ON BODY
Translation © George MacDonald Ross, 1975–1999
PART II: FIRST PHILOSOPHY
Chapter 7: Space and Time
7.1. Non-existent things are intelligible and can be reasoned about by virtue of their names.
[81] As I have already shown above, the best way to begin an exposition of the philosophy of nature is to start from absence of being — that is, to imagine that the universe has ceased to exist. Let us suppose that everything has been annihilated, apart from yourself, who have escaped the holocaust. It might well be asked what is left for you to philosophise about, or to reason about at all, or what you could give a name to in order to reason about it.
What I say is that you would retain the ideas of the world and of all the bodies which you had looked at with your eyes, or perceived with your other senses, before they were annihilated — in other words, a memory and imagination of [82] sizes, motions, sounds, colours, etc., and also of their order and parts. Even if all these are only ideas or phantasms, and internal accidents of yourself who is imagining them, nevertheless they will appear to you as external, and independent of the powers of the mind. So you would give them names, and subtract and add them. Since the supposition is that everything else has been annihilated, and nothing is left but yourself, as a person who thinks, imagines, and remembers, the only things you can think about are things which no longer exist.
But even if things have not been annihilated, if we pay careful attention to what we have in mind when we reason, the only things we make our calculations about are our phantasms. If we make calculations about the sizes and motions of the heavens and the earth, it is not as if we literally travelled into the sky to divide the sky itself into its parts or measure its motions. Rather, we do it without moving anywhere, and in our studies, or with our eyes closed.
So there are two ways in which things can be considered, or brought into account: as internal accidents of the mind (as they are considered when it is a question of the faculties of the mind), or as species of external things — that is, not as existing, but as appearing to exist or to be external to ourselves. I shall now consider them in the second way.
7.2. What space is.
If we remember, or have a phantasm of something which existed before the supposed annihilation of external things; and if we have no interest in what sort of thing it was, but are only interested in the fact that it was external to the mind; then we have what is called space. It is certainly imaginary, since it is a mere phantasm; but it is precisely what everyone calls ‘space’. No-one calls a space a space because it is filled by something, but because it can be. Again, no-one thinks that bodies can transport their positions with them when they move, but that different bodies successively occupy one and the same space. [83] This would not be possible if the space which a body originally occupied always travelled with it.
This is so obvious that I would not have deemed it worthy of explanation if I had not come across philosophers who have given false definitions of space. One concludes directly that the world is infinite. He identifies space with the extension of bodies themselves, and since space can be infinitely extended, he says that this is also true of bodies themselves. Another, starting out from the same definition, draws the rash conclusion that it is impossible even for God to create more than one world. He says that if another world was to be created, since there is nothing outside this world, and therefore no space (according to his definition), it would have to be situated in nothing. But nothing, he says, can be situated in nothing; however, he does not explain why something cannot be situated in nothing. In fact the opposite is the truth: nothing additional can be situated where something already exists, but empty space is as capable of accommodating new bodies as a plenum is not. I say all this for the benefit of these two philosophers and their followers.
I now return to the topic in hand, and say that the definition of space is as follows: space is the phantasm of an existing thing in so far as it exists — that is, with no consideration of any of the thing’s accidents other than the fact that it appears external to the person imagining it.
7.3. Time.
Just as a body leaves a phantasm of its size in the mind, so a moving body leaves a phantasm of its motion in the mind, namely the idea of the body passing through a continuous succession of spaces. Such an idea or phantasm is what I call time; and this is in accordance with ordinary language, and not much different from Aristotle’s definition. Since everybody agrees that a year is a time, but nobody thinks that a year is an accident, or affection, or mode of a body, they must necessarily [84] agree that it is to be found, not in things themselves, but in our mental thinking. And when people talk of the times of their ancestors, do they think that, now that they are dead, their times can exist anywhere but in the memories of those who think about them?
Some people identify days, years, and months with the motions of the sun and the moon. But since for a motion to have happened in the past is for it to have ceased to exist, and for it to be going to happen in the future is for it not yet to exist, they do not mean what they say, because this implies that no time ever exists, whether now, or in the past, or in the future. For if it can be said of something that it was or it will be, it could also have been said at some time in the past, or at some time in the future, that it is. So where are days, months, and years, unless they are the names of calculations performed in the mind? It follows that time is a phantasm, and that it is the phantasm of motion. For when we want to know how many units of time have passed, we use some motion or other, such as the motion of the sun, or of a clock, or of an hour-glass; or we draw a line along which we imagine something to be moving. This is the only way we can become aware of time.
However, to say that time is the phantasm of motion is not a sufficient definition, since we use the word ‘time’ to denote before and after, or the successiveness of a body in motion, in so far as it exists first in one place, and then in another. So a full definition of time is like this: time is the phantasm of motion, in so far as we imagine before and after, or succession, in the motion. This is consistent with the Aristotelian definition, namely that time is the quantification of motion with respect to before and after. It is consistent with it because quantification is an act of the mind, and there is no difference between saying time is the quantification of motion with respect to before and after, and time is the phantasm of quantified motion. But to say that time is the measure of motion is incorrect, since we measure time by means of motion, not motion by means of time.
7.4. Part.
[85] A space or a time is called a part of another space or time, when the former is included in the latter, and more besides. From this it follows that it is wrong to call anything a part, except relative to something else in which it is included.
7.5. Division.
So to make parts of, or partition, or divide space or time is nothing other than to consider different bits of them separately. So whenever you divide space or time, you have as many distinct concepts as the number of parts you have made, plus one more, namely the original concept of what was to be divided. You have the original concept, then the concept of the first part, then of the second part, and so on indefinitely as long as you continue dividing.
However, it should be noted that, in this context, ‘division’ does not mean literally splitting up or tearing apart one space or time from another — who could believe that parts of spaces or times could be literally separated, say two halves of a sphere, or one hour from the next? Rather, it means considering separately, so that division is the job of the mind, not of the hands.
7.6. One.
When a space or time is considered along with other spaces or times, it is called one, that is, one of them. If it were not possible for one space or time to be added to or subtracted from another space or time, it would be enough to say simply ‘space’ or ‘time’, since it would be redundant to say one space or one time, if the existence of another one was unintelligible. The common definition that one is what is undivided is vitiated by an absurd consequence, since it follows that whatever is divided is many, or in other words that one divided thing is many divided things, which is self-contradictory.
7.7. Number.
Number is one and one, or one and one and one, and so on. That is, one and one is the number two, and and one and one is the number three, [86] and similarly for all other numbers. It is the same as if we were to say that number is unities.
7.8. Composition.
To compose a space out of spaces, or a time out of times, is to consider them first one after the other, and then all together, as one. For example, if you first enumerate separately the head, the legs, the arms, and the torso, and then put them all together into a single account, the result is a person. So that which stands for everything it consists of is called a whole, and these individual items (when they are once more considered separately, through division of the whole) are its parts. Consequently, the whole and all the parts taken together are absolutely the same thing. But just as I mentioned when talking about division, that there is no need for the parts to be literally separated, similarly it must be understood that in composition a whole can be formed without the parts physically moving together or touching each other. All that is necessary is for the mind to add them together into a single sum. Thus all human beings considered together constitute the human race, even though they are scattered over space and time. And twelve hours add up to the single number twelve, even if they are hours from different days.
7.9. Whole.
Now that all this is understood, it is obvious that it is wrong to call anything a ‘whole’ unless it is understood as being composed of parts, and divisible into its parts. So if we say that something cannot be divided and has no parts, we are saying that it is not a whole. For example, if we say that the soul cannot have any parts, we are also saying that no soul is a whole. It is also obvious that nothing has any parts until it has been divided; and that when it has been divided, it has only as many parts as the number of times it has been divided. Similarly, a part of a part is a part of the whole — for example, a part of four, such as two, is also a part of eight. This is because four consists of two and two, [87] and eight consists of two and two and four; so two, which was a part of a part, namely four, is also a part of the whole, namely eight.
7.10. Contiguous and continuous spaces and times.
Two spaces are said to be contiguous with one another when there is no other space between them. But two times between which there is no other time are called immediate —for example, AB and BC.
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Both spaces and times are said to be continuous with one another when two of them have a part in common — for example, AC and BD have the part BC in common.
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And more than two spaces or times are continuous with one another when every pair of adjacent parts is continuous.
7.11. Beginning, end, route, finite, infinite.
A part which comes between two other parts is called a middle part, and one which is not positioned between two other parts is called an extreme part. The extreme part which is counted first is the beginning; the one which is counted last is the end; and the ones in the middle all taken together are the route. Extreme parts and limits are the same thing. It obviously follows that beginning and end depend on the order in which we count things; and for space or time to be finite or limited is the same as for us to imagine its beginning and end; and anything is either finite [n.82] or infinite depending on whether or not we have imagined bounds to it in any direction. The limits of number are unities; and the one we begin counting from is the beginning, and the one we stop at is the end. A number is said to be infinite if it has not been said what it is. If it has been called two, three, a thousand, etc., it is always finite; but if all that has been said is that it is an infinite number, this should be taken as meaning the same as if it had been said that this name ‘number’ is an undefined name.
7.12. What potential infinity is; that nothing infinite can be called ‘whole’ or ‘one’; that there are not infinitely many spaces or times.
A space or time is called potentially finite, or limitable, when it is possible to specify a number of finite spaces or times (e.g. paces or hours), such that there can be no greater number of them in that space or time. It is potentially infinite if there can be a larger number of such paces or hours than any given number. It should be noted that even in the case of potentially infinite space or time, although more paces or hours can be counted than any specifiable number, their number will still be finite, since every number is finite.
Consequently, it is fallacious to try to prove that the world is finite by using the following argument: if the world is infinite, then you can specify some part of it which is an infinite number of paces away from us; but no such part can be specified; therefore the world is not infinite. The consequence drawn from the major premise is false, since, even if space is infinite, whatever we specify, or assign mentally, will be at a finite distance from us. By the very act of assigning it a position, we set that as a limit to the space of which we ourselves are the beginning; and we limit, [n.83] that is to say make finite, whatever we mentally cut off from infinity at both ends.
It cannot be said of unlimitable space or time that it is a whole, or one. Not a whole, because it cannot be composed of any parts, since however many parts there are, they will each be finite, and taken together they will also make a finite whole. Not one, because nothing is called ‘one’ except relative to something else, and it makes no sense to say that there are two infinite spaces, or two infinite times. Finally, if we ask [89] whether the world is finite or infinite, we have nothing in our minds corresponding to the word ‘world’, since whatever we imagine is finite simply by virtue of being imagined, whether we count as far as the fixed stars, or to the ninth, the tenth or even the thousandth sphere. [n.84] The only question is whether God has actually added as much body to body, as we can add space to space.
7.13. There is no minimum divisible.
It is usually said that space and time are infinitely divisible. But this should not be taken as meaning that some infinite or eternal division has actually taken place. The meaning of the statement is better expressed as follows: whatever is divided is divided into parts which can be further divided; or alternatively, there is no minimum divisible; or as most geometricians put it, it is always possible to specify a smaller quantity than any given quantity. It can easily be proved, as follows. Let any given space or time considered to be a minimum divisible be divided into two equal parts, A and B. I say that either part (say, A) can be divided again. Let us suppose that part A is contiguous to part B on the one side, and to another space equal to B on the other side. Then the whole of this space (which is larger than the one originally given) is also divisible. But if it is divided into two equal parts, then A, which is in the middle, will also be divided into two equal parts. Consequently A was divisible.
Chapter 8: Body and accident
8.1. The definition of ‘body’.
[90] We now understand the nature of imaginary space, in which we suppose nothing external to exist, but only the pure absence of the things which, when they existed, left their images in the mind. Let us next suppose that one of these things is put back again, or re-created. It is therefore necessary for that re-created or replaced thing not only to occupy some part of the said space (i.e. to coincide and be coextensive with it), but also to be something which does not depend on our imagination. But this is the very thing which [91] is customarily called body on account of its extension; self-subsistent on account of its independence from our thought; existent because it subsists outside us; and finally substance or subject because it seems to support and underlie imaginary space, so that it is not by the senses, but only by reason that we understand that something is there. So the definition of body is something like this: Body is whatever coincides or is coextensive with a part of space, and does not depend on our thought.
The nature of an accident can be explained more easily by examples than by a definition. If we think of a body occupying a particular space, i.e. being coextensive with it, this coextension is not the coextended body itself; similarly, if we think of the same body being moved, this motion is not the moved body itself; or if we think of the body not being moved, this lack of motion, or state of rest, is not the body at rest itself. What are they, then? They are the accidents of the body — but this is just what we are asking when we ask, ‘What is an accident?’; i.e., we are asking something we already understand, and are failing to ask what we ought to have asked. For who is there who does not always have the same understanding of someone who says that something is extended, or moves, or does not move? But most people want to be told that an accident is something, i.e. some part of the real world, even though in fact it is not part of it. The best way of satisfying such people as far as possible is to reply that an accident is to be defined as the mode of a body through which it is conceived; which is equivalent to saying that an accident is a capacity of a body to impress a concept of itself upon us. Even if this definition does not correspond to what was asked, it does at least correspond to what [92] ought to have been asked. In other words, if it is asked, ‘Why does it happen [n.85] that one part of a body is perceptible in one place, another in another?’, the correct answer will be, ‘Because of its extension;’ or if it is asked, ‘How is it that the whole body is continuously perceived now here, now there?’, the answer will be, ‘Because of its motion;’ or if again it is asked, ‘How is it that the same space is seen to be occupied for some time?’, the answer should be, ‘Because the body has not been moved.’ For if the question, ‘What is it?’ is asked about the name of a body, i.e. a concrete name, the appropriate reply will take the form of a definition, since the question is only about the meaning of a word. But if the question, ‘What is it?’ is asked about an abstract name, what is asked is the cause of something’s appearing in this or that mode. So, if it is asked what a solid thing is, the answer will be, ‘A solid thing is that of which a part gives way only if the whole gives way;’ but if it is asked, ‘What is solidity?’, we need to indicate the cause of the part not giving way unless the whole gives way. So we shall define an accident as a mode of conceiving a body.
When it is said that an accident is in a body, this must not be construed as if something were contained in the body — as if, for example, redness were in blood in the way that blood is in a blood-stained garment, i.e. as a part in a whole. If this were so, an accident would also be a body; but every accident should be understood as being in its subject in the way that size, rest, or motion is in that which is big, at rest, or moved (and everybody understands how this is to be understood). Even Aristotle made this point, if negatively, when he said, ‘An accident is in its subject not as a part, but in such a way that the subject can survive without it.’ This is right, except that there are certain accidents [93] which a body cannot survive without, since a body cannot be conceived without any extension or shape at all. But other accidents, which are not common to all bodies, but are exclusive only to some (e.g. being at rest, moving, colour, solidity, etc.) continually disappear with others taking their place, but without any body ever disappearing. It might be thought that not all accidents are in their bodies in the way that extension, motion, rest, or shape are; e.g. that colour, heat, smell, virtue, vice, etc. are in their bodies in a different way — that, as they say, they inhere in them. But I would prefer anyone who thinks that way to suspend judgment for the present, and wait a while until it has been investigated by reasoning whether such accidents are not also motions of certain sorts, whether of the mind forming its images, or of the sensed bodies themselves. Finding an answer to this question is one of the major tasks of natural philosophy.
The extension of a body is that same as its magnitude, or what some call its ‘real space’. This space does not depend on our thought, as imaginary space does, since the latter is the effect of the former, and magnitude is the cause. Imaginary space is an accident of the mind, and real space is an accident of a body existing outside the mind.
