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Chapter 4 Space and its Three Dimensions
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Chapter 4
Space and its Three Dimensions
1. The Group of Displacements1
 
Let us sum up briefly3 the results obtained. We proposed to investigate what was meant in saying that space has three dimensions and we have asked first what is a physical continuum and when it may be said to have n dimensions. If we consider different systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in saying that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain4 besides that two of these systems can sometimes be discriminated5 from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum C. And each of these systems is called an element of the continuum C.
 
How many dimensions has this continuum? Take first two elements A and B of C, and suppose there exists a series Σ of elements, all belonging to the continuum C, of such a sort that A and B are the two extreme terms of this series and that each term of the series is indistinguishable from the preceding. If such a series Σ can be found, we say that A and B are joined to one another; and if any two elements of C are joined to one another, we say that C is all of one piece.
 
Now take on the continuum C a certain number of elements in a way altogether arbitrary. The aggregate6 of these elements will be called a cut. Among the various series Σ which join A to B, we shall distinguish those of which an element is indistinguishable from one of the elements of the cut (we shall say that these are they which cut the cut) and those of which all the elements are distinguishable from all those of the cut. If all the series Σ which join A to B cut the cut, we shall say that A and B are separated by the cut, and that the cut divides C. If we can not find on C two elements which are separated by the cut, we shall say that the cut does not divide C.
 
These definitions laid down, if the continuum C can be divided by cuts which do not themselves form a continuum, this continuum C has only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divide C, C will have 2 dimensions; if a cut forming a continuum of 2 dimensions suffices, C will have 3 dimensions, etc. Thanks to these definitions, we can always recognize how many dimensions any physical continuum has. It only remains7 to find a physical continuum which is, so to speak, equivalent to space, of such a sort that to every point of space corresponds an element of this continuum, and that to points of space very near one another correspond indistinguishable elements. Space will have then as many dimensions as this continuum.
 
The intermediation of this physical continuum, capable of representation, is indispensable; because we can not represent space to ourselves, and that for a multitude of reasons. Space is a mathematical continuum, it is infinite, and we can represent to ourselves only physical continua and finite objects. The different elements of space, which we call points, are all alike, and, to apply our definition, it is necessary that we know how to distinguish the elements from one another, at least if they are not too close. Finally absolute space is nonsense, and it is necessary for us to begin by referring space to a system of axes invariably bound to our body (which we must always suppose put back in the initial attitude).
 
Then I have sought to form with our visual sensations a physical continuum equivalent to space; that certainly is easy and this example is particularly appropriate for the discussion of the number of dimensions; this discussion has enabled us to see in what measure it is allowable to say that ‘visual space’ has three dimensions. Only this solution is incomplete and artificial. I have explained why, and it is not on visual space but on motor space that it is necessary to bring our efforts to bear. I have then recalled what is the origin of the distinction we make between changes of position and changes of state. Among the changes which occur in our impressions, we distinguish, first the internal changes, voluntary and accompanied by muscular sensations, and the external changes, having opposite characteristics. We ascertain that it may happen that an external change may be corrected by an internal change which reestablishes the primitive8 sensations. The external changes, capable of being corrected by an internal change are called changes of position, those not capable of it are called changes of state. The internal changes capable of correcting an external change are called displacements of the whole body; the others are called changes of attitude.
 
Now let α and β be two external changes, α′ and β′ two internal changes. Suppose that a may be corrected either by α′ or by β’, and that α′ can correct either α or β; experience tells us then that β′ can likewise correct β. In this case we say that α and β correspond to the same displacement2 and also that α′ and β′ correspond to the same displacement. That postulated9, we can imagine a physical continuum which we shall call the continuum or group of displacements and which we shall define in the following manner. The elements of this continuum shall be the internal changes capable of correcting an external change. Two of these internal changes α′ and β′ shall be regarded as indistinguishable: (1) if they are so naturally, that is, if they are too close to one another; (2) if α′ is capable of correcting the same external change as a third internal change naturally indistinguishable from β’. In this second case, they will be, so to speak, indistinguishable by convention, I mean by agreeing to disregard circumstances which might distinguish them.
 
Our continuum is now entirely10 defined, since we know its elements and have fixed11 under what conditions they may be regarded as indistinguishable. We thus have all that is necessary to apply our definition and determine how many dimensions this continuum has. We shall recognize that it has six. The continuum of displacements is, therefore, not equivalent to space, since the number of dimensions is not the same; it is only related to space. Now how do we know that this continuum of displacements has six dimensions? We know it by experience.
 
It would be easy to describe the experiments by which we could arrive at this result. It would be seen that in this continuum cuts can be made which divide it and which are continua; that these cuts themselves can be divided by other cuts of the second order which yet are continua, and that this would stop only after cuts of the sixth order which would no longer be continua. From our definitions that would mean that the group of displacements has six dimensions.
 
That would be easy, I have said, but that would be rather long; and would it not be a little superficial? This group of displacements, we have seen, is related to space, and space could be deduced from it, but it is not equivalent to space, since it has not the same number of dimensions; and when we shall have shown how the notion of this continuum can be formed and how that of space may be deduced from it, it might always be asked why space of three dimensions is much more familiar to us than this continuum of six dimensions, and consequently doubted whether it was by this detour12 that the notion of space was formed in the human mind.
2. Identity of Two Points
 
What is a point? How do we know whether two points of space are identical or different? Or, in other words, when I say: The object A occupied at the instant α the point which the object B occupies at the instant β, what does that mean?
 
Such is the problem we set ourselves in the preceding chapter, §4. As I have explained it, it is not a question of comparing the positions of the objects A and B in absolute space; the question then would manifestly have no meaning. It is a question of comparing the positions of these two objects with regard to axes invariably bound to my body, supposing always this body replaced in the same attitude.
 
I suppose that between the instants α and β I have moved neither my body nor my eye, as I know from my muscular sense. Nor have I moved either my head, my arm or my hand. I ascertain that at the instant α impressions that I attributed to the object A were transmitted to me, some by one of the fibers13 of my optic nerve, the others by one of the sensitive tactile15 nerves of my finger; I ascertain that at the instant β other impressions which I attribute to the object B are transmitted to me, some by this same fiber14 of the optic nerve, the others by this same tactile nerve.
 
Here I must pause for an explanation; how am I told that this impression which I attribute to A, and that which I attribute to B, impressions which are qualitatively16 different, are transmitted to me by the same nerve? Must we suppose, to take for example the visual sensations, that A produces two simultaneous sensations, a sensation purely17 luminous18 a and a colored sensation a′, that B produces in the same way simultaneously19 a luminous sensation b and a colored sensation b′, that if these different sensations are transmitted to me by the same retinal fiber, a is identical with b, but that in general the colored sensations a′ and b′ produced by different bodies are different? In that case it would be the identity of the sensation a which accompanies a′ with the sensation b which accompanies b′, which would tell that all these sensations are transmitted to me by the same fiber.
 
However it may be with this hypothesis and although I am led to prefer to it others considerably20 more complicated, it is certain that we are told in some way that there is something in common between these sensations a + a′ and b +b′, without which we should have no means of recognizing that the object B has taken the place of the object A.
 
Therefore I do not further insist and I recall the hypothesis I have just made: I suppose that I have ascertained21 that the impressions which I attribute to B are transmitted to me at the instant β by the same fibers, optic as well as tactile, which, at the instant α, had transmitted to me the impressions that I attributed to A. If it is so, we shall not hesitate to declare that the point occupied by B at the instant β is identical with the point occupied by A at the instant α.
 
I have just enunciated22 two conditions for these points being identical; one is relative to sight, the other to touch. Let us consider them separately. The first is necessary, but is not sufficient. The second is at once necessary and sufficient. A person knowing geometry could easily explain this in the following manner: Let O be the point of the retina where is formed at the instant α the image of the body A; let M be the point of space occupied at the instant α by this body A; let M′ be the point of space occupied at the instant β by the body B. For this body B to form its image in O, it is not necessary that the points M and M′ coincide; since vision acts at a distance, it suffices for the three points O M M′ to be in a straight line. This condition that the two objects form their image on O is therefore necessary, but not sufficient for the points M and M′ to coincide. Let now P be the point occupied by my finger and where it remains, since it does not budge23. As touch does not act at a distance, if the body A touches my finger at the instant α, it is because M and P coincide; if B touches my finger at the instant β, it is because M′ and P coincide. Therefore M and M′ coincide. Thus this condition that if A touches my finger at the instant α, B touches it at the instant β, is at once necessary and sufficient for M and M′ to coincide.
 
