CHAPTER II.
EXTENSION NOT PERCEPTIBLE AS THE DIRECT AND IMMEDIATE OBJECT OF SENSATIONS
9. Extension has the remarkable peculiarity of being perceived by different senses. As regards sight and touch this is evident; it is also true as far as concerns the other senses. We perceive taste in different parts of the palate, and we refer sound and smell to distinct points in space, and this involves the idea of extension.
But what is more strange is, that although extension is the indispensable basis of all sensations and therefore perceived by all the senses, it is, in itself, and separated from every other quality, imperceptible to them all. The eye perceives only light, and the ear sound, the palate taste, the smell odor, and the object of touch is that which is warm or cold, moist or dry, solid or liquid, etc. None of these objects is extension, none in particular is necessary for the perception of extension; for we constantly find it separated from each of these qualities, and yet it is still perceptible. No one in particular is necessary for the perceptibility of extension, but some one is indispensable; for, unless accompanied by some one of them, it is imperceptible to the senses.
Hence, extension is a necessary condition of our sensations, but is not itself perceived by the senses. Still it is not therefore unknown, and this brings me to some other reflections which take us out of the phenomenal into the transcendental order, and give rise to very serious and difficult questions, which have hitherto been insolvable, and it is to be feared must ever remain so.
10. We have seen that extension in itself is not the direct object of sensation. What, then, is it? What is its nature?
There are two things which may be considered in the idea of extension: that which it is in us, and that which it represents to us; or, in other words, its relation to the subject, and its relation to the object. The first being subject to immediate observation, inasmuch as it exists within us, is difficult but not impossible to explain. The second is more difficult, and almost impossible to explain, because it is a very abstract and transcendental idea, and also requires a series of arguments, the thread of which may be broken without the one who reasons perceiving it.
11. Extension in us is not a sensation, but an idea. Sometimes we imagine it under a sensible form, confounding it with a determinate object; at other times we picture it to ourselves as a vague obscurity in which bodies are placed; but these are only fictions of the imagination. A man born blind can have none of these internal representations, and yet he forms a very good conception of extension. We ourselves in thinking of extension abstract all these forms under which we imagine it.
Two different sensations, those of sight and touch, produce the same idea of extension. This is conclusive proof that extension is rather intelligible than sensible.
Whatever may be the relation of extension to sensation, we cannot deny that it is an idea if we reflect that it is the foundation of the whole science of geometry. Thus, although we form various images of extension, they are only the particular forms with which the mind clothes the idea, if we may use the expression, according to the circumstances of the case. That which is fundamental and essential in the idea, is of a different and higher order, and has nothing in common with the applications which the mind makes in order to explain and apply it. This idea includes dimensions, but not determined or applied; they are mere conceptions which represent nothing in particular.
12. The idea of extension is a primitive fact of our mind. It is not produced by sensations, but precedes them, if not in time, at least in the order of being. There is no ground for asserting that the idea of extension exists in the mind prior to the first impression of the senses, but unless extension serves as their basis these impressions are inconceivable. Whether this idea is innate or developed, or produced in the mind by the impressions, there can be no doubt that it is distinct from them, necessary to them, and independent of any one of them in particular.
It may be that when these impressions are first received extension may not be known as a separate idea; but it is certain that it is afterwards separated and stripped of the corporeal form, and spiritualized, and that this phenomenon may be occasioned but not caused by the sensation.
In sight, abstracting extension, there is color, but we cannot discover in it any thing from which we can produce so fruitful an idea as that of extension. Even at first we see that the color itself is not perceptible without extension, and so far from extension being produced by color, it is on the contrary an indispensable condition without which color cannot be perceived.
Colors as the objects of sensation are only individual phenomena, which have no connection with one another nor with the general idea of extension. What has been said of them will equally apply to all the impressions of touch.
CHAPTER III.
SCIENTIFIC FRUITFULNESS OF THE IDEA OF EXTENSION
13. In order to understand the superiority of the idea of extension over mere sensations; or rather, in order to understand that there is a true idea of extension considered in itself, and that there is no such idea of the direct and immediate objects of sensation, I wish to call attention to the fact that among all the objects of the senses, extension alone gives origin to a science.
