed and precisely executed experiments provided solid empirical evidence to back up those theoretical conclusions. However, contrary to popular belief this does not make him the first experimental physicist.
Galileo opens Day III with a definition of what he terms steady or uniform motion:
By steady or uniform motion, I mean one in which the distances travelled by a moving particle during any equal interval of time, are themselves equal.[1]
This is followed by four axioms which basically explain what is meant by greater and lesser distances and greater and lesser speeds. Which in turn are followed by four theorems and four propositions dealing with cases of speeds and distances.
Having dealt fairly comprehensively with steady or uniform motion, Galileo now moves on to naturally accelerated motion, i.e. the case of free fall. Here we have a protracted discussion in which various definitions naturally accelerated motion are proposed by one discussion partner and then shown to be inadequate by another, till they arrive at the following consensus:
But now continuing the thread of our talk, it would seem that up to the present we have established the definition of uniformly accelerated motion which is expressed as follows:
A motion is said to be equally or uniformly accelerated when starting from rest its momentum (celeritatis momenta) receives equal increments in equal times.
Salv. This definition established, the Author makes a single assumption, namely,
The speeds acquired by one and the same body moving down planes of different inclination are equal when the heights of those planes are equal.
By height of an inclined plane we mean the perpendicular let fall from the upper end of the plane upon a horizontal line drawn through the lower end of the same plane.[2]
Here we have Galileo the mathematical physicist at his best introducing for the first time his inclined plane concept. Galileo turned to inclined planes for his experiments because he realised the difficulty of trying to measure times of objects in free fall. Following another protracted discussion of this definition, which delivers some more substantiation for it, Galileo now delivers Theorem I, Proposition I, the mean speed theorem:
The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed and the speed just before acceleration began.[3]
Galileo illustrates and proves this theorem with a graph.
This leads on to Theorem II, Proposition II
The spaces described by a body falling from rest with uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances.[4]
Again, illustrated with a diagram and followed by a derivation.
There Follows Corollary I
Hence it is clear that if we take and equal intervals of time whatever, counting from the beginning of the motion, such as AD, DE, EF, FG, in which the spaces HL, LM, MN, NI are traversed, these spaces will bear to one another the same ratio as the series of odd numbers, 1, 3, 5, 7; for this is the ration of the differences of the squares of the lines [which represent time], differences which exceed one another by equal amounts this excess being equal to the smallest line [viz. the one representing a single time-interval]: or we may say [that this ratio] of the differences of the squares of the natural numbers beginning with unity.
While therefore, during equal intervals of time the velocities increase as the natural numbers, the increments of distances traversed during these equal time-intervals are to one another as the odd numbers beginning with unity.[5]
Another diagram another derivation.
After another discussion we finally arrive at the detailed description of Galileo famous inclined plane experiments:[6]
I have attempted in the following manner to assure myself that the acceleration actually experienced by falling bodies is that described above.
A piece of wood moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this grove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time require to make the descent. We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that of the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces travelled were to each other as the squares of the time, and this was true for all inclinations of the plane, i.e., of the channel, along which we rolled the ball. We also observed that the times of descent, for various inclinations of the plane, bore to one another precisely that ratio which, as we shall see later, the Author had predicted and demonstrated for them.
For the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and of the times, and this with such accuracy, that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.
The French historian of early modern science Alexander Koyré (1892–1964) was convinced that with this experimental set up, in particular his method of measuring time with a water clock, Galileo would not be able to achieve the accurate results he claimed to have accomplished. In this he quoted Marin Mersenne (1588–1648), who had questioned the feasibility of reproducing Galileo’s results. Koyré thought that they were merely thought experiments intended to illustrate his deductions. In 1961, Thomas B. Settle, as a graduate student atCornell University, succeeded in reproducing Galileo’s experiments with inclined planes using the methods and technologies described in Galileo’s writing. Since them several others have also repeated the experiments and confirmed Settle’s finding. (You can find a description of one such reconstruction here)
After sixty pages, in the original, of analysis of sixteen problems of uniformly accelerated motion with solutions, we arrive at a scholium in which Galileo gives the false solution to the brachistochrone problem:
In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) ‘shortest time’), or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. (Wikipedia)
Galileo:
From the preceding it is possible to infer that the path of quickest descent [lationem omnium velocissimam] from one point to another is not the shortest path, namely, a straight line, bur the arc of a circle.[7]
Shown once again by a diagram and derivation:
Day VI Theorem I, Proposition I open with the parabola law:
A projectile which is carried by a uniform horizonal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola.[8]
As we saw in an earlier episode in this series this law was discovered by Galileo and Guidobaldo dal Monte (1545–1607) one afternoon in 1592 whilst drinking wine and discussing projectile motion in the latter’s garden so, at least half the credit for this law should go to dal Monte.
Following the statement of the parabola law is a quite extensive discussion of the Conics of Apollonius for the benefit of Galileo’s readers, who at this time in history might not have been fully aware of this important piece of mathematics. Following this we get Theorem II, Proposition II:
When the motion of a body is the resultant of two uniform motions, one horizontal, the other perpendicular, the square of the resultant momentum is equal to the sum of the squares of the two component momenta.[9]
As on Day III we get a lot of related theorems as well as problems and their solutions. Towards the end of Day VI, Galileo makes another error as with the brachistochrone problem in that he fails to recognise the catenary, although he senses that it is not actually a parabola.
In physics and geometr, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superficially similar in appearance to a parabola, which it is not. (Wikipedia)
Galileo write:
Besides I must tell you something which will both surprise and please you, namely that a cord stretched more or less tightly assumes a curve which closely approximates the parabola.[10]
Although the Discorsi is generally recognised as Galileo’s most important work and his major contribution to the history of physics nobody has up till now tried to determine its impact on developments in the seventeenth century. We do know that Marin Mersenne expended a lot of effort trying to persuade people to read the Discorsi, which would suggest that it was at least to a certain extent ignored. However, it must have had some impact, as in his Principia Newton, of course, gives Galileo credit every time that he uses either the laws of fall or the parabola law of projectile motion.
[1] Galileo Galilei, Dialogues Concerning Two New Sciences, Translated by Henry Crew and Alfonso de Savio, Dover, 1954 p. 154
[2] Dialogues Concerning Two New Sciences p. 169
[3] Dialogues Concerning Two New Sciences p. 173
[4] Dialogues Concerning Two New Sciences p. 174
[5] Dialogues Concerning Two New Sciences pp. 175–176.
[6] Dialogues Concerning Two New Sciences pp. 178–179
[7] Dialogues Concerning Two New Sciences p. 239
[8] Dialogues Concerning Two New Sciences p. 245
[9] Dialogues Concerning Two New Sciences p. 257
[10] Dialogues Concerning Two New Sciences p. 290