Section Three: Measure
Abstractly expressed, in measure quality and quantity are united. Being as such is an immediate identity of the determinateness with itself. This immediacy of the determinateness has sublated itself. Quantity is being which has returned into itself in such a manner that it is a simple self-identity as indifference to the determinateness.
But this indifference is only the externality of having the determinateness not in its own self but in an other. Thirdly, we now have self-related externality; as self-related it is also a sublated externality and has within itself the difference from itself-the difference which, as an externality is the quantitative, and as taken back into itself is the qualitative, moment.
In transcendental idealism the categories of quantity and quality are followed, after the insertion of relation, by modality, which may therefore be mentioned here. This category has there the meaning of being the relation of the object to thought. According to that idealism thought generally is essentially external to the thing-in-itself. In so far as the other categories have only the transcendental character of belonging to consciousness, but to the objective element of it, so modality as the category of relation to the subject, to this extent contains relatively the determination of reflection-into-self; i.e. the objectivity which belongs to the other categories is lacking in the categories of modality; these, according to Kant, do not in the least add to the concept as a determination of the object but only express the relation to the faculty of cognition. The categories which Kant groups under modality — namely, possibility, actuality and necessity will occur later in their proper place; Kant did not apply the infinitely important form of triplicity — with him it manifested itself at first only as a formal spark of light — to the genera of his categories (quantity, quality, etc.), but only to their species which, too, alone he called categories. Consequently he was unable to hit on the third to quality and quantity.
With Spinoza, the mode is likewise the third after substance and attribute; he explains it to be the affections of substance, or that element which is in an other through which it is comprehended. According to this concept, this third is only externality as such; as has already been mentioned, with. Spinoza generally, the rigid nature of substance lacks the return into itself.
The observation here made extends generally to those systems of pantheism which have been partially developed by thought. The first is being, the one, substance, the infinite, essence; in contrast to this abstraction the second, namely, all determinateness in general, what is only finite, accidental, perishable, non-essential, etc. can equally abstractly be grouped together; and this is what usually happens as the next step in quite formal thinking. But the connection of this second with the first is so evident that one cannot avoid grasping it as also in a unity with the latter; thus with Spinoza, the attribute is the whole substance, but is apprehended by the intellect which is itself a limitation or mode; but in this way the mode, the non-substantial generally, which can only be grasped through an other, constitutes the other extreme to substance, the third generally. Indian pantheism, too, in its monstrous fantasies has in an abstract way received this development which runs like a moderating thread through its extravagances; a point of some interest in the development is that Brahma, the one of abstract thought, progresses through the shape of Vishnu, particularly in the form of Krishna, to a third form, that of Siva. The determination of this third is the mode, alteration, coming-to-be and ceasing-to-be-the field of externality in general. This Indian trinity has misled to a comparison with the Christian and it is true that in them a common element of the nature of the Notion can be recognised; but it is essential to gain a more precise consciousness of the difference between them; for not only is this difference infinite, but it is the true, the genuine infinite which constitutes it. This third principle is, according to its determination, the dispersal of the unity of substance into its opposite, not the return of the unity to itself — not spirit but rather the non-spiritual. In the true trinity there is not only unity but union, the conclusion of the syllogism is a unity possessing content and actuality, a unity which in its wholly concrete determination is spirit. This principle of the mode and of alteration does not, it is true, altogether exclude the unity; in Spinozism, for example, it is precisely the mode as such which is untrue; substance alone is true and to it everything must be brought back. But this is only to submerge all content in the void, in a merely formal unity lacking all content. Thus Siva, too, is again the great whole, not distinct from Brahma, but Brahma himself. In other words, the difference and the determinateness only vanish again but are not preserved, are not sublated, and the unity does not become a concrete unity, neither is the disunity reconciled. The supreme goal for man placed in the sphere of coming-to-be and ceasing-to-be, of modality generally, is submergence in unconsciousness, unity with Brahma, annihilation; the Buddhist Nirvana, Nibbana etc., is the same.
Now although the mode as such is abstract externality, indifference to qualitative and quantitative determinations, and in essence the external and unessential elements are not supposed to count, it is still, on the other hand, admitted in many cases that everything depends on the kind and manner of the mode; such an admission means that the mode itself is declared to belong essentially to the substantial nature of a thing, a very indefinite connection but one which at least implies that this external element is not so abstractly an externality.
Here the mode has the specific meaning of measure. Spinoza's mode, like the Indian principle of change, is the measureless. The Greek awareness, itself still indeterminate, that everything has a measure — even Parmenides, after abstract being, introduced necessity as the ancient limit by which all things are bounded — is the beginning of a much higher conception than that contained in substance and in the difference of the mode from substance.
Measure in its more developed, more reflected form is necessity; fate, Nemesis, was restricted in general to the specific nature of measure, namely, that what is presumptuous, what makes itself too great, too high, is reduced to the other extreme of being brought to nothing, so that the mean of measure, mediocrity is restored. 'The absolute, God, is the measure of all things' is not more intensely pantheistic than the definition: 'The absolute, God, is being,' but it is infinitely truer. Measure, it is true, is an external kind and manner of determinateness, a more or less, but at the same time it is equally reflected into itself, a determinateness which is not indifferent and external but intrinsic; it is thus the concrete truth of being. That is why mankind has revered measure as something inviolable and sacred.
The Idea of essence, namely, to be self-identical in the immediacy of its determined being, is already immanent in measure; so that the immediacy is thus reduced by this self-identity to something mediated, which equally is mediated only through this externality, but is a mediation with itself — that is, reflection, the determinations of which are, but in this being are nothing more than moments of their negative unity. In measure, the qualitative moment is quantitative; the determinateness or difference is indifferent and so is no difference, is sublated. This nature of quantity as a return-into-self in which it is qualitative constitutes that being-in-and-for-itself which is essence. But measure is only in itself or in its Notion essence; this Notion of measure is not yet posited. Measure, still as such, is itself the immediate [seiende] unity of quality and quantity; its moments are determinately present as a quality, and quanta thereof; these moments are at first inseparable only in principle [an sich], but do not yet have the significance of this reflected determination. The development of measure contains the differentiation of these moments, but at the same time their relation, so that the identity which they are in themselves becomes their relation to each other, i.e. is posited. The significance of this development is the realisation of measure in which it posits itself as in relation with itself, and hence as a moment. Through this mediation it is determined as sublated; its immediacy and that of its moments vanishes; they are reflected. Measure, having thus realised its own Notion, has passed into essence.
At first, measure is only an immediate unity of quality and quantity, so that: (1), we have a quantum with a qualitative significance, a measure. The progressive determining of this consists in explicating what is only implicit in it, namely, the difference of its moments, of its qualitatively and quantitatively determined being. These moments further develop themselves into wholes of measure which as such are self-subsistent. These are essentially in relationship with each other, and so measure becomes (2), a ratio of specific quanta having the form of self-subsistent measures. But their self-subsistence also rests essentially on quantitative relation and quantitative difference; and so their self-subsistence becomes a transition of each into the other, with the result that measure perishes in the measureless. But this beyond of measure is the negativity of measure only in principle; this results (3), in the positing of the indifference of the determinations of measure, and the positing of real measure — real through the negativity contained in the indifference — as an inverse ratio of measures which, as self-subsistent qualities, are essentially based only on their quantity and on their negative relation to one another, thereby demonstrating themselves to be only moments of their truly self-subsistent unity which is their reflection-into-self and the positing thereof, essence.