But we who, as yet, do not know geometry can not reason thus; all that we can do is to ascertain experimentally that the first condition relative to sight may be fulfilled without the second, which is relative to touch, but that the second can not be fulfilled without the first.
 
Suppose experience had taught us the contrary, as might well be; this hypothesis contains nothing absurd. Suppose, therefore, that we had ascertained experimentally that the condition relative to touch may be fulfilled without that of sight being fulfilled and that, on the contrary, that of sight can not be fulfilled without that of touch being also. It is clear that if this were so we should conclude that it is touch which may be exercised at a distance, and that sight does not operate at a distance.
 
But this is not all; up to this time I have supposed that to determine the place of an object I have made use only of my eye and a single finger; but I could just as well have employed other means, for example, all my other fingers.
 
I suppose that my first finger receives at the instant α a tactile impression which I attribute to the object A. I make a series of movements, corresponding to a series S of muscular sensations. After these movements, at the instant α’, my second finger receives a tactile impression that I attribute likewise to A. Afterward24, at the instant β, without my having budged25, as my muscular sense tells me, this same second finger transmits to me anew a tactile impression which I attribute this time to the object B; I then make a series of movements, corresponding to a series S′ of muscular sensations. I know that this series S′ is the inverse26 of the series S and corresponds to contrary movements. I know this because many previous experiences have shown me that if I made successively the two series of movements corresponding to S and to S′, the primitive impressions would be reestablished, in other words, that the two series mutually compensate27. That settled, should I expect that at the instant β’, when the second series of movements is ended, my first finger would feel a tactile impression attributable to the object B?
 
To answer this question, those already knowing geometry would reason as follows: There are chances that the object A has not budged, between the instants α and α’, nor the object B between the instants β and β’; assume this. At the instant α, the object A occupied a certain point M of space. Now at this instant it touched my first finger, and as touch does not operate at a distance, my first finger was likewise at the point M. I afterward made the series S of movements and at the end of this series, at the instant α’, I ascertained that the object A touched my second finger. I thence conclude that this second finger was then at M, that is, that the movements S had the result of bringing the second finger to the place of the first. At the instant β the object B has come in contact with my second finger: as I have not budged, this second finger has remained at M; therefore the object B has come to M; by hypothesis it does not budge up to the instant β’. But between the instants β and β’ I have made the movements S′; as these movements are the inverse of the movements S, they must have for effect bringing the first finger in the place of the second. At the instant β′ this first finger will, therefore, be at M; and as the object B is likewise at M, this object B will touch my first finger. To the question put, the answer should therefore be yes.
 
We who do not yet know geometry can not reason thus; but we ascertain that this anticipation28 is ordinarily realized; and we can always explain the exceptions by saying that the object A has moved between the instants α and α’, or the object B between the instants β and β’.
 
But could not experience have given a contrary result? Would this contrary result have been absurd in itself? Evidently not. What should we have done then if experience had given this contrary result? Would all geometry thus have become impossible? Not the least in the world. We should have contented29 ourselves with concluding that touch can operate at a distance.
 
When I say, touch does not operate at a distance, but sight operates at a distance, this assertion has only one meaning, which is as follows: To recognize whether B occupies at the instant β the point occupied by A at the instant α, I can use a multitude of different criteria30. In one my eye intervenes, in another my first finger, in another my second finger, etc. Well, it is sufficient for the criterion relative to one of my fingers to be satisfied in order that all the others should be satisfied, but it is not sufficient that the criterion relative to the eye should be. This is the sense of my assertion. I content myself with affirming an experimental fact which is ordinarily verified.
 
At the end of the preceding chapter we analyzed32 visual space; we saw that to engender33 this space it is necessary to bring in the retinal sensations, the sensation of convergence and the sensation of accommodation; that if these last two were not always in accord, visual space would have four dimensions in place of three; we also saw that if we brought in only the retinal sensations, we should obtain ‘simple visual space,’ of only two dimensions. On the other hand, consider tactile space, limiting ourselves to the sensations of a single finger, that is in sum to the assemblage of positions this finger can occupy. This tactile space that we shall analyze31 in the following section and which consequently I ask permission not to consider further for the moment, this tactile space, I say, has three dimensions. Why has space properly so called as many dimensions as tactile space and more than simple visual space? It is because touch does not operate at a distance, while vision does operate at a distance. These two assertions have the same meaning and we have just seen what this is.
 
Now I return to a point over which I passed rapidly in order not to interrupt the discussion. How do we know that the impressions made on our retina by A at the instant α and B at the instant β are transmitted by the same retinal fiber, although these impressions are qualitatively different? I have suggested a simple hypothesis, while adding that other hypotheses, decidedly more complex, would seem to me more probably true. Here then are these hypotheses, of which I have already said a word. How do we know that the impressions produced by the red object A at the instant α, and by the blue object B at the instant β, if these two objects have been imaged on the same point of the retina, have something in common? The simple hypothesis above made may be rejected and we may suppose that these two impressions, qualitatively different, are transmitted by two different though contiguous nervous fibers. What means have I then of knowing that these fibers are contiguous? It is probable that we should have none, if the eye were immovable. It is the movements of the eye which have told us that there is the same relation between the sensation of blue at the point A and the sensation of blue at the point B of the retina as between the sensation of red at the point A and the sensation of red at the point B. They have shown us, in fact, that the same movements, corresponding to the same muscular sensations, carry us from the first to the second, or from the third to the fourth. I do not emphasize these considerations, which belong, as one sees, to the question of local signs raised by Lotze.
3. Tactile Space
 
Thus I know how to recognize the identity of two points, the point occupied by A at the instant α and the point occupied by B at the instant β, but only on one condition, namely, that I have not budged between the instants α and β. That does not suffice for our object. Suppose, therefore, that I have moved in any manner in the interval34 between these two instants, how shall I know whether the point occupied by A at the instant α is identical with the point occupied by B at the instant β? I suppose that at the instant α, the object A was in contact with my first finger and that in the same way, at the instant β, the object B touches this first finger; but at the same time my muscular sense has told me that in the interval my body has moved. I have considered above two series of muscular sensations S and S′, and I have said it sometimes happens that we are led to consider two such series S and S′ as inverse one of the other, because we have often observed that when these two series succeed one another our primitive impressions are reestablished.
 
If then my muscular sense tells me that I have moved between the two instants α and β, but so as to feel successively the two series of muscular sensations S and S′ that I consider inverses35, I shall still conclude, just as if I had not budged, that the points occupied by A at the instant α and by B at the instant β are identical, if I ascertain that my first finger touches A at the instant α, and B at the instant β.
 
This solution is not yet completely satisfactory, as one will see. Let us see, in fact, how many dimensions it would make us attribute to space. I wish to compare the two points occupied by A and B at the instants α and β, or (what amounts to the same thing since I suppose that my finger touches A at the instant α and B at the instant β) I wish to compare the two points occupied by my finger at the two instants α and β. The sole means I use for this comparison is the series Σ of muscular sensations which have accompanied the movements of my body between these two instants. The different imaginable series Σ form evidently a physical continuum of which the number of dimensions is very great. Let us agree, as I have done, not to consider as distinct the two series Σ and Σ + S + S′, when S and S′ are inverses one of the other in the sense above given to this word; in spite of this agreement, the aggregate of distinct series Σ will still form a physical continuum and the number of dimensions will be less but still very great.
 
To each of these series Σ corresponds a point of space; to two series Σ and Σ′ thus correspond two points M and M′. The means we have hitherto used enable us to recognize that M and M′ are not distinct in two cases: (1) if Σ is identical with Σ′; (2) if Σ′ = Σ + S + S′, S and S′ being inverses one of the other. If in all the other cases we should regard M and M′ as distinct, the manifold of points would have as many dimensions as the aggregate of distinct series Σ, that is, much more than three.
 
For those who already know geometry, the following explanation would be easily comprehensible. Among the imaginable series of muscular sensations, there are those which correspond to series of movements where the finger does not budge. I say that if one does not consider as distinct the series Σ and Σ + σ, where the series σ corresponds to movements where the finger does not budge, the aggregate of series will constitute a continuum of three dimensions, but that if one regards as distinct two series Σ and Σ′ unless Σ′ = Σ + S + S′, S and S′ being inverses, the aggregate of series will constitute a continuum of more than three dimensions.
 