The space (and I always use this word to mean imaginary space) which coincides with the magnitude of any body, is called the place of that body; and then the body itself is said to be placed. Place differs from the magnitude of the thing which is placed, in a number of ways. Firstly, the same body always retains the same magnitude, both when it is at rest and when it is in motion; but when it is in motion, it does not retain the same place. Secondly, the place of any body [94] is a phantasm of a body of a particular size and shape, whereas the magnitude of any body is an accident belonging solely to that body. This is because something which is placed can have different places at different times, but it cannot have different magnitudes. Thirdly, there is no place external to the mind, and no magnitude internal to it. Again, place is imaginary extension, and magnitude is real extension; and a placed body is extended, but is not extension.
Furthermore, place is immobile. What we mean by motion is that something is transported from one place to another. But if a place were to move, a place would also be transported from one place to another; from which it follows that there would have to be a place of the place, and again another place of the place in which the place is — which is completely absurd.
As for those who accept that place is indeed immobile, but identify it with real space, they too make place a phantasm, even if they do not notice that this is what they are doing. One approach is to say that place is immobile because what is being considered in a place is space in general. But this is to forget that nothing is general or universal apart from names; from which it follows that the space which is said to be considered in general is nothing other than a phantasm or memory of some body or other with a particular magnitude and shape, and that it resides in the mind. Another approach is to say that real space is made immobile by the understanding. Take running water, for example. If we think about the fact that the water moving at the surface is always being replaced, the surface itself is kept stationary by the understanding, and it is this surface which is the immobile place of the river. But this is nothing other than to make place imaginary (which it is), although in obscure and convoluted language.
Finally, the nature of place does not consist in the surface of what surrounds a space, but in the solid space. The whole placed body [95] is coextensive with the whole place, and its parts with the parts of the place. But since the placed body is a solid, it is meaningless to say that it is coextensive with a surface. Besides, how can a whole body move, unless all its individual parts move at the same time? And how can the parts belonging to it move without leaving their places? But a part belonging to the body cannot move away from the contiguous part of the surrounding surface. From this it follows that, if place is the surface of what surrounds the body, the parts of that which moves do not move — in other words, that which moves does not move.
A space (or place) which is filled with body is called a plenum, and one which is empty of body is called a vacuum.
There are various names which provide an answer to the question ‘Where is it?’, such as ‘here’, ‘there’, ‘in the country’, or ‘in the city’. However, they are not names of the place itself, nor are they sufficient to bring to mind the precise place which is asked about.
‘Here’ and ‘there’ mean nothing unless a particular thing is simultaneously pointed at with a finger or something else. But when the eye of the questioner is directed towards the thing asked about by a finger or some other pointer, the place is not defined by the answerer, but discovered by the questioner.
The other ways of identifying a place are purely verbal. For example, when the answer is like ‘in the country’, or ‘in the city’, the answer can be more or less specific — as, ‘in the country’, ‘in the city’, ‘in a particular area of the city’, ‘in a particular house’, ‘in a particular room’, or ‘in a particular bed’. Although these ways of identifying it progressively direct the questioner more and more specifically to a specific place, they do not determine the place precisely. They merely restrict it to a more narrowly specified place, and signify that the place of the thing is within the particular space designated by the words in question, as a part is within a whole. The word ‘somewhere’ is the highest genus of all names such as these, which answer to the question ‘where?’. It follows that whatever [6] is somewhere is in some specific place. In other words, it is in a place which is part of a larger space signified by an expression such as ‘in the country’, ‘in the city’, and so on.
A body, its magnitude, and its place are divided by one and the same act of the mind, since dividing an extended body, and its extension, and the idea of extension (its place) is the same as dividing any one of them, since they coincide, and it can only be carried out by the mind, that is, by the division of space. It obviously follows that two bodies cannot be in the same place at the same time, and that one body cannot be in two places at the same time. Two bodies cannot be in the same place because, when a body which occupies the whole of a space is divided into two, the space itself is divided into two, so there are two places. One body cannot be in two places because if the space which is occupied by a body (i.e. its place) is divided into two, the body placed there is also divided into two (since, as I have said, a place and the body in it are both divided by a single act), so there are two bodies.
Two bodies are said to be contiguous or continuous with each other on the same basis as two spaces, namely that things are contiguous if there is no space between them. As before, by ‘space’ I mean the idea or phantasm of a body. So even if there is no other body between two bodies (and consequently no magnitude, or ‘real space’ as it is called), the two bodies are not contiguous if there could be a body between them — in other words, if there is a space between them which is capable of holding a body. This is so easy to understand that I would be amazed that anyone skilled in philosophy could ever think otherwise, if I had not observed [97] that most people who pretend to a certain skill in metaphysics are led astray by the outward show of words, as if by will-o’-the-wisps. No-one using their common sense would ever think that two bodies must necessarily be in contact with one another, simply because there was no other body between them; or that there is no vacuum because a vacuum is nothing, or a non-being. This is as childish as arguing that no-one can fast, because fasting is to eat nothing, but nothing cannot be eaten.
Two bodies are continuous with each other if they have a part in common; and more than two are continuous with each other if every pair of adjacent parts is continuous. This is exactly the same definition as I gave of continuous spaces above.
Motion is the continuous leaving of one place and acquiring of another place; and the place which is left is usually called the ‘origin’, and the place which is acquired, the ‘destination’. [n.86] I say ‘continuous’, because however small a body may be, it is impossible for the whole of it to leave the earlier place at the same time, in such a way that no part of it will be in a part which is common both to the place it leaves and the place it acquires. For example, let there be a body in the place ABCD. It cannot arrive at the place BDEF unless it is first in GHIK, of which the part GHBD is common to both the place ABCD and the place GHIK, and the part BDIK is common to both the place GHIK and the place BDEF.
It is inconceivable that anything should move except in time. For time, by definition, is a phantasm, namely conceived motion. Therefore to conceive something as moving other than in time would be to conceive motion without motion being conceived, which is impossible.
[98] That which is in the same place for a certain time is said to be at rest; that which was previously in a place other than where it is now (depending on whether it is now moving or at rest) is said to be moving, or to have moved. These definitions have three implications. The first is that whatever is moving has moved, since if what is moving is in the same place as before, it is at rest — in other words, it is not moving, in accordance with the definition of being at rest. But if it is in another place, it has moved, in accordance with the definition of having moved. Secondly, whatever moves will continue moving, since whatever is moving is leaving the place where it is, so it will acquire another place, and will therefore continue moving. Thirdly, whatever moves is not in one place for any period of time, however short, since, in accordance with the definition of rest, that which is in one place for any period of time is at rest.
There is a certain sophistical argument against the reality of motion which seems to arise from ignorance of this last proposition. It is said that, if a body moves, it moves either in the place where it is, or in a place where it is not; but both these alternatives are false; therefore nothing moves. But it is the major premise which is false, since whatever moves does not move either in the place where it is, or in a place where it is not, but from the place where it is, to a place where it is not. it cannot be denied that whatever moves moves somewhere — in other words, it moves within a certain space. But the place of the body is not the whole of that space, but only a part of it, as I said above in Article 7.
So I have demonstrated that whatever moves not only has moved, but will also continue to move. A further consequence of this is that motion cannot be conceived without a conception of both the past and the future.
Every body in motion always has some specific magnitude. But if its magnitude is left out of consideration, the path along which it travels is said to be a line, or a single and simple dimension; [99] the space it passes through is called its length; and the body itself is called a point — in the sense in which the earth is usually treated as a point, and the path of its annual rotation is treated as an ecliptic line. [n.87] But if the moving body is now considered as having length, and is assumed to be moving in such a way that its individual parts are conceived as tracing separate lines, [n.88] the path of each line is called the breadth of the body, and the space which is traced out is called its surface, consisting of the two dimensions of length and breadth. The whole breadth of the body is connected to each individual part of its length, and vice versa.
Again, if the body is now considered as having a surface, and is conceived as moving in such a way that each individual part traces a line, the path of each of its parts is called the thickness or depth of the body, and the space which is formed is called a solid. It is made up of three dimensions, the whole of any two of which are connected to the individual parts of the third.
But if a body is now considered as a solid, it is impossible for its individual parts to trace out individual lines. Whatever direction it moves in, the path of a part at the back will coincide with the path of a part in front of it, and the solid which is generated will be the same as the one which had previously been generated by a surface alone. So it is impossible for there to be any other dimension of body as such, apart from the three already mentioned. Nevertheless, as I shall say later, when speed (which is a function of motion and distance) is applied to all the parts of a solid, it forms the quantity of motion, which consists of four dimensions — rather as the quality of each part of a piece of gold added together makes up its value.
Bodies are said to be equal to one another if they can occupy the same place. A body can occupy the same place [100] as another body occupies even if it does not have the same shape, provided that it can be conceived as being reduced to the same shape by bending or transposing its parts.
One body is larger than another when part of the former is equal to the whole of the latter. It is smaller when the whole of the former is equal to part of the latter. A magnitude is said to be equal to, larger than, or smaller than another magnitude on the same basis — that is to say, when the bodies of which they are the magnitudes are larger, equal, or smaller.
The magnitude of one and the same body is always one and the same. For a body and its magnitude and place can only be understood by the mind as coinciding with each other. But suppose a body is understood as being at rest, i.e. as remaining in the same place for a period of time. And suppose also that its magnitude is larger during one part of that time, and smaller during another part of it. Then it follows that its place, which remains the same, will sometimes coincide with the larger magnitude and sometimes with the smaller one — in other words, one and the same place will be both larger and smaller than itself, which is impossible.
The matter is so self-evident that there would be no need to demonstrate it, if I had not observed that some people think of a body and its magnitude as if one and the same body could exist separately from its magnitude, and be endowed with more or less magnitude at different times. This principle is used to explain why bodies can vary in density.
Motion, in so far as a specific distance can be covered in a specific time, is called speed. We nearly always describe something as speedy relative to something which is slower or less speedy, just as ‘large’ is relative to what is smaller. But among philosophers, just as ‘magnitude’ is taken in an absolute sense as meaning ‘extension’, so ‘speed’ can stand for motion over a distance in an absolute sense.
[101] We say that a number of motions are completed in equal times, when each of them begins simultaneously with some other motion, and finishes simultaneously; or if it began simultaneously, it also finished simultaneously. For time is the phantasm of motion, and it is measured only by some perceptible motion. This is done by the motion of the sun in the case of a sundial, or of the hands in the case of a clock; so that if two or more motions start and finish simultaneously with the motion of the clock, they are judged to have taken equal times. From this it can also easily be understood what it is to move for a longer time, or more slowly; and for a shorter time, or more quickly: namely a motion is longer if it started together but finished later, or started earlier and finished together.
Motions are said to be equally fast, if equal distances are covered in equal times; and a motion is faster if a greater distance is covered in an equal time, or if an equal distance is covered in less time. A speed is called a uniform speed or motion if equal distances are covered in equal parts of the time. Non-uniform motions are those in which the speed gets faster or slower in equal parts of the time; and if the increases or decreases in speed are always equal, they are said to be uniformly accelerated, or uniformly decelerated.
Motion is not called more, or less, or equal simply in proportion to the distance covered in a specific period of time, i.e. in proportion to speed alone, but in proportion to speed applied to each particle of magnitude. For when a body moves, each part of it, however small, also moves. If you consider two halves of it, say, then their motions will have the same speed as each other, and as the motion of the whole. [102] But the motion of the whole will be equal to the sum of the two motions, of which each has the same speed as the motion of the whole. So for two motions to be equal to each other is different from their having the same speed. This is obvious from the example of a pair of horses, where the motion of both horses taken together has the same speed as either of them, but the motion of both is greater than that that of one of them, in fact twice as great. So we say that motions considered in themselves are equal when the speed of the one multiplied by the whole of its magnitude is equal to the speed of the other, likewise multiplied by the whole of its magnitude. And one motion is greater than another when the speed of the former, multiplied in the above manner, is greater than the speed of the latter similarly multiplied. And it is less when less. Finally, the quantity of motion calculated in the way I have just specified is precisely what we usually call force.
It is clear that whatever is at rest always remains at rest, unless there is some other body, distinct from itself, which prevents it from remaining at rest by taking its place. Let us suppose that there exists some finite body at rest, and imagine that the whole of the rest of space is empty. Now if this body begins to move, it will obviously move along a particular path. But the reason why it moves along this particular path must lie outside it, since everything in the body itself has disposed it towards rest. Similarly, if it moved along any other path, the reason for its moving along that path must also lie outside the body itself. But since it has been assumed that there is nothing outside the body itself, there is as much reason for motion along any one path as along any other. Consequently, it could move along all paths simultaneously — but this is impossible.
Similarly, it is clear that whatever is in motion always continues in motion [103] unless there is something external to it which causes it to come to rest. For if we suppose that there is nothing external to it, there will be no reason why it should come to rest now rather than at any other time. Consequently, its motion would cease at every point of time simultaneously, which is unintelligible.
When we say that an animal, a tree, or any other named body comes in or out of being, even though they are bodies, it is not meant that something which is not a body turns into a body, or that a body turns into something which is not a body. Rather, an animal turns into something which is not an animal, a tree turns into something which is not a tree, and so on. In other words, the accidents by virtue of which we call one thing an animal, another a tree, and another something else do indeed come in and out of being, so that these names no longer apply to the things they previously applied to. But the magnitude on account of which we call something a body does not come in or out of being. In our minds we can certainly imagine a point swelling into a large mass, and collapsing back again into a point; and this is to imagine something coming into being out of nothing, and nothing coming into being out of something. However, it is impossible for our minds to understand how that could happen in the real world.
Consequently, philosophers (who are bound by natural reason) make it a presupposition that body cannot come in or out of being. Rather, body is revealed to us in different ways through its various manifestations, and hence is given a variety of names. So what it was once appropriate to call ‘human’, it is now appropriate to call ‘non-human’; but it what it was once appropriate to call ‘body’, cannot now be called ‘non-body’.
It is obvious that all accidents, apart from magnitude or extension, can come in or out of being. For example, when white turns into black, the whiteness which once existed no longer exists, and the blackness which did not exist comes into being. Consequently, the difference between bodies and accidents, [104] through which bodies reveal themselves to us in different ways, is that bodies are things which do not come into being, whereas accidents do indeed come into being, but they are not things.
So, given that a thing reveals itself to us in different ways because of its different accidents, it must not be thought that an accident passes from one subject into another. As I said above, accidents are not in their subjects in the way that a part is in a whole, or a content is in a container, or a head of household is in a house. Rather, one accident passes away, and another one comes into being. For example, when a moving hand moves a pen, the motion of the hand does not pass into the pen, otherwise the flow of the hand’s writing would stop. What happens is that a new motion is generated in the pen, and it is the pen’s own motion.
So it is equally improper to say that an accident is moved. For example, we should not say ‘A body carries its shape away with it,’ but ‘Its shape is an accident of a body which is carried away.’
An accident on account of which we impose a definite name on a body, i.e. an accident which denominates its subject, is usually called an essence. For example, rationality is said to be the essence of a human, whiteness of something white, and extension of a body. And the same essence, in so far as it has actually come into being, is called a form.
Conversely, a body is called a subject relative to any of its accidents. Relative to form, it is named matter.
Again, the coming in or out of being of any accident means that its subject is said to change, and it is only if the form has come in or out of being that the subject itself is said to have come in or out of being. However, in all coming into being and change, the name ‘matter’ always remains. For a table made out of wood is called ‘wood’ as well as ‘wooden’, and a statue made of brass is called ‘brass’ as well as ‘brazen’. However, in the Metaphysics, Aristotle [105] says that what is made is not ‘that’, but ‘thaten’, [n.89] so he thinks that what is made of wood should not be called ‘wood’, but ‘wooden’.
