Galilei, Galileo, De Motu Antiquiora

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Author: Galilei, Galileo
Title: De Motu Antiquiora
Translator: Raymond Fredette

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Copyright: Max Planck Institute for the History of Science (unless stated otherwise)
License: CC-BY-SA (unless stated otherwise)
1
Older Works On Motion
As we will explain later that all natural motion of translation {1}, whether it be upward or downward, is the result of the proper {2} heaviness or lightness of the mobile, we have thought it in accordance with reason to bring forth for every one to see how it should be said that a thing is lighter or heavier than another, or equally heavy.Indeed, it is necessary to determine this: for it often happens that things that are lighter are called heavier, and conversely.Thus, at times we say of a large piece of wood that it is heavier than a small piece of lead, even though, purely and simply,lead is heavier than wood; {1}and of a large piece of lead, we say that it is heavier than a small one, even though lead is not heavier than lead. For this reason, in order that we may escape pitfalls of this kind, those things will have to be said to be equally heavy to one another which, when they are equal in size (1), will also be equal in heaviness: thus, if we take two pieces of lead, which are equal in size, and they are also congruent in heaviness, they will have to be said to really weigh the same.{2}Thus, it is clear that wood and lead must not be said to be equally heavy: for a piece of wood, which weighs the same thing as a piece of lead, will considerably exceed the latter in size.Moreover, a thing should be called heavier than another, if when a piece of it is taken, equal to a piece of the other, it is found to be heavier than the piece of the other: as, for example, if we take two pieces, one of lead and one of wood, which are equal to one another, and the piece of lead is heavier than the piece of wood, then we shall surely be justified in asserting that lead is heavier than wood.That is why, if we find a piece of wood which weighs the same as a piece of lead, wood and lead should certainly not be deemed to be equally heavy; for, we will find
252 that the size of the lead is considerably exceeded by the size of the wood.Finally, the converse declaration must be made about things that are lighter: for one thing must be deemed lighter, if when a piece of it is taken, equal in size to a piece of the other, it is found to be less in heaviness; as, if we take two pieces, one of wood, the other of lead, which are equal in size, but the piece of wood exerts less weight than the piece of lead, then, rightly, it must be declared that wood is lighter than lead.
That it has been established by nature that heavy things are in a lower place, and light things in a high place, and why.
Since things that are moved naturally are moved towards their proper places, and since the things that are moved are either heavy or light, it must be understood which are the places of heavy things, and which are those of light ones, and why.Now, every day we observe with our senses, that the places of heavy things are those that come near the centre of the world, and the places of light things are those that are farther distant from it; consequently, that such determined places were prescribed for them by nature is not something that we may doubt; but it can be called into question, on the other hand, why prudent nature has observed such an arrangment in distributing places, and not the opposite one.Now, from what I have read, no other cause of this distribution is adduced by philosophers, except that all things had to be arranged in a certain order, and it pleased Supreme Providence to distribute them in this one; and it seems that Aristotle also adduces this cause in Physics, book VIII, text 32, [255b15-17], when, asking why heavy things and light ones are moved towards their proper places, he supposes that the cause is because they are by nature suited to be carried somewhere, that is, the light upward, and the heavy downward. {1} And yet, if we examine the matter more attentively, we certainly shall not be able to think that there was no necessity or utility in nature's making such a distribution, but that it acted only according to whim or some kind of chance.Since I considered carefully that it was quite impossible to think this of provident nature, I scrupulously tried from time to time to imagine some cause, which would be, if not necessary, at least appropriate and useful: and indeed, I have discovered that it is not without the highest justification and the greatest prudence that nature has chosen this distribution.Indeed, since, as it has pleased more ancient {1} philosophers to assert, there is but one matter of all bodies, and those bodies are heavier that contain a greater number of particles of this matter in a narrower space {2} - as these same philosophers, who were perhaps unjustly
253 refuted by Aristotle in book IV of his De Caelo, [308b-309b] asserted {3} - assuredly it was in accordance with reason that bodies that enclosed more matter in a narrower place, should also occupy narrower places, such as are those that come nearer the center.If, for example, we understand that nature, at the time of the original construction of the world, divided all the common matter of the elements into four equal {1} parts, and then assigned to the form of earth its own matter, and in the same way to the form of air its own matter, and that the form of earth caused its matter to be concentrated in a very narrow place, and that the form of air caused its matter to be placed in a very wide place, was it not fitting that nature should assign to air a grand space, and to earth a lesser one?Now, in a sphere, places are narrower the nearer we come to the center, and they are more ample the more we recede from it: hence, it is with both prudence and fairness that nature decreed that the place of earth was that which is narrower than the others, that is, near the center, and that for the remaining elements the places were more ample, the rarer was their matter.I would not say, however, that the quantity of the matter of water is as great as that of the matter of earth, and that for this reason water, since it is rarer than earth, occupies greater places; but only that, if we take a part of water which weighs the same as a part of earth, and for this reason there is as much aqueous matter as earth [under consideration], assuredly this part of earth will then occupy a smaller place than the water, for which reason, justly, it will have to be placed in a narrower space, that is nearer the center. {1} And thus, by proceeding in a similar fashion with the other elements, we will find a certain suitability, not to say a necessity, in such a distribution of the heavy and the light.
That natural motions of translation are made by heaviness or lightness.
Since we have determined in the previous {1} chapter, and have presupposed it as very well known, that it has been established by nature that, indeed, heavier things remain under lighter ones, it must now be considered how the things that are carried downward are moved by heaviness, and how those that are carried upward are moved by lightness.For since heavy things, in virtue of their heaviness, are such as to remain under lighter ones (for inasmuch as they are heavy, they were placed by nature under the lighter things) {1}, in virtue of the same heaviness, they will be such as to be carried under lighter things, if they are placed over them, lest, contrary to the distribution of nature, lighter things should remain under heavier ones.And similarly, in virtue of their lightness, light things will be carried upward, when they are placed under heavier ones: for if in virtue of their lightness they are such as to remain above heavier things, in virtue of the same lightness they will be such as not to remain under heavier things, unless they are hindered.Now from this it is evident that, in the case of motion, consideration must be taken not only of the lightness or heaviness of the mobile, but also of the heaviness and lightness of the medium through which the motion takes place: for if water were not lighter than stone, then a stone would not go down in water.But since a difficulty could arise here concerning why a stone projected into the sea naturally proceeds downward, despite the fact that the water of the sea is heavier by far than the projected stone, it must be remembered what we have noticed in chapter [1]: in fact, the stone is indeed heavier than the water of the sea, if we take an amount of water as great in size as the size of the stone; and thus, the stone, inasmuch as it is heavier than the water, will be carried down in the water.But, again, a difficulty will arise concerning why what must be taken into consideration is the stone and an amount of water as great in size as is the proper size of the stone, and not the whole sea.In order that we may remove this difficulty, I have decided to adduce some demonstrations on which depends not only the solution to this difficulty, but also the whole of the present work.Although in truth the media through which motions occur are several, that is, fire, air, water, etc, and in all of them the same reckoning must be made, we will presuppose that the medium in which motion is to take place is water: and first of all we will demonstrate that those bodies that are equally as heavy as water itself, when they are let down into water, are completely submerged, but that then, however, they are no more carried downward than upward; and secondly, we will show that bodies that are lighter than water not only do not go down into the water, but even cannot be completely submerged; thirdly, we will demonstrate that bodies that are heavier than water are necessarily carried downward.
Chapter 4 First demonstration, where it is proved that those things that are equally as heavy as the medium are carried neither upward nor downward.
And so, coming to the demonstrations, first of all let us consider a certain magnitude equally as heavy as water, that is, whose heaviness is equal to the heaviness of an amount of water the size of which is equal to the size of the said magnitude; let ef be such a magnitude: then it must be shown that the magnitude ef, if let down into water, is completely submerged, and is no more carried upward than downward.And let the position {1} of the water, before the magnitude
255 is let down into this water, be abcd; and let the magnitude ef, when it has been let down into the water, not be completely submerged, if this can be done, but let a certain part protrude, namely e; and let only part f be submerged.Then it is necessary that, while the magnitude f is being submerged, the water should be raised: accordingly, let the surface of the water ao be raised up to the surface st.It is consequently manifest that the size of water so is as great as the size of the submerged part of the magnitude, namely f: for it is necessary that the place, into which the magnitude enters, should be evacuated of water, and that an amount of water should be removed that is as great in size as the size of the magnitude that is being submerged.And so the size of water so is equal to the size of the submerged magnitude, namely this f; hence also the heaviness of this same f will be equal to the heaviness of water so.And since water so strives by its heaviness to return to its former position, but it cannot achieve this unless solid ef is first removed from the water and raised by the water; and the solid, so as not to be raised, resists with all its proper heaviness; and both the solid magnitude and the water are assumed to be standing still in this position; therefore it is necessary that the heaviness of water so, by which it strives to raise the solid upward, be equal to the heaviness with which the solid resists and exerts pressure downward (for if the heaviness of water so were greater than the heaviness of solid ef, ef would be raised and expelled by the water; but if the heaviness of the solid ef were greater, the water, on the other hand, would be raised: yet all these things are assumed to be standing still as they are.Consequently the heaviness of water so is equal to the heaviness of the whole of ef: which is unacceptable; for the heaviness of the same so is equal to the heaviness of the part f.It is consequently manifest that no part of the solid magnitude ef will protrude, but that it will be completely submerged.
This is the complete demonstration, which I have explained by means of a rather lengthy account in this way in order that those who have come upon it for the first time may be able to understand it more easily; but it could also have been better explained by means of a briefer exposition, in such a way that the complete core of the demonstration would be as follows.It must be demonstrated that the magnitude ef, which is assumed to be equally as heavy as water, is completely submerged.For, if it is not completely submerged, let a certain part of it protrude: let e protrude; and let the water be raised up to the surface st; and, if such a thing can be done, let both the water and the magnitude remain in this position. Since, consequently, the magnitude ef by its heaviness exerts pressure on and raises water so; and water so, so as not to be raised further, resists with its heaviness; it is necessary that the heaviness of ef that exerts pressure be as great as
256 the heaviness of water so that resists: for since they are assumed to remain as they are, the pressure will not be greater than the resistance, nor vice versa.Consequently, the heaviness of water so is equal to the heaviness of the magnitude ef: which is unacceptable; for since the size of ef taken as whole is greater than the size of the same water so, the heaviness of the magnitude ef also will be greater than the heaviness of water so. It is consequently manifest that magnitudes equally as heavy as water are completely submerged in water: in addition, I say that they are no more carried upward than downward, but that, wherever they are placed, there they remain.For there is no cause in virtue of which they should go down or up: for since they are assumed to be equally as heavy as water, to say that they go down in water would be the same as if we were to say that water, in water, goes down under water, and that on the other hand the water that rises above the first water then goes down again, and that water thus continues without end going alternately down and up; which is unacceptable.
Chapter 5 Second demonstration, in which it is proved that those things that are lighter than water cannot be completely submerged.
256.17 Now since in the preceding {1} chapter those things have been demonstrated that concern the state of rest, we must now consider things that pertain to motion upward.I say, then, that magnitudes lighter than water, when let down into water, are not completely submerged, but that a certain part protrudes.
Accordingly let the first position of the water, before the magnitude is let down, be along surface ef; and let magnitude a, lighter than water, when let down into the water, be completely submerged, if this can be done, and let the water be raised up to surface cd; and, if it is possible, let both the water and the magnitude remain in this position.Now, the heaviness, with which the magnitude exerts pressure and raises water cf, will be equal to the heaviness with which water cf exerts pressure in order to raise magnitude a.But the size of water cf is equal to the size of magnitude a.There are thus two magnitudes, one which is a, the other which is water cf; and the heaviness of this a is equal to that of this cf, and the size a is also equal to the size of this water cf: therefore, the magnitude a is equally as heavy as water: which is surely absurd; for it has been assumed that the magnitude is lighter than water.Consequently, magnitude a will not remain completely submerged under the water; therefore it will necessarily be carried upward.
257 It is evident, then, why and how upward motion comes from lightness: and, from the things that have been conveyed in this chapter as well as in the preceding one, it can easily be concluded that things that are heavier than water are completely submerged and are necessarily carried downward.That they are completely submerged is necessary: for if they were not completely submerged, they would then , contrary to what has been presupposed, be lighter than water; for that things that are not completely submerged are lighter than water is evident from the converse of the demonstration just adduced. In addition, these things must be carried downward.For if they were not, either they would be at rest, or they would be moved upward: but they would not be at rest; for it has been demonstrated in the preceding chapter that things that are equally as heavy as water are at rest and are no more carried upward than downward: and it has just become apparent that things lighter than water are carried upward.Consequently, from all these considerations, since it is necessary that things that are moved downward be heavier than the medium through which they are carried, it can adequately be grasped how heavier things are moved downward by heaviness; and how in the case of a stone thrown into the sea, the reckoning must be made not with all the water of the sea, but only with that very small part which must be removed from the place into which the stone enters.But, because all these things that have been conveyed in the two preceding chapters can be made clear in a manner still less mathematical and more physical, by reducing them to a consideration of the scale pan, I have decided in the following chapter to explain the correspondence that these natural mobiles observe with the weights of an equal-armed balance{1}: and the purpose of this is to attain a richer knowledge of the things that will be conveyed and more exact knowledge on the part of my readers.
Chapter 6
257.23-260.4 In which is explained the correspondence that natural mobiles have with the weights of a balance.
Thus we will first examine the things that happen in the scale pan, so that we may then show that all these things happen in the case of natural mobiles.
Thus let line ab be understood to be an equal-armed balance, whose center, above which motion takes place, is c, precisely dividing line ab in two; and let two weights, e and o, be suspended from points a and b. Accordingly in the case of weight e three things can happen: either it is at rest, or it is moved upward, or it is moved downward. Consequently if weight e is heavier
258 than weight o, then e will be carried downward: but if it is less heavy, it will surely be moved upward; and this is so, not because it does not have heaviness, but because the heaviness of o is greater. From this it is evident that in the scale pan both motion upward as well as motion downward come from heaviness, but in a different manner: for motion upward will take place for e because of the heaviness of o, but motion downward because of its own heaviness. But if the heaviness of weight e is equal to the heaviness of this o, then e certainly will not be moved either upward or downward: for e will not be moved downward, unless the weight that it must raise, namely o, is less heavy: nor will this same e be carried upward, unless weight o, by which it must be pulled by force, is heavier.
Having examined these things in the case of the scale pan, returning to natural mobiles, we can put forward the following as a general proposition: namely, that the heavier cannot be raised by the less heavy. {1} With this presupposed, it is easy to understand why solids that are lighter than water are not completely submerged. For if, for example, we let down a beam into water, then, if the beam is to be submerged, it is necessary that water move out of the place into which the beam enters, and be raised upward, that is be moved away from the center of the world. Consequently if the water, which is to be raised, is heavier than the beam itself, then surely it will not be able to be raised by the beam: but if the beam is completely submerged, then it is necessary that from the place, into which the beam enters, an amount of water as great in size as the size of the beam itself should be removed: but an amount of water as great in size, as is the size of the beam, is heavier then the beam (for it is assumed that the beam is lighter than water): therefore it will not be possible for the beam to be completely submerged. And this corresponds to what has been said in the case of the scale pan, namely that a lesser weight cannot raise a greater one. But if the beam were equally as heavy as water, that is if the water, which is raised by the beam that is to be submerged, is not heavier than but equally as heavy as the beam, then the beam surely will be completely submerged, since it does not have resistance from the water that is to be raised; but in addition, when it is completely under water, it will be carried no more upward than downward: and this corresponds by analogy to what has been said in the case of the scale pan concerning equal weights, of which neither is carried either upward or downward.But if, on the other hand, the beam is heavier than that water which is to be raised by the beam, that is if the beam is heavier than an amount of water as great in size, as its own proper size (for there is raised by the submerged beam, as has often been said, an amount of water as great in size as its own size), then certainly the beam will be carried downward: which indeed corresponds by analogy to what has been said in the case of the scale pan,
259 namely that one weight is then carried downward and raises the other, when it is heavier than the other. Besides, in the case of natural mobiles, just as in the case of the weights of a balance, the cause of all motions, upward as well as downward, can be reduced to heaviness alone. For when a thing is carried upward, it is at that time raised by the heaviness of the medium; just as, if a beam lighter than water were being kept by force under water, then, because the submerged beam has lifted a quantity of water equal to its own size, and an amount of water as great in size as the size of the beam is heavier than the beam, then, doubtless, the beam will be raised by the heaviness of that water, and it will be impelled upward: and in this way upward motion will be brought about by the heaviness of the medium and the lightness of the mobile{1}; and downward motion by the heaviness of the mobile and the lightness of the medium. {2}And from this, contrary to Aristotle in text #89 of Book I of the <i>De Caelo</i>, someone will easily be able to conclude, in what way things that are moved, are moved, somehow, by force and through the extrusion of the medium: for water extrudes violently the beam that has been submerged by force, when, in going down, it returns towards its proper region, and does not want to suffer that, that which is lighter than it, should remain under it; and, in the same way, the stone is extruded {1} and impelled downward because it is heavier than the medium. It is consequently evident that such a motion can be called forced; although wood, in water, is commonly said to be carried upward naturally, and stone downward. Nor is Aristotle's argument valid, when he says {1}, If it were forced it would be weakened at the end, and not be increased, as it is: for forced motion is weakened only when the mobile will have been outside the hand of the mover, and not while it is linked to the mover. {2}
It is consequently evident, in what way the motion of natural mobiles may be suitably reduced to the motion of weights in a balance: namely in such a way that the natural mobile plays the role of one weight in the balance; and an amount of the medium as great in size as the size of the mobile represents the other weight in the balance. So that, if an amount of the medium as great in size as the size of the mobile will be heavier than the mobile, and the mobile will be lighter, then the mobile, being the lighter weight, will be carried upward: but if the mobile will be heavier than the same amount of the medium then, being the heavier weight, it will go down: and if, finally, the said amount of the medium will be equal in heaviness to the mobile, the mobile will be carried neither upward nor downward: just as in the balance weights that are equal to one other are neither pressed down nor raised. And because natural mobiles are very appropriately compared
260 to the weights in an equal-arms balance, we will bring this correspondence to light in all the ensuing things that will be said concerning natural motion: which will surely contribute not a little to the understanding of these things.
Chapter 7
260.4-262.18 That by which is caused the swiftness and slowness of natural motion. {1}
Since it has been quite abundently explained in the preceding [chapters], how natural motions come from heaviness and lightness, now it must be considered whence the greater or lesser swiftness of this motion comes about. In order that we be able to accomplish this more easily, the following distinction must be made: namely that inequalities in the slowness and swiftness of motion occur in two ways: for either the same mobile is moved in different media; or the medium is the same, but the mobiles are different. We will demonstrate shortly {1} that in both cases of motion the slowness and swiftness depend on the same cause, namely, the greater or lesser heaviness of the media and of the mobiles; {2} but first we will show that the cause of such an effect which has been conveyed by Aristotle is insufficient.
Thus Aristotle in Book IV of the <i>Physics</i>, text #71, has written{1}, that the same mobile is moved more swiftly in a more subtle medium than in a thicker one, and, therefore, that the cause of the slowness of motion is the thickness of the medium, and that of the speed, its subtlety; and he has confirmed this by appeal to no other reason than experience, namely, because we see that a mobile is moved faster in air than in water. But it will be easy to demonstrate that this cause is not sufficient. For if the speed of motion comes from the subtlety of the medium, the same mobile will always be moved more swiftly
261 through more subtle media: which is certainly false; for many are the mobiles which are moved with a natural motion faster in thicker media than in more subtle ones, as, for example, in water than in air. For if we take as an example a very thin inflated bladder, it will in air go down slowly with a natural motion; but if it is let go from deep in water, it will fly very swiftly upward, again with a natural {1} motion. But at this point I know that someone may retort that the bladder in air certainly is moved and is carried fast downward; but in water, not only it does not go down more swiftly, but it does not even go down. To this I would reply, on the contrary, that the bladder in water is carried upward very swiftly, but then in air it is not moved. But, in order not to prolong the dispute, I say that in more subtle media it is not every motion that comes about faster, but only motion downward; and motion upward is swifter in thicker media. And it is certainly reasonable that this happens: for it is necessary that where motion downward takes place with difficulty, motion upward takes place with ease. It is consequently manifest that Aristotle's statement that the slowness of natural motion comes about because of the thickness of the medium was inadequate. Consequently, having abandoned his opinion, in order that we may bring forth the true cause of the slowness and the swiftness of motion, care must be taken that swiftness is not separated from motion: for he who assumes motion, necessarily assumes swiftness; and slowness is nothing other than lesser swiftness. Consequently, swiftness comes from the same thing as does motion: and so, since motion comes from heaviness and lightness, it is necessary that slowness or swiftness come from the same thing; from a greater heaviness of the mobile comes a greater swiftness of that motion which happens because of the heaviness of the mobile, that is, motion downward; and from a lesser heaviness, a slowness of that same motion; and, on the other hand, from a greater lightness of the mobile will emanate a greater swiftness of that motion which happens because of the lightness of the mobile, namely motion upward. It is consequently manifest how diversity in swiftness and in slowness of motion comes about in different mobiles in motion in the same medium: for if the motion is downward, what is heavier will be moved more swiftly than what is less heavy: but if the motion is upward, what is lighter will be moved faster. But whether two mobiles carried in the same medium observe the same ratio in the swiftness of their motions as there is between their heavinesses, as Aristotle believed, will be examined below. {1} Next, concerning the swiftness and slowness of the same mobile in different media, it happens, similarly, that a mobile is moved more swiftly [262] downward in that medium in which it is heavier, than in another in which it is less heavy; and it goes up more swiftly in that medium in which it is lighter, than in another in which it is less light. {1} Hence it is manifest that, if we find in what media the same mobile will be heavier, the media will have been found in which it will go down more swiftly; but if, on the other hand, we demonstrate how much heavier the same mobile is in this medium than in that one, it will also have been demonstrated how much more swiftly it will be moved downward in this medium than in that one: and, by examining the light in converse fashion, when we find in which medium the same mobile will be lighter, a medium will have been found in which the mobile will go up more swiftly; but if we discover by how much the same mobile is lighter in this medium than in that one, it will then have been discovered how much more swiftly the mobile will go up in this medium than in that one. But, in order that all these things may be grasped with more exactitude in the case of any particular motion, treating first of all those motions that are made by different mobiles in the same medium, we will show what ratio their motions observe with one another, with respect to slowness and swiftness; next, inquiring about motions that are made by the same mobile in different media, we will similarly demonstrate what ratio they observe in motions of this sort.
Chapter 8 [262.19-273.31] In which it is demonstrated that different mobiles moving in the same medium observe another ratio than the one attributed to them by Aristotle.
In order, then, that the things that must be thoroughly studied may be dealt with more easily, one must, in the first place, consider that {1} difference between two mobiles can happen in two ways: for either they are of the same species, as, for example, both lead, or both iron; and they differ in size: or they are of different species, e.g. one iron, the other wood; they then differ from one another either in size and heaviness, or in heaviness and not in size, or in size and not in heaviness. Concerning those mobiles that are of the same species Aristotle has said, that
263 the larger is moved faster: and that in Book IV of <i>De Caelo</i>, text #26 [311a 19-22], where he has written that any magnitude of fire is carried upward, and that which is larger, faster; and also that any magnitude of earth is moved downward, and similarly, that which is larger, faster.
And again, in Book III of <i>De Caelo</i>, text #26 [301a 26-32], he says: Let there be a heavy mobile b, and let it be carried along line ce, which is divided at point d; if then mobile be is divided according to the proportion in which line ce is divided at point d, it is manifest that, during the time in which the whole of it is carried along the totality of line ce, in the same time this part is moved along line cd. From this it is overtly established that Aristotle wants mobiles of the same genus to observe between themselves in the speed of motion the ratio of the sizes that these mobiles have: and he says that very openly in Book IV of the <i>De Caelo</i>, text #16 [309b 14], by affirming that a large piece of gold is carried more swiftly than a small one. {1} How ridiculous this opinion is, is clearer than daylight: for who will ever believe that if, for example, two lead balls were released from the sphere of the Moon, one being a hundred times larger than the other, if the larger took an hour to come to Earth, the smaller would use in its motion a space of time of a hundred hours? or, if from a high tower {1}, two stones, one being double the size of the other, were thrown at the same moment, that, when the smaller was at mid-tower, the larger would already have reached the ground? Or, again, if from the depth of the sea a very large beam and a small piece of the same beam begin to go up at the same time, in such a way that the beam is a hundred times larger than that piece of wood, who would ever say, that the beam will have to go up to the surface a hundred times faster?{1}
But, in order that we may always make more use of reasons than of examples (for we are seeking the causes of effects, which are not reported by experience), we will bring forth our way of thinking, whose confirmation will result in the downfall of Aristotle's opinion. We say, then, that mobiles of the same species (let those things be said to be of the same species that are constituted of the same material, such as lead or wood, etc.), though they may differ in size, are however moved with the same swiftness, and a larger stone does not go down more swiftly than a smaller one. Those who are surprised by this conclusion will also be surprised that a very large beam can float on water, just as well as a small piece
264 of wood: for the reasoning is the same; thus, if we conceive in our mind that the water, on which a beam and a small piece of the same beam float, becomes imperceptibly and progressively lighter, in such a way that in the end the water gets to be lighter than the wood and the pieces of wood start slowly to go down, who would ever say that the beam would go down first or more swiftly than the small piece of wood? For although a large beam may be heavier than a small piece of wood, the beam must be put into relation with the great quantity of water that must be raised by it, and the small piece of wood with the small quantity of water [that must be raised by it]: and since an amount of water as great in size as the beam itself must be raised by the beam, and similarly for the small piece of wood, these two amounts of water, namely those that are raised by the pieces of wood, will have the same ratio in heaviness to one another as their sizes have (for the parts of homogeneous things are to one another in heaviness as they are in size, something which should be demonstrated{1}), that is, the ratio that the sizes of the beam and the small piece of wood have to one another: hence the heaviness of the beam will have the same ratio to the heaviness of the water that must be raised by it as the heaviness of the small piece has to the heaviness of the water that must be raised by it: and the reluctance of the large quantity of water [to be raised] will be surpassed by the large beam with the same facility as the resistance of a little water will be overcome by the small piece of wood. And if, on the other hand, we conceive in our mind, for example, an amount of wax of considerable size floating on water, and we mix the wax with either sand or some other heavier thing, in such a way that in the end it comes to be heavier than water and it just barely begins to go down very slowly, who would ever believe, if we took a particle of such wax, say one hundredth of it, either that it would not go down or that it would go down a hundred times more slowly than the totality of the wax? Surely no one.And it will be possible to experience the same thing in the balance: for if very large, equal weights are placed on each side, and then to one of them something heavy, but only modestly so, is added, the heavier will then go down, but not any more swiftly than if the weights had been small. And the same reasoning holds in water: for the beam corresponds to one of the weights of the balance, while the other weight is represented by an amount of water as great in size as the size of the beam: if this amount of water weighs the same as the beam, then the beam will not go down; if {1} the beam is made slightly heavier in such a way that it goes down, it will not go down more swiftly than a small piece of the same wood, which weighed the same as an [equally] small part of the water, and then was made slightly heavier.
But it is pleasing to confirm this by another argument. And first, let the following be presupposed: namely, if there are two mobiles, one of which is moved faster than
265 the other, the combination of the two is moved more slowly than that part which was moved faster than the other, but more swiftly than the remaining part, which, alone, was carried more slowly than the other: {1} as, for example, if we understand two mobiles, such as a piece of wax and an inflated bladder, both of which are carried upward from deep water, but the wax more slowly than the bladder, we ask that it be conceded, that if they are combined, the combination will go up more slowly than the bladder alone, but more swiftly than the wax alone. Indeed this is very clear: for who doubts that the slowness of the wax will be diminished by the speed of the bladder, and, on the other hand, that the speed of the bladder will be retarded by the slowness of the wax, and that a certain motion intermediate between the slowness of the wax and the speed of the bladder will result? Similarly, if on the other hand two mobiles go down, one of which is carried more slowly than the other, as, for example, if one is wood, the other a bladder, which go down in air, the wood more swiftly than the bladder, we presuppose this: if they are combined, the combination will go down more slowly than the wood alone, but more swiftly than the bladder alone. For it is manifest that the swiftness of the wood will be retarded by the slowness of the bladder, while the slowness of the bladder will be accelerated by the speed of the wood; and similarly a certain motion intermediate between the slowness of the bladder and the swiftness of the wood will result. This having been presupposed, I argue as follows: by proving that mobiles of the same species, of unequal sizes, are carried with the same swiftness.{1}
Let there be two mobiles of the same species, the larger a, and the smaller b; and, if it can be done, as our adversaries hold, let a be moved more swiftly than b. There are then two mobiles one of which is moved more swiftly than the other; hence, according to what has been presupposed, the combination of the two will be moved more slowly than the part, which alone, was moved more swiftly than the other. If then a and b are combined, the combination will be moved more slowly than a alone: but the combination of a and b is larger than a alone: hence, contrary to our adversaries' view, the larger mobile will be moved more slowly than the smaller; which would certainly be unsuitable. {1}What clearer indication do we require of the falsehood of Aristotle's opinion? But, I ask, who will not recognize the truth of this on the spot, when he examines it in a pure and simple and natural way?For if we presuppose that the mobiles a and b are equal and that they are very near each other, then, by the consensus of all, they will be moved with equal swiftness: and if we understand that while they are being moved, they are joined, why, I ask, will they double the swiftness of their motion, as Aristotle held, or increase it? Accordingly, let it be sufficiently confirmed that there exists no cause, per se, why mobiles of the same species should be moved with unequal speeds, but there certainly is one why they should be moved with equal speed. But if there were some accidental cause, such as, for example, the shape of the mobile, it must not be classified amongst the causes per se: and moreover, the shape helps or hinders the motion but little, as we shall show in the proper place {1}. Also, one must not , as many are in the habit of doing, immediately revert to extremes, {1} by taking, for example, a piece of lead of very large size, and, on the other hand, a tiny blade or a leaf of the same lead, which sometimes will even float on water: for since there is a certain cohesion and (so to speak) a tenacity and a viscosity of the parts of air as well as of water, this cannot be overcome by a very small heaviness. Accordingly, the conclusion must be understood as concerning those mobiles where the heaviness and size of the smaller one is large enough that it is not hindered by that small tenacity of the medium; such as, for example, a leaden ball of one pound. Moreover, as for scoffers of this kind, who perhaps persuade themselves that they can defend Aristotle, what happens to them if they revert to extremes, is this: the greater the difference between the mobiles that they choose, the more they will have to toil. For if they take a mobile that is a thousand times larger than the other, before they show that one surpasses the other a thousand times in speed, it will undoubtedly cost them sweat and toil.
But, in order to come to the things that remain, it follows now that we should see what ratio mobiles of different species, moved in the same medium, observe in their motions. Such mobiles can differ from one another in three ways; for either they differ in size and not in heaviness, or they differ in heaviness and not in size, or they differ in size and in heaviness; it is necessary to inquire only about those that differ in heaviness and not in size. For the ratios of those that differ in the other two ways can be reduced to this one: thus if the mobiles differ in size and not in heaviness, if from the larger a part is taken that is equal to the smaller, then the mobiles will differ in heaviness and not in size; and this part, taken away from the larger mobile, will observe the same ratio with the other mobile, as the ratio that the whole
267 of the intact one will also observe (for it has been demonstrated that mobiles of the same genus, however much they may differ in size, are moved with the same speed): and similarly, if the mobiles differ in size and in heaviness, having taken hold, from the larger, of a part that is equal to the smaller mobile, we will again have two mobiles which differ in heaviness and not in size; and this part will observe the same ratio with the other mobile in its motion, as the whole of the other intact mobile (for, again, it is with the same swiftness that the part and the whole of mobiles of the same species are moved). Thus is it evident how, if the ratio of the motions of those mobiles that differ only in heaviness and not in size is given, the ratios of those that differ in any other way are also given. And so {1}, in order that we may find this ratio and, against Aristotle's way of thinking, show that in no way do mobiles observe the ratio of their heavinesses, even if they are of different species, we will demonstrate things on which depends the answer not only to this investigation , but also to the investigation of the ratio of the motions of the same mobile in different media; and we will examine both questions simultaneously.
And so let us come to the investigation
268 of the ratio of the same mobile in different media: and first let us examine whether or not Aristotle's way of thinking concerning this is more truthful than the one set forth above.
Now Aristotle believed that the motions of the same mobile in different media observe the same ratio with one another in swiftness, as the ratio that the subtleties of the media have with one another; and this he has openly written in Book IV of the Physics, text #71 [215b 1-11], when he has said: A medium hinders more, because it is thicker; thus a will be moved through a space b in time c, but through a space d, when it is more subtle, in time e, according to the ratio of the hindrance, if the length is equal; so that, if b is water, but d air, then to the extent that air is more subtle than water, to that extent a will be moved more swiftly through d than through b. Therefore speed has to speed the ratio that air has to water: and so, if air is twice as subtle as water, a will travel along line b in twice the time as line d; and time c will be twice time e. These are Aristotle's words, which surely contain a false way of thinking: now in order that this may appear clearer than daylight, I will give form to the following demonstration.
If speed has the same ratio to speed as subtlety of medium to subtlety, let there be a mobile o, and medium a, whose subtlety is of 4, and let that be, for example, water; but let the subtlety of medium b be 16, namely greater than the subtlety of a, and let b, for example, be air; and let mobile o be such that it does not go down in water; but let the swiftness of this same mobile in medium b be 8. {1} Therefore, since the swiftness of mobile o in medium b is 8, but it is zero in medium a, surely some medium can be found in which the swiftness of mobile o is 1. Let such a medium be c. Thus since o is moved more swiftly in medium b than in medium c, it is necessary that the subtlety of this medium c be less than the subtlety of b, and, according to our adversaries, that it be
269 as much less as the swiftness in medium c is less than the swiftness in medium b: now the swiftness of medium b has been assumed to be eight times that of the swiftness in medium c: hence the subtlety of medium b also will be eight times that of the subtlety of medium c: that is why the subtlety of this c will be 2. Therefore mobile o is moved with swiftness 1 in the subtlety of medium c, which is of 2; but it has been assumed that it is not moved in the subtlety of medium a, which is of 4: hence mobile o will not be moved in the greater subtlety, although it is moved in a lesser subtlety: which is most absurd. It is thus evident that, the speeds of motions do not observe with one another the ratios of the subtleties of the media. But, apart from any other demonstration, can anyone fail to see the falsehood of Aristotle's opinion? For if the motions observe the ratio of the media, then, conversely, the media will also observe the ratio of the motions: thus since wood goes down in air but in water not at all, and, consequently, motion in air has no ratio to motion in water, therefore also the rareness of air will have no ratio to the rareness of water: what can be more absurd than this? But lest someone might think he had sufficiently answered my argument, if he said "Though wood is not moved downward in water, it is however moved upward, and the ratio that the upward motion has in water to the downward motion in air, is the same as that of the rareness of water to the rareness of air", and with this he might think he had cleverly saved Aristotle -- we will do away with this subterfuge also: namely, by taking a body which in water is moved neither upward or downward, such as, for example, water itself, which however is moved quite fast in air.
Thus then, Aristotle's way of thinking having been with reason ranked in second place, let us now inquire about the ratio that the motions of the same mobile done in different media observe; and, in the first place, let us show concerning upward motion, that solid magnitudes lighter than water, having been impelled into water, are carried upward with as much force, as that by which a quantity of water, whose size is equal to the size of the submerged magnitude, will be heavier than that magnitude.
And thus let the first position of the water, before the magnitude is submerged in it, be along surface ab; and let the solid magnitude cd be forcibly submerged in it; and let the water be raised to the surface ef: and since water eb, which is raised, has a size equal {1} to the size of the whole submerged magnitude, and the magnitude is assumed to be lighter than water, the heaviness of water eb will be greater than the heaviness of cd. Then let it be understood that tb is that part of the water, whose heaviness is equal to the heaviness of magnitude cd: accordingly it must be demonstrated that, magnitude cd
270 is carried upward with a force as great as the heaviness of water tf (for it is by this heaviness that water eb is heavier than the heaviness of water tb, that is, than the heaviness of magnitude cd). Now since the heaviness of water tb is equal to the heaviness of cd, water tb will exert an upward pressure so as to raise the magnitude with as much force as that with which the magnitude will resist being raised. And thus the heaviness of the part of water exerting pressure, namely tb, is equal to the resistance of the solid magnitude: but the heaviness of all the water that exerts pressure, eb, surpasses the heaviness of water tb by the heaviness of water tf: hence the heaviness of all the water eb will surpass the resistance of solid cd by the heaviness of water tf. And so the heaviness of all the water exerting pressure will impel the solid magnitude upward with as much force as the heaviness of part tf of the water: which was to be demontrated.
From this demonstration it is evident, first, how, as was said above, upward motion comes about from heaviness, but the heaviness of the medium, not of the mobile: second, the aim of our research is concluded. For since we are investigating how much more swiftly the same mobile goes up through one medium than another, whenever we know with how much swiftness it is carried through each one of them, we will also know the interval between the swiftnesses: and that is what we seek. If then this piece of wood, for example, whose heaviness is 4, is carried upward in water, and the heaviness of an amount of water as great in size as the size of the wood is 6, then the wood will be carried with a swiftness of 2: but if, on the contrary, this same piece of wood is carried upward in a medium heavier than water, such that the heaviness of an amount of this second medium as great in size as the size of the wood is 10, then the wood will be carried upward in it with a swiftness of 6. But it was carried in the other medium with a swiftness of 2: hence these two swiftnesses will be to one another as 6 and 2, not as the heavinesses or the thicknesses of the media, as Aristotle wanted it, which are to one another as 10 and 6. Thus it is evident that, in all cases, the swiftnesses of the upward motions are to one another as the excess of heaviness of one medium over the heaviness of the mobile is to the excess of heaviness of the other medium over the heaviness of the same mobile. For that reason, if on the spot we want to know the swiftnesses of the same mobile in two media, let us take from each medium two amounts equal in size to the size of the mobile, and let the heaviness of the mobile be substracted from the heaviness of each one of the media; the remaining numbers will be to one another {2} as
271 the swiftnesses of the motions. {1} And an answer to the other question is also obtained, namely, what ratio different mobiles, equal in size, and unequal in heaviness, observe in the swiftness of their motions. For if each of them is carried upward with as much force as that by which an amount of the medium as great in size as the size of the mobile is heavier than the mobile itself, then, having subtracted the heavinesses of the mobiles from the heaviness of the aforesaid amount of the medium, the remaining numbers will observe with one another the ratio of the swiftnesses: if, for example, the heaviness of one mobile is 4, of the other 6, and of the medium 8, then the swiftness of the mobile whose heaviness is 4, will be 4, but the swiftness of the other will be 2. Now, these swiftnesses, 4 and 2, are not to one another as the lightnesses of the mobiles, which are 6 and 4 {1}: for the excesses of one number over two others will never be to one another as those two numbers; nor will the excesses of two numbers over another number be to one another as the exceeding numbers. {2}It is therefore very clear that in motion upward, the motions of different mobiles are not to one another as the lightnesses of the mobiles.
It thus remains for us to show that also in the motion of mobiles downward the swiftnesses are not to one another as the heavinesses of the mobiles and, at the same time, to show what ratio the swiftnesses of the same mobile observe in different media; all these things will easily be drawn from the following demonstration. I say, then, that a solid magnitude heavier than water is carried downward with as much force as that by which a quantity of water, having a size equal to the size of the same magnitude, is lighter than this magnitude.