The development of measure which has been attempted in the following chapters is extremely difficult. Starting from immediate, external measure it should, on the one hand, go on to develop the abstract determination of the quantitative aspects of natural objects (a mathematics of nature), and on the other hand, to indicate the connection between this determination of measure and the qualities of natural objects, at least in general; for the specific proof, derived from the Notion of the concrete object, of the connection between its qualitative and quantitative aspects, belongs to the special science of the concrete. Examples of this kind concerning the law of falling bodies and free, celestial motion will be found in the Encyclopedia of the Phil. Sciences, 3rd ed., Sections 267 and 270, Remark. In this connection the general observation may be made that the different forms in which measure is realised belong also to different spheres of natural reality. The complete, abstract indifference of developed measure, i.e. the laws of measure, can only be manifested in the sphere of mechanics in which the concrete bodily factor is itself only abstract matter; the qualitative differences of such matter are essentially quantitatively determined; space and time are the purest forms of externality, and the multitude of matters, masses, intensity of weight, are similarly external determinations which have their characteristic determinateness in the quantitative element. On the other hand, such quantitative determinateness of abstract matter is deranged simply by the plurality of conflicting qualities in the inorganic sphere and still more even in the organic world. But here there is involved not merely a conflict of qualities, for measure here is subordinated to higher relationships and the immanent development of measure tends to be reduced to the simple form of immediate measure. The limbs of the animal organism have a measure which, as a simple quantum, stands in a ratio to the other quanta of the other limbs; the proportions of the human body are the fixed ratio of such quanta. Natural science is still far from possessing an insight into the connection between such quantities and the organic functions on which they wholly depend. But the readiest example of the reduction of an immanent measure to a merely externally determined magnitude is motion. In the celestial bodies it is free motion, a motion which is determined solely by the Notion and whose quantitative elements therefore equally depend solely on the Notion (see above); but such free motion is reduced by the living creature to arbitrary or mechanically regular, i.e. a wholly abstract, formal motion.
And in the realm of spirit there is still less to be found a characteristic, free development of measure. It is quite evident, for example, that a republican constitution like that of Athens, or an aristocratic constitution tempered by democracy, is suitable only for States of a certain size, and that in a developed civil society the numbers of individuals belonging to different occupations stand in a certain relations to one another; but all this yields neither laws of measure nor characteristic forms of it. In the spiritual sphere as such there occur differences of intensity of character, strength of imagination, sensations, general ideas, and so on; but the determination does not go beyond the indefiniteness of strength or weakness. How insipid and completely empty the so-called laws turn out to be which have been laid down about the relation of strength and weakness of sensations, general ideas, and so on, comes home to one on reading the psychologies which occupy themselves with such laws.
Chapter 1: Specific Quantity
Qualitative quantity in the first place an immediate, specific quantum. Secondly, this quantum as relating itself to another becomes a quantitative specifying, a sublating of the indifferent quantum. This measure is so far a rule and contains the two moments of measure distinguished; namely, the intrinsic quantitative determinateness and the external quantum. In this distinction, however, these two sides become qualities and the rule becomes a relation between them; consequently measure exhibits itself thirdly, as a relation of qualities. These at first have a single measure, but this is further specified within itself into distinct measures.
A. THE SPECIFIC QUANTUM
1. Measure is the simple relation of the quantum to itself, its own determinateness within itself; the quantum is thus qualitative. At first, as an immediate measure it is an immediate quantum, hence just some specific quantum or other; equally immediate is the quality belonging to it, some specific quality or other. The quantum as this no longer indifferent limit but as a self-related externality, is thus itself quality, and although distinguished from it does not transcend it, neither does the quality transcend the quantum. It is thus the determinateness which has returned into simple identity with itself, one with the specific determinate being, just as this latter is one with its quantum.
If it is desired to make a proposition out of the determination in question, it can be expressed thus: all that exists has a measure. Everything that exists has a magnitude and this magnitude belongs to the nature of the something itself ; it constitutes its specific nature and its being-within-self. Something is not indifferent to this magnitude, so that if this were altered it would continue to be what it is; on the contrary, an alteration of the magnitude would alter the quality of the something. Quantum, as measure, has ceased to be a limit which is no limit; it is now the determination of the thing, which is destroyed if it is increased or diminished beyond this quantum.
A measure taken as a standard in the usual meaning of the word is a quantum which is arbitrarily assumed as the intrinsically determinate unit relatively to an external amount. Such a unit can, it is true, also be in fact an intrinsically determinate unit, like a foot and suchlike original measures; but in so far as it is also used as a standard for other things it is in regard to them only an external measure, not their original measure. Thus the diameter of the earth or the length of a pendulum may be taken, each on its own account, as a specific quantum; but the selection of a particular fraction of the earth's diameter or of the length of the pendulum, as well as the degree of latitude under which the latter is to be taken for use as a standard, is a matter of choice. But for other things such a standard is still more something external. These have further specified the general specific quantum in a particular way and have thereby become particular things. It is therefore foolish to speak of a natural standard of things. Moreover, a universal standard ought only to serve for external comparison; in this most superficial sense in which it is taken as a universal measure it is a matter of complete indifference what is used for this purpose. It ought not to be a fundamental measure in the sense that it forms a scale on which the natural measures of particular things could be represented and from which, by means of a rule, they could be grasped as specifications of a universal measure, i.e. of the measure of their universal body. Without this meaning, however, an absolute measure is interesting and significant only as a common element, and as such is a universal not in itself but only by agreement.
This immediate measure is a simple quantitative determination as, for example, the size of organic beings, of their limbs etc. But everything that exists has a size which makes it what it is, and in general enables it to have an external reality. As a quantum it is an indifferent magnitude open to external determination and capable of increase and decrease. But as a measure it is also distinguished from itself as a quantum, as such an indifferent determination, and is a limitation of that indifferent fluctuation about a limit.
Since the quantitative determinateness of anything is thus twofold — namely, it is that to which the quality is tied and also that which can be varied without affecting the quality — it follows that the destruction of anything which has a measure takes place through the alteration of its quantum. On the one hand this destruction appears as unexpected, in so far as the quantum can be changed without altering the measure and the quality of the thing; but on the other hand, it is made into something quite easy to understand through the idea of gradualness.
The reason why such ready use is made of this category to render conceivable or to explain the disappearance of a duality or of something, is that it seems to make it possible almost to watch the disappearing with one's eyes, because quantum is posited as the external limit which is by its nature alterable, and so alteration (of quantum only) requires no explanation. But in fact nothing is explained thereby; the alteration is at the same time essentially the transition of one quality into another, or the more abstract transition of an existence into a negation of the existence; this implies another determination than that of gradualness which is only a decrease or an increase and is a one-sided holding fast to quantity.