This is a very important fact; – to explain it as it deserves, I shall establish the following propositions:
FIRST PROPOSITION.
Extension is the basis of geometry.
SECOND PROPOSITION.
Not only is extension the basis of geometry, but all that we know of the nature of bodies may be reduced to the manifestations, applications, and modifications of extension, with the addition of the ideas of number and time.
THIRD PROPOSITION.
Whatever we know of sensations that deserves the name of science is included in the modifications of extension.
FOURTH PROPOSITION.
We can form no fixed idea of corporeal objects, nor make any observation on the sensible world, unless we are guided by the rule of extension.
These four propositions are nothing more than the enunciation of certain facts, the mere exposition of which is a sufficient demonstration.
14. Extension is the basis of geometry. This is evident, since geometry treats only of dimensions, and the idea of dimension is essential to extension.
When geometry treats of figures, it is still extension which it is treating of; for figures are only extension with certain limitations. The quadrilateral contains two triangles. To distinguish them, it is only necessary to draw their limit, which is the diagonal. The idea of figure is merely the idea of limited extension, and the figure is of this or that kind according to the nature of its limits. Consequently, the idea of figure is nothing new superadded to extension; but merely its application.
Moreover, limit or termination is not a positive idea; it is a pure negation. If I have extension and wish to form all the figures possible, I need not conceive any thing new, but only abstract what I have already; I do not add, but take away. Thus in the quadrilateral I obtain the conception of the triangle by abstracting one of the two equal parts into which it is divided by the diagonal. In the same manner I deduce the quadrilateral from a pentagon by abstracting the triangle formed by a line drawn from one of its angles to either of the opposite angles. These observations apply to all geometrical figures.
The idea of extension is like an immense ground on which we have only to draw limits in order to obtain whatever we want.
It does not follow from this that the understanding cannot proceed by addition or the synthetic method; for, just as the subtraction of one of the parts of the quadrilateral formed a triangle, so also the addition of two triangles with an equal side will produce a quadrilateral. And in the same way points produce lines, lines surfaces, and surfaces solids. In all these cases the idea of figure is that of limited extension, since the quantities which constitute it are merely extension with certain limitations.
15. An observation here presents itself to my mind, which I think must throw great light upon the question which we are now discussing. If we compare the two methods by which the idea of figure is obtained; the synthetic, or that of composition or addition, and the analytic, or that of subtraction or limitation, we shall find that the second is more natural than the other; because that which the analytic method produces is permanent in the figure and essential to it, whilst the synthetic only seems to constitute it, and as soon as it is thus constituted the marks of its formation are obliterated.
An example will make this clearer. In order to conceive a rectangle I have only to limit indefinite space by four lines in a rectangular position; that is, to affirm a part, and deny the rest. The lines are nothing in themselves, and represent only the limit beyond which the space included in the rectangle cannot pass. To abstract this limitation or denial of all that is not contained in the surface of the rectangle, would be to destroy the rectangle. Therefore, the denial in which this method consists is always permanent, the manner of the production of the idea is inseparable from the idea itself.
But if, on the other hand, I proceed to form the rectangle by addition or by joining the hypotheneuse of two right-angle triangles, the ideas of the two component parts are not necessary to the idea of the rectangle after its formation. I can conceive the rectangle even abstracting the diagonal.
Thus, then, it is demonstrated that the idea of extension is the only basis of geometry, and that this idea is an immense field on which, by means of limitation or abstraction, we can obtain all the figures which form the object of geometry. Figures are only extension limited, a positive extension accompanied by a negation, and consequently whatever is positive in geometry is extension.
16. We cannot doubt that, whatever we know of the nature of bodies, may be reduced to certain modifications or properties of extension, if we observe that the entire object of the natural sciences is the knowledge of the motion or of the different relations of things in space, which is nothing more than the knowledge of the different kinds of extension.