Thus, let the first position of the water be along surface de; let the solid magnitude bl be released into the water, and let the water be raised up to surface ab; also let water ae be such that it has a size equal to the size of the same magnitude: and since the solid magnitude is assumed to be heavier than water, the heaviness of the water will be less than the heaviness of the solid magnitude. Then let ao be understood to be an amount of water that has a heaviness equal to the heaviness of bl: and since water ae is lighter than ao by the heaviness of do, it must be demonstrated that the magnitude bl is carried downward with a force as great as the heaviness of water do. Let another solid body be imagined, lighter than water, joined
272 to the first, with a size equal to that of water ao, and let its heaviness be equal to that of water ae; and let the said magnitude be lm: and since size bl is equal to size ae, and size lm is equal to size ao, the size of the combined magnitudes bl, lm is equal to the size of the combined amounts of water ea, ao. But the heaviness of the magnitude of water ae is equal to the heaviness of magnitude lm, while the heaviness of water ao is equal to the heaviness of magnitude bl: hence the total heaviness of both magnitudes bl, lm is equal to the heaviness of water oa, ae. But in addition it has been demonstrated that the size of the magnitudes is equal to the size of water oa, ae; hence, by the first proposition, the magnitudes so combined will be carried neither upward nor downward. Therefore the force of the magnitude bl exerting pressure downward will be as great as the force of magnitude lm, which impels upward: but, by the previous proposition, magnitude lm impels upward with as much force as the quantity of heaviness of water do: therefore magnitude bl will be carried downward with a force as great as the heaviness of water do. Which is what was to be demonstrated.
Now, if this demonstration has been grasped, the answer to these questions can easily be discerned.For it is clear that that the same mobile going down in different media, observes in the swiftness of its motions, the ratio to one another of the excesses of its own heaviness over the heavinesses of the media: thus if the heaviness of the mobile is 8, but the heaviness of a size of one medium, equal to that of the mobile, is 6, then the swiftness of this body will be 2; if the heaviness of an amount of the other medium, equal to the size of the mobile, is 4, then the swiftness of the mobile, in this medium, will be 4. It is therefore evident that these swiftnesses will be to one another as 2 and 4; and not as the thicknesses or the heavinesses of the media, which is what Aristotle wanted, which are to one another as 6 and 4. {1} Similarly the answer to the other question is evident : namely, what ratio the speeds of mobiles equal in size, but unequal in heaviness, observe with one another in the same medium. For the speeds of such mobiles will be to one another as the excesses by which the heavinesses of the mobiles exceed the heaviness of the medium: thus, for example, if two mobiles are equal in size, but unequal in heaviness, the heaviness of one being 8, and of the other 6, but the heaviness of an amount of the medium, equal in size to the size of one of the two mobiles, is 4, then the swiftness of the former mobile will be 4, and that of the latter will be 2. Hence these speeds will observe the ratio of 4 to 2; and not that which is between the heavinesses, namely 8 to 6. {1}And from all the things that have been conveyed here, it will not be difficult to apprehend also the ratio that will be observed by mobiles of different species in different media. For one should scrutinize what ratio the two observe, in swiftness, in the same medium; how this is to be done is evident from the preceding: {1} next, one should investigate what swiftness the other has in the other medium, also by means of what has been conveyed above: and we will have what is sought. Thus, for example, if there are two mobiles, equal in size, but different in heaviness, and the heaviness of one is 12, and of the other 8, and we seek the ratio between the swiftness of the one whose heaviness is 12, going down in water, and the swiftness of the one whose heaviness is 8, going down in air; let us see, first, how much faster 12 goes down in water than 8, next how much more swiftly 8 is carried in air than in water: and we will have what we are aiming at; or, alternatively, let us see how much more swiftly 12 goes down in air than 8, then how much more slowly the 12 is carried in water than in air.
These, then, are the general rules of the ratios of the motions of mobiles, whether of the same species or not, whether in the same medium or in different media, whether moved upward or downward. But it must be noted that a very great difficulty arises here: it will be found that these ratios are not observed by one who has made a test. For if one takes two different mobiles, which have such properties that one is carried twice as swiftly as the other, and then releases them from the top of a tower, it will certainly not hit the ground faster, twice as swiftly: what is more, if one makes the observation, the one which is lighter at the beginning of the motion will precede the heavier and will be faster. This is not the place to inquire into how these differences and, so to speak, prodigies come about (for they are accidental): for it is necessary first to examine certain things which have not yet been inspected. For it is necessary, first, to see why natural motion is slower at the beginning.
Chapter 9 [274.1-276.23]
274 In which all the things that have been demonstrated above {1} are considered from a physical point of view, and natural mobiles are reduced to the weights of a balance.
When someone has obtained the truth about a certain matter, and has acquired it with very great labour, then, when he inspects his discoveries more carefully, he often recognizes how the things that he has found with great effort could have been grasped very easily. For truth has the property that it does not lie hidden to the extent that many people have believed; but its traces shine brightly in different places, and many are the paths by which one approaches it: yet it often happens that we do not notice things that are nearer and more clear. And we have a manifest example of this at hand: for all the things that have been demonstrated and made clear above in a rather laboroius way are exposed to us so openly and manifestly by nature that nothing could be clearer or more open.
That this may be apparent to anyone, let us consider, first, how and why things that are carried upward, are carried with a force as great as as the amount by which an amount of the medium, through which they are carried, that is as great in size as the size of the mobile, is heavier than the mobile itself. Thus let us consider a piece of wood which goes up in water and floats on water: now it is manifest that the piece of wood is carried upward with as much force, as is necessary to submerge it by force under water. Hence if we find how great a force is necessary to hold the wood under water, we will have what is sought: but if the wood were not lighter than water, that is, if it were as heavy as an amount of water equal to its own size, then surely it would be submerged, and would not be raised above the water: therefore, a force that is as great as the heaviness by which the heaviness of the wood is surpassed by the heaviness of the said amount of water, suffices to submerge the wood. Therefore, the amount of heaviness required to submerge the piece of wood has been found: but it has just been determined that the piece of wood is carried upward with as much force as is required to submerge it; now, what is required to submerge it is the heaviness just found: hence the piece of wood is carried upward with a force as great as the heaviness by which an amount of water, as great in size as the size of the wood, exceeds the heaviness of the wood: which is what was sought.
One must reason about downward motion in the same manner. Thus we ask with what force a lead sphere is carried downward in water. Now it is evident, firstly,
275 that the lead sphere is carried downward with as much force, as would be required to pull it upward: but if the lead sphere were of water, no force would be necessary to pull it upward, or certainly the least of forces: thus as much heaviness resists, that the sphere might not be pulled upward, as that by which a lead sphere exceeds a sphere of water equal to it. But the lead sphere is carried downward with the same force, as that with which it resists being pulled upward: hence the lead sphere is carried downward with a force as great as the heaviness by which it exceeds the heaviness of a sphere of water. Now it is possible to observe the same thing in the weights of a balance. Indeed if there are some equally ponderous weights, and something heavy is put on one of them, in this case it will be carried downward; but not according to its whole heaviness, but only according to the heaviness by which it exceeds the other weight: it is the same thing as if we said, this weight is carried downward with a force as great as the amount by which the other weight is lighter than it. And for the same reason the other weight will be carried upward with as much force, as the amount by which the other weight is heavier than it.
From the things that have been conveyed in this and in the preceding chapter, the general conclusion can be drawn that mobiles of different species observe in the swiftnesses of their motions the ratio that the heavinesses of these same mobiles have to one another, provided they are equal in size, and this not purely and simply, but weighed in the media in which motion is to take place.
Like, for example, let there be two mobiles, a, b, equal in size, unequal in heaviness; and let the heaviness of a in air be 8, but let the heaviness of b in air be 6: the swiftnesses of these mobiles in water will not observe, as has already been said, the ratio which is 8 to 6. For if we take an amount of water c, which is equal to the size of one or the other mobile; let its heaviness be 4; therefore the swiftness of mobile a will be as 4, but the swiftness of b will be as 2; these swiftnesses will be to one another in a ratio of 2/1, and not in one of 4/3, as are the heavinesses of the mobiles weighed in air. However the heavinesses of these same mobiles in water also will be in a ratio of 2/1; for the heaviness of a in water is only of 4. Which is evident as follows. If the heaviness of a were as 4, in water it would be zero. For then a would be equally as heavy as water, since it has been assumed that the heaviness in air of an amount of water as great in size as the size of a, namely c, is 4; now the heaviness of c in water would be zero;
276 for it would not be carried either upward or downward: hence the heaviness of a in water also would be zero, if in air it were as 4. But since in air it is 8, in water it will be 4: and, for the same reason, the heaviness of b in water would be 2: for which reason their heavinesses would be in a ratio of 2/1, just as the swiftnesses of the motions. And one must proceed by similar reasoning concerning the light.And the conclusion is drawn that, given the heavinesses of the two weights in air, immediately their heavinesses in water can be known: for once the heaviness of an amount of water as great in size as the sizes of the weights has been subtracted from each of them, their heavinesses in water will remain. And similarly with other media. And, from what has been said, it can be manifest to anyone, that we have of no thing its own proper heaviness {1}: for if, for example, two weights are weighed in water, who will say that the heavinesses that we will see then are the true heavinesses of these weights, whose heavinesses, when the weights are weighed in air, will show themselves to be different from these, and will observe with one another another ratio? If again they could be weighed in another medium, for example, fire, the heavinesses would again be different, having to one another another ratio: and in this way they will always vary, according to the difference in the media.And if they could be weighed in the void, in this case surely, where no heaviness of the medium would diminish the heaviness of the weights, we would perceive their exact heavinesses. But since the Peripatetics, with their leader, have said, that in the void no motions can come about, and consequently that all things weigh equally, perhaps it will not be inappropriate to examine this opinion, and to consider with care its foundations and its demonstrations: for this question is one of those that have to do with motion.
Chapter 10 [276.24-284.29]Where, against Aristotle, it is demonstrated that, if there were a void, motion [in it] would not happen in an instant, but in time. {1}
Aristotle, in Book IV of the Physics, trying to abolish the void, makes numerous arguments; those that are found beginning with text #64 [214b 10 et sqq.] are drawn from motion. For since he assumes that motion cannot come about in an instant, he tries to demonstrate that, if a void were given, motion in it would happen in an instant: and since assuredly this is impossible, he necessarily concludes that it is impossible for void to exist. Now, since we are dealing with motion, we have decided to inquire whether it is true
277 that, if a void were given, motion in it would take place in an instant: and since we are just about to determine that, in a void motion takes place in time, we will examine first the contrary opinion and its arguments.
Now, first of all, of the arguments alleged by Aristotle, there is surely none that has necessity; there is however one which, at first sight, seems to have necessity: and this is the one which is written in texts #71 and 72 [215a 24 (-215b 22) - 216a 7], where he deduces the following unacceptable consequence, namely that, if motion in the void took place in time, the same mobile will be moved in a plenum and in a void in the same time. In order that we may be better able to refute this argument, we have decided now to bring it forward for everyone to see. Thus, first, when he saw that the same mobile is carried more swiftly through more subtle media than through thicker ones, Aristotle assumed this: the ratio of the speed of the motion in one medium to the speed of the other motion in the other medium is the same as the ratio of the subtlety of one medium to the sublety of the other.
He then argued as follows: Let mobile a cross medium b in time c; but let it cross a more subtle medium, namely d, in time e: it is manifest that, as the thickness of b is to the thickness of d, thus time c is to time e.Then let f be a void; and let mobile a, if that can happen, cross this f, not in an instant, but in time g; and let the thickness of medium d be to the thickness of another medium as time e is to time g.Now, from the things that have been established, mobile a will be moved through this medium that has just been found in time g, since medium d has the same ratio to this medium that has just been found as time e to time g; but, in this same time g, a is also moved through void f: hence a, in the same time, will be moved through two equal distances, one of which is a plenum, but the other a void; which assuredly is impossible. Therefore the mobile will not be moved through the void in time; therefore [the motion will take place] in an instant.
This is Aristotle's demonstration: to be sure it would have concluded very much to the point and from necessity, if Aristotle had demonstrated the things he took for granted, or, if they had not been demonstrated, if they had at least been true ; but he has been deceived {1} in this, that these things, which he took for granted as well known axioms things which are not only not
278 manifest to the senses, but have never been demonstrated , and are further not demonstrable, because they are totally false. {2}For he has assumed that the motions of the same mobile in different media observe the same ratio to one another, in swiftness, as that which the subtilties of the media have: that this is assuredly false, has been abundantly demonstrated above. {1}In confirmation of which I shall add this one thing: if the subtlety of air has the same ratio to the subtlety of water as the swiftness of a mobile in air to its swiftness in water, then when a drop or any other part of water goes down fast in air, but in water is not moved downward at all, since the swiftness in air has no ratio to the swiftness in water, then, according to Aristotle himself, the subtlety in air will not observe any ratio to the subtlety of water: which is ridiculous. {1}Thus it is evident that, when one argues this way, Aristotle should be answered in the following manner: for in the first place it is false, as has been shown above {1}, that the difference of slowness and speed of the same mobile comes from the greater or lesser thickness and subtlety of the medium; and even if that were conceded, again it is false that a mobile in its motions observes the ratio that the subtleties of the media have.
And as for what Aristotle has written in the same section {1}, that it is impossible for a number to have the same ratio to a number than a number to nothing, this surely is true for a geometric ratio, and not only for numbers but in every quantity. In the case of geometric ratios, since it is necessary that a lesser quantity could be augmented so many times that it exceeds any quantity whatever, the said quantity must be something and not nothing; for nothing augmented by itself again and again will nevertheless exceed no quantity. However, this is not necessary in the case of arithmetic ratios: for in the case of these, a number can have to a number the ratio that a number has to nothing. For since numbers are in the same arithmetic ratio when the excesses of the larger ones over the smaller ones are equal, it will be perfectly possible for a number to have the same ratio to a number as another number has to nothing: as if we were saying, 20 to 12 is as 8 to 0;
279 for the excess of 20 over 12, which is 8, is the same as the excess of 8 over 0. That is why, if, as Aristotle wanted, the motions had to one another the ratio, geometrically, of subtlety to subtlety, he would correctly have concluded that in the void motion could not happen in time; for the time in the plenum to the time in the void cannot have the ratio of the subtlety of the plenum to the subtlety of the void, since the subtlety of the void is null: but if swiftness observed to swiftness the said ratio not geometrically but arithmetically, nothing absurd would follow. But it is certainly true that swiftness observes to swiftness, arithmetically, the ratio of the lightness of the medium to the lightness of the medium; since since the swiftness in relation to the swiftness is, not as the lightness of the medium to the lightness of the medium, but, as has been demonstrated {1}, as the excess of the heaviness of the mobile over the heaviness of this medium to the excess of the heaviness of the same mobile over the heaviness of the other medium.
In order that this be more apparent, here is an example.
Let there be mobile a, whose heaviness is 20; let there be also two media, bc, de, unequal in heaviness; and let the size of b be equal to the size of a, and the size of d be similarly equal to the size of a; and, since we are now talking about downward motion taking place in the void, let the media be lighter than the mobile, and let the heaviness of b be 12, but the heaviness of d be 6: it is therefore manifest, from what has been demonstrated above, that the swiftness of mobile a in medium bc to the swiftness of the same mobile in medium de will be as the excess of the heaviness of this a over the heaviness of b to the excess of the heaviness of this a over the heaviness of d, that is as 8 to 14. Hence let the swiftness of a in medium bc be as 8, but let the swiftness of the same a in medium de be 14: {1}it is then apparent that swiftness 14 to swiftness 8 does not observe geometrically the ratio of the lightnesses of the media. For the lightness of medium de is double that of the lightness of medium bc (for since the heaviness of b is 12, but the heaviness of d is 6, that is since the heaviness of b is double the heaviness of d, the lightness of d will be
280 double the lightness of b); however the swiftness 14 is less than double the swiftness 8.But swiftness 14 certainly has the same ratio, arithmetically, to swiftness 8 as the lightness of d to the lightness of b; since the excess of 14 over 8 is 6, and 6 is also the excess of the lightness of d, 12, over the lightness of b, 6. But if medium de should be lighter, in such a way that the heaviness of this d is 5, then the swiftness f will be 15 (for the excess of the heaviness of mobile a over the heaviness of medium d will be 15); and, again, swiftness 15 will be to swiftness 8 in the same ratio as the heaviness of medium b, 12, to the heaviness of medium d, 5, that is, the lightness of d to the lightness of b: for the excess in both cases will be 7. If, on the other hand, the heaviness of d is only 4, the swiftness f will be 16; and swiftness 16 will be to swiftness 8 (whose excess is 8) similarly in the same arithmetic ratio as the heaviness of b, 12, to the heaviness of d, 4, that is, the lightness of d to the lightness of b, whose excess is similarly 8. Now if, again, medium de is lighter and the heaviness of d is only 3, then the swiftness f will be 17; and the swiftness f, 17, will be to swiftness 8 (whose excess is 9) in the same arithmetic ratio as the heaviness of b, 12, to the heaviness of d, 3, that is, the lightness of d to the lightness of b. And if, again, medium de is lighter and the heaviness of d is only 2, then the swiftness f will be 18; and its arithmetic ratio to the swiftness 8 will be the same as that of the heaviness of b, 12, to the heaviness of d, 2, that is, of the lightness of d to the lightness of b: for the excess in both cases will be 10. If, again, medium de is lighter and the heaviness of d is only 1, then the swiftness f will be 19; it will have the same arithmetic ratio to swiftness 8, as the heaviness of b, 12, has to the heaviness of d, 1, that is, the lightness of d to the lightness of b: for the excess in both cases will be 11. And if, finally, the heaviness of d is 0, so that the excess of the heaviness of mobile a over medium d is 20, the swiftness f will be 20; and swiftness f, 20, will be to swiftness 8 in the same ratio, arithmetically, as that of the heaviness of b, 12, over the heaviness of d, 0: for in both cases the excess will be 12.
It is therefore evident how it is not geometrically but arithmetically that swiftness observes to swiftness the ratio that the lightness of the medium has to the lightness of the medium: and since it is not absurd, in an arithmetic ratio, that a quantity be to a quantity as a quantity to nothing, it will
281 undoubtably similarly not be absurd, that swiftness to swiftness can be arithmetically just as rareness to nothing. Thus, in a void also a mobile will be moved in the same way as in a plenum. For in a plenum a mobile is moved swiftly according to the excess of its own heaviness over the heaviness of the medium through which it is moved; and thus in a void it will be moved according to the excess of its heaviness over the heaviness of the void: since this is null, the excess of the heaviness of the mobile over the heaviness of the void will be the total heaviness of this same mobile; thus it will be moved swiftly according to its own total heaviness. But in no plenum will it be able to be moved as swiftly, since the excess of the heaviness of the mobile over the heaviness of the medium is less than the total heaviness of the mobile: thus the swiftness will also be less, than if it were moved according to its own total heaviness.
From this it can manifestly be concluded, how in a plenum, as is the case around us, in no way do things weigh according to their proper and natural heaviness; but always they will be lighter to the extent that they are in a heavier medium, and they will be lighter by just as much as the heaviness, in the void, of a size of such a medium equal to the size of that thing: so that a lead sphere in water will be lighter than in a void by just as much as the heaviness of a sphere of water, equal to the lead sphere, in a void; and likewise a lead sphere in air is lighter than in a void by just as much as the heaviness of an airy sphere, equal in size to the lead sphere, in the void; and likewise in fire, and in other media. And since the swiftness of motion follows from the heaviness that the mobile has in the medium in which it is moved, its motion will be swifter, the heavier the mobile is in accordance with the variety of the media. However, the following argument is invalid: The void is a medium infinitely lighter than any plenum; hence in it motion will happen infinitely more swiftly than in a plenum; therefore it will happen in an instant. For it is true that the void is infinitely lighter than any medium whatever; one must not, however, say that such a medium is of infinite heaviness; instead what must be understood is that between the lightness of, say, air and the void there can exist an infinity of media, lighter than air, but heavier than a void. And if the matter is understood this way, between swiftness in air and swiftness in the void there can also exist an infinity of swiftnesses, greater than that which is found in air, but smaller than the swiftness in the void: so also between the heaviness of a mobile in air and its heaviness in the void there can exist an infinity of intermediate heavinesses, greater to be sure than the heaviness in air, but smaller
282 than the heaviness in [any] medium. {1}
And this happens in every continuum: as between lines a, b, of which a is greater, there can exist an infinity of intermediate lines, smaller than a, but greater than b (since indeed the excess by which a surpasses b is a line, it will be infinitely divisible): on must however not say that line a infinitely exceeds line b, in such a way , even if b were infinitely augmented, that in the end it would not make up a greater line than a. And thus, by similar reasoning, if we understand that a is the swiftness in a void, but b the swiftness in air, assuredly between a and b there will be able to exist an infinity of swiftnesses, greater than b and smaller than a: one must however not conclude that a exceeds infinitely this b, in such a way that the time during which swiftness a takes place, having been augmented by itself as many times as one wants, can still never exceed the time of swiftness b, and, consequently, that the swiftness of time a is instantaneous. It is therefore evident how this must be understood: The lightness of a void exceeds infinitely the lightness of a medium, hence the swiftness in a void will infinitely exceed the swiftness in a plenum. All that is conceded. That, therefore, the swiftness in the void will be in an instant, is denied.For it can exist in time, but assuredly one briefer than the time of swiftness in a plenum; so that between the time in the plenum and the time in the void there can stand an infinity of times, greater than the latter, but smaller than the former: and thus it is not necessary that motion in the void takes place in an instant, but in a time smaller than is the time of motion in any plenum whatsoever.
Thus, to put it briefly, my whole intent is the following: if there is a heavy thing a, whose proper and natural heaviness is 1000, in any plenum whatever its heaviness will be less than a thousand, and, consequently, the swiftness of its motion in any plenum whatever will be less than a thousand. And if we take a medium, such that the heaviness of a size of it equal to the size of a is only 1, the heaviness of a in this medium will be 999; thus its swiftness also will be 999: and the swiftness of this same a will only be a thousand in the medium where its heaviness is one thousand; and that will be nowhere but in the void.
This is the solution of Aristotle's argument: from this it can be understood quite readily, how in the void it is in no way required that motion be instantaneous. The other arguments of Aristotle are of no soundness and have no necessity. Now to say {1}, for example, that in the void a mobile will no more be moved towards here than towards there, or upward than downward, because the void does not give way more upward or downward but equally on all sides, is childish
283 : for I will say the same thing concerning air; for when a stone is in the air, how does it give way more downward than upward, or to the left rather than to the right, if the rariness of the air is everywhere the same?Here perhaps someone might say, following Aristotle {1}, that air exerts weight in its own region, and because of that it helps downward motion more: but we will examine these chimaeras in the next chapter, where we will inquire whether the elements exert weight in their proper place. Similarly also when they say {1}, In the void there is no upward nor downward; who has dreamt this up? Is it not the case that, if the air were a void, the void in the vicinity of earth would be nearer to the center than the void which would be in the vicinity of fire? Further, as to the argument Aristotle makes concerning projectiles, saying: Projectiles cannot be moved in the void, for projectiles, when they are beyond the hand of the mover, are moved by the air or by another corporeal medium, which surrounds them and is moved, which is surely lacking in the void; this is similarly of no importance whatsoever: for he assumes that projectiles are borne by the medium; that this is false, we will demonstrate in its proper place. {1} Similarly false is what he adds {1} to the argument, concerning different mobiles in the same medium: for he assumes that in a plenum heavier things are carried faster, because they cleave the medium more strongly, and that this is the only cause of the swiftness; since this resistance does not exist in a void , he infers that all motions in the void will be in the same time and with the same swiftness {2}: and this he asserts to be impossible.And yet, in the first place, Aristotle sins in this, that he does not show how it is absurd, that in the void different mobiles are moved with the same swiftness: but he sins even more when he assumes that the swiftnesses of motions of different mobiles depend on the fact that heavier mobiles divide the medium better. For the swiftness of mobiles is not to be sought from that, as has been demonstrated above, but from the greater excess of the heaviness of mobiles over the heaviness of the medium; for the swiftnesses follow the ratio of such excesses: but the excess of the heaviness of different mobiles over the heaviness of the same medium is not the same (for the mobiles would be equally heavy): thus neither will the swiftnesses be equal. Thus in the case of a mobile whose heaviness is 8, the excess over the heaviness of the void, which is null, is 8; thus its swiftness will be 8: but the excess of a mobile whose heaviness is 4 over the void will similarly be 4; and thus its swiftness is 4. Finally, by using in the case of the void the same demonstration that we have put forward in
284 the case of the plenum, we will demonstrate that mobiles of the same species, but of different size, are moved with the same swiftness in the void.And that is enough on this subject.
The {1} force of truth is such, that the most learned men, and even Peripatetics, have recognized the fallacy of this way of thinking of Aristotle, although none of them has been able to refute Aristotle's argument in an appropriate way.And certainly none of them has ever been able to knock down the argument which is written in Book IV of the Physics, texts #71 and #72 [215a24 (-215b22)-216a3]: for up to now the fallacy in it has never been observed: and although Scotus, the divine Thomas, Philoponus {1} and some others hold a way of thinking contrary to Aristotle, they have, however, arrived at the truth by belief more than via true demonstration, or by having answered Aristotle. And, as a matter of fact, there is not one of them who could hope to be able to answer Aristotle and knock down his demonstration, if he concedes the ratio which is assumed by him between the speeds of the same mobile in different media. For he assumes that the speed in one medium is so related to the speed in another, as the subtlety of one medium to the subtlety in the other: this no one up to now has dared to deny. {1} Nor is there any soundness in what is assumed by the authors mentioned above, namely a double resistance of the mobile to motion -- one, extrinsic, coming from the thickness of the medium; the other, intrinsic, by reason of the determinate heaviness of the mobile. For this is somehow a fictitious thing: for if we consider the thing accurately, these two resistances do not differ from one another. For, as has been made clear above, the thicknesses or ( to better express myself) the heaviness of the medium makes the lightness of the mobile, and the lightness of the medium provides the heaviness of the mobile; and the same mobile is now heavier now lighter, according as it is in a lighter or a heavier medium. Consequently they add nothing new by assuming this double resistance; since it is only increased and diminished according to the diminution or the increase of the heaviness or the thickness of the medium. On the other hand, if they concede that it is increased and diminished in the ratio in which the heavinesses of the medium vary, it is in vain that they will try to knock down Aristotle's argument.
285 Chapter [11] In which Aristotle's error, in saying, that air exerts weight in its own place, is made manifest.
The method which we shall observe in this treatise will be that the things that must be said always depend on those that have been said; and that (as much as this will be possible) I never presuppose as true those that must be made clear. As a matter of fact my masters in mathematics have taught me this method: but it is not sufficently observed by certain philosophers {1}, who quite often, in teaching the elements of physics, presuppose things that have been reported either in the books De Anima, or in the books De Caelo, and even in the Metaphysics; and not only that, but even, in teaching logic itself, they constantly mouth words that have been reported in the last books of Aristotle; so that, while they teach pupils the first rudiments, they presuppose that these pupils know everything, and they hand down {2} their teaching not from things better known, but from things purely and simply unknown and unheard of. Now what happens to those who learn this way is that they never know anything by its causes, but they only believe as by faith, that is because Aristotle has said so. There are only a few who inquire whether what Aristotle said is true: for it suffices for them that they will have the reputation of being more learned, the more passages of Aristotle they have at hand. But, leaving this aside, returning to our subject, it must be considered whether air and water really have weight in their proper places: for this question can be explained presupposing only the things that have been reported
Aristotle has written in Book IV of the De Caelo, text #30 [311b8-10] {1} that not only is water heavy in its own place, but air also: saying that, except for fire, everything has weight in its own region, even air itself.Now concerning air he immediatly verifies this with the aid of a sign; saying, since a bladder pulls by force more when inflated than when it is not, this is thus a sign that the air in the bladder has weight.He repeats the same thing in text #39 of the same Book [312b2-19], saying that all things that have heaviness and lightness, have heaviness in their own region: for he assumes that air and water in relation to the other elements are sometimes heavy yet sometimes light, but they exert weight absolutely only in their proper region.Now some more recent philosophers {1}, having noted what Aristotle has said in Book III of the De Caelo, text #28 [301b22-26] {2}, namely that air helps both motions, that is, to the extent it is light, it helps motion
286 upward, but to the extent it is heavy, it helps motion downward, have given form to another argument, saying: Air, because it carries heavy things downward more easily, helps downward motion more than upward motion, because it carries light things upward with more difficulty.They have concluded that air must necessarily be deemed heavy in its own region. However, it will soon be known that this is entirely false: and we will demonstrate that air and water in their own region are neither heavy nor light; {1} we will subsequently demonstrate that the argument of the more recent philosophers leads to a conclusion purely and simply the opposite of that which they laboriously try to prove, and that they could not have found an argument which is more at variance with themselves.
And, to begin with, it seems entirely inconceivable how air and water exert weight in their proper place. For any portion of water in the place of air, that is in air itself, exerts weight, and indeed is carried downward because it exerts weight; but who will ever conceive that a part of water goes down in water? For if it goes down, when it will be at the bottom, it is necessary that the place, into which it enters, be then evacuated by the other water, which will be forced to go up to where the other has receded from; and thus this portion of water will then be light in its own place. Secondly, if any portion of water is heavy in water, let it be called, for example, a: hence since the portion of water a in water is heavy and goes down, if we then take another portion of water which is equal in size to this same a, a will necessarily be heavier than this portion of water, and thus water will be heavier than water: what more foolish thing could be imagined? And to Aristotle's example concerning the bladder, I answer that, if the opening of the inflated bladder or bag is opened, in such a way that air is retained in the bag, not compressed by force, then the bladder will not be heavier than when not inflated: but if much air is compressed by force in it, who will doubt that it will exert weight?For the air, condensed by force, is heavier than free, roaming air: just as if a bladder is filled with wool, but then another equal quantity of wool is added, compressing it by force, who will be so irresolute as to doubt whether or not the bladder will become heavier?
By similar reasoning, if, for example, we consider a portion of air in which is a, and that another portion of air, in which is b, is twice as large as a, in this case air b in the place of fire, for instance, will be twice as heavy as air a: if therefore {1} air b is contracted by force
287, in such a way that its size becomes equal to the size of a, air b will then somehow be another species of air heavier than air a is; what then is so astonishing if air b goes down in the air of which a is a portion? Thus the reason why the inflated bladder pulls by force more is evident: for the air which is in it is heavier than the surrounding air, to the extent that it contains more of the same matter in a smaller place. It is thus manifest that the argument concerning the bladder has no soundness; since, wanting to show that air that is free and rare, as is its nature, is heavy, he then takes in his example air condensed by force and compressed in a narrow place.
As for the argument of those who then say that air is heavy since it carries more easily heavy things downward than light ones upward, I answer that this form of arguing goes diametrically against those who argue it. For if that medium is to be thought heavy because it carries more easily heavy things downward, air will then be heavier than water: for things that are carried downward, go down more easily and more swiftly in air than in water. Add this: it has been demonstrated above {1} that heavy things which are carried downward in water, go down with as much force, as that by which their heaviness exceeds the heaviness of a size of water equal to their size.
If consequently there were a certain heavy body, as, for instance, a body a, whose heaviness is 8, and the heaviness of water b, whose size is equal to the size of a, is 4, then solid a in water will be swiftly and easily carried downward as 4; but if subsequently the same body is carried through a medium lighter than medium b, so that a size of such a medium as great as the size of this same b would have a heaviness of only 3, then a in such a medium would be moved swiftly and easily as 5. Therefore it is evident that this same body a is moved more easily downward through lighter media than through heavier ones: hence it follows necessarily that a medium is to be thought lighter, the more easily heavy things are moved downward in it; of which they were asserting the contrary.To whom, then, is it not now perfectly clear that, if air were still lighter, heavy things would be moved downward more easily? And if this is so, it follows that air is light for this reason, because heavy things are easily carried downward in it. By reasoning in the opposite manner about light things, we will conclude that that medium is to be thought heavy, through which light things are more easily carried upward; but that medium is to be thought light, through which light things rise with difficulty. Therefore, both because in air light things are moved upward with more difficulty, and because in it heavy things are moved downward more easily, it follows that air
288 is to be thought more light than heavy. But this is the only thing I would conclude by their manner of arguing; if it is a good manner, let them see for themselves what is concluded: as for myself however I would say that the elements in their own places are neither heavy nor light.For if a portion of water in water were heavy, it would go down; which it does not: and if it were heavy, how, swimming in the deep, would we not feel the heaviness of such a vast size of water? To this they would answer: because the parts of water adhere to the parts below, as the bricks of the wall lean on the bricks below; whence, they say, it happens that the mouse which lives in a wall does not feel the weight of the stones. {1}As a matter of fact this comparison does not seem quite appropriate. For first they compare fluid and falling water to a solid and fixed wall: then, there is a sign that the bricks do not sit on the mouse's shoulder in that, when the mouse is taken away, the hole where the mouse was remains, and the bricks do not fall in it; but, when a fish or a man is taken out of water, the place where the man was does not remain, but is immediately filled with water, which indicates that water rests on the fish or the men. How, then, is the problem to be untangled, if not by our saying that water and air do not exert weight in their own regions? So that the whole explanation of the problem is as follows: we are said to be weighed down, when a certain weight which tends downward by its heaviness rests on us, and we need to resist by our force in order that it does not go down any further; now this resisting is what we call being weighed down.But since it has been demonstrated {1}, that bodies heavier than water, let down into water, go down, and are, indeed, heavy in water, but less heavy than in air; and it has been shown that things lighter than water, impelled by force under water, are raised upward; but that those that are equally as heavy as water are carried neither upward nor downward, but stay where they are placed, provided that they are completely under water; from this it is evident that if when we are under water a certain body heavier than water leans on us, such as a stone, we will indeed be weighed down, but less than if we were in air, since the stone in water is less heavy than in air: but if, staying in water, a body lighter than water is fastened to us, not only would we not be weighed down, but we would even be raised by it; as it is evident when swimming with a gourd, even though otherwise, being in air, we are weighed down by the gourd; this is because the gourd impelled into water is carried upward and lifts, but in air it is carried downward and exerts weight: now if when we are in water a certain body equally heavy with water hangs over us, we will not
289 be weighed down by it nor will we be raised, since such a body will be carried neither upward nor downward. But no body can be found which is more equal to water in heaviness than water itself: it is therefore not astonishing if water in water does not go down and does not exert weight; for we have said that to be weighed down means to resist with our force a body that is inclining downward. And exactly the same reasoning should be used concerning air.
This, in my judgment, whatever others may say, is the true explication of the problem. Therefore, since neither air nor water in their own regions is carried downward or upward, they are not to be called either heavy or light; since heavy things may be defined to be those that are carried downward, and light things those that are carried upward. And when we speak of motion, one must always take account not only of the heaviness or the lightness of the mobile, but also of the heaviness and the lightness of the medium: a heavy thing will not be moved downward, unless it will be heavier than the medium through which it must be carried; and a light thing will not go up, unless it is lighter than the medium through which it is moved. This being so, water will not go down in water, since water is not heavier than water; and since it does not go down, water will not be heavy in water. Now if it is asked, not as Aristotle considered them, but in themselves, purely and simply and absolutely, regardless of anything else, whether the elements are heavy, we answer that, not only water or earth or air, but also fire, and whatever is lighter than fire, have heaviness, and in short all things that have quantity and matter linked with their substance. But since Aristotle says the contrary of this, assuming something purely and simply light which nowhere exerts weight, we judge that such an opinion must be examined: which is what we will develop in the next chapter.
Chapter
12 ... in which it is concluded against Aristotle that the existence of the purely and simply light and of the purely and simply heavy should not be assumed: which even if they were given, will not be earth and fire as he believed.
Those who came before Aristotle have considered the heavy and the light only by comparison with bodies less heavy or less light; and in my opinion, this is perfectly right: now, Aristotle, in Book IV of the De Caelo, endeavours to refute the opinion of the ancients, and to confirm his own which is contrary to it.Now since in this matter we are going to follow the opinion of the ancients, we will examine Aristotle's refutations as well as his confirmations, confirming his refutations, and refuting his confirmations; and we will carry this out after we have set forth Aristotle's opinion.
In the first place, then, Aristotle says, by way of definition, that he calls purely and simply heaviest that which stands under all else and is always carried towards the center; and he calls the lightest, that which rises above all else and is always moved upward, never downward: and he writes that in texts 26[311a16-18] and 31[311b16-18] of De Caelo Book IV[+308a29-31].He then says that the heaviest is earth, and the lightest is fire: and he says that in text 32 [311b19-29] and in other places.So, against those who assume a certain heaviness in fire, he argues thus: If fire has a certain heaviness, hence it will stand under something; now that is not seen; hence [etc.].This argument is not conclusiveFor, in order for something to hang over something else, it is sufficient for it to be less heavy than that over which it must hang; but it is not necessary that it be deprived of all heaviness; just as for wood to float on water it is not required that it necessarily be deprived of all heaviness, but it is sufficient that it be less heavy than water; and thus, by similar reasoning, for fire to hang over air, it is sufficient that it be less heavy than air, and it is not necessary that it be deprived of all heaviness.For this reason it is evident that this argument has no necessity.
He also argues in the following manner: If fire has a certain heaviness, then a lot of fire will be heavier than a little bit; and for this reason a lot of fire will go up in air more slowly than a little bit of fire: and similarly, if earth has a certain lightness, a lot of earth, which will have more lightness, will go down more slowly than a little bit of earth: experience however shows the contrary; for we see that a lot of fire goes up faster than a little bit, just as also a lot of earth goes down faster: this is therefore a sign that in fire there is only lightness; and since in a lot of fire there is more lightness, it goes up faster.This argument also is weak.And, in the first place, it certainly oversteps the bounds.What follows has no value, If fire considered absolutely has heaviness, therefore
291 a lot of fire in air will be heavier than a little bit: for fire in air has no heaviness.But, one should argue this way: Fire, considered absolutely, has heaviness: hence, where fire has heaviness, a lot of fire will have a lot of heaviness; and where fire has lightness, as in air, there a lot of fire will have a lot of lightness, and a little bit, will have little.It is thus certain that there is an error of Aristotle in arguing. Secondly, what he asserts is false, namely, that a lot of fire goes up more swiftly than a little bit, or that a lot earth goes down faster than a little bit; as we have demonstrated above {1}.
Thirdly, he argues: If fire has heaviness, then a lot of fire will be heavier than a little bit of air; which he assumes to be the greatest absurdity, as if we were saying, If earth has a certain lightness, a certain part of earth will be lighter than a certain part of water: which, says he, is false, because we see, that any part of earth goes down in water, and any portion of fire in air is carried upward.Now this assuredly is a weaker argument than all the others: who is so stupid as not to believe that a lot of water is heavier than a little bit of earth, and a lot of air than a little bit of water, and a lot of fire than a little bit of air?And what Aristotle says does not stand in the way: We see earth go down in water.For, when he says this, he is not self-consistent: that is, when we say that water has heaviness, we are not saying that it has heaviness in its own region, where, as has been demonstrated, it has neither heaviness nor lightness; but we are saying that, a lot of water is heavier than a little bit of earth in a place where water also has heaviness, as, for instance, in air.For if this way of arguing were valid, I could also conclude that a little bit of lead is heavier than a very large beam, because lead goes down in water, whereas the beam will not: but what happens is that a little bit of lead is heavier than a beam in a place where the beam has no heaviness; but if we want to talk of the heaviness of the beam, the beam must be assumed to be in a place where it has heaviness.Similarly, when he says, Any particle of water in air goes down, hence as much air as one wants is lighter than a particle of water; this will be true in a place, where air has no heaviness, but water has: but this will not be to talk of absolute heaviness, as we are doing here.For if we assume a lot of air in a place where air also exerts weight, as in fire or in a void, there surely it will be heavier than a little bit of water. Let it not be concluded, Hence it will go down faster: for someone who concluded thus would
292 show that he is ignorant of where the slowness and speed of motion comes from. For this is not valid, A bag filled with tow is heavier in air than a little bit of lead, hence in air it will go down faster: for an idiot would not say this, nor any one who has understood the things that have been said above.Concerning fire one must reason in the following way: for a lot of fire will be heavier than a little bit of air; not however in air, where fire has no heaviness, but in another place where fire also exerts weight, as would be the case in a void or in a medium lighter than fire.
