# In Opposition to Aristotle: Contrary Motions Can Be Continuous Without an Intervening Moment of Rest

Introduction by Edward Grant

2. Galileo Galilei

Translated by I. E. Drabkin^{35}

Annotated by Edward Grant

**CHAPTER [20]**

*In which, in opposition to Aristotle and the general view, it is shown that at the turning point [an interval of] rest does not occur*.

Aristotle and his followers believed that two contrary motions—he defines contrary motions as those which tend toward opposite goals—could in no way be continuous with each other. And therefore they believed that, when a stone is projected upward and then falls back over the same path, it must necessarily remain at rest at the turning point. The chief argument with which Aristotle tries to prove this is as follows. "Whatever moves by approaching some point and then moving away from that same point, using it as an end and as a beginning, will not move away from it, unless it has first stopped at it. But that which moves to the farthest point of a line and then moves back from that point, uses that point both as an end and as a beginning. It must therefore remain stationary between the motion toward and the motion away from the point."^{36} Aristotle proves his major premise thus: "Whoever treats something both as beginning and as end, makes what is one in number two in logic, just as the person who in thought takes a point, which is one and the same numerically, and makes it two in logic, namely the end of one thing and the beginning of a second thing. But if something uses one thing as two it must necessarily remain stationary there; for there is [an interval of] time between the two."

Such is Aristotle’s argument. But how weak it is will soon be clear. For, as he himself holds, the moving body makes use of a point on the line of its motion, i.e., one point, numerically, for what are two things in logic, for a beginning and an end. Yet there is no line between these two things, since they are only one in number. And why, similarly, will the same body not use the same instant (one, in number) as two in logic, namely, for the end of the time of moving toward [the turning point] and for the beginning of the time of moving away [from it], so that between these instants that are two in logic, no time intervenes, since they are only one in number?

There is no compelling reason why this should not be the case, especially since Aristotle himself holds that what is true of a line is true also of time and motion. If, then, on the same line the same point, numerically, is both the end of one motion and the beginning of a second, and if, nevertheless, it is not necessary that a line form a connection between this beginning and that end, then, in the same way, the same instant numerically, will, in logic, be the end of one time and the beginning of a second time, and it will not be necessary for time to intervene between the two. It is clear, then, that the refutation of Aristotle’s argument can be neatly derived from the propositions of that same argument. Hence, since the argument no longer has compelling force for us, let us see whether we can construct arguments for the opposite view that are more sharply convincing.

So much in opposition to Aristotle. But in order for us to show by other arguments that [an interval of] rest does not occur at the turning point, and that there need not be such rest between contrary motions, consider these additional arguments.

Secondly, suppose that some continuum, such as the whole of line *ab,* moves in the direction of *b* in a motion like a forced motion which becomes continuously slower. And while the line so moves, suppose that a body, say *c,* moves on the same line in the opposite direction, from *b* to *a*. But let this motion be like a natural motion, that is, one that is accelerated. And let the motion of the line at the beginning be faster than the motion of *c* at the beginning. Now it is clear that at the beginning *c* will move in the same direction as that in which the line moves, because the motion by which it is carried in the opposite direction is slower than the motion of the line. And yet, since the motion of the line becomes slower and the [leftward component of the] motion of *c* becomes faster, at some moment *c* will actually move toward the left, and will thus make the change from rightward to leftward motion over the same line. And yet it will not be at rest for any [interval of] time at the point where the change occurs. And the reason for this is that it cannot be at rest unless the line moves to the right at the same speed as body *c* moves to the left. But it will never happen that this equality will continue over any interval of time, since the speed of one motion is continuously diminished, and that of the other continuously increased. Hence it follows that *c* will change from one motion to its contrary with no intervening state of rest.^{37}

My third argument can be drawn from a certain rectilinear motion which Nicholas Copernicus in his *De Revolutionibus* compounds from two circular motions.^{38} There are two circles [the center of] each of which is carried on the circumference of the other. When one circle moves more swiftly than the other, a point on the circumference of the first circle moves in a straight line continuously back and forth over the same path. And yet it cannot be said that the point is at rest at the extremities, since it is carried continuously by the circumference of the circle.^{39}

My fourth argument is the well-known one about a large stone falling from a tower.^{40} A little pebble is forcibly thrown up from below against it, but the stone will not be sufficiently blocked by the pebble so as to allow the pebble to be at rest for any interval of time. Hence the pebble will surely not remain at rest at the farthest point of its upward motion, and, despite what Aristotle said, it will use that farthest point as two termini, namely, of upward motion and of downward motion. And the last instant is taken twice, viz., as the end of one interval of time and as the beginning of the other.

But in order to escape from this argument my adversaries declare that the large stone is at rest, and so they believe that they have answered the argument.^{41} But, so that they may not believe this in the future (unless they are thoroughly obstinate), I shall add the following to my argument. Suppose that those stones which move with contrary motions move not up and down but on a plane surface parallel to the horizon, one with great impetus, and the other more slowly. And suppose that they move in opposite directions from opposite parts and meet in the middle in an interacting motion. In that case, there is no doubt that the weaker will be thrust back by the stronger and forced to move back. But how can they say that at that point of impact an interval of rest occurs? For if once they remained at rest, they would thereafter always be at rest, since they would not have reason for moving. In the case of the large stone falling from a high point, even if it were stopped by the pebble, yet after the interval of rest, both would fall together, moving down by reason of their own weight. But when they are in a plane parallel to the horizon there exists no cause of motion after the [supposed] interval of rest.