In fact, let there be in space a surface A, on this surface a line B, on this line a point M. Let C0 be the aggregate of all series Σ. Let C1 be the aggregate of all the series Σ, such that at the end of corresponding movements the finger is found upon the surface A, and C2 or C3 the aggregate of series Σ such that at the end the finger is found on B, or at M. It is clear, first that C1 will constitute a cut which will divide C0, that C2 will be a cut which will divide C1, and C3 a cut which will divide C2. Thence it results, in accordance with our definitions, that if C3 is a continuum of n dimensions, C0 will be a physical continuum of n + 3 dimensions.
 
Therefore, let Σ and Σ′ = Σ + σ be two series forming part of C3; for both, at the end of the movements, the finger is found at M; thence results that at the beginning and at the end of the series σ the finger is at the same point M. This series σ is therefore one of those which correspond to movements where the finger does not budge. If Σ and Σ + σ are not regarded as distinct, all the series of C3 blend into one; therefore C3 will have 0 dimension, and C0 will have 3, as I wished to prove. If, on the contrary, I do not regard Σ and Σ + σ as blending (unless σ = S + S′, S and S′ being inverses), it is clear that C3 will contain a great number of series of distinct sensations; because, without the finger budging36, the body may take a multitude of different attitudes. Then C3 will form a continuum and C0 will have more than three dimensions, and this also I wished to prove.
 
We who do not yet know geometry can not reason in this way; we can only verify. But then a question arises; how, before knowing geometry, have we been led to distinguish from the others these series σ where the finger does not budge? It is, in fact, only after having made this distinction that we could be led to regard Σ and Σ + σ as identical, and it is on this condition alone, as we have just seen, that we can arrive at space of three dimensions.
 
We are led to distinguish the series σ, because it often happens that when we have executed the movements which correspond to these series σ of muscular sensations, the tactile sensations which are transmitted to us by the nerve of the finger that we have called the first finger, persist and are not altered by these movements. Experience alone tells us that and it alone could tell us.
 
If we have distinguished37 the series of muscular sensations S + S′ formed by the union of two inverse series, it is because they preserve the totality of our impressions; if now we distinguish the series σ, it is because they preserve certain of our impressions. (When I say that a series of muscular sensations S ‘preserves’ one of our impressions A, I mean that we ascertain that if we feel the impression A, then the muscular sensations S, we still feel the impression A after these sensations S.)
 
I have said above it often happens that the series σ do not alter the tactile impressions felt by our first finger; I said often, I did not say always. This it is that we express in our ordinary language by saying that the tactile impressions would not be altered if the finger has not moved, on the condition that neither has the object A, which was in contact with this finger, moved. Before knowing geometry, we could not give this explanation; all we could do is to ascertain that the impression often persists, but not always.
 
But that the impression often continues is enough to make the series σ appear remarkable38 to us, to lead us to put in the same class the series Σ and Σ + σ, and hence not regard them as distinct. Under these conditions we have seen that they will engender a physical continuum of three dimensions.
 
Behold39 then a space of three dimensions engendered40 by my first finger. Each of my fingers will create one like it. It remains to consider how we are led to regard them as identical with visual space, as identical with geometric space.
 
But one reflection before going further; according to the foregoing, we know the points of space, or more generally the final situation of our body, only by the series of muscular sensations revealing to us the movements which have carried us from a certain initial situation to this final situation. But it is clear that this final situation will depend, on the one hand, upon these movements and, on the other hand, upon the initial situation from which we set out. Now these movements are revealed to us by our muscular sensations; but nothing tells us the initial situation; nothing can distinguish it for us from all the other possible situations. This puts well in evidence the essential relativity of space.
4. Identity of the Different Spaces
 
We are therefore led to compare the two continua C and C′ engendered, for instance, one by my first finger D, the other by my second finger D′. These two physical continua both have three dimensions. To each element of the continuum C, or, if you prefer, to each point of the first tactile space, corresponds a series of muscular sensations Σ, which carry me from a certain initial situation to a certain final situation.8 Moreover, the same point of this first space will correspond to Σ and Σ + σ, if σ is a series of which we know that it does not make the finger D move.
 
8 In place of saying that we refer space to axes rigidly41 bound to our body, perhaps it would be better to say, in conformity43 to what precedes, that we refer it to axes rigidly bound to the initial situation of our body.
 
Similarly to each element of the continuum C′, or to each point of the second tactile space, corresponds a series of sensations Σ′, and the same point will correspond to Σ′ and to Σ′ + σ′, if σ′ is a series which does not make the finger D′ move.
 
What makes us distinguish the various series designated σ from those called σ′ is that the first do not alter the tactile impressions felt by the finger D and the second preserve those the finger D′ feels.
 
Now see what we ascertain: in the beginning my finger D′ feels a sensation A′; I make movements which produce muscular sensations S; my finger D feels the impression A; I make movements which produce a series of sensations σ; my finger D continues to feel the impression A, since this is the characteristic property of the series σ; I then make movements which produce the series S′ of muscular sensations, inverse to S in the sense above given to this word. I ascertain then that my finger D′ feels anew the impression A′. (It is of course understood that S has been suitably chosen.)
 
This means that the series S + σ + S′, preserving the tactile impressions of the finger D′, is one of the series I have called σ′. Inversely44, if one takes any series σ′, S′ + σ′ + S will be one of the series that we call σ′.
 
Thus if S is suitably chosen, S + σ + S′ will be a series σ′, and by making σ vary in all possible ways, we shall obtain all the possible series σ′.
 
Not yet knowing geometry, we limit ourselves to verifying all that, but here is how those who know geometry would explain the fact. In the beginning my finger D′ is at the point M, in contact with the object a, which makes it feel the impression A′. I make the movements corresponding to the series S; I have said that this series should be suitably chosen, I should so make this choice that these movements carry the finger D to the point originally occupied by the finger D′, that is, to the point M; this finger D will thus be in contact with the object a, which will make it feel the impression A.
 
I then make the movements corresponding to the series σ; in these movements, by hypothesis, the position of the finger D does not change, this finger therefore remains in contact with the object a and continues to feel the impression A. Finally I make the movements corresponding to the series S′. As S′ is inverse to S, these movements carry the finger D′ to the point previously45 occupied by the finger D, that is, to the point M. If, as may be supposed, the object a has not budged, this finger D′ will be in contact with this object and will feel anew the impression A′. . . . Q.E.D.
 
Let us see the consequences. I consider a series of muscular sensations Σ. To this series will correspond a point M of the first tactile space. Now take again the two series S and S′, inverses of one another, of which we have just spoken. To the series S + Σ + S′ will correspond a point N of the second tactile space, since to any series of muscular sensations corresponds, as we have said, a point, whether in the first space or in the second.
 
I am going to consider the two points N and M, thus defined, as corresponding. What authorizes46 me so to do? For this correspondence to be admissible, it is necessary that if two points M and M′, corresponding in the first space to two series Σ and Σ′, are identical, so also are the two corresponding points of the second space N and N′, that is, the two points which correspond to the two series S + Σ + S′ and S + Σ′ + S′. Now we shall see that this condition is fulfilled.
 
First a remark. As S and S′ are inverses of one another, we shall have S + S′ = 0, and consequently S + S′ + Σ = Σ + S + S′ = Σ, or again Σ + S + S′ + Σ′ = Σ + Σ′; but it does not follow that we have S + Σ + S′ = Σ; because, though we have used the addition sign to represent the succession of our sensations, it is clear that the order of this succession is not indifferent: we can not, therefore, as in ordinary addition, invert47 the order of the terms; to use abridged48 language, our operations are associative, but not commutative.
 
That fixed, in order that Σ and Σ′ should correspond to the same point M = M′ of the first space, it is necessary and sufficient for us to have Σ′ = Σ + σ. We shall then have: S + Σ′ + S′ = S + Σ + σ + S′ = S + Σ + S′ + S + σ + S′.
 
But we have just ascertained that S + σ + S′ was one of the series σ′. We shall therefore have: S + Σ′ + S′ = S + Σ + S′ + σ′, which means that the series S + Σ′ + S′ and S + Σ + S′ correspond to the same point N = N′ of the second space. Q.E.D.
 
Our two spaces therefore correspond point for point; they can be ‘transformed’ one into the other; they are isomorphic. How are we led to conclude thence that they are identical?
 
