(Part II, Paragraphs 24-54; Part III, Paragraphs 56-59; Letter to Clerselier)

Translation Copyright ©1977, 1995 M.S. Mahoney


24. What motion is, according to the commonly accepted sense.

But motion (that is, local motion, for no other occurs to my thinking, nor, therefore, do I think that any other should be established in the nature of things), motion, I say, as it is commonly taken, is nothing other than the action by which some body is transferred from one place to another. And therefore, just as we have warned [monuimus] above, that the same thing at the same time can be said to change place and not to change [place], so the same thing can also be said to be moved and not to be moved. For example, someone sitting in a boat while it sails from port thinks himself to be moving if he looks at the shore and considers it to be unmoved; but not if [he looks] at the boat itself, among the parts of which he always maintains the same location. Indeed, insofar as we commonly think action to be in every motion, and the cessation of action in rest, he is then more properly said to be at rest than to be moved, because he feels no action in himself.

25. What motion is, properly taken.

But if we were to consider not so much from common usage as from the truth of the matter what should be understood by motion, in order that some determinate nature be assigned to it, we can say it is the translation of one part of matter, or of one body, from the vicinity of those bodies that are in direct contact with it and are viewed as at rest to the vicinity of others. Where by 'one body' or 'one part of matter' I understand everything that is transferred at the same time, even if this itself might again consist of many parts which have other motions in themselves. And I say that translation is not the force or action that transfers, as I shall show that this [motion] is always in the mobile, not in the mover, because these two are not usually distinguished with sufficient care, and that it is only its mode, not some thing subsisting [in it], just as shape is the mode of the shaped thing and rest [the mode] of the thing at rest.

26. No more action is required for motion than for rest.

In fact, it should be noted that we labor under a great prejudice in this, in that we judge that more action is required for motion than for rest. We have persuaded ourselves from the beginning of time that our body is wont to be moved by our will, of which we are intimately conscious, and to be at rest only because it adheres to the earth by gravity, the force of which we do not feel. Moreover, because this gravity and many other causes not known to us resist the motions we want to produce in our members, and cause us to become tired, we think that greater action or greater force is necessary to produce motion than to arrest it, taking action to be that tendency [conatus] which we use to move our members and, with their aid, other bodies. Nevertheless, we will easily set aside that prejudice, if we consider that we need conatus not only to move external bodies but often also to arrest their motions when they are not arrested by gravity or some other cause. For example, we do not use greater action to impel a boat at rest in standing water than to stop the same boat suddenly when it is moving, or certainly not much greater. For, one should measure the weight of the water supported by it and the viscosity [lentor] of the water, by which it could be arrested gradually.

27. Motion and rest are only different modes of the moved body.

But since here it is not a matter of that action which is understood to be in the mover, or in that which arrests motion, but of translation alone and of the absence of translation, or rest, it is clear that this translation cannot be outside the moved body and that this body is in one mode when it is translated and in another when it is not translated, or when it is at rest; with the result that the motion and rest in it are nothing other than two different modes.

28. Motion properly taken can only be referred to the bodies contiguous to that which is moved.

I have furthermore added that translation takes place 'from the vicinity of contiguous bodies to the vicinity of others', but not 'from one place to another', because, as I have explained above, the meaning of 'place' is varied and depends on our thinking. But, when we understand by motion that translation which takes place from the vicinity of contiguous bodies, since only one [group of] bodies can be contiguous to the same mobile at the same moment of time, we cannot attribute to this mobile many motions at the same time, but only one.

29. Nor can it be referred except to those contiguous bodies which are viewed as at rest.

I have added finally that this translation takes place from the vicinity not of any contiguous bodies, but only 'of those which are viewed as at rest.' For this translation is reciprocal, and body AB cannot be understood to be transferred from the vicinity of body CD unless it is understood at the same time that body CD is also transferred from the vicinity of body AB; and clearly the same force and action is required from the one part as from the other. Whence, if we should want to assign to motion a nature altogether its own and not related to something else, we should say that, when two contiguous bodies are transferred, one in one direction, the other in another direction, and thus are mutually separated, there is as much motion in the one as in the other. But this is very much incompatible with the common way of speaking; for, since we are accustomed to stand on the earth and consider it as at rest, even though we see some of its parts contiguous to other smaller bodies transferred from the vicinity of those bodies, we do not, however, therefore think the earth to be moved.