2. The sudden conversion into a change of quality of a change which was apparently merely quantitative had already attracted the attention of the ancients who illustrated in popular examples the contradiction arising from ignorance of this fact; they are familiar under the names of ‘the bald’ and ‘the heap’.
These elenchi are, according to Aristotle's explanation, ways in which one is compelled to say the opposite of what one had previously asserted. The question was asked: does the pulling out of a single hair from the head or from a horse's tail produce baldness, or does a heap cease to be a heap if a grain is removed? An answer in the negative can be given without hesitation since such a removal constitutes only a quantitative difference, a difference moreover which is itself quite insignificant; thus a hair, a grain, is removed and this is repeated, only one of them being removed each time in accordance with the answer given. At last the qualitative change is revealed; the head or the tail is bald, the heap has disappeared. In giving the said answer, what was forgotten was not only the repetition, but the fact that the individually insignificant quantities (like the individually insignificant disbursements from a fortune) add up and the total constitutes the qualitative whole, so that finally this whole has vanished; the head is bald, the purse is empty.
The dilemma, the contradiction which results therefrom, is not a sophism in the usual sense of the word; for such contradiction is not a sham or a deception. The real mistake is committed by the assumed Other (i.e. our ordinary consciousness), the mistake, namely, of assuming a quantity to be only an indifferent limit, i.e. of assuming that it is just a quantity in the specific sense of quantity. This assumption is refuted by the truth to which it is brought — to wit, that quantity is a moment of measure and is connected with quality.
What is refuted is the error of one-sidedly holding fast to the abstract determinateness of quantum. Therefore these examples, too, are not a pointless or pedantic joke but have their own correctness; they are the product of a mentality which is interested in the phenomena which occur in thinking.
Quantum, when it is taken as an indifferent limit, is the aspect of an existence which leaves it open to unsuspected attack and destruction. It is the cunning of the Notion to seize on this aspect of a reality where its quality does not seem to come into play; and such is its cunning that the aggrandisement of a State or of a fortune, etc., which leads finally to disaster for the State to for the owner, even appears at first to be their good fortune.
3. Measure in its immediacy is an ordinary quality with a specific magnitude attaching to it. Now that aspect of the quantum according to which it is an indifferent limit which can be exceeded without altering the quality, is also distinguished from its other aspect according to which it is qualitative and specific. Both are quantitative determinations of one and the same thing; but because of the initial immediacy of measure, this distinction is also to be taken as immediate, and therefore both aspects also have a distinct existence. The existence of measure, then, which is intrinsically determinate magnitude, is in its behaviour towards the existence of the alterable, external aspect, a sublating of its indifference, a specifying of measure.
B. SPECIFYING MEASURE
This is first a rule, a measure which is external with reference to mere quantum. Secondly it is a specific quantity which determines the external quantum, and thirdly both sides, as qualities of a specific quantitative determinateness, are related to one another as one measure.
(a) The Rule
The rule or standard, which has already been mentioned, is in the first place an intrinsically determinate magnitude which is a unit with reference to a quantum having a particular existence in a something other than the something of the rule; this other something is measured by the rule, i.e. is determined as an amount of the said unit. This comparison is an external act, the unit itself being an arbitrary magnitude which in turn can equally be treated as an amount (the foot as an amount of inches). But measure is not only an external rule; as a specifying measure its nature is to be related in its own self to an other which is a quantum.
(b) Specifying Measure
Measure is a specific determining of the external, i.e. indifferent magnitude which is now posited by some other existence in general in the measurable something; true, this latter is itself a quantum but, as distinguished from it, it is the qualitative side determining the merely indifferent, external quantum. The measurable something has in it this side of being-for-other to which the indifferent increasing and decreasing is proper. This immanent measuring standard is a quality of the something to which is opposed the same quality in another something; but in the latter something the quality is at first only relative, the quantum having no significance as a measure in relation to the other something which is determined as the standard of measuring.
Something, in so far as it is a measure within itself, has the magnitude of its quality altered from outside itself; it does not accept this externally imposed alteration as an arithmetical amount: its measure reacts against it, behaves towards the amount as an intensive quantum and assimilates it in a characteristic way; it alters the externally imposed alteration, makes this quantum into a different one and through this specifying shows itself to be self-determined in this externality. This specifically assimilated amount is itself a quantum which is also dependent on the other, or is for it only an external amount. Consequently the specified amount is also alterable, but is not therefore a quantum as such but the external quantum specified in a constant manner. The determinate being of measure is thus a ratio, the specific element of which is in general the exponent of this ratio.
When considering intensive, and extensive quantum we found that it is the same quantum which is present, once in the form of intensity and again in the form of extension. In this difference the quantum lying at the base suffers no alteration, the difference being only an outer form. In the specifying measure, on the contrary, the quantum is taken in the first instance in its immediate magnitude, but in the second instance it is taken through the exponent of the ratio in another amount.
The exponent which constitutes the specific element can at first seem to be a fixed quantum, as a quotient of the ratio between the external and the qualitatively determined quantum. But as such, it would be nothing but an external quantum; the exponent here must be understood as nothing else but the qualitative moment itself which specifies the quantum as such. As we have already seen, the strictly immanent qualitative form of the quantum is solely its determination as a power. It must be such a determination which constitutes the ratio and which here, as the intrinsic determination of the quantum, confronts the quantum as externally constituted. The principle of the quantum is the numerical one which constitutes its intrinsic determinedness, and the mode of relation of the numerical one is external; and the alteration which is specified only by the nature of the immediate quantum as such, consists by itself in the addition of such a numerical one and then another and so on. If in this way the alteration of the external quantum is an arithmetical progression, the specifying reaction of the qualitative nature of measure produces another series which is related to the first, increases and decreases with it, but not in a ratio determined by a numerical exponent but in a number of incommensurable ratios, according to a determination of powers.
Remark
To cite an example, temperature is a quality in which these two sides of external and specified quantum are distinguished. As a quantum it is an external temperature, and that too, of a body as a general medium, and it is assumed that the alteration of the temperature proceeds on the scale of an arithmetical progression, increasing or decreasing uniformly. On the other hand, the particular bodies in the medium differ in the way they absorb the temperature, for through their immanent measure they determine it as received from outside themselves and the change of temperature in any one of them does not correspond in a direct ratio with that of the medium or of the other bodies among themselves. Different bodies compared at one and the same temperature give the numerical ratios of their specific heats, of their thermal capacities. But the thermal capacities of bodies vary in different temperatures and associated with this is a change in the specific shape. Thus a particular specification is manifested in the increase or decrease of temperature. The ratio of the temperature, taken as external, to the temperature of a specific body which is at the same time dependent on the former temperature, has no fixed exponent; the increase or decrease of this heat does not proceed uniformly with the increase or decrease of the external heat. Here a temperature is assumed which is purely external and whose changes are merely external or purely quantitative. But the temperature is itself the temperature of the air or some other specific temperature. The ratio, therefore, if looked at more closely, would strictly speaking have to be taken not as the ratio of a merely quantitative quantum to a qualitative one, but as the ratio of two specific quanta. In fact, the determining of the specifying ratio has now advanced to the stage where the moments of measure not only consist of a quantitative side and a side qualifying the quantum, both being sides of one and the same quality, but are related to each other as two qualities which are in themselves measures.