Consider the two series σ and S + σ + S′ = σ′. I have said that often, but not always, the series σ preserves the tactile impression A felt by the finger D; and similarly it often happens, but not always, that the series σ′ preserves the tactile impression A′ felt by the finger D′. Now I ascertain that it happens very often (that is, much more often than what I have just called ‘often’) that when the series σ has preserved the impression A of the finger D, the series σ′ preserves at the same time the impression A′ of the finger D′; and, inversely, that if the first impression is altered, the second is likewise. That happens very often, but not always.
 
We interpret this experimental fact by saying that the unknown object a which gives the impression A to the finger D is identical with the unknown object a′ which gives the impression A′ to the finger D′. And in fact when the first object moves, which the disappearance49 of the impression A tells us, the second likewise moves, since the impression A′ disappears likewise. When the first object remains motionless, the second remains motionless. If these two objects are identical, as the first is at the point M of the first space and the second at the point N of the second space, these two points are identical. This is how we are led to regard these two spaces as identical; or better, this is what we mean when we say that they are identical.
 
What we have just said of the identity of the two tactile spaces makes unnecessary our discussing the question of the identity of tactile space and visual space, which could be treated in the same way.
5. Space and Empiricism
 
It seems that I am about to be led to conclusions in conformity with empiristic ideas. I have, in fact, sought to put in evidence the r?le of experience and to analyze the experimental facts which intervene in the genesis of space of three dimensions. But whatever may be the importance of these facts, there is one thing we must not forget and to which besides I have more than once called attention. These experimental facts are often verified but not always. That evidently does not mean that space has often three dimensions, but not always.
 
I know well that it is easy to save oneself and that, if the facts do not verify, it will be easily explained by saying that the exterior50 objects have moved. If experience succeeds, we say that it teaches us about space; if it does not succeed, we hie to exterior objects which we accuse of having moved; in other words, if it does not succeed, it is given a fillip.
 
These fillips are legitimate51; I do not refuse to admit them; but they suffice to tell us that the properties of space are not experimental truths, properly so called. If we had wished to verify other laws, we could have succeeded also, by giving other analogous52 fillips. Should we not always have been able to justify53 these fillips by the same reasons? One could at most have said to us: ‘Your fillips are doubtless legitimate, but you abuse them; why move the exterior objects so often?’
 
To sum up, experience does not prove to us that space has three dimensions; it only proves to us that it is convenient to attribute three to it, because thus the number of fillips is reduced to a minimum.
 
I will add that experience brings us into contact only with representative space, which is a physical continuum, never with geometric space, which is a mathematical continuum. At the very most it would appear to tell us that it is convenient to give to geometric space three dimensions, so that it may have as many as representative space.
 
The empiric question may be put under another form. Is it impossible to conceive physical phenomena54, the mechanical phenomena, for example, otherwise than in space of three dimensions? We should thus have an objective experimental proof, so to speak, independent of our physiology55, of our modes of representation.
 
But it is not so; I shall not here discuss the question completely, I shall confine myself to recalling the striking example given us by the mechanics of Hertz. You know that the great physicist56 did not believe in the existence of forces, properly so called; he supposed that visible material points are subjected to certain invisible bonds which join them to other invisible points and that it is the effect of these invisible bonds that we attribute to forces.
 
But that is only a part of his ideas. Suppose a system formed of n material points, visible or not; that will give in all 3n coordinates57; let us regard them as the coordinates of a single point in space of 3n dimensions. This single point would be constrained58 to remain upon a surface (of any number of dimensions < 3n) in virtue59 of the bonds of which we have just spoken; to go on this surface from one point to another, it would always take the shortest way; this would be the single principle which would sum up all mechanics.
 
Whatever should be thought of this hypothesis, whether we be allured60 by its simplicity61, or repelled62 by its artificial character, the simple fact that Hertz was able to conceive it, and to regard it as more convenient than our habitual63 hypotheses, suffices to prove that our ordinary ideas, and, in particular, the three dimensions of space, are in no wise imposed upon mechanics with an invincible64 force.
6. Mind and Space
 
Experience, therefore, has played only a single r?le, it has served as occasion. But this r?le was none the less very important; and I have thought it necessary to give it prominence65. This r?le would have been useless if there existed an a priori form imposing66 itself upon our sensitivity, and which was space of three dimensions.
 
Does this form exist, or, if you choose, can we represent to ourselves space of more than three dimensions? And first what does this question mean? In the true sense of the word, it is clear that we can not represent to ourselves space of four, nor space of three, dimensions; we can not first represent them to ourselves empty, and no more can we represent to ourselves an object either in space of four, or in space of three, dimensions: (1) Because these spaces are both infinite and we can not represent to ourselves a figure in space, that is, the part in the whole, without representing the whole, and that is impossible, because it is infinite; (2) because these spaces are both mathematical continua, and we can represent to ourselves only the physical continuum; (3) because these spaces are both homogeneous, and the frames in which we enclose our sensations, being limited, can not be homogeneous.
 
Thus the question put can only be understood in one way; is it possible to imagine that, the results of the experiences related above having been different, we might have been led to attribute to space more than three dimensions; to imagine, for instance, that the sensation of accommodation might not be constantly in accord with the sensation of convergence of the eyes; or indeed that the experiences of which we have spoken in § 2, and of which we express the result by saying ‘that touch does not operate at a distance,’ might have led us to an inverse conclusion.
 
And then yes evidently that is possible; from the moment one imagines an experience, one imagines just thereby67 the two contrary results it may give. That is possible, but that is difficult, because we have to overcome a multitude of associations of ideas, which are the fruit of a long personal experience and of the still longer experience of the race. Is it these associations (or at least those of them that we have inherited from our ancestors), which constitute this a priori form of which it is said that we have pure intuition? Then I do not see why one should declare it refractory68 to analysis and should deny me the right of investigating its origin.
 
When it is said that our sensations are ‘extended’ only one thing can be meant, that is that they are always associated with the idea of certain muscular sensations, corresponding to the movements which enable us to reach the object which causes them, which enable us, in other words, to defend ourselves against it. And it is just because this association is useful for the defense69 of the organism, that it is so old in the history of the species and that it seems to us indestructible. Nevertheless, it is only an association and we can conceive that it may be broken; so that we may not say that sensation can not enter consciousness without entering in space, but that in fact it does not enter consciousness without entering in space, which means, without being entangled70 in this association.
 
No more can I understand one’s saying that the idea of time is logically subsequent to space, since we can represent it to ourselves only under the form of a straight line; as well say that time is logically subsequent to the cultivation71 of the prairies, since it is usually represented armed with a scythe72. That one can not represent to himself simultaneously the different parts of time, goes without saying, since the essential character of these parts is precisely73 not to be simultaneous. That does not mean that we have not the intuition of time. So far as that goes, no more should we have that of space, because neither can we represent it, in the proper sense of the word, for the reasons I have mentioned. What we represent to ourselves under the name of straight is a crude image which as ill resembles the geometric straight as it does time itself.
 
Why has it been said that every attempt to give a fourth dimension to space always carries this one back to one of the other three? It is easy to understand. Consider our muscular sensations and the ‘series’ they may form. In consequence of numerous experiences, the ideas of these series are associated together in a very complex woof, our series are classed. Allow me, for convenience of language, to express my thought in a way altogether crude and even inexact by saying that our series of muscular sensations are classed in three classes corresponding to the three dimensions of space. Of course this classification is much more complicated than that, but that will suffice to make my reasoning understood. If I wish to imagine a fourth dimension, I shall suppose another series of muscular sensations, making part of a fourth class. But as all my muscular sensations have already been classed in one of the three pre-existent classes, I can only represent to myself a series belonging to one of these three classes, so that my fourth dimension is carried back to one of the other three.
 
What does that prove? This: that it would have been necessary first to destroy the old classification and replace it by a new one in which the series of muscular sensations should have been distributed into four classes. The difficulty would have disappeared.
 
It is presented sometimes under a more striking form. Suppose I am enclosed in a chamber74 between the six impassable boundaries formed by the four walls, the floor and the ceiling; it will be impossible for me to get out and to imagine my getting out. Pardon, can you not imagine that the door opens, or that two of these walls separate? But of course, you answer, one must suppose that these walls remain immovable. Yes, but it is evident that I have the right to move; and then the walls that we suppose absolutely at rest will be in motion with regard to me. Yes, but such a relative motion can not be arbitrary; when objects are at rest, their relative motion with regard to any axes is that of a rigid42 solid; now, the apparent motions that you imagine are not in conformity with the laws of motion of a rigid solid. Yes, but it is experience which has taught us the laws of motion of a rigid solid; nothing would prevent our imagining them different. To sum up, for me to imagine that I get out of my prison, I have only to imagine that the walls seem to open, when I move.
 