30. Why of two contiguous bodies which are separated from one another one is said to be moved rather than the other.

The chief reason for this is that motion is understood to be of the whole body that is moved, and thus it cannot be of the whole earth in the case of the translation of some of its parts from the vicinity of smaller bodies to which they are contiguous, since one may often show many translations of this sort, mutually contrary, on the earth. For example, if body EFGH is the earth and on it at the same time body AB is transferred from E toward F and CD from H toward G, although the parts of the earth contiguous to this body AB are transferred from B toward A, and the action in them should not be less nor of another nature for this translation than in the body AB, we do not therefore, understand the earth to be moved from B toward A, or from west to east, because, by the same argument, in the case where its parts contiguous to body CD are transferred from C to D, it would be understood that it was also moved in the other direction, that is, from east to west, which two [motions] contradict one another. Thus, lest we draw back too much from the common way of speaking, we should not say here that the earth is moved, but only bodies AB and CD; and thus for other bodies. But in the meantime we will remember that everything that is real and positive in bodies that are moved, according to which they are said to be moved, is also found in others contiguous to them, which nevertheless are only viewed as at rest.

31. How innumerable different motions can be in the same body.

But even if any body has only one motion proper to it, since it is understood to recede from only one [group of] bodies contiguous to it and at rest, it can nevertheless also participate in innumerable others, if, for example, it is a part of other bodies having other motions. For example, if someone walking in a boat carries a watch in his pocket, the wheels of his watch will be moved with only a single motion proper to them, but they will participate also in another insofar as, joined to the walking man, they compose with him one part of matter; and in another insofar as they are joined to the boat in a heaving sea; and in another insofar as they are joined to this sea; and finally in another insofar as they are joined to the earth itself, if indeed the whole earth be moved. And all these motions are really [revera] in these wheels; but, because so many [motions] cannot be understood at the same time, nor also can all be known, it will suffice to consider in itself that single [motion] that is proper to each body.

32. How also the motion properly taken, which is unique in any body, can be taken for many.

Furthermore, that single motion of any body that is proper to it can be considered as many: for example, when we distinguish in the wheels of chariots two different [motions], to wit, one circular [motion] about their axis and another along the length of the path through which they are borne. But that therefore such motions are not in fact distinct is clear from the fact that any point of a body that is moved describes only some one line. It does not matter that this line is often quite contorted and therefore seems to have been generated by many different motions, because we can imagine in the same way any line, even a straight line, which is the simplest of all, to result from an infinite number of different motions. For example, if line AB is moved toward CD and in the same time point A is moved toward B, the straight line AD, which this point A describes, will depend no less on the two rectilinear motions, from A to B and from AB to CD, than the curved line, which is described by any point of the wheel, depends on the rectilinear and circular motion. And, consequently, although it is often useful to separate in this way one motion into many parts, to perceive it more easily, nevertheless, absolutely speaking, one should number only one motion in any body.

33. How in every motion a whole circle of bodies is moved at the same time.

But, from what has been demonstrated above, that all places are filled with bodies and that the same parts of matter are always coequated with equal places, it follows that no body can be moved unless along a circle, i.e. such that it expels some other body from the place into which it enters, and this [other body] again another, and another, up to the last, which enters into the place left behind by the first in the same moment of time in which it is left behind. And this we easily understand in a perfect circle, because we see that no vacuum, and no rarefaction or condensation,is required in order that part A of the circle be moved toward B, provided that in the same time part B is moved toward C, C toward D, and D toward A. But the same can also be understood in a non-perfect circle, however irregular, provided that it is made clear how all inequalities of places can be compensated for by unequal speed of motions. Thus, the whole matter contained in space EFGH can be moved circularly without any condensation or vacuum, and at the same time its part that is toward E can move over toward G, and that which is toward G can move over toward E, provided only that as the space at G is supposed to be four times wider than at E and twice as wide as at F and H, so also it is moved four times faster at E than at G and twice as fast as at F or H. And thus in all other places the speed of motion compensates for the narrowness of the place. For, in this manner, in any determinate time, as much matter will move through one part of this circle as through another.