Following Aristotle, philosophers usually call the matter which is common to all things primary matter. But it is not a body distinct from other bodies, nor is it one of those bodies. So what is it? It is nothing but a name. But it is not a meaningless name, since it signifies that body is being considered without taking into account any form or accident, except only magnitude (or extension), and the capacity for receiving form or accident. So whenever we need to use the expression ‘body taken generally,’ it is perfectly in order for us to use the expression ‘primary matter’.
Suppose you did not know whether water or ice existed first, and you wondered which was the matter of both; then you would be compelled to assume that there was some third matter, which was the matter of neither of them. Similarly, if you wondered what the matter of everything was, you would have to assume that it was quite different from the matter of all the things which actually exist. Consequently, primary matter is not a thing, and this is why people do not attribute any form to it, or any accident other than that of quantity. However, since all individual things are endowed with their forms and particular accidents, it follows that primary matter is universal body; in other words, it is body considered universally. This does not mean that it does not have any forms or accidents, but that no account is taken of forms or accidents, apart from quantity; and when I say that ‘no account is taken of them,’ I mean that they are not referred to in any reasoning about it.
8.25. That a whole is greater than its part, and the demonstration of this.
Drawing on what has been said above, it is possible to demonstrate the axioms about the equality and inequality of magnitudes, which Euclid merely assumes at the beginning of Book I of the Elements. [106] Here I shall demonstrate just one of them, namely that a whole is greater than its part. My purpose is that my readers should know scientifically that these axioms are not indemonstrable, and consequently that they are not the first starting points of demonstration. The outcome should be that they will be careful not to accept anything as a starting point unless it is at least as clear is this.
So the definition of greater is that of which a part is equal to another whole. Now if we take a whole A, and its part B, since the whole B is equal to itself, and B is part of the whole A, it follows that a part of A is equal to the whole B. Therefore, by the definition of greater, A is greater than B, which is what was to be proved.
Chapter 9: Cause and Effect
9.1. What action and passion are.
A body is said to act on a body in which it either brings into being or destroys some accident; and it is said to be acted upon [n.90] by one which brings about or destroys some accident in itself. So a body which brings about motion in another body by propelling it is called the agent, and the one [107] in which the motion is brought into being by propulsion is called the patient. For example, a fire warming the hand is called the agent, and the hand which becomes warm, the patient. The accident which comes into being in the patient is called the effect.
When the agent and the patient are contiguous with each other, their action and passion are said to be immediate — in other cases mediated. The body which constitutes the medium between the agent and the patient by being contiguous with both of them, is both an agent and a patient: agent in relation to the following body which it acts upon, and patient in relation to the previous body by which it is acted upon. Likewise, if a number of bodies are lined up so that each is contiguous with its neighbour, they are all mediums between the first and the last, and agents and patients. The first only acts, and the last is only acted upon.
The agent is understood to produce its specific effect in the patient in a specific way, or through a specific accident or accidents which belong to both it and the patient — that is, not because they are bodies, but because they are bodies of a particular sort, or moving in a particular way. Otherwise all agents would produce similar effects in all patients, since they are all equally bodies. So, for example, fire does not heat things because it is a body, but because it is hot; and one body does not impel another because it is a body, but because it moves into its place. So the causes of all effects consist in specific accidents of the agents and the patient. When they are all present, the effect is produced; and if any one them is absent, the effect is not produced.
But an accident, whether of the agent or of the patient, without which the effect cannot be produced, is called the cause ‘without which not’, and which is necessary by hypothesis; and the prerequisite for the effect to be produced. A cause without qualification, or a complete cause [108] is the totality of all the accidents, both of the agents (however many there may be) and of the patient, such that assuming all to be present, it is inconceivable that the effect should not be produced together with it; and assuming one of them to be absent, it is inconceivable that the effect should be produced.
When an effect is produced, the totality of the accidents required for the effect which are in the agent or agents is called its efficient cause. The totality of those which are in the patient when an effect is produced is usually called the material cause. I say ‘when an effect is produced,’ because when there is no effect, there is no cause either; for nothing can be called a cause, when there is nothing which could be called an effect. The efficient cause and the material cause are partial causes, or parts of the cause which we called complete just above. From this it immediately follows that the effect which we expect when there are appropriate agents can be frustrated for lack of an appropriate patient, and when there is an appropriate patient, it can be frustrated for lack of appropriate agents.
The complete cause is always sufficient for producing its effect, provided only that the effect is possible in every respect. Suppose someone expects a certain effect to be produced: if it is produced, then it is obvious that the cause which produced it was sufficient; and if it is not produced, and yet was possible, then clearly one of the agents or the patient lacked something without which it could not be produced — in other words, some accident which was required for its production was missing. Therefore its cause was not complete, contrary to what was supposed.
From this it also follows that at the very instant the cause becomes [109] complete, the effect is produced at the same instant. For if it is not produced, there still lacks something required for its production, and so it was not the complete cause, as was supposed.
If a necessary cause is defined as one such that, if it is supposed, the effect cannot fail to follow, it can also be concluded that whatever effect is ever produced, is produced by a necessary cause. For what is produced, by the very fact that it is has been produced, had a complete cause; that is, all those things such that, once they are supposed, it is inconceivable that the effect should not follow — and that is a necessary cause. By the same reasoning, it can be shown that whatever effects will ever happen in the future will have a necessary cause, and hence that whatever either will be or has been produced had its necessity in things preceding it.
From the fact that an effect is produced at exactly the same instant as its cause becomes complete, it is also obvious that the causation and the production of effects consists in a certain continuous progression, in such a way that, corresponding to the continuous change in the agent or agents which comes from other things acting on them, there is also continuous change in the patient which they act upon. For example, as a fire gets hotter and hotter, so too its effect continuously increases more and more at the same time, namely the heat of the bodies next to it, and then that of the bodies next to them (which is already a major argument that change consists solely in motion, which I show to be true elsewhere).
But in this progression of causation (that is, of action and passion), if you hold a part of it in your imagination, and then divide it into parts, you can consider the first part (its beginning) only as an action [110] or cause; since if you also consider it as an effect or passion, you must necessarily consider something else before it as an action and its cause. But this is impossible, since there is nothing before a beginning. Similarly, you can consider the last part only as an effect, since you can consider it as a cause only relative to something after it; but nothing follows after the last. This is why in action, beginning and cause are held to be the same. But each of the intermediate parts are both action and passion, or cause and effect, depending on whether they are related to the previous part or to the one which follows.
The cause of motion can only be in a body which is contiguous and moving. Take any two non-contiguous bodies, and suppose that the space between them is either a vacuum, or, if it is filled, that it is filled with body at rest. Let us also suppose that one of the first two bodies is at rest. I say that it will always remain at rest; since if it moves, the cause of its motion will be in a body external to it (cf. Chapter 8, Article 19). So, if there is empty space between it and the body external to it, we can see that, whatever the state of the external bodies or of the patient itself (except that we are supposing it is now at rest), it will remain at rest as long as it is not touched by them. But since a cause (by definition) is the totality of accidents, such that, if all of them are supposed to be present, it is inconceivable that the effect should fail to follow, then neither the accidents which are in the external objects, nor those which are in the patient itself can be the cause of future motion. Similarly, because it is conceivable that what is already at rest will still continue at rest, even if it is contiguous with another body, provided only that the second body is at rest, then there will be no cause of motion in the contiguous body at rest. [111] Therefore there is no cause of motion in body other than in a contiguous and moving body.
By the same reasoning, it can be proved that whatever moves, will always continue in the same direction and at the same speed, unless it is obstructed by another moving body touching it; and hence that no bodies can bring about, extinguish, or diminish motion in another body if they are at rest, or if there is intervening empty space. Someone has written that bodies in motion are resisted more by bodies at rest than by bodies moving in the opposite direction, because he thought that rest was more contrary to motion than motion. But this is a linguistic mistake, since although the words ‘rest’ and ‘motion’ are contradictories, when we are talking about things, motion is resisted by contrary motion, not by rest.
Let us suppose that one body acts on another at one time, and then the same body acts on the same body at another time. Let us also suppose either that the agent as a whole and each of its parts is at rest, or that, if it is in motion, both the agent as a whole and each of its parts is moving in the same way as before. And finally let us suppose that what has been said of the agent also applies to the patient, so that the only difference is one of time — that is, that one action is earlier in time, the other later. All this being supposed, it is self-evident that the effects will be equal and similar, [n.91] and that they will differ only in time. And just as effects themselves arise from their causes, so also differences between effects depend on differences between their causes.
It necessarily follows from the above that change is nothing other than the motion of the parts of the body undergoing change. Firstly, we do not say that anything has changed, except when it appears to our senses differently from how it appeared before. Secondly, both these appearances are effects produced in the sentient being; so, if they are different, it is necessary (as was shown in the previous article) either that some part [112] of the agent which was previously at rest is now in motion (in which case the change consists in that motion); or that a part which was previously in motion is now moving differently (and in this case too, change consists in a new motion); or that a part which was previously in motion is now at rest; which, as I demonstrated above, can only happen as the result of motion (so again change is motion); or, finally, that one of these alternatives happens to the patient or one of their parts, so that, one way or another, the change will consist in a motion of the parts of the body which is sensed, or of the sentient being, or of both. Therefore change is motion (namely of the parts of the agent or patient), which is what I set out to demonstrate. It is a consequence of this that rest is not the cause of anything, and that no action whatever is performed by it — so it is not the cause of any motion or change.
Accidents are called contingent, relative to other accidents which precede them or are earlier in time, if they do not depend on them as their causes. I stress that I am referring only to their relation to accidents by which they were not brought into being. In relation to their causes, all things happen with equal necessity; for if they did not happen necessarily, they would not have causes — something which is unintelligible in the case of things which have come into being.
Chapter 10: Power and Action
10.1. Power and cause are the same.
[113] Power and action correspond to cause and effect. Indeed, power is the same thing as cause, and action is the same thing as effect, even though they are called by different names, depending on how they are considered. When any agent contains all the accidents which are necessarily required on the part of the agent for producing a certain effect in any patient, then we say that this agent can [n.92] produce this effect, provided that it comes into contact with a patient. But in the last chapter, I pointed out that these same accidents constitute the efficient cause; so the same accidents constitute both the efficient cause and the power of the agent. Therefore the power of the agent and the efficient cause are one and the same thing, but considered differently. It is called a cause in relation to an effect which has already been produced, and a power in relation to the same effect as to be produced; so a cause relates to the past, and a power relates to the future. The power of the agent is often called its active power.
Similarly, whenever any patient contains all the accidents which are required on its part for a certain effect to be produced in it by any agent whatever, we say that this effect can be produced in this patient, [114] provided it comes into contact with a suitable agent. But as defined in the previous chapter, these same accidents constitute the material cause. So the power of the patient (or passive power, as it is also commonly called) is the same as the material cause. However, they are considered differently, in that a cause relates to the past, whereas a power relates to the future.
So the power of the agent and the power of the patient taken together (which can be called the complete or full power), is the same as the complete cause. Both consist in the totality of all the accidents which are simultaneously required, both in the agent and in the patient, for producing an effect. Finally, just as the accident which is produced is called an effect relative to its cause, so it is called an action in relation to the power to produce it.
Just as an effect is produced at the precise instant when its cause becomes complete, so also an action which could be produced by a certain power, is produced at the same instant as the power becomes complete. And just as no effect can happen unless it is produced by a sufficient and necessary cause, similarly no action can be produced unless it is produced by a sufficient power, and one by which it could not fail to be produced.
Just as I have shown that, in themselves, the efficient cause and the material cause are only parts of the complete cause, and can only produce an effect if they are brought together; similarly too, active and passive power are only parts of the full or complete power, and cannot lead to an action without being brought together. So, as I said in Article 1, these powers are called ‘powers’ only with the following qualification, namely that the agent can bring about the action, only provided that it comes into contact with a patient; and [115] the patient can be acted upon, only provided that it comes into contact with an agent; otherwise neither would be able to perform any action. Consequently, the accidents which they contain cannot properly be called powers; nor can any action be called ‘possible’ simply by virtue of the power of the agent alone or of the patient alone.
An action is impossible if the full power for producing it will never exist. Given that a full power is one which combines everything which is required for producing an action, if the full power will never exist, one of the necessary requirements for producing the action will always be absent; therefore this action could never be produced — in other words, this action is impossible.
A possible action is one which is not impossible. Consequently, every possible action will be produced at some time or other; for if it is supposed that it will never be produced, it will never be the case that all the requirements of its production will come together; therefore (by definition) this action is impossible, which is contrary to what was supposed.
An action is necessary if it is impossible for it not to take place. Consequently, any action which is going to take place will necessarily take place, since it is impossible for it not to take place, because (as has just been demonstrated) every possible action will be produced at some time or other. Indeed, the proposition that what will be will be is no less a necessary proposition than a human is a human.
Here you might ask whether so-called future contingents are necessary. My global answer is that everything which happens contingently is contingent upon necessary causes, as I have shown in the previous chapter; and events are called ‘contingent’ only in relation to other events which they do not depend on. For example, tomorrow’s rain will be produced necessarily (i.e. by necessary causes); but we think and say that this rain happens by chance, [116] because we have not yet seen its causes, which already exist. People call something a ‘chance’ or ‘contingent’ event when they cannot see through to its necessary cause. They habitually talk about past events in the same way, when they say that it is ‘possible’ that something did not happen, when they mean that they do not know whether it happened or not.
So every proposition about a future contingent (e.g. ‘It will rain tomorrow’) or a future non-contingent (e.g. ‘The sun will rise tomorrow’) is necessarily true or necessarily false. But when we do not yet know scientifically whether a proposition is true or false, we call it ‘contingent’ on that account — even though its truth does not depend on our scientific knowledge, but on preceding causes.
However, there are people who agree that the whole proposition ‘Tomorrow it will rain or it will not rain’ is necessarily true, but maintain that neither of its separate components (‘Tomorrow it will rain,’ or ‘Tomorrow it will not rain’) is true. They say that this is because neither the one nor the other is ‘determinately’ true. But what can ‘determinately true’ mean, other than known to be true, or evidently true? So they are saying the same as what I say, namely that it is not yet known scientifically whether the proposition is true or not. However, they say it more obscurely, since the words they use in their attempt to hide their own scientific ignorance, at the same time conceal the obvious truth about the matter in question.
In Article 9 of the preceding chapter, I showed that the efficient cause of all motion and change consists in the motion of the agent or agents; and in Article 1 of the present chapter, I showed that the power of the agent is the same thing as the efficient cause. From this it follows that all active power also consists in motion, and that power is not some accident completely distinct from action, but an action (namely, motion), which is called ‘power’ because it subsequently produces another action distinct from itself. For example, if the first of three bodies propels the second, [117] and the second propels the third, the motion of the second body is its action relative to the action of the first body which produced it, and it is its active power relative to the action of the third body.
In addition to efficient and material causes, metaphysicians list two other types of cause, namely essence (which some call the formal cause), and purpose, [n.93] or final cause. However, both are really efficient causes. The essence of a thing is said to be its cause, as if being rational were the cause of a human being. But this is unintelligible, since it is the same as if we said that being a human was the cause of a human, which is an abuse of language. On the other hand, knowledge of a thing’s essence is the cause of knowledge of the thing, since if I already know that something is rational, I know from this that it is a human. But the way in which it is a cause makes it nothing other than an efficient cause. Final causes have no place except in relation to things which have sense and will; and I shall show that these are also efficient causes in the proper place.