I believe, therefore, that if by space is understood a mathematical continuum of three dimensions, were it otherwise amorphous75, it is the mind which constructs it, but it does not construct it out of nothing; it needs materials and models. These materials, like these models, preexist within it. But there is not a single model which is imposed upon it; it has choice; it may choose, for instance, between space of four and space of three dimensions. What then is the r?le of experience? It gives the indications following which the choice is made.
 
Another thing: whence does space get its quantitative76 character? It comes from the r?le which the series of muscular sensations play in its genesis. These are series which may repeat themselves, and it is from their repetition that number comes; it is because they can repeat themselves indefinitely that space is infinite. And finally we have seen, at the end of section 3, that it is also because of this that space is relative. So it is repetition which has given to space its essential characteristics; now, repetition supposes time; this is enough to tell that time is logically anterior77 to space.
7. R?le of the Semicircular Canals
 
I have not hitherto spoken of the r?le of certain organs to which the physiologists79 attribute with reason a capital importance, I mean the semicircular canals. Numerous experiments have sufficiently80 shown that these canals are necessary to our sense of orientation81; but the physiologists are not entirely in accord; two opposing theories have been proposed, that of Mach-Delage and that of M. de Cyon.
 
M. de Cyon is a physiologist78 who has made his name illustrious by important discoveries on the innervation of the heart; I can not, however, agree with his ideas on the question before us. Not being a physiologist, I hesitate to criticize the experiments he has directed against the adverse82 theory of Mach-Delage; it seems to me, however, that they are not convincing, because in many of them the total pressure was made to vary in one of the canals, while, physiologically83, what varies is the difference between the pressures on the two extremities84 of the canal; in others the organs were subjected to profound lesions, which must alter their functions.
 
Besides, this is not important; the experiments, if they were irreproachable85, might be convincing against the old theory. They would not be convincing for the new theory. In fact, if I have rightly understood the theory, my explaining it will be enough for one to understand that it is impossible to conceive of an experiment confirming it.
 
The three pairs of canals would have as sole function to tell us that space has three dimensions. Japanese mice have only two pairs of canals; they believe, it would seem, that space has only two dimensions, and they manifest this opinion in the strangest way; they put themselves in a circle, and, so ordered, they spin rapidly around. The lampreys, having only one pair of canals, believe that space has only one dimension, but their manifestations86 are less turbulent.
 
It is evident that such a theory is inadmissible. The sense-organs are designed to tell us of changes which happen in the exterior world. We could not understand why the Creator should have given us organs destined87 to cry without cease: Remember that space has three dimensions, since the number of these three dimensions is not subject to change.
 
We must, therefore, come back to the theory of Mach-Delage. What the nerves of the canals can tell us is the difference of pressure on the two extremities of the same canal, and thereby: (1) the direction of the vertical88 with regard to three axes rigidly bound to the head; (2) the three components89 of the acceleration90 of translation of the center of gravity of the head; (3) the centrifugal forces developed by the rotation91 of the head; (4) the acceleration of the motion of rotation of the head.
 
It follows from the experiments of M. Delage that it is this last indication which is much the most important; doubtless because the nerves are less sensible to the difference of pressure itself than to the brusque variations of this difference. The first three indications may thus be neglected.
 
Knowing the acceleration of the motion of rotation of the head at each instant, we deduce from it, by an unconscious integration92, the final orientation of the head, referred to a certain initial orientation taken as origin. The circular canals contribute, therefore, to inform us of the movements that we have executed, and that on the same ground as the muscular sensations. When, therefore, above we speak of the series S or of the series Σ, we should say, not that these were series of muscular sensations alone, but that they were series at the same time of muscular sensations and of sensations due to the semicircular canals. Apart from this addition, we should have nothing to change in what precedes.
 
In the series S and Σ, these sensations of the semicircular canals evidently hold a very important place. Yet alone they would not suffice, because they can tell us only of the movements of the head; they tell us nothing of the relative movements of the body or of the members in regard to the head. And more, it seems that they tell us only of the rotations93 of the head and not of the translations it may undergo.