34. From this follows the division of matter into truly indefinite particles even though they be incomprehensible to us.

Nevertheless, it should be said that something is found in this motion that our mind perceives to be true, but nevertheless does not comprehend in what manner it happens: to wit, the division of some particles of matter ad infinitum or indefinitely, and into so many parts that we cannot determine by thought any [part] so small that we do not understand it to be divided in fact into others still smaller. For it cannot happen that the matter that now fills space G will successively fill all the spaces smaller by innumerable degrees that are between G and E, unless some part of it accommodates its shape to the innumerable measures of those spaces. In order for that to happen, it is necessary that all imaginable particles of it, which are in fact without number [innumerae], separate from each other a slight amount and such extremely small separation [quantulacunque remotio] is true division.

35. How this division takes place; and that it should not be doubted that it takes place, even if it is not comprehended.

But it should be noted that I am not talking here about all of the matter but only about some part of it. For, although we suppose that two or three of its parts at G are as wide as the space E and that, further, there are many other smaller parts that remain undivided, nevertheless it can be understood that they are moved circularly toward E, provided that some others are mixed in with them that in some way turn in [se inflectant] and so change their shape that, joined to those not so changing their shapes but only accommodating their speed of motion in proportion to the place to be occupied, all angles which these others do not occupy are completely filled. And, although we cannot comprehend by thought how this indefinite division takes place, we should not, however, doubt that it happens, because we clearly perceive that it necessarily follows from the nature of matter most evidently known to us, and we also perceive that it is of the species of things that cannot be understood by our mind, being finite.

36. God is the primary cause of motion and always conserves the same quantity of motion in the universe.

The nature of motion being thus understood, it is necessary to consider its cause, and that in two ways: that is, first its universal and primary cause, which is the general cause of all motions in the world, and then its particular cause, by which it happens that individual parts of matter acquire motions that they did not have before. As regards the general cause, it seems clear to me that it is nothing other than God Himself, who in the beginning created matter together with motion and rest and now conserves just as much motion and rest as a whole as He then posited. Now, although this motion in moved matter is nothing other than its mode, nevertheless it has a certain and determinate quantity, which we easily understand to be able to be always the same in the whole universe of things, even though it be changed in its individual parts. So it is evident, as we think, that when one part of matter is moved twice as fast as another, and this second [part of matter] is twice as large as the first, there is as much motion in the smaller as in the larger; and by as much as the motion of one part is made slower, the motion of some other equal to it is made faster. We also understand perfection to be in God, not only that He is immutable in Himself, but that he works in a most constant and immutable way, such that, save those changes that clear experience or divine revelation renders certain and that we believe or perceive to be made without any change in the Creator, we should suppose no other [changes] in His works, lest one then argue an inconstancy in Him. Whence it follows that it is most wholly in accord with reason that we think on this basis alone that God moved the parts of matter in various ways when He first created them and that He now conserves all of this matter clearly in the same way and for the same reason that He formerly created, and that He also conserves the same amount [tantundem] of motion in it always.

37. The first law of nature: that any object, in and of itself, always perseveres in the same state; and thus what is moved once always continues to be moved.

