The Essential Galileo (64 page)

I proceed to the consideration of motions through the air, since it is with these that we are now especially concerned. The resistance of the air exhibits itself in two ways: first by offering greater impedance to less dense than to very dense bodies, and second by offering greater resistance to a body in rapid motion than to the same body in slow motion.

Regarding the first of these, consider the case of two balls having the same dimensions, but one weighing ten or twelve times as much as the other; one, say, of lead, the other of oak, both allowed to fall from an elevation of 150 or 200 cubits. Experiment shows that they will reach the ground with a slight difference in speed, showing us that in both cases the retardation caused by the air is small. For if both balls start at the same moment and at the same elevation, and if the leaden one be slightly retarded and the wooden one greatly retarded, then [277] the former ought to reach the earth a considerable distance in advance of the latter, since it is ten times as heavy; but this does not happen; instead, the gain in distance of one over the other does not amount to the hundredth part of the entire fall. And in the case of a ball of stone weighing only a third or half as much as one of lead, the difference in their times of reaching the ground will be scarcely noticeable. Now, the impetus acquired by a leaden ball in falling from a height of 200 cubits (which is such that if its motion became uniform the ball would traverse 400 cubits in a time interval equal to that of the fall) is very considerable in comparison with the speeds which we are able to give to our projectiles by the use of bows or other machines (except firearms); so it follows that we may, without noticeable error, regard as absolutely true those propositions which we are about to prove without considering the resistance of the medium.

Passing now to the second case, where we have to show that the resistance of the air for a rapidly moving body is not very much greater than for one moving slowly, ample proof is given by the following experiment. Attach to two threads of equal length—say four or five cubits—two equal leaden balls and suspend them from the ceiling; now pull them aside from the perpendicular, one through 80 or more degrees, the other through not more than 4 or 5 degrees; so that when set free, the first falls, passes through the perpendicular, and describes large but slowly decreasing arcs of 160, 150, 140 degrees, etc., and the other swings through small but also diminishing arcs of ten, eight, six degrees, etc. Here it must be remarked first of all that the first passes through its arcs of 180, 160 degrees, etc., in the same time that the other swings through its ten, eight degrees, etc.; from this it follows that the speed of the first ball is sixteen and eighteen times greater than that of the second; accordingly, if the air offers more resistance to the high speed than to the low, the frequency of oscillation in the large arcs of 180 or 160 degrees, etc., ought to be less than in the very small arcs of ten, eight, four, two degrees, or even one. But this prediction conflicts with experiment. For if two persons start to count the oscillations, one the large and the other the small, they will discover that after counting tens and even hundreds they will not differ by a single oscillation, not even by a fraction of one. This observation justifies the two following propositions, [278] namely, that oscillations of very large and very small amplitude all take the same time, and that the resistance of the air does not affect motions of high speed more than those of low speed, contrary to the opinion which we ourselves entertained earlier.

S
AGR.
    On the other hand, we cannot deny that the air hinders both of these motions since both become slower and finally vanish; so we have to admit that the retardation occurs in the same proportion in each case. But why? Because insofar as the greater resistance offered to one body than to the other originates from the greater impetus and speed of one body as compared to the other, then the speed with which a body moves is at once a cause and a measure of the resistance it meets; therefore, all motions, fast or slow, are hindered and diminished in the same proportion. And this is a result, it seems to me, of no small importance.

S
ALV.
    Thus, in this second case too, we can say that the errors in the conclusions that will be demonstrated by neglecting external accidents are of little concern in our operations; these involve great speeds for the most part and distances that are negligible in comparison with the radius of the earth or one of its great circles.

S
IMP.
    I would like to hear your reason for separating the projectiles from firearms, i.e., those from the force of gunpowder, and the other projectiles from bows, slings, and crossbows, insofar as they are not equally subject to change and resistance from the air.

S
ALV.
    I am led to this view by the excessive and, so to speak, supernatural violence with which the former projectiles are launched; indeed, it appears to me that without exaggeration one might say that the speed of a ball fired either from a musket or from a piece of artillery is supernatural. For if such a ball be allowed to fall from some great height, its speed will not go on increasing indefinitely, owing to the resistance of the air; what happens to bodies of small density in falling through short distances—I mean the reduction of their motion to uniformity—will also happen to a ball of iron or lead after [279] it has fallen a few thousand cubits; this terminal or final speed is the maximum which such a heavy body can naturally acquire in falling through the air. This speed I estimate to be much smaller than that impressed upon the ball by the burning gunpowder.

An appropriate experiment will serve to demonstrate this fact. From a height of one hundred or more cubits fire a rifle loaded with a lead bullet, vertically downwards upon a stone pavement; then with the same rifle shoot against a similar stone from a distance of one or two cubits; and observe which of the two balls is the more flattened. Now, if the ball that has come from the great height is found to be the less flattened of the two, this will show that the air has hindered and diminished the speed initially imparted to the bullet by the powder, and that the air will not permit a bullet to acquire too great a speed, no matter from what height it falls; but if the speed impressed upon the ball by the fire does not exceed that acquired by it in falling freely, then its downward blow ought to be greater rather than less. I have not performed this experiment, but I am of the opinion that a musket ball or cannon shot, falling from a height as great as you please, will not deliver so strong a blow as it would if fired into a wall only a few cubits away, i.e., at such a short range that the splitting or cutting of the air will not be sufficient to rob the shot of that excess of supernatural violence given it by the powder.

The enormous impetus of these violent shots may cause some deformation of the trajectory, making the beginning of the parabola flatter and less curved than the end. But, as far as our Author is concerned, this is a matter of small consequence in practical operations. The main one of these is the preparation of a table of ranges for shots of high elevation, giving the distance attained by the ball as a function of the angle of elevation. And since shots of this kind are fired from mortars using small charges and imparting no supernatural impetus, they follow their prescribed paths very exactly.