点击收听单词发音收听单词发音  

1 displacements 9e66611008a27467702e6346e1664419     
n.取代( displacement的名词复数 );替代;移位;免职
参考例句:
  • The laws of physics are symmetrical for translational displacements. 物理定律对平移是对称的。 来自辞典例句
  • We encounter only displacements of the first type. 我们只遇到第一类的驱替。 来自辞典例句
2 displacement T98yU     
n.移置,取代,位移,排水量
参考例句:
  • They said that time is the feeling of spatial displacement.他们说时间是空间位移的感觉。
  • The displacement of all my energy into caring for the baby.我所有精力都放在了照顾宝宝上。
3 briefly 9Styo     
adv.简单地,简短地
参考例句:
  • I want to touch briefly on another aspect of the problem.我想简单地谈一下这个问题的另一方面。
  • He was kidnapped and briefly detained by a terrorist group.他被一个恐怖组织绑架并短暂拘禁。
4 ascertain WNVyN     
vt.发现,确定,查明,弄清
参考例句:
  • It's difficult to ascertain the coal deposits.煤储量很难探明。
  • We must ascertain the responsibility in light of different situtations.我们必须根据不同情况判定责任。
5 discriminated 94ae098f37db4e0c2240e83d29b5005a     
分别,辨别,区分( discriminate的过去式和过去分词 ); 歧视,有差别地对待
参考例句:
  • His great size discriminated him from his followers. 他的宽广身材使他不同于他的部下。
  • Should be a person that has second liver virus discriminated against? 一个患有乙肝病毒的人是不是就应该被人歧视?
6 aggregate cKOyE     
adj.总计的,集合的;n.总数;v.合计;集合
参考例句:
  • The football team had a low goal aggregate last season.这支足球队上个赛季的进球总数很少。
  • The money collected will aggregate a thousand dollars.进帐总额将达一千美元。
7 remains 1kMzTy     
n.剩余物,残留物;遗体,遗迹
参考例句:
  • He ate the remains of food hungrily.他狼吞虎咽地吃剩余的食物。
  • The remains of the meal were fed to the dog.残羹剩饭喂狗了。
8 primitive vSwz0     
adj.原始的;简单的;n.原(始)人,原始事物
参考例句:
  • It is a primitive instinct to flee a place of danger.逃离危险的地方是一种原始本能。
  • His book describes the march of the civilization of a primitive society.他的著作描述了一个原始社会的开化过程。
9 postulated 28ea70fa3a37cd78c20423a907408aaa     
v.假定,假设( postulate的过去式和过去分词 )
参考例句:
  • They postulated a 500-year lifespan for a plastic container. 他们假定塑料容器的寿命为500年。
  • Freud postulated that we all have a death instinct as well as a life instinct. 弗洛伊德曾假定我们所有人都有生存本能和死亡本能。 来自辞典例句
10 entirely entirely     
ad.全部地,完整地;完全地,彻底地
参考例句:
  • The fire was entirely caused by their neglect of duty. 那场火灾完全是由于他们失职而引起的。
  • His life was entirely given up to the educational work. 他的一生统统献给了教育工作。
11 fixed JsKzzj     
adj.固定的,不变的,准备好的;(计算机)固定的
参考例句:
  • Have you two fixed on a date for the wedding yet?你们俩选定婚期了吗?
  • Once the aim is fixed,we should not change it arbitrarily.目标一旦确定,我们就不应该随意改变。
12 detour blSzz     
n.绕行的路,迂回路;v.迂回,绕道
参考例句:
  • We made a detour to avoid the heavy traffic.我们绕道走,避开繁忙的交通。
  • He did not take the direct route to his home,but made a detour around the outskirts of the city.他没有直接回家,而是绕到市郊兜了个圈子。
13 fibers 421d63991f1d1fc8826d6e71d5e15f53     
光纤( fiber的名词复数 ); (织物的)质地; 纤维,纤维物质
参考例句:
  • Thesolution of collagen-PVA was wet spined with the sodium sulfate as coagulant and collagen-PVA composite fibers were prepared. 在此基础上,以硫酸钠为凝固剂,对胶原-PVA共混溶液进行湿法纺丝,制备了胶原-PVA复合纤维。
  • Sympathetic fibers are distributed to all regions of the heart. 交感神经纤维分布于心脏的所有部分。
14 fiber NzAye     
n.纤维,纤维质
参考例句:
  • The basic structural unit of yarn is the fiber.纤维是纱的基本结构单元。
  • The material must be free of fiber clumps.这种材料必须无纤维块。
15 tactile bGkyv     
adj.触觉的,有触觉的,能触知的
参考例句:
  • Norris is an expert in the tactile and the tangible.诺里斯创作最精到之处便是,他描绘的人物使人看得见摸得着。
  • Tactile communication uses touch rather than sight or hearing.触觉交流,是用触摸感觉,而不是用看或听来感觉。
16 qualitatively 5ca9292f7a0c1ddbef340e3c76a7c17b     
质量上
参考例句:
  • In other words, you are to analyze them quantitatively and qualitatively. 换句话说,你们要对它们进行量和质的分析。
  • Electric charge may be detected qualitatively by sprinkling or blowing indicating powders. 静电荷可以用撒布指示粉剂的方法,予以探测。
17 purely 8Sqxf     
adv.纯粹地,完全地
参考例句:
  • I helped him purely and simply out of friendship.我帮他纯粹是出于友情。
  • This disproves the theory that children are purely imitative.这证明认为儿童只会单纯地模仿的理论是站不住脚的。
18 luminous 98ez5     
adj.发光的,发亮的;光明的;明白易懂的;有启发的
参考例句:
  • There are luminous knobs on all the doors in my house.我家所有门上都安有夜光把手。
  • Most clocks and watches in this shop are in luminous paint.这家商店出售的大多数钟表都涂了发光漆。
19 simultaneously 4iBz1o     
adv.同时发生地,同时进行地
参考例句:
  • The radar beam can track a number of targets almost simultaneously.雷达波几乎可以同时追着多个目标。
  • The Windows allow a computer user to execute multiple programs simultaneously.Windows允许计算机用户同时运行多个程序。
20 considerably 0YWyQ     
adv.极大地;相当大地;在很大程度上
参考例句:
  • The economic situation has changed considerably.经济形势已发生了相当大的变化。
  • The gap has narrowed considerably.分歧大大缩小了。
21 ascertained e6de5c3a87917771a9555db9cf4de019     
v.弄清,确定,查明( ascertain的过去式和过去分词 )
参考例句:
  • The previously unidentified objects have now been definitely ascertained as being satellites. 原来所说的不明飞行物现在已证实是卫星。 来自《简明英汉词典》
  • I ascertained that she was dead. 我断定她已经死了。 来自《简明英汉词典》
22 enunciated 2f41d5ea8e829724adf2361074d6f0f9     
v.(清晰地)发音( enunciate的过去式和过去分词 );确切地说明
参考例句:
  • She enunciated each word slowly and carefully. 她每个字都念得又慢又仔细。
  • His voice, cold and perfectly enunciated, switched them like a birch branch. 他的话口气冰冷,一字一板,有如给了他们劈面一鞭。 来自辞典例句
23 budge eSRy5     
v.移动一点儿;改变立场
参考例句:
  • We tried to lift the rock but it wouldn't budge.我们试图把大石头抬起来,但它连动都没动一下。
  • She wouldn't budge on the issue.她在这个问题上不肯让步。
24 afterward fK6y3     
adv.后来;以后
参考例句:
  • Let's go to the theatre first and eat afterward. 让我们先去看戏,然后吃饭。
  • Afterward,the boy became a very famous artist.后来,这男孩成为一个很有名的艺术家。
25 budged acd2fdcd1af9cf1b3478f896dc0484cf     
v.(使)稍微移动( budge的过去式和过去分词 );(使)改变主意,(使)让步
参考例句:
  • Old Bosc had never budged an inch--he was totally indifferent. 老包斯克一直连动也没有动,他全然无所谓。 来自辞典例句
  • Nobody budged you an inch. 别人一丁点儿都算计不了你。 来自辞典例句
26 inverse GR6zs     
adj.相反的,倒转的,反转的;n.相反之物;v.倒转
参考例句:
  • Evil is the inverse of good.恶是善的反面。
  • When the direct approach failed he tried the inverse.当直接方法失败时,他尝试相反的做法。
27 compensate AXky7     
vt.补偿,赔偿;酬报 vi.弥补;补偿;抵消
参考例句:
  • She used her good looks to compensate her lack of intelligence. 她利用她漂亮的外表来弥补智力的不足。
  • Nothing can compensate for the loss of one's health. 一个人失去了键康是不可弥补的。
28 anticipation iMTyh     
n.预期,预料,期望
参考例句:
  • We waited at the station in anticipation of her arrival.我们在车站等着,期待她的到来。
  • The animals grew restless as if in anticipation of an earthquake.各种动物都变得焦躁不安,像是感到了地震即将发生。
29 contented Gvxzof     
adj.满意的,安心的,知足的
参考例句:
  • He won't be contented until he's upset everyone in the office.不把办公室里的每个人弄得心烦意乱他就不会满足。
  • The people are making a good living and are contented,each in his station.人民安居乐业。
30 criteria vafyC     
n.标准
参考例句:
  • The main criterion is value for money.主要的标准是钱要用得划算。
  • There are strict criteria for inclusion in the competition.参赛的标准很严格。
31 analyze RwUzm     
vt.分析,解析 (=analyse)
参考例句:
  • We should analyze the cause and effect of this event.我们应该分析这场事变的因果。
  • The teacher tried to analyze the cause of our failure.老师设法分析我们失败的原因。
32 analyzed 483f1acae53789fbee273a644fdcda80     