Statics is occupied in determining the laws of the equilibrium of bodies, but in what way? Does it penetrate into the nature of the causes? No; it only determines the conditions to which the phenomenon is subject, and the only ideas which enter into these conditions are the direction of the force, that is to say, a line in space, and the velocity, which is the relation of space to time.
The idea of time is the only idea which is here joined with that of extension. In another place I shall prove that time, separated from things, is nothing, and consequently, although this idea is here joined to that of extension, it does not interfere with the truth of what I have established. In statics, all that relates to other sensations is counted as nothing; in order to solve the problems of the composition and decomposition of forces, we abstract all color, smell, and other sensible qualities of bodies in motion. What has been said of statics applies equally to dynamics, hydrostatics, hydraulics, astronomy, and to all sciences which regard motion.
17. Here an objection may be made. That with the ideas of time and space, we seem to combine another which is distinct from them, and necessary, in order to complete the idea of motion, and this is the idea of a body moved. It is not time, nor is it space, for space is not moved, therefore it is distinct from them.
To this I reply, first, that I am speaking of extension, and not of space alone, which it is important to remember, for what I shall afterwards say; and secondly, that science regards the thing moved as a point, and this is sufficient for all its purposes. Thus in the systems of forces there is a point of application for each of the component forces, and another for the resultant. This point is not regarded as having any properties, but is in relation to motion what the centre is in relation to a circle. Every thing is related to it, yet it is nothing in itself, except inasmuch as it occupies a definite position in space. It may change according to the quantity and direction of the forces, it may run over or describe a line in space with greater or less velocity, and the line may be of this or that class, and accompanied by various conditions. If a body be impelled by two forces, B and C, acting upon a point A, science considers in the body only the point through which the resultant of the forces B and C passes, and abstracts all the other points of the body which, being joined to the point A, move with it.
18. When I say that the natural sciences go no farther than the consideration of extension, I only mean to exclude the other sensations, but not ideas; for it is clear that the ideas of time and number are combined with the idea of extension. This is so true in mechanics, in this sense at least, that all its theorems and problems are reduced to geometrical expressions, and even the idea of time is expressed by lines.
In every force there are three things to be considered: the direction, point of application, and intensity. The direction is represented by a line, and the point of application by a point in space. The intensity is represented only in the effect which it can produce, and this is expressed by a line, the length of which expresses the intensity of the force. The effect of the intensity which is represented by a line includes the time also; for the measure of a motion cannot be determined until we know its velocity, which is merely the relation of space to time. Therefore, although the idea of time is combined with that of extension, the result is expressed by lines, that is, by extension.
19. There is another circumstance still which shows the fruitfulness of the idea of extension. It is that in the expression of the laws of nature, it reaches cases which are beyond the idea of number. If we suppose two equal rectangular forces, AB and AC, acting on the point A, the resultant will be AR. Now, if we consider AR to be the hypotheneuse of a right-angled triangle, AR2 = AB2 + AC2, extracting the square root AR = √(AB2 + AC2). If we suppose each of the component forces equal to 1, AR = √(12 + 12) = √2, a value which can neither be expressed in whole numbers nor in fractions, but which is represented by the hypotheneuse.
20. In the physical sciences, such words as force, cause, agent, etc., are frequently used, but the ideas which these terms express are a part of science only inasmuch as they are represented by effects. This is not because true philosophy confounds the cause with the effect, but as physical science regards only the phenomenon in all that relates to the cause, it limits itself to the abstract idea of causality, which presents nothing determinate, and consequently is not the object of its scientific labors. The system of universal attraction has immortalized the name of Newton, and he begins by confessing his ignorance of the cause of the effect which he explains. When we go beyond the phenomena and the calculations to which they give rise, we enter the field of metaphysics.
21. The natural sciences consider certain qualities of bodies which have no relation to extension, as, for example, heat and light, and this might seem to be a refutation of what we have said of extension. Still this objection disappears when we examine in what manner science takes note of these qualities, and instead of overthrowing our thesis, the result will strengthen, extend, and explain it.