But now let us proceed with the reading of the treatise, at the point where the Author invites us to the study and investigation of the impetus of a body that moves with a motion compounded of two others. Next is the case in which the two components are uniform, one horizontal and the other vertical.

1.
For the historical background, see the Introduction, especially §0.2 and the end of §0.9.

2.
Galilei 1890–1909, 8: 49–54; translated by Crew and De Salvio (1914, 1–6); revised by Finocchiaro for this volume.

3.
Here I follow Drake (1974, 15) in translating
ingrossamento della materia
as increase of the
size
of material, rather than increase of the
amount
of material, as Crew and De Salvio (1914, 6) have it.

4.
Galilei 1890–1909, 8: 105–13; translated by Crew and De Salvio (1914, 61–68); revised by Finocchiaro for this volume.

5.
Aristotle,
Physics
, IV, 6–9, 213a11–216b21.

6.
Galilei 1890–1909, 8: 127–41; translated by Crew and De Salvio (1914, 83–98); revised by Finocchiaro for this volume.

7.
Here I am omitting the passage in Galilei 1890–1909, 8: 134.33–139.7;

Crew and De Salvio 1914, 91–95.

8.
Galilei 1890–1909, 8: 151–71; translated by Crew and De Salvio (1914, 109–33); revised by Finocchiaro for this volume.

9.
Pseudo-Aristotle,
Questions of Mechanics
, no. 3.

10.
Archimedes,
On the Equilibrium of Planes
, book 1, propositions 6–7.

11.
Here and in the rest of this chapter,
equidistance of ratios
translates Galileo's phrase
egual proporzione
. With this rendition, I am adopting Drake's (1974, xxxii, 111) translation, thus revising Crew and De Salvio's
equating ratios
, as well as the traditional
ratio ex aequali
. This notion comes from Euclid,
Elements
, book 5, definition 17 and proposition 22. For the meaning, see the Glossary.

12.
Ellipsis in the original, to indicate Salviati's interruption of Sagredo's speech.

13.
Here and in the rest of this chapter,
perturbed equidistance of ratios
translates Galileo's phrase
proporzione perturbata
. With this rendition, I am adopting

Drake's (1974, xxxiii, 114) translation, thus revising Crew and De Salvio's translation (which mostly uses the Latin phrase
ex aequali in proportione perturbata
), as well as the traditional
equality in perturbed proportion
. This notion again comes from Euclid,
Elements
, definition 18 and proposition 23. For the meaning, see the Glossary.

14.
Pseudo-Aristotle,
Questions of Mechanics
, no. 27.

15.
Giovanni di Guevara (1561–1641), Bishop of Teano, author of a commentary on the pseudo-Aristotelian
Questions of Mechanics
entitled
In Aristotelis mechanicas commentarii
(Rome, 1627).

16.
Ludovico Ariosto (1474–1533),
Orlando Furioso
, XVII, 30.

17.
Galilei 1890–1909, 8: 190; translated by Crew and De Salvio (1914, 153–54); revised by Finocchiaro for this volume.

18.
Here the original Latin reads simply
comperio
, and so I have dropped the phrase
by experiment
, which Crew and De Salvio (1914, 153) add immediately after the word
discovered
. As Koyré (1943, 209–10) pointed out, this unjustified addition is a sign of Crew and De Salvio's empiricist leanings.

19.
Galilei 1890–1909, 8: 196.23–205.6; translated by Crew and De Salvio (1914, 160–69); revised by Finocchiaro for this volume.

20.
In this clause I am changing Crew and De Salvio's (1914, 168)
velocity
in the singular to the plural. This is in accordance with the literal meaning of Galileo's original Italian
le velocità
and with suggestions made by Drake (1970, 231; 1974, 160). For the significance of this difference, see Drake (1970, 229–37; 1973); Finocchiaro (1972; 1973).

21.
Galilei 1890–1909, 8: 205.7–219.33; translated by Crew and De Salvio (1914, 169–85); revised by Finocchiaro for this volume.

22.
That is, the first section, on uniform motion, of the treatise
On Local Motion
, presented at the beginning of Day III of
Two New Sciences
and omitted here. See Galilei 1890–1909, 8: 194; Crew and De Salvio 1914, 157.

23.
Here the Italian text does indeed read
scienziato
.

24.
The dialogue that follows did not appear in the original 1638 edition of
Two New Sciences
. It was composed in 1639 jointly by Galileo and his pupil Vincenzio Viviani, and it was intended to be added to future editions. In including it, I am following Crew and De Salvio (1914, 180–85), as well as Drake (1974, 171–75).

25.
As Crew and De Salvio (1914, 183) note, this is an approximation to the principle of virtual work elaborated by Jean Bernoulli in 1717.

26.
Galileo did not draw
FI
in the previous diagram, but he must have been thinking of a line from
F
to a point somewhere along the line
AC.

27.
Crew and De Salvio (1914, 185) note that in modern notation this argument would read as follows: AC = 1/
₂
gt
_c
2; AD = 1/2 (AC/AB)gt
_d
2; since AC2 = AB*AD, it follows that t
_d
= t
_c
.

28.
Galilei 1890–1909, 8: 268–79; translated by Crew and De Salvio (1914, 244–57); revised by Finocchiaro for this volume.

29.
Apollonius of Perga (c. 262–c. 200 B.C.), Greek mathematician, author of the classical treatise on conic sections (parabola, hyperbola, and ellipse).

30.
Euclid,
Elements
, book 2, proposition 5.