Chapter 11: The Same and the Different
11.1. What it is for one thing to differ from another.
So far I have discussed body in itself, and the accidents common to all bodies, namely magnitude, motion, rest, action, passion, power, possibility, etc. [118] It would now be time to go down to the level of the accidents by which one body is distinguished from another, except that I must first explain what it is to be distinct or not distinct, in other words, what the same and the different are. After all, it is also common to all bodies that one can be distinguished, or different from another. So two bodies are said to be different from each other when something is true of one of them which cannot at the same time be true of the other.
11.2. What it is to differ in number, magnitude, species, and genus.
First of all, it is obvious that two bodies are not the same. For if they are two, they are in two places at the same time; whereas that which is the same, is in the same place at the same time. Therefore all bodies are numerically different from each other, or differ as one and another; so that ‘the same’ and ‘different in number’ are contradictorily opposed names.
Things differ in magnitude if one is larger than the other, for example ‘one cubit long’ and ‘two cubits long’, or ‘two pounds in weight’ and ‘three pounds in weight’. The opposite of these is equal.
Things which differ more than in magnitude are dissimilar, and things which differ only in magnitude are usually called similar. But some things which are dissimilar are said to differ specifically and others generically. They differ specifically if their differentia is perceived by the same sense — for example, white and black. They differ generically if their differentia can only be perceived by different senses — for example, white and hot.
11.3. Relation, ratio, and things related.
The similarity or dissimilarity, or equality or inequality of any body to any other body is called its relation; and the bodies themselves are called related or correlated in respect of it. Aristotle calls them ‘the to what’, and the first is usually called the antecedent, and the second [119] the consequent. The relation of antecedent to consequent in respect of magnitude (that is, its equality, excess, or short-fall) is called the ratio or proportion of the antecedent to the consequent. So a ratio is nothing other than the equality or inequality of the antecedent compared to the consequent in respect of magnitude. For example, the ratio of three to two is simply that three exceeds two by one; and the ratio of two to five is simply that two falls short of five by three. Therefore in the ratio of unequals, the ratio of the smaller to the larger is called its short-fall, and the ratio of the larger to the smaller is called its excess.
11.4. Proportionals.
Further, some unequals are more unequal, some are less unequal, and some are equally unequal. So as well as ratios of magnitudes, there are ratios of ratios, namely when two unequals have a relation to two other unequals. For example, when the inequality between 2 and 3 is compared with the inequality between 4 and 5. In such as comparison, there are always four magnitudes; or if there are only three, the middle one (which has the same value) is counted twice. If the ratio of the first to the second is equal to the ratio of the third to the fourth, all four are said to be proportionals, or that the third is to the fourth as the first is to the second; but this is the only case in which they are called ‘proportionals’.
11.5. The ratio between magnitudes.
The ratio between the antecedent and the consequent consists in the difference, that is, in the part of the larger which separates the smaller from it, in other words, in the remainder in the larger after the smaller has been subtracted. But the difference is not absolute, since it is relative to one or other of the related quantities. So the ratio of two to five is the three by which five is larger than two, not absolutely, but [120] in so far as it is compared with two or five. For even though the difference between two and five is the same as the difference between nine and twelve (namely three), the inequality is not the same, and hence the ratio between two and five, and nine and twelve is not the same either.
11.6. A relation is not a new accident, but something which was in the things related before they were related, or before the comparison was made. Similarly, the causes of the accidents of each related thing are the cause of the relation.
However, a relation is not to be thought of as if it were an accident distinct from the other accidents of the body to which it is related. Rather, it is one of them, namely the one by reference to which the comparison is made. For example, the similarity of something white with something else which is white, or its dissimilarity with something black, is the same accident as the whiteness itself and equality or inequality is the same accident as the magnitude of the thing which is compared. But the accidents have different names, since that which is called ‘white’ or ‘of such-and-such a size’ when it is not being compared with something else, the very same thing is called ‘similar’, ‘dissimilar’, ‘equal’, or ‘unequal’ when it is being compared.
It also follows from this that the causes of the accidents which belong to the related bodies are also the causes of similitude, dissimilitude, equality, and inequality. That is, if you make two unequal bodies, you also make their inequality; and if you make a rule and perform an action, you yourself are the cause of its conformity (if the action is in accordance with the rule), and of its non-conformity if it is not.
This is all I have to say about the comparison between one body and another.
11.7. The principle of individuation.
However, one and the same thing can be compared with itself, although only at different times. This has given rise to a major controversy among philosophers over the principle of individuation. In other words, in what sense should a body sometimes be deemed to be the same body, and sometimes a different one from what it was before? For example, is an old man the same man as he was when young, [121] or a different one? Or is a nation the same nation or different in different centuries? Some make the unity of matter the principle of individuation; some make it the unity of form; and some even say that it consists in the unity of the sum of all accidents taken together.
The argument in favour of matter being the principle is that a piece of wax is the same piece of wax whether its shape is spherical or cubic, since the matter is the same. The argument in favour of form being the principle is that a person is numerically the same person from infancy to old age, even though their matter is not the same; so, since their identity cannot be attributed to their matter, it seems that it must be attributed to their form. It is impossible to give an example of the sum of accidents constituting the principle of individuation; but since we usually give something a different name when a new accident has come into being, anyone who made this the cause of individuation would therefore have to say that the thing itself was also different.
The first opinion implies that, because of the perpetual change of matter in the human body, the person who commits a crime would not be the same person as is punished; nor would it be the same nation which passes laws in one century, and repeals them in the next — which would undermine the whole institution of human law-giving.
The second opinion implies that two bodies could be numerically one and the same while existing simultaneously. Take the well-known example of Theseus’s ship, which the sophists of Athens argued about long ago. The argument was about the difference between the original ship, and the one which was gradually remade through the continuous replacement of old planks by new ones. After all the planks had been replaced, was it numerically identical with the original ship? But if someone had preserved the old planks as soon as they were removed, and had later made a new ship by putting the preserved planks back again in the same arrangement, there is no doubt that this would be numerically identical with the original ship. We would then have two numerically identical ships, which is absolutely absurd.
[122] The third opinion implies that nothing at all is identical with what it was before, so that a person who is standing is not the same as the person who was sitting, and the water in the jug is not the same as the water which is going to be poured out of it.
So the principle of individuation is not always to be found in matter alone, or form alone.
What needs to be considered is the name we use to describe something when its identity is in question. There is a great difference between asking whether Socrates is the same person, or whether he is the same body. Simply because of the change in size, he cannot be the same body when a child and when an old man, since one and the same body always has one and the same size. But this does not prevent him from being one and the same person.
So, when we are enquiring whether something is the same as it was before, if the name we use was imposed on it because of its matter alone, it is the same individual if the matter is the same. For example, the water which was in the sea is the same as the water which is subsequently in a cloud; and it is always the same body, whether compacted into ice, or rarefied into a liquid. But if it was given its name because of a form which is capable of functioning as a principle of motion, the individual will remain the same as long as the principle is present. For example, a person will be the same person as long as all their actions and thoughts flow from one and the same principle (namely the one which was there when they were born). Similarly, it is one and the same river which flows from one and the same source, whether or not the water is the same, or even whether it is water at all. Again, a nation is one nation if its acts continuously flow from one and the same set of institutions, whether or not the same individual people are involved.
Finally, in cases where a name has been given to a thing by virtue of an accident, its identity will depend on its matter, since, as parts of its matter come and go, some accidents disappear, and new ones, which are not numerically identical, come in to being. For example, since the word ‘ship’ [123] means matter of a certain shape, the ship will be the same as long as all its matter is the same. If no part of its matter is the same, it is totally numerically different; and if part of its matter is still there and part has gone, the ship will be partly the same and partly different.
Chapter 12: Quantity
12.1. Definition of quantity.
In Chapter 8 above, I said what dimension is, and that there are three dimensions: line (or length), surface, and solid. Each of these is usually called a quantity if it is ‘determined’, that is, if its terms or limits become known. Everyone takes ‘quantity’ as being whatever is meant by any word which provides an appropriate answer to the question ‘How much?’. [n.94] So, for example, whenever someone asks ‘How long is the journey?’, or ‘How big is the estate?’, or ‘How large is the rock?’, it would be inappropriate to give the indefinite answer ‘Length,’ or ‘Area,’ or ‘Solid.’ Instead, the answer will be determinate, such as ‘The journey is a hundred miles,’ or ‘The estate is a hundred acres,’ or ‘The rock is a hundred cubic feet,’ or at least something which will enable the mind to grasp the size of the thing in question within definite limits. So the only way quantity can be defined is that it is a determined dimension, [124] or a dimension of which the limits are known, either by virtue of their place, or by virtue of some relation.
12.2. Explanation of what quantity is.
A quantity can be determined in two ways. One is by reference to sensation, and the determination is made be means of a sensible object, as when we have before our eyes a line, or a surface, or a solid, with a foot or a cubit physically marked out on it. This kind of determination is called display, and the quantity known this way is said to be displayed. The other way is by reference to memory, and the determination is made by comparison with a displayed quantity.
If someone asks how large a thing is, an answer of the first kind would be ‘The size you see displayed.’ As for the second kind, the enquirer will be satisfied only by a comparison with something displayed. So if someone asks how long the road is, the answer will be ‘So many thousand paces.’ In other words, the road is compared with a pace, or some other known measure determined by display. Alternatively, its quantity is related to some other quantity known by display as the diagonal of a square to its side, or in some other similar way. However, the displayed quantity must either be permanent (as when it is marked out on solid material), or capable of being sensed again; otherwise no comparison with it can be made.
When the comparison of one magnitude with another is made in the way I discussed in the previous chapter,
Since it follows from what I said in the previous chapter that comparison of one magnitude with another is precisely what is called a ‘ratio’, it is obvious that a quantity determined in the second way is nothing other than the ratio of a dimension which is not displayed to one which is displayed — that is, its equality or inequality compared with a displayed dimension.
12.3. How lines, surfaces, and solids are displayed.
The first way in which lines, surfaces, and solids are displayed is through motion, namely in the way I said (in Chapter 8) that they were brought into being, except that the traces of such motions remain as a line on paper, or cut into solid material. [125] The second way is by juxtaposition, as when a line is laid against a line (i.e. a length against a length), or a breadth against a breadth, or a thickness against a thickness. This is to draw a line by means of points, a surface by lines, and a solid by surfaces, except that here points are to be taken as very short lines, and surfaces as very thin solids. The third way in which lines and surfaces can be displayed is through sections, since a line is displayed by cutting a surface, and a surface is displayed by cutting a solid.
12.4. How time is displayed.
Time is displayed when, in addition to a line, something mobile is displayed moving along it uniformly, or assumed to be moving in this way. Since time is the image of motion in so far as before and after (i.e. succession) are considered in it, the drawing of a line is not sufficient for displaying time, but the mind must also have an imagination of something mobile travelling along the line with a uniform motion, so that time can be divided and added as often as necessary. So when philosophers draw a line in their demonstrations, and say ‘Let this line be time,’ they should be understood as it they had said, ‘Let time by the conception of uniform motion along this line.’ For although the circle on a sundial or the face of a clock is a line, it is useless for telling the time without the motion of the sun’s shadow or the hands, whether actual or assumed.
12.5. How number is displayed.
Number is displayed through the displaying of points, or of the names of numbers, such as ‘one’, ‘two’, ‘three’, etc. In the case of points, they must not be contiguous with each other, so that there is nothing to tell them apart by. They must be positioned in such a way that they can be discerned from each other. This is why number is called a discrete [n.95] quantity, whereas [126] every quantity which is generated by motion [n.96] is called continuous. In order for names of numbers to display number, they must be recited in order from memory, as ‘one, two, three,’ etc. If you merely say ‘one, one, one,’ etc., you do not know any number — except perhaps the numbers two or three, which are small enough to be held in the memory, although in the form of a figure, and not as a number.
12.6. How speed is displayed.
By definition, speed is the motion by which a specific distance is covered in a specific time. So for it to be displayed, there must be displayed not only time, but also the distance which is to be travelled by the moving body whose speed we want to determine, and over which the body is understood as moving. So two lines need to be displayed. On is the line along which the uniform motion is understood to take place, so that the time is definite. The other is the one along which the speed is calculated. Thus, if we want to display the speed of the mobile body A, we draw two lines, AB and CD, and we also place a body at C. Then we say that the speed of the mobile body A is such that it traverses the line AB in the same time as the mobile body B traverses the line CD with uniform motion.
12.7. How weight is displayed.
Weight is displayed by means of any heavy body, consisting of any material, provided it is always equally heavy.
12.8. How the ratio of magnitudes is displayed.
The ratio of two magnitudes is displayed when the magnitudes themselves are displayed — that is, the ratio of equals when the displayed magnitudes are equal, and the ratio of unequals when the displayed magnitudes are unequal. Since, as I said in Article 5 of the previous chapter, the ratio of unequals consists in the difference between them compared with [127] any magnitude whatever, if the unequals are displayed, the difference between them is also displayed, since the displayed magnitudes have a ratio, and it is necessarily the case that the ratio itself is also displayed. Similarly the ratio of equals (which consists in the fact that there is no difference between the magnitudes) is displayed along with the displayed equals. For example, if the displayed lines AB and CD are equal, the ratio of equality is displayed. Again, if the unequals EF and EG are displayed, then both the ratio of EF to EG and the ratio of EG to EF are displayed, since both they and their difference (which is GF) are displayed. The ratio of unequals is a quantity, since it consists in the difference GF, which is a quantity. But the ratio of equality is not a quantity, since there is no difference between the things which are equal. Nor is one equality greater than another, by contrast with inequalities, where one can be greater than another.
12.9. How the ratio of times and speeds is displayed.
The ratio of two times, or of two uniform speeds, is displayed when two lines are displayed, along which two moving bodies are understood to travel uniformly. So the same two lines display their own ratio, and that of times and speeds, depending on whether the are considered as displaying their own magnitudes, or times, or speeds. Suppose there are two displayed lines, A and B. From what has just been said, their ratio is displayed. Then consider them as being drawn at the same uniform speed. Since the times are longer, equal, or shorter, depending on whether the same distances are covered in a longer, equal, or shorter time, the lines will display the equality or inequality of A and B — that is, the ratio of times.
Finally, consider the same lines A [128] and B as being drawn over the same period of time. Since the speeds are faster, equal, or slower, depending on whether the bodies travel along longer, equal, or shorter lines during the same period of time, these lines will display the equality or inequality (that is, the ratio) of their speeds.
PART III: THE RATIOS OF MOTIONS AND MAGNITUDES
Chapter 15: The Nature and Properties of Motion and Conation, and Various Other Considerations.
15.2. Some additional principles.
[177] . . . Here I shall add the following to these principles. First, I shall define conation as a motion through space and time which is less than any given quantity, i.e. a quantity which is determined, whether by being displayed, or by being assigned a number; in other words, it is a motion through a point. In order to explain this definition, I must remind you that by ‘point’ I do not mean that which has no quantity, or which cannot conceivably be divided, since there is no such thing in the real world. Rather, it is that the quantity of which is entirely disregarded, in other words, that of which neither the quantity nor any part of it enters into the calculation for the purposes of demonstration. So a point should not be taken as indivisible, but as undivided.
In the same way, an instant [178] should be taken as an undivided period of time, and not as an indivisible one.
Similarly, conation should be understood as certainly being a motion, but such that, in any demonstration, the quantity of the time over which it happens and of the distance it travels have no relation to the quantity of the time or distance of which it is a part. Nevertheless, just as a point can be compared to a point, so a conation can be compared to a conation, and one might be found to be larger or smaller than another. If the points at the vertices of two angles are compared, they will be equal or unequal in the same ratio as the angles themselves; again, if a straight line cuts a number of circumferences of concentric circles, the points of intersection in the same ratio as the circumferences themselves. In the same way, if there are two motions which start and stop at the same time, their conations will be equal or unequal in the same ratio as their speeds — just as we see a lead ball falling with a larger conation than one made of wool.