v.分析( analyze的过去式和过去分词 );分解;解释;对…进行心理分析
参考例句:
  • The doctors analyzed the blood sample for anemia. 医生们分析了贫血的血样。 来自《简明英汉词典》
  • The young man did not analyze the process of his captivation and enrapturement, for love to him was a mystery and could not be analyzed. 这年轻人没有分析自己蛊惑著迷的过程,因为对他来说,爱是个不可分析的迷。 来自《简明英汉词典》
33 engender 3miyT     
v.产生,引起
参考例句:
  • A policy like that tends to engender a sense of acceptance,and the research literature suggests this leads to greater innovation.一个能够使员工产生认同感的政策,研究表明这会走向更伟大的创新。
  • The sense of injustice they engender is a threat to economic and political security.它们造成的不公平感是对经济和政治安全的威胁。
34 interval 85kxY     
n.间隔,间距;幕间休息,中场休息
参考例句:
  • The interval between the two trees measures 40 feet.这两棵树的间隔是40英尺。
  • There was a long interval before he anwsered the telephone.隔了好久他才回了电话。
35 inverses 27aaf468caa744923b6a11a67a48ba65     
vt.使倒转(inverse的第三人称单数形式)
参考例句:
  • Meanwhile,the Drazin inverses of PQ and P-Q are established. 同时,给出正交投影的积PQ和差P-Q的Drazin逆的表达式。 来自互联网
  • Watch the spreads i. e. don't be bullish if inverses are narrowing. 注意多空套做。也就是说假如顺好在加大不要看多。 来自互联网
36 budging 7d6a7b3c5d687a6190de9841c520110b     
v.(使)稍微移动( budge的现在分词 );(使)改变主意,(使)让步
参考例句:
  • Give it up, plumber. She's not budging. 别费劲了,水管工。她不会改变主意的。 来自互联网
  • I wondered how Albert who showed no intention of budging, felt about Leopold's desertion. 对于从未有迁徙打算的艾伯特来说,我不知道它会怎样看待利奥波德这样弃它而去呢。 来自互联网
37 distinguished wu9z3v     
adj.卓越的,杰出的,著名的
参考例句:
  • Elephants are distinguished from other animals by their long noses.大象以其长长的鼻子显示出与其他动物的不同。
  • A banquet was given in honor of the distinguished guests.宴会是为了向贵宾们致敬而举行的。
38 remarkable 8Vbx6     
adj.显著的,异常的,非凡的,值得注意的
参考例句:
  • She has made remarkable headway in her writing skills.她在写作技巧方面有了长足进步。
  • These cars are remarkable for the quietness of their engines.这些汽车因发动机没有噪音而不同凡响。
39 behold jQKy9     
v.看,注视,看到
参考例句:
  • The industry of these little ants is wonderful to behold.这些小蚂蚁辛勤劳动的样子看上去真令人惊叹。
  • The sunrise at the seaside was quite a sight to behold.海滨日出真是个奇景。
40 engendered 9ea62fba28ee7e2bac621ac2c571239e     
v.产生(某形势或状况),造成,引起( engender的过去式和过去分词 )
参考例句:
  • The issue engendered controversy. 这个问题引起了争论。
  • The meeting engendered several quarrels. 这次会议发生了几次争吵。 来自《简明英汉词典》
41 rigidly hjezpo     
adv.刻板地,僵化地
参考例句:
  • Life today is rigidly compartmentalized into work and leisure. 当今的生活被严格划分为工作和休闲两部分。
  • The curriculum is rigidly prescribed from an early age. 自儿童时起即已开始有严格的课程设置。
42 rigid jDPyf     
adj.严格的,死板的;刚硬的,僵硬的
参考例句:
  • She became as rigid as adamant.她变得如顽石般的固执。
  • The examination was so rigid that nearly all aspirants were ruled out.考试很严,几乎所有的考生都被淘汰了。
43 conformity Hpuz9     
n.一致,遵从,顺从
参考例句:
  • Was his action in conformity with the law?他的行动是否合法?
  • The plan was made in conformity with his views.计划仍按他的意见制定。
44 inversely t4Sx6     
adj.相反的
参考例句:
  • Pressure varies directly with temperature and inversely with volume. 压力随温度成正比例变化,与容积成反比例变化。 来自《简明英汉词典》
  • The amount of force needed is inversely proportional to the rigidity of the material. 需要的力度与材料的硬度成反比。 来自《简明英汉词典》
45 previously bkzzzC     
adv.以前,先前(地)
参考例句:
  • The bicycle tyre blew out at a previously damaged point.自行车胎在以前损坏过的地方又爆开了。
  • Let me digress for a moment and explain what had happened previously.让我岔开一会儿,解释原先发生了什么。
46 authorizes 716083de28a1fe3e0ba0233e695bce8c     
授权,批准,委托( authorize的名词复数 )
参考例句:
  • The dictionary authorizes the two spellings 'traveler' and 'traveller'. 字典裁定traveler和traveller两种拼法都对。
  • The dictionary authorizes the two spellings "honor" and "honour.". 字典裁定 honor 及 honour 两种拼法均可。
47 invert HRuzr     
vt.使反转,使颠倒,使转化
参考例句:
  • She catch the insect by invert her cup over it.她把杯子倒扣在昆虫上,将它逮住了。
  • Invert the cake onto a cooling rack.把蛋糕倒扣在冷却架上。
48 abridged 47f00a3da9b4a6df1c48709a41fd43e5     
削减的,删节的
参考例句:
  • The rights of citizens must not be abridged without proper cause. 没有正当理由,不能擅自剥夺公民的权利。
  • The play was abridged for TV. 剧本经过节略,以拍摄电视片。
49 disappearance ouEx5     
n.消失,消散,失踪
参考例句:
  • He was hard put to it to explain her disappearance.他难以说明她为什么不见了。
  • Her disappearance gave rise to the wildest rumours.她失踪一事引起了各种流言蜚语。
50 exterior LlYyr     
adj.外部的,外在的;表面的
参考例句:
  • The seed has a hard exterior covering.这种子外壳很硬。
  • We are painting the exterior wall of the house.我们正在给房子的外墙涂漆。
51 legitimate L9ZzJ     
adj.合法的,合理的,合乎逻辑的;v.使合法
参考例句:
  • Sickness is a legitimate reason for asking for leave.生病是请假的一个正当的理由。
  • That's a perfectly legitimate fear.怀有这种恐惧完全在情理之中。
52 analogous aLdyQ     
adj.相似的;类似的
参考例句:
  • The two situations are roughly analogous.两种情況大致相似。
  • The company is in a position closely analogous to that of its main rival.该公司与主要竞争对手的处境极为相似。
53 justify j3DxR     
vt.证明…正当(或有理),为…辩护
参考例句:
  • He tried to justify his absence with lame excuses.他想用站不住脚的借口为自己的缺席辩解。
  • Can you justify your rude behavior to me?你能向我证明你的粗野行为是有道理的吗?
54 phenomena 8N9xp     
n.现象
参考例句:
  • Ade couldn't relate the phenomena with any theory he knew.艾德无法用他所知道的任何理论来解释这种现象。
  • The object of these experiments was to find the connection,if any,between the two phenomena.这些实验的目的就是探索这两种现象之间的联系,如果存在着任何联系的话。
55 physiology uAfyL     
n.生理学,生理机能
参考例句:
  • He bought a book about physiology.他买了一本生理学方面的书。
  • He was awarded the Nobel Prize for achievements in physiology.他因生理学方面的建树而被授予诺贝尔奖。
56 physicist oNqx4     
n.物理学家,研究物理学的人
参考例句:
  • He is a physicist of the first rank.他是一流的物理学家。
  • The successful physicist never puts on airs.这位卓有成就的物理学家从不摆架子。
57 coordinates 8387d77faaaa65484f5631d9f9d20bfc     
n.相配之衣物;坐标( coordinate的名词复数 );(颜色协调的)配套服装;[复数]女套服;同等重要的人(或物)v.使协调,使调和( coordinate的第三人称单数 );协调;协同;成为同等
参考例句:
  • The town coordinates on this map are 695037. 该镇在这幅地图上的坐标是695037。 来自《简明英汉词典》
  • The UN Office for the Coordination of Humanitarian Affairs, headed by the Emergency Relief Coordinator, coordinates all UN emergency relief. 联合国人道主义事务协调厅在紧急救济协调员领导下,负责协调联合国的所有紧急救济工作。 来自《简明英汉词典》
58 constrained YvbzqU     
adj.束缚的,节制的
参考例句:
  • The evidence was so compelling that he felt constrained to accept it. 证据是那样的令人折服,他觉得不得不接受。
  • I feel constrained to write and ask for your forgiveness. 我不得不写信请你原谅。
59 virtue BpqyH     
n.德行,美德;贞操;优点;功效,效力
参考例句:
  • He was considered to be a paragon of virtue.他被认为是品德尽善尽美的典范。
  • You need to decorate your mind with virtue.你应该用德行美化心灵。
60 allured 20660ad1de0bc3cf3f242f7df8641b3e     
诱引,吸引( allure的过去式和过去分词 )
参考例句:
  • They allured her into a snare. 他们诱她落入圈套。
  • Many settlers were allured by promises of easy wealth. 很多安家落户的人都是受了诱惑,以为转眼就能发财而来的。
61 simplicity Vryyv     
n.简单,简易;朴素;直率,单纯
参考例句:
  • She dressed with elegant simplicity.她穿着朴素高雅。
  • The beauty of this plan is its simplicity.简明扼要是这个计划的一大特点。
62 repelled 1f6f5c5c87abe7bd26a5c5deddd88c92     
v.击退( repel的过去式和过去分词 );使厌恶;排斥;推开
参考例句:
  • They repelled the enemy. 他们击退了敌军。 来自《简明英汉词典》
  • The minister tremulously, but decidedly, repelled the old man's arm. 而丁梅斯代尔牧师却哆里哆嗦地断然推开了那老人的胳臂。 来自英汉文学 - 红字