By Hercules! it is disgusting and shameful that words must be spouted to untangle such childish arguments as well as such crass subtleties as those which Aristotle crams into the whole of Book IV of the De caelo against the ancients: they are without strength, without learning, without elegance, without appeal, and anyone who has understood the things that have been said above, will recognize their fallacies.As when he says, We see that earth is below all things, but fire is above everything; Aristotle must have had the eyes of Lynceus{1}, if he has seen whether or not there is in the bowels of the earth something heavier than earth, and whether above fire their is a lighter body. But, without the eyes of Lynceus, someone who is blind will be able to see that there are many things heavier than earth, such as all the metals, on which, when they are liquified, earth floats, as on the one called quick-silver; and not only is earth lighter than quick-silver, it is more than ten times as light. How, then, can the metals take their heaviness from earth, if they are much heavier than earth; and moreover, if they were constituted from earth, water, air, and fire, would they not have to be much lighter than earth alone?It is therefore evident that there exist many things heavier than earth.Therefore when he says: The contrary places are two, the center and the extremity, taking as the extremity the concave sphere of the moon; hence it must be that the things in those places are contraries; which will not be the case, unless earth be assumed to be deprived of all lightness, and fire to be devoid of all heaviness: an argument that has no neccessity whatsoever; and even if it had any, it is in the same way that the concave sphere of water and of air is in opposition to the center, as is the concave sphere of the Moon; and yet, the things that are under the concave sphere of air are not deprived of all heaviness.{1}But what he says concerning the lightness of fire, saying {1} that, if air is removed from underneath, fire will not go down, like air when water is removed from underneath, is in need of demonstration: because Aristotle has not proved it, unless you say what he has said {2}, Just as earth does not go up in the small cupping glasses of physicians because it is very heavy, so fire will not go down because it is very light.But the ratio has no worth: for, it is not because
293 earth is very heavy, that it does not go up, but because it is not fluid; for neither would wood go up, although it is lighter than water, which does go up; but mercury would go up, although it is heavier than earth, because it is fluid; and thus fire would go down, because it is held to be not solid but flowing.{1}But, I will be grateful to know, if the elements, as he himself wants it, are transmuted into one another, when fire is made from heavy air, what happens to that heaviness of the air?Could it be that it is annihilated?But, if it is annihilated, when in its turn earth is made from fire, where does the heaviness come from? Perhaps heaviness, which is something, comes from non-heaviness, which is nothing?Furthermore, if fire is deprived of all heaviness, it will hence be deprived of all density; for the dense follows the heavy: but what is deprived of all density, is a void: hence fire is a void.What could be more absurd?But, finally, how could anyone ever imagine fire, a substance linked to quantity, not having any heaviness?This surely seems entirely unreasonable.And when we say, that earth is the heaviest of all things, because it stands under all things, we are compelled, whether we like it or not, to say that earth is the heaviest, in comparison with other things, because it stands under all other things.For to stand under all things and to be the heaviest of all, is the same thing: and that is evident; for, if it is the heaviest because it stands under all things, if all things are removed, it will no longer be able to be called the heaviest, since it stands under none of them.It is therefore said of it that it is heaviest in comparison with things that are less heavy, under which it stands; and the same must be said concerning the lightness of fire.Hence we conclude that it is not possible for anything to be called the heaviest without relation to other things that are less heavy than it, since the heaviest cannot be defined or mentally conceived except in so far as it stands under less heavy things; and similarly, it is not possible for anything to be called the lightest except in comparison with less light things, over which it rises; nor is the lightest body that which is deprived of all heaviness, for this is a void, not a body, but that which is less heavy than those things that have heaviness.However, I would not say that there cannot be found in the nature of things something that is heavier than all, and something that is lighter than all, that is less heavy; but we only deny these two things: that this could be considered absolutely, without comparison with other things, and that earth and fire are such things.For there are many things heavier than earth
294, and we surely see them: and also certain things lighter than fire could exist, like certain vapors, which rise above fire; but we cannot confidently affirm this because we have not been above fire.But even if it is fire, it is however not deprived of all heaviness; for that belongs to the void: that is why fire also, if air is removed from under it, will go down, if void or any other medium less heavy than fire is left under it.For all things go down, provided that they are heavier than the medium through which they must be carried, as has been shown above {1}; and there is nothing against motion taking place in the void, as was also made clear {2}.But in fact fire does not go down because the air, through which it would have to be carried, is heavier than fire itself, and not because fire has no heaviness: just as air does not go down, because it would have to be carried through water, which, since it is heavier than air, does not permit this; and it must not be said that air is deprived of all heaviness because it does not go down.
Chapter [13] In {1} which it is demonstrated against Aristotle and Themistius, that it is only in the void that the differences of heavinesses and motions can be discerned with exactness.
Themistius, following Aristotle's opinion, in discussing the void, on text #74 of Book IV of the Physics [216a9-21] has written this: Since, then, the void yields uniformly, but as a matter of fact it yields in no way (for since it is nothing, the man who can imagine the void yielding is clever indeed), it so happens that the differences between heavy things and light things, that is the variations of things, are suppressed, and, in consequence, the speed of all things that are moved comes to be equal and indiscriminate. Now how false these words are will soon be known, when we have made clear how it is only in the void that the true differences between heavinesses and motions can be given, and how in a plenum these can in no way be found.
And, to begin with, just as among philosophers different opinions on the same subject indicate with sure evidence that none of them has discovered the truth (for if it had once been found by someone, immediately and without controversy, being what it is by its nature, it would have allowed itself to be seen and known by all), so, in the same way, the different ratios of the heavinesses of the same bodies in different media prove with a strong argument, that the true natural weights are not determined by any medium. For the heavier a medium is, the greater is the difference between the heavinesses of solids.In order that this may be understood still more easily, let those things
295 which have been demonstrated above be called back into memory.
It has been demonstrated {1}, for example, that a certain solid weighs as much less in water than in air, as the heaviness in air of a size of water equal to the size of the solid: so that if there are two solids a, b, and the heaviness of a in air is 8, while the heaviness of b is 6, and their sizes are equal, and the size of water c, whose heaviness in air is 3, is equal to their size, it is evident, from the things said above, that the heaviness of a in water is 5, but the heaviness of b 3. Consequently in water the difference [in terms of ratios] between the heavinesses of a and b will be greater, as between 5 and 3 the discrepancy is greater than between 8 and 6. But if, on the other hand, there were a certain medium heavier than water, whose heaviness is, for instance, 5, the heaviness of a in it will be 3, but the heaviness of b 1. And thus it is evident how in the heavier media the difference between the heavinesses is always greater; for in air the heaviness of a is 4/3 of the heaviness of b; in water, 5/3, and in another heavier medium, 3/1: but who will say that the true heavinesses of the solids are in this medium rather than that one? No one, surely: but it will certainly be truer to say that in none of them are the exact weights found. For since in every medium the heavinesses of heavy things are diminished, by the amount that a portion of that medium equal in size to the solid would weigh, it is evident that the entire and undiminished heavinesses of the solids will only be found in that medium whose heaviness would be null: but only the void is such. But in other media heavy things weigh and exert weight only to the extent that they are heavier than those media (for if they were equally as heavy as a certain medium, they would exert no weight in such a medium): and since in a void, similarly, solids exert weight only to the extent that their heavinesses surpass the heaviness of the void; they surpass it according to the their own total heaviness, since the heaviness of the void is null; it thus follows, necessarily, that it is only in the void that the true heavinesses of heavy things can be found: that is why the differences of such heavinesses will be present only there.
Similar considerations also hold concerning the speeds of motions and of their ratios. For who will say that they are found in plenum media, if a mobile has one speed in this medium, another in that, still another in another one, and in still another one, null, like wood in water? and similarly, if there is one ratio of speeds in air, another in water, another in a heavier medium, another in a lighter one; as anyone will easily be able to find from these things that have been written above? And finally, since the speeds of the mobiles, in the medium in which they are moved, follow the heavinesses; and, consequently, the ratios of the speeds follow the ratios of the heavinesses; and it happens that these are not given except in the void; it must be asserted beyond any doubt that the true and natural differences of speeds also occur in the void only. {1}
Chapter 1 of Book II [293.5-302.18][Drabkin ch.14]In which the matter in question concerns the ratios of the motions of the same mobile on different inclined planes.
The question that we are about to explain has been treated thoroughly {1} by no philosopher, as far as I know: yet, since it concerns motion, it seems that it must necessarily be examined by those who profess to hand on a treatment concerning motion that is not incomplete. Now this question is no less necessary, than it is elegant and clever. For what is asked is why the same heavy mobile, in descending naturally along planes inclined to the plane of the horizon, is moved more easily and more swiftly on those that will maintain angles nearer a right angle with the horizon; and, furthermore, the ratio of such motions made on diverse inclinations is sought.The answer to this question, when first I had tried to investigate it, seemed to have explanations that were not entirely easy: yet, when I examined the thing more carefully and tried to resolve its demonstration into its principles, I finally discovered that its demonstration, like that of others that at first glance seem very difficult, relied on known and manifest principles of nature. These ideas, as they are necessary for the explanation of it, we will now expound first.
And to begin with, in order that all these things may be better understood, let us make the problem clear by means of an example.
Thus let there be a line ab, directed toward the center of the world, which is perpendicular to a plane parallel to the horizon; and let line bc be in the plane parallel to the horizon; and from point b let there be drawn any number of lines maintaining acute angles with line bc, and let them be lines bd, be. It is then asked why a mobile, in going down, goes down most swiftly on line ab; and on line bd more swiftly than on line be, however more slowly than on ba; and on line be, more slowly
297 than on bd: it is also asked how much faster the mobile goes down on ba than on bd, and how much faster on bd than on be. In order then that we may be able to attain this, what we noted above {1} must first be taken into consideration: namely, that it is manifest that what is heavy is carried downward with as much force, as would be necessary for pulling it upward by force; that is, it is carried downward with as much force as that with which it resists going up.
If then we find with how much less force the heavy thing is pulled up by force on line bd than on line ba, it will then have been found with how much more force the same heavy thing goes down on line ab than on line bd; and, similarly, if we find how much more force is required in order to impel the mobile upwards on line bd than on be, it will then have been discovered with how much more force it will go down on bd than on be. But we will know how much less force is required to pull the mobile up by force on bd than on be, when we will have found out how much greater will be the heaviness of this same mobile on the plane along line bd, than on the plane along line be.Let us proceed then to a careful investigation of this heaviness. And {1} let it be undestood a balance cd, whose center is a, and at point c a weight equal to another weight which is at point d. If then we consider that the line ad, remaining fixed at point a, is moved towards b, the descent of the mobile at the initial point d will be as if on line ef; that is why the descent of the mobile on line ef will be according to the heaviness of the mobile at point d. But then again, when the mobile will be at point s, its descent at the initial point s will be as if on line gh; that is why the motion of the mobile on line gh will be according to the heaviness the mobile has at point sAnd again, when the mobile will be at point r, then its descent at the initial point r wll be as if on line tn; that is why the mobile will be moved on line tn according to the heaviness it has at point r. If,
298 then, we show that the mobile at point s is less heavy than at point d, it will then be manifest that its motion will be slower on line gh than on ef: and if, again, we show that at r the mobile is even less heavy than at point s, it will then be manifest that the motion will be slower on line nt than on gh. Now it is already manifest, that the mobile at point r exerts less weight than at point s; and less in s, than in d. For the weight at point d weighs equally with the weight at point c, since the distances ca and ad are equal: but the weight at point s does not weigh equally with the weight in c. For if a line is drawn from point s perpendicular to cd, the weight at s, as compared with the weight at c, is as if it were suspended from p; but the weight at p exerts less weight than the weight at c, since the distance pa is smaller than the distance ac. And, similarly, a weight at r exerts less weight than a weight at s: which will likewise be evident if a perpendicular is drawn from r on ad, which will intersect this same ad between points a and p. Consequently it is manifest that the mobile will go down with a greater force on line ef than on line gh. and on gh than on nt. But with how much greater force it is moved on ef than on gh will be known thus: if line ad is extended up to outside the circle, it stands to reason that it intersects line gh at point q. And since the mobile goes down more easily on line ef than on line gh, by as much as it is heavier at point d than at point s; and it is heavier at point d than at s, by as much as line da is longer than line ap; hence the mobile will go down more easily on line ef than on gh, by as much as line da is longer than this same pa. Therefore the swiftness along ef will have the same ratio to the swiftness along gh, as line da has to line pa. Now as da is to ap, so is qs to sp, that is the oblique descent to the right descent: it is hence certain that the same weight is pulled by force with a smaller force upward along an inclined ascent than along a right one, by as much as the right ascent is smaller than the oblique; and, consequently, the same heavy thing goes down with a greater force along a right descent than along an inclined one, by as much as the inclined descent is greater than the right one. But it must be understood of this demonstration that there exists no accidental resistance (roughness either of the mobile or of the inclined plane; or because of the shape of the mobile): but it must be presupposed that the plane is somehow incorporeal, or at least very carefully smoothed and hard, so that, while the mobile exerts weight on the plane, it may not cause the plane to bend, and somehow come to rest on it, as in a trap.It is also necessary that the mobile be perfectly smooth, and of a shape which does not resist motion, like
299 a perfectly spherical shape would be, and, of the hardest material, or else fluid like water.If all these things are thus disposed, any mobile on a plane parallel to the horizon will be moved by a minimal force, indeed by a force smaller than any given force. And this, since it seems quite difficult to believe, will be demonstrated by the following demonstration.
And so let there be a circle, whose center is a, and a balance bc that can move about its center a, and which is parallel to the horizon; and let the perpendicular ad be drawn from the center a, directed towards the center of the world; and let us imagine that some weight is suspended from point d. It is then manifest that the weight at d, as it is moved in the direction of c, goes up necessarily. I say, then, that any force applied on point b can move the weight at d, and that it necessarily moves it.For consider a weight, however small, suspended from point b, and as the weight at d is to the weight at b, let line ba be to another line, to which it is assumed that line ae is equal. Now if d is suspended from point e, then it will weigh equally with the weight in b; neither one will be moved by the other, nor will the balance incline. [see my notes below]But the weight at d suspended from a is lighter than if it is suspended from e, since it is weighed not only nearer the center, but suspended from the center itself: it is necessary that the weight at d, suspended from a, is moved by the weight at b, and that the balance inclines on the side of b and that d goes up. Therefore, if by any force any weight at d not only is moved, but even is raised, what wonder is it, that the same weight d should be moved, on a non ascending plane, by the same force or even a smaller one, than the force at b?Furthermore: a mobile, having no extrinsic resistance, will go down naturally on a plane inclined no matter how little below the horizon, with no extrinsic force applied; as is evident in the case of water: and the same mobile does not go up on a plane erected no matter how little above the horizon except violently: it therefore remains that on the plane of the horizon itself it is moved neither naturally nor violently. Now if it is not moved violently, hence it will be able to be moved with the minimum of all possible forces. We can also demonstrate this differently: namely, that any mobile, subject to no extrinsic resistance, can be moved {1} on a plane which is directed neither upward nor downward by a force smaller than any proposed force whatever. To demonstrate that we presuppose this: namely, that any heavy mobile can be
300 moved on a plane parallel to the horizon, by a force smaller than on a plane inclined above the horizon.
Thus let the plane parallel to the horizon be along line ab, to which bc is at right angles; and let the mobile be the sphere e; and let f be any force whatever: I say that the sphere e, having no extrinsic and accidental resistance, can be moved on plane ab by a force smaller than force f. Let n be the force that can pull weight e upward by force; and as the force n is to the force f, thus let line ad be to line dbThen from the things that have been demonstrated above {1}, the sphere e will be able to be pulled by force upward on the plane ad by the force f: hence on plane ab, the sphere e will be moved by a force smaller than f. Which was to be demonstrated. {1}
On the other hand, I am not unaware that someone here may object that I have presupposed as true for these demonstrations what is false: namely, that the weights suspended from the balance maintain right angles with the balance; even though the weights, since they tend to the center, would be convergent. To these people I would answer that I cover myself with the protecting wings of the superhuman Archimedes (whom I never mention without admiration).For he has presupposed the same thing in his Quadrature of the Parabola {1}; and this, perhaps, in order to show that he was so far ahead of the others that he could draw the true from the false: and yet, one must not be in doubt, as if he had concluded falsely, since he had proven the same conclusion previously by another geometric demonstration. That is why, either one must say that the suspended weights really maintain right angles with the balance, or it is of no importance whether or not they maintain right angles, but it suffices that they are equal; which perhaps will be more probable: unless we wish to say rather that this is a case of geometric licence; as when the same Archimedes presupposes that surfaces have heaviness, and that one is heavier than another, even though in fact they are all without heaviness. And the things we have demonstrated, as we have said above {1}, must be understood concerning mobiles free from all
301 extrinsic resistance: but since it is perhaps impossible to find such mobiles in the realm of matter, let no one be astonished, in putting these things to the test {2}, if the experiment is disappointing, and a large sphere, even if it is on an horizontal plane, cannot be moved by a minimal force. For to the causes already mentioned {1}, there is added this: namely, that a plane cannot be really parallel to the horizon. For the surface of the Earth is spherical, to which a plane cannot be parallel: that is why, since a plane touches a sphere in only one point, if we recede from such a point, it is necessary to go up: that is why the sphere, with reason, will not be able to be removed from such a point by any minimal force.
And from the things that have been demonstrated, it will be easy to obtain the solution of certain problems: such as the following. First: given two inclined planes, whose right descent is the same, to find the ratio of the swiftnesses of the same mobile on them.
{1} Cf. noteFor let the right descent be ab, and the plane of the horizon bd, and let ac, ad be the oblique descents: it is now asked, what ratio the swiftness on ca has to the swiftness on ad. And, since the slowness on ad is to the slowness on ab as the line da is to the line ab, as was shown above {1}; and just as line ab is to line ac, so the slowness on ab is to the slowness on ac; it will follow, ex aequali, that just as the slowness on ad is to the slowness on ac, so line da is to line ac: hence as the swiftness on ac is to the swiftness on ad, so line da is to line ac. It is therefore certain that, the swiftnesses of the same mobile on different inclines are to each other inversely as the lengths of the oblique descents, provided that these hold equal right descents. Furthermore, we can find inclined planes on which the same mobile observes a given ratio in its swiftnesses.
For let the given ratio be that which line e has to f; and let da to ac in the previous figure be made as e to f: what was asked will then be resolved. Also other similar problems {1} can be resolved: as, being given two mobiles of different genus, equal in size, to set up a plane inclined in such a way that the mobile which in vertical motion was moved faster than the other goes down on this plane at the same speed as that of the other in its vertical motion. But since these things and others similar can easily be found by those who
302 have understood the things said above, we deliberately omit them: only noting the following, that, just as has been said above concerning vertical motion, so also in the case of these motions on planes it happens that the ratios that we have put forward are not observed, sometimes for the reasons of the causes just now alleged, sometimes -- and this is accidental -- because in the beginning of its motion a lighter mobile goes down more swiftly than a heavier one: why this happens, we will make clear in its proper place {2}; for this question depends on the one in which it is asked why the swiftness of natural motion is augmented {3}.But, as we have often said, these demonstrations presuppose that there are no extrinsic impediments, either from the shape of the mobile, or else from the roughness of the plane or the mobile, or from the motion of the medium in the opposite or in the same direction [as that of the body], or else from an extrinsic moving force quickening or retarding the motion, and other similar things: for concerning these accidents, since they can happen in countless ways, rules cannot be given. One must consider in a similar fashion the case of upward motion.
Let these things suffice concerning motion on inclined planes. Now it remains that we say something in the next chapter concerning circular motion: asking, first, whether or not it has a ratio to rectilinear motion, and whether it is forced or natural motion. {1}
Older Works on Motion, Book II, chapter 2 [15]
302.19-304.7In which it is concluded against Aristotle that rectilinear and circular motions have a ratio to each other
That Aristotle was little versed in geometry appears in a number of places of his philosophical work; but above all in the one where he asserts {1} that circular motion does not have a ratio to rectilinear motion, because, of course, a straight line is not in a ratio nor comparable to a curve: this falsehood (for it is not worthy of the name opinion) demonstrates that Aristotle was ignorant, not only of the profound and more abstruse discoveries of geometry, but even of the most elementary principles of this science. For how could he say that the circular and the rectilinear have no ratio, if for quantities to have a ratio to one another it suffices that the smaller could be augmented so many times that it exceeds the other? Or perhaps the chord of an arc, which
303 is less than the arc, augmented by itself often enough, will not exceed the length of the arc? But if it will exceed it, why is it said by Aristotle that the arc and the chord do not have a ratio?
Yet there have not been lacking those who try to save Aristotle, by saying, Aristotle only wanted the following, namely, that the curved and the straight are not comparable to each other. But these people {1} are by far more ignorant than Aristotle in geometry, since, while they try to show that he himself did not err, they attribute to him an error which is far more serious than that from which they try to cleanse him. And first of all, where in geometry have they found mention made of ratio or lack of ratio of the curved and the straight, when a ratio is not found except where there is greater and lesser, that is where there is quantity? But who has ever proclaimed the curved and the straight to be quantities? But what greater foolishness could Aristotle ever have imagined, than to say that the curved and the straight are without ratio to each other or comparable?For this would be as if one were to say that the triangle and the square are not comparable since the triangle has only three angles, whereas the square has four. But what is the point of this when Aristotle did not want what they themselves want?For he says the following words in Phys., Book VII, text #24 [248b4-6]: If the straight and the curved are comparable, it turns out that a straight line is equal to a circle; but these are not comparable. {1} These are his words. But in order that I convince them so that they will never be able to escape, I will argue in the following way. Surely they will not deny that a plane surface has a ratio to some part of itself: if this is so, I already have achieved my intent. For a circle, inscribed in a square, is a certain part of that square; hence the square has some ratio to the circle: but the square is to the inscribed circle as the perimeter of the square is to the circumference of the circle: that is why the perimeter of the square, which consists of straight lines, has a ratio to the curved circumference of the circle. But why do I go further? Aristotle casually says: "A straight line equal to the circumference of a circle is not given": that this is false is demonstrated by the divine Archimedes in his treatise On Spirals, proposition [n¼ XX], where a straight line is found equal to the circumference of a circle around the spiral of first revolution. {1} And do not say: This has escaped Aristotle's attention, because Archimedes is much more recent than Aristotle.For, if the demonstration of finding a straight line equal to a curved one has escaped Aristotle's attention, the demonstration proving that a straight line equal to a curved one is not given has also escaped his attention
304; that is why he should not have casually asserted that such a straight line is not given But what is more: who is so blind that it could escape his attention that, if there are two equal straight lines, one of which is bent into a curve, that curve will be equal to the straight line? Or, if a circle is traced around a straight line, who will doubt, that in one revolution it passes through a straight line equal to its circumference? Consequently, let us have no more doubts that there exists a rectilinear motion equal to a circular motion and in a relation of any ratio to it.
Older Works on Motion, Book II, chapter 3 (Drabkin16)
304.8-307.23In which it is asked concerning circular motion whether it is natural or forced.
Concerning circular motion, about which we are about to have a certain discussion, we will at the very beginning make the following distinctions. Crcular motion takes place either around the center of the world, or outside. Now let us see if that which takes place about the center of the world is forced or not; as, for example, if a marble sphere were in the center of the world, in such a way that its center would not differ from the center of the world.We will have the solution to this question if it is made clear what is natural motion and what is forced motion.
Now a motion is natural as long as the mobiles, in moving forward, come near their proper places; but it is forced as long as the mobiles, which are moved, recede from their proper place. . Since these things are so, it is manifest that the sphere rotating around the center of the world is moved with a motion that is neither natural nor forced. For since the sphere is heavy, and the place of heavy things is the center, and heavy things are moved in keeping with their center of heaviness; if then the center of heaviness of the sphere were at the center of the world, in which it would remain while the sphere is rotating; it is manifest that it would be moved neither naturally nor forcibly, since it would neither come near to nor recede from its proper place.It must be noted here that, if the sphere were made of totally similar parts, in such a way that the center of heaviness and the center of magnitude were the same, then its center would not differ from the center of the world; but if it were made of dissimilar parts, in such a way that its center of heaviness would differ from the center of magnitude, then the center of heaviness would be the same as the center of the world, but the center of magnitude would be different. But, however this would happen, so long as the center of heaviness was the same as the center of the world, the sphere at the center of the world would rotate neither naturally nor forcibly
305 Indeed each one of these things that are moved, is moved in keeping with its center of heaviness: that if the center of heaviness does not differ from the center of the world, then the sphere, rotating around its center of heaviness, will be moved neither naturally nor forcibly.
But here two things can be asked: first, whether a sphere of heterogeneous parts, whose center of magnitude was at the center of the world, but whose center of heaviness was at a distance from the center of the world, whether, I say, such a sphere would be moved forcibly or not; secondly, if a sphere was at the center of the world, and would rotate neither naturally nor forcibly, it is asked whether, having received a start of motion from an external mover, it would be moved perpetually or not. For if it is not moved contrary to nature, it seems that it should be moved perpetually; but if it is not moved according to nature, it seems that it should finally come to rest.
Returning, then, to the first question, we say that a sphere of heterogeneous parts, whose center of heaviness differs from its center of magnitude, would contrary to nature and by force remain at rest in such a way that its center of magnitude was at the center of the world, but its center of heaviness was at a distance; however it is not contrary to nature and by force that it would be moved.
As, for example, let there be a sphere, whose center of magnitude is a, which does not differ from the center of the world; but let it be heterogeneous, as, for example, if the sphere was of wood, yet at part o there were a piece of lead; also the center of heaviness of such a sphere would be between center a and o, as, say, at c. It is manifest that it will not be retained in such a place except forcibly. For since heavy things seek the center, and are moved towards it in keeping with their center of heaviness, hence the center of heaviness c of the sphere would be moved naturally towards a, the center of the world: that is why it will be kept outside this center only forcibly. However it is not forcibly that it would rotate outside the center of the world: for, in such a circular motion, the center of heaviness would describe a circle around the center of the world, neither coming near to it nor receding from it. {1}This being the case, it will rotate around itself neither naturally nor forcibly: for it would then be moved naturally, as we also have said above, when in its motion it came near the center of the world; but forcibly, if, while it is moved, it receded from this same center. From this is evident the error of those {1} who say, If a star were added to the heavens, the motion of the heavens would either cease or become
306 slower. For neither of these things would happen: for since, even on their own way of thinking, the rotation of the heavens takes place {1} around the center of the world, an added star or any other heavy weight added as well will neither help nor retard the motion; since such a weight, in such a circular motion, neither acquires nor loses nearness to or receding from the center, towards which it would be carried by reason of its heaviness. Hence they who say such things are deceived in the following way: because, in the first place, they decree that the force of the motive intelligence is in such a ratio to the resistance of the heavens, that it could itself move it with the swiftness at which it now does, and not at a greater one: but if, by the addition of some star, the resistance of the heavens is increased, then, they say, the motion of the heavens by the same motive force will be caused to be slower.Now they are led to believe this, in my opinion, because they see around us that someone who moves a large wheel, if a great weight is added on the other side, will then work more, or else the motion will become slower: but they do not note that the cause of such an effect is that the wheel is moved outside the center of the world; so that, when the added weight must be carried from the lowest point on the wheel to the highest, it is then moved contrary to nature, since it tends upward, receding from the center of the world. But if the wheel were rotating around the center of the world, who would ever say that it is hindered by the weight, since the weight in the circular motion does not come near nor does it recede from the center of the world?It is in a similar fashion that one must judge concerning the heavens. For a star will be able to retard the motion only when it will be moved away from the place toward which it would naturally tend: but this never happens in a circular motion taking place around the center of the world, since it is never moved upward and never downward: hence the motion will not be retarded by the addition of a star.
As for the second [of the questions] asked above {1}, this is not the place to answer it; for one must first see by what things that are not moved naturally are moved. {2}
And similarly in the case of the circular motion that takes place outside the center of the world, a distinction must be made concerning the mobile, whether, that is, it is made of parts that are totally similar, or else different. And if the mobile is of totally similar parts, as, for example, a marble sphere, which is moved on an axis, this motion will be neither natural nor forced; since the center of heaviness of the sphere neither comes near nor recedes from the center of the world, and the heaviness of the parts of the sphere that go up is equal to that of the parts that go down, so that the sphere is always in equilibrium. And yet, by accident, such a circular motion is
307 forced; because, of course, there is resistance exerted at the pivots of the axis. For since it happens that the sphere is outside its proper place, it happens also that it exerts weight and is in need of support; that is why the extremities of the axis of the sphere, by exerting weight on the pivots, hinder the motion. But the more the ends of the axis are polished and thin, the less they will suffer resistance: so that, if we imagine them to be indivisibles {1}, then no resistance will develop from them. It also happens that such a motion is retarded by the surface of the sphere, if the surface is rough and unevenly cut: for the air flowing round and retained in the cavities of the surface will hinder the motion, and will not help it, as someone has believed; this will be explained in its place {1}. But if the sphere is heterogeneous, in such a way as to have a center of heaviness outside the center of size [i.e. magnitude], but it rotates around the center of magnitude, in this case, independently of the other accidental causes adduced above, there will be in addition a cause per se why such a motion is not, as the other was, neither natural neither forced, but sometimes natural, sometimes forced. For since the center of heaviness in such a circular motion describes a circle about the center of magnitude, when it goes up from the lowest point towards the highest, it will be moved by force, since it recedes from the center of the world; but when it tends from the highest to the lowest, it will surely be carried by nature. But since it cannot be raised by the impetus {1} it has received as much as it descends by nature (the cause of this will be explained in its own place) {2}, consequently the difficulty in ascent is greater than the propensity in descent: from this it follows, both because of this and because of the other accidental causes, that it smacks more of the nature of the forced than of the natural.
Older Works on Motion, Book II, chapter 4 [17] [307.24-314.25]By what projectiles are moved.
Aristotle, as in almost all he has written concerning local motion, has written the opposite of the truth on this question also: and this is surely not astonishing, for who concludes true things from the false? Aristotle could not defend the view that that the mover must be in contact with the mobile, unless by saying that projectiles are moved by the air.And he has given testimony of this opinon of his in many places, {1}
308 which, because we must refute it, we will bring forth for everyone to see; but we will only touch upon it briefly here (for it is made clear at considerable length by the commentators).
Aristotle, then, wants that the mover, as, for example, one throwing a stone, before he lets go of the stone, sets in motion the contiguous parts of the air, which, he says, similarly move other parts, and these still others, and so on in progression; and that the stone, having been let go by the thrower, is then carried by those parts of the air; and thus that the motion of the stone becomes, somehow, discontinuous, and is not a single motion but several. {1}Aristotle and his followers, who have not been able to get it thouroughly into their minds how a mobile could be moved by an impressed force, or what that force could be, attempted to take refuge in this view. But, in order that the other opinion may be known to be most true, we will first try to knock down to its foundations that of Aristotle; then, as far as is possible, we will make clear and illustrate with examples the other opinion, which concerns the impressed force.
And so, I argue against Aristotle as follows. Let A, B, C, D, E, be the parts of the air which move the mobile; of these, let A be contiguous to the mover. Now either all these parts are moved simultaneously, or one after the other: if A, B, C, D, E, are all moved simultaneously, I ask, by what are they moved once the mover is at rest; and in that case one must come to the notion of an impressed force: if A is moved before B, in like manner I ask, once A is at rest, by what is B moved?Furthermore, according to Aristotle himself {1}, violent motion is faster in the middle than at the beginning: hence part C of the air, impelled by B, is moved faster than B; hence likewise C will impel D faster than A, B, C, D, E themselves will have been impelled by B; hence D will also impel E faster than it has been impelled by C; and so on in succession: hence violent motion will always be increased. Secondly: there is the argument concerning the arrow impelled by the bowstring, which, even against a blowing north wind, still flies very swiftly. To this argument the adversaries have nothing else to reply, except that, however hard the wind may blow, air is nevertheless carried against the wind, having received an impetus from the bow; and it is without shame that they utter such childish things. But what will they say to the following similar argument? When a trireme is impelled by oars against the current of a river, and, the oars having been taken out of the water, the ship is still brought along over a great distance against the course of the water, who is so blind as not to see that the water rushes with a very great impetus in the opposite direction? and that, I say, the water contiguous to the boat
309 is not deflected even an inch from its natural course by the impetus of the vessel?Thirdly: if it is the medium that carries mobiles from one place to another, how does it come about that, when one throws with the same shot of the cannon an iron ball, with which, however, is also carried some wood or tow or something light, but in such a way that the heavy thing comes out first -- how, I say, does it happen that the iron is put in motion along a very long distance, while the tow, after it has come along with the iron for some distance, is stopped and falls to the ground? If then it is the medium which carries them both, why does it carry the lead or the iron very far, but not the tow? Is it easier for the air to move the very heavy iron than the very light tow or the wood? Fourthly: it seems that Aristotle is not fully self-consistent. For, in De Caelo, Book III, ch.2, text #27 [301b1-4] he says: If what is moved is neither heavy nor light, it will be moved by force; and what is moved by force, having no resistance of heaviness or of lightness, is moved without end. Now he says in the next passage that projectiles are carried by the medium: hence since air has neither heaviness nor lightness, once moved by a thrower it will be moved without end, and always at the same speed; hence it will carry projectiles without end, and will never tire itself, since it is always moved with the same force. However experience teaches the opposite of this. Fifthly: consider a marble sphere, perfectly round and smooth, which can be moved on an axis resting on two pivots; then let a mover come near who twists both ends of the axis with his finger tips. Surely in that case the sphere will rotate for a long time: and yet the air has not been put into motion by the mover; nor can the air act upon the sphere, by impelling it, since it never changes place and, since it is smooth, it has no cavities into which the air can rush: indeed, the air around the sphere will stay entirely motionless, as is evident if a flame is brought near to the sphere, as it will neither be extinguished nor moved.
These are the reasons by which we believe to have refuted enough and more than enough that absurd way of thinking, which those who cannot thoroughly get into their minds what an impressed force is try to defend.Now on the other hand, in order that we may explain our way of thinking, let us inquire, first, what is that motive force, which is impressed in the projectile by the thrower. We say, then, that it is
310 a deprivation of heaviness, when the mobile is impelled upward; but when impelled downward, it is a deprivation of lightness. But a person will not marvel at how a thrower can, by directing a heavy thing upward, deprive it of heaviness and render it light, if he does not marvel at how fire can deprive iron of cold, by introducing heat. A mobile, then, is moved upward by the thrower, so long as it is in the latter's hand and is deprived of heaviness; similarly, the iron is moved, in an alterative motion, towards heat, so long as the iron is in the fire and it is deprived by it of cold: the motive force, namely lightness, is conserved in the stone, when what has moved it no longer touches it; the heat is conserved in the iron, when it is removed from the fire: the impressed force is progressively weakened in the projectile when it is apart from the projector; the heat is diminished in the iron, when fire is no longer present: the stone finally is reduced to rest; similarly, the iron comes back to natural cold: motion is impressed by the same force more in a mobile that is resisting than in one which resists less, as in a stone more than in light pumice: and heat, similarly, is impressed in a more penetrating way by the same quantity of fire in very hard and very cold iron than in tenuous and less cold wood. He would be ridiculous, who would say that the air previously heated by the fire, once the fire is out or removed to a distance, conserves the heat in the iron; since iron is brought to white-heat even in very cold air: and he is even more ridiculous, who would believe that mobility would be conserved in the projectile by air which is motionless or often enough blows in the opposite direction. And who will not say that iron is cooled more swiftly in cold air by the coldness of the air? and who of sound mind will not say that air, either by staying at rest or by blowing in the opposite direction, hinders motion?
But let an even more beautiful example be given. You are astonished by what comes out of the hand of the thrower and is impressed in the projectile; and you are not astonished by what comes out of the hammer and is transferred into the bell of the clock, and how it comes about that so great a sound is transported from a silent hammer into a silent bell, and how it is conserved in the latter, in the absence of what has struck it. The bell is struck by a striking thing; the stone is moved by a moving thing: the bell is deprived of silence; the stone is deprived of rest: a sonorous quality is introduced in the bell contrary to its natural silence; a motive quality is introduced in the stone contrary to its state of rest: the sound is conserved in the bell, in the absence of what has struck it; the motion is conserved in the stone, in the absence of what has moved it: the sonorous quality is progressively weakened in the bell; the motive quality is gradually weakened
311 in the stone. And who of sound mind will say that it is the air that continuously strikes the bell? For, in the first place, it is only a single small part of the air which is moved by the hammer; but if someone puts his hand on the bell, even on the side opposite to the hammer, he will immediatly feel running through all the metal a certain sharp, stinging numbness.In the second place: if the air strikes and causes the bell to sound, why is it silent in the presence of a wind blowing very hard? Can it be that the south wind, which turns the sea topsy-turvy, toppling towers and fortifications, whips more softly than the small hammer, which is hardly moved? In the third place: if it were the air which reverberated in the bronze, and not the bronze in the air, then all bells of the same shape would emit the same sound; what is more, a wood bell or at least a leaden or marble one, would make as much noise as a bronze one. But, finally, let those keep silent who say that it is air which reverberates or carries the sound from one thing to another: for the bell shakes, so long as it emits a sound, and, in the absence of a striking thing, the shaking motion and the sound stays in it and is conserved; surely to attribute this to air, claiming that the latter moves such a size when it has scarcely been moved by the hammer, exceeds all sense. Hence, returning to our subject, why are they astonished at how a motive quality can be impressed in a mobile, and not at how a sound and a certain motion of trepidation can be impressed by a hammer in a bell?
But, what is more, they say that they cannot conceive how a very heavy stone can get to be light, by a motive force received from a thrower: which force, since it is lightness, inhering in the mobile, will render it light: however these same people say that it is absolutely ridiculous to believe that a stone after an upward motion has ended up light and that it weighs less than before. But these people do not judge those things according to a sane and rational attitude. For I too would not say that a stone, after its motion, has become light, but that it retains its original natural heaviness: just as white-hot iron is deprived of cold; but after the heat, it resumes the same coldness that is its own. And there is no reason for us to be astonished that, the stone as long as it is moved, is light: for between a stone in that act of motion and any other light thing, it will not be possible to assign any difference; for since we call light that which is carried upward, and the stone is carried upward, therefore the stone is light as long as it is carried upward. {1}But you will say that that is light which is carried upward naturally; and not that which is carried by force. Now I would say that that is naturally light which is carried upward naturally {1}; but also that that is light contrary to nature or by accident or by force, which is carried upward contrary to nature, by accident, and
312 by force. But such is the stone impelled by force: and in the stone its own innate and intrinsic heaviness is lost in the same manner as it is also lost when it is placed in media heavier than itself. For a stone which floats on mercury, for example, and does not go down in it, loses all its own heaviness; and it even loses its heaviness so much and dons lightness, that it also resists energetically a lot of heaviness coming to it from outside (as if someone exerts pressure on it): and wood also becomes so light in water, that it cannot be kept down unless by force. And yet neither the stone nor the wood lose their own natural heaviness, but, taken out of those heavier media, they resume their proper heaviness: in the same way a projectile, freed from the force projecting it, displays, by going down, its own real, intrinsic heaviness. {1}
Furthermore, those who defend points of view opposed to ours ask themselves in which part of the mobile that force is received: on the surface, or at the center, or in some other part. I answer briefly this, that they should first make clear to me in what part of iron heat is received; and I will then tell them where the motive force is received and I will place it where they place heat: and if the heat is received only on the surface, I will say that the force is received only on the surface; and if it is at the center, at the center; and if they should say that the heat is received where the cold was before, I will say that the extrinsic lightness has entered those parts in which formerly the native heaviness resided.