(c) Relation of the two Sides as Qualities
1. The qualitative, intrinsically determinate side of the quantum exists only as a relation to the externally quantitative side; as a specifying of the latter it is a sublating of its externality through which quantum as such is. This qualitative side thus has a quantum for its presupposition and its starting point. But this quantum is also qualitatively distinguished from the quality itself; this difference between them is now to be posited in the immediacy of being as such, in which determination measure still is. The two sides are thus qualitatively related and each is on its own account a qualitative determinate being; and the one quantum which at first was only a formal quantum indeterminate in itself, is the quantum of a something and of its quality, and also — now that the connection between them is determined as a measure — the specific magnitude of these qualities. These qualities are related to each other according to their determination as measures which determination is their exponent. But they are already implicitly related to each other in the being-for-self of measure; the quantum in its dual character is both external and specific so that each of the distinct quantities possesses this twofold determination and is at the same time inseparably linked with the other; it is in this way alone that the qualities are determined. They are therefore not only simply determinate beings existing for each other but they are posited as inseparable and the specific magnitude connected with them is a qualitative unity — a single determination of measure in which, in accordance with their Notion, they are implicitly bound up with each other. Measure is thus the immanent quantitative relationship of two qualities to each other.
2. In measure there enters the essential determination of variable magnitude, for measure is quantum as sublated, and therefore no longer what, as quantum, it is supposed to be, but quantum and something else; this something else is the qualitative element and, as we have seen, is nothing else than its relation of powers. In immediate measure this alteration is not yet posited; it is only an arbitrary, single quantum to which a quality is bound. In the specifying of measure (the preceding determination), which is an alteration of the merely external quantum by the qualitative element, there is posited the distinction between the two specific magnitudes and hence generally the plurality of measures in a common, external quantum. It is in this distinguishedness of the quantum from itself that it first shows itself to be a real measure; for it now appears as a determinate being which is both one and the same (e.g. the constant temperature of the medium), and also quantitatively varied (in the different temperatures of the bodies present in the medium). This distinguishedness of the quantum in the different qualities (the different bodies) gives a further form of measure, that in which the two sides are mutually related as qualitatively determinate quanta. This can be called realised measure.
Magnitude, simply as magnitude, is alterable, for its determinateness is a limit which is at the same time no limit, so that the alteration concerns only a particular quantum, the place of which is taken by another. But the genuine alteration is that of the quantum as such; when understood in this way, we have the interesting determination of the variable magnitude in higher mathematics; here we must not stop short at the merely formal determination of variability in general, neither must we introduce any determination other than the simple determination of the Notion according to which the other of quantum is only quality. The genuine determination, therefore, of real variable magnitude is that it is magnitude qualitatively determined, that is, as has been sufficiently demonstrated, determined by a ratio of powers. In this variable magnitude the fact is posited that what counts is not quantum as such but quantum determined in accordance with its other, i.e. qualitatively determined.
The two sides thus related have, in keeping with their abstract aspect as qualities generally, some particular significance — for example, space and time. Taken at first simply as specific magnitudes in their ratio as measures, one of them is an amount which increases and decreases in an external arithmetical progression, the other is an amount which is specifically determined by the first, which for it is unit. Now if each were only just a particular quality, that element of difference would be lacking which, with regard to their character as quantities, would indicate which of them was to be taken as merely externally quantitative and which as varying according to the specifying of its magnitude. If, for example, they are related as root and square, it is immaterial which is regarded as increasing or decreasing merely externally in arithmetical progression, and which, on the other hand, as specifically determining the other quantum.
But the difference between the qualities is not undefined, for as moments of measure the specifying of measure must be present in them. The next determinateness of the qualities themselves is that one is extensive — is in its own self externality — the other is intensive, the being-within-self or negative relatively to the other. Accordingly, the former of the quantitative moments is amount, and the latter is unit; in the simple direct ratio the former is to be taken as the dividend, the latter as divisor, and in the specifying ratio the former as the power or the becoming-other, the latter as the root. In so far as we still count here, i.e. still reflect on the external quantum (which is thus the quite contingent specific magnitude, the empirical amount), the alteration, too, being likewise taken as an external, arithmetical progression, then this falls on the side of the unit, of the intensive quality; the external, extensive side, on the other hand, is to be represented as altering in the specified series. But the direct ratio (like velocity as such, s/t) is here reduced to the merely formal determination which has no existence except as an intellectual abstraction; and even though in the ratio of root and square (as in s = at2) the root is to be taken as an empirical quantum varying in an arithmetical progression, and the other side is to be taken as specified, yet the higher realisation of the qualifying of the quantitative, a realisation more in harmony with the Notion, is that in which both sides are related to each other in higher determinations of powers (as is the case in s3 = at2
Remark
The exposition here of the connection between the qualitative nature of something and its quantitative determination has its application in the already indicated example of motion. First of all, in velocity as the direct ratio of space traversed and time elapsed, the magnitude of time is taken as denominator while that of space is taken as numerator. If velocity as such is only a ratio of the space and time in a motion, it is immaterial which of the two moments is to be considered as amount or as unit. Space, however, like weight in specific gravity, is an external, real whole as such — hence amount — whereas time, like volume, is the ideal, negative factor, the side of unity. But here there essentially belongs the more important ratio, that which holds between the magnitudes of space and time in free motion; at first, in the still conditioned motion of a falling body where the time factor is determined as a root and the space factor as a square, or in the absolutely free motion of the celestial bodies where the period of revolution is lower by one power than the distance from the sun, the former being a square and the latter a cube. Fundamental relationships of this kind rest on the nature of the interrelated qualities of space and time and on the kind of relation in which they stand, either as a mechanical motion, i.e. as an unfree motion which is not determined by the Notion of the moments of space and time, or as the descent of a falling body, i.e. as a conditionally free motion, or as the absolutely free celestial motion. These kinds of motion, no less than their laws, rest on the development of the Notion of their moments, of space and time, since these qualities as such (space and time) prove to be in themselves, i.e. in their Notion, inseparable and their quantitative relationship is the being-for-self of measure, is only one measure-determination.
In regard to the absolute relations of measure, it is well to bear in mind that the mathematics of nature, if it is to be worthy of the name of science, must be essentially the science of measures — a science for which it is true much has been done empirically, but little as yet from a strictly scientific, that is, philosophical point of view. Mathematical principles of natural philosophy-as Newton called his work-if they are to fulfil this description in a profounder sense than that accorded to them by Newton and by the entire Baconian species of philosophy and science, must contain things of quite a different character in order to bring light into these still obscure regions which are, however, worthy in the highest degree of consideration.