Indeed, from the same immutability of God can be known certain rules or laws of nature, which are the secondary and particular causes of the diverse motions that we perceive in individual bodies. The first of these is that any object, insofar as it is simple and undivided, remains, in and of itself, always in the same state and is never changed, unless by external causes. Thus, if some part of matter is square, we may easily persuade ourselves that it will continue perpetually to be square, unless something should come from elsewhere that changes its shape. If it were at rest, we do not believe it would ever begin to be moved, unless it were impelled to do so by some cause. Nor is there any greater reason, if it were moved, why we should think that it would ever of its own accord, and impeded by nothing else, interrupt its own motion. And therefore one should conclude that that which is moved is, in and of itself, always moved. But, because we are here talking about the earth, the constitution of which is such that all motions that take place near to it are shortly halted, and often due to causes that are hidden from our senses, we have often from earliest times judged that these motions, which were so halted by causes unknown to us, cease of their own accord. And then we are inclined to posit of all what we seem to have experienced in many, namely that these [motions] by their nature cease, or tend toward rest. Actually, it is wholly in opposition to the laws of nature; for rest is contrary to motion, and nothing can be moved to its contrary, or to its own destruction, by its own nature.

38. On the motion of projectiles.

Certainly, everyday experience of things that are thrown wholly confirms our rule. For there is no other reason why thrown [bodies] should continue in motion for any time after they have been separated from the thrower than that once moved they continue to be moved, until they are slowed by contrary bodies. And it is manifest that they usually are gradually retarded by the air, or some other fluid bodies in which they are moved, and hence their motion cannot last long. For we can experience air resisting the motions of other bodies by our sense of touch if we strike it with a fan; the flight of birds also confirms the same thing. And there is no other fluid which does not, even more manifestly than air, resist the motions of projectiles.

39. The second law of nature: that every motion of itself is rectilinear; and hence what is moved circularly tends always to recede from the center of the circle it describes.

The second law of nature is that any part of matter, considered apart, never tends to continue to be moved along any oblique lines, but only along straight lines, even if many are often forced to deflect due to the collision of others, and, as has been said shortly before, in any motion a circle is somehow made from all the matter moved at the same time. The cause of this rule is the same as that of the one preceding, namely the immutability and simplicity of the operation by which God conserves motion in matter. For He does not conserve it other than precisely the way it is in the moment of time in which He conserves, with no relation to what perhaps was shortly before. Although no motion occurs instantaneously, it is nevertheless manifest that everything that is moved, in the single instants that can be designated while it is moved, is determined to continue its motion toward some direction along a straight line, and never along any curved line. For example, stone A, rotated in sling EA around circle ABF, at the instant in which it is at point A is determined to motion in some direction, namely along a straight line toward C, such that the straight line AC is tangent to the circle. But one cannot arrange that it be determined to any curved motion; for, even if it previously came from L to A along a curved line, nevertheless nothing of this curvity can be understood to remain in it when it is at point A. This is also confirmed by experience, because if it then left the sling it would not continue to be moved toward B, but toward C. From which it follows that everybody that is moved circularly, perpetually tends to recede from the center of the circle it describes. We experience this by tactile sense in a stone that we move in a circle with a sling. And, because we will often use this consideration in the things that follow, it should be diligently understood and will be expounded in more detail below.

40. Third law: that a body, in colliding with another larger one, loses nothing of its motion; but, in colliding with a smaller one, loses as much as it transfers to that one.

The third law of nature is this: where a body that is moved meets another, if it has less force [vis] to continue along a straight line than the other has to resist it, then it is deflected in another direction and, retaining its motion, loses only the determination of motion; if it has greater force, then it moves the other body with it and gives it as much of its motion as it loses. Thus we learn by experience that any hard bodies that, when thrown, strike against another hard body do not therefore cease from motion, but are reflected in the opposite direction. On the contrary, however, when they meet a soft body, they are then immediately brought to rest, because they easily transmit all of their motion to that body. Indeed, all particular causes of the changes that befall bodies are contained in this third law, at least those that are themselves corporeal; for whether, and in what way, human or angelic minds have the force to move bodies, we do not now inquire but reserve for our treatise On Man.