Finally, my adversaries are astonished at how the same hand has the ability to impress, sometimes lightness, sometimes heaviness, but also sometimes what seems to be neither heavy nor light. But then why are they not instead astonished at how they now want something, and a little latter they do not want that same thing; and how they believe something, and then on the same subject they hesitate and have doubts about it, and sometimes even disbelieve it? But if, as in these cases, it depends on one's will that he can now raise his arm, then lower it, then move it in different directions, and if the arm, governed by the will, has the ability now to exert weight, now to lift; why should we be astonished that what is weighed down by the arm receives heaviness, but what is lifted is clothed in lightness?
But, because this does not diverge from our subject, let me not pass over in silence a certain quite common error: namely, that of those who believe that air and water because they are fluids, can be moved more easily and more swiftly, especially air; led by this, they have believed that the thrower moves the air more than he does the projectile, and that the air carries projectiles along.But the matter is quite different: as even they themselves sometimes admit, those who, following their master Aristotle, sometimes say that the air, in order that it may carry along projectiles, is moved very swiftly because of its lightness, since it has practically no resistance; but sometimes they say that things which have neither heaviness nor lightness cannot be moved, since it is necessary that what is moved resists: and, by talking this way, they sometimes believe and sometimes deny the same things, according to what suits their intentions better. However the fact is that the lighter a mobile is, the more easily it is moved while it is linked to the mover, but, once released by it, it retains the received impetus for only a short time: as is evident if one throws a feather, applying as much force as if one had to throw a pound of lead; for surely he will move the feather more easily than the lead; but the impressed force will be conserved for a longer time in the lead than in the feather, and it <the lead> will be thrown much farther.Now if it were the air that carried the projectile along, who would ever believe that the air would carry the lead more easily than the feather? We see, then, that the ligher a thing is, the more easily it is moved, but the less it retains the impetus it has received: hence, since the air, as has been demonstrated above {1}, has no heaviness in its own proper place, it will indeed be moved easily, but it will however conserve only minimally the impetus it has received. Now we will demonstrate below why light things do not retain the impetus. {1}
Nor is there any impact in the example they hand down of a pebble projected in a lake; they say that by it, water is moved in a circle over a very great distance. {1}For, first, it is false that the water is moved: as is evident if pieces of wood or straw are floating on the water: these will not at all be moved from their place by the eddies of water, but will only be raised a little by the wavelets and will not follow the circumferences of the circles. Second: the comparison does not hold good in the case of air, whose surface is not moved by the thrower, as it is only the surface of the water which is moved by the pebble; and this topmost surface of the water is raised and lowered only because it offers resistance to being raised and carried into the place of air: but in the middle of the air the motive force cannot be impressed, for then the air offers no resistance, since it is not driven away from its place towards the place of another medium. And this would also happen in the middle of the water which would not conserve the impetus received, since its motion would have no inclination; for it would not have any natural one, because it would not be moved towards its own place,
314 since it would already be there; nor a violent one, since it would not be driven towards the place of another medium.
This has been the common error of those who have said that projectiles are moved by the medium. Now it happens sometimes that certain opinions, however false they may be , continue to exist for a very long time among people; because at first sight they offer some appearance of truth, and because of that no one bothers himself to seak to know if it is worthy of belief. An illustration of such a thing is what is believed about things that are under water, concerning which common opinion asserts that they appear larger than they really are. {1}Now when I was not able to find a cause for such an effect, finally, having recourse to {1} experience, I found that a coin resting in deep water in no way does appear larger, but rather smaller: hence I judge that the one who first has put forth this way of thinking had been led to this opinion during the summer, when plums and other fruits are sometimes put in a glass vessel full of water, whose form recalls the surface of a conoid; they surely would appear far larger than they are to those who view them in such a way that the rays of light pass through the glass. But it is the form of the cup, not the water, which is the cause of such an effect; as we have made clear more extensively in the commentaries to the Almagest of Ptolemy, which (with God's help) will soon be published. {1} Now a sign of this is that, if the eye is placed above the water, so as to be able to observe the plum without the intervention of the glass medium, the plum does not appear larger.
Consequently, let us conclude finally that projectiles are in no way moved by the medium, but by the motive force impressed by the thrower. But now let us go on to show that this force is progressively diminished; and that in a violent motion no two points can be assigned in which the motive force is the same.
Older Essay on Motion, Book II, chapter 5, (D18), [314.26-315.25]In which it is shown that the motive force in a mobile is progressively weakened.
Thus, since, in the preceding chapter, it has been determined that projectiles are moved by an impressed force, it is evident that violent motion is one and continuous,
315 and not many and discontinuous, as Aristotle believed. And since this is so, and since violent motion is not without end, it necessarily follows that that force, impressed by the thrower, is continuously weakened in the projectile; and that there cannot be given in that motion two points of time in which the motive force is the same and is not weakened. In order that this may be even more clearly apparent, I shall use this demonstration: by first presupposing that the same mobile, in the same medium and along the same line, is moved by the same force with the same speed.
This having been presupposed, let the line on which the motion takes place be line ab; and let the motion take place from a to b; and, if it can be done, let there be found on line ab two points at which the impelling force is the same, and let them be c,d. Since, then, at c and at d the motive force is the same, the medium is the same, the mobile is the same, and the line along which the motion takes place is the same, therefore the mobile will be moved from point d at the same speed as it has been moved from point c: but from point c it has been moved so as to be carried from point c to d always at the same speed, and the force has not ended up to be weaker: therefore from point d also it will be moved at the same speed along a line equal to cd, the impressed force having stayed the same. For there is no greater reason why the force should stay the same from c to d, but not from d along a line toward b equal to line cd, since the force is the same, the mobile is the same, the medium is the same, and the line of motion is the same. Hence, repeating the same argumentation, it will be demonstrated that violent motion is never weakened, but is carried always at the same speed and without end, with the motive force always staying the same: which is truly very absurd: hence it is not true that in violent motion two points can be assigned in which the impelling force stays the same. Which is what had to be demonstrated.
DMA, Book II, chapter 6[Drabkin 19] [315.26-323.18]In which is brought forth for everyone to see the cause of the acceleration of natural motion at the end, a cause far different from that which the Aristotelians assign.
It surely is more difficult to discover than to explain the cause why the speed of natural motion is increased at the end; and either no one has yet found it or else, if from time to time someone has alluded to it, he has reported it in an incomplete and defective way, and in addition it has not been received by the community
316 of philosophers. However, while I was on and off investigating the cause of this effect, which I would not call astonishing but necessary (for the cause which is reported by Aristotle never appealed to me), for a long time I was troubled, and I found nothing that was fully satisfying. But, once the very true cause (at least in my own judgment) had been found, at first I rejoiced: but, when I examined it with more application, I was suspicious because it appeared free from any difficulty: but, finally, all difficulty having been progressively cleared away with the passage of time, I will now bring out for everyone to see which is assuredly finished and very certain. But first of all, according to our custom, we will consider with care what strength there is in the cause which is expressed by Arsitotle.
And, in the first place, it must be known that certain recent authors assert that Aristotle attributes this cause to the parts of the air, which hit the back of the mobile, rushing back to fill in the void: by this striking, they themselves say, the natural motion is increased. But that Aristotle did not think this can clearly be concluded from the things that are read in De Caelo, Book I, [ch.8], text #89, [277b1-9], where he says in clear words: Natural motion is not helped by extrusion, as some have believed; for in that case it would be a violent motion, which is weakened at the end, not increased, like the natural.It is thus evident that Aristotle not only does not hold this opinion, but that he rejects it: and indeed it deserves to be rejected. For, from what they say concerniing the void, either a void is left behind in back of the mobile, or not; if not, why do they say that the air rushes back to fill in the void? and if it is left, why do they not say that the mobile also moves back to fill in the void, and that thus the motion is rather retarded by such a cause, and not helped?In the second place: let a mobile be taken, concerning which there can be no doubt that it is not impelled from behind by the air; such as a solid rhombus made up of two cones, both tapering out into a very sharp point. Surely this thing will not be able to be impelled by the air, since its shape has nothing against which the air could hit. In the third place, things that are moved by violence, are not moved more swiftly than that by which they are moved; but the air, while it is moved towards the back of the mobile, is moved by force (for in its natural region it is at rest): hence it cannot be moved more swiftly than that by which it is moved. Now it is moved by that body going down; hence the air will not be moved faster than the mobile going down: but if it is not moved faster, it certainly will not be able to impel that body;
317 for, if someone wants to impel someone else running in the same direction as himself, it is necessary for him to run faster than the other and to be rapidly flung in the direction in which he runs.
But this does not happen in the case of a natural mobile; on the contrary, the air is moved in the opposite direction: as, if a sphere abc goes down, the surrounding air, rushing back from parts b, c towards a, the back of the mobile, will be moved upward with respect to the downward motion of the sphere. This is also admitted by them, when they say that the medium resists motion, since it must be split: hence if the medium must be split, surely it will not be moved in the same direction as the mobile. Therefore, either it will be at rest, or it will be moved in the opposite direction to the mobile, or, if we want it to be moved in the same direction, it will at least be moved more slowly: but since this is so, in what way will it help motion? In the fourth place, they do not seek a cause per se of the acceleration of motion, but they only bring up an accidental cause; for it is by accident that a mobile is moved in a plenum, and that its speed is either hindered or helped by the medium: but we are asking why a natural mobile, when it is moved naturally by its proper heaviness, without any consideration of the medium, is moved faster at the end than in the middle, and faster in the middle that at the beginning; and how from a consideration of motion, it is necessary that it be weaker at the beginning. So much for those who hang on to this opinion.
Others have said that the mobile is moved faster at the end because the parts of the medium which must be split by it are less numerous; and for this reason, since it has lesser resistance, they have believed that it is carried faster. But this way of thinking is not only false, but ridiculous: for, if it were true, it would follow that a stone going down from a very high tower would be moved more slowly at mid-tower, than if the same stone were falling to the ground from a very low place, and for this reason the mobile [falling from a greater height] would also make a lesser impact.
In order that this may be more clearly understood, let there be a line abc, and let ac be much longer than cb: I say, then, that if the stone were going down from a, it would be moved more slowly when it was around c, than if the same stone were released from c, near b; because, of course, there would be fewer parts of the air left to be split by the mobile when it was around b, having been released from c, than when it was around c, coming from a. It can also be added that the stone in going down from a would precipitate itself to the ground with the same impetus, as if it were going down from c: and the reason is that, in going down from a, when it was a bit under c, it was not moved faster than when it was a bit under c, in going down from c, since in the former case no fewer parts of the air remained to be split than in the latter case; and they say that it is on such
318 splitting that the swiftness of motion depends. Now there is no one who can ignore how incongruous all these things are.
But, leaving aside the ways of thinking of the others, in order that we may track down what we believe to be the true cause of this effect, we will make use of the following resolutive {1} method. Since, then, a heavy mobile (let us however speak about natural downward motion, coming from heaviness: for, that being known, we will judge the case of upward motion by proceeding in reverse) in going down is moved more slowly at the beginning, it is therefore necessary that it be less heavy at the beginning of its motion than in the middle or at the end; for we know with certainty, from the things demonstrated in the first book {1}, that speed and slowness follow heaviness and lightness. If, then, it is found out how and why a mobile is less heavy at the beginning, the cause for which it goes down more slowly will certainly have been found. But the natural and intrinsic heaviness of the mobile is certainly not diminished, since neither its size nor its density is diminished: it remains, therefore, that that diminution of heaviness is against nature and accidental. Hence if we have found in what way the heaviness of the mobile is diminished against nature and extrinsically, what we need will surely have been found. But that heaviness is not diminished by the heaviness of the medium, for the medium is the same at the beginning of motion as at the middle: it remains, therefore, that the heaviness of the mobile is diminshed by some violence that is extrinsic and comes from outside (for it is only in these two ways that a mobile gets to be light by accident). {1} If then, again, we find out how a mobile could be lifted by an extrinsic force, the cause of slowness, again, will have been found. Now the force impressed by a thrower not only at times diminishes the heaviness of a heavy thing, but it often even renders it so light that it flies upward with great speed: hence let us see and search attentively, whether perhaps this force is the cause of the diminishing of the heaviness of the mobile at the beginning of its motion. And, I say, it certainly is that force impressed by the thrower which renders natural motion weaker at the beginning: so let us hasten to make clear by what method
319 it is able to accomplish this. For a heavy mobile to be able to be moved upward violently, an impelling force greater than the resisting heaviness is necessary; otherwise the resisting heaviness would not be able to be overcome, and, consequently, the heavy thing would not be able to be carried upward. Hence, the mobile is carried upward, provided the motive impressed force is greater than the resisting heaviness. Now since this force, as has been demonstrated {1}, is continuously diminished, it will finally become so diminished that it will no longer overcome the heaviness of the mobile, and then it will not impel the mobile any further: but that impressed force will not therefore have been annihilated at the end of the violent motion, but it will only be diminished to the point that it no longer surpasses the heaviness of the mobile, but it will be equal to it; and, to put it in a word, the impelling force, which is lightness, will no longer dominate in the mobile, but will have been reduced to parity with the heaviness of the mobile: and then, at the ultimate point of the violent motion, the mobile will be neither heavy nor light. But, moreover, as the impressed force diminishes in its own way, the heaviness of the mobile begins to predominate; and hence the mobile starts to go down. But since at the beginning of such a descent a great deal of force that impels the body upward, which is lightness, still remains (even though it is no longer greater than the heaviness of the mobile), it comes about that the proper heaviness of the mobile is diminished by this lightness, and, consequently, that the motion at the beginning is slower. And, moreover, since that extrinsic force is further weakened, the heaviness of the mobile is increased by having less resistance, and the mobile is moved still faster.
I think that this is the true cause of the acceleration of motion: when I had thought it out, and, two months later, happened to be reading the things written by Alexander {1} on this subject, I learned from him that this had also been the way of thinking of that very great man of learning, who is praised by the very learned Ptolemy -- namely, Hipparchus, who is greatly esteemed and extolled with the highest praises by Ptolemy throughout the whole of his Almagest. {2} According to Alexander {1}, Hipparchus also believed that this was the cause of the acceleration of natural motion: but, since he added nothing beyond what we have said above, this opinion seemed defective, and was thought to deserve rejection by philosophers; for it seemed to apply only in the case of those natural motions that were preceded by a violent motion, and that it could not be attributed to a motion which does not follow a violent motion. {2}But they were not content to reject it as defective, but actually as false, and not even true in the case of a motion preceded by a violent [motion]. But we will add the things that have not been explained by Hipparcus, by showing how even in a motion that is not preceded by a violent [motion], the same cause applies; and we will try to free our explanation from all malicious criticisms. I would not however say that Hipparcus was wholly undeserving of reproach; for he has left undetected a difficulty of great importance: but I will only add the things that are missing, and will reveal the splendor of the truth.
I say, then, that it is for the same reason that motions, which a violent motion does not precede, are also moved more slowly at the beginning. For even in motions which a violent [motion] does not precede, the mobile begins to be moved, from a state {1} of rest, and not from a violent motion. Thus a stone projected upward, when it begins to be moved downward from that extreme {1} point at which there is equality between the impelling force and the resisting heaviness, which is a state of rest, begins to go down; this is the same thing as if it were falling from someone's hand. For even when the stone falls from someone's hand, with no force impelling it upward having been impressed on it, it leaves with a quantity of impressed force equal to its heaviness. For when the stone is at rest in someone's hand, one must not say that in that case he who holds it impresses no force on the stone: for since the stone exerts pressure downward by its heaviness, it is necessary that it be impelled upward by the hand with an equal quantity of force, neither larger nor smaller. For if the force, by which the hand impels the stone upward, was greater than the heaviness of the stone, the resisting stone would be raised by the hand; and it would not be at rest, as we are presupposing: but, on the other hand, if the stone exerted more weight than the hand lifted, the stone would proceed downwards; now we are presupposing that the stone is at rest in the hand: hence there is impressed in the stone, by the hand or by whatever else by which it is controlled, as much force impelling upward as there is heaviness of the stone tending downward; nevertheless the stone is not raised, because, as we have said, that impelling force cannot surpass the resisting heaviness, since it is not greater than that heaviness. It is thus evident how, when the stone comes out of the hand, it leaves with a quantity of impressed force equal to its heaviness: which is not different from what happens when the stone starts to proceed downward, having completed its upward motion; for in that case too, when it recedes from the state of rest, it leaves with a quantity of force equal to its heaviness: hence for the same reason it is moved more slowly at the beginning in the case of that motion, just as in the case of this one.
321 But, in order that this whole affair may be explained even more clearly, I will bring forth for everyone to see a particular example.
Let there be a line ab, along which a violent motion takes place from a to b, but a natural motion from b to a; and let there be a mobile c, whose heaviness is 4. It is thus necessary, in order that mobile c be moved upward, that a motive force greater than its heaviness be impressed in it: for by an equal force it would not be moved; for it would be neither heavy nor light, since its heaviness would be equal to the impressed force, which is lightness. Hence let the force which can impel c up to b be 8: and since the motive force, as we have demonstrated above, is continuously weakened, and it could not move c unless it is greater than the heaviness of the mobile, it is evident that when c will be at b, the impressed force will be equal to the heaviness of c. For it will not be smaller, since then it would not have impelled it up to b; nor greater, since it would have impelled it even further: it remains, therefore, that it is necessarily equal. Hence when c is in b, it has a quantity of impressed force equal to its heaviness, namely, 4: since this force continues to be progressively weakened, and immediately begins to be diminished, c changes itself over to downward motion. Hence when it begins to go down from the first point b, c recedes with a quantity of impressed force which is equal to its heaviness; thus it will be moved very slowly at the beginning of such a motion: but the more the contrary force is weakened, and, consequently, the heaviness is increased, the faster the motion that takes place. Again, if an impressed force of 8 throws c up to b, it is evident that there can be impressed in it a quantity of force which impels it only up to d: this force will certainly be smaller than 8, but greater than 4; for that is the heaviness of the resisting c. Furthermore, there can even be impressed a quantity of force which throws c only up to e: this force, again, will be smaller than that which impels it up to d, but it will be greater than 4; for by a force equal to 4 c is not moved. And, similarly, it will be possible for as much force to be impressed as will impel c upward, taken along line ae, through any distance no matter how small. But nonetheless this force, since it moves upward, will always be greater than 4: and every force that is smaller than 4 not only will not impel c upward, but cannot stand in its way so that it does not go down, since it is surpassed by a greater heaviness: it necessarily remains, therefore, that an impressed force which is 4 just sustains c:
322 hence, when c is at rest, there will be in it an impressed force impelling it upward, which will be 4.But if it is abandoned by the impressing force, it will recede with an impressed force of 4: hence after such a retreat it will not be moved upward; but it will go down very slowly at the beginning, and afterwards it will be moved more swiftly, according as the contrary force will be weakened.
We assert without doubt that this is the true, proper, and most important cause by which natural motion at the beginning is slower; no doubt those who will examine it properly and fairly will embrace it and will follow it as very true. But could these same things be obtained by an easier method? Indeed they could, and they are easily known to one proceeding from natural considerations: for, pray, do not the two following propositions coincide: "Forced motion is slow at the end", "Therefore natural motion is slow at the beginning"? For natural motion follows violent motion, and the end of violent motion is linked with the beginning of the natural: but the cause of the slowness of violent motion at the end is the small excess by which the impelling force exceeds the resisting heaviness, that is, by which the cause of violent motion exceeds the cause of natural motion: hence, in the same way, one must deem that the cause of the slowness of natural motion at the beginning is the small excess by which the cause of natural motion surpasses the cause of violent motion, i.e. by which the heaviness which exerts pressure downward exceeds the lightness, that is the impressed force, impelling upward. You can therefore see how well true things suit one another. Now from this way of considering things anyone will easily be able to understand how these two motions are not truly contraries, but rather a certain motion composed of the violent and the natural {1}: for these local motions somehow depend on certain other alterative motions, so long as the proper heaviness and the extrinsic lightness (for we will hereafter call the impressed force {2} lightness) are mixed in the mobile. From such a mixture it follows, and in a sense by accident, that a mobile is moved sometimes upward sometimes downward: for when in the mixture there is more lightness than heaviness, from it an effect of lightness, namely an upward motion, will arise; but if in the mobile, lightness having been diminished, there is more heaviness, an effect of heaviness, namely a downward motion, will emanate. But this alterative motion, while the mobile is moved from lightness to heaviness, is a single and continuous motion: as when hot water by accident becomes cold, it is moved with a single motion towards cold, and its motion from hot to tepid is not something else than its motion from tepid to cold: thus, when from light something becomes neither heavy nor light, the motion is not separate from the motion during which from neither heavy nor light it becomes heavy.
323 Hence, these motions are so far from being contraries, that they are actually only one, continuous, and having the same limit: and thus the effects that emanate from these causes must not be truly called contraries, since contrary effects depend on contrary causes: thus we cannot truly call an upward motion contrary to the following downward motion -- both of which motions emanate from a motion of the mixture of lightness and heaviness. And from this it can also easily be deduced how a state of rest does not intervene at the turning point.For, if there were then a rest, it would be necessary that a rest happen also in that motion of the mixture of heaviness and lightness, when the lightness had come into equality with the heaviness; for only then can the mobile be at rest, when the impelling force neither overcomes nor is overcome: but, as we have already made clear, that motion, when from light it becomes heavy, is one and continuous, as when hot becomes cold, which does not come to rest in time: hence also local motions, which emanate from that motion, will be one and continuous.But since this way of thinking runs against common opinion (it is commonly believed that there is a rest at the turning point), it will be deferred to the next chapter; there, the opposed way of thinking will first be examined and refuted, and our opinion will be rendered still stronger.
Older Works on Motion, Book II, chapter 7[323.19-328.10]In which it is demonstrated against Aristotle and the common way of thinking, that at the turning point no rest is given.
Aristotle and those who believe Aristotle believed that two contrary motions (he calls contraries those things that tend towards contrary limits) can in no way be continuous; and, for that reason, that when a stone is impelled upward and then comes back along the same line, at the turning point it is necessarily at rest. Now the most important argument, by which Aristotle tries to prove this {1}, is the following: That which {2} is moved by coming near a certain point and
324 by receding from it, and by making use of it as an end and a beginning, will not recede from it unless it has stopped at it: but what is moved to the extreme point of a line and comes back from it makes us of it as an end and as a beginning: it is therefore necessary that it stand still between the approach and the receeding. Aristotle proves the major proposition like this: For he who makes use of something as beginning and end, makes what is one in number two in logic; just as someone who in thought makes a point which is one and the same in number two in logic, namely the end of one thing, but the beginning of another: but if something makes use of one thing as two, it is necessary that it stands still, for between the two there is some time.
Such is Aristotle's argumentation; how weak it
325 is will soon be apparent. For, as he himself wants it, what is moved makes use on the line of its motion of a point, one in number, for what are two things in logic, for a beginning and for an end; and yet between these two no line intervenes, since they are only one in number: also, why will the same mobile, in like manner, not make use in the time of its motion of the same instant, one in number, for two, in logic, namely for the end of the time of approach and for the beginning of the time of receding, so that between these two instants, in logic, there would be no time, since they are only one in number?There is no compelling reason why this should not be so; and all the more so since Aristotle himself teaches that things which are valid for a line also apply to time and the motion. Therefore, if on the same line the same point, in number, is both the end of this motion and the beginning of another, and, however, it is not necessary that a line be in the middle between this beginning and that end; then, in the same way, the same instant in number, will in logic be the end of this time and the beginning of the other, and yet it will not be necessary for time to intervene. It is thus evident how the solution to Aristotle's argument can be conveniently drawn from the propositions of the argument itself: hence, since it no longer compels us, let us see if we can construct arguments more sharply compelling in favor of the contrary.
So much in opposition to Aristotle: but, in order that we may show by other arguments that at the turning point no rest intervenes, and that it is not necessary that there be rest between contrary motions, here are these other arguments.
See my note below.Secondly, let a certain continuum, such as the whole of line ab, be moved towards b in a motion like a violent one, which continuously weakens: and while the line is thus carried, let a certain mobile, say c, be moved on the same line by a contrary motion, from b to a; but let this motion be like a natural one, namely, one that is increased: but let the motion of the line at the beginning be faster than the motion of c at the beginning. It is thus manifest that at the beginning c will be moved in the same direction as that in which the line is carried, because its motion, by which it is carried in the opposite direction, is slower than the motion of the line; however, since the motion of the line weakens, but the motion of c increases, at some time c will actually be moved toward the left, and thus will make the switch from a rightward motion to a leftward motion, and along the same line; yet at the point of the receding there will be no rest for any time whatever. And the reason is that it cannot be at rest, unless the line is moved towards the right at a speed equal to that with which mobile c is carried towards the left;
326 but it will never happen that this equality lasts for any length of time whatever, since the one motion is continuously weakened but the other is continuously intensified: hence, necessarily, c will switch from one motion to its contrary, without the intervention of any rest. A third argument can be taken from a certain rectilinear motion that Nicholas Copernicus in his De Revolutionibus compounds from two circular motions. For they are two circles, one of which is carried on the circumference of the other, and when one is moved more swiftly than the other, a mark on the circumference (of the slower?) is carried along a straight line and runs back and forth along it continuously; and yet it cannot be said that it is at rest at the extremities, since it is continually carried around by the circumference of the circle. {1} [see also my notes below]There is a fourth well-known argument concerning a large stone going down from a tower, which will not be sufficiently blocked by a pebble impelled upward by force, so as to permit the pebble to be at rest for any time: hence surely the pebble will not be at rest at the ultimate point of its upward motion, and Aristotle notwithstanding, it will make use of the ultimate point for the two limits, namely of upward motion and of downward motion; and the ultimate instant is taken twice, namely, for the end of one time and for the beginning of the other. But the adversaries, in order to escape from this, say that the large stone is at rest, and thus they persuade themselves that they have done enough for the argument. But, so that in the future (unless they should be downright obstinate) they may not believe this, I shall add this to the argument: let these stones, which are moved by contrary motions, be carried, not upward and downward, but on a plane surface parallel to the horizon, one with great impetus, but the other more slowly, and let them be moved from contrary positions in contrary directions; and let them converge in the middle in an interacting motion: in that case the weaker will undoubtedly be thrust back by the stronger and will be compelled to be carried in the opposite direction; but how will they say that at that point of impact a rest intervenes? For if only once they were at rest, they would thereafter always be at rest, since they would have no cause for moving, as in the case of that large stone, coming from on high: if it were stopped by the pebble, still after the rest, they would both go down in concordance, moved by their proper heaviness; but when they are on a plane parallel to the horizon, after the rest, there is no cause for motion after the rest. Here is a last argument: before its explication let these two things be presupposed. First, I presuppose that a mobile can be at rest outside its proper place only when the force impeding its descent is equal to its heaviness, which exerts pressure downward: which surely is manifest; for if the impressed force were greater than the resisting heaviness, the mobile
327 would continue to be moved upward; but if it were less, it would descend. Second, I presuppose that the same mobile can be sustained by equal forces for equal lenghts of time in the same place. In such a case I go on like this: if at the turning point, as when the stone turns from violent upward motion to downward motion, there is a rest which lasts for a certain length of time, there will also be equality through the same time between the impelling force and the resisting heaviness: which is impossible, since it has been demonstrated in the preceding chapter that the impelling force is continuously weakened; for the motion by which the stone is moved accidentally from being light towards heaviness, is a unique and continuous motion, as when iron is moved from hot to cold: hence the stone will not be able to be at rest.
Furthermore: let a stone be moved from a to b violently, from b to a naturally; if, then, the stone is at rest for a certain length of time at b, let the extreme moments of such a time be cd. If, then, the mobile is at rest during time cd, then during {1} time cd the extrinsic {2} impelling force is equal to the heaviness of the mobile: but the natural heaviness is always the same; therefore the force at moment c is equal to the force at moment d. Now it is the same stone, and the same place; hence it will be sustained during equal lengths of time by equal forces: but the force at moment c sustains it during time cd: hence the force at moment d will sustain the same stone during a length of time equal to the length cd. The stone, therefore, will be at rest during twice time cd: which is unacceptable; for it was supposed to be at rest only during time cd. Indeed, by observing the same way of arguing, it will even be demonstrated that the stone would always be at rest in b. And do not be upset by this, namely, the argument that if the heaviness and the impelling force are sometimes equal, the mobile must also sometimes be at rest: for it is one thing to say that the heaviness of the mobile sometimes comes into equality with the impelling force; but it is another thing to say that it remains in such a state of equality for a length of time. Now this becomes clear from the following: for, while the mobile is moved, it being given (as has been said) that the impelling force is always weakened, but the intrinsic heaviness remains always the same, it follows necessarily that, before they have reached this ratio of equality, countless other ratios intervene: and yet it is impossible that the force and the heaviness remain in any of these ratios during a certain length of time; because it has been demonstrated that the impelling force never stops during a certain length of time in the same state,
328 since it is always weakened. So it is true that the force and the heaviness will pass through ratios like 2/1, 3/2, 3/4 and countless others; but that they remain in any one of them for a certain time is false and impossible: thus at some time they even arrive at equality, but they do not remain in equality. Since this is so, and since local upward and downward motion is a consequence of that alterative motion of the transformation from light by accident into heavy per se, in such a way that that upward motion follows from an excess of impressed force, downward from its deficiency, and rest from equality; since equality does not last for a length of time, it is necessary that the rest does not last.
Older Works on Motion, Book II, chapter 8 [328.11-333.13]In which it is proved against Aristotle that, if natural motion could be extended without end, it would not become faster without end.
Aristotle thought (as can be seen in De Caelo I, text #88 [277a 27-33]) {1}, not only that natural motion is always accelerated until the mobile has reached its proper place, but also that, if the motion could continue without end, the heaviness of the mobile and the swiftness of its motion would also be increased without end. For, in trying to show that things that are moved are carried towards a certain definite place, he has written this: If earth, while it is moved downward, were not moved towards a definite place, but without end, its heaviness and its speed would also be increased without end; but a heaviness without end and a speed without end cannot be given; hence what is carried downward is not moved without end. This, then, is Aristotle's opinion: now we will make it manifest that the truth is diametrically opposed to this; namely, we will show that the speed is not always increased, and that even if it were always increased and the motion could be extended without end, [the mobile] would not necessarily attain a heaviness without end and a swiftness without end.
Thus as to what concerns the first point, anyone will easily understand from the things that have been written above {1}, having grasped the cause of the acceleration of natural motion at the end, why such an acceleration must finally cease. For since the mobile is accelerated because the contrary force weakens continuously,
329 and the natural heaviness is being acquired, it will stand to reason that all the contrary force will finally be lost, and that the natural heaviness will be resumed, and that, therefore, the cause having been removed, the acceleration will cease. And yet I would not say that the whole contrary force is consumed because I may happen to think that whatever is always diminished must finally be annihilated (for I am not unaware that this is not necessary, as will be said below); but I will only say that it is consumed because experience seems to me to show this. {1}For, in the first place, if we look at something that is not at all heavy coming from on high, a ball of wool or a feather or some such thing, we will see that it is moved more slowly at the beginning, but that nevertheless, a little latter, it will observe a uniform motion. Now the reason why this appears more manifest in things that are less heavy is that, when these things begin to be moved, since they have an amount of the contrary force equal to their proper heaviness, and they themselves are not very heavy, hence, the contrary impressed force will also be small, and thus it will be consumed more swiftly; when it has been consumed, these things will be moved with a uniform motion: and since they are moved slowly, it will be easier to observe the uniformity of such a motion in them than in the case of those that go down very swiftly. Now in the case of heavier things, since a great amount of contrary force must be consumed in their descent, a greater time will be required for it to be consumed; in which time, since they are carried swiftly, they will descend a great distance: since we cannot avail ourselves of such great distances from which to release heavy things, it is not astonishing if the stone, released from merely the height of a tower, will seem to accelerate all the way to the ground; for this short distance and short time of motion are not sufficient to destroy the whole contrary force.A second experience can be taken from other alterative motions in which at the end the contrary quality is completely lost; as when, after being white-hot, iron becomes very cold, and all the heat is entirely annihilated. One must, therefore, judge in the same way in the case of the stone which from light becomes heavy, that it loses all of its extrinsic lightness; when this happens, the intensification of the speed will cease. Thirdly, it can be confirmed by both reason and experience, not only that motion is not always accelerated when a mobile has receded from a state of rest; but also that, if at the beginning of the motion a great force which impels it downward is impressed by an external mover, it too will be destroyed. And the reason is because then the heaviness of the mobile would have the effect of lightness, since the heaviness by itself free and simple would go down more slowly than when joined with an impetus; and thus the proper and natural slowness [of the mobile] in going down would resist
330 a violence impelling downward. {1} There is also a manifest example which happens quite often to divers and swimmers. For their natural heaviness is great enough for them to go down, if they want to, to the bottom of the sea, and then it is only by the action of their proper heaviness that they sink: but if they are impelled downward by an external mover, with no matter how great a force, e.g. if they are thrown from a very high place, like the top of a ship's mast, at the beginning their motion in water will surely be very excited and much beyond the natural; and yet this motion will be retarded by their proper absolute heaviness, which then, in comparison to the heaviness joined with the impetus received, amounts to lightness, and it will be retarded until in going down it has reached its natural slowness; and moreover if the water is deep enough, [the diver] will not suffer any greater injury at the bottom than if he had gone down by his proper natural motion from the surface of the water. And from this the following argumentation can be drawn: for if [a mobile] in going down were always accelerated in its motion, it would be capable of every swiftness whatever, so that no swiftness would be beyond the natural for it; and therefore it would not lose or reject the downward impetus received from the external mover, since at the end it would even have arrived naturally at this impetus: but experience teaches that the contrary occurs: hence it is evident to everyone that a definite and established swiftness is prescribed to natural motion in going down. {1}
But even if the swiftness were always intensified and the length of the motion were without end, it would not however follow that the motion would finally attain a swiftness without end and the mobile a heaviness without end -- which will not be difficult to understand for those who are versed in mathematics. For this is like what seems impossible to nearly all who are incapable of following the demonstration: namely that two lines can be found which, being prolonged without end, always come closer, yet never meet; so that the distance between them always diminishes without end, yet never is consumed. But that such lines are given is known to all who have come upon either the asymptotes of the hyperbola in the Conics of Apollonius of Perga, or with the first conchoid curve of Nicomedes in the commentary of Eutocius of Asca
331lon on incomparable Archimedes' Sphere and Cylinder, Book II {1}: for these are cases of two lines (and numerous others could also be imagined), which, being prolonged without end, always come nearer, but it is impossible that they finally meet; hence the distance between them is always diminished, yet it is never consumed. And if a line is drawn at right angles to the straight line that lies under the conchoid, or at right angles to the asymptote, and we assume that this line, always remaining at right angles, is moved without end in the direction in which the lines that do not meet are extended without end; on this perpendicular line, the point at which it is intersected by the hyperbola or the conchoid will always be moved towards the other extremity by coming near it, but will never reach its end point. What happens concerning the swiftness is something similar: indeed the slowness of motion can always be diminished and, therefore, the swiftness increased, and yet never be finally consumed. As, for example, let ab be the slowness, so that if the mobile consumed it entirely, the motion would happen in an instant: I say that, even if it is always diminished without end, it is not necessary that in the end it is consumed.
For let a motion begin which can be intensified without end: but let it be such that in the first distance of a mile it is accelerated to such an extent that it diminishes the slowness ab by an eighth part, let us say, ac; but at the end of the second mile let it diminish it by the eighth part of the remainder cb; and at the end of another mile let it diminish it by an eighth part of the next remainder.And thus this diminution will be able to be produced without end, since the seven-eighths that remain can always be divided into eight equal parts; and the mobile will be able to be moved over miles without end, by consuming at each mile some of its slowness: and yet it is not necessary that this slowness be completely consumed.
What is more: those who with Aristotle have believed that if slowness is always diminished, the mobile must finally reach a swiftness without end -- what will they say if it is shown to them that, not only is it not necessary for that swiftness to come to be without end, but it is even demonstrated that a mobile can be always accelerated, yet without its swiftness being intensified to the extent that it is equal to, much less in excess of, a certain finite swiftness?
And, to express it more clearly, let something be moved whose swiftness at the beginning of the motion is ab; now let there be another swiftness cd greater than ab; I say that the mobile, moving without end, can increase without end its swiftness ab, which nevertheless, increased
332 without end, will never be as great as the swiftness cd. As would be the case if a mobile, receding from rest, acquired after the first mile of its motion the swiftness ab, which is two-thirds of swiftness cd; and if after the second mile its swiftness were increased by one-third of swiftness ab; if after the third it were increased by one-third of one-third of swiftness ab; after the fourth by one-third of one-third of one-third of ab; and if in this way without end there came about an increase for each mile of one third of the increase in the preceding mile: and the swiftness will always, surely, be increased, yet it will never be as great as cd, but always one-half {1} of the last increase will be wanting. {2}Now here is the demonstration of this.
Let there be any number of continuous swiftnesses, each one three times its successor, ab, bc, cd; let the greatest be ab, whose sesquialter {1} is ea.I say that the sum of all the magnitudes ab, bc, cd, together with one half of cd, is equal to ea itself. For since ea = 3 ab/2,ab + 1/2 ab = ae.And since ab = 3 bc, bc + 1/2 bc = 1/2 ab: but it has been demonstrated that ab + 1/2 ab = ae.Therefore, abc + 1/2 bc = ae.Similarly, since bc = 3 cd, cd + 1/2 cd = 1/2 bc. but it has been demonstrated that ac + 1/2 bc = ae:Therefore, ad + 1/2 dc = ae.And the same demonstration being always repeated, it will be demonstrated that any number of swiftnesses, each one three times its successor, taken together with one-half the smallest, are equal to a swiftness which is 3/2 that of the greatest swiftness among them. And if this is so, it is evident that the sum of the swiftnesses, taken together, each one three times its successor, is less than a swiftness 3/2 that of the greatest, since they always fall short of it by half the smallest swiftness. It is thus evident how the swiftness ab could always be increased without end; and yet it would never be equal to swiftness ae. Consequently let us conclude that, in the case of a mobile, for the reasons adduced above, the speed is not always increased; but it reaches a certain motion, beyond which its determinate hea
333viness does not naturally allow a greater swiftness: but even if it were conceded that its speed is always intensified without end, it would still not reach a swiftness without end.
From the things that have thus far been written, it will be easy for anyone to find the cause for which heavy things in their natural motions do not observe those ratios which we had assigned to them when we discussed the matter {1}; namely, the ratios of their heavinesses, which they have in the medium through which they are moved.For since at the beginning of their motion they are not moved in accordance with their heaviness, because they are impeded by a contrary force, it will surely not be astonishing if the swiftnesses do not observe the ratios of the heavinesses; on the contrary, and this surely seems remarkable, lighter things go down more swiftly than heavier ones at the beginning. Others, too, {1} have tried to assign a cause for this remarkable effect; since they have not been successful, we will in the next chapter refute them and endeavor to bring forth the true cause.