Secondly, I define impetus as speed itself, but considered as at any given point of the time during which the motion takes place. Consequently, impetus is nothing other than the quantity or speed of the conation itself. . . . [n.97]
Thirdly, I define resistance, when two moving bodies come into contact, as a conation contrary to a conation, whether wholly, or in a particular part. By ‘contrary’ I mean when these two moving bodies exercise conation along the same straight line from its opposite ends. By ‘contrary in a part,’ I mean when both of them exercise conation along lines leading away from the ends of the straight line they started from.
Fourthly, to define what it is to push: We say that one of two moving bodies pushes the other, when [179] its conation brings it about that the whole or part of the other leaves its place.
Fifthly, we say that a body which has been pushed without being dislodged reconstitutes itself, when, once the body pushing it has been removed, because of its internal constitution, all its parts which have been moved return to their original places. We see this happen in springs, balloons, and in many other bodies. At first their parts, to a greater or lesser extent, give way to the conation applied by the body pushing them; then, after the pushing body has been removed, they reconstitute themselves by some internal force, and restore the body as a whole to its former shape.
Sixthly, I define force as impetus multiplied either by itself, or by the magnitude of the moving body, by virtue of which a moving body exerts more or less action on a body resisting it. . . .
15.7. Every conation is propagated to infinity.
[182] Every conation, whether strong or weak, is propagated to infinity. This is because it is a motion. So if it takes place in a vacuum, the moving object will always continue to move at the same speed, since to suppose that it is a vacuum is to suppose [183] that there is nothing to resist its motion. Therefore (by Chapter 9, Article 7) it will always continue in the same direction and at the same speed. If it takes place in a plenum, since conation is a motion, the closest thing standing in its way will be moved back and exert its own conation, which in its turn will move back the closest thing standing in its way, and so on to infinity. So the generation of conation from one part of the plenum to another continues to infinity.
It also reaches any distance, however great, in an instant, since at the very instant when the first part of the intervening plenum moves back the part next to it, the second part in its turn moves back the part standing next to it. So every conation, whether in a vacuum or in a plenum, travels not only to any distance whatever, but also in the smallest possible time — that is, in an instant. Nor is it relevant that the conation becomes weaker the further it gets, so that eventually it becomes completely insensible, since it can happen that a motion is too small to stimulate sensation. And here we are dealing with the reasons for things, and not with sensation and experience. . . .
Chapter 20: The Dimension of the Circle, and the Section of Arcs or Angles [n.99]
Propositions 1–6.
What analysis and synthesis are.
[251] When I was discussing method in Chapter 6, I thought I should postpone what might have been said at the time about the analytic method in geometry, [252] since I had not yet mentioned lines, surfaces, solids, equality, or inequality, and I could not have made myself understood. So I shall now say what I think about it.
Analysis starts from the definitions of the terms of a proposition which we assume to be true. Then it defines the terms used in those definitions, and repeats the process of reasoning until it arrives at things which are known. Putting them together again is the demonstration of the truth or falsehood of the proposition assumed to be true. This putting together or demonstration is precisely what is called ‘synthesis’. So the analytic method is the method of reasoning from something assumed to principles, that is, to as many primary propositions, or propositions demonstrated from primary propositions, as are sufficient for demonstrating the truth or falsehood of the assumed proposition. The synthetic method is nothing other than that of demonstrating it.
So the only difference between synthesis and analysis is the difference between forwards and backwards; and both are included in logistic. Consequently, in the analysis and synthesis of any question or problem, the terms of all the propositions must be convertible, or if the question is expressed hypothetically, it is necessary not only that the truth of the consequent should be inferred from the truth of the antecedent, but also that the truth of the antecedent should be inferred from the truth of the consequent. Otherwise, when analysis has arrived at principles, synthesis could not get us back to the matter in question. The terms which come first in analysis will come last in synthesis. For example, when in analysis we say that two rectangles are equal, therefore their sides are proportional to each other; in synthesis it will be necessary for us to say that since their sides are proportional to each other, therefore the rectangles themselves are equal. But we could not say this [253] unless ‘having sides proportional to each other’ and ‘being equal rectangles’ were convertible terms.
In every analysis of two quantities, we are looking for a ratio by means of which the unknown quantity can be displayed in sensation by drawing a figure. This displaying is the goal to be arrived at, and it consists either in answering a question, or in performing a construction set as a problem.
In so far as analysis is reasoning from something assumed to principles (that is, to definitions or theorems which are already known), since this reasoning ultimately tends towards some equation or other, there is no end to analysis until one arrives at the very causes of the equality or inequality, or at enough theorems previously demonstrated from these causes to enable one to demonstrate the answer to the question.
In so far as the goal of analytic is either to perform a construction which has been set as a problem (if it is possible), or to discover its impossibility; if it is possible, the analyst must not stop before reaching whatever contains the efficient cause of what is to be constructed. It necessarily consists in primary propositions, and these are definitions. So the efficient cause of the construction must be contained in these definitions. But note that I say ‘construction’, and not a ‘demonstrated conclusion.’ The cause of a conclusion is indeed contained in the premises, in that the truth of the assertion which is proved is in the assertions which prove it. Whereas the cause of a construction is in the things themselves, and consists in a motion or a combination of motions. So the propositions in which analysis comes to an end are definitions, but of a sort which signify the way in which the thing itself is constructed or brought into being. Otherwise, when in synthesis we retrace our steps to the problem which was to be proved, there will be no demonstration. For there is no demonstration [254] which does not generate scientific knowledge. But it does not generate scientific knowledge unless it starts out from knowledge of the causes by which the construction set as a problem is carried out.
To summarise the above briefly: analysis is reasoning from a construction which is either assumed to be possible or which has actually been carried out, to the efficient cause or combination of efficient causes of the actual or assumed construction. Similarly, synthesis is continuous reasoning from the primary causes of a construction through intermediate stages to the actual construction itself.
However, there are many ways in which one and the same thing can be brought into being, or a construction set as a problem can be carried out. Consequently different geometricians use different methods, and sometimes even the same geometrician at different times. For example, if you are told to construct a quantity equal to another given quantity, you might try to find out whether this is possible by imagining some motion or other. For the equality or inequality of various quantities can be shown from motion and time no less than by showing that they are congruent. It can happen that a certain motion makes two quantities coincide or be congruent in extension, whether they are lines or surfaces, and even if one is straight and the other curved. This is the method which Archimedes used when dealing with spirals.
Equality and inequality can also be discovered and demonstrated by considering weights, as Archimedes also did in the quadrature of the parabola.
Again, equality and inequality are often discovered by dividing each quantity into parts which are considered as indivisible, as Archimedes did in many places, and as Bonaventura Cavalieri [n.100] has done in our own times.
Finally, the same can be done by considering the powers of lines, or the roots of powers, by multiplication, division, addition, subtraction, and the extraction of roots from powers, or by discovering where straight lines with the same ratio terminate. For example, [255] the ends of all the straight lines, as many as can exist, drawn from a straight line and passing through a single point, if their parts on either side of that point are in the same ratio, they will also end in a straight line. The same is true if you take a point between two circles. [n.101] Consequently, the loci of all these points [n.102] are either straight lines or circumferences of circles, and are called ‘plane loci’.
Similarly, if the ends of parallel straight lines are applied to the same straight line, and if the parts [n.103] of the line they are applied to have a ratio which is twice that of adjacent applied lines, then all their ends will be in conic section, and the conic section itself is the locus of the ends. It is called a ‘solid’ locus, since it is useful for solving equations consisting of three dimensions.
So there are three methods of investigating the cause of the equality or inequality of two given quantities: firstly by making calculations about motions, since equal distances are covered if the motion and the time are equal (and weighing is a motion); [n.104] secondly by means of indivisibles, since all parts taken together are equal to the whole; and thirdly by means of powers, since roots are equal when their powers are equal, and, conversely, powers are equal when their roots are equal.
However, in questions involving any degree of complexity, it is impossible to establish a definite rule, in any of these methods, for deciding which unknown factor it is best to take as the initial assumption for beginning the analysis. Nor can we tell which of the equations which first emerge we should best pursue. Rather, any success has to be attributed to native wit, previously acquired scientific knowledge, and even partly to luck. You are subsequently as good an analyst as you were previously a good geometrician. Nor do the rules of analysis make [256] a good geometrician; whereas synthesis does, when it begins with the elements themselves, and accompanies the elements with their logical use. The correct method for teaching geometry is the synthetic method, as systematically laid down by Euclid. If you have Euclid as your master, you can become a geometrician without Vieta. [n.105] Although Vieta was clearly an admirable geometrician, if you have Vieta as your master without Euclid, you cannot become a geometrician.
Some geometricians (though not first-class ones) have thought that the branch of analytic which proceeds by means of powers is suitable for solving all problems whatever. But in fact it is rather limited in scope. It is entirely restricted to the theory of rectangles, and solids bounded by rectangles. So much so, that even if they arrive at an equation which determines the required quantity, they cannot ever have any technique for displaying that quantity on a plane surface. They can only do it by means of a conic section, which is (as the geometricians say) a mechanical and not a geometrical technique. They call problems of this sort ‘solid’, and when they cannot display the quantity even by means of a conic section, they call the problem ‘linear’. Consequently, analysis through powers is of no use at all for the quantities of angles or arcs of circles.
This is why the ancients decreed that the only way it was possible to display sections of angles on a plane surface was mechanically (apart from bisections, and bisections of bisections). Thus, Pappus [n.106] says (before Proposition 31 of Book 4): some are called ‘plane’, some ‘solid’, and some ‘linear’. So those which can be solved by straight lines and the circumference of a circle (i.e., which can be drawn with a ruler and compass, without using any other instrument) are rightly called [257] ‘plane’, since the lines by means of which the answers to such problems are found are generated on a plane surface. Those which are solved by using one or more conic sections in their construction are called ‘solid’, since it is necessary to use the surfaces of solid figures (namely conics) in their construction. There remains the third kind, which is called ‘linear’, since in their construction, use is made of lines other than the ones already mentioned, etc. Then a little further on: this kind includes spirals, quadratics, conchoids, and cissoids. But geometricians consider it a serious fault when someone solves a problem involving a plane surface by using solids or linears.
Pappus classifies the trisection of an angle as a solid problem, and its quinquesection as a linear one. But what then? Were the ancient geometricians wrong when they used a quadratic in order to find a straight line equal to an arc of a circle? And was Pappus himself wrong to discover the trisection of a given angle by means of a hyperbola? Absolutely not. The ancients made use of this analysis by powers. The fault was to solve a problem by using higher powers when lower ones were sufficient; and the argument was that it betrayed an inadequate understanding of the nature of the matter in question.
The virtue of analytic through powers consists in exchanging, rotating, and distorting rectangles and analogisms; and the skill of the analyst is logic, by which everything which lies concealed in the subject and predicate of the conclusion to be proved can be methodically discovered. And the symbolic method (which many use today in the belief that it is the analytic method) is neither analytic nor synthetic, but the art of arithmetical calculation. As such it is true enough, but when used for geometrical calculations it is a false short-hand. It is wholly inappropriate for either learning or teaching geometry, and its only use is for keeping a quick and brief record of geometrical discoveries. [258] Even though symbols make it easier to go through a long chain of propositions, I doubt that the process should be considered of much use, since it takes place without any ideas of the things themselves.
Chapter 22: Other Kinds of Motion
22.12. All pulling is pushing.
[280] Another distinction between kinds of motion is that between pushing and pulling. As I have already defined it, pushing is when that which is moved is in front of that which moves it. By contrast, pulling is when the mover is in front of the moved. However, if you consider more carefully, you will see that pulling is really pushing. If you consider two parts of a hard body, the one in front pushes before it the medium in which the motion takes place, and the part it pushes pushes another part, and this pushes another one, and so on. In this action, assuming there is no vacuum, it is necessarily the case that, since the pushing is continuous (i.e. the medium closes up round the object as it goes through), that which is doing the moving is behind the part which originally seemed to be pulled rather than pushed. So since that which is pulled is now in front of the body by which it is moved, it is pushed, not pulled.
22.20. Habit.
[284] Having discussed motion, I should like to add something about habit. Habit is one of the ways in which motion can be generated — not motion taken in itself, but the effortless behaviour of a body moving on a particular and predetermined course. Since habits are formed by the weakening of conations which divert the body in motion from that course, such conations must be gradually weakened. This can only be brought about by sustained action, or the repetition of actions. So it is familiarity which gives rise to the ease of acting which is rightly called a ‘habit’ in ordinary language. It can be defined as follows: A habit is a motion made more fluent by familiarity, that is, by a continuous conation, or repeated conations, against conations resisting it, on a particular course which is different from the one on which it had originally begun to embark.
In order to make this clearer by an example, we observe that when people who do not know how to play the guitar put their hand on the instrument, after the first note, their hand does not go to the desired position for the second note. [285] Instead, they have to draw it back with a new conation, and, starting again as it were, they pass from the first note to the second. Nor will this new conation move the hand to the third position, but they have to withdraw the hand again, and so on repeatedly, with a different conation for every note. It is only after this has been done many times, and a sequence of discrete motions or conations has been synthesised into a single, smooth conation, that the hand can eventually complete the predetermined route from the beginning with fluency.
But habits can be observed in inanimate bodies as well as in animals. Take, for example, the string of a cross-bow which has been stretched back with much force, so that if it were released, it would spring back with great force. We observe that, after being left stretched for a long time, it acquires a habit, such that, when it is released and left to its own devices, it does not only fail to spring back, but even that it takes just as much force as before to bring it back to a state of tension.
PART IV: PHYSICS, OR THE PHENOMENA OF NATURE
Chapter 25: Animal Sensation and Motion
25.1. The connection between what has already been said, and what follows.
[315] In Chapter 1, I defined philosophy as knowledge of effects, acquired by correct reasoning from knowledge of how they came into being; and knowledge of a possible way of coming into being, from known effects or phenomena. [n.108]
Consequently, there are two methods of philosophising. One is from the way something comes into being to its possible effects, and the other is from effects revealed in experience [n.109] to a way they possibly came into being. [316] In the first of these, it is we ourselves who are responsible for the truth of the primary principles of reasoning (namely definitions), by agreeing on the names of things. I have carried out this first part in the preceding chapters; and, unless I am mistaken, I have asserted nothing other than definitions and what follows from them. That is to say, I have asserted nothing which I have not sufficiently demonstrated to those who agree with me as to the usage of words — and these are the only people I have business with.
I now embark on the second part, which proceeds from the phenomena or effects of nature known to us by sensation, to the investigation of some way in which, I do not say they did come into being, but in which they could have come into being. Consequently, the starting points which are the basis of what follows are not created by ourselves, nor are they asserted universally, as definitions are. Rather, we observe what has been placed in things themselves by the author of nature, and we employ singular rather than universal propositions. Again, they do not amount to the necessity characteristic of a theorem, but only show the possibility of some particular way of coming into being — though not without the universal propositions already demonstrated above.
I have given the title Physics, or the Phenomena of Nature to this part, because the knowledge which is dealt with here has its beginnings in the phenomena of nature, and results in a sort of science of natural causes. By ‘phenomena’ is meant anything which appears to us, or is revealed to us by nature.