41. Proof of the first part of this rule.

The first part of this law is demonstrated on the basis that there is a difference between motion considered in itself and its determination in a certain direction, by which [difference] it happens that this determination can be changed, the motion remaining unchanged [integer]. For, since, as was said before, whatever [the nature of] the motion of any thing that is not composite but simple, it continues to be [such], as long as it is not destroyed by any external cause; and, in the collision with a hard body, it appears as the cause that impedes the motion of the other body, which it meets, from remaining determined toward the same direction, but not a [cause] that takes away or diminishes that motion, because motion is not contrary to motion, whence it follows therefore that it cannot be diminished.

42. Proof of the second part.

Furthermore, the second part is demonstrated from the immutability of the operation of God, now continually conserving the world by the same action by which He formerly created. For, since all things are filled with bodies and, nevertheless, the motion of any body tends in a straight line, it is most clear that, from the beginning, God, in creating the world, not only moved its various parts in different ways but at the same time also brought it to pass that some would impel others and transfer their motions to them; in order that now, in conserving that [world] by the same action and by the same laws by which He created, He conserves motion not always fixed in the same parts of matter but passing from some parts into others according as they collide with one another. And thus this continuous change of things created is itself to be argued of the immutability of God.

43. In what the force of any body to act or to resist consists.

Here certainly one should diligently show in what the force of any body to act on another or to resist the action of another consists. Truly [it consists] in this one thing: any thing tends, in and of itself, to remain in the same state in which it is, according to the law posited in the first place. For, from this, that which is conjoined to another has some force to impede its being disjoined; that which is disjoined, [some force] to remain disjoint; that which is at rest, to persevere in its rest, and consequently to resist all those things which can change it; that which is moved, to persevere in its motion, i.e. in motion of the same speed and in the same direction. And this force should be measured [aestimari] both by the magnitude of the body in which it is, and the surface along which this body is disjoined from another, and by the speed of the motion, and the nature and contrariety of the way in which various bodies collide with one another.

44. Motion is not contrary to motion, but to rest; and determination in one direction to determination in the opposite direction.

It should also be noted that one motion is in no way contrary to another equally fast motion, but that properly only a twofold contrariety is found here: one between motion and rest, or also between swiftness and slowness of motion, namely insofar as this slowness participates in the nature of rest; the other between the determination of motion in some direction and the collision with a body [located] in that direction, either at rest or otherwise moved. And, with respect to the direction in which the body colliding with another is moved, this contrariety is greater or less.

45. How one may determine how much the motion of any body is changed by the collision of other bodies; and that by the following rules.

In order that we may determine from these things in what manner [quo pacto] individual bodies increase or diminish their motions, or turn in other directions, due to the collisions of other bodies, it is necessary only to calculate [calculo subducere] how much force is in each, either to move or to resist motion, and to establish as a certainty that that which is stronger will always achieve its effect. And this may be easily calculated if only two bodies collide with one another, and they are perfectly hard and are separated from all others in such a way that their motions are neither impeded nor aided by any others lying about. For they will observe the following rules.

46. First rule.

First, if these two bodies, say B and C, were wholly equal and were moved equally fast, B from the right toward the left and C on a line with it [illi in directum] from the left toward the right, when they collided with one another, they would be reflected and afterward would continue to be moved, B toward the right and C toward the left, no part of their speed having been lost.

47. Second rule.

Secondly, if B were just slightly [tantillo] larger than C, other things being posited as before, then only C would be reflected, and both would be moved toward the left at the same speed.

48. Third rule.

Thirdly, if they were equal in mass [mole], but B were moved just slightly faster than C, not only would both continue to be moved toward the left, but also the half part of the speed by which C is exceeded by B would be transferred from B to C. That is, if before there were six degrees of speed in B and only four in C, after mutual collision each would tend toward the left with five degrees of speed.

49. Fourth rule.

Fourthly, if body C were wholly at rest and were slightly larger than B, whatever the speed at which B were moved toward C, it would never move this C, but would repelled from it in the contrary direction; because a body at rest resists a great speed more than a small one, and this in proportion to the excess of the one over the other, and, therefore, there would always be a greater force in C to resist than in B to impel.