It is a great service to ascertain the empirical numbers of nature, e.g., the distances of the planets from one another; but it is an infinitely greater service when the empirical quanta are made to disappear and they are raised into a universal form of determinations of quantity so that they become moments of a law or of measure — immortal services which Galileo for the descent of falling bodies and Kepler for the motion of the celestial bodies, have achieved. The laws they discovered they have proved in this sense, that they have shown the whole compass of the particulars of observation to correspond to them. But yet a still higher proof is required for these laws; nothing else, that is, than that their quantitative relations be known from the qualities or specific Notions of time and space that are correlated.
Of this kind of proof there is still no trace in the said mathematical principles of natural philosophy, neither is there in the subsequent works of this kind. It has already been remarked in connection with the show of mathematical proofs of certain relationships in nature, a show based on the misuse of the infinitely small, that it is absurd to try todemonstrate such proofs on a strictly mathematical basis, i.e. neither empirically nor from the standpoint of the Notion. These proofs presuppose their theorems, those very laws, from experience; what they succeed in doing is to reduce them to abstract expressions and convenient formulae.
Undoubtedly the time will come when, with a clearer understanding of what mathematics can accomplish and has accomplished, the entire, real merit of Newton as against Kepler — the sham scaffolding of proofs being discarded — will clearly be seen to be restricted to the said transformation of Kepler's formula and to the elementary analytical treatment accorded to it.
C. BEING-FOR-SELF IN MEASURE
1. In the form of specified measure just considered, the quantitative element of both sides is qualitatively determined (both in the ratio of powers); hence they are moments of one measure-determinateness of qualitative nature. At the same time, however, the two sides are so far posited only as immediate, merely different qualities, which do not themselves stand in the same relationship as their quantitative determinatenesses; that is, they cannot be said to have no meaning or existence outside that relationship, as is the case in the determinateness of quantity as a ratio of powers. The qualitative element thus masks itself, specifying not itself but the quantitative determinateness; only in the latter is it posited, remaining on its own account an immediate quality as such which, beside the fact that it explicitly differentiates the magnitude and beside its relation to its other, still has an independent determinate being of its own. Thus space and time, apart from that specification contained in their quantitative determinateness in the descent of a falling body, or in the absolutely free motion, count as space in general and time in general, space having an enduring existence of its own apart from and without time, and time flowing on its own independently of space.
This immediacy of the qualitative element as against its specific measure relation is, however, just as much bound up with a quantitative immediacy and with the indifference of this quantitative aspect in it towards this its relation; the immediate quality has also a merely immediate quantum. Consequently, the specific measure has also a side which is, to begin with, subject to external alteration in the sense of a merely arithmetical progression unaffected by the specific measure and in which falls the external, hence only empirical, determinateness of magnitude. Quality and quantum as thus also appearing outside the specific measure are at the same time correlated with it; immediacy is a moment of those sides which themselves belong to measure. Thus the immediate qualities also belong to measure, are likewise in relation and stand in a quantitative relationship which, as outside the specified determination, the ratio of powers, is itself only a direct ratio and an immediate measure. This conclusion and its import is to be indicated in more detail.
2. Even though the immediately determined quantum as such is, in virtue of its being a moment of measure, established as in itself determinable by the Notion, it is still with reference to the specific measure an externally given quantum. But the immediacy which is thus posited is the negation of the qualitative determination of measure; this has already been demonstrated in respect of the sides of this determination of measure which for that reason appeared as independent qualities. Such negation and the return to the immediate determination of quantity lies in the qualitatively determined relation, for the relation of distinct terms as such implies their correlation as one determinateness which latter, in distinction from the determination of the relation, is here in the sphere of quantity a quantum. As a negation of the distinct qualitatively determined sides, this exponent is a being-for-self or absolute determinedness; but it is such being-for-self only in principle [an sich]; as a determinate being it is a simple immediate quantum which is the quotient or exponent of a direct ratio between the sides of the measure, but in general is the unit appearing as empirical, in the quantitative side of measure. In the motion of failing bodies the spaces traversed are proportional to the squares of the elapsed times, s = at2. This is the specifically determined relation of space and time, a ratio of powers. The other, the direct ratio, would concern space and time as mutually indifferent qualities; it is supposed to be the ratio of the space traversed to the first unit of time. The same coefficient a remains in all the following units of time — the unit being an ordinary quantum, while the amount is determined by the specifying measure. This unit at the same time counts as the exponent of that direct ratio which belongs to the imaginary, spurious velocity i.e. the merely formal velocity which is not specifically determined by the Notion. Such a velocity does not exist here, any more than the one previously mentioned which is supposed to accrue to the falling body at the end of a unit of time. That velocity was ascribed to the first unit of time in the motion of a falling body; but this so-called unit of time is itself only an assumed unit and has as such atomic point no real being.
The beginning of the motion — the alleged smallness of it could make no difference — is straightway a magnitude and one specified by the law of descent of a falling body. The said empirical quantum is attributed to the force of gravity, on the supposition that this force itself has no connection with the actual specification, with the ratio of powers characteristic of a determination of measure. The immediate moment, that in the motion of a falling body the amount of some fifteen spatial units, taken as feet, is traversed in a unit — the so-called first unit — of time, a second, is an immediate measure, like the measurements of the limbs of the human body, the distances and diameters of the planets and so forth. The determination of such a measure falls elsewhere than in the qualitative measure determination itself, here in the law of descent of a falling body; however, the concrete sciences have so far failed to throw any light on the basis of determination of such numbers, which are only immediate and consequently are the empirical embodiment of a measure. Here we are concerned only with the determinateness specified by the Notion, namely, that the said empirical coefficient constitutes the moment of being-for-self in the measure-determination, and that too only in so far as this moment is unexplicated [an sich], and hence an immediacy. The second moment is the developed side of this being-for-self, the specific measure determinateness of the sides. In the motion of falling, a motion which is still half-conditioned and half-free, gravity, according to this second moment, is to be regarded as a force of nature, so that the relationship expressed by the law of descent of a falling body is determined by the nature of space and time, and consequently the said specification, namely, the ratio of powers, falls within gravity; the above-mentioned simple direct ratio expresses only a mechanical relationship between space and time in the [merely] formal velocity which is externally produced and determined.