Of all the phenomena which exist by us, [n.110] the phenomenon of appearing [n.111] is itself the most remarkable: among the various bodies in nature, some have within themselves copies of almost all things, and others of none. So if phenomena are the starting points of knowledge of other things, it must be said that sensation is the starting point of knowing these starting points, and that all scientific knowledge is derived from it. [317] Consequently, the investigation of its causes cannot start out from any other phenomenon than sensation itself. But, you will ask, by what sense will we be contemplating sensation? My answer is, by sense itself; or more precisely, by the memory of other sensible things, which remains for a certain time after they have gone. This is because sensing that you have sensed is the same as remembering.
First of all, then, we must investigate the causes of sensation (that is, of the ideas or phantasms which we experience as continuously arising in us when we are sensing), and the process by which it occurs. Right from the start, it will help the investigation to note that our phantasms do not retain their identity for ever, but new ones repeatedly come into being and old ones disappear, as our sense organs are turned towards different objects. So since they come in and out of being, it follows that they are some sort of change in the state in the body which is sensing.
In Chapter 9, [n.112] Article 9, I showed that all change is some sort of motion or conation (and conation is also a motion) in the inner parts of the thing which changes. The reason was that, as long as even the smallest parts of any body maintain their same relative positions, nothing new happens to it (except perhaps that the body as a whole can move), so that in fact it both is and is seen to be the same as it previously was and was seen to be. Therefore in a sentient being, sensation cannot be anything other than a motion of various internal parts existing inside the sentient being, and these moving parts are parts of the organs by means of which we sense. The parts of the body by which sensation is carried out are those which we commonly call ‘sensory organs.’ So we now have the subject of sensation, namely that which phantasms are in; [318] and also a partial account of its nature, namely that there is some sort of internal motion in the sentient being.
I have also shown (Chapter 9, [n.113] Article 7) that motion can be brought about only by something which is moving and contiguous. From this it follows that the immediate cause of sensation is in that which first touches and presses against the sensory organ. If the outermost part of the organ is pressed, the part which is immediately next to it on the inward side will also be pressed by it as it moves back, and in this way the pressure or motion will be propagated through all the parts of the organ right to the innermost. In the same way, the pressure of the outermost part also results from a pressure from a more distant body, and so on continuously, until we come to that which we judge the phantasm itself (which is brought into being by sensation) to have come from, as if flowing from its primary source. [n.114] And this is what we usually call the ‘object’, whatever thing it belongs to. So sensation is an internal motion in a sentient being, generated by a motion of the internal parts of the object, and propagated through the medium to the innermost part of the organ. With these words, we have nearly defined what sensation is.
I have also shown (Chapter 15, Article 2) that resistance is always a conation contrary to a conation, that is, a reaction. So, when a motion is propagated by the object through the medium to the innermost part of the organ, the whole organ resists it, or reacts against it, through the natural internal motion of the organ itself. Consequently there occurs a conation from the organ which is contrary to the conation from the object. The conation inwards is that last of the series of events which constitute the act of sensation, and it is only then that the reaction, which lasts a little time, gives rise to the phantasm itself. This phantasm always seems (appears) [n.115] as if it were something situated outside the organ, because of the conation outwards. So I propose the following as the complete definition of sensation, consisting of an explanation of its causes and [319] the order of its coming into being: sensation is a phantasm brought into being through reaction by an outward conation from the sensory organ, which in turn is brought about by an inward conation from the object, and which lasts a certain time.
The subject of sensation is the sentient being itself, namely an animal; and it is more correct to say that the animal sees, than that the eye sees. The object is that which is sensed; and so it is more accurate to say that we see the sun, than that we see light. This is because light, and colour, and heat, and sound, and other qualities which we usually call sensible qualities, are not objects, but phantasms of sentient beings.
A phantasm is the act of sensing, and the only difference between a phantasm and a sensation is that one is coming into being, and the other has come into being — and in the case of things which come into being instantaneously, this is no difference at all. And phantasms come into being in an instant. In every motion propagated through continuous body, the part which moves first moves the second, the second moves the third, and so on to the last part, however distant. And in the point of time at which the first or earlier part advances into the place of the second and pushes it forward, at the very same point of time the last but one takes the place of the last one, which makes way for it. And the last one reacts at the same instant; and if the reaction is strong enough, it generates a phantasm; and if a phantasm is brought into being, sensation occurs simultaneously.
Since the organs of sensation are in the sentient being, they are parts of that being; and if they are damaged, the power of generating images is lost, even if no other part is damaged. Now in most animals these parts are found to be the spirits; and the membranes which come from the pia mater [n.116] and cover the brain and all the nerves; and also the brain itself, and the arteries which are in the brain; and anything else which communicates motion to the heart, which is the origin of all sensation. [320] Wherever the action of the object touches the body of the sentient being, the action is propagated by a nerve to the brain. If the nerve leading there is damaged or obstructed, so that the motion cannot be propagated any further, no sensation follows. Again, if the same motion between the brain and the heart is cut off because of a defect in an organ carrying it there, there will be no sensation of the object.
Even though, as I have said, all sensation occurs by reaction, it is not necessarily the case that whatever reacts has a sensation. I know there have been philosophers, and learned ones at that, who have held that all bodies were endowed with sensation; nor do I see how they could be refuted, if the nature of sensation consisted in reaction alone. But even if bodies other than sentient ones had some sort of phantasm whenever they reacted, it would cease as soon as the object went away. Unless they had organs capable of retaining an impressed motion even after the object had gone away, as animals have, their sensation would be unaccompanied by any memory that they had sensed — and this has nothing to do with the sort of sensation we are talking about here. By ‘sensation’, we customarily mean a sort of judgment, based on phantasms, about the things which are the objects of sensation. More specifically, we compare and distinguish phantasms; and this is possible only if the motion in the organ which gave rise to a phantasm continues for some time, and the phantasm itself is brought back during that time. So the sort of sensation I am talking about (which is what is meant by ‘sensation’ in ordinary language) is necessarily accompanied by a memory, which enables us to compare earlier sensations with later ones, and to distinguish one from another.
Therefore it is also integral to sensation properly so called, that it permanently involves a variety of phantasms, [321] in such a way that one can be distinguished from another. Suppose you were put together with well-functioning eyes, and all the other organs of sight in good order, but endowed with no other sense. Suppose also that you were confronted with one and the same thing, and that it always manifested the same colour and form, without any (or at least hardly any) variation. Whatever anyone else might say, it certainly seems to me that you would no more see, than I seem to be able to sense the bones in my arms by my organs of touch. Yet these bones are permanently surrounded on all sides by the most sensitive of tissues. I might say that you were absorbed in the thing, and perhaps that you were looking at it; but that you were oblivious. I would not say that you saw it. So it comes to the same thing whether you sense one thing all the time, or do not sense anything at all.
The nature of sensation does not allow more than one thing to be sensed at the same time. Since the nature of sensation consists in motion, as long as the sense organs are occupied with one body, they cannot be moved by another in such a way that the motion of each brings into being an unadulterated phantasm of each. There would not be two phantasms of two objects, but a single one conflated out of the action of both objects.
Besides, as I made clear in Chapter 7, body and place can only be divided together, so that if you count a number of bodies, you will also necessarily count the same number of places, and vice versa. The same is true of the division of motion and time: the more motions you say there are, the more times there are understood to be; and the more times you say there are, the more motions there are understood to be. Even when we see a variegated object, it is a single variegated object, not a variety of objects.
Here is another point. Consider the organs which are common to all the senses, such as (in the case of humans) all the parts of every organ which run from the roots of the nerves to the heart. [322] As long as they are agitated by a strong sensation of one object, they are less capable of receiving the action of any other objects, whatever sense they impinge on, because things in motion are resistant to receiving a new motion. This is why the study of one object prevents any sensation of other objects, even when the object is present. By ‘study’ I mean nothing other than ‘occupation of the mind’ — in other words, a motion of the sense organs made forceful by one particular object, which makes them oblivious to everything else as long as it lasts. As Terence said, ‘The audience was oblivious, since its mind was occupied by the study of a tightrope walker.’ [n.117] And what is obliviousness other than anaesthesis, [n.118] or ceasing to sense other things? So only one single object is perceived by sensation at any given time. For example, when we are reading, we see one letter after another, not all of them together. Even if the whole page is in good light, and each letter is very clearly written, if we look at the page with a single gaze, we cannot read anything.
It follows from this that not every outward conation of an organ is to be called a sensation, but only the one which is predominant and more forceful than the rest at any given time. It eliminates the phantasms of the other things just as the light of the sun eliminates the light of the other stars — not by impeding their action, but by obscuring and hiding them through its excess of brightness.
The motion of the organ which gives rise to a phantasm is usually called ‘sensation’ only when the object is present. When the object is removed or passed by, but a phantasm remains, it is called ‘phantasy’, or ‘imagination’ in Latin; but since not all phantasms are images, it does not correspond perfectly to phantasy taken in its general sense. [323] But we are safe enough using the word ‘imagination’, provided it is taken in the sense of the Greek ‘phantasy’.
So imagination is in fact nothing other than a sensation which is decaying or weakening because its object has been removed. But what could be the cause of its weakening? Is motion weakened by the removal of the object? If it were, the phantasms of the imagination would always and necessarily be weaker than those of sensation. But this is not true, because in dreams (which are the imaginations of people who are asleep), they are no less clear than they are in someone who is actually sensing. However, the phantasms which people have of past things when they are awake are more obscure than those of present things, because the organs, which are simultaneously being set in motion by the presence of the objects, make the phantasms of past things less predominant. But in sleep, external action is blocked off, and cannot interfere with internal motion.
If this is true, we must next consider whether it is possible to discover any cause, which, if present, would necessarily block external objects of sensation from access to the inside of the organ. Suppose, then, that the organ becomes tired, because the repeated action of objects makes it necessarily respond with a reaction (especially the spirits). This tiredness means that it becomes painful for the spirits to move the parts of the organ. So they abandon and let go of the nerves, and withdraw to their source, which is located in the cavities either of the brain or of the heart. Consequently, the action which previously flowed through the nerves is necessarily cut off. For action on a patient which is retreating before it is at first less effective, and then gradually reduces to nothing as the nerves become slacker. So all reaction, i.e. sensation, ceases until the organ has been refreshed by rest, and the supply of animal spirits has been replenished, and the sleeper wakes up. It seems obvious that this is always what happens, except when some other cause somehow overrides [324] the rhythm of animal nature — for example, when tiredness or some disease causes an internal heat, which buffets the spirits and other parts of the organ more vigorously than usual.
In this variety of phantasms, some arise out of others; and the same phantasms sometimes bring similar ones to the mind, and sometimes dissimilar ones. But it is not without cause, nor is it as random as many people may think. In the motion of the parts of a continuous body, one part follows another by cohesion. So when we turn our eyes, or the organs of the other senses, onto a number of objects in succession, as long as the motion set up by each of them remains, their phantasms are regenerated whenever any of them has a more forceful motion than the rest; and they predominate in the same order as they were brought into being through sensation at some earlier time. Consequently, after we have sensed very many things as we get older, any thought can arise from almost any other thought; and this is why it can seem a matter of chance which will follow which. However, there is generally less uncertainty when we are awake, since the thought or phantasm of a desired end conjures up the phantasms of the means conducive to that end — and in the analytic order: from the last of the means to the first, and then back again from the beginning to the end. But this presupposes both appetite and a judgment about the relation of the means to the end, which are the outcome of experience. And experience is abundance of phantasms resulting from sensations of many things.
The only difference between having a phantasm [n.119] and remembering is that remembering implies a time in the past, whereas having a phantasm does not. In memory, phantasms are considered as worn down by time, whereas in phantasy they are considered as they are. This distinction is not [325] a distinction between things, but between different ways in which a sentient being considers things. What happens in memory is similar to what happens when we look at things from a great distance. Just as in the latter case the smaller parts of bodies are indiscernible because the distance is too great, in the former case many accidents, and places, and parts of things which were once perceived by the senses are obliterated by time.
The perpetual arising of phantasms in both sentient and thinking beings is precisely what is usually called the ‘discourse of the mind’. It is common to humans and animals, since thinking consists in comparing phantasms as they pass, that is, in noticing the similarities and dissimilarities between them. Generally, someone who is quick at spotting similarities between things which have different natures, or are at a great distance from each other, is praised for having a good phantasy; and someone who can discriminate between similar things is praised for having good judgment. But this capacity to discriminate does not depend on some special sense (the so-called ‘common sense’) giving rise to distinct sensations over and above those properly so called. Rather, it is a memory of differences between particular phantasms which remain for a certain time; just as, for example, the distinction between the hot and the bright is nothing other than the memory of a hot object and of a bright object.
The phantasms of people who are asleep are dreams; and we learn five things about them from experience. First, they are mostly disordered and incoherent. Secondly, the contents of our dreams are put together from combinations of phantasms of past sensations, and nothing else. Thirdly, they sometimes arise from discontinuities in the degenerate phantasms of people who are gradually falling asleep through drowsiness, and they sometimes arise in the middle of sleep. Fourthly, they are more forceful than the imaginations of people who are awake, but not than their sensations; however, they are as clear as sensations themselves. Fifthly, when we are dreaming, we are completely unsurprised by where things are or, what they are like. [326] Given what has already been said, it is not difficult to identify possible causes of these phenomena.
As for the first, all order and coherence arises from ‘considering the end’, [n.120] or having a plan. But since in sleep we lose all thought of our plans, it necessarily follows that our phantasms no longer follow each another in an order which tends towards an end, but contingently. It is just like when the objects of sight are observed in no particular order, because we have no preference for one visible object over another, and what we see depends on the mere fact that our eyes are open, rather than on our will.
The second arises from the fact that, after sensation has ceased, the objects can give rise to no new motion. Therefore there can be no new phantasm, unless we describe as ‘new’ a phantasm which is a compound of old ones, such as a chimaera, a golden mountain, and so on.As for the third, there is an obvious reason why a dream should sometimes seem like the continuation of sensation, when phantasms are fractured by illness, for example. This is that sensation continues in some organs, but not in others. However, it is more difficult to say how some phantasms are resuscitated when all the external organs are shut down. Nevertheless, what I have already said provides the cause for this as well. Anything which stimulates the pia mater excites certain of the phantasms whose motions still continue in the brain; and the motion which predominates over the rest gives rise to a phantasm, provided only that the pia mater is stimulated by an internal motion of the heart. These motions of the heart are appetites and aversions, about which I shall say more shortly. Just as appetite and avoidance are generated by phantasms, so phantasms in turn are generated by appetite and avoidance. For example, anger and fighting give rise to heat in the heart; and conversely, [25] heat in the heart, even if it has come from some other source, excites anger and the image of an enemy in sleep. Again, just as love and a beautiful physical appearance arouse heat in certain organs, so heat in the same organs, even when it has a different source, sometimes excites desire and an image of a submissive beauty. Finally, if people get cold while they are asleep, the coldness gives them a horrifying image or a phantasm of danger, and makes them frightened, in the same way as people go cold with fear when they are awake. All this shows how close a correspondence there is between the motion of the heart and the motion of the brain.
As for the fourth, there are two causes which bring it about that the things we seem to see or feel in sleep are as clear as in actual sensation. The first is that, when external sensation ceases, the motion responsible for the phantasm is predominant, as being the only one present. The second is that the parts of phantasms which have faded away over time are made good by being replaced with imaginary ones.
Finally, the reason why we are not surprised by previously unknown places and appearances of things when we are asleep, is because surprise presupposes that a new and unfamiliar thing is seen; but this cannot happen unless you remember an earlier appearance to compare it with; and in sleep everything is experienced as in the present.