Older Works on Motion, Book II, chapter 9 [333.14-337.10]In which is assigned the cause for which at the beginning of their natural motion things that are less heavy are moved faster than heavier ones.
This question, surely, is no less elegant than difficult: others also, like Averroes and those who follow him, have tried to explain its solution; and, in my opinion, they have worked in vain, by laying down certain unattractive hypotheses. For, they say {1} that air in its proper region is heavy; from this it follows that, those things that will have more air, in the place of air are more heavy (and this is also Aristotle's opinion): the result of this is that , for example, since a sphere of wood has within itself more air than one of lead, it has 3 weight exerting elements[gravantia:see my note below], namely air, water and earth; but lead, since within itself it is weak in air, has, as it were, only 2 weight exerting elements[?gravia cf. ms;see my notes below], the wood sphere will go down more swiftly than the lead one. {2}Not content with this, they also say that rare lead is heavier in air than dense iron because there are more parts of air in rare lead than in the dense iron. There is no one who could fail to see
334 how many and how great are the difficulities that this solution involves. In the first place, who is ignorant of the fact that air in its own region is neither heavy nor light, {1}and that therefore it is carried neither upward nor downward? For this has been demonstrated above. {1}Secondly, if the speed of the motion of the mobile follows its heaviness, as everyone wants, and the sphere of lead has earth and water in place of the parts of air which there are in the woodden sphere, and earth and water are heavier than air, as we can readily believe, will not the lead be heavier, and will it not go down faster? And as for what they say about iron and lead, in order to show that air exerts weight, if lead is heavier because it has more air, then wood will be heavier than both iron and lead, since it is has more air than each one of them. Thirdly, if the great quantity of air which is in wood makes it faster, then it will always be moved faster, as long as it is in air. Yet experience shows the contrary: for it is true that wood at the beginning of its motion is carried more speedily than lead; but a little later the motion of lead is so accelerated that it leaves the wood behind, and, if they are released from a high tower, the lead gets ahead of it by a large distance: and I have often put this to the test. {1}Consequently we must try to draw a sounder cause from sounder hypotheses.
Oh, how easily are true demonstrations derived from true principles. If it is true, as we have said {1}, that mobiles, as they recede from a state of rest, recede with an amount of contrary impressed force equal to their heaviness, then those that are heavier will recede in conjunction with a greater contrary force: but if heavier ones must consume more force impelling them in the contrary direction, it will surely be necessary that heavier mobiles are moved more slowly, since they undergo a greater resistance. And if, on the other hand, these things are true, it follows that heavier things, after they have consumed so much of the contrary resistance that they are no longer hindered by as much as lighter things are, must go down more speedily: which, again, experience surely shows.
But then again one must not silently pass over a great difficulty that arises hereFor, though heavier mobiles have more of the contrary quality to consume than lighter mobiles, yet they also have a greater heaviness, with which they can destroy it; since this is so, it seems reasonable that they should be moved at the beginning only at equal speed: and the cause
335 for which less heavy things must be moved more swiftly is not yet apparent. This objection is surely of great importance; but nevertheless it is not so powerful that it can obscure the splendor of the truth. Now in order that we may do away with it, it must be noted that the contrary quality in a mobile is not weakened because it is assailed by the heaviness of the mobile; for the heaviness cannot accomplish this, since in a mobile filled with the contrary quality it is entirely nullified; but it is by itself that that quality is weakened and leaves the mobile: as is also the case when white-hot iron grows cold: the heat in it weakens not because contrary coldness fights against it; for there is no cold in the iron at that time; but by its own nature it recedes from the iron bit by bit. Secondly, it must be noted that the contrary quality recedes more easily and the more swiftly, the lighter the mobile in which it has been impressed: this can be confirmed by many examples. Thus, if from the same cannon at the same time two acorn-sized balls, one of lead, the other of wood, are shot, then, beyond any doubt, the same force will be impressed in both; yet it will be kept more penetratingly and for a longer time in the lead than in the wood: an indication of this is the fact that the lead will be moved in violent motion farther and for a longer time. And the same thing is again evident, if someone projects, with the same hand, at the same moment, together, 2 pieces, one of wood, the other of iron; the iron or lead will be moved on a longer distance: which presumably indicates that, the motive force hangs on more penetratingly in the iron and is kept for a longer time in it than in the wood. {1}The same thing is evident if two weights, one of wood, the other of lead, are suspended from two equal threads and, when they have received an impetus from an equal distance from the perpendicular, they are released; of the two, the lead will certainly be moved back and forth for a longer interval of time. {1}And finally, it is manifest in all cases that all contrary qualities are kept longer, the heavier and the denser and the more contrary to them is the material in which they have been impressed. For if wood and lead are heated, in such a way that both of them are equally hot at the beginning, nevertheless it will be kept longer in the lead, even though the greater coldness of the lead is more opposed to the heat than the modest coldness of the wood. And this is manifestly evident in air, which, when it has been warmed directly and strongly by fire, if the fire is removed or covered with ashes, immediatly becomes cold: if on the other hand water is warmed by fire, I would not say to the point of boiling, but only so that it is as warm
336 as the air was, it will undeniably keep the heat for a long period of time; even though the coldness of water has far more distaste for heat than air. It is also evident that during summer stones or iron become much warmer than air, since hands are hardly able to stand the heat of the stone, and they keep this heat for a long time; but in winter these same stones come to be far colder than the air itself. And thus, on the basis of all these examples, it is evident that all contrary qualities inhere more strongly in heavier and denser material, and recede from it more slowly. Now with these things established, the solution of the problem is evident. For if in the wood and in the lead the contrary qualities weakened according to the same ratio, it would indeed be true that at the beginning of their natural motion they would be moved with the same speed: but, since a contrary quality is consumed more easily and more swiftly in less heavy material, it results from this that in the wood the impressed force is lost and recedes from the wood more swiftly; since this is so, it will necessarily be moved faster. But because the wood does not acquire as much heaviness as the lead, once the contrary quality has been lost, since lead, considered of and by itself, is heavier than wood, it results from this that the lead subsequently catches up with the wood and leaves it far behind.
But these things may be made clear even more easily by an example.
Let there be two mobiles, equal in size; but let one be of wood, the other of lead: and let the heaviness of the lead be 20, but of the wood 4; and let them be held up by line ab. In the first place, then, it is manifest that these same mobiles exert pressure downward with as much force as the force with which line ab impels upward. For, if they exerted more pressure, line ab would not curb them, but, in spite of the line, they would be carried downward: but in fact these same mobiles are not carried downward in air, because they do not exert weight on air, through which they must be carried (it has been demonstrated {1}, that nothing is carried downward, unless it is heavier than the medium, through which it must be carried), on the contrary, it is on ab that they exert weight; and, since they do not exert more weight on it than it exerts to hold them up, they are necessarily at rest. Now when they are abandoned by the line, at their first point of receding they still retain the contrary impressed quality impelling upward, which does not weaken in an instant but progressively; the lead has 20 units of the contrary quality to consume, and the wood 4. If these qualities weakened equally in each of them, so that, when 1 unit of lead receded, in the wood
337 1 unit also went away, and, consequently, they had recovered one unit of heaviness, both, indisputably, would be moved with the same speed: but, since in the time during which one unit of the quality has receded from the lead, more than one has departed from the wood, and, therefore, when the lead has recovered only one unit of heaviness, the wood has acquired more than one, it results from this that the wood is then moved faster; and, again, since when 2 units of the quality had receded from the wood, in the lead less than 2 had gone away, it is because of this that the lead is then moved more slowly. But, because in the end the lead acquires more heaviness than the wood, it follows that the lead is then carried much faster.
Older Works on Motion, Book II, chapter 10, the last one [337.11-340.16]Why projectiles moved by the same force are carried farther in a straight line {1}, the less acute are the angles they make with the plane of the horizon.
Anyone could raise this not inconsiderable difficulty on the basis of the things we have written above {1}, when we were discussing motion on different inclines to the horizon. For it is concluded from those things that, the more acute the angle that the plane over which the motion takes place will maintain with the horizon, the more easily a heavy thing will be able to be impelled upward; now we seem to be alleging the contrary; this rightly provides anyone with grounds for doubting. In order to make this difficulty disappear, in so far as there seems to be disagreement with what was said above, the following must be noted: when it was said above that heavy things are moved upward more easily, the more inclined the plane over which the motion takes place, this must be understood as applying to things that are carried over a solid plane; now, however, it applies to things which, not being supported by something else, but suspended in the air, are supported and impelled only by the impressed force; as when iron balls are thrust forth from cannons used for battering walls; these, it is certainly clear, are carried in the same straight line over a longer distance, according as the line of motion makes less acute angles with the horizon.In order that we may find the true cause of this effect, no matter what others say, it must be noted that, as we have also noted above {1}, an impelling force is impressed far more penetratingly in that which resists more, provided that the force does not fail through weakness: but if we could find some reason that the same heavy thing would resist more now
338 than before, it will now, undoubtedly, be moved more strongly by the force. But that which strives against something resists more than that which is either at rest or carried in the same direction: the force, then, is impressed more compactly in that which strives against it; those who play ball know this from experience; they want the ball to be thrown hard at them by someone, so that more motive force may be impressed in it, as it then opposes and resists more. But, as we have just said {1}, only those can accomplish this who are equipped with a strong and vigorous arm: but those with a languishing strudiness, and who are not able to strive against the impetus, put the ball in motion in the other direction when it is at rest or not moving in a contrary direction: but if the ball is moved in the same direction, as everyone knows, it is impelled only a little. As a matter of fact the cause of this effect is that a thing at rest that is struck with a very great force is moved before all the force is impressed, because its mobility {1} does not wait for such a great force to be impressed on it: this does not happen in the case of what is put in motion in the contrary directions {2} ; for, since its resistance has been increased by the movement of its heaviness {cf. my notes1below}, it resists more, and does not move back before all the force has been impressed. We all experience the same thing when we want to project a stone forward: for we first carry it swiftly backward with our hand so that, having been moved in a contrary direction, not only the stone but also the hand may resist more the force that must be impressed: but if, when it has been carried backward, we stopped the hand and that the stone, having been moved backward, came to rest, it could be thrown much less far, as is evident to everyone. It is thus necessary that the stone, to be thrown with a greater impetus, not come to rest at the turning point after it has been carried back.And the same thing is evident in the case of those who project a stone with a sling: for they first whirl the sling around two or three times in a circle, so that it may be moved faster; and finally they change this motion to a backward one, so that then a greater force may be impressed in the resisting stone.
These things having been noted, I bring forth a twofold cause for the present question: first I say that, even though the motive force in the cannon is the same, yet more of this same force is impressed in the iron ball, the more erect the cannon is set.The cause of this is that the ball then resists the force more: for it is moved with more difficulty in the ball chamber when it is to be impelled upward, as though on a more erect plane, than when it is more inclined; hence it also resists more the force that must be impressed on it. For when the cannon is low nearly falling, the ball then does not wait for the force that must be impressed, but it is shot out before the complete
339 impression: on the contrary, when the cannon will have been set upright, the ball exerts pressure on the black powder with its heaviness. and stands more in the way of motion on an erect plane, and it waits for the impression of a great quantity of force before departing. And one must not fear that the impelling force may be listless and weak: for it is so great that if the ball is so tightly squeezed in the ball chamber that it is incapable of being shot out, the cannon will be blown to bits; for such an amount of fire, compressed in such a narrow space, would destroy not only those bronze chambers, but also those a hundred times stronger; as is evident when it blows walls, ramparts, and whole fortifications sky-high. Now how great an amount of fire is then enclosed in a narrow chamber, one is able to judge, on the basis of the quantity of black powder, whose weight sometimes comes to 8 or 10 pounds, and that this is totally converted into fire, leaving nearly no ashes or any flame residues. How great is the amount of fire whose heaviness comes to 10 pounds, let those people imagine who assume that in fire there is no heaviness. So much for the first cause.
The second cause is that when the ball is carried upward perpendicularly to the horizon, it cannot deviate from that straight line, so as to turn back as is necessary through the same line, unless the upward impelling quality has first receded completely{1}: this does not happen when it is carried along a line inclined to the horizon. For in that case, it is not necessary that the impelling force be consumed when the ball begins to be deflected from the straight line: for it is enough for the violently impelling force that it removes the mobile from the starting point of its motion; this it can well accomplish so long as the mobile is carried along a line inclined to the horizon, even though in its motion it may be only inclined a little. For at the moment when the ball begins to drop, its motion is not contrary to a straight motion; hence the mobile will be able to change itself over to that other motion without the receding of the impelling force: but this cannot take place while the mobile is put in motion upward along the perpendicular, because the line of incline is the same as the line of violent motion. When, then, the mobile in its inclination does not come near to the place from which it is expelled by the impressed force, the force permits the mobile to drop {1}; for it is enough for it that it removes the mobile from the point from which it has receded; and it will more readily permit it to drop, the less that drop impedes the receding from the point of departure.
But if it is carried along the perpendicular line ab, the mobile can in no way deviate from that line, unless by going back over the same line, it comes near to the point from which it receded; but as long as the impelling force will subsist, it will never
340 tolerate that: when, on the other hand, the mobile is carried along line ac, since the downward inclination still tends toward the point of departure, the motive force will not permit it unless it is largely weakened: but when it is carried along line ae nearly parallel to the horizon, the mobile can begin to be inclined downward extremely swiftly {1}; for this inclination does not impede the receding from the point [of departure]. But the opposite of this takes place when the mobile is moved on various inclined planes: for on planes that are more inclined the mobile is moved farther by the same force on a straight line, than in planes more upright. The reason is that, the more the plane is inclined, the less the mobile exerts weight on it, because a part of its heaviness is sustained by the plane; from this it results that it can be moved more easily by the mover: and, since on the said planes the mobile cannot deviate from its motion except by turning back along the same line, it will be moved farther in a straight line on those planes on which the heaviness of the mobile that must be impelled is less.
Version II {1}
341.1 {1} As we will explain later that all natural motion of translation is the result <u>of an excess or a deficiency of heaviness,</u> we have thought it in accordance with reason first to bring forth for everyone to see how it should be said that a thing is <u>more, less </u>or equally heavy than another.Indeed, it is necessary to determine this: for it often happens that things that are <u>less heavy</u> are called heavier, <u>while things that are more heavy are called less heavy</u>.Thus, at times we say of a large piece of wood that it is heavier than a small piece of lead, even though, purely and simply, lead is heavier than wood; {1} and of a large piece of lead, we say that it is heavier than a small one, even though lead is not heavier than lead.For this reason, in order that we may escape pitfalls of this kind, those things will have to be said to be equally heavy to one another which, when they are equal in size (1), will also be equal in heaviness: thus, if we take two pieces of lead, which, equal in size, are also congruent in heaviness, they will have to be said to really weigh the same.Thus, it is clear that it must not be said of wood and lead that they exert weight equally: for a piece of wood which weighs the same thing as a piece of lead will considerably exceed the latter in size.Moreover, a thing should be called heavier than another, if when a piece of it is taken, equal to a piece of the other, it is found to be heavier than the piece of the other: as, for example, if we take two pieces, one of lead and one of wood, equal to one another, and the piece of lead is heavier, then we shall surely be justified in asserting that lead is heavier than wood.That is why, if we <u>take</u> a piece of wood which weighs the same as a piece of lead, <u>still</u> wood and lead should not be deemed to be equally heavy; for we will find that the size of the lead is considerably exceeded by the size of the wood.Finally, the converse declaration must be made about things that are <u>lighter{1}/less heavy</u>: it must be deemed that a thing is less heavy, if when a piece of it is taken, equal in size to a piece of the other, it turns out to be less in heaviness; as, if we take two <u>solids</u>, one of wood, the other of lead, which are equal in size, but the piece of wood exerts less weight than the piece of lead, then it must be deemed that wood is <u>less heavy</u> than lead.
342 That it has been established by nature that <u>heavier</u> things are nearer the center, and <u>less heavy</u> things farther away from the center, and why. {1}
<u>Up to now we have said "heavy" and "less heavy", and not "heavy" and "light"; likewise, we have said "nearer the center" and "farther from the center", not "downward" and "upward": for we are going to explain below that there is nothing light, that is, deprived of weight, and that there does not exist a place which is only up, and not also down.</u> {1}<u>And yet if sometimes, in order to express myself according to common usage (indeed a war of words offers very little interest for our endeavour), I say "heavy" and "light", and "upward" and "downward", let this be understood as standing for "less and more heavy", and "nearer the center and farther from the center"; until, when the occasion arrives, it is possible to determine more qualifications on this subject.</u><u>But as for what concerns the present matter</u>, {1} since things that are moved naturally are moved towards their proper places, and since things that are moved are either heavy or light, it must be understood which are the places of heavy things, and which are those of light ones, and why.<u>Of course, concerning the first,</u> the places of heavy things are those that come nearer the center, while the places of light things are those that are more at a distance from it, as we observe every day with our senses; consequently, that such determined places were prescribed for them by nature is not something that we may doubt: but it can be called into question why prudent nature has observed such an arrangment, <u>and not the opposite one</u>, in its distribution of places.
Now, from what I have read, no other cause of this distribution is adduced by philosophers, except that all things had to be arranged in a certain order, <u>and to distribute them in this one was pleasing to Supreme Providence</u>; and Aristotle seems to adduce a <u>similar [cause]</u> in Physics, book VIII, text 32, [255b15-17], when, asking why heavy and light things are moved towards their proper places, he supposes that the cause is because they are by nature suited to be carried somewhere, that is, the light upward, and the heavy downward. On the other hand, Ptolemy, at the beginning of chapter 7 of book I of his Syntaxis, says that it is futile to inquire about why heavy things are carried towards the center, after having demonstrated that the Earth, towards which they are carried, is in the center. {1}<u>But these [arguments] do not remove the difficulty: for given that they are carried towards the center because they are carried towards the earth, we are seeking, on the other hand, why the earth has been placed in the center and not in the place of fire. </u><u>But</u> if we examine the matter more attentively, we certainly shall not be able to think that there was no necessi
343ty or utility in nature's making such a distribution, but that it acted only according to whim or some kind of chance.Since I considered carefully that it was quite impossible to think this of provident nature, <u> from time to time I scrupulously tried to imagine some cause, which would be, if not necessary, at least appropriate and useful</u>: and indeed, I have discovered that it is not without the highest justification and the greatest prudence that nature has chosen this distribution.Indeed, since {1} there is but one matter of all bodies, and those bodies are heavier that contain a greater number of particles of this matter in a narrower space, assuredly it was in accordance with reason that bodies that would enclose more matter in a narrower place, should also occupy narrower places, such as are those that come nearer the center.If, for example, we understand that nature, when things were first being gathered together, had divided all the common matter of the elements in four {1} parts, and then assigned its matter to the form of earth, and that she has done the same for the form of air, and that the form of earth has made it so that its matter has been concentrated in a very narrow place, and that the form of air has made it so that its matter has been placed in a very wide place, was it not becoming that nature had assigned to air a grand space, and to earth a lesser one?Now, in a sphere, places are narrower the nearer <u>they</u> come to the center, and they are more ample the more <u>they</u> are at a distance from it: hence, it is with both prudence and fairness that nature decreed that the place of earth was that which is narrower than the others, that is, near the center, and that for the remaining elements the places were more ample, the rarer was their matter.I would not say, however, that the quantity of the matter of water is as great as that of the matter of earth, and that for this reason water, since it is rarer than earth, occupies greater places; but only that, if we take a part of water which weighs the same as a part of earth, and for this reason there is as much aqueous matter as earth [under consideration], assuredly this part of earth will then occupy a smaller place than the water, for which reason, justly, it will have to be placed in a narrower space. {1} <u>And, similarly</u>, the form of air filled a very large space with the same amount of matter as the form of earth enclosed in a narrow place: therefore nature must have assigned to air a larger space than to earth; <u>therefore, one farther from the center.</u>And thus, by proceeding in a similar fashion <u>in the case of fire</u>, we will find a certain suitability, not to say a necessity, to such a distribution of the heavy and the light.
Version III {1}
344 That it has been established by nature that heavier bodies are nearer the center, and those that are less heavy are more remote from the center, and why. {1}
After the marvelous assemblage of the most immense celestial sphere, the divine Creator, in order not to offend the eyes of the immortal and blessed spirits, forcefully drove away its excrements to the center of this very ball and hid them there: but, since its very dense and very heavy matter did not fill up by its size the ample and wide space left under the concave surface of the ultimate sphere, in order that this large space should not go unoccupied and be void, He tore apart that ponderous and confused mass, which, under the pressure of its heaviness, had confined itself within narrow limits; and from its innumerable particles, more or less rarefied, He formed those four bodies that we have later called the elements. <u>Of these, what was heaviest and most dense remained as it had been before; He did not remove it from the place where it had previously taken refuge. Thus earth was left in the center and, and according to a similar disposition, the things that were denser were placed nearer to earth.</u><u>And of the bodies which have been constituted from this matter, those were called denser which, being of the same size, contained a greater number of particles of this matter; and the denser were the heavier.</u>
<u>And so, that bodies were distributed in this arrangement by nature, that is, in such a way that those which were heavier remained nearer the center, constant experience declares to us:</u> but it can be called into question why prudent nature has observed such an arrangment, and not the opposite one, in its distribution of places.Now, from what I have read, no other cause of this distribution is adduced by philosophers, except that all things had to be <u>disposed</u> in a certain order, and to distribute them in this one <u>would be</u> pleasing to Supreme Providence.And Aristotle seems to adduce a similar one in Physics, book VIII, text 32, [255b15-17], when, asking why heavy and light things are moved towards their proper places, he supposes that the cause is because they are by nature suited to be carried somewhere, that is, the light upward, and the heavy downward.On the other hand, Ptolemy, at the beginning of chapter 7 of book I of his Syntaxis, says that it is futile to inquire about why heavy things are carried towards the center, after having demonstrated that the Earth, towards which they are carried, is in the center. But these [arguments] do not remove the difficulty: for given that they are carried towards the center because they are carried towards the earth, we are seeking, on the other hand, why the earth has been placed in the center and not in the place of fire.But if we examine the matter more attentively, we certainly shall not be able to think that there was no necessity or <u>at least</u> utility in nature's making such a distribution, but that it acted only according to whim or some kind of chance.Since I considered carefully that it was quite impossible to think this of provident nature, from time to time I scrupulously tried to imagine some cause, which would be, if not necessary, at least appropriate and useful: and indeed, I have discovered that it is not without the highest justification and the greatest prudence that nature has chosen this distribution.Indeed, since there is but one matter of all bodies, and those bodies are heavier that contain a greater number of particles of this matter in a narrower space, assuredly it was in accordance with reason that bodies that <u>would maintain</u> more matter in a narrower place, should also occupy narrower places, such as are those that come nearer the center.If, for example, we understand that nature, at the time of the original construction of the world, divided all the common matter of the elements into four equal parts, and then assigned to the form of earth its own matter, and in the same way to the form of air its own matter, and that the form of earth caused its matter to be concentrated in a very narrow place, and that the form of air caused its matter to be placed in a very wide place, was it not fitting that nature should assign to air a greater space, and to earth a lesser one?Now, in a sphere, places are narrower the nearer <u>they are to</u> the center, and they are more ample the more they are at a distance from it: hence, it is with both prudence and fairness that nature decreed that the place of earth was that which is narrower than the others, that is, near the center, and that for the remaining elements the places were more ample, the rarer was their matter.I would not say, however, <u>(as Aristotle believed)</u> {1} that the quantity of the matter of water is as great as that of the matter of earth, and that for this reason water, since it is rarer than earth, occupies greater places; but only that, if we take a part of water which weighs the same as a part of earth, and for this reason there is as much aqueous matter as earth [under consideration], assuredly this part of earth
346 will then occupy a smaller place than the water, for which reason, justly, it will have to be placed in a narrower space.And, similarly, the form of air filled a very large space with the same amount of matter as the form of earth comprehended in a narrow place: therefore nature must have assigned to air a larger space than to earth; therefore, one farther from the center.And thus, by proceeding in a similar fashion in the case of fire, we will find a certain suitability, not to say a necessity, to such a distribution of the heavy and the light. {1}
From this it may be concluded that no importance can be attributed to the argument of Aristotle by which he tries to establish that the [amount of] matter of the [various] elements is equal, when he says: If the matter of fire exceeded that of air and of water, then air and water, having been completely consumed by fire, would have transformed themselves into fire. {1}For even if we assume that fire exceeds air a thousand times, there will be no fear that air could be transformed into the nature of fire: for since all the place under the concave sphere of the Moon is already full, and, if air got to be fire, it would need a far more ample place than the one it now occupies, it is evident that it cannot acquire a fiery nature, because it is deprived of a space in which to be located.And the same must be deemed of the other elements.
346.18 That things that are moved downward naturally are moved by the excess of their weight over the weight of the medium. {1}
I will really be convinced of having shown that the most appropriate cause of natural descent is the excess of the heaviness of the mobile over that of the medium through which it must be carried, when the two following propositions have been demonstrated: first, that it is impossible for any bodies, if they are heavier than a certain medium, not to descend in it (if they are not impededed); second, that no body can exist which, if it is less heavy than the medium in which it has been placed, could descend naturally in it.The confirmation of these things can easily be drawn from the things that have been made clear and presupposed in the preceding chapter.For since it has been established by nature that heavier bodies remain under less heavy ones, it is therefore insofar as they are heavier that they remain at rest under less heavy ones: therefore the cause why heavier things are under less heavy ones is the excess of their heaviness.But that which results in remaining under things less heavy, is also what results in not remaining over things less heavy: and it is the same thing not to remain over less heavy things as to be carried under less heavy things: hence, the excess of the heaviness of the mobile over the heaviness of the medium is the cause of natural descent; and any things heavier than the medium through which they must be carried will, unless they are hindered, descend under it, lest, contrary to what has been instituted by nature, heavier things should remain over those that are less heavy.Since these things are so, it follows necessarily that any things that are less heavy than the medium cannot descend naturally: for things that are moved naturally are moved [to a place] where they rest naturally: but less heavy things cannot rest naturally under heavier ones: therefore, they do not descend naturally either.And so, from these things it is established that, from the assumption of an excess of heaviness, motion downward always follows (in the absence of external hinderances), and, if the same excess is removed, natural descent is also taken away: it thus follows that the said excess of heaviness is the cause of natural descent.
But here someone could rightly raise a doubt about how a mobile which is carried downward naturally is always heavier than the medium through which it must be carried; especially since we see that a small pebble descends in a great amount of water, the pebble surely being much less heavy. Therefore, in order that we may do away with this and any other difficulties, and so that what has been said may be more clearly explained, we are going to write some demonstrations through which the solution of the whole matter will be evident.Thus we will first explain the terms that need explanation, and I will presuppose the axioms which will be necessary for the demonstrations; then, we shall hurry on to the demonstrations themselves. {1}
347.24 Which things must be said to be more, which less, and which equally heavy. {1}
Let us first bring forth for everyone to see how it should be said that a thing is more, less, or equally heavy.Indeed, it is necessary to determine this: for it often happens that things that are less heavy are called heavier, while things that are more heavy are called less heavy.Thus, at times we say of a large piece of wood that it is heavier than a small piece of lead, even though, purely and simply, lead is heavier than wood; and of a large piece of lead, we say that it is heavier than a small one, even though lead is not heavier than lead.For this reason, in order that we may escape pitfalls of this kind, those things will have to be said to be equally heavy to one another which, when they are equal in size, will also be equal in heaviness: thus, if we take two pieces, <u>one of silver, the other of steel, </u>which, equal in size, are also congruent in heaviness, they will have to be said to really weigh the same.Thus, wood and lead<u> must not be said to</u> exert weight equally: for a piece of wood which weighs the same thing as a piece of lead will considerably exceed the latter in size. [see note below]Moreover, a thing should be called heavier than another, if when a piece of it is taken, equal to a piece of the other, it is found to be heavier than the piece of the other: as, for example, if we take two pieces, one of lead and one of wood, equal to one another, and the piece of lead is heavier, then we shall surely be justified in asserting that lead is heavier than wood. [see note]That is why, if we <u>assume</u> a piece of wood which weighs the same as a piece of lead, still wood and lead should not be deemed to be equally heavy; for we will find that the size of the lead is considerably exceeded by the size of the wood. [see note below]Finally, the converse must be <u>deemed</u> to hold of things that are less heavy: it must be <u>decreed</u> that a thing is less heavy, if when a piece of it is taken, equal in size to a piece of the other, it turns out to be less in heaviness; as, if we <u>take</u> two solids, one of wood, the other of lead, which are equal in size, but the piece of wood exerts less weight than the piece of lead, then it must be <u>asserted</u> that wood is less heavy than lead. [see note]
<u>These are the things that had to be said concerning the definitions of terms.</u><u>Now, in order that we may more easily come to the things that have to be demonstrated, let the following axiom be assumed: namely, that what is heavier cannot be lifted by what is less heavy, other things being equal.</u>But for the things that have to be said we also need the following lemma.
Lemma for what follows.{1}
Heavinesses of unequal sizes of equally heavy bodies have to one another the ratio which the sizes have.
Thus let a, b be unequal sizes of equally heavy bodies, and let a be greater; a will then be heavier than b.Let c be the heaviness of a and let d be the heaviness of b: I say that heaviness c has the same ratio to heaviness d as size a has to size b.For let the sizes a, b be augmented by any number of multiples whatever: let size efg be a multiple of size a, and let size hk be a multiple of size b, but in such a way that size efg exceeds size hk: and let heaviness nop be as many multiples of heaviness c, as size efg is of size a; and let heaviness lm be taken to be as many multiples of heaviness d, as size hk is of size b.Now, because size efg and weight nop are equal multiples between, respectively, a and c, as much size their is of efg, equal in size to a, as much will their be of weight nop, equal in weight to c: and because weight c is equal to the weight of size a, and that weight c is equal to the weight n, and size a to size g, the weight of n will be equal to the weight of size g.Similarly it will be shown that weight o is equal to the weight of size f, and the weight p with the weight of size e: consequently, the weight of size efg taken as a whole will be nop.Now, it will also be similarly shown, that the weight lm is equal to the weight of size hk.Now, it has been assumed as true that size efg is greater than size hk: hence, the weight of this same efg, which is nop, will be greater than the weight of size hk, which is lm.Similarly we will show, for any multiple whatsoever, that if size efg has been greater than size hk, weight nop also is greater than weight lm; and if efg had been smaller or equal to hk, in the same way would nop have been smaller or equal to hk: and efg are equal multiples of a, also nop, respectively to c; and, for any multiple whatever, hk is an equal multiple of b, also lm for d: accordingly, by definition of an equal ratio, as size a is to size b, weight c is to weight d.Which was what was to be demonstrated.
These things having thus been inspected, let us come near now to explain how the cause of downward motion is the excess of the weight of the mobile over the weight of the medium: because, as we have already made clear earlier {1} it will then be obvious, when that will have been shown, that nothing can naturally be moved downward, less their is such an excess of weight over that of the medium; and that nothing, in a medium which is exceding in weight, can go down in it, unless it is hindered.Now, because the media, through which motions are made, are varied, and because water is a very suitable medium through which can be understood motions, upwards as well as downwards, we will put ourselves in a situation to make observations concerning motions as happening in such a medium: and, in the first place, we will demonstrate, that any solid bodies whatever, equally heavy with water, let down in it, are surely totally submerged; but they are not yet moved downward in water.Now, let it be remembered what was said earlier, that is, those bodies, which between
350 themselves, on the one hand are equally heavy, and on the other present themselves as equal in size, weight equally: that is why, if there were a certain body equal in size with water, which would weight equally, this body will weight equally with water.
Any solid bodies whatever, equally heavy with water, that will have been let down in water, certainly submerged, but are not yet carried downward.
Thus, let it be understood a certain body equally heavy with water, and that this body ef be in water; let also water abcd, whose surface ao is that which it had before body ef is sent in it: I say that solid ef, sent in water, is then totally submerged, but is not yet carried downward.Thus, that it be sent; and, if such a thing can be done, that it not be totally submerged, but that a certain part pops out of the water, and let that part be e.It is then necessary, while solid ef is submerged, that the water be raised: indeed, it is a must, in order that the place, in which the solid enters, be evacuated of water.That is why, while the solid is submerged, water is raised up to surface st; and, if such a thing can be done, let both water and solid remain in this state of rest. And since the solid ef pressing with its heaviness has lifted water so, water so will not be heavier than the solid ef; indeed, it has been assumed{1} that heavier things cannot be raised by things lighter{2}: thus it is true that solid ef will not be heavier than water so; because, if it is heavier, it would press and raise some more, and, consequently, it would be submerged more, whereas it is on the other hand assumed to stand in this state of rest: hence it resists in such a way that as much as there is heaviness of water so that resists in order not to be raised any more, as much is there of heaviness of solid ef that presses and raises the water.Now, the solid exerts pressure with the totallity of its heaviness, whereas the water similarly resists with the totallity of its heaviness; hence the heaviness of solid ef is equal to the heaviness of water so.But then again: the size of water so is less than the size of the totallity of solid ef; because it is equal to as much size as there is of submerged size under water.This indeed is evident: because as much as there is of size of water of the place, in which the solid has entered, that have been expelled, as much is there of size of the solid that has been submerged; hence the size of water so is equal to this part of the size of the submerged solid, namely part f.Thus then are 2 bodies, one which is water so, the other solid ef,
351 and it has been demonstrated that the heaviness of water so is equal to the heaviness of the whole of solid ef: now, the size of solid ef is greater than the size of water so: hence body ef is less heavy than water ( indeed of two bodies equal in heaviness, but unequal in size, the greater in size is the less heavy).Now this is contrary to the assumption; indeed it has been assumed that the solid ef be equally heavy with water: that is why no part of this same ef will pop out of the water: hence il will be totally submerged.Which has been what had to be shown.I say in addition, that, being totally in water, it is not yet that it is carried downward.Since indeed it is equally heavy with water, to say of it that it comes down in water would be like if we were saying that, water in water comes down under water, and that water coming back on its tracks, first goes up above, and comes down again downward, and that thus water is alternatively going without end coming down and going up again; which would be unsuitable.
After we have inspected that bodies equally heavy with the medium through which they should be carried do not come down, it follows that we should show that things that are less heavy, similarly in no way can be moved downward: and thus the first part of our proposition will have been demonstrated; namely, that it is impossible for a thing which does not exceed in heaviness the medium, through which it must be carried, to be carried downward.
351.20 [#6] Bodies, whatever they are, less heavy than the medium, that will have been let fall in it, not only are not carried downward, but cannot even be totally submerged.{1}
Let the first state of rest of water, before the body is sent in it, be along the surface ef; and let a certain body a, less heavy than water, which is sent in the water, if such a thing can be done, be totally submerged, and let the water be raised up to the surface cd.And thus since body a, exerting pressure with its heaviness, could raise water cf upward, solid a will not be less heavy than water cf ( it is indeed assumed , that less heavy things cannot raise heavier ones); but size a is equal with size cf: thus there are two bodies, a and cf, and the heaviness of this a is not less than the heaviness of this cf, now, size a is to be taken as equal with size cf: hence body a will
352 not be less heavy than water: which is contrary to the hypothesis; indeed we have assumed that body a is less heavy than water.That is why it is evident, it is impossible for bodies less heavy than the medium that they can be totally submerged; that is why they will even less be able to be moved downward.
This having thus been demonstrated, we can unquestionably claim the first part of the proposition of which we had to substantiate the demonstration, namely, it is impossible for something to be moved downward, if it does not exceed in heaviness the medium through which it must be carried.Now, it is with statements of a similar kind that we will have confirmed the second part, namely, that bodies, whatever they are, heavier than the medium, that will have been let fall in it, necessarily go down, unless hindered.First of all, indeed, it will be confirmed in the following manner: those are the ones that are moved downward, as we daily see it; and things that are moved downward, necessarily exceed the medium in heaviness: hence, conversly, any of these things that exceed the medium in heaviness, will be moved downward.Secondly: if these things that exceed the medium in heaviness are not moved downward, either they will float on the surface, or they will be submerged in such a way as not to go down yet.If they are not submerged, they will , contrarily to the hypothesis, be less heavy than the medium: and if they are totally submerged, in such a way however that they be not yet carried downward, they will be equally heavy with water; which similarly is contrary to what has been assumed: hence what is left is that they necessarily are carried downward.All these things find themselves such that, by us, they reasonably attest that the cause of natural motion downward is the excess of the heaviness of the mobile over the heaviness of the medium through which it is carried: which is what we had to substantiate the confirmation.
After that up till now, as long as that has been possible, we have tracked and disentangled the cause of natural motion downward, there remains to undertake the discovery of the cause of upward motion.But, because our ways of thinking on the subject of upward motion are far different from those that have been reported on this regard by Aristotle and the Peripatetics, before we inquire about its cause, we will first demonstrate, against the opinion of the Peripatetics, that any upward motion is counter-natural.
352.31 [section #7]There exists no natural motion upward.
In oder for us to be in a position to explain and confirm more appropriately this opinion of ours, it is necessary to inspect what are the conditions required by this motion for it to be called natural: for as much as they can be found in an upward motion, it will be natural; if not, it will not be natural.Now, there are two of these conditions: one of course which is taken from the side of the sole motion, in as much as it is a motion totally unrelated either to the mobile or to the medium; the other is taken from the side of the mobile. The condition on the side of the motion, in as much as it is pure motion, is that it cannot exist without end and up to the indeterminate, but that it be finite and delimited: {1} because things that are moved by nature, are carried towards a certain limit, in which naturally in can be at rest.The condition on the side of the mobile is that it be moved by a cause not external, but internal. {2}None of these conditions exist in the motion that happens by the medium; hence it cannot be called natural. We will first seriously examine what concerns the first condition, and a little latter what concerns the second.
And thus I say that motion upward, by the fact that it is a certain receding from the center, cannot be natural: and I prove this in the following manner.There exists a certain limit to natural motion: but no limit can be found to upward motion: hence upward motion is not natural.The major is evident: nature indeed does not move what it can never reach; hence, in a certain limit.The minor, namely that there does not exist a limit to upward motion, is proved this way: the limit of any motion is the one which this same motion of which it is the limit cannot be receded from; but in upward motion cannot be assigned a limit, from which this same motion could not be removed, namely upwards; hence there exists no limit to this upward motion.The major is evident: if indeed we could have receded with this same motion by walking, then there would be no limit to this motion; as, for example, it will not be said of Rome that it is the limit of southward motion , because, after we arrive in Rome in a southward motion , we can by going with the same motion recede from this city, namely with a southward motion. The minor, similarly, is as true as can be: for, whatever place is assigned in this upward [motion], we can depart from it by proceeding with the same motion, namely upward motion, receding from the center; for there is no distance from the center that is so great that a distance greater
354 than it cannot be conceived.And thus upward is deprived of a limit; hence nothing can be moved naturally upward.
But now that we deny to upward motion the character of being natural, let us consider how appropriately it applies to downward motion.Indeed in downward motion there exists a limit, namely the center, which something cannot be receded from during the same species of motion as the one it had when it had come near to it: indeed it had come near by a downward motion; and if it wishes to be deviated, it will be carried upward.Remoteness from the center is indeterminate and without end; but closeness is limited, namely by the center itself: hence something which will be endowed with such an ability, in such a way as to flee the center, this thing for sure will be suited to move without end; could anything be more absurd ?And thus we will say reasonably that there exists a natural motion towards the center, as for the one away from the center, it is contrary to nature.