Heat is not measured by the sensation which it produces in us. If we enter a room where the temperature is very high, we experience a strong sensation of heat, which gradually grows weaker, while the temperature remains the same. If we reach our hand to a friend we experience a sensation of heat or cold, in proportion as his hand is warmer or colder than our own.
Heat and cold are measured, not in themselves, nor in relation to our sensations, but in the effect which they produce. These effects are included in the modification of extension; for the thermometer marks the temperature by a greater or less elevation of the mercury in a line. Its degrees are expressed by parts of a line, on which they are marked.
I know that what is measured is distinct from extension; but, its measurement is only possible by relation to extension, and by attending to effects which are modifications of extension. Thus, the temperature at which water boils is 212°, and this is discovered by the motion of the water, and has relation to extension. So, also, the rarefaction and condensation of bodies are modifications of extension, since these states consist in the occupation of greater or less space, or in the increase or diminution of their dimensions.
22. All that science teaches us of light and colors relates to the different directions and combinations of the rays of light. Our observation goes no farther than sensation. We know that we can combine the rays in different manners, and direct them, so as to modify our sensation, but this is nothing more than the scientific knowledge of extension in the medium which we make use of, and of the sensation experienced in consequence. All beyond this is entirely unknown.
23. We may say the same of all other sensations, that of touch included. What is that quality of bodies which we call hardness? the resistance which we encounter when we touch them? But abstracting sensation, which only produces the consciousness of itself, what do we find? Impenetrability. And what do we understand by impenetrability? The impossibility of two bodies occupying the same space at the same time. Here, then, we meet with extension. If, by hardness, we mean the cohesion of molecules, in what does cohesion consist? In the juxtaposition of parts in such manner that they cannot, without difficulty, be separated. But, to be separated, is to be made to occupy a place different from that which was before occupied. Here, too, we find the idea of extension.
Of sound we know nothing scientifically, except as relates to extension and motion. The musical scale is expressed by a series of fractional numbers representing the vibrations of the air.
24. These examples demonstrate the third of the above propositions, that whatever we know of sensations that deserves the name of science, is included in the modifications of extension.
25. It is the same with the fourth proposition, that without the idea of extension, we can have no fixed idea of any thing corporeal, no fixed rule in relation to phenomena, but are like blind men. If, for an instant, we abstract the idea of extension, it is impossible for us to take a step in advance. The examples already adduced in order to demonstrate the second proposition, render further explanation here unnecessary.
26. Although extension is essentially composed of parts, there is in it something fixed, unalterable, and, in some manner, simple. There may be more or less extension, but not different kinds. One right line may be longer or shorter than another, but its length is not of a different species. One surface may be larger than another, a solid of a certain kind greater than another of the same kind, but not in a different manner.
When I say that in the idea of extension objectively considered there is a certain sort of simplicity, I do not mean that there is any thing entirely simple; for I have just said that its object is essentially composite. Neither do I abstract its essential elements, which are the three dimensions, nor any idea which it involves, as its limitability, or capacity to be limited in various ways. All I wish to show is that in all the different figures these fundamental notions are sufficient, that they are never modified, but always present the same thing to the mind.
Let us compare a right line with a curve. A right line is a direction which is always constant; the curve a direction which is always varied. A direction always varied is a collection of right directions infinitely small. Therefore, the circumference of a circle is considered as a polygon of an infinite number of sides. The curve is therefore formed by the variety of directions reduced to infinitesimal values. This theory which explains the difference of the right line and the curve, is evidently applicable to surfaces and solids.
Let us compare a quadrilateral with a pentagon; all that the second has which the first has not is one side more in perimeter, and in area the space contained in the triangle formed by a line drawn from one of its angles to either of the opposite angles. The lines are of the same kind, the surfaces differ only in the ways in which they terminate. But termination is the same as limitation. Therefore, all that is essential to the idea of extension, that is, direction and limitability, remain always the same and unchangeable.
This intrinsical constancy is indispensable to science. That which is mutable, may be the object of perception, but not of scientific perception.