3. Measure has now acquired the character of a specified quantitative relation which, as qualitative, has in it the ordinary external quantum; but this is not a quantum in general, but essentially a determinant of the relation as such; hence in the sense of an exponent, and by virtue of the immediacy of its determination, of a fixed exponent, namely that of the already mentioned direct ratio between the same qualities whose quantitative relationship is at the same time specifically determined by the ratio. This direct ratio is, so to speak, anticipated and assumed as given in the example used of a measure, namely the law of descent of a falling body; but still, as we remarked, it does not exist in this motion. The fact, however, that the two sides of measure are themselves measures, the one immediate and external, and the other immanently specified, both being contained within the unity of measure itself, means that measure is now further determined, is realised. As this unity, measure contains the relation in which the magnitudes are determined and posited as differently specified by the nature of the qualities; its determinateness is accordingly wholly immanent and self-subsistent, and has at the same time collapsed into the being-for-self of an immediate quantum, the exponent of a direct ratio. The self-determination of the relation is thus negated, for in this its other it has its final, explicit determinateness; and conversely, the immediate measure which is supposed to be in its own self qualitative, possesses in truth such qualitative determinateness only in the other side of the relation. This negative unity is a real being-for-self, the category of a something as a unity of qualities which are related as measures — a completely self-subsistent something. The two sides which have presented themselves as distinct relations also immediately possess a twofold existence; or, to put it more explicitly, a self-subsistent whole of this kind, just because it is a real being-for-self, is at the same time a repulsion into distinct self-subsistent somethings whose qualitative nature and subsistence (materiality) lies in their measure determinateness.
Chapter 2 Real Measure
Measure is now determined as a correlation of measures which constitute the quality of distinct self-subsistent somethings — or things. The relations of measure just considered concern abstract qualities like space and time; those now about to be considered are exemplified in specific gravity and later on in chemical properties, i.e. in determinations characteristic of material existence. Space and time are also moments of such measures but their relationship no longer depends simply on their own nature because they are now subordinated to further determinations. Among the determining moments in sound, for example, there is the time in which a number of vibrations occur, and the spatial element of length and thickness of the vibrating body; but the magnitudes of those ideal moments are determined externally. Space and time are no longer in a relation of powers but in the ordinary direct relation, harmony being reduced to a quite external simplicity of numbers whose relations can be grasped with the utmost ease and hence afford a satisfaction falling entirely within the element of sensation since there is an absence for spirit of figurate conception, fantasy, abstract thought, and the like. In that the sides which now constitute the measure relation are themselves measures, but at the same time real somethings, their measures are, in the first place, immediate measures and as regards their relations, direct relations. It is the inter-relationship of such relations that is now to be considered in its progressive determination.
Measure, as now real measure, is first, a self-subsistent measure of a material thing which is related to others and in this relation specifies them and with them their self-subsistent materiality. This specification as an external relating to a plurality of others in general, produces other relations and hence other measures; and the specific self-subsistence does not continue as a single direct relation but passes over into a specific determinateness which is a series of measures. Secondly, the direct relations thus produced are in themselves determinate and exclusive measures (elective affinities); but because the difference between them is also only quantitative, the progressive determination of the relations presents itself as in part a merely externally quantitative development which, however, is also interrupted by qualitative relationships and forms a nodal line of specific self-subsistent measure relations. Thirdly, however, in this development measure gives place to the measureless as such, and more specifically to the infinity of measure. In this, the self-exclusive and self-subsistent measures are one with each other, and the self-subsistent measure enters into a negative relation with itself.
A. THE RELATION OF SELF-SUBSISTENT MEASURES
Measures now are no longer merely immediate but self-subsistent, because they have become in themselves relations of measures which are themselves specified; and thus in this being-for-self are physical somethings, in the first instance, material things. The whole, which is a relation of such measures is, however, (a) first, itself immediate; thus the two sides which are determined as such self-subsistent measures exist apart in particular things and their combination is effected externally; (b) but what the self-subsistent material things are qualitatively, they are only in virtue of their quantitative determination as measures; hence through this same quantitative connection with others they are determined as differently specified in regard to them (so-called affinity), namely, as members of a series of such quantitative relationships; (c) at the same time, this indifferent, manifold interrelationship finishes by converting itself into exclusive being-for-self-so-called elective affinity.
(a) Combination of Two Measures
Something is immanently determined as a measure relation of quanta which also possess qualities and the something is the connection of these qualities. One of them is the being-within-self or inwardness of the something by virtue of which it is a real being-for-self, a material thing (such as, taken intensively, weight, or in its extensive aspect, the multiplicity of material parts); the other quality is the externality of this inwardness (the abstract, ideal element of space). These qualities are quantitatively determined and their correlation constitutes the qualitative nature of the material something — e.g. the ratio of weight to volume: specific gravity. The volume, the ideal aspect, is to be taken as unit, but the intensive aspect, which manifests quantitatively and in comparison with the former as an extensive magnitude, as a plurality of independent ones, is to be taken as amount. The purely qualitative relation of the two specific magnitudes, that is, as a ratio of powers, has vanished, because with the self-subsistence of the material thing immediacy has returned and in this the specific magnitude is an ordinary quantum whose relation to the other side is likewise determined as the ordinary exponent of a direct ratio.
This exponent is the specific quantum of the something, but it is an immediate quantum and this is determined and with it the specific nature of such something — only in the comparison with other exponents of such ratios. The exponent constitutes the specific intrinsic determinedness, the inner characteristic measure of something; but because this its measure rests on a quantum, it too is only an external, indifferent determinateness, with the consequence that the something, in spite of its inner determination as a measure, is alterable. The other to which it can, as alterable, enter into relation is not a material plurality, quantum in general — this it can withstand through its specific intrinsic determinedness -but it is a quantum which is at the same time also an exponent of such a specific ratio. Two things with different internal measures stand in relation and enter into combination — such as two metals of different specific gravities (the combination in question would not be, for example, of a metal and water); but what other kind of homogeneity is required to make possible such a combination will not be considered here. Now on the one hand, each of the two measures, just because it is a measure, preserves itself in the alteration which it ought to suffer through the externality of the quantum, but on the other hand this self-preservation is itself a negative relation towards this quantum, a specification of it; and since the quantum is the exponent of the measure relation, the self-preservation is an alteration of the measure itself and moreover a reciprocal specification.
As purely quantitatively determined, the compound would be a mere addition of the two magnitudes of the one quality and the two magnitudes of the other, e.g., the sum of the two weights and of the two volumes in the case of a compound of two material substances of different specific gravities; thus not only the weight of the mixture would remain equal to the said sum, but also the space occupied by the mixture would be equal to the sum of the said spaces. But this is true only of the weight of the mixture, which is equal to the sum of the weights before the combination; addition applies to that quality which, as a real being-for-self, has acquired a fixed determinate being with a permanent immediate quantum — the weight of the material thing, or, what counts as the same from the quantitative point of view, the number or amount of material parts. The exponents however are subject to alteration since they are the expression of the qualitative aspect of the compound, of its being-for-self in the form of measure relations; and since the quantum as such suffers contingent external alteration by an increase which is summed, the being-for-self at the same time displays itself as negating this externality. Because this immanent determining of the quantitative element cannot, as we have seen, be manifested in the weight, it displays itself in the other quality which is the ideal side of the relation. From the point of view of sense perception it may appear remarkable that the mixing of two specifically different material substances should be followed by an alteration — usually a diminution-in the sum of the two volumes, for it is space itself which constitutes the subsistence of matter in its external separated existence. But this subsistence in face of the negativity immanent in the being-for-self lacks intrinsic being, it is subject to alteration. In this manner space is posited as what it truly is, an ideal being.