And do not serve me the objection of the widespread axiom: Once assumed a contrary in the nature of things, also is assumed the remnant: now, a natural motion downward is given by nature; hence will also be given a natural upward motion. Because, to this given axiom, I retort, in the first place, it is one thing to say, Given in nature a contrary, and is given its opposite; it is another thing if we say, One of the contraries exists in nature, and hence concerning the remnant, similarly, it is necessary that it be according to nature.We concede the first, we deny the second.Thus downward motion is given in nature; is also given in nature an upward motion, its contrary.If you then add, The downward motion exists according to nature, hence the upward motion will be according to nature; that is denied.On the contrary, by turning the argument around, we will reason in this manner: Given a contrary in nature, is given also its opposite; but in nature is given a natural motion downward; hence will be given the opposite of this, which will be the conternatural motion upward. {1}
These are the things with which it can be made evident, that motion away from the center, in as much as it is a motion, which has no relation whatsoever either with the mobile or the medium, through which the motion occurs, cannot be natural.Now, for us to be able to explain the arguments with which the things on the side of the mobile are fitted, it is first necassary to consider that bodies have heaviness.
Against the opinion of Aristotle, their exists no body deprived of heaviness.{1}
Up till now we have not even said the word light, but only heavy and less heavy; that is why this place offers the oportunity to examine whether it was righly or not that we have done such a thing.Thus then if Aristotle and the other philosophers had been satisfied with taking for light what we call less heavy, we would not find it heavy to admit, also, this appellation of light: on the contrary, because they have wanted (not satisfying themselves with understanding by light what is less heavy) that be given, in addition, a certain light body, which would purely and simply be such and would be deprived of all heaviness, abhorring this more than a dog a snake, we are in the obligation to knock down altogether and from its foundation even the light itself.For this reason, following in this the opinion of the ancients, that Aristotle tries vainly to destroy in Book IV of the <i>De Caelo</i> <u>we will examine not only Aristotle's refutations that are there, but also his confirmations, confirming on one hand what has been refuted, refuting on the other what has been confirmed; and we will accomplish this when we will have expounded Aristotle's opinion.</u>
Thus Aristotle has wanted that be given a certain body that is purely and simply the heaviest, of which it could never be said that it can be light, and heavier than which nothing could ever exist; similarly, that be given its opposite, something that would be purely and simply light, that has in it no heaviness whatsoever, and lighter than which nothing could be found.And now, in the first place, defining the heaviest, <u>the purely
356 and simply heaviest, says he, is what we say stands under all things and which always is carried towards the center; in the second place, he calls the lightest, that which stands above all things and is always upward, and which besides is never moved downward: and he has written that in texts #26 [311a 16-18] and #31 [311b 16-18] of Book IV of the De Caelo.</u><u>He then says that earth is what is the heaviest, and that the lightest is fire: and that is in text #32 [311b 19-29] and in others places</u><u>Now, against those who assume a certain heaviness in fire, he argues thus: If fire has a certain heaviness, it will stand under something; well then, that is not seen; hence [etc.] </u><u>This argument is not conclusive</u><u> Because for something to be above something else, it is sufficient for it, to be less heavy than the one above which it must be; now, it is not necessary that it be deprived of all heaviness: like in the case of wood which floats in water, it is not required that it be deprived of all heaviness, but is enough that it be less heavy than water; and thus, by a similar reasoning, in the case of fire, for fire to stand above the air, it is enough that it be less heavy than air, and it is no way necessary that it be deprived of all heaviness.</u><u>That is why it is evident that this argument bears no necessity</u>
<u>He also argues in the following manner: If fire has a certain heaviness, then a lot of fire will be heavier than a little bit; and for this reason a lot of fire will go up in air more slowly that a little bit of fire: hence, if earth has a certain lightness, a lot of earth, which will have more lightness, will go down more slowly than a little bit: experience however shows the contrary: indeed we see that, a lot of fire goes up more swiftly than a little bit, as a lot of earth goes down more swiftly: this is thus a sign that in fire there is only lightness: and since in a lot of fire there is more lightness, it goes up more swiftly.</u><u>This argument also is very weak</u>First, indeed, Aristotle is not self-consistent: talking indeed of absolute heaviness and lightness, without it having any relation with something else, he produces an example from which nothing can be concluded, unless it be that fire is less heavy than air, and that earth is heavier than water or air. <u> What follows is not good, If fire considered absolutely had heaviness, a lot of fire in air would be heavier than a little bit : indeed we are not saying that fire is heavy in air, but only that it is heavy.</u><u>But, one should argue this way: Fire, considered absolutely, has heaviness: hence, where fire has heaviness, a lot of fire will have a lot of heaviness; and where fire has lightness, as in air, there a lot of fire will have a lot of lightness, and a little bit, will have little. It is thus certain that there is an error of Aristotle in arguing. </u><u>It is thus certain that there is an error of Aristotle in arguing. </u>For if this mode of argumentation had any worth, we too could demonstrate, that any piece of wood has no heaviness, by staging a scene like the following: If wood has a certain heaviness, therefore a big piece of wood will be heavier than a small one, that is why a big piece of wood will go up more slowly in water than a small one: of which however experience shows the opposite; indeed a big piece of wood even springs out of water with an impetus upward greater than a small one.And who has ever said that wood, in so far as it is wood, without having any relation with the medium in which it goes up, is deprived of all heaviness?It is thus that while wood is above water, not because it is absolutely deprived of heaviness, but only because it is less heavy than water; {1}then, in a similar way, fire is above air, not because it has purely and simply no heaviness whatsoever, but because it is less heavy than this very air.<u>Second: {1} since it is presupposed as very true by Aristotle, namely that a lot of fire goes up more swiftly than a little bit, or that a lot of earth goes down faster than a little bit, perhaps this is false, as we will demonstrate in its proper place{2}.</u>
<u>Thirdly, he argues: If fire has heaviness, then a lot of fire will be heavier than a little bit of air; which is very absurd, as if we were saying, If earth has a certain lightness, a certain part of earth will be lighter than a certain part of water: which, says he, is false, because we see, that any part of earth goes down in water, and any portion of fire in air is carried upward. </u><u> On the contrary this argument is weaker than the rest of the others: now who has such a disposition of mind not to believe, that, a lot of water is heavier than a little bit of earth, and a lot of air than a little bit of water, and a lot of fire than a little bit of air?</u><u>And what Aristotle says does not hold: We see earth go down in water. </u><u>For, when he says this, he is not self-consistent: </u> because, talking of absolute heaviness, he assumes an example concerning heaviness in relation to the lesser heaviness of the medium. <u>That is, when we say that water has heaviness, </u>, and, then, that an importante size of water is heavier than a little bit of earth, <u>we are not saying that it has heaviness in its own region, </u> nor that a lot of water is heavier than a little bit of earth in water, <u>there where water has no heaviness, as will be demonstrated below{1}: we assert, that a lot of water is heavier than a little bit of earth in a place where water also has heaviness, as, for instance, in air. </u>Moreover, what follows is not deduced correctly, Any particle of earth goes down in water, hence it is necessary that this particle of earth be heavier than any size of water: for, as has been demonstrated above, in the case of a particle of earth, for it to go down in water, it is sufficient that it stays heavier than a size of water equal to its own size.And the same must be said of fire: a great portion of which will be heavier than a little bit of air, but
358 not in the place of air, where fire cannot exert its heaviness, but in a place where fire also weights.<u>For if Aristotle's argument had any necessity, I could also conclude, a little bit of lead is heavier than a very large beam, because, it can be seen, that lead in water has a certain heaviness and is carried downward, , whereas the beam never: and what happens is the contrary, a little bit of lead is heavier than a beam in a place where the beam has no heaviness, but if we want to talk of the heaviness of the beam, the beam must be assumed in a place where its heaviness could show itself. </u><u>Similarly, when he says, Any particle of water in air goes down, hence as much air as one wants is lighter than a particle of water; this will be true in this place, where air has no heaviness, whereas water has: but this will not be talking of absolute heaviness, as we have been doing here. </u><u>For if we assume a lot of air in a place where air weights, as in fire or in the void, there surely it will be heavier than a little bit of water. </u>And not what Aristotle infers from this, Hence a lot of air goes down faster than a little bit of water.What follows indeed has no worth, This is in any case heavier than that, hence it will go down faster: for a big inflated bladder will be heavier in air than a little bit of lead; however it will not go down faster.But, [we will talk) more extensively on this subject, where we will deal with the cause of greater or lesser speed.{1}Also in the same manner, it is not because a lot of fire is heavier than a little bit of air, that we shall say that fire must go down faster.
Fourth, he argues: <u>Two are the contrary places, the center and the extremity, taking as the extremity the concave sphere of the moon; hence it must be from these that the contraries exist; which will not be the case, unless earth be assumed to be deprived of all lightness, as fire be devoid of all heaviness</u>. For a great number of reasons the argument is of no strength whatsoever. In the first place, indeed, neither is earth in the center nor fire in the concave sphere of the Moon; indeed the center is not a place, since it is an indivisible point; as for fire only the convex surface is on the concave sphere of the Moon: that is why from this nothing can be concluded, if not only that the center of the earth be the contrary of the convex sphere of fire: and the center of the earth is in no way a part of this same earth, as the convex surface of fire is in no way a part of fire: hence from this nothing whatever could be infered concerning earth and fire.Moreover: earth is no more in the center than on the concave surface of water or air, no more than fire is on the convex surface of air: that is why he should show in what way the concave surface of air could be in opposition to the convex.But if this is so, there will be air which is maintained between both surfaces in contrary places; and the air (if Aristotle's way of arguing holds) which is under its own convex
359 surface will be contrary to the air which is above its concave surface. Moreover: as Plato has well written in the <i>Timaeus</i>[62-63], the concave [of the sphere] of water and air is also in opposition to the center in the same way as the concave of the sphere of the Moon; nevertheless, the things that are under the concave [sphere] of air are not in opposition to those that are around the center.It is thus evident that such an argument is of no moment {1}.
Fifth, Aristotle asserts{1}: <u>If air is removed from underneath it, fire will not go down, any more than air if water is removed from underneath it</u>; hence this is a sign that fire does not have heaviness. The antecedent lacks a demonstration: <u>Aristotle has not proved it, unless you say what he has said, Just as earth does not go up in the physicians' cups because it is very heavy, so fire will not go down because it is very light.</u><u>But the ratio has no worth: for, it is not because it is very heavy, that earth does not go up, but because it is not fluid; for neither wood would go up, although it is lighter than water, which does go up; however mercury would go up, although it is heavier than earth, because it is fluid; and thus fire would go down, because it is not solid and hard but flowing.</u>
These things [concern] Aristotle against the ancients, and us in favor of the ancients; but let us now proceed against Aristotle himself.First, then, Aristotle defines heavy and light through downward and upward: if then the purely and simply heavy and the purely and simply light, which is deprived of all heaviness, are granted, then the purely and simply downward and the purely and simply upward, than which nothing can be more upward, must also be granted. Now the purely and simply upward, than which nothing is more upward, and which could not exist downward, not only is not found in fact, but it cannot even be conceived by the mind: concerning the downward, though there may be something that is downward in such a way that there could not be anything else more downward, still it is not such that any body could be in it, since it is an indivisible point. Hence since these things are not given, something that is so heavy that nothing else heavier than it could be given, will not be given either -- nor something that is lacking in all heaviness.<u>Second: if the elements, as he himself wants it {1}, are transmuted into one another, when fire is made from heavy air, what happens to that heaviness of the air?</u><u>Could it be that it is annihilated? </u>But, if it is annihilated, when in its turn earth is made from fire, where does the heaviness come from? Perhaps heaviness, which is something, comes from non-heaviness, which is nothing?<u>Third, if fire is deprived of all heaviness, it will hence be deprived of all density ; the dense indeed follows the heavy: but what is deprived of all density, is the void: hence fire is the void. </u><u>What could be more absurd? </u><u>But, precisely, how can anyone ever
360 imagine fire, a substance linked to quantity, not having any heaviness? </u><u>This surely seems totally unreasonable. </u>And when we say that fire is the lightest of all things and that earth is the heaviest of all things, we are compelled, whether we like it or not, to say that earth is the heaviest, in comparison with all other things, because it stands under all other things.<u>For to stand under all things and to be the heaviest of all, is the same thing: and that is evident; for, if it is the heaviest because it stands under all things, if all things are removed, it will no longer be able to be called the heaviest, since it stands under none of them.</u><u>It is therefore said of it that it is heaviest in comparison with things that are less heavy, under which it stands; and the same must be said concerning the lightness of fire. {1}</u>
Let us conclude, then, that no body is deprived of heaviness, but that all things are heavy, some more, others less, according as their matter is more concentrated and compressed, or else expanded and spread out: from this it follows that it cannot be said that fire is purely and simply light, that is, that it is deprived of all heaviness; for this belongs to the void.<u>However, I would not say that there cannot be found in the nature of things a body, heavier than which no other one can be found, and similarly another, than which there is nothing less heavy</u> for we concede this, since we know as a matter of fact that there is not an infinite number of bodies, one heavier than another: but we say that they are not such that there could not be others still more and still less heavy, and for that reason they cannot be called purely and simply heavy or purely and simply light, as being deprived of all heaviness.We also say that, those things, which are more or less heavy than others, are perhaps not earth and fire.For concerning earth experience already teaches that it is not the heaviest of all things: for it floats on all liquified metals, as on what is called quicksilver; from this it is evident that there exist metals heavier than earth itself. Similarly, perhaps there will be certain vapors, floating above fire, less heavy than fire is: but we do not confidently affirm this, for we have not been above fire.But if comets are burning exhalations
361 , as the Peripatetics themselves testify, one thing is sure, that these vapors themselves must have flown above fire; since several comets far above {1} what they assume, have been observed higher than the region of air.
Now that these things have been explained, let us return to those which remained to be exposed at the end of the preceding question.
361.6-363.4 Chapter #9. It is proved that there cannot exist, on the part of the mobile, a natural upward motion. {1}
On the part of the mobile, no one doubts that upward motions cannot be natural in all cases which have an observable external cause, as when a stone expelled by force is carried upward in air: for it is evident that such a motion is not natural, because it does not tend towards [a place] where the mobile may be at rest; for the mobile immediately changes itself spontaneously to the downward motion that is proper to it.And so, the problem to be solved concerns that upward motion when the mobile is carried to [a place] where it is at rest; as when wood or air are carried upward in water, and fire in air: on this subject, even if it goes against the way of thinking of all the Peripatetics, we have resolved that we should assert that it cannot be said to be truly natural; we have confirmed this conclusion, as much as was possible, on the part of the motion itself, independently of any relation either to the mobile or to the medium; in a similar way we will now try hard to confirm it on the part of the mobile.
FIrst, then, that which has no intrinsic cause of its own motion, but is in need of an external one, cannot be said to be moved naturally: but such are the bodies that are moved upward: hence they are moved contrary to nature. The major is evident: for things that are moved accidentally, by an external cause, are moved by something else and not from their own nature.The minor is also evident: for since every body has an internal cause of downward motion, namely its heaviness, it is impossible for it to have the contrary cause of the contrary motion.And do not say that, since the excess of heaviness of the mobile over the heaviness of the medium is the <i>per se</i> and intrinsic cause of downward motion, thus the deficiency of heaviness of the mobile, compared to the heaviness of the medium, is the <i>per se</i> and intrinsic cause of upward motion: for the <i>per se</i> cause of downward motion is the absolute heaviness of the mobile; but it is by accident that the said heaviness must exceed the heaviness of the medium, just as it is by accident
362 that a mobile is moved downward in a medium that has some heaviness.For a heavy body -- though the medium may have no heaviness, and because of that its own heaviness would not exceed the heaviness of the medium -- would neverthless be moved downward, because it has an intrinsic cause of descent: but it is impossible to make such a judgment concerning a deficiency of heaviness; for since deficiency of heaviness, namely the non heavy itself, is nothing, a medium heavier than the mobile is necessarily required for the mobile to be able to be called deficient in heaviness.Since, then, the mobile cannot be nonheavy, except through the presence of a heavier medium (for no body by itself is without heaviness), it follows that the nonheaviness of the mobile depends entirely on the heaviness of the medium (for if the heavy medium is not present, the mobile will no longer be nonheavy, but it will remain heavy): since this is so, the very nonheaviness will be external to the mobile, as it both comes from something else and requires an external heaviness.Hence, if this nonheaviness is the cause of upward motion, it will be extrinsic and will come to the mobile from some other thing: since the mobile does not have an internal cause of motion, it will be impossible for it to be moved according to nature.
And so there is a dissimilarity between upward and downward motion: because in downward motion the mobile does not need a medium, from which to receive the cause of motion; for it has an intrinsic heaviness as the cause of downward motion; its motion in fact is hindered by the medium, since its own heaviness is diminished by the medium, as will be demonstrated below {1}: but the cause of upward motion depends so much on a heavy medium that in a nonheavy medium nothing whatsoever could be moved upward, since the nonheaviness of the mobile comes entirely from the heaviness of the medium. {2}But what need is there of more [words]?There is a single matter of all bodies, and in all of them it is heavy: but of this same heaviness there cannot be contrary natural inclinations: hence, if there is one natural inclination, it is necessary that its contrary be contrary to nature: but the natural inclination of heaviness is towards the center: hence, it is necessary that the one which is away from the center is contrary to nature.
I can, however, believe that the error of those who have thought that motion away from the center is natural owes its origin to this: that they have not been able to find the external cause by which mobiles were moved; and thus have been compelled to assume an intrinsic one, and to call it lightness.Because of this, in order that we may get rid of an error of this kind, let us now hasten to explain how things that are carried upward are moved by an extrinsic cause, namely by the medium itself, through extrusion.
Chapter #10 [363.5]. That the things that up to now have been said to be moved upward naturally are moved, not by an internal cause, but an external one, namely by the medium itself, through extrusion. {1}
If, then, things that are moved upward are moved contrary to nature, it is necessary that they have an external cause of their own motion.Now we say that this cause is the extrusion of the medium: which happens as follows.
First, then, it is necessary that things that are moved upward be less heavy than the medium through which they are carried.For it has been demonstrated that things which are heavier than a certain medium are carried downward in it; and things which are equally heavy are moved neither upward nor downward, but remain at rest in it: therefore it necessarily follows that bodies which are carried upward are less heavy [than the medium].And so, when a certain body less heavy than a certain medium has been submerged in it, the parts of the surrounding medium, exerting pressure with their heaviness, try to expel the body from below, in order that they themselves may occupy the lower regions.Now if the resistance they encounter in that body is less than the force with which they themselves exert pressure, they overcome it and extrude it: but the resistance of the mobile against being raised will be less, whenever its own heaviness is less than the heaviness of the medium exerting pressure: hence it will then be extruded.
But in order that the whole matter may be better understood, let us bring forth an example for everyone to see.Thus, consider a certain body a, which is less heavy than the medium, let us say, water; and let the surface of the water, before body a is submerged, be along the line bc; when a is submerged, let the water be lifted up to the surface de.Then it is clear that, if body a could not be held there, water dc would drop back into its place.Now water dc exerts pressure, trying with its own heaviness to expel body a, in order that it may itself occupy a lower place; and a resists being raised with
364 its own heaviness. But if the heaviness of water dc is greater than the heaviness of a, it will overcome it, and it will extrude a: but since the size of water dc is equal to the size of a and body a is assumed to be less heavy than water, the heaviness of water dc will be greater than the heaviness of a: hence a will be extruded by the water and expelled upward.
Such is the manner in which the medium can force out and extrude bodies that are less heavy than itself.But against what we have said, certain objections can be raised.And so, we will raise these objections, in order that from their solutions the truth of what we have said may appear with more precision .First, then, if the matter is as we said, why {1} is a mass of lead not extruded from the depths of the sea, since the sea is far heavier than it?To this I respond: for a plate of lead to be extruded from the sea, it is necessary that an amount of water equal in size to the size of the lead be heavier than the lead.For the portion of water that can extrude the lead can only be the one whose place the lead occupies, and it is this portion which, by an extruding action, can enter the place where the lead previously was: for the water which has suffered injustice at the hands of the lead, while its place was occupied by the lead, is only of a size equal to the size of the lead; and only this portion, by an extruding action, is what can enter into the place left by the lead. Now if the heaviness of this water does not surpass the heaviness of the lead, it will certainly not be strong enough to raise it: but it is presupposed that the lead is heavier: it is thus not astonishing if it is not extruded.But if the body had been less heavy than water - as if, for instance, you consider a sphere of wood at the bottom of a well full of water - then the parts of water surrounding the sphere will compress it, and will try to enter in the place where the sphere now finds itself: now only as much of the water will try this as can be held in the place left by the sphere when it goes up: and this is an amount equal in size to the size of the sphere.But if the water exerting pressure encountered a resistance in the sphere less than its own force, with which it presses, (which in this case will happen when the sphere is less heavy than water), indisputably it will expel the sphere and it will extrude it.
Secondly, you will object: If the elements do not have heaviness in their own region (as I have promised above {1} that I would demonstrate), therefore a heavy body will not be able to be extruded by the heaviness of the medium.I reply, it is one thing for the elements not to have heaviness in their own region; it is another for them not to be able to exert heaviness. The first is false; for the heaviness of the same bodies is always
365 the same: as for the second, it is true, for heavy bodies cannot always exert their heaviness.Now this is evident in the case of wood; which surely is always heavy, but, if it is placed in water, then it cannot exert its heaviness, that is be carried downward, because it is hindered by the greater heaviness of the water.Accordingly the elements cannot exert heaviness in their own region: because no part of them can go down, since the place into which it would have to be carried is already occupied by some other water, which is not less heavy than the part which is higher; and although the higher parts exert pressure on the lower ones, they nevertheless do not extrude them, because they resist with as much heaviness as that with which they are impelled.But it happens otherwise if another body less heavy than water is in water.In this case, as much water as the place that the body occupies would take up, being heavier than the body, provided that it is above it, will already be outside its own region (for only those things that are not above less heavy things are in their own region; since, as has been said {1}, it has been established by nature that heavy bodies remain under less heavy ones): hence, by exerting its own heaviness, encountering, in that body, a lesser heaviness as well as a lesser resistance {2}, it will extrude and will put to flight that body in the manner explained a moment ago.
Thirdly, you will object, with Aristotle in Book I of the <i>De Caelo</i>, in text #89 [277b 1-8] {1}: Things that are moved by extrusion are moved by violence; and things that are moved by violence are moved more slowly at the end of their motion: therefore, if things that are carried upward were moved by extrusion, they would be moved more slowly at the end: which however is not the case.I reply, it is not in the case of all things that are moved by violence that it is necessary that they be moved more slowly at the end, but only in the case of those things that are carried violently after being separated from what has put them into motion: as, when a stone is thrown upward by a man, its motion is weakened at the end, after it has been disconnected and separated from the thrower; {1} but if the thrower did not release it from his hand, it could move faster even at the end.But things that are moved upward through extrusion are not seperated from the mover in their motion; but what drives it is always joined to it: hence it is not necessary that the motion be weakened at the end.You will say: Though it may not be weakened, it should not however be
366 intensified; but things that, like fire, are moved upward, are moved faster at the end.To this I reply that what they assume is false, namely that the speed of upward motion is increased: for it is always uniform; and I cannot comprehend how Aristotle could have been able to observe that things that are moved upward go faster at the end.
[Dialogue on Motion between Alessandro and Domenico] {1}
367 AL. Where [are you going] on such a swift foot, dearest Domenico?
DO. Ah, greetings, dear master {1}
AL. Stop for a moment, I pray you; for, in running to catch up with you, I have so tired myself out that I can hardly supply my overheated heart with as much vital breath as it ardently desires.
DO. As for myself, though I am walking with a swift pace, I nevertheless cannot overcome the resistant cold; and that old saying, Motion is the cause of heat, is hardly confirmed in my case. Let us, then, walk as slowly as you please, and, according to our custom, let us set out for a walk outside the city, where I had intended to go myself in any case. But what will we discuss this morning?
AL. Let us discuss the first thing that occurs to you or to me, so long as it is not unpleasant to talk about.
DO. Then let our subject be what I mentioned a moment ago.
AL. And what was that?
DO. Well, I did recite that trite saying... {1}
AL. Oh, yes, now this has come back to memory.
DO. I think, then, that a discussion about motion would not be unpleasant. {1}However, a discussion of all motion in general, and about its essence and each one of its accidents, would be too long, not to say superfluous as well.For when I want a thorough treatment of this subject, I will consult Aristotle himself, in his <i>Physics</i>, and all his commentators. So I will constrain myself to only one species of motion, namely to the motion of things heavy and light. {1}But still, since this too has been treated with thoroughness by many, and most thoroughly by Girolamo Borro {1}, it will be pleasant to know your way of thinking concerning certain particuliar
368 things, omitting all others, or to hear your solution of certain problems, concerning which my mind is not as satisfied as it should be with regard to the opinion and solution of others. These are of the following sort {1}:
First, whether you believe it true that at the turning point of motion a rest is required.
Second, what cause do you allege for this, namely that, if two bodies equal in size, of which one is, for example, of wood, the other of iron, so that one is heavier than the other, are let down at the same time from a certain very high place, the wood is carried through the air more swiftly than the iron, that is, the lighter more swiftly than the heavier -- if, indeed, you admit this as true.
Third, how it happens that natural motion is faster at the end than at the middle or at the beginning, but violent motion is faster at the beginning than at the middle, and faster there than at the end.
Fourth, why the same body goes down more swiftly in air than in water; and, indeed, why certain bodies go down in air, which are not submerged in water.
Fifth (this is a request of our very dear friend Dionisio Font, the most worthy knight) {1} what cause do you give for the fact that guns, both cannons used against fortifications as well as manual arms, throw lead spheres farther along a straight line if they project them at right angles to the horizon, than if on a line parallel to the same horizon, although the first motion is more opposed to natural motion.
Sixth, why the same guns throw heavier balls more swiftly and farther than lighter ones, as those of iron in comparison with wooden ones, although the lighter ones resist the impelling force less.
It will be very pleasant to hear your way of thinking on these topics and on similar ones which depend on them: for I know that on this subject you will either say nothing or bring forth something new and very near the truth itself. Now since you have grown accustomed to very reliable, very clear and also very subtle mathematical demonstrations, as those of the divine Ptolemy and the most divine Archimedes, you cannot in any way give your approval to cruder arguments: and since these things which I have proposed to you are not very far removed from mathematical considerations, it is with eager ears that I expect something beautiful from you.
369 AL. Our Domenico could not bring forward something unworthy of his elevated mind.
But in order that I may be able to go over my opinion about these problems, certain things must first be assumed.Thus in the case of motion, so far as the present affair is concerned, three things must be taken into consideration: these are the mover, the mobile, and the medium through which motion takes place. {1}The last two are the same in both natural and violent motion; the first, namely the mover, is not the same in both motions: for in natural motion it is the proper heaviness or lightness; in violent motion it is a certain force impressed by the mover; in a medium....
DO. Wait, wait: we must proceed gradually, lest perchance the building, erected on unstable foundations, collapse entirely when you wanted to place the roof on it. Thus you have just said that in natural motion the mobile is moved by heaviness or by lightness, but in violent [motion] by an impressed force: these two things, before they are conceded or believed by me, are in need of clarification. And in the first place, on what reasoning do you rely when you audaciously assert that in violent motion the mobile is moved by a force impressed by the mover, despite the fact that Aristotle alleges another cause of this motion, namely, that it is moved by the medium? {1}Do you think, then, that Aristotle's way of thinking on this is false?
AL. There is no necessity that I must also follow that way of thinking, which they attribute to Aristotle, and which many defend or rather try to defend, since there is another opinion which has its adherents, and even very learned ones. And yet, if you wish to hear by what arguments I have been moved to reject Aristotle's way of thinking, I will adduce for everyone to see many reasonings that destroy it, which are not imagined and do not depend on even greater chimeras, but are drawn from the senses themselves.
DO. It will be no less pleasant to hear these things than the solutions to the problems; which, if time will not be sufficent to complete them before the dinner hour, we shall postpone until tomorrow. But perhaps it will not be inopportune for us to walk to the seashore, and there take from the fishermen and the sailors something appropriate to eat, and stay for most of the day; especially since as the sun climbs higher above the horizon, the air warms up so much that it makes the chill of winter less unpleasant: and this way, with an adequate amount of leisure, I will be able to hear from you what I want.
370 AL. Let us then go wherever it pleases.
DO. As for yourself, in the meantime, do not hesitate to state your arguments.
AL. Before I come to my arguments, it will be useful first to make clear the way of thinking of those who follow Aristotle's opinion. {1}For they say: when a mobile is projected by a thrower, for example, a stone by a man, then the air contiguous to it is first impelled by the hand of the mover, and [this air] moves the other parts of the air; later, when the mobile is abandoned by the hand in the air, the air already in motion carries the mobile along with it. .This is Aristotle's cause: some {1} also add that when the mobile is moved, parts of air come up to fill the void which it leaves behind; by the collision of these, {2} the rear parts of the mobile are impelled.However, I will try to show by the following arguments that this way of thinking is entirely false.
Let the first argument {1} be the following: if the mobile is moved by the medium, the mobile will necessarily be moved in the same direction as the medium; now we often see the contrary of this; hence [etc.].The minor is evident: for if, while a very strong wind is blowing, a mobile is projected against it, however much the wind blows from the south, it is nevertheless towards the south, if it is directed there, that the mobile will be carried; therefore it is evident that the mobile is moved, not by the medium, but by another mover. And one must not say. although we observe the wind to be carried in the contrary direction, nevertheless the parts of the air which are very near the mobile are carried in the same direction as the mobile: for here is a very manifest example concerning this {1}. Dont you see that little boat which is impelled by a single sailor towards Pisa, against the direction of the current? For we see that the boat, having once been impelled by the sailor, is then driven along a certain distance against the advancing current: yet it is manifestly clear to the eyes that the parts of water touching the boat are carried in the contrary direction. And do not by chance believe that the last parts of water, which touch only the stern of the boat, impel the boat, although the other parts of the water are seen to be moved in the contrary direction: for, apart from the fact that this is ridiculous, experience also teaches the contrary.For if someone, remaining over the stern, suspends a piece of wood, as small as may be, from a cord and lets it down into the water, he will clearly see
371 that it is carried away in the direction contrary [to that of the boat], and that it resists being pulled by the cord: yet the contrary of this would follow if the water were moved in the same direction as that in which the little boat is impelled.
Second: if it is the medium that carries mobiles from one place to another, how does it come about that, when one throws with the same shot of the cannon an iron ball, with which, however, is also carried some wood or tow or something light, but in such a way that the heavy thing comes out first -- how, I say, does it happen that the iron is put in motion along a very long distance, while the tow, after it has come along with the iron for some distance, is stopped and falls to the ground? {1}If then it is the medium which carries them both, why does it carry the lead or the iron very far, but not the tow? Or perhaps it will be easier for the air to move the very heavy iron than the very light tow or the wood?
Third: if the mobile is moved by the medium, that mobile which has a greater number of parts impelling it will be carried more swiftly and for a longer distance: yet experience demonstrates the contrary of this. For if a very thin arrow sharpened at both ends is impelled by a bow, it will be carried along a longer distance than another, thick, piece of wood, thrown by the same bow, though it be of the same weight as the arrow: and yet fewer parts of air impel the sharp point of the arrow.
DO. To this there would be an easy reply: for, since the arrow is sharp, it is more fit for cleaving the medium than the blunt wood; hence the air will resist it less.
AL. Perhaps you believe that the other adherents of this way of thinking would also reply to the argument in this way?
DO. Yes, indeed; and it seems to me also the best solution.
AL. Then how will you not admit that the medium is not moving in the same direction as the parts towards which the mobile is carried?For if it were carried in the same direction, the air would not have to be cleaved by the mobile. It is thus clearer than daylight that the medium must be cleaved, from the fact that sharp things are carried more easily than blunt ones: from which it then follows that the medium is not moving towards [the place] where the mobile is tending. It only remains, then, to demonstrate that the mobile is also not moved by the medium because the medium, in taking the place of the parts that have been vacated, impels it. For if the parts of air are moved to take the place of the void left by the mobile, why does the mobile not in like manner step backward to take the place of that same void? For example, let what is moved be part of a cylinder, with one
372 of its bases going in front; and let it be moved toward the north. It is clear that some parts of air will rush into the void from the east, and others from the west, but none from the south. For the place into which the forward parts of the cylinder enter must be emptied of air, lest there be given a case of interpenetration of bodies; these parts of air must enter the place which is abandoned by the cylinder: but the place which the cylinder empties of air is always equal to the place which the cylinder abandons behind itself: hence to fill it the parts which at that moment were going in front of the cylinder suffice; these parts, entering into the void from the outer circumference of the the base of the cylinder, all come either from the east or from the west.If this is the case, how will they impel the cylinder toward the north?To these arguments add another: if we consider that a cone is being moved, whose base goes in front, but whose point follows behind, then no parts of air will be able to be moved to fill the void. Finally, and let this be the strongest argument, consider a marble or iron sphere, perfectly round and smooth, which can be moved on an axis at rest on two pivots {1}; then let a mover come near who twists both ends of the axis with his finger tips: surely in that case the sphere will rotate for a long time: and yet the air has not been put into motion by the mover, nor can the medium ever come up into the parts abandoned by the mobile, since the sphere never changes place. What, then, is to be said of this violent motion? by what will the sphere be moved when it is outside the hand of the mover? what is to be said, except that it is moved by an impressed force?
DO. I cannot not surrender to your arguments: and yet concerning this last one, which you seem to deem outstanding, there is something which I could doubt.For those who defend the contrary viewpoints could perhaps reply to an argument of this kind by saying that that motion is not violent, since it is circular. For since natural motion is contrary to violent motion, but to circular motion there is no contrary, circular motion will in no way be violent: and since it is not violent, the consequence that you deduce from the motion of the sphere, will be of no moment.
AL. It will be no difficulty to dispute this reply. For when they say that violent motion is brought about by the medium, they do not consider this [to hold] only of that motion which is diametrically opposed to the natural, but of any kind of motion which is not natural, that is of the violent and the mixed. For it would be entirely childish and ridiculous to say,
373 for example, that a stone projected at right angles to the horizon, because the motion will be diametrically opposed to the natural, is moved, when out of the hand, by the medium; but if the same stone is projected at an inclined angle, then it is moved by a different mover: and yet the second motion will be a mixture of the natural and the violent. Hence all motions, however different they may be from the natural, are included in the motion of projectiles; but the motion of such a sphere as we are talking about is not natural, but {1} mixed; hence [etc.].Let it be proved that it is mixed: a mixed motion is one which is composed of natural and violent; but such is the motion of the sphere; hence [etc.]. Now the rotation of the sphere is composed of a natural and a violent motion: because certain of its parts recede from the center of the world, toward which the sphere naturally is carried; certain others approach it.And do not tell me that the parts of the sphere which are moved downward pull upward the parts that go up; and this because, since the sphere is in equilibrium, the parts that go up do not resist the parts that go down.Indeed this is of no worth: for if the parts of the sphere are all equally heavy, there will be no more reason for the parts on the right to raise the parts on the left, rather than the contrary. Next, take a sphere whose parts do not weigh equally: in its motion you will see that heavier parts are raised by lighter ones, and yet there will be a resistance in that motion.And then? is there not always in such motion the resistance of the axes, which, weighed down by the weight of the sphere on the pivots, resist the motion?Finally, I would like you to note this: that, when it is said that a circular motion is not violent, this is understood of that circular motion which takes place around the center of the world, such as is the motion of the heaven. {1}Thus if there were a marble sphere at the center of the world, so that the center of the world and the center of the sphere were the same, and then a beginning of motion of the sphere were given by an external motor, perhaps then the sphere would not be moved by a violent motion but by a natural one; since there there would be no resistance of the axes, and the parts of the sphere would neither approach nor recede from the center of the world. Now I have said, perhaps: because if such a motion were not violent, it would endure forever; but that eternity of motion seems far removed from the nature of earth itself, to which rest seems to be more pleasant than motion. {1}And so
374 from all the reasons adduced it seems quite clear that the medium, far from helping motion, rather, on the contrary, opposes it. And thus it must be concluded that the mobile, when it is moved by a motion against nature, is moved by a force impressed by a motor: but what that force is is hidden from us. {1}
DO. Let that opinion be at last sufficiently refuted; and, now that I have been convinced by your reasons, let it be conceded that a mobile is moved not by the medium but by an impressed force. But now, before you say anything about the medium, I would like you to explain to me in what way you understand that in natural motion the mobile is moved by heaviness or by lightness.
AL. The things that must be said about the natural mover and the medium, so far as they concern the present affair, can conveniently be made clear at the same time. Let it,then, be presupposed, in the first place, that it has been established by nature that heavier things remain under lighter ones: that this in fact is the case is very manifest to the senses.
DO. We grasp by our senses that this thing which you presuppose is very true; but I would like to understand the cause why nature would preserve such an order, and not the contrary instead.
AL. To produce the cause of such an order cannot in any way be useful for our aims, since it is manifest that it is such in fact {1}; to produce the most important cause would probably be very difficult: and I could not produce any, except that things had to be disposed according to some order, and it has pleased nature to dispose them in this one. {2}Unless perhaps we would want to say that heavier things are nearer the center than lighter ones, because those things somehow seem to be heavier which contain more matter in a narrower place: as, for example, if there is a bag full of wool, which was paked in it with no force, and then much more wool is compressed in the same bag with much violence, then it will be heavier than before, because more matter will be stuffed in the same space. {1}Since, then, the spaces which are nearer to the center of the world are always narrower than those which are more distant from the center{1}, it was in accordance with reason that they should be filled with matter whose heaviness, greater than that of other matter, would occupy the narrower spaces.
375 DO. That reason, although one must not think that it is the most important one for such a disposition of the elements, nevertheless has in it some appearance of truth, to which the mind readily gives assent: so that, both because what you ask for is in itself very clear, and because the things you have just said on this same subject in a way offer a cause, I will be quite content to concede that heavier things have been placed under lighter ones by nature. Therefore, if you please, turn to the things which remain to be said.
AL. Now on the other hand one must take note that heavy or light are not said except in comparison. {1}
DO. Stop, I pray you; we must proceed step by step. For you say that the absolutely heavy or light are not given, but only the heavier and the lighter, in comparison: this first thing will not be conceded by me, especially because Aristotle's opinion throughout all of <i>De Caelo, Book IV</i>, is opposed to it; there he shows, against the way of thinking of the Ancients, that earth is purely and simply heavy, fire purely and simply light. Therefore, unless you first do away with what is assumed by Aristotle, I will never embrace your opinion.
AL. Our conversation will be too long, if Aristotle will have to be refuted with reasons in the case of all the things that will be proposed by me against Aristotle's way of thinking.
DO. Our conversation will be too brief, if perchance you want to lay foundations by bringing your opinions forth for everyone to see without justification: for then
376 I don't care to hear any more from you. Therefore, unless you will explain by what reasoning you have been led to reject Aristotle, you can put an end to your speaking, for (as the proverb goes) you would be lecturing to a deaf man.
