The Essential Galileo
Page 47
SAGR. 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.
SALV. 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.
SIMP. 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.
SALV. 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 Sal
vio (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 = 1/2gtc 2; AD = 1/2 (AC/AB)gtd 2; since AC2 = AB*AD, it follows that td = tc.
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.