50. Fifth rule.

Fifthly, if the body C at rest were less than B, then, however slowly B were moved toward C, B would move C with it, by transferring such a part of its motion to C that afterward both would be moved equally fast. That is, if B were twice as large as C, it would transfer to it the third part of its motion, because that one third part would move C as quickly as the two other remaining [parts would move] B [which is ] twice as large. And thus, after B had collided with this C, it would be moved a third part more slowly than before, would require as much time to be moved through a distance of two feet as before to be moved through a distance of three. In the same way, if B were three times as large as C, it would transfer to it the fourth part of its motion, and so on.

51. Sixth rule.

Sixthly, if body C at rest were most accurately equal to body B moved toward it, it would be partly impelled by B and would partly repel it in the contrary direction. That is, if B were to approach C with four degrees of speed, it would communicate to C one degree and with the three remaining would be reflected in the opposite direction.

52. Seventh rule.

Finally, if B and C were moved in the same direction, C more slowly and B pursuing it more quickly, such that it finally reached it, and C were larger than B, but the excess of speed in B were greater than the excess of magnitude in C, then B would transfer so much of its motion to C that both would be moved afterward equally fast and in the same direction. But if, on the contrary, the excess of speed in B were less than the excess of magnitude in C, B would be reflected in the contrary direction and would retain all of its motion. And these excesses are thus computed: if C were twice as large as B and B were not moved twice as fast as C, B would not impel C but would be reflected in the contrary direction. But if it were moved more than twice as fast, it would impel C. That is, if C had only two degrees of speed and B had five, from B would be taken two degrees which, transferred to C, would make up only one degree, because C is twice as large as B. Whence it would happen that the two bodies B and C would afterward be moved with three degrees of speed. And thus one should evaluate other cases. Nor do these things require proof, because they are manifest in themselves. [And the demonstrations are so certain that, even if experience seemed to show us the contrary, we would nevertheless be obliged to place more faith in our reason than in our senses.]*

53. The use of these rules is difficult, for the reason that each body is touched by many at the same time.

But, because no bodies in the world can be so separated from all the others and no bodies around us are wont to be completely hard, it is therefore the more difficult to enter into calculation to determine how much the motion of any body will be changed due to collision with others. For, one must have knowledge at the same time of all those that touch it on all sides, and these have very different effects with respect to it [quantum ad hoc], according as they are hard or fluid. Therefore, one must here inquire in what their diversity consists. [In fact, it often happens that experience can at first seem to contradict the rules that I have just set out, but the reason for that is evident. For they presuppose that the two bodies B and C are perfectly hard, and separated from all the others in such a way that there is none around them that might aid or impede their motion; but we see no such bodies in this world. That is why, before one can judge if [the rules] are observed or not, it does not suffice to know how two bodies such as B and C can act against one another when they meet, but beyond that one must consider how all the bodies that surround them can increase or decrease their action. And because there is nothing which causes them to have different effects here except the distance between them --in that some are liquid and soft, and others hard -- we must examine at this point what constitutes these two qualities of being hard and of being liquid.]

54 What bodies are hard, what fluid.

That is, according to the senses [sensu teste], we know no other [difference] than that the parts of fluids recede easily from their places and that therefore they do not resist our hands moving against them. But, on the contrary, the parts of hard bodies so cohere mutually that they cannot be separated without a force that suffices to overcome their coherence. Investigating further what causes some bodies to relinquish their places to other bodies without any difficulty and some not, we easily see that those which are already in motion do not impede others from occupying their places, which they cede of their own accord [sponte deferunt], but that those which are at rest cannot be extruded from their places without some force. Whence one may gather that bodies divided into many small particles, agitated by motions different from one another, are fluids, and those all of whose particles are mutually at rest next to one another are hard bodies.
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(Part II, Paragraphs 24-54; Part III, Paragraphs 56-59; Letter to Clerselier)


56. What tendency to motion should be understood in inanimate things.

When I say that the globules of the second element tend to recede from the center about which they are revolved, one should not think therefore that I affix to them any [power of] thought from which this tendency [conatus] proceeds, but only that they are so placed and incited to motion that in fact they will go in that direction unless they are impeded by some other cause.