AL. Now that I have undertaken this mission, in order to gratify you I will bring forth, as concisely as I can, what causes have impelled me to reject with disdain Aristotle's way of thinking. {1}Now, in the first place, Aristotle assumed that earth is the heaviest of all things: for he has written, in the <i>De Caelo, Book IV</i>, text #29 [311b 1-8], that all things have lightness except earth, but that mixed things have more heaviness the more earth they will contain. Therefore if earth is the heaviest of all things, it is manifest that no mixed thing will be heavier than earth itself; since they are composed also of water, air and fire, which have a lesser heaviness than earth. Now this is false: for to whom is it not evident that all metals are heavier than earth itself, as, for example, mercury, on which earth floats? and that that which rises to the surface of something else, according to Aristotle himself, is lighter than that on which it sits. And who will doubt that a pot full of lead is heavier than one filled up with earth?How, then, is earth the heaviest of all things?Aristotle also adduces another indication of the earth's heaviness, by saying {1}: If air or water is removed, earth will never go up in the place of air or of water; as is evident in physicians' cups, which attract water and flesh, but earth not at all: hence earth is the heaviest thing. But can anything more childish be imagined?For if earth is not a fluid body, how will a part of it be raised over another?This therefore does not come from the absolute heaviness of the earth, but from its solidity: for even frozen water, which is not earth, will not be raised, nor swell up in the cup {1}; on the other hand mercury will be raised, although it is far heavier than earth, because it is fluid. To say next, The center is the contrary of the extremity; therefore it is in accordance with reason that the things which are at the center be the contraries of those which are at the extremities, and this will not be the case unless earth is assumed to be purely and simply the heaviest thing, and fire the lightest{1}; now this is reasoning which not only bears no necessity, but, in my opinion, is of little moment. For if he takes as an extremity the concave sphere of the Moon {1}, but as the middle the center of the world, then surely the concave sphere of the Moon will no more be in opposition to the center of the world, than the concave sphere of air and water: {2}
377 and if it is in this way that the contrariety of places must be taken, earth itself will be in contrary places, since it is at the center and at the concave surface of air and water. Aristotle argues in the same manner concerning fire, saying <i>De Caelo, IV</i>, 312b5-11 that, if air is removed from underneath it, fire will not go down, as air [would if] water [was] removed from underneath it. {1}This needs a demonstration; Aristotle has not proved it. Unless perhaps you believe that what he assumes in text #39 of the same book [<i>De Caelo, Book IV</i>, 312b 15], saying that fire will not go down because it has no heaviness, is a proof: but that would be to prove a thing by the same thing if, while he tries to demonstrate that fire has no heaviness because it does not go down, he were to prove that it does not go down because it has no heaviness. And so this lacks a demonstration; and all the more because Aristotle himself, I believe, has not made a test [to determine] whether, air having been removed, fire does go down. Now on what grounds did Aristotle know that there exists nothing lighter than fire? Can there not be certain exhalations which would fly above fire? {1} But, finally, how could anyone ever imagine that fire, a substance linked to quantity, does not have heaviness? {1} This truly seems entirely unreasonable to me.And when we say that earth is the heaviest of all things because it stands under all things, we are compelled, whether we like it or not, to say that earth is the heaviest, in comparison with other things, because it stands under all other things; <u>and it stands under all other things, because it is heavier than the things under which it stands.</u> {1}And this is evident; for, if it is the heaviest because it stands under all things, if all things are removed, it will no longer be able to be said to be the heaviest, since it stands under none of them.It is therefore said to be the heaviest in comparison with what is less heavy; and the same should be said concerning {1} fire.And do not say, If fire had heaviness, it would go down. {1}For does not air have heaviness? and yet it does not go down below water: thus fire also has heaviness, yet it does not go down below air, because it has less heaviness than air. {1}Therefore, to confine the thing to a few words, I say that in the nature of things there exists something which is the heaviest of all, and something which is the lightest of all, that is possessing the least heaviness: but I deny that those things are necessarily earth and fire; similarly I also deny that one can speak of the purely and simply heaviest or lightest, without any relation to the less heavy or light, but only that it is the heaviest of all things that are heavy, and not of [all] the things that could possibly exist. And that is what I would like to say briefly to confirm my opinion.
DO. Your reasons are entirely satisfactory, and all the more because this question is of little moment, namely whether earth is absolutely or relatively heavy; and because I believe, whatever assumption one makes, it does not contribute much to the present affair. {1}Thus let us proceed to the remaining things [to be examined].
AL. I say, then, that one does not speak of the heavy and the light except in a comparison. Now that comparison happens in two ways: for either we compare to one another two bodies which find themselves in the same medium, or we compare a certain body with the medium in which it is moved. In the first comparison, those things are said to be equally heavy which, when they are equal in size and are in the same medium, will be of the same heaviness. From this it is evident that if we take two pieces, for example, one of wood, the other of iron, which are equal in heaviness, those things should not however be called equally heavy: for the piece of wood will be much greater in size than the piece of iron. Next, that body will be said to be heavier than another, of which an amount equal in size to the size of the other is heavier than the latter, if they are weighed in the same medium. On the other hand, that body which reveals itself as lighter than another, if an amount of it equal in size to the size of the other body is taken, and the two bodies are weighed in the same medium, is called lighter. Further, by similar reasoning media also will be called heavier or lighter based on comparison to one another; and solid bodies will similarly be called heavier or lighter based on comparison with the medium in which they are moved. Now the media through which motion can take place are all the elements with the exception of earth, which, since it is the most solid, cannot be divided by another body: but the rest of the elements, namely water, air and fire, since they are fluids, allow motion to take place in them. {1}
These things having been made clear, it will be easy to conceive how heavy things are moved by heaviness and light things, on the other hand, by lightness. For those bodies which are heavier than the medium through which they are moved are moved downward: for it has been established by nature that heavier things remain under lighter ones; but if a certain body heavier than water were to remain above water, then what is lighter would be under what is heavier. Therefore heavy things are moved downward, insofar as they are heavier than the medium through which they are moved: hence their heaviness with respect to the medium, is the cause of such a downward motion. {1} And again it is by similar reasoning that one must reach an understanding about things lighter than the medium.
DO. The things you have just said are not yet entirely satisfying; and I have a reason to doubt [them].For if, for example, we take a very small pebble and project it into the sea, it will, no doubt, be carried downward through the medium of water: now I cannot in any way grasp by what reason one must think that the pebble
379 is heavier than the water of the sea; particularly since this same water of the sea is certainly heavier than practically any number of pebbles.
AL. Have you so quickly forgotten the things I have said just now?Have I not said that a body is heavier than another body if an amount of the one body equal in size to the size of the other is heavier than the other?If, then, we take a portion of water whose size is equal to the size of this same pebble, we will subsequently find that the pebble is heavier than the water: since this is so, it is not surprising if the pebble goes down in the medium of water.
DO. This is entirely true: but nevertheless I still do not understand for what reason, in the case of the pebble in water, one must only take into consideration an amount of water as great in size as the size of the pebble, and not all the water.
AL. I am, at last, unable to avoid demonstrating to you some theorems, from the comprehension of which you will understand most clearly not only what you are asking for, but also what ratio bodies have, both heavy ones and light ones, with regard to the swifness or the slowness of their motion, as well as what the ratio is of the heavinesses and lightnesses of one and the same body, if we were to weigh it in different media: all these things had to be demonstrated when I tried to find the real reason by which we could, in a mixture of two metals, assign to each individual metal a very precise share. {1}Of these theorems (though they are not different from those demonstrated by Archimedes) I will bring for every one to see demonstrations that are less mathematical and more physical; I will make use of assumptions that are clearer and more obvious to the senses than those that Archimedes has taken.
DO. Are you asserting, then, that you also found out the true reason by which Archimedes discovered the goldsmith's fraud in the golden crown? Haven't the same things been written by many authors, and in particular by Vitruvius?
AL. I could demonstrate that that popular method of those who talk of a bowl full of water, etc., is most unreliable; by contrast, the one I have discovered is very accurate, and I think it is the same as that of Archimedes, both because it is so elegant, and because it rests on Archimedes' own demonstrations.
DO. But if that beautifull invention of yours relies on the demonstrations you are about to explain, it will be very pleasant, time permitting, to here of it also.Now on the other hand, if your demonstrations
380 require diagrams, here is a smooth and wide surface of the finest sand, on which you will be able to draw them deftly with this rod.
AL. Before I come to the description, it is necessary to take heed of the following: even though, as I have said, there are three principal media through which motion takes place, still, since [the region of] fire is too distant from us, and since in the case of air we do not have ready at hand things that go upward, I will bring forth the demonstrations with water as the medium: and no one will deny that the things that are demonstrated of water are also true of the other media. {1}I say, then, that in the first place, a solid magnitude equally as heavy as a certain portion of water whose size is equal to the size of the said magnitude, when it has been let down into water, is entirely submerged; yet, when it is entirely under water, it is carried no more upward than downward: this is the same as if we said that solid magnitudes equally as heavy as water, when they have been let down into water, are entirely submerged, yet they are [then] carried no more downward than upward. {1}Thus let the first position of the water, before the magnitude is let down into it, be ced; and let magnitude ab, which is equally as heavy as water, when it has been let down into the water, not be completely submerged, if this can happen, but let a certain part of it, namely a, protrude from the water; and let the surface of the water cd be raised to fg, and let both the water and the magnitude stand still in this position. It is then manifest that the amount of water
381 contained by the surfaces fg and cd, which is raised, is equal to the size of the part of the magnitude which is submerged under water, namely b. For this is very clear: because the said amount of water cannot be smaller, for then there would be a case of interpenetration of bodies; nor can it be greater, lest an empty place be left behind. And so, since the water contained by the surfaces fg and cd strives by its own heaviness to return downward to its original position, but it cannot achieve this unless solid ab is first taken out of the water and raised by it; now the solid, in order not to be raised, resists with the whole of its proper heaviness; and it is assumed that both the solid magnitude and the water remain in this position; hence is it necessary that the heaviness of the water fgcd, by which it strives to raise the solid upward, is equal to the heaviness by which the solid resists and exerts pressure {1} downward (for if the heaviness of the water fgcd were greater than the heaviness of the solid magnitude ab, the solid ab would be raised and expelled by the water; if, on the other hand, the heaviness of the solid ab were greater, the water would, in turn, be raised: however all these things are assumed to remain thus at rest).Thus if the heaviness of the portion of water fgcd is equal to the heaviness of the whole of solid ab, then the size of the portion of water fgcd will also be equal to the size of the whole of solid ab: now this assuredly cannot be the case. For the size of the water fgcd is equal to the size of that portion of the solid which is under water, namely to portion b; now b is smaller then ba itself.Hence it is not true that a certain part of the solid magnitude will protrude outside the water; hence it will be totally submerged. However, when it is entirely in the water, it will not go downward; but it will remain where it is placed.For there is no cause in virtue of which it must go down: for since it is assumed to be equally as heavy as water, to say it goes down in water would be the same as if we were to say that water in water goes down under water, and that on the other hand the water that rises above the first water then goes down again, and that water thus continues without end going alternately down and up; which is unacceptable. {1}
Thus, this having been demonstrated, it follows that we show that solid magnitudes lighter than water, having been let down into water, are not completely submerged, but a certain part of them protrudes out of the water. {1}It is therefore necessary to recall that a solid magnitude is then said to be lighter than another, if, being equal to it in size, it is lighter than it in heaviness. Let the first position of the water, before the solid magnitude is let down into it, be ef; and let ab, the magnitude let down into the water, be lighter than water: I then say that this same
382 solid magnifude is not completely submerged. For if this can be done, let it be totally submerged, and let the water be raised up to surface cd; therefore the size of water cf will be equal to the size of magnitude ab: and, if this can be done, let both the magnitude and the water remain in this position. And since the heaviness with which water cf exerts pressure downward to raise the solid magnitude is the same as the heaviness with which magnitude ad, exerting pressure downward, resists being raised (for they are assumed to stay at rest); and, furthermore, the size of water cf is equal to the size of magnitude ab; consequently the solid magnitude is equally as heavy as water: which is unacceptable; for it was assumed to be lighter.Hence the magnitude ab will not be totally submerged, but a certain part of it will protrude out of the water.
And so, it having been demonstrated that solid magnitudes lighter than water are not completely submerged, it is now useful to show what parts of them are submerged. I say, then, that solid magnitudes lighter than water, having been let down into water, are submerged up to the point where an amount of water as great in size as the size of the submerged part of the magnitude has the same heaviness {1} as the whole magnitude. Let the first position of the water be surface ab; then let the solid magnitude cd, lighter than water, be let down into the water: it is then manifest from what has previously been said that it will not be completely submerged. Thus let d be the part which is submerged, and let the water be raised up to the surface et: I say that an amount of water as great in size as the size of the submerged part of the magnitude has the same heaviness {1} as the whole of the magnitude. For since water eb exerts pressure with a heaviness equal to the [heaviness] with which magnitude cd resists (for they are assumed to remain thus at rest), therefore the heaviness of water eb is equal to the heaviness of the whole magnitude cd: but the size of water eb equals the size of the submerged part of the magnitude, namely d: hence an amount of water as great in size as the size of the submerged part of the magnitude has the same heaviness as the whole of the magnitude. Which was to be demonstrated.
But now, before we come to the demonstration for solids heavier than water, it must be demonstrated by how much force a solid magnitude lighter than water is carried upward if, by force, it is completely submerged under water. {1}I say, then, that solid magnitudes lighter than water, having been impelled into water, are carried
383 upward with a force equal to the amount by which an amount of water whose size is equal to the size of the submerged magnitude will be heavier than this same magnitude. {1}Thus let the first position of the water, before the solid magnitude is let down into it, be along surface ab; and let the solid magnitude cd be let down into it by force; and let the water be raised up to surface ef: and since water eb, which is raised, has a size equal to the size of the whole submerged magnitude, and the magnitude is asumed to be lighter than water, the heaviness of water eb will be greater than the heaviness of solid cd. Then let tb be understood to be the part of the water whose heaviness is equal to the heaviness of solid cd: it must therefore be demonstrated that the solid magnitude is carried upward with an amount of force equal to the heaviness of water tf (for it is according to this heaviness that water eb is heavier than the heaviness of water tb, that is than the heaviness of solid cd). Since then the heaviness of water tb is equal to the heaviness of the solid magnitude cd, water tb will exert pressure downward {1} to raise the solid with a force equal to that with which the solid resists being raised. Thus the heaviness of the part of the water which exerts pressure, namely tb, is equal to the resistance of the solid magnitude: but the heaviness of all the water eb which exerts pressure surpasses the heaviness of water tb by the heaviness of water tf: hence the heaviness of all the water eb will surpass the resistance of solid cd by the heaviness of water tf. Consequently the heaviness of all the water exerting pressure will impel the solid magnitude upward with an amount of force equal to the heaviness of the part of water tf: which was to be demonstrated.
Now, from these things that have been demonstrated it is sufficiently clear that solid magnitudes heavier than water are carried downward, if they are let down into water. For, if they are not carried downward, either a certain part of them will protrude, or they will remain at rest under water, and will be carried neither upward nor downward. But no part of them will protrude; for, as has been demonstrated, they would [then] be lighter than water: nor will they remain at rest in the water, since they would [then] be equally as heavy as water: it remains, therefore, that they are carried downward. Let us now show with what amount of force they are carried downward. I say, then, that solid magnitudes heavier than water, having been let down into water, are carried downward with an amount of force equal to the amount by which water having a size equal to the size of the same magnitude is lighter than that magnitude.Thus let the first position of the water be along surface de; and let solid magnitude bl, heavier than water, be let down into the water, and let the water be raised to surface ab; let ae be water which has a size
384 equal to the size of the magnitude: and since the solid magnitude is assumed to be heavier than water, the heaviness of water ae will be less than the heaviness of the solid magnitude.Then let ao be understood to be an amount of water which has a heaviness equal to the heaviness of bl: and since water ae is lighter than ao according to the heaviness of do, it must be demonstrated that the solid magnitude bl is carried downward with an amount of force equal to the heaviness of water do. Consider another solid magnitude lighter than water, joined to the first, whose size is equal to the size of the water ao, but whose heaviness is equal to the heaviness of water ae; and let lm be the said magnitude: and since the size of bl is equal to the size of ae, but the size of lm is equal to the size of ao, hence the size of the combined magnitudes bl and lm is equal to the size of the combined water ea and ao. But the heaviness of the magnitude of water ae is equal to the heaviness of magnitude lm: and the heaviness of water ao is equal to the heaviness of magnitude bl: hence the total heaviness of both magnitudes bl, lm is equal to the heaviness of water oa, ae. But it has also been demonstrated that the size of magnitudes ml, lb is equal to the size of water oa,ae; therefore, by the first proposition, the magnitudes so combined will be carried neither upward nor downward. Therefore the force of magnitude bl exerting pressure downward will be equal to the force of magnitude ml impelling upward: but, by the previous proposition, magnitude ml impels upward with an amount of force equal to the heaviness of water do: hence magnitude bl will be carried downward with an amount of force equal to the heaviness of water do. And from this it can manifestly be concluded that a solid heavier than water is lighter in water than in air by an amount equal to the heaviness in air of an amount of water with a size equal to the size of this solid. For solid bl in air is carried downward by its heaviness, which is assumed to be equal to the heaviness of water ao; now in water it is carried downward by an amount of heaviness equal to the heaviness of do: but the heaviness of ao exceeds the heaviness of do by the heaviness of ae: which is the heaviness of an amount of water having a size equal to the size of the solid bl. Moreover all these things that have been demonstrated concerning water must also be understood of air.From this it is evident in general that those bodies which are heavier than the medium through which they must be carried are carried downward; but those which are lighter than the medium through which they must be carried are carried upward.
385 DO. From these very certain and very manifest demonstrations I have acquired so perfect and excellent an understanding of these motions that I will never again have any doubt about those things concerning which I previously did have doubt, since I had never understood them except in a sort of confused manner.
AL. It follows also that heavy things are carried downward more easily, the lighter the medium through which they must be moved; but lighter things are carried upward more easily, the heavier the medium through which they are carried.
DO. Therefore what nearly all philosophers assert will be false, when, trying to demonstrate that air is more heavy than light, they say that one must think that air is more heavy than light because it carries heavy things downward more easily than light things upward. {1}
AL. O ridiculous chimeras! O inept ways of thinking of men, which not only do not approach truth, but are opposed to truth itself!But, heavens!, how, I ask you, are we to believe the chimeras of those people, with which they profess to explain the most hidden secrets of nature, if in the case of things that are, as it were, completely open to the senses they rashly assert the opposite of the truth?And who, I ask you, dreamed up the idea that those media are more heavy than light which transport heavy things downward more easily than light things upward, when in fact it is entirely the reverse which happens? {1}For if this consequence were sound, air really would be heavier than water: for, whatever heavy thing there is that is moved downward, it will always be carried more easily and more swiftly in air than in water; moreover, many are the bodies which are moved very swiftly and very easily downward in air, which in water not only are not carried downward, but float on the water, and, having been impelled into water by force, are carried upward. Now let this be evident from an example: for a gourd, for example, is easily carried downward in air, but subsequently in water will not be impelled donward except with very great difficulty and violence; hence, since air carries bodies downward more easily than water, it will have to be thought heavier than water.O what absurdity! O what absurdity!But listen, I pray you.It has been demonstrated that heavy things that are carried downward in water, go down with as much force as the amount by which their heaviness excedes the heaviness of an amount of water equal in size to their size. {1}Therefore, if there were a certain heavy body, as, for instance, a body a, whose heaviness
386 is, for example, 8, but the heaviness of water b, whose size is equal to the size of a, is 4, then solid a will be carried downward as swiftly and easily as 4; but if subsequently the same body a is carried through a lighter medium, so that the heaviness of an amount c of such a medium, equal in size to size a, would be as 2, then surely solid a in this second medium would be carried downward easily and swiftly as 6. It is thus evident that the same body a is moved downward more easily through lighter media than through heavier ones: therefore it follows that a medium must be thought lighter, to the extent that heavy things in it are moved downward more easily; yet general opinion asserts the contrary of this.To whom, then, is it not very clear that, if air were even lighter, heavy things would be moved downward more easily?And if this is so, it follows that it is because air is very light that heavy things are easily carried downward in it. But it is in the opposite way to this that one must reason concerning light things, which are moved upward: and it will be concluded that one must think that medium heavy, through which light things are more easily carried upward, but that medium light, through which light things are moved upward with difficulty. Therefore, both because in air light things are moved upward with more difficulty, and because, in it, heavy things are moved more easily downward, it follows that one must think that air is more light than heavy.However, what I have said, I would like to have been said only for the refutation of the opinion of those who have said that air is more heavy than light. {1}
DO. What, then, is your way of thinking concerning the heaviness or the lightness of the elements? {1}
AL. If we speak of absolute heaviness or lightness, I say that all bodies, whether they are mixed or not mixed, have heaviness: on the other hand if we are having a discussion about relative heaviness or lightness, I say that all the same all bodies have heaviness, but some greater, others lesser; now it is this lesser heaviness which we call lightness. And thus we say that fire is lighter than air, not because it is deprived of heaviness, but because it has less heaviness than air has; now of air, in the same manner, we say that it is lighter than water.
DO. But, I ask you, if you asume heaviness in fire, by what method will you defend [the idea] that fire does not go down, since heavy things are those which are carried downward?
387 Yet it is thought impossible by everyone for fire to go down, even if air were to be removed from under it.
AL. Ah, ah! New chimeras, new fictions. Those things are carried downward which are heavier than the medium through which they must be carried: but fire is not heavier than air; and because of that it cannot be carried downward. But if air were removed from under it, so that void would be left under fire, who doubts that fire would go down into the place of air?For, since in the void there is nothing, that which is something is heavier than nothing: therefore since fire is something, it would indisputably go down below nothing; for it has been established by nature that heavier things remain under lighter ones. And yet you should not believe that fire will go down in order to fill the void: for, when it went down, it would leave a void under the concave surface of the sphere of the Moon {1}; to fill this it could not go up further, since heavier things do not go up above lighter things.
DO. But if you assert that all elements, air, as also fire and water, are heavy, how can we sustain the weight of air? just as we also sustain, when swimming under water, an extremely vast amount of water, by which we feel not at all weighed down?
AL. The solution of this uncertainty is easy, and depends on the things that we have demonstrated above; the solutions given by others, on the contrary, are not entirely satisfying.
For some say {1} that, fish and men in water are like a mouse in a wall, where they do not feel the weight of the bricks, because the bricks adhere upon bricks, and not upon the mice: how silly a solution this is, is clearer than daylight. For, I ask you, what relation is there between solid and firm bodies, like stones, and liquid and fluid bodies, like water?Now, if we remove the mice from the wall, the opening in which the mouse was will remain there; from this it becomes manifest that the stones do not rest upon the mice, but upon other stones: but if we remove a fish from water, do you think the place in which it was will remain there? will not water immediatly flow into it? Which is the clearest indication that the water rests on the fishes.
Now others say that the elements in their proper place are neither heavy nor light, and that therefore swmmers in water are not weighed down by it. Those who say this do not resolve the uncertainty.For, in the first place, there will have to be a demonstration of what they presuppose, namely that the elements in their proper
388 place are neither heavy nor light.But if this is so, why do they then say that air is more heavy than light?If that is what they want to defend, by what rational means will they defend [the claim] that water is not very heavy?Then, even if this {1} is conceded, the doubt is still not yet resolved. For if the elements in their proper place are neither heavy nor light, I ask them what the proper place of water is.They will answer (I think), under air.But if we go up a very high tower, on the top of which there is a bathtub, the same thing will happen to us when in it as if we were to go into the sea: for we will not be weighed down by the water, even though the latter, having air underneath it, is outside its proper place. That is, their whole error has resulted from the fact that they have not taken into account the heaviness of the medium through which heavy things must be carried, but only the proper heaviness or lightness of the mobiles. But, in order that I may satisfy your request, listen to my solution.
Now, we are said to be weighed down when a certain weight, which tends downward by its heaviness, rests upon us, and we must resist with our force so that it does not go down any further; it is this resisting that we call being weighed down. But since it has been demonstrated that bodies which are heavier than water, having been let down into water, go down, and that they are, to be sure, heavy in water, though less heavy than in air, <u>by the heaviness of an amount of water equal in size to the size of the body</u>; and it has been shown that things lighter than water, having been impelled by force under water, are raised upward with an amount of force equal to the amount by which an amount of water equal in size to the size of the body is heavier than that body in air; <u>and </u> it has been demonstrated that those that are equally as heavy as water, when they have been <u>submerged in water, are</u> carried neither upward nor downward, but remain where they are placed, provided that they are completely under water; from this it is evident that if a certain body heavier than water leans on us when we are under water, {1}, we will surely be weighed down, but less than if we were in air, since the stone in water is less heavy than in air: but if a body lighter than water were fastened to us, and we remained in the water, far from being weighed down, we would even be raised by it; as is evident when swimming with a gourd, even though otherwise, in air, we are weighed down by the gourd; and this is because the gourd, having been impelled into water is carried upward and lifts, but in air it is carried downward and exerts weight: now if a certain body equally as heavy as water hangs over us in water, we will not
289 be weighed down by it nor will we be raised, since such a body will be carried neither upward nor downward. {2}But no body can be found which is more equal to water in heaviness or lightness than water itself: it is thus not astonishing if water in water does not go down and exert weight; indeed we have said that to be weighed down is to resist with our force a body that is heading downward. {1}And one must reason in exactly the same way concerning air.
DO. What a beautifull discovery! How very accurate and very true a solution!With your words, you have dispersed all clouds of ignorance to such an extent that you leave no room for any further doubt concerning these matters. But what about my problems? {1}
AL. From these considerations which we have set down concerning media, mobiles and motions, the solution to at least one of your problems is manifestly apparent; as for the explanations of the others, they will be known soon, both from the established considerations and from those still to be established.Thus as for the question why the same mobile goes down more swiftly in air than in water, the answer is evident. For the ratio is greater between the heaviness of the mobile and the heaviness of air, than between the heaviness of the same mobile and the heaviness of water; for water is heavier than air: from this it follows that the same mobile goes down with a greater force in air than in water. {1}
DO. I had already grasped perfectly the solution to this one from the considerations established and demonstrated by you.But what about the question of the turning point of the motion....{1}
AL. Before I come to the explanation of my way of thinking about this, certain things must be examined.First, then, it being assumed, as we have sufficiently confirmed above [cf. my notes below], that a mobile, as long as it is moved in a non-natural motion, is moved by a force impressed by the mover, let it be presupposed that the same mobile, having been projected by equal forces, along straight lines, maintaining equal angles with the horizon, is always moved through equal spaces. {1} Let it be presupposed, secondly, that a mobile, moved by a finite impressed force, cannot be moved in violent motion through a distance without end. {1} Let it be presupposed, thirdly, that a mobile is not moved in a violent motion unless the impelling force is greater than the resistance of the proper heaviness of this mobile; but if the resisting heaviness is greater than
390 the impelling force, then the mobile is no longer moved in a violent motion, but changes itself to a natural one. From this it clearly follows that, when the body is at rest, the proper heaviness is equal to the impelling force: for if the heaviness were greater, then the body would go down; but if the impelling force were greater, then the body would be moved with a violent motion. From this it is manifest that the mobile is at rest as long as equality subsists between the resisting heaviness and the impelling force.Let it be presupposed, fourthly, that the same heavy body is sustained [see my notes below] by equal forces through equal intervals of time {1}Now, these things having been assumed, it is demonstrated that the force impressed by the motor is successively weakened in the case of a violent motion, and that in the same motion no two points can be assigned at which the impelling force is the same. Let ab be a line on which a violent motion from a to b is brought about by a finite force {1}; and since such a motion, by assumption, cannot be without end, let it terminate at point b, and let the mobile be moved no farther. I say, then, that the impelling force is always weakened in the case of such a motion, and that there cannot be assigned on line ab two points at which the impelling force is of the same strength. For, if this can be done, let there be two points c, d and let the force at d be no weaker than at c: it will, therefore, be either the same or stronger.Let it first be the same.Thus, since the mobile is the same, and the impelling force is the same at d as at c, but the line on which the motion takes place, since it is the same, maintains the same angle with the horizon, consequently the mobile will be moved through equal distances from points c and d: but it is moved from c as far as b: therefore it will be moved from d beyond b: which is absurd; for it has been assumed that it is not moved farther than b. And an even greater absurdity would follow if we were to say that the force at d was greater than at c. Thus if in motion ab two points cannot be assigned at which the impelling force is equal, it is evident that in the time during which such a motion takes place, it will not be possible for two moments to be assigned during which the impelling force is equal. These things having been assumed and demonstrated, it follows necessarily that no rest is given at the turning point. {1} For, if there is rest for a certain interval of time, there will also be equality between the heaviness of the mobile and the impelling force, enduring for a certain interval of time: but it has been demonstrated that the impelling force is always and
391 successively weakened: it is thus impossible that for a certain interval of time the impelling force remains in a state of equality with the heaviness of the mobile; and, because of this, it is impossible that the mobile remains at rest for a certain interval of time.
This will appear more clearly from the following demonstration, described with the same figure as above. {1}For, if the mobile, when it is at b, stays at rest for a certain interval of time, let the end moments of such an interval be cd. Therefore if the mobile is at rest during time cd, the impelling force is equal to the heaviness of the mobile during the entire {1} time cd: but the heaviness of the mobile is always the same; hence the force at moment c is equal to the force at d. Now it is the same heavy body; therefore it will be sustained during equal intervals of time by equal forces: but the force at moment c sustains it during time cd: hence the force at moment d will sustain the same body during an interval of time equal to the interval cd. Therefore the stone will be at rest during twice time cd: which is unacceptable; for it was assumed to be at rest [only] during interval cd. Indeed, by pursuing the same way of arguing, it will even be demonstrated that the stone will forever be at rest at b; although in fact it is assumed to change into a natural motion. {1}
DO. I cannot but acknowledge that these demonstrations lead to necessary conclusions, since they depend on very manifest and very certain principles, which can in no way be denied. And yet something that I do not know still upsets my mind: for if you assume that sometimes the impelling force is equal to the resisting heaviness, how will you not also assume that sometimes the body is at rest?
AL. It will be easy to remove this doubt.For it is one thing to say that the heaviness of the mobile sometimes comes into equality with the impelling force; but quite another to say that it remains in such a state of equality during an interval of time. This will be made manifest through an example. For, when a mobile is moved, because (as has been demonstrated) the impelling force is always weakened, but the heaviness always remains the same, it followa necessarily that, before they reach a ratio of equality, innumerable other ratios intervene: and yet it is impossible that the force and the heaviness remain in any of these ratios for any interval of time; since it has been demonstrated that the impelling force never remains for any inteval of time in the same state, since it is always weakened. And so it is true that the force and the heaviness pass, for example, through the ratio of 2/1, 3/2, 4/3, and innumerable others; but it is absolutely false that they remain in any one of them for any time whatsoever.As, for example, if we imagine that a certain mobile is moved on a certain surface, it then will touch all the lines of the surface, and will cross each one of them: but it is false [to say] that it remains in any one of them for any interval of time; since it is never at rest, but is always moved. Also by similar reasoning it is true that the force and the heaviness sometimes come to a ratio of equality, as to innumerable other [ratios]; yet it does not follow that because of that they stay at rest in such a ratio and remain for an interval of time, just as they do not in the other ratios.
DO. All doubt is now removed, and I am compelled to approve of your demonstrations.But what about Aristotle's arguments?
AL. Aristotle's most powerful argument was this one.For he said: If at the turning point no rest is given, it follows that two contrary motions are continuous, and, because of that, they are only one, since they have only one limit; which would be absurd. {1}Now to this the response is that such motions are contiguous, but not continuous; from this no error ensues; and that nothing intervenes between the limit of the upward motion and the limit of the downward motion; for such is, according to Aristotle himself, the nature of contiguous things. But still someone has at times objected, on the basis of Aristotle's own way of thinking, saying that air is also a cause why a heavy thing is at rest at the turning point: for since the mobile at such a point is very much weakened, it suffers resistance from the air, which resists its motion. Now to this I say that it is not my part to respond: since, even if at the turning point rest would occur in virtue of such a cause alone, one would not, however, have to say because of that that rest is necessary at the said point; this cause would be by accident. But if Aristotle had thought that contrary motions have, in virtue of the nature of contraries, the property that they could be joined together and therefore that it was a result of their nature that at the turning point an intervening rest was not required, he would certainly have said that rest is absolutely not given at the turning point: but if a rest
393 then occurred through the intercession and resistance of the air, he would have placed the latter at the level of causes by accident, and, when he discussed the nature of contrary motions, he would have disregarded it completely.However, and so that no one might perhaps believe that through this cause a rest really occurs at the turning point, I have decided to remove it completely from the dispute, by making clear that a rest is not required even by accident. First, the inconsistency of the adversaries would be in opposition: for, according to their needs, they say that motion is helped by the medium, and that this same motion is hindered by the medium, because, of course, the medium is in the way. But if the medium helps the motion, how does it destroy the latter at the turning point?Secondly: as motion is to motion, so is resistance of the medium to resistance; so that the more the motion is fast and impetuous, the more the medium will resist, since it must be split more swiftly: therefore by an altering of the ratio, the motion will always have precisely the same ratio to the resistance of the medium; so that the slower the motion is, the smaller the resistance [of the medium] will be. Thus, since at the turning point the motion is slowest, the resistance of the medium will also be the smallest; and just as the swiftest motion overcomes the greatest resistance, so the smallest motion will surpass the smallest resistance, since the motion is always in the same ratio to the resistance. Thirdly: note, please, how weak the adversaries' manner of arguing is; for while they attempt to remove motion, they presuppose it necessarily. For either the air resists the mobile while it is at rest, or while it is moved.But certainly not while it is at rest: for since to resist is for air itself to be in a way affected, air assuredly is in no way affected by a body which is at rest, but by a body which is moved; hence air will not resist motion, unless it is while the motion is taking place.For if the air resisted, and it resisted before the motion, the motion would certainly never take place: hence one must say that the air resists while the mobile is moved. It thus follows necessarily that those who say that at the turning point the medium resists the mobile presuppose a mobile that is in motion; and thus they both attempt to remove motion, and concede it: what could be more foolish?I could also frame arguments from Aristotle's own way of thinking: he who assuredly assumes that a motion without resistance from the medium cannot take place; so that he denies motion in the void, since there there is no resistance of the medium, by saying, If there were a void, motion in it would certainly take place instantaneously. But since my opinion is different, I shall content myself with the arguments that have been brought forth.
DO. Your way of thinking concerning the turning point has been sufficiently and more than sufficiently
394 confirmed, both through your demonstrations, and through your rejection of the contrary arguments: but now that you have alluded to the void, do not be reluctant to say something concerning this.
AL. On the subject of the void I could bring forth many things for everyone to see; things which I will, however, keep silent about, in order that such a dispute may not distract us from our purpose, and I will bring forth only that which depends on what has preceded. I say, then, that in a void motion would not take place in an instant: which is known on the basis of the things that have been demonstrated. For it has been shown that the swiftness of things that are moved is equal to the amount of heaviness by which they exceed the medium through which they are moved. {1}Thus, if a is the size of a mobile, and b, the amount of the medium through which it must be carried, is equal to the size of a, and the heaviness of a is 8, and of b as 3, surely then the speed with which it will go down will be as 5: but if the heaviness of b were 2, the speed of a would be 6: but if the heaviness of b were 1, the speed would be 7: and if the heaviness of b were zero, the speed of a would be 8, and not infinite.For the heaviness of a exceeds a heaviness which is zero by its whole heaviness, which is finite: but the swiftness of the motion is as the excess heaviness: therefore the speed of the motion will be delimited, and not infinite.That is, the speed would be infinite and instantaneous, the moment the heaviness would be infinite, and then, the motion would be instantaneous, as much in the void as in the plenum; at least, in a penetrable plenum and of non infinite resistance. Thus we will with utmost rightness say that an infinite heaviness, wherever it may be moved, is moved in an instant; but a finite heaviness, in whatever place it is moved, is put into motion with a finite swiftness. Thus, just as those who are against us argue as follows, In a void, if a motion took place, it would take place in an instant; but motion does not take place in an instant, therefore in a void motion would not take place; so we, in a converse manner, will conclude like this: In a void, if motion occurred, it would certainly not take place in an instant; but motion occurs, provided it takes place in time; therefore in a void motion will take place. From this it can also be concluded that the medium does not help motion, but directly hinders it; since where there was no medium, a more impetuous motion would take place.Secondly: one may even elicit from the argument of the adversaries themselves, by which they attempt to destroy [the idea] that there is a void, that very thing: namely that in a void motion takes place in time. For they themselves say that if you take two perfectly polished stones, whose surfaces
395 are so well matched when they have been fitted together that nothing of a different kind is left between them, then if you attempt in turn to separate them, but in such a way that they are always equidistant, your effort will be in vain; for nature greatly abhors the void that would at some time be left between them: from which they conclude that a void cannot exist. But if this is true, as it certainly is, then I argue as follows: The stones cannot be separated; therefore motion in an instant does not take place in a void. For if the stones cannot be separated, lest a void place be left [between them], then they will be able to be separated when a void is not left [between them]: for the surrounding air will rush towards the void in an instant, and thus there will never be a void place.However, because the stones are still not separated, it is an indication that for a certain time a void would be left between them: and this void, because it would persist during a certain time, demonstrates sufficiently, and more than sufficiently, that motion in it does not take place instantaneously but successsively. {1} On the other hand, I want these things concerning the void to be considered somewhat as a rough estimate, since they do not especially contribute to our primary aim.
DO. And they have also been very pleasing to hear, since in my judgment they reach the truth. {1}Thus, I pray you, do not be reluctant
396 to spend a little more time discussing this. For I wish to hear from you concerning a question I have.If, then, your opinion is true, as your demonstrations seem to bear out, the contrary opinion will necessarily be false: from this it necessarily follows that Aristotle fell into some error when he tried to demonstrate the contrary; which he did, of course, in Book IV of the <i>Physics</i>.And since he made use of a kind of geometric demonstration, I am surprised that sophisms are found in it: and for that reason I would like to ask you again and again to make this error visible.
AL. If we want to examine with precision Aristotle's demonstration, the discussion will be too long, carrying us beyond our primary aim. But, in accordance with your wish, I will uncover completely the sophisms in it; and so that they may appear more clearly, I will bring forth his demonstration for everyone to see.In the first place, then, he puts this forward: that the slowness and the swiftness of motion depend on a twofold cause; namely, either on the mobile itself, or on the medium itself. On the mobile itself, to be sure: for a heavier mobile will be moved through the same medium more swiftly than a less heavy one.On the other hand, he says that by reason of the medium slowness or swiftness happens in two ways: first when, in the case of the same medium, either it remains at rest, either it rushs in the contrary or in the same directions with the mobile: for the motion of the same mobile will be faster if the medium is carried in the same directions than if it remains at rest and [faster] if it remains at rest than if it is moved in the contrary directions: second, in the case of different media, the mobile shall be swifter when it will be moved through one which is more subtle than a thicker one, as through air than through water. These things having first been pointed out, since he saw that the same weight is carried more swiftly through more subtle media than through thicker ones, he presupposed, in second place, that the speed of motion [in one medium] observes the same ratio to the speed [of motion in another] as the subtlety of the first medium to the subtlety of the other medium. {1} These things having been established, turning to the demonstration he argued as follows: Let mobile a cross medium b in time c; but let it cross a more subtle medium, namely d, in time e: it is manifest that, as the thickness of b is to the thickness of d, thus time c is to time e.Then let f be a void; and let mobile a, if that can happen, cross this f in time g: the void will thus have the same ratio
397 to a plenum as g has to e.Let us then consider another medium more subtle than d, to whose thickness the thickness of d has the same ratio as time e has to time g.Now, from the things that have been established, mobile a will be moved through this medium that has just been found in time g, since medium d has the same ratio to this medium that has just been found as time e to time g; but, in this same time g, a is also moved through void f: hence a, in the same time, will be moved through two equal distances, one of which is a plenum, but the other a void; which assuredly is unacceptable.
This is Aristotle's demonstration: to be sure it would have reached a sound and necessary conclusion, as far as the form of the demonstration is concerned {1}, if Aristotle had demonstrated the things he took for granted, or, if they had not been demonstrated, if they had at least been true; but Aristotle has (in my opinion) been deceived in this, that he assumed things as known axioms which not only are not known to the senses, but have never been demonstrated either, nor are they even demonstrable, since they are completely false. In order that this may appear clearer than daylight, we will examine his hypotheses one by one.