63 habitual x5Pyp     
adj.习惯性的;通常的,惯常的
参考例句:
  • He is a habitual criminal.他是一个惯犯。
  • They are habitual visitors to our house.他们是我家的常客。
64 invincible 9xMyc     
adj.不可征服的,难以制服的
参考例句:
  • This football team was once reputed to be invincible.这支足球队曾被誉为无敌的劲旅。
  • The workers are invincible as long as they hold together.只要工人团结一致,他们就是不可战胜的。
65 prominence a0Mzw     
n.突出;显著;杰出;重要
参考例句:
  • He came to prominence during the World Cup in Italy.他在意大利的世界杯赛中声名鹊起。
  • This young fashion designer is rising to prominence.这位年轻的时装设计师的声望越来越高。
66 imposing 8q9zcB     
adj.使人难忘的,壮丽的,堂皇的,雄伟的
参考例句:
  • The fortress is an imposing building.这座城堡是一座宏伟的建筑。
  • He has lost his imposing appearance.他已失去堂堂仪表。
67 thereby Sokwv     
adv.因此,从而
参考例句:
  • I have never been to that city,,ereby I don't know much about it.我从未去过那座城市,因此对它不怎么熟悉。
  • He became a British citizen,thereby gaining the right to vote.他成了英国公民,因而得到了投票权。
68 refractory GCOyK     
adj.倔强的,难驾驭的
参考例句:
  • He is a very refractory child.他是一个很倔强的孩子。
  • Silicate minerals are characteristically refractory and difficult to break down.硅酸盐矿物的特点是耐熔和难以分离。
69 defense AxbxB     
n.防御,保卫;[pl.]防务工事;辩护,答辩
参考例句:
  • The accused has the right to defense.被告人有权获得辩护。
  • The war has impacted the area with military and defense workers.战争使那个地区挤满了军队和防御工程人员。
70 entangled e3d30c3c857155b7a602a9ac53ade890     
adj.卷入的;陷入的;被缠住的;缠在一起的v.使某人(某物/自己)缠绕,纠缠于(某物中),使某人(自己)陷入(困难或复杂的环境中)( entangle的过去式和过去分词 )
参考例句:
  • The bird had become entangled in the wire netting. 那只小鸟被铁丝网缠住了。
  • Some military observers fear the US could get entangled in another war. 一些军事观察家担心美国会卷入另一场战争。 来自《简明英汉词典》
71 cultivation cnfzl     
n.耕作,培养,栽培(法),养成
参考例句:
  • The cultivation in good taste is our main objective.培养高雅情趣是我们的主要目标。
  • The land is not fertile enough to repay cultivation.这块土地不够肥沃,不值得耕种。
72 scythe GDez1     
n. 长柄的大镰刀,战车镰; v. 以大镰刀割
参考例句:
  • He's cutting grass with a scythe.他正在用一把大镰刀割草。
  • Two men were attempting to scythe the long grass.两个人正试图割掉疯长的草。
73 precisely zlWzUb     
adv.恰好,正好,精确地,细致地
参考例句:
  • It's precisely that sort of slick sales-talk that I mistrust.我不相信的正是那种油腔滑调的推销宣传。
  • The man adjusted very precisely.那个人调得很准。
74 chamber wnky9     
n.房间,寝室;会议厅;议院;会所
参考例句:
  • For many,the dentist's surgery remains a torture chamber.对许多人来说,牙医的治疗室一直是间受刑室。
  • The chamber was ablaze with light.会议厅里灯火辉煌。
75 amorphous nouy5     
adj.无定形的
参考例句:
  • There was a weakening of the intermolecular bonds,primarily in the amorphous region of the polymer.分子间键合减弱,尤其在聚合物的无定形区内更为明显。
  • It is an amorphous colorless or white powder.它是一种无定形的无色或白色粉末。
76 quantitative TCpyg     
adj.数量的,定量的
参考例句:
  • He said it was only a quantitative difference.他说这仅仅是数量上的差别。
  • We need to do some quantitative analysis of the drugs.我们对药物要进行定量分析。
77 anterior mecyi     
adj.较早的;在前的
参考例句:
  • We've already finished the work anterior to the schedule.我们已经提前完成了工作。
  • The anterior part of a fish contains the head and gills.鱼的前部包括头和鳃。
78 physiologist 5NUx2     
n.生理学家
参考例句:
  • Russian physiologist who observed conditioned salivary responses in dogs (1849-1936). (1849-1936)苏联生理学家,在狗身上观察到唾液条件反射,曾获1904年诺贝尔生理学-医学奖。
  • The physiologist recently studied indicated that evening exercises beneficially. 生理学家新近研究表明,傍晚锻炼最为有益。
79 physiologists c2a885ea249ea80fd0b5bfd528aedac0     
n.生理学者( physiologist的名词复数 );生理学( physiology的名词复数 );生理机能
参考例句:
  • Quite unexpectedly, vertebrate physiologists and microbial biochemists had found a common ground. 出乎意外,脊椎动物生理学家和微生物生化学家找到了共同阵地。 来自辞典例句
  • Physiologists are interested in the workings of the human body. 生理学家对人体的功能感兴趣。 来自辞典例句
80 sufficiently 0htzMB     
adv.足够地,充分地
参考例句:
  • It turned out he had not insured the house sufficiently.原来他没有给房屋投足保险。
  • The new policy was sufficiently elastic to accommodate both views.新政策充分灵活地适用两种观点。
81 orientation IJ4xo     
n.方向,目标;熟悉,适应,情况介绍
参考例句:
  • Children need some orientation when they go to school.小孩子上学时需要适应。
  • The traveller found his orientation with the aid of a good map.旅行者借助一幅好地图得知自己的方向。
82 adverse 5xBzs     
adj.不利的;有害的;敌对的,不友好的
参考例句:
  • He is adverse to going abroad.他反对出国。
  • The improper use of medicine could lead to severe adverse reactions.用药不当会产生严重的不良反应。
83 physiologically QNfx3     
ad.生理上,在生理学上
参考例句:
  • Therefore, the liver and gallbladder cannot be completely separated physiologically and pathologically. 因此,肝胆在生理和病理上不能完全分离。
  • Therefore, the liver and gallbladder are closely related physiologically and pathologically. 因此,肝胆在生理和病理上紧密联系。
84 extremities AtOzAr     
n.端点( extremity的名词复数 );尽头;手和足;极窘迫的境地
参考例句:
  • She was most noticeable, I thought, in respect of her extremities. 我觉得她那副穷极可怜的样子实在太惹人注目。 来自辞典例句
  • Winters may be quite cool at the northwestern extremities. 西北边区的冬天也可能会相当凉。 来自辞典例句
85 irreproachable yaZzj     
adj.不可指责的,无过失的
参考例句:
  • It emerged that his past behavior was far from irreproachable.事实表明,他过去的行为绝非无可非议。
  • She welcomed her unexpected visitor with irreproachable politeness.她以无可指责的礼仪接待了不速之客。
86 manifestations 630b7ac2a729f8638c572ec034f8688f     
n.表示,显示(manifestation的复数形式)
参考例句:
  • These were manifestations of the darker side of his character. 这些是他性格阴暗面的表现。 来自《简明英汉词典》
  • To be wordly-wise and play safe is one of the manifestations of liberalism. 明哲保身是自由主义的表现之一。 来自《现代汉英综合大词典》
87 destined Dunznz     
adj.命中注定的;(for)以…为目的地的
参考例句:
  • It was destined that they would marry.他们结婚是缘分。
  • The shipment is destined for America.这批货物将运往美国。
88 vertical ZiywU     
adj.垂直的,顶点的,纵向的;n.垂直物,垂直的位置
参考例句:
  • The northern side of the mountain is almost vertical.这座山的北坡几乎是垂直的。
  • Vertical air motions are not measured by this system.垂直气流的运动不用这种系统来测量。
89 components 4725dcf446a342f1473a8228e42dfa48     
(机器、设备等的)构成要素,零件,成分; 成分( component的名词复数 ); [物理化学]组分; [数学]分量; (混合物的)组成部分
参考例句:
  • the components of a machine 机器部件
  • Our chemistry teacher often reduces a compound to its components in lab. 在实验室中化学老师常把化合物分解为各种成分。
90 acceleration ff8ya     
n.加速,加速度
参考例句:
  • All spacemen must be able to bear acceleration.所有太空人都应能承受加速度。
  • He has also called for an acceleration of political reforms.他同时呼吁加快政治改革的步伐。
91 rotation LXmxE     
n.旋转;循环,轮流
参考例句:
  • Crop rotation helps prevent soil erosion.农作物轮作有助于防止水土流失。
  • The workers in this workshop do day and night shifts in weekly rotation.这个车间的工人上白班和上夜班每周轮换一次。
92 integration G5Pxk     
n.一体化,联合,结合
参考例句:
  • We are working to bring about closer political integration in the EU.我们正在努力实现欧盟內部更加紧密的政治一体化。
  • This was the greatest event in the annals of European integration.这是欧洲统一史上最重大的事件。
93 rotations d52e30a99086786b005c11c05b280215     
旋转( rotation的名词复数 ); 转动; 轮流; 轮换
参考例句:
  • Farmers traditionally used long-term rotations of hay, pasture, and corn. 农民以往长期实行干草、牧草和玉米轮作。
  • The crankshaft makes three rotations for each rotation of the rotor. 转子每转一周,曲轴转3周。


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