Before expounding my last argument, I make these two assumptions. My first assumption is that only then is it possible for a body to be at rest outside its proper place, when the force that opposes its fall is equal to its weight, which exerts pressure downward. Surely this is clear: for if the impressed force was greater than the resistant weight, the body would continue to move upward; and if it were smaller, the body would fall. Secondly, I assume that the same body can be sustained in the same place over equal intervals of time by equal forces.

I then urge the following. If a state of rest lasting for some interval of time occurs at the turning point, e.g., when a stone changes from forced upward motion to [natural] downward motion, then over the same interval of time there will exist equality between the projecting force and the resisting weight. But this is impossible, since it was proved in the previous chapter that the projecting force is continuously diminished.^{42} For the motion in which the stone changes from accidental lightness to heaviness is one and continuous, as when iron moves [i.e., changes] from heat to coldness. Therefore the stone will not be able to remain at rest.

Furthermore, suppose that the stone moves forcibly from *a* to *b,* and naturally from *b* to *a*. If, then, the stone is at rest at *b* for some interval of time, suppose that this time has as its end moments *c* and *d*. If, then, the body is at rest for time *cd,* the external projecting force will, through time *cd,* be equal to the weight of the body. But the natural weight is always the same. Therefore the [projecting] force at moment *c* is equal to that force at moment *d*. Now it is the same stone and the same place: hence the stone will be held there over equal intervals of time by equal forces. But the force at moment *c* sustains the body throughout time *cd*. Hence the force at moment *d* will sustain the same stone throughout an interval of time equal to interval *cd*. The body will therefore be at rest throughout twice time *cd*. But this is inconsistent: for it was assumed to be at rest only through time *cd*. Indeed, by continuing the same form of argument, we could prove that the stone would always be at rest at *b*.

But do not be confused by the argument that, if the weight and projecting force are equal at some time, then the body must be at rest for some time. For it is one thing to say that the weight of the body at some time comes to be equal to the projecting force; but it is another thing to say that it remains in this state of equality over an interval of time. This becomes clear from the following consideration. While the body is in motion, since (as has been shown) the projecting force is always being diminished, but the intrinsic weight always remains the same, it must follow that, before they arrive at a relation of equality, countless other ratios occur. Yet it is impossible for the force and the weight to remain in any of these ratios over any interval of time. For it has been proved that the projecting force never remains at the same level over an interval of time, since it is always diminishing.

And so, it is true that the [projecting] force and the weight pass through ratios of, let us say, 2 to 1, 3 to 2, 4 to 3, and countless other ratios; but it is false and impossible that they should remain for any interval of time in any one of these ratios. So, too, they arrive at equality at some moment, but they do not remain at equality. This being so, since local motion upward and downward is a consequence of that alterative motion of change from light *per accidens* to heavy *per se,* in such a way that upward motion flows from an excess of impressed force, downward motion from a deficiency thereof, and rest from equality, and since this equality does not persist over an interval of time, it follows that neither does the state of rest persist.

^{35.} Reprinted with permission of the Regents of the University of Wisconsin from Galileo Galilei "On Motion" and "On Mechanics" Comprising "De Motu" (ca. 1590), translated with introduction and notes by I. E. Drabkin, and "Le Meccaniche" (ca. 1600), translated with introduction and notes by Stillman Drake (Madison Wis.: University of Wisconsin Press, 1960), pp. 94–100. The basic similarity between some of Galileo’s arguments and those of Marsilius of Inghen in the preceding selection will be obvious. On this question, Galileo merely continues and elaborates a well-established medieval tradition.

^{36.}*Physics* VIII.8.262b.2–8 and also the quotation at the beginning of this selection.

^{37.} This argument is fundamentally the same as Marsilius of Inghen’s, in the preceding section.

^{38.} Book III, chapter 4.

^{39.} Drabkin notes (p. 97, n. 5) that "there is an analogous and simpler example in Benedetti, *Diversarum speculationum. . .liber,* p. 183 (cited by A. Koyré, *Études Galiléennes,* I, 51), in which one end of a rigid bar is attached to a point (other than the center) of a continuously rotating circle, and the other end moves back and forth along the same straight line."

had formulated much the same argument long before (see my n. 19).

^{40.} See Inghen section and note 18.

^{41.} Probably a reference to the type of argument cited by Marsilius of Inghen but formulated long before by Avicenna (see Inghen section and n. 20). However, Galileo’s version is distorted. It was not claimed that the large stone would rest, but rather that the pebble would have had its upward direction reversed (by the force of the air pushed downward by the large falling stone) before the large stone reached it. Perhaps Galileo’s version represents a Renaissance variation on the traditional arguments.

^{42.} Actually demonstrated in Chapter 18 ("In which it is shown that the motive force is gradually weakened in the moving body"). In chapter 19 upward deceleration and downward acceleration of bodies is explained in terms of a self-expending impressed force. (Also see Drabkin, p. 89). "In Galileo’s exposition, the notion of a residual force plays the leading role. Initially, the mover imparts an impressed force to a stone that is hurled aloft. As the force diminishes, the body gradually decreases its upward speed until the impressed force is counterbalanced by the weight of the stone at which moment the stone commences to fall, slowly at first and then more quickly as the impressed force diminishes and gradually dissipates itself. The acceleration arises from the continual increase of the difference between the weight of the stone and the diminishing impressed force." (Edward Grant, "Bradwardine and Galileo: Equality of Velocities in the Void," *Archive for History of Exact Sciences,* II [1965], 360.)