57. How tendencies to different motions can be in the same body at the same time.

Indeed, because many different causes frequently act together on the same body, and some impede the effects of others, according as we look at these or those, we can say that it tends, or tries to go [ire conari], in different directions at the same time. For example, stone A, rotated in sling EA about center E, tends from A toward B, if all the causes that are present to determine its motion are viewed together, because in fact it is moved in that direction. But if we look only at the force of motion that is in the stone itself, we will say that, when it is at point A, it tends toward C, in accordance with the law of motion set forth above, assuming, of course, that line AC is the straight line that is tangent to the circle at point A. For, if the stone were to leave the sling at the moment of time in which coming from L it arrived at point A, in fact it would move from A toward C, not toward B. And, although the sling impedes this effect, it does not, however, impede the tendency. If, finally, we do not look at the whole of this force of motion, but only at that part of it that is impeded by the sling, distinguishing it from some other part of it that has its own effect [sortitur suum effectum], we will say that this stone, while it is at point A, tends only toward D, or tries to recede from center E along the straight line EAD.

58. How those things that are moved circularly tend to recede from the center of their motion.

To understand this clearly, let us compare the motion by which the stone, being at point A, is moved toward C, if it is not impeded by some other force, with the motion by which a bee located at the same point A would also moved toward C, if line Ey were a rod along which it proceeded in a straight line from A toward Y while, in the meantime, this rod were rotated about center E, point A of the same rod described circle ABF, and these two motions were so timed together [ita inter se contemperati] that the bee arrived at X when the rod was at C and at Y when the rod was at G, and thus the bee always remained on the straight line ACG. And then let us compare also the force by which the same stone, moved in the sling along the circular line ABF, tends to recede from center E along the straight lines AD, BC, FG, with the tendency that would remain in the bee if it were detained at point A on rod EY by some fastening or glue while, in the meantime, this rod carried it about center E along the circular line ABF, and the bee tried with all its strength [viribus] to go toward Y and thus to move away from center E along the straight lines EAY, EBY, and so on.

59. How much the force of this tendency is.

I know that the motion of this bee will be most slow at the beginning, and therefore its tendency, if one refers only to the beginning of motion, cannot seem to be great; but certainly it is not merely nothing, and while it takes effect it increases, so that the motion proceeding from it can be rather fast. Now, to use yet another example, if EY is a tube [canalis] in which ball [globulus] A is contained, in the first moment of time in which this tube is moved in a circle about center E, the ball A will proceed with only a very slow motion toward Y; but, in the second moment it will move slightly faster, for it will retain its previous force and acquire beyond that a new force from the new tendency to recede from center E, because, as long as the circular motion lasts, this tendency lasts and is, as it were, renewed in each moment. And experience confirms this: for, if the tube EY were moved rather quickly about center E, in a short time the ball in it would arrive at Y from A. We also experience the same thing in a sling; for the faster the stone is rotated in it the more the string is stretched, and this tension, resulting only from the force by which the stone tends to recede from the center of its motion, shows us the quantity of this force.

Source: Principia philosophiae. Text of the 1644 Amsterdam first edition, printed in vol. VIII-l of Oeuvres de Descartes, ed. Adam and Tannery, 2nd ed., Paris, 1964. 

(Part II, Paragraphs 24-54; Part III, Paragraphs 56-59; Letter to Clerselier)



Translation Copyright ©1977, 1995 M.S. Mahoney

Egmond, 17 February 1645


The reason why I said that a body that is without motion can never be moved by another smaller than it, whatever the speed at which this smaller body can move, is that it is a law of nature that a body that moves another must have more force to move it than the other body has to resist. But this [force to resist] can depend on nothing more than its magnitude, for that which is without motion has as many degrees of resistance as the other, which is moved, has of speed. The reason for this is that, if it is moved by a body that is moving twice as fast as another, it should receive from that body twice as much motion; but it resists this twice as much motion twice as much more.