Not only, then, is one of the qualitative sides posited as alterable but measure itself — and so too the qualitative nature of the something based on it — has shown that it is unstable in its own self and, like the quantum as such, has its determinateness in other measure relations.
(b) Measure as a Series of Measure Relations
(1) If two things forming a compound body owed their respective specific natures only to a simple qualitative determination, they would only destroy each other when combined. But a thing which is an immanent measure relation is self-subsistent; it is therefore also capable of combining with another such thing. But in being reduced to an element of this unity, it preserves itself through the persistence of its indifferent, quantitative character and at the same time functions as a specifying moment of a new measure relation. Its quality is masked in the quantitative element and is thus also indifferent towards the other measure, continuing itself in it and in the newly formed measure. The exponent of the new measure is itself only some quantum or other, an external determinateness, and its indifference finds expression in the fact that the specifically determined thing effects, in association with other such measures, precisely similar neutralizations of the reciprocal measure relations; it is in only one measure relation formed by itself and another specifically determined thing that its specific peculiarity is not expressed.
(2) This combination with a number of others which are likewise measures within themselves, yields different ratios which therefore have different exponents. The self-subsistent measure has the exponent of its intrinsic determinedness only in the comparison with others; its neutrality with the others, however, constitutes its real comparison with them; it is its comparison with them through its own self. But the exponents of these ratios differ and the qualitative exponent of the self-subsistent measure is thus represented as the series of these different amounts of which it is the unit, i.e. as a series whose members are in a specific relationship to others. The qualitative exponent, as one immediate quantum, expresses only one relation. The distinctive character of the self-subsistent measure finds its true expression in the characteristic series of exponents which it, taken as unit, forms with other such self-subsistent measures; for one of these measures when brought into relation with the rest of them and taken as unit forms another series. Now it is the interrelationship of the members of such a series that constitutes the qualitative aspect of the self-subsistent measure.
At first, it seems that a self-subsistent measure which forms a series of exponents with a series of such measures, is distinguished from another measure — outside this series — with which it is compared, by the fact that this other measure forms another series of exponents with the members of the first series. But in this way these two measures would not be comparable, because each is thus regarded as unit with respect to its exponents, and between the two series arising from this relation there is no specified difference. The two measures which, as self-subsistent, are supposed to be compared are at first contra-distinguished only as quanta; in order to determine their relation, an independent unit common to them both is required. This specific unit is to be sought, as has been indicated, only in that feature which embodies the specific determinate being of the measures to be compared, i.e. in the ratio which the exponents of the ratios of the members of the series have to each other. This ratio of the exponents themselves is, however, such independent and actually specific unit only in so far as the members of the series together have it as a constant ratio to both measures; in that way it can be their common unit. It is this alone, therefore, which makes it possible to compare the two self-subsistent measures which were assumed to be not reciprocally neutralising measures, but indifferent to each other. Each of them taken separately and apart from the comparison is the unit of the ratio it forms with the opposite members, which are the amounts relatively to that unit and which thus represent the series of exponents. But conversely, this series is the unit for the two measures which when compared with each other are related as quanta; as such they are themselves different amounts of the unit just indicated.
But further, those measures which together with the two, or rather indefinitely many self-subsistent measures of the first series — measures which are compared only with each other — yield a series of exponents of the ratios between the members of that series, are similarly in themselves self-subsistent measures, each being a specific something with its own intrinsic measure ratio. Each of these then is similarly to be taken as unit so that they have a series of exponents in the two, or rather indefinitely many members of the first series which are compared merely among themselves, and these exponents are the numbers resulting from comparison among themselves of the measures just named; and conversely, the comparative numbers of the measures which are now also to be taken singly as self-subsistent are the series of exponents for the members of the first series. In this way, both sides are series in which each number is firstly simply a unit to the opposite series in which it has its self-determined character as a series of exponents; secondly, each number is itself one of the exponents for each member of the opposite series; and thirdly, it is a comparative number for the other numbers of its series and as such amount, which belongs to it also as an exponent, it has its own specifically determined unit in the opposite series.
3. In this form of relationship there is a return to the particular kind of way in which quantum is posited as self-determined, i.e., as degree: namely, it is simple or unitary, but it has its quantitative determinateness in a quantum existing outside it, which is a circle of quanta. In measure, however, this external aspect is not merely a quantum and a circle of quanta, but a series of numerical ratios and it is in the entirety of these that the self-determinedness of measure lies. As is the case in the being-for-self of quantum as degree, the nature of the self-subsistent measure is converted into this externality of itself. Its self-relation is in the first place an immediate relation and therefore its indifference to an other consists only in the quantum. It is into this externality therefore that its qualitative side falls and its relationship to its other becomes that quantitative mode of relationship which constitutes the specific determination of this self-subsistent measure, a mode which is determined just as much by the other as by the measure itself; and this other is a series of quanta and the measure is reciprocally a quantum. But this relation in which two specific measures specify themselves in a third something, the exponent, also implies that the one has not passed into the other; that therefore there is not only one negation, but that both are posited as negative in the relation, and since in this each preserves itself as indifferent towards the other its negation is also in turn negated. This their qualitative unity is thus a self-subsistent exclusive unit. It is only as exclusive that the exponents, which are primarily comparative numbers of the members of the series, have in them their genuinely specific determinateness relatively to one another, and their difference thus acquires a qualitative nature. But this difference has a quantitative basis: first, the self-subsistent measure is related to a plurality of its qualitatively other side only because in this relationship it is, at the same time, indifferent; secondly, the neutral relationship, in virtue of its quantitative aspect, is now not simply an alteration but is posited as a negation of the negation and as an exclusive unit. Consequently, the affinity of a self-subsistent measure with the measures of the other side is no longer an indifferent relationship but an elective affinity.
(c) Elective Affinity
The expression elective affinity used here and the terms neutrality and affinity employed in the preceding paragraphs, refer to the chemical relationship. For a chemical substance has its specific determinateness essentially in its relation to its other and exists only as this difference from it. Furthermore, this specific relation is bound up with quantity and is at the same time the relation, not only to a single other but to a series of specifically different others opposed to it. The combinations with this series are based on a so-called affinity with every member of the series; but along with this indifference each member is at the same time exclusive towards the others and it is this correlation of opposed determinations which we have still to consider. It is, however, not only in the sphere of chemistry that the specific relation is represented in a circle of combinations; the individual note, too, only has meaning in relationship and combination with another note and with a series of others. The harmony or disharmony in such a circle of combinations constitutes its qualitative nature which is at the same time based on quantitative ratios; these form a series of exponents and are the ratios of the two specific ratios which each of the combined notes is in its own self. The individual note is the key of a system, but again it is equally an individual member in the system of every other key. The harmonies are exclusive elective affinities whose characteristic quality is equally dissolved again in the externality of a merely quantitative progression. What it is, however, that constitutes the principle of a measure for those affinities which (whether chemical or musical or what else) are elective affinities between and in opposition to the others, will be the subject of a further Remark in connection with chemical affinity; but this higher question is very closely bound up with the specific nature of the strictly qualitative aspect and belongs to the particular concrete natural sciences.