It should be noted that some dreams were once (and even now) not considered to be dreams. This is especially true of those which occur to people who are only half asleep, or to people who are ignorant of the nature of dreaming, and are superstitious. They used to think that ghosts, and voices which they seemed to hear in their sleep, were not phantasms, but were themselves actual objects subsisting in the real world outside the dreamer. In some people, fear itself can conjure up terrifying phantasms in the mind, not only when they are asleep, but even when they are awake. This is especially the case if they feel guilty about a crime, or in the night, or in holy places — and stories about such apparitions help a little too. [328] They used to impose (and still do) the names of ‘ghosts’ and ‘incorporeal substances on such phantasms, as if they were genuine things.
In most animals there are observed to be five kinds of sensation, distinguished both by the organ of sense, and by the kind of phantasm. Sight, hearing, smell, taste, and touch have organs which are partly exclusive to themselves, and partly common to all.
The organ of sight is partly animate, and partly inanimate. The inanimate part consists of three humours. The first is the watery humour which is contained within the membrane called the ‘uvea’, which has a hole in the middle called the ‘pupil’. On the one side it is kept back by the innermost concave surface of the eye, and on the other side by the ‘ciliary processes’, and the covering of the crystalline humour. The crystalline humour is the second, and it is suspended in the middle of the ciliary processes. It is approximately spherical in shape; it has a thicker consistency; and it is bounded all round by its own transparent skin. The third is the glassy humour, which fills the rest of the eye. It is thicker than the watery humour, but thinner than the crystalline humour.
The animate part consists, first, of the choroid membrane. It is part of the pia mater, except that it is covered with a coat which comes out from the medulla of the optic nerve, and which is called the ‘retina’. And since the choroid is part of the pia mater, it stretches without break right up to the beginning of the spinal medulla, which is inside the skull, and where all the nerves within the skull have their roots. So whatever animal spirits are pumped into the nerves enter them at this point, since it is unthinkable that they should get in anywhere else.
So, since sensation is nothing other than the action of objects propagated to the furthest parts of the organ; and since animal spirits are nothing other than vital spirits from the heart with are transported and purified through the arteries; given all this, it necessarily follows that the action flows from the heart through certain arteries to the roots of the nerves which are in [329] the head. These arteries could be either the netiform plexus, or any of the other arteries which are connected to the substance of the brain. So these arteries are the remaining part of what makes up the visual organ as a whole. But this last part is the common organ of all the senses, while the first part, which extends from the eye to the roots of the nerves, is proper to vision.
The organ proper to hearing is the ear drum, and its own nerve running from it; and the other nerves to the heart are common. The organ proper to taste is the skin and its internal nerve in the palate and the tongue; and that of smell, the skin and nerve in the nostrils; and everything beyond the beginnings of these nerves is common. Finally, the organ of touch is the skin and nerves distributed over the whole body, and they too start from the root of the nerves. Everything else in common between all the senses seems to be the function of the arteries, and not the nerves.
The phantasm proper to vision is light. The name ‘light’ also includes colour, which is confused light. So the phantasm of a shining body is light, and the phantasm of a coloured body is colour. Properly speaking, the object of vision is not light or colour, but the body itself which is shining, or lit up, or coloured. Light and colour are in the sentient being, since they are phantasms; and they are not accidents of the thing which is sensed. This is obvious enough, because visible things often appear in places where we know scientifically they certainly are not; and because they appear with a different colour in different places, and can appear to be in a number of places at the same time. Motion, rest, magnitude, and figure are common to vision and touch. But there can be no light or colour without shape. This whole, consisting of shape and light or colour, is usually called an eidos, or an eidolon or an idea in Greek, and a species or image in Latin — [330] and all these words have the same meaning, which is ‘aspect’. [n.121]
The phantasm which comes from hearing is a sound; that from smell, an odour; and that from taste, a flavour. But touch gives rise to many phantasms which are easily distinguishable in sensation, but not by words — such as hardness and softness, heat and cold, wetness, oiliness, and so on. Lightness, roughness, tenuity, and density refer to figure, and hence are common to touch and vision. Again, the objects of hearing, smell, taste, and touch are not sounds, odours, flavours, hardness, etc.; rather, they are the bodies which give rise to sounds, odours, flavours, hardness, etc. I shall leave it till later to talk about their causes and how they are produced.
So these phantasms are the effects in sentient subjects, of objects acting on their organs. Nevertheless, there are other effects produced by the same objects in the same organs, namely various motions arising from sensation, and these are called animal motions. Since in every sensation of external things there is a mutual action and reaction (that is, two opposite conations against each other), it is obvious that the motion resulting from both together will continue in every direction, especially within the confines of each body. When this happens in an internal organ, the conation is directed outwards through a solid angle which is proportional to the force of the impression, and this is why some ideas are larger than others.
This provides the physical cause both of why things are seen as larger when, other things being equal, they are seen under a larger angle; and of why more fixed stars are visible on a calm, moonless, and cold night, than at any other time. This is because the action of the stars is less impeded through calm air, and is less hidden or overshadowed when the moon is inactive; and cold [331] calms the air, and helps or strengthens the action of the stars on the eye, so that we can now see stars which are otherwise invisible.
I have now said enough about sensation taken in a general sense, as occurring through the reaction of the organ. There are still other matters to be discussed, which we experience in ourselves when sensing, such as the location of images, and visual illusions. But since they largely depend on the specific structure of the human eye, it will be appropriate to leave them to Section II: The Human Being.
However, there is an entirely different kind of sensation, which I am going to say something about now. This is the sensation of pleasure or pain, which arises, not through an outward reaction of the heart, but through a continuous action from the outermost part of the organs towards the heart. Since the principle of life is in the heart, it is necessarily the case that a motion propagated by the sentient being to the heart will change or divert the vital motion in some way or other — more specifically, by making it easier or more difficult, or by helping or impeding it. If it helps it, this gives rise to pleasure; if it impedes it, it gives rise to pain, harm, or sickness. And just as phantasms seem to exist outside us because of the outward conation, so when pleasure and pain are sensed, they seem to be within us, because of the inward conation of the organ. Indeed, they seem to be where the first cause of the pleasure or pain is — for example, if we are suffering pain from a wound, the pain seems to be where the wound itself is.
Now the vital motion is the motion of the blood, which continuously circulates through the veins and arteries, as has been shown with abundant and absolutely certain evidence by our fellow-countryman Harvey, who was the first to observe the fact. If this motion is impeded by a motion brought about by the action of sensible objects, it will be restored again by the bending or straightening of parts of the body (that is, as the spirits are forced into one or other set of nerves), until all the harm is eliminated, as far as is possible. [332] If, on the other hand, the vital motion is helped by the motion arising from sensation, the parts of the organ will be disposed in such a way as to control the spirits so that, with the help of the nerves, the motion is preserved and increased as much as possible. This is the primary conation in animal motion, and it is even found in an unborn baby, since, if there is ever any harm to avoid or pleasure to pursue, it moves its limbs in its mother’s womb with a voluntary motion. And this primary conation is called appetite of approach when it is directed towards experiencing things which are known to be pleasant, and when unpleasant things are shunned, it is called aversion or avoidance.
At first, when little infants have only just been born, they desire few things and avoid few things, because they have no experience or memory; and this is why they do not have the same variety of animal motions as we see in adults. They cannot know whether objects will give them pleasure or pain without sensory knowledge (i.e. experience and memory) of a wide range of things. However, there is some scope for making conjectures based on the way things look, and depending on their degree of ignorance as to what will be good or bad for them, they hesitantly sometimes approach and sometimes avoid one and the same thing. But later, with practice, they gradually become better at knowing what to pursue and what to avoid, and also better at controlling their nerves and other organs when approaching some things and avoiding others. So appetite and avoidance, or mental aversion, is the primary conation of animal motion.
After primary conation, there follows an injection of animal spirits into the nerves, and their subsequent extraction; so there must be some sort of container or place for them near the roots of the nerves. This motion or conation must necessarily be followed by a swelling or relaxation of the muscles; and that in turn must be followed by a contraction or extension of the limbs — which is animal motion.
[333] There are various different ways of considering appetite and avoidance. When animals alternately seek and avoid one and the same thing, depending on whether they think it will please them or harm them; as long as the alternation between appetite and avoidance continues, there exists the sequence of thoughts which is called deliberation. It lasts as long as it is in their power, either to obtain what pleases them, or to avoid what displeases them. So appetite and avoidance as they are in themselves (i.e. without any previous deliberation) are simply called ‘appetite’ and ‘avoidance’. But if they have been preceded by deliberation, than, if the last act in the deliberation is an appetite, it is called willing or volition; and if it is avoidance, it is called refusing. [n.122] Consequently, the same thing is called both ‘will’ and ‘appetite’, but by virtue of different ways of considering it, namely whether it was before or after deliberation. What takes place inside human beings when they will something, is not dissimilar to what takes place in other animals, when they seek something after first deliberating about it.
Nor is the freedom of willing to do something or not to do it any greater in humans than in other animals. When a being has an appetite, the complete cause of the appetite will have preceded it, and therefore, as I showed in Chapter 9, Article 5, the appetition itself could not have failed to follow; in other words, it followed necessarily. So if freedom means freedom from necessity, then it is incompatible with the will of humans as well as that of other animals. But if by ‘liberty’ we mean the capacity, not of willing, but of doing what we will, then liberty in that sense can be accepted in both cases; and when it is present, it is present equally in both humans and other animals.
Again, when appetite and aversion follow each other in quick succession, the whole sequence they constitute takes its name sometimes from the one, and sometimes from the other. When the name is taken from appetite, the sequence is called ‘hope’, and when it is taken from avoidance, it is called ‘fear’, depending on whether the deliberation inclines more towards the one than towards the other. [334] For if there is no hope at all, it should not be called ‘fear’, but hatred; and if there is no fear, it should not be called ‘hope’, but love.
Finally, all the so-called passions of the mind consist of appetite and aversion (apart from pure pleasure and pain, which are the actual enjoyment of what is good, or suffering of what is bad). So, for example, anger is avoidance of an impending evil, but combined with the desire to escape from it by force.
But since there are innumerable passions and disturbances of the mind, and many of them are undetectable except in humans, I shall discuss them at greater length in Section II: The Human Being. As for objects which do not arouse the mind at all (if there are any such), we are said to ignore them.
So far I have dealt with sensations in general. I shall now turn to the objects of sensation.
Chapter 26: The Universe and the Heavens
26.1. The size and duration of the universe are inscrutable.
[334] Contemplation of sensation is followed by contemplation of bodies, which are the efficient causes, or objects of sensing. Every object in the whole world is either a part of it, or an aggregate of parts. The largest of all bodies or sensible objects [335] is the world itself, which is perceptible to us looking around in all directions on this point of it which we call the earth. There are very few questions we can ask about it as a single aggregate of many parts, and none can be given a definitive answer. The only questions we can ask about the whole world are how large it is, how long-lived it is, and whether it is the only one — but nothing else. This is because space and time (or size and duration) are the only phantasms of body as such (or taken absolutely generally), as I showed in Chapter 7; whereas all other phantasms are of bodies (or their objects) in so far as they are different from each other — the colour of things which are coloured, the sound of things which are heard, and so on.
The questions about the magnitude of the world are whether it is finite or infinite, and whether it is a plenum or not. The questions about its duration are whether it had a beginning, and whether it will exist to eternity. The question about its number is whether it is one or many — although in the case of number, if the world is infinite in magnitude, there can be no argument. Also, if it had a beginning, what was its cause, and what material was it made out of? Again, there will be further questions about the origins of this cause and this material, until we arrive at some eternal cause, whether one or many.
Definitive answers to all these questions would have to be given by anyone who claimed to contain within themselves a complete philosophy, if it is possible for us to know as much as we can ask. However, knowledge of the infinite is inaccessible to the finite enquirer. Whatever we humans know, we learned from our phantasms; but there is no phantasm of the infinite, whether the infinite in magnitude or the infinite in time. No human, nor any other thing (apart from anything which is itself infinite) can have any conception of the infinite. Even if you can progress, by absolutely sound reasoning, from any effect whatever, to its immediate cause, and from there to a more remote cause, and so on continuously, [336] you will never be able to carry on to eternity. Eventually you will have to give up through exhaustion, and you will not even know whether you could have got any further or not.
However, there are no absurd consequences whether we decide that the world is finite or infinite, since whichever was decided by the creator of the world, it could make no difference to anything we now see. Again, even if it follows from the fact that nothing can move itself, that there is some prime mover which has existed from eternity, it does not follow (as many suppose) that the prime mover is unmoved, but that on the contrary it is eternally in motion. This is because, just as it is true that nothing is moved by itself, it is also true that nothing is moved except by something which is in motion.
Consequently, questions about the magnitude and origin of the world are not to be decided by philosophers, but by those who have legal responsibility for managing the worship of God. After God, the Best and Greatest, had led his people into Judaea, he assigned to the priests his own right to the first-fruits. [n.123] In the same way, when he had handed over the world he had made to the disputations of humans, he willed that opinions about the nature of infinity and eternity, which were known only to himself, and were so to speak the first-fruits of wisdom, should be judged by those he willed to use as his ministers in managing religious affairs.
I cannot commend those who boast of having demonstrated, by their own reasoning from things in nature, that there was some first beginning of the world. They are rightly condemned, both by the laity because they do not understand what they are saying, and by the learned who do understand. Who would ever commend anyone who used this type of demonstration? They would argue: ‘If the world was eternal, then the number of days (or any other measure of time you like) preceding the birth of Abraham would be infinite. But the birth of Abraham preceded the birth of Isaac; consequently one infinity is larger than another infinity, or one eternity than another eternity — [337] which is absurd.’ This is similar to the argument that, since there are infinitely many even numbers, there are as many even numbers as there are numbers as such, i.e. even and odd numbers taken together.
And surely those who deny the eternity of the world in this way, thereby also deny the eternity of the creator of the world? So from this absurdity, they fall into another one, since they are forced to call eternity a ‘continued now’, and an infinite number of numbers a ‘unity’, which is much more absurd. For why should eternity be called a ‘continued now’ rather than a ‘continued then’? So either there will be a multiplicity of eternities, or else ‘now’ and ‘then’ will have the same meaning. It is impossible for us to begin to argue with people like this, who are not even speaking the same language [n.124] in their demonstrations.
But what is absolutely unforgivable is that the people who argue so absurdly are not mere amateurs, but geometricians, who set themselves up as strict judges of other people’s demonstrations, even beyond their sphere of competence. The reason is because, once they become entangled by the words ‘infinite’ and ‘eternal’, which are not backed up by any idea of anything in the mind, apart from its own lack of comprehension, then either they have to say something absurd, or they have to shut up (which is even less to their liking). Geometry is rather like wine: when it is new it makes you flatulent; [n.125] but when it has stopped fermenting, it is less sweet, but it is good for you. Similarly, if you are new to geometry, you think you can demonstrate anything that is true; but not once you have settled down.
So I deliberately pass over questions about the infinite and the eternal. I am satisfied with what the Holy Scriptures teach about the magnitude and origin of the world, and I accept them, along with the stories of miracles which confirm them, and the traditions of my country, and the respect owed to the law. I now move on to other matters, which it is not sacrilegious to argue about.
26.2. There is no empty space in the universe.
[38] Another common question about the world is whether or not its parts are contiguous with each other in such a way that leaves no room for even the smallest vacuum between them. There are quite plausible arguments on both sides.