First, then, he assumes that the cause of the swiftness and the slowness of the motion is the subtlety or the thickness of the medium, through which the motion takes place: this supposition is assuredly false, both because of what has been demonstrated above {1}; where it was shown that it is the heaviness of the medium, and not its thickness or its subtlety, which is the cause of the slowness or swiftness of motion; and also because of the things that we shall add now. For, I pray you, if the subtlety of the medium is the cause of the swiftness of the motion, no doubt in a more subtle medium motion will take place faster: now, according to Aristotle himself, air is more subtle than water: yet certain things are moved faster in water than in air. {1}Just as if, for example, we take an inflated bladder, it will be moved with natural motion more slowly in air than in water: for if it were restrained and tied down in deep water, then, released from its bonds, the bladder would fly very swiftly upward; now if we took a still lighter body, it would be moved more slowly in air, but faster in water; so that we would be able to arrive at something which in air would hardly be moved, but in water [would be moved] very swiftly. Moreover I will add this: for if we consider a certain body so light that it goes up in air, no doubt it will be lighter than the bladder; if then such a body is held by force under water, and is then released, who will doubt that it will go up far more swiftly in water than
398 afterwards in air, although air is more subtle than water?How, then, will it ever be true that natural motion is necessarily faster in a more subtle medium than in a thicker one? {1}
Now here it does not escape me that there is a great number of more recent philosophers who profess to know things which they know more by relying on the authority of others than by demonstration: these persons, if they heard such things, would immediately undertake to reply, and they would be satisfied to throw in a couple of words, even if they had nothing to do with the subject; for a little latter they would add, Such an opinion has been more than sufficiently and fully refuted before; and it is with similar turgid terms that they persuade themselves and their listeners, who are even more ignorant than they, of their opinions.For to these people, if they were to hear my arguments, and retorted that my reasoning is not conclusive, because I speak now of upward motion and now of downward, which is contrary to Aristotle's intention, or brought forward similar remarks without any soundness, {1} to these people, I say, it would seem that my opinion has been more than sufficiently refuted. But so much for them: let it be sufficient for me that I would have forestalled such a response of theirs, when I added a second example {1}, which only talks of one motion.It must therefore be concluded that the following is entirely false, namely that the slowness or the swiftness of motion results from the thickness or the subtlety of the medium.
Secondly: Aristotle has taken for granted as if it were known, that the speed of one motion has the same ratio to the speed of a second motion, as the subtlety of the first medium has to the subtlety of the second medium. This, Aristotle has not demonstrated, and he even skillfully avoided it: for to have taken pains to do so would have been in vain, since it is not demonstrable, and, not only not demonstrable, but actually false. For, even if it is conceded that subtlety is the cause of swiftness, it will follow, to be sure, that in a [medium of] greater subtlety, the swiftness will be greater; but it still will not follow that the swiftnesses and the subtleties increase in the same ratio. And, to use Aristotle's way of speaking, the subtlety of air would have no ratio to the subtlety of water: for, as an example, {1} wood goes down in air, but not in water; hence the swiftness in air will have no ratio to the swiftness in water. {2}In order that this may appear clearer than daylight,
399 I will first show, by a demonstration entirely similar to the one by which Aristotle tried to demonstrate that motion in a void would happen in an instant, that motions do not observe with one another, with regard to speed, the ratio of the subtleties of the media; then, I will also show what ratio they do observe, so that the truth be more clearly known.
Thus if, as he himself said, the speed [in one medium] to the speed [in another] has the same ratio as the subtlety of [the first] medium to the subtlety of the other, let there be a mobile o; and let there be two media a, b, of which a is, for example, water, and b air; {1} also let the subtlety of air, which is 8, be greater than the subtlety of water, which is 2; and let the mobile float, not sink in water, but in air let its swiftness be as 4; and as the subtlety of air b, which is 8, is to the subtlety of water a, which is 2, let the swiftness in air, which has been assumed to be 4, be to another swiftness, which, of course, will not be zero, but 1.And so, since mobile o in subtlety b is moved with a swiftness of 4; and the swiftness of 4 is to the swiftness of 1, just as the subtlety of b is to the subtlety of a; it is thus manifest that the swiftness of mobile o in subtlety a will be as 1: however it has been assumed to be zero; which is unacceptable.Therefore speed will not be to speed as subtlety is to subtlety, as Aristotle assumed.
But I think it would be good to bring forth another demonstration. For let the conditions be the same as above; and let the subtlety of b be as 16, and the subtlety of a as 4; and let mobile o not be moved in a, but let it float; and let the swiftness of the same mobile in medium b be as 8. Again let there be another speed, which is as 1; and let the subtlety of b have the same ratio to the subtlety of another medium, c, as the swiftness 8 is to the swiftness 1: the subtlety of c will then be as 2.And since speed 8 is to speed 1 as the subtlety of b is to the subtlety of c, mobile o in subtlety b is moved with swiftness 8; therefore the same mobile o in the subtlety c will be moved with a swiftness as 1. That is, o will be moved in medium c: but medium c is thicker than medium a (for the subtlety of medium a, 4, is greater than the subtlety of c, which is as 2); now it has been assumed that in medium a mobile o is not moved: consequently mobile o will be moved through a thicker medium, but not at all through a more subtle one: which is most absurd and entirely unworthy of Aristotle.
It is thus evident that the speeds of motions do not observe with one another the ratios of the subtleties of the media. But in order that it may be known what ratios they observe, let us take {1} the true cause of the slowness and the swiftness of motion of the same mobile; which, as we have demonstrated above, is the lightness or heaviness of the medium with respect to that of the mobile. Let there be a mobile a, and a medium c, twice as light as medium b: certainly the time during which a is moved through b will not be twice the time during which a is moved through c; that is, the swiftness in [medium] c will not be twice the swiftness in b. {1}For let {1} d be the time in which a is moved through b itself; and let e be the time in which a is moved through c.And since the swiftness of a in the space of b will be equal to the excess by which the heaviness of a exceeds the heaviness of b, as demonstrated above; if the heaviness a is 20, but the heaviness of b is 8, the swiftness d {1} will certainly be 12: but, for the same reason, if the heaviness of c is 4, the swiftness e will be as 16: hence the swiftness e will not be twice the swiftness d. Therefore, since spaces b, c are equal in length, the time d will not be twice the time e.It is thus evident that the speed in medium c is not as great as Aristotle wanted it to be, but that it is much smaller: for, according to his way of thinking, the speed e would have to be, in consideration of speed d, 24; actually it is 16. It is therefore manifest that the lighter the medium, the faster the motion due to heaviness will be. But because speed always stands to speed in a lesser ratio than rareness to rareness, it follows that, when the rareness of the plenum stands to the rareness of the vacuum in the greatest of all ratios, it is not the case that the speed in the plenum must observe the same ratio with the speed in a void, as Aristotle falsely thought.
401 From the things that have been set out above, it follows that swiftness to swiftness always observes the ratio of the lightness of one medium to another arithmetically, not geometrically. {1}For if, again, the heaviness of c is 2, so that its lightness is four times the lightness of b, the swiftness of e certainly will not be four times the swiftness of d, but will be 3/2; for the swiftness e will be as 18: and yet it will have the same arithmetic ratio [to d] as c has to b {1}, since the excesses are equal, namely 6. And if, again, the heaviness of c is 1, so that the lightness of c is eight times the heaviness {1} of b, the swiftness of e certainly will not be eight times that of d, but by far smaller than eight times, namely 19/12; for the swiftness of e will be as 19: and the arithmetic ratio [of the swftnesses] will be the same as that of the lightness [of one medium] to the lightness [of the other], since the excess is the same. namely 7. Now if the heaviness of c is zero, so that the lightness of c has no ratio to the lightness of b, the swiftness of e will be as 20, having the same arithmetic ratio to d as 8 to 0: for the excess of swiftness 20 to swiftness 12 is the same as that by which 8 exceeds 0, namely 8.And thus, contrary to what Aristotle says, it is not unacceptable that a number has to another number the same ratio as a third number has to zero, provided that we speak of an arithmetic ratio: 20 to 12 has the same ratio as 8 has to 0; for the excess of 20 over 12 is the same as the excess of 8 to 0.
DO. Oh! what a subtle discovery, oh! how beautifully imagined!Let them remain silent, silent, those who assert that they can pursue philosophy without a knowledge of divine mathematics. And will anyone ever deny that only with it as guide can the true be distinguished from the false, that with its aid keenness of mind is stimulated, and that, finally, with it as guide whatever is really known among us mortals can be apprehended and understood? {1}
AL. Listen, I pray you. On the basis of his hypotheses, Aristotle drew a second argument: namely, that if motion in a void took place in time, then lighter and heavier things would be moved with the same swiftness, since for both lighter and heavier things there would be no resistance from the medium {1}; which is unacceptable. In this argument Aristotle has been deceived in a similar way, in that he assumed that the swiftness and the slowness of motion arise only from the resistence of the medium, whereas in fact the whole affair depends
402 on the heaviness or the lightness of the medium and the mobile.{1}Thus I say that, in a void heavier things would go down more swiftly than lighter ones; since the excess [of heaviness] of the heavier things over [the heaviness] of the medium would be greater than the excess [of heaviness] of lighter things. {1}
And moreover what Aristotle has said concerning the ratio of the motions as compared with the heavinesses of the mobiles is not true: namely that, for example, the ratio of swiftness to swiftness [of two mobiles] in the same medium is the same as the ratio of the heaviness [of one] to the heaviness [of the other]; as if, for example, a were twice as heavy as b, the swiftness of a also would be twice the swiftness of b. For we shall demonstrate this in the same manner as above.For if the heaviness of a is 4, but the heaviness of b is 2; and a goes down in the medium water, and its swiftness is 2, but b does not go down; it is then clearly evident that the swiftness of a will not be twice the swiftness of b, since b is not moved. But here also an arithmetic ratio will be observed by the swiftnesses, that is in accordance with the excess of the heavinesses over the heaviness of the medium; thus if a is 4, but b 2, and the medium is 1 in heaviness, then the swiftness of a will be 3 as compared with the swiftness of b, which will be as 1. Therefore, to summarize I will say, that the slowness and the speed of all downward {1} motion comes, in the first place and per se, from the proper heaviness of the mobiles; motion comes to be weaker because this heaviness then comes to be less than the heaviness of media. But if the heaviness of the medium is equal to that of the mobiles, then, since the mobiles in such a medium have no heaviness, no motion takes place; but if, on the other hand, the heaviness of the medium is greater [than that of the mobiles], then the heaviness of the mobiles in comparison with the heaviness of the medium becomes lightness, and the mobile is carried upward; but if the heaviness of the medium is zero, then the mobiles will be moved according to their pure heaviness, and they will observe in their motion the ratio that their proper heavinesses have to each other. And from this another serious error is made evident; Aristotle's way of thinking on this matter is precisely the opposite it what it should have been. For he was saying that it is in a plenum that heavy things observed in their motions the ratio of the heavinesses; and in a void, not at all, but they are all moved in the same time. But, on the contrary, in a void they will observe the ratios of their heavinesses, since the excesses over [the heaviness of] the medium are the whole heavinesses of the mobiles; in the plenum, on the other hand, they will not observe this ratio, as has been demonstrated above. But, as has often been said above, it is necessary that you always understand and
403 assume that the different mobiles of which we speak differ only in heaviness, though they are equal in size; so that you would not perchance say: Let there be, for example, a mobile [a] whose heaviness [in air {1}] is 8; and let the heaviness of an amount of water equal in size to the said mobile be 3; then it is manifest, from the things that have been said, that the swiftness [in water] of the said mobile will be as 5: but if you take another mobile, such as c, whose size were twice that of a, and whose heaviness [in air] were less than twice the heaviness of a, namely 12, and the heaviness of an amount of water equal in size to c were 6, then the swiftness [in water] of c would be as 6.However, it must not be said that c will go down faster than a; because then the ratio would no longer hold good, since the mobiles differ in size. Bu if we want the ratio to hold good, let us take half of c, so that its size is equal to the size of a; then the heaviness of c [in air] will be 6: but the heaviness [of an amount] of water [equal to] half [the size of c] will be 3, so that the speed of half of c will be 3, as compared with the speed of a, which will be 5: therefore a will go down faster than the whole of c (for the whole is moved with the same swiftness as its half). Therefore it will be true to say, that the swiftness of a is 5 as compared to the swiftness of half of c, which will be 3; or else that the swiftness of the whole of c is 6 as compared to the swiftness of the double of a, which will be 10. Hence it is evident that when we reason on the subject of the swiftness or the slowness of mobiles, it is necessary to have in mind those mobiles whose difference depends solely on heaviness, although they are equal in size. Then when the ratio of the speeds to one another has been found, the same ratio will be observed between mobiles of the same kind, no matter how much they may differ in size {1}: for a piece of lead whose heaviness is 10 pounds goes down with the same speed as a piece of lead whose heaviness is 100 pounds.
DO. This seems truly astonishing, and is contrary to Aristotle's opinion. It will indeed be difficult for me to believe it, unless you somehow convince me.
AL. The demonstrations brought forth above should be sufficient to convince you of this; {1} even if they do not explicitly demonstrate this, it nevertheless depends on them: but, if they are not sufficient for you, I will bring forth
404 others for everyone to see. In order that I may achieve my purpose, I ask you to grant me the following: namely, that if there are two mobiles, one of which is moved more swiftly than the other, these mobiles joined together will be moved on the one hand more swiftly than the one which alone was moved more slowly, but more slowly than the one which was moved more swiftly. As if, for example, mobile a is moved faster than b, I say that the mobile composed of both a and b will go down more slowly than a alone, but more swiftly than b alone. {1}This is clearer than daylight: for who doubts that the slowness of b will retard the swiftness of a, and, on the contrary, that the swiftness of a will intensify the motion of b, and that in this way a motion intermediate between the swiftness of this a and the slowness of b will take place?
DO. I shall never dare deny this.
AL. This having been presupposed, if for my adversaries such a thing can be done, let a great amount of something be moved more swiftly than a small amount (let them be of the same matter); and let a on the one hand be the great amount and b the small amount. If therefore b is moved more slowly than a, therefore, from these things that have been assumed above, the mobile composed of both a, b will be moved more slowly than a alone: and a, b are of the same matter: therefore the larger amount of the same matter will be moved more slowly than the smaller amount of the same matter; which is surely diametrically opposed to their view, and is contrary to what has been presupposed.It is therefore not true, that a great amount is moved more swiftly than a lesser, if they are of the same matter; although Aristotle, against the Ancients, has presupposed this as something known, throughout Book IV of the <i>De Caelo</i>.Consider, thus, how firm the foundations are on which rests his refutation of the opinion of those who did not assume (as indeed should not be assumed) the purely and simply heavy and the purely and simply light, but only the comparatively lighter and heavier: and, consequently, you can see the strength of the reasons by which he tried to attribute to earth and to fire absolute heaviness and absolute lightness, and even to water and to air heaviness in their proper region.But, having put these things aside, returning to our subject I say that the same reasoning must also be made in the case of the void; namely, mobiles which are of the same matter, though unequal in size, are carried with the same swiftness: which will be demonstrated exactly the same way as in the case of a plenum.
DO. You can return to the solution of the remaining problems {1}, which I await with eager ears.
AL. Receive now my solution to one problem, which can be known solely on the basis of the things assumed above: namely, the problem in which
405 the cause was sought why natural motion is faster at the end than at the middle, and faster in the middle than at the beginning. Thus we must recall what was made clear above; namely that a mobile, when it is moved with a violent motion, is moved up to the point where the force impressed by the motor is greater than the resisting heaviness: from this it follows that when a heavy thing ceases to go up, the force impressed in it is equal to its heaviness; from this it clearly follows that, when a heavy thing begins to be moved downward, it is not moved with a motion that is purely and simply natural. For at the beginning of such a motion there is still in this mobile some of the impressed force which was impelling it upward: this force, because it is smaller than the heaviness of the mobile, does not impel it any farther upward; however it still resists the heavy thing that is heading downward, because it has not yet been annihilated.For it has been demonstrated that it is successively weakened: and thus it happens that the mobile at the beginning of its own natural motion is moved slowly: but afterwards, since the contrary force is weakened and diminished, the moving thing, encountering lesser resistance, is moved more swiftly. As if, for example, we imagine a mobile which is moved with a violent motion from a to b, whose heaviness is 4, it is manifest that the force which impels it will be greater than 4 at any point of the line ab described by the forced motion: but at b itself it will not be greater than 4 (for if it were greater, the same mobile would be impelled by it beyond b); also it will not be smaller (for it would have been equal to it before [the body was at] b; but it has been demonstrated that it has always been greater); hence the force at b will be equal to the heaviness of the mobile, namely it will be 4.Therefore, when the mobile recedes from b, the force which was as 4 begins to weaken, and, on account of this, the mobile begins to have a lesser resistance to its own heaviness; because this resistance is continuously weakened, it results that the natural motion is continuously intensified.
DO. This solution more than pleases me: however it seems to have its place only in the case of a natural motion preceded by a violent one. But when someone having a stone in hand does not impel it upward, but only lets go of it, in this motion, which is not preceded by a violent [motion], what will be the cause of the intensification? {1}
AL. This very doubt had also come to my mind when I was working out an explanation to this problem; but when I examined the matter more carefully, I discovered that it was of little importance. Thus the intensification occurs by the same cause in each of the two motions [downward], as much in that which is preceded by a state of rest, than in that [of a violent motion {1}]
406 which has preceded a natural motion: for even in the case of a natural motion which is preceded by a violent one, the mobile recedes as from a ratio of equality {2}, which is the ratio of rest. Now pay attention in order to understand this more clearly. {1}Let o be a mobile whose heaviness is 4: and let the line along which the violent motion takes place be oe. Thus it is manifest that in mobile o there can be impressed a force great enough to move {1} it as far as r; this force will necessarily be greater than 4, which is the heaviness of the mobile: again a force can be impressed which moves it only as far as t; again it will be greater than 4, and smaller than that which impelled it as far as r: again, a force can be impressed that is great enough to move the piece only as far as s; however it will be greater than 4, but smaller than that which impelled it as far as t: and so on, indefinitely, a force can always be impressed which will impel the mobile over any distance, however small; and yet this force will always be greater than 4. It remains, therefore, that that force is 4 which impels the mobile with a violent motion over no distance at all: from this it is evident that when mobile o recedes from the hand, it recedes with a force that is as great as 4; which then, since it is successively consumed by the heaviness, is the cause of the intensification of the motion. And what I have said will appear even more lucid if we consider that, when a heavy thing is at rest in the hand {1}, since by its heaviness it exerts pressure downward, it is necessary that it be impelled upward by something, namely the hand, with a force equal to it own heaviness, which exerts pressure downward: otherwise, unless it were hindered by another force, as great, impelling it upward, it would head downward, if the resistance were smaller; but upward, it it were greater. Therefore it is evident that, when it is abandoned by what sustains it, a heavy thing goes down with an impressed force equal to its proper heaviness; from which it follows, etc.
DO. What you say is quite satisfying; and yet there still remains something I do not understand, which troubles my mind. {1}For if the slowness of the natural motion at the beginning occurs by the resistance of the impressed force, this force will eventually be consumed, since you assert that it is continuously weakened; therefore the natural motion, when the said force will be annihilated, will no longer be subject to any more violent motion: which, however, is opposed to the opinion of many.
AL. That this is opposed to the opinion of many, is of no concern to me, so long as it is coherent with reason and experience, even though sometimes experience points rather to the contrary. For if a stone goes down from a high tower, its swiftness seems always to be increased: yet this happens because the stone, in comparison
407 with the medium through which it is carried, namely air, is very heavy; and since it goes away with an amount of impressed force as great as its heaviness, it assuredly goes away with a great impressed force, which the motion from the height of a tower is not sufficient to consume, so that the swiftness is always intensified all through the height of a single tower.Now if we were to take a certain heavy thing, whose heaviness would not so very far surpass the heaviness of air, assuredly we would then see with our own eyes that this thing, a little after the beginning of its motion, would always be moved uniformly, provided however that the air was very calm. And we would observe the same thing happen in the case of a stone, if it were let down from a very high place, and we were so placed as always to observe the line of motion according to the same disposition. {1}For even our position hinders us from apprehending the uniformity of the motion. For let there be a uniform motion from b to f, and let bc, cd, de, ef be equal distances; and let the eye of the observer be at a, and let lines of sight ab, ac, ad, ae, af be drawn: and since the motion is assumed to be uniform, and bc, cd, de, ef, are equal distances, therefore the mobile will go over them in equal times. Hence the time of transit from b to c will be equal to the time of transit from c to d: the motion from c to d, however, will appear faster to the observer, since distance cd also appears greater than distance bc (for it is seen under a greater angle {1}). And thus the motion from d to e will appear faster than that from c to d, since distance de appears greater than cd, and it is traversed by the mobile in an equal time: and for the same reason, the motion from e to f will appear faster than the motion from d to e. Hence the whole of motion bf will appear difform, and always faster at the end, although it is assumed to be uniform. It is thus necessary for distinguishing the uniformity and the difformity of motion that the distance be extended enough that the mobile is able to consume the resisting force in it, and that the eye be disposed in such a way that it is deceived as little as possible by the disparity of the angles.
DO. I have more than adequately understood your most elegant explanation.And so, I have but one further thing to inquire about on this subject; and that is whether you believe that a heavy thing which is sent downward and forcibly thrown by a mover, is accelerated in its motion just like
408 that heavy thing which, in going down, has not received from the mover any force impelling it downward.
AL. From the things that have been assumed above it is evident that a heavy thing which goes down, receding from a state of rest, is accelerated in its motion until the resisting impressed force is annihilated ; but if that force is extinguished by an external motor, then it will no longer be accelerated in its motion. As, for example, if a heavy thing whose heaviness is 4 goes down from a state of rest, assuredly it will recede with a resistance of 4; since this must be destroyed by the heaviness of the mobile, the natural motion at the beginning will be weaker {1}: but if the said force as 4 is consumed by an external motor, by impressing on the mobile a force which exerts pressure downward as 4, then without a doubt the heavy thing will no longer be accelerated in its motion, since at the beginning it would not be retarded by any force resisting [the downward motion]. But if the force which is impressed by the external motor impelling downward is smaller than 4, that is, less than the force which had been impressed in the mobile while it was in a state of rest, assuredly the mobile will be accelerated [in its motion downward]; because there is a certain amount of contrary force to be consumed that had not been completely removed by the external motor. Now if there is impressed on the mobile a force propelling downward that is greater than 4, then the natural motion will be faster at the beginning: for it will be moved with a motion that is greater than natural, and which exceeds the motion required by its own proper heaviness {1}. So that its heaviness would then have the effect of lightness {1}, since the heaviness by itself, free and simple, would go down more slowly than when joined with the impetus; and thus the proper and natural slowness of the descending [mobile] would resist the violence impelling {2} [it] downward. And this will be rendered manifest with an example of what happens often to any swimmer.For, when a man is in the water, if he wishes, his heaviness is great enough for him to go down to the very bottom of the water, if he wants to; and then, pulled by his proper heaviness, he will be submerged with a uniform motion: now if he is impelled downward by an external motor with a force as great as can be, as if he were to be cast down from a very lofty place, at the beginning his motion in the water will be very vivacious and greater than natural: and yet his motion will be retarded by his proper absolute {1} heaviness, which then, in comparison to the heaviness which has been joined with the impetus received, is as lightness, and [it will be retarded] until in going down he reaches his natural slowness; and in such a way that, if the water is deep enough, he will not suffer any greater injury at the bottom than if he had gone down from the surface of the water with his proper natural motion.
409 In order that you may not, by any chance, believe that Aristotle thought that the purely and simply heavy is not earth but in general that which is under everything else, see <i>De Caelo</i>, Book IV, text #29 [311b5], where he says that everything except earth has lightness.
What Aristotle says in Book IV of the <i>De Caelo</i>, text #32 [ch.4, 311b22-24] has no value: It is impossible for fire to have heaviness, because it would stand under some other thing. If the elements are transmuted into one another, how will fire not have heaviness, when it is made from heavy air?
Aristotle, in Book III of the <i>De Caelo</i>, text #27 [301b11-13], says that in a violent motion, as the greater mobile is to the smaller, so is the swiftness of the smaller to the swiftness of the greater, if they are impelled by the same force.
It is foolish to raise a doubt about how a projectile is moved by an impressed force: for it is moved just as it happens in the case of other motions: as, for instance, iron is moved in an alterative motion and made hot by fire. For fire impresses heat; then when the iron has been removed from the fire, the heat remains, but not by the surrounding force and heat, if it is transported into very cold air; then little by little the iron is moved toward coldness, as long as it cools off: in a similar way a stone is moved by a human being: and when it has been released by a human being it is still moved, the surrounding air being unchanged, as long as it tends toward rest. Again similarly: someone strikes a bell with a hammer, and deprives it of silence; the bell is then moved, the striking thing having been removed, and the sonorous quality proceeds
410 through it, with the medium unmoved (nor does it do anything to the sound, even if the medium were moved; for no matter how strong the wind is blowing the bell is silent); the sound in the bell is progressively weakened, and by itself it returns to silence.
And just as heat as well is impressed in a more penetrating way in dense and very cold matter, like iron, than in matter which is rare and less cold, even if both of them become warm through the same heat, so a heavier thing is sometimes moved more, farther, and more swiftly by the same force.
It must not be said that in the bell it is the air which sounds: indeed this is the saying of fools; for the sound of different bells would be the same, and a wooden or leaden bell would sound just as would a bronze one.
There is a threefold classification of motion: first, with respect to spaces: thus, one straight, another circular: another, with respect to points of arrival; thus one upward, another downward: a third, with respect to primary efficient causes; thus, one natural, another violent.
Aristotle, in Book III of <i>De Caelo</i>, text #27 [301b1-4], asserts that things that are moved must be either heavy or light; for if they were neither heavy nor light, when moved by force they would be moved without end: and in the following text [301b23-27]{my notes} he says that air helps both motions, namely upward and downward, and that projectiles are carried by the medium: hence if the medium air, impelled by force, is moved without end, since it is neither heavy nor light, it will carry projectiles without end.
Philoponus, Avempace, Avicenna, the divine Thomas, Duns Scotus, and others, who try to defend [the view that] motion in a void takes place in time, do not proceed well by assuming in the mobile a double resistance, one, namely, accidental, stemming from the medium, the other intrinsic, from the proper heaviness. For these two resistances are but one, as is evident; for the same medium both resists a heavier thing more, and makes a mobile lighter. {1}
Averro‘s, in his Commentary on <i>De Caelo</i>, Book I, text #32 says {1} that a sphere is not physically tangent at a point.
411 Alexander thought he had well refuted Hipparchus' opinion concerning the acceleration at the end of natural motion, after he adduced in opposition a natural motion that was not preceded by a violent motion: but surely even Hipparchus did not notice that a violent motion precedes every case of natural motion, as we have made clear. {1}
One must crticize the false opinion of those who say that when a pebble has been projected into water, the water is later, by itself, moved in circle.
Burley, in his <i>Commentary on the Physics of Aristotle</i>, Book VIII, text #76, and Contarini, <i>De Elementis</i>, Book I, assign the cause of acceleration of natural motion at the end to the parts of the air, both those that precede and those that follow.
See Aristotle, <i>De Caelo</i> Book I, text #88 [277a27-33], where he says that the speed of natural motion is always increased, and, if [a body] were moved without end, speed would also be increased without end.
And see text #89 [277b4-8], where he says that a greater portion of earth is carried more swiftly than a smaller one, and that natural motion is not accelerated because of the extrusion of the air: for in this way a greater portion of earth would be moved more slowly than a small one, because it would be extruded with more difficulty; and it would not be accelerated in its motion, because it would be a violent motion, which is weakened.
Benedict Pereira, toward the end of chapter 3 of Book 14 {1}, writes thus: I would say without doubt that if the space of the air through which the rock is carried downward were infinite, the rock's motion would always be swifter and more vehement, and yet no addition would be made to its weight.Now notice the error of Pereira in the following words.For he says {1}: Aristotle does not conclude correctly, Because speed grows in natural motion, heaviness must also grow in the mobile: for if a stone is moved through a certain distance, whose first part is denser and thicker, while the last part is more tenuous and more rare, without doubt the motion will be faster at the end, and that will not happen on account of an increment in heaviness.
Light things are not moved faster by a greater force. Just as straw or tow is not made hotter by the greatest and most burning fire, because it does not await so much heat, but is previously burned up by a lesser one, so light things do not resist [motion] up until a great force is impressed in them, but are moved before that.
412 Julius Scaliger, in his work against Cardano, in exercice #28 {1}, brings forth certain arguments against those who say that projectiles are moved by the air.
Themistius, on the subject of text #74 of Physics Book IV [216a9-21] says {1}: It is thus that since the void yields uniformly, but in fact it does not yield in any way (as indeed it is nothing, very clever the man who can fancy the void yielding), thus the result is that the differences of heavy things and things light, that is the variations of things, are suppressed, and what follows is that the speed of all things that are moved comes to be equal and indiscriminate. So much for the words of Themistius: which in fact are not only false, but in fact it is their contrary that is true; for it is only in the void that the heavinesses show their difference precisely and naturally, and it is only there that the swiftnesses of their motions show their difference to the greatest extent. Thus let a chapter be written {1}, in which these things are demonstrated.
A motor impresses contrary qualities in a projectile, namely upward and downward: because the beginning of the motion depends on the will, which has the power to move the arm upward or downward; {1} and the impelling force upward is not different from the one impelling downward. There is the example of the steel spring in clocks, which moves [the arms of] the clock either upward or downward, either forward or backward, depending on [the direction] it is turned: its intention is to unwind and straighten itself, just as the intention of the arm is to remove the stone from itself.
There will be many who, after they have read my writings, will turn their mind, not to consider whether the things I have said are true, but only to seek in what way, whether rightly or wrongly, they could undermine my opinions.
It is better to say that things that are moved naturally are moved by the medium than [to say that] things [that are moved] by violence [are moved by the medium].
Aristotle, in <i>De Caelo</i> Book I, text #89 [277b1-2], says: Things that are moved are not moved by another thing, as through extrusion. The contrary of which, however, could be defended.
413 That a contrary force is impressed more strongly in heavier things is evident from the things which, after being suspended by a thread, are put into a motion of going back and forth: for the heavier they are, the longer they will be moved.
Things that are more solid, more heavy and more dense conserve all contrary qualities longer, more sharply and more easily; such as stones, which in winter become far colder than air, but in summer warmer.
Aristotle says {1}: Things that are moved naturally are not moved through extrusion; because in this way the motion would be violent and it would be weakened at the end, although we see that it is increased. To that it is answered that violent motion is weakened when the mobile is out of the hand of the mover; but while it is linked to the mover -- as if we should say, of what is carried toward its proper place {1}, that it is moved by air through extrusion -- it is not necessary that at the end the motion becomes weaker.
The definition of the heavy and the light through motion handed down by tradition is not a good one {1}: for when a heavy or light thing is being moved, it is neither heavy nor light. For that thing is heavy which exerts weight on something; but what exerts weight on something else is resisted by that thing; hence a heavy thing, when it exerts weight, is not moved: as is evident if you have a stone in hand, which then will exert weight when the hand resists its heaviness; but if it is moved downward with the stone, the stone will not then exert weight on the hand. Hence the definition will better be: That thing is heavier which remains under things that are lighter. {1}For if we should say, Heavy is what remains downward, and light what remains upward, we would not be defining well, since upward and downward are not distinguished in fact, but only in logic. {1}
It must be considered whether, if there were a void above water, things that are moved in water would be moved more slowly or more swiftly; and whether different mobiles would observe the same ratio in their motions.
Downward motion is more natural by far than upward motion. {1}For upward motion entirely depends on the heaviness of the medium, which assigns an accidental
414 lightness to the mobile: but downward motion results from the intrinsic heaviness of the mobile. Taking no account of the medium, all things will be moved downward.Upward motion takes place through extrusion by a heavy medium: as in the balance a less heavy thing is moved violently upward by a heavier one, so a mobile is extruded violently upward by a heavier medium. {1}
It is evident that the difficulty of cleaving the medium is not the cause why wood does not go down in water: for if [in this case] this difficulty were overcome by the form of the mobile, the wood would descend, if, for example, it was given the form of a cone or an arrow; and yet this floats no less than a flat board.
A fragment of Euclid proves that there exists a mathematical treatment of the heavy and the light. {1}
Telesio says that the cause of the acceleration of motion at the end is that matter accelerates motion because it is wearied of going down. {1}
It is proved that there exists no natural upward motion. {1}What is moved naturally, without hindrance, is moved toward a limit in which it is naturally at rest and from which it cannot back out of unless it be violently. Hence if wood goes up naturally in water, hence it it is moved to [a place] from which it will not recede except by force: but wood is carried without hindrance toward a limit which is near the surface of the water: hence from there it will not recede except by force. Which however is false; for if the water is removed from underneath it, the wood will retreat and descend naturally. And do not tell me that the limit of the natural motion of wood upward is the surface of the water itself, and because of that, if the limit is moved, what was in it is [also] moved: for this is ridiculous. For the limit of natural motion is not some body, but must be something indivisible and immobile: now only the center is such a thing. Hence it is only towards the center that something is carried naturally; at which it naturally remains at rest, and from which
415 it cannot be removed except by force: but it does not set out towards the center except by going down. Furthermore: what is moved naturally must be moved towards something determinate: but in what is upward, there is nothing of which we could say, This is the upward limit; on the contrary there can be infinite limits upward, and upward is extended without end: hence nothing can be moved naturally without end, towards the indeterminate -- hence, upward. Now concerning downward the same cannot be said: for there is a certain limit of the downward, one, finite, and even indivisible, from which a thing cannot be removed even by the width of a fingernail by seeking the downward; and such is the center. And do not tell me that there exists an upward limit, namely the concave sphere of the Moon {1}: for this is false. For the limit of a certain motion must be such that what recedes from it is no longer moved by the motion of which it was the limit: but the concave sphere of the Moon is not such; for the limit of upward motion is not such that what recedes from there cannot be moved further upward. But the center is so much the limit of downward motion that nothing, receding from there, can be moved downward more.
Remoteness from the center is infinite; but proximity is finite and determined by the center itself: hence if there is something endowed with the ability of fleeing from the center, surely this thing will be suited to be moved without end. What could be more absurd than this?
A motion to which a limit cannot be assigned cannot be natural: but to upward motion a limit cannot be assigned: hence upward motion is not natural. The minor is proved: a motion is delimited at a place from which, by the same motion, it is not possible to recede (for if by the same motion one could progress farther, the limit would not be there): but motion upward is nowhere delimited in such a way that from that place by the same motion, namely upward, it would be impossible to be brought farther: hence upward motion is nowhere delimited; hence it is deprived of a limit; hence it cannot be natural. Now it is evident that there is nowhere that a limit of upward motion can be placed, from which by the same motion it is impossible to be brought farther: for if any place is assigned [as a limit], it is possible to recede from it by going up, and another can be assigned which is more distant from the center than it. {1}
The disparity between upward motion and downward motion is as great as can be. For, in addition to the things that have just been said, there is also the following difference: the fact is that it never happens that downward motion is helped by the medium, but is always
416 hindered [by it]; for since the medium diminishes the heaviness of the mobile, it hinders its motion: upward motion in fact never takes place, unless it is helped by the medium.
To a positive effect there must exist a positive cause; hence the cause of motion cannot be lightness, which is a privation. {1}Hence it remains that it be heaviness: and that things that are moved upward are moved by heaviness.
We call local motion that [motion] in which the center of heaviness of the mobile is moved; hence we will not speak of the local motions of the celestial orbs since their center of heaviness, which is also their center of magnitude, always remains immobile.
If the corpulence and the density of water were the cause why wood is not submerged, undoubtably by the same cause, after it was submerged by something else, it would be hindered from coming back upward.
Aristotle has shown the falsity of Plato's excessive devotion to geometry, in Book I of De generatione et corruptione [ch. 2, 315b25-316a17]. {1}
Aristotle, in Book III, section 8 of the <i>Metaphysics</i> [ch. 2, 997a35-998a4], writes thus: For the lines perceptible to our senses are not such as geometry presupposes: for nothing that is perceptible to our senses is straight and curved in such a way: for the circle does not touch the rule at a point, but [it touches it] just as Protagoras said, in showing the falsity of the geometers.
Aristotle, in Book VII, text #10 of <i>Physics</i> [243a12-15], says that for the naturalness of motion an internal cause of motion, and not an external one, is required.
Arstotle, in Book III, text #72 of <i>De Caelo</i> [307a13-22], says: If it is because of the triangles that fire warms, it would follow that mathematical bodies would warm.
417 Aristotle, in Book I, text #44 of <i>De Caelo</i> [273a9-13], says that if one of two contraries is delimited it is thus necessary that the other be delimited as well; and because the center, which is the limit of downward motion, is limited, it is necessary that upward [motion] itself be delimited . And the same thing is concluded from text #58 [274b9-15]: {1} read {2} the text.
Aristotle, in Book I, text #51 of <i>De Caelo</i> [273b32-274a2] {1} says, Speed is to speed as heaviness is to heaviness.
Aristotle, by text #89 of Book I, of <i>De Caelo</i> [277a33-277b2], shows that things that are moved naturally are not moved by force and through extrusion. {1}
Aristotle, in Book I, text #96 of <i>De Caelo</i> [278b14-15], writes: For as a rule we have called the outermost thing and what is uppermost the heaven. And in text #21 [270b6-8] he says: And the place which is above that is assigned to God. {1}
Things that are moved upward, go up more violently than naturally: for ascent has an external cause, but descent an internal one.
Aristotle, in Book I, text #5 of <i>De Caelo</i> [268b21-22], says: Now I call upward what is away from the center, but downward what is toward the center. {1}
[Program of a work on motion]{1}
It may be asked whether heavy things are truely moved toward the center; on this, Ptolemy, <i>Almagest</i>, Book I, chapter 7.
Whether the impressed force will be consumed by time or by the heaviness of the mobile.
By what natural motion takes place.
By what violent motion takes place.
Whether a medium is necessary for motion.
Whether the purely and simply heavy and purely and simply [light {1}] are given.
Whether the elements in their proper place are heavy or light.
Concerning the ratio of motions of the same mobiles in different media.
Concerning the ratio of the motions of different mobiles in the same medium.
Concerning the cause of the slowness and the speed of motion.
Whether there a rest at the turning point.
Whether natural motion is always intensified, and why it is intensified.
Whether the slowness and the swiftness of natural motion is due to the rareness or the de....of the me.... .
In motion 3 things are considered: the mobile, the medium, and the mover.
How the form of mobiles helps or hinders motion.
Concerning the ratio of the heavinesses of the same heavy thing in different media, on which depends the question of the ratio of its motions.
Given the heaviness of the medium and the speed of the mobile, is also given the heaviness of the mo.... .
Given the heaviness of the mobile and of the medium, the speed of motion is given.
Given the speed and the heaviness of the mobile, the heaviness of the medium is given. {1}
Concerning circular motion.
One must consider the ratio of motions on different inclined planes, and whether perchance lighter things go down more swiftly at the beginning; just as in the balance, the smaller the weights, the more easily the motion takes place.
The medium retards natural motion in the following manner: as, when a bell goes down, it is so to speak a solid body consisting of air enclosed .... by the metal, and because of this it is lighter than if air were not present.
Lighter mobiles are moved more easily, as long as they are linked.... to what moves them; but, outside the hand of the mover, ....[they conserve] for a short time the impetus.... .
The argument of those who say that motion is accelerated at the end since.... {1} .End see my notes below.