For example, body B cannot push body C unless it causes C to move as fast as it itself is moved after having pushed C. That is, if B is to C as 5 is to 4, B must transfer to C 4 of the 9 degrees of motion that are in B, in order to cause C to go as fast as it; which is easy for B, since it has the force to transfer up to 4 and a half (i.e. half of all it has) before [plutôtque] reflecting its motion in the other direction. But if B is to C as 4 to 5, B cannot move C unless it transfers to C 5 of its 9 degrees of motion, which is more than half of what it has; in consequence of which body C resists more than B has force to act. That is why B should be reflected in the other direction, before moving C. Otherwise, no body would ever be reflected by impact with another.

Furthermore, I am pleased that the first and principal difficulty that you have found in my Principles concerns the rules according to which the motion of bodies that collide with one another changes, for I judge from that that you have found no difficulty in what precedes them and that you will also not find much in what follows. Nor will you have any more difficulty with these rules once you have taken note that they depend only on a single principle, which is that when two bodies, which have in them incompatible modes, collide some change must certainly take place in these modes to render them compatible, but that this change is always the least possible. That is to say, if, when a certain quantity of these modes has been changed, they can become compatible, a greater quantity will not be changed. One must consider two different modes in motion: one is motion alone, or speed, and the other is the determination of that motion in a certain direction. These two modes are changed with equal difficulty.

Hence, to understand the fourth, fifth, and sixth rules, where the motion of body B and the rest of body C are incompatible, one must take note that they can become compatible in two ways, to wit, if B changes the whole determination of its motion, or if it changes the rest of body C by transferring to C such a portion of its motion that it can chase C before it as fast as it itself will go. In these three rules I have said nothing other than that, when C is greater than B, it is the first of these two ways that obtains; and, when it is smaller, it is the second; and finally, when they are equal, this change occurs half in the one way and half in the other. For, when C is the greater, B cannot push C before it unless it transfers to C more than half its speed and at the same time more than half its determination to go from right to left, inasmuch as this determination is joined to its speed; instead, being reflected without moving C it changes only the whole of its determination, which is less of a change than that which results from more than half of this same determination and, in addition, more than half of the speed. If, on the contrary, C is less than B, it should be pushed by B, for then B gives it less than half its speed and less than half the determination that is joined to it, which makes less than the whole of this determination that would have to change if it were reflected.

And this does not contradict experience; for, in these rules, I understand by a body that is without motion a body that is not in the act of separating its surface from those of other bodies that surround it, and consequently that it constitutes part of another hard body that is greater. For I have said further that, when the surfaces of two bodies are separated, all that is positive in the nature of motion is just as much in that which is commonly said not to be moved as in that which is said to be moved, and I have explained, after that, how a body suspended in the air can be moved by the least force.

Nevertheless, I must here admit to you that these rules are not without difficulty, and I will try to elucidate them further if I have the opportunity; but, because I have occupied my mind with other thought I will wait, if you don't mind, until another time to send you my opinion in greater length.

I am much obliged to you for the victories you have gained for me on various occasions, and your solution of the argument that The pagans had a notion of many gods, etc. is quite correct. For, even if the idea of God is so imprinted on the human mind that there is no one who does not have in himself the faculty for knowing it, this does not prevent many persons from having been able to live their whole lives without ever representing to themselves distinctly this idea. And, in fact, those who think they have the idea of many gods have nothing of the sort. For it leads to a contradiction to conceive of many sovereignly perfect beings, as you have quite correctly noted; and when the Ancients named many gods they did not understand thereby many all-powerful beings, but only many powerful beings, above whom they imagined a single Jupiter as sovereign, to whom alone, consequently, they applied the idea of the true God, which was presented confusedly to them.

I am, etc.

Source: Oeuvres de Descartes, ed. Adam and Tannery, Vol. IV, pp.183-188.