Against the vacuum I shall adduce just one piece of experimental evidence. It is very familiar, but entirely conclusive in my opinion. Suppose you have a watering jar [n.126] like the sort gardeners use for watering gardens with, which has its base perforated with many little holes, and a mouth at the top which is small enough to be closed with a finger when necessary. Now, if you first block the hole at the top, fill it with water, and turn it upright, as long as the mouth is still closed, the water will not flow out of any of the holes in its base. But if you take your finger away, so that air can get in from above, the water flows through all of them; and if you close it again with your finger, the flow will immediately cease. It would seem that the only possible cause of this phenomenon is that the water cannot push down the air underneath it by its own natural downward conation, because there is no place to receive the air being pushed down. There are only two alternative possibilities. One is that by pushing the air next to it, there is a continuous conation as far as the mouth of the jar, and there the air enters it, and takes the place of the water flowing out. The second is that, in resisting the downward conation of the water, the air passes through the water itself. But the first is impossible, since the mouth is blocked. Nor can the second happen, unless the holes are large enough for the weight of the water flowing out to force the air into the jar through the same holes and at the same time. This would be the same as what happens when we have a vessel with a reasonably wide mouth, and quickly invert it in order to pour water out of it; in this case, the weight of the water forces the air through the circumference of the mouth so that it gets into the vessel, as is evidenced by the way the water gurgles, [n.127] and keeps stopping and starting. So for me it is an indication of the plenitude [339] of the universe that, unless it is accepted, it cannot be explained why the natural motion of water, specifically that of a heavy body, should be impeded.
26.3. Lucretius’s arguments for a vacuum are invalid.
On the other side, many quite plausible arguments and items of experiential evidence are adduced in favour of a vacuum. However, in each of them there is always seen to be something missing, which is required for the conclusion to be firmly established. Some of the arguments in favour of a vacuum are those employed by the followers of Epicurus. [n.128] He taught that the universe consists of tiny places with no body inside them, interspersed with tiny bodies with no vacuum inside them, which he called ‘atoms’ because of their hardness. The arguments of the Epicureans, as expounded by Lucretius, [n.129] are as follows.
First, he claims that nothing could move without a vacuum. For, he says, it is the function of body to obstruct and impede motion. So if everything was filled with body, there would be an obstacle to motion everywhere. Therefore there could be no beginning of motion, and therefore there could be no motion at all. [n.130]
Now it is true that it is impossible for there to be a beginning of motion in a plenum, if all its parts are at rest; but there is nothing here to prove the existence of a vacuum. Even if it were conceded that there is a vacuum, if the bodies interspersed in it were all at once and together at rest, they would still never move. As I demonstrated above (Chapter 9, Article 7), nothing can move unless it is moved by something which is in motion and contiguous with it. So since the supposition is that everything is together at rest, there will be no contiguous, moving body, and there will therefore never be a beginning of any motion.
But to deny the beginning of motion, does not mean that there is no motion now, unless it also implies that body has no beginning. After all, motion could have been co-eternal with body, or created with it. It seems to be no more necessarily the case that bodies were first [340] at rest, and set in motion only subsequently, than that they were first in motion, and subsequently at rest (if they ever were at rest). Nor does there appear to be any more reason why the matter of the world should be dotted with empty spaces in order to make motion possible, than that they should be full of matter, though fluid. Finally, there is no knowable reason why it should not be the motion of some interspersed liquid which forces these hard atoms together into the large masses of compound bodies as we see them. Consequently, the only conclusion which can be drawn from this argument is that motion is co-eternal with, or has existed for as long as, the things which are capable of moving. But neither of these is compatible with the teaching of Epicurus, who denies that either the world or motion has any beginning. Consequently, the necessity for a vacuum has not yet been demonstrated.
As far as I can gather from conversations with people who believe in the vacuum, their reason is that, when they think about the nature of a fluid, they imagine it as consisting in little granules of hard matter, rather as grinding corn results in fluid flour. However, it is perfectly possible to conceive of a fluid as fluid by its own nature, and just as homogeneous as an atom or a vacuum itself.
The second argument is taken from weight, and it is contained in the following lines from Lucretius:
Since it is the function of body to press everything downwards, whereas the vacuum remains by nature without weight, it follows without any doubt, that anything which has the same size but is perceived to be lighter, declares itself to contain more vacuum. I.362–365. [n.131]
I shall overlook the fact that he is wrong to assume that bodies have a downward conation, because ‘downwards’ has nothing to do with the nature of things, but is a figment of our imagination. I shall also overlook the fact that, if everything were heading towards a single lowest point, then either nothing would coalesce together, or everything would be forced into the same place. What destroys the force of this argument is that, in the thing he is referring to, [341] air interspersed among the atoms would have precisely the same effect as he thinks the interspersed vacuum has.
The third argument is from the fact that lightning, sound, heat, and cold are observed to penetrate all bodies, however solid, except atoms themselves. [n.132] But this reasoning is invalid, unless it has first been demonstrated that none of these things can happen by a continuous generation of motion, without any vacuum. But they can, as I shall demonstrate at the appropriate places.
Finally, Lucretius sets out the fourth argument in the following lines:
If two flat bodies in contact with each other are suddenly pulled apart, air must necessarily fill all the vacuum which is created between the bodies. But however quickly it rushes round them, it cannot fill the whole space in a single instant, since it must occupy the first space first, and then all the rest in succession. I.384–390. [n.133]
However, this is rather more inconsistent with Epicurus’s own opinion, even than to deny the existence of a vacuum. What is true is that if two bodies were infinitely hard, and if they had exactly plane surfaces in perfect contact, then it would be impossible to pull them apart, precisely because this cannot be done unless there is instantaneous motion. However, just as there is no largest magnitude or fastest speed, so there is no maximum hardness. Consequently, it could happen that, given a sufficient force, the air could enter progressively, because the outer parts of the bodies joined together were peeled apart before the inner ones. [n.134] So Lucretius should first have shown that there are things which are as hard as could possibly be — in other words, not merely in comparison with softer things, but absolutely, that is, infinitely hard. But this is not true.
On the other hand, let us suppose [342] that Epicurus is right in saying that there are indivisible atoms, and that they have their own minute surfaces. [n.135] If two bodies form a compound by virtue of the fact that some of their minute surfaces, or indeed only one surface of each is in contact, Lucretius’s argument will constitute a knock-down demonstration that bodies which are built up from atoms (as he assumes all bodies are) can never be broken apart by any force whatever — which is in complete conflict with everyday experience.
26.4. Certain other invalid arguments for supposing that there is a vacuum.
. . . . [346] Nevertheless, it is only reasonable to expect someone who denies the vacuum to [347] provide alternative causes for the above phenomena, which do not involve a vacuum, and which are at least as probable, if not more probable. This will be done in the following chapters, each in its proper place. But first I must state the most general hypotheses which apply to physics as a whole.
Since hypotheses are assumed as the true causes of perceived effects, it is necessary for a hypothesis to consist in some assumed possible motion, otherwise it will be absurd. Rest cannot be the efficient cause of any thing. But motion presupposes mobile bodies, of which there are three kinds: fluids, solids, and mixtures of the two. Fluids are those whose parts can be separated from each other by the slightest conation; and solids are those which require a greater force to tear their parts away from each other. There are different degrees of solidity, and these degrees are called ‘softness’ or ‘hardness’, depending on whether they are relatively more or less solid. A fluid is always divisible into parts which are equally fluid (just as a quantity is always divisible into quantities); and soft things, whatever their degree of softness, are always divisible into parts which have the same degree of softness.
However, many people seem to believe that the only way there can be a difference between degrees of fluidity is if it comes from a difference in the size of parts. So on this account, it seems that even diamond dust can be described as a fluid. But on my understanding, fluidity is naturally present in every part of a fluid substance altogether. It is not analogous to the way dust flows, otherwise you would have to call a house a fluid if it was about to fall down. Rather, it is as we see water flow apart into parts which are always themselves fluid.
Now that this is understood, I come to my hypotheses.
26.5. Six hypotheses for explaining the phenomena of nature.
First, I assume that the vast expanse which we call the world is an aggregate of bodies. Some of these are solid and visible, such as the earth and the stars; others are invisible, tiny [348] atoms, which are scattered throughout the space between the earth and the stars; and finally there is a perfectly fluid ether, which fills all the remaining space in the universe, so that no place is left empty.
Second, I assume (with Copernicus) that the largest bodies in the world are solid and permanent, and that they are in the following order: first the Sun, second Mercury, third Venus, fourth the Earth (with the Moon revolving round it), fifth Mars, sixth Jupiter (together with its satellite), seventh Saturn, and finally the fixed stars at different distances from the Sun.
Third, I assume that not only the sun, but also the earth and each of the planets has, and always has had, a simple circular motion, which has always accompanied their natures.
Fourth, I assume that that there are some bodies which are interspersed with ethereal body. They are not fluid, but too small to be sensed. They move with their own simple motion, and some are harder or more solid than others.
Fifth, I assume (with Kepler) that the ratio between the distance of the sun from the earth and the distance of the moon from the earth is the same as the ratio between the distance of the moon from the earth and the earth’s radius.
As for the size of the circular orbits, and the time it takes the bodies on them to complete them, I assume them to be whatever seems most consistent with the actual phenomena in question.
Chapter 27: Light, Heat, and Colours
[362] Apart from the stars, whose motions I have just discussed, all the remaining bodies in the universe can be included under the single name of ‘interstellar’. [n.136] By hypothesis, these are either absolutely fluid ether, or bodies whose parts have a certain cohesion. And these latter differ from each other in consistency, magnitude, motion, and shape. As for consistency, I assume that some are harder and some are softer, through all the intervening degrees of cohesion; and as for magnitude, some of them are larger than others, but most of them are ineffably tiny.
Bearing in mind that any quantity is always divisible by the understanding into quantities which are further divisible, if you could do as much with your hands as you can do with your understanding, you would be able to remove from any given magnitude a part which was smaller than any given magnitude. But the omnipotent author of the universe can actually separate off a part of anything whatever, which is as small as anything we can understand it as being divisible into. Therefore there is no minimum possible size of a body. So why should it not also be probable that there is no minimum actual size? We know that there are some little animals which are so tiny that we can hardly discern their whole bodies; but they too have their babies, their little veins and other vessels, and their little eyes which are too small to be seen under any microscope. Consequently, we cannot assume that there is any magnitude so small, that nature itself cannot go further than our assumed magnitude. Moreover, today’s microscopes commonly make things seen through them appear more than a hundred thousand times larger than they are seen by the naked eye. [n.137] Nor can it be doubted that, as microscopes become more powerful (and the only limit to increasing their power is the purity of the material and the skill of the technician), any of those one-hundred-thousandth parts would appear another hundred thousand times larger than before.
Nor is the smallness of some bodies any more surprising than the enormous size of others. The same infinite power can make things infinitely larger as well as infinitely smaller. It is equally within the power of the Author of Nature, both to bring it about that the great orbit (namely that of which the radius extends from the earth to the sun) is like a point in relation to the distance of the sun from the fixed stars, and, on the other hand, to make a body so small, that it is smaller than anything visible, by the same proportion. The former, concerning the immense distance of the fixed stars, was for a long time considered incredible, yet is now believed by nearly all educated people. So why should not the latter, concerning the smallness of some bodies, become credible at some time in the future? The Divine Majesty is the same in matters small and large, and it [364] equally transcends the human senses in the minuteness of the parts of the universe, as much as in its size. For the existence of primary elements of compounds, of the first beginnings of actions, and of the first moments of periods of time is no less credible than what we now believe about the distance of the fixed stars. Mortals accept that there are some large but finite things, since they see them as such; they also accept that things they do not see can be of infinite magnitude; but it takes much time and learning for people to realise that there is a half-way house between the infinite, and the largest of the things they see or think about. But after many such remarkable facts have become more familiar through meditation and contemplation, then we believe in them, and transfer our wonder from created things to the Creator.
But however small some bodies can be, I shall not assume that they are any smaller than is required for explaining the phenomena themselves. I shall make similar assumptions about their motion (by which I mean how quickly or slowly they move), and about how many different shapes they have — that is, I shall assume that they are just as fast, or just as many, as is required for the explanation of natural causes.
Finally, I assume that there is no motion of the parts of pure ether (as if it were primary matter), other than that which it acquires from the non-liquid bodies moving in it.
Having made the above assumptions, let us now start talking about causes. First we shall investigate the cause of the light of the sun. Since the solar body has a simple circular motion, it pushes away the etherial substance surrounding it in all directions. The parts of it next to the sun are moved by the sun itself, and they in turn push the next nearest, so that necessarily, if there is an eye at whatever distance, there is eventually a pressure on its front part. Then the pressure on this part propagates [365] a motion to the heart, which is the innermost part of the visual organ. Finally, the heart reacts with a motion, which gives rise to a conation, which goes back again along the same route, until it ends in an outward conation of the membrane which is called the ‘retina’. But this outward conation (as defined in Chapter 25, above) is precisely what is called ‘light’, or a ‘phantasm of the shining’, since it is on account of this phantasm that the object is called ‘shining’. So we now have a possible cause of the light of the sun, which is what we undertook to discover.
The generation of warmth accompanies the generation of the light of the sun. Now everybody knows what warmth is, by sensing it when they are warm; but in other things we know about it only through reasoning. For it is one thing to be warm, but quite another thing to warm. So we perceive that fire or the sun warm, but we do not perceive that they are warm. We infer by reasoning from their being like us, that when animals warm things, they also are warm; but the conclusion does not follow necessarily. For even if it is correct to say ‘An animal warms, therefore it is warm,’ it does not follow that it is right to infer ‘Fire warms, therefore fire is warm,’ any more than ‘Fire causes pain, therefore it is in pain.’ Consequently, the only thing which is properly called ‘warm’ is that which, by virtue of sensing it, we necessarily feel warm. . . . .
Chapter 29: Sound, Smell, Taste, and Touch
29.18. The primary organ of touch, and how we know objects which are common to the sense of touch and to other senses.
[412] Warm and cold things are sensed by touch, even at a distance; but everything else, such as the hard, the soft, the rough, and the smooth, are sensed only when they are in contact. The organ of touch consists of all the continuous membranes of the pia mater which spread throughout the whole body, in such a way that no part of the body can be pressed without the pia mater also being pressed. So things which press it are sensed as hard or soft, in other words, as more or less hard.As for the sensation of roughness, it is nothing other than innumerable sensations of hardness and hardness following each other at extremely short intervals of both time and space. Consequently, the rough and the smooth, just like magnitude and shape, cannot be recognised by touch alone, but require memory. For touching [413] occurs at a specific point, whereas the rough, the smooth, quantity, and shape cannot be sensed without the flux of a point, that is, without time; and memory is required for us to have any sensation of time.
Chapter 30: Gravity
30.15. The cause of magnetic power.
[430] . . . . Up to now, I have been discussing the nature of body in general; and this constitutes Section I of The Elements of Philosophy. In Parts I–III of this section, the foundations of reasoning are embedded in our own understanding, that is, in the rule-governed use of words — a usage which we ourselves have created. Unless I am mistaken, I have demonstrated all the theorems in accordance with these rules.
Part IV depends [431] on hypotheses. Consequently, since we do not know whether these hypotheses are true or not, it is impossible to demonstrate that the causes of things are actually those which I have set down. However, I have assumed as a hypothesis nothing which is not both possibly true and easy to understand, and I have also reasoned validly from my assumptions. Therefore I have succeeded in demonstrating that these could have been the actual causes of things, which is the purpose of thinking about physics.
But if anyone can demonstrate what I have demonstrated, or more, by assuming a different set of hypotheses, then we shall owe that person more thanks than I claim for myself — provided the hypotheses they use are intelligible. For nothing at all will have been said by anyone who says that anything can be moved or brought into being by itself, by species, by power, by a substantial form, by an incorporeal substance, by an instinct, by antiperistatis, by antipathy, by sympathy, by an occult quality, or by any other empty words of university philosophers.
I now move to the phenomena of the human body, where I shall treat of optics, and of the causes of the wits, affects, and customs of human beings — if God wills that I live long enough.
THE END
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