Inasmuch as the member of a series has its qualitative unity in its relation to all the members of an opposite series, whose members however are distinguished only by the quantum required for neutralising them with the member of the first series, the more specific determinateness in this multiple affinity is likewise only quantitative. In elective affinity as an exclusive, qualitative correlation, the relationship is rid of this quantitative difference. The next determination which offers itself is this; that in accordance with the difference of the amounts, that is, of extensive magnitude, of the substances of the one series required for the neutralisation of a substance in the other series, the elective affinity of the latter substance would also be directed towards the substances of the first series with all of which it has an affinity. The exclusion which would thereby be established in the form of a firmer holding together against other possibilities of combination, would appear, thus transformed, in a proportionately greater intensity in virtue of the previously demonstrated identity of the forms of extensive and intensive magnitude, the quantitative determinateness being one and the same in both forms. However, this sudden conversion of the one-sided form of extensive magnitude into its other, intensive form, makes no difference to the nature of the fundamental determination, which is one and the same quantum; consequently, no real exclusion would, in fact, result therefrom and there could take place equally well either only one combination, or a combination of an indefinite number of substances, provided that the portions of them entering into the combination corresponded to the required quantum in accordance with the ratios existing between them.
The combination which we have also called neutralisation, however, is not only the form of intensity; the exponent is essentially a measure determination and therefore exclusive. In this aspect of exclusive relations, numbers have lost their continuity with one another and their fluid combinatory nature; the relationship is one of more or less, which acquires a negative character and the preference which one exponent has over another does not remain confined to the quantitative determinateness. But equally, too, there co-exists this other aspect which again makes it a matter of indifference that a substance receives the neutralising quantum from several substances of the opposed series and from each according to its specific ratio relatively to the others; the exclusive, negative relation is thus at the same time adversely affected by the quantitative aspect. The effect of this is an actual conversion of an indifferent, merely quantitative relationship into a qualitative one, and conversely, a transition from a specifically determined relation into a merely external one — a series of relations which are sometimes of a merely quantitative nature and sometimes are specific relations and measures.
Remark: Berthollet on Chemical Affinity and Berzelius's Theory of it
Chemical substances are the most characteristic examples of those measures which, as moments of a measure, are characterised solely by their relationship to other such measure moments. Acids and alkalis or bases generally, appear to be intrinsically determinate things just as they are; but the fact is that they are incomplete elements of bodies, constituents which strictly speaking do not exist for themselves but only as a tendency to get rid of their isolatedness by combining with another constituent. Further, the difference in virtue of which they are self-subsistent, does not consist in this immediate quality but in the peculiar quantitative mode of the relationship. This, namely, is not restricted to the chemical opposition of acid and alkali or base in general, but is a specific measure of saturation and consists in the specific determinateness of the quantity of the substances which neutralise each other. This specific quantity required for saturation constitutes the qualitative nature of a substance; it makes it what it is on its own account and the number which expresses this is essentially one of several exponents for an opposed unit. A substance of this kind has a so-called affinity with another. If this connection remained of a purely qualitative nature, then, as in the case of the magnetic poles or positive and negative electricity, the one determinateness would be only the negative of the other and both sides would not at the same time show themselves to be indifferent towards each other. But because the connection has also a quantitative side, each of these substances is capable of neutralising itself with more than one and is not restricted only to the one to which it is opposed. It is not only an acid and an alkali or base which are in relation, but acids and alkalis or bases which are related to one another. They are specifically distinguished from each other primarily according to whether one acid, for example, requires a greater amount of an alkali for its saturation than another acid does. But the independent, self-determined character of the substances is displayed in the exclusiveness of the relation between the affinities, one having a preference over another in that one acid can by itself enter into combination with any alkali, and conversely. Thus the cardinal difference between two acids consists in one of them having a closer affinity to a base than the other, i.e., in a so-called elective affinity.
The law of the chemical affinities of acids and alkalis has been discovered and it states that if two neutral solutions are mixed resulting in dissociation followed by two new compounds, these products, too, are neutral. From this it follows that the amounts of two alkaline bases required for the saturation of an acid must be in the same ratio to saturate another acid; and in general, when for one alkali, taken as unit, the series of numerical ratios has been determined in which the various acids saturate it, then this series is the same for any other alkali, though the different alkalis must be taken in different amounts relatively to one another-amounts which again on their part form a similar fixed series of exponents for each of the opposite acids since they are related to any one acid in the same ratio as to any other. Fischer was the first to extract these series in their simplicity from the works of Richter and Berthollet. Since that was written, our knowledge of the numerical ratios of mixtures of chemical elements has been greatly expanded in every direction and to consider it here would be a digression, all the more so because this empirical expansion which is in part only hypothetical remains confined within the same group of concepts. We may however add a few remarks on the categories employed here and also on the views about chemical elective affinity itself and its relation to the quantitative aspect as well as the attempt to base this on specific physical qualities.
It is well known that Berthollet modified the general conception of elective affinity by the concept of the activity of a chemical mass. This modification does not affect the quantitative ratios of the chemical laws of saturation themselves, but its effect on the qualitative moment of exclusive elective affinity as such is not only to weaken it but rather to eliminate it, and this is a point that must not be overlooked. If two acids act on an alkali and the one which has a greater affinity for it is also present in the requisite amount for saturating it, then according to the concept of elective affinity this is the only saturation which occurs; the other acid remains quite inactive and is excluded from the neutral combination. According to the concept of the activity of a chemical mass, on the other hand, each of the two acids is active in a proportion which is composed of the amounts of the acids present and their saturation capacity or so-called affinity. Berthollet's investigations have indicated in greater detail the circumstances in which the activity of the chemical mass is nullified and one acid (the one with a stronger affinity) appears to expel and to exclude the action of the other acid (with a weaker affinity) that is, appears to be active in the sense of elective affinity. He has shown that this exclusion takes place in certain circumstances, such as strength of cohesion, or the insolubility in water of the salts formed, but the qualitative nature of the agents as such does not come into play; and the action of these circumstances can in turn be nullified by other circumstances, for example, by temperature. With the removal of these obstacles the chemical mass enters unimpeded into activity and what appeared as a purely qualitative exclusion, as an elective affinity, proves to depend only on external modifications.
It is Berzelius principally who should be heard further on this subject although in his Textbook of Chemistry he does not put forward anything original or more specific on the matter. The views of Berthollet are taken up and repeated literally, only decked out in the metaphysics peculiar to an uncritical reflection, the categories of which are therefore all that offer themselves for a more detailed consideration. The theory goes beyond the limits of experience and, on the one hand, invents sensuous images such as are not given in experience and on the other hand, applies categories of thought, in both cases making itself a subject for logical criticism. We propose therefore to hear what he has to say about the theory in the Textbook itself. Now there we read 'that one must imagine that in a uniformly mixed liquid, each atom of the dissolved substance is surrounded by an equal number of atoms of the solvent; and if several substances are dissolved together they must share between them the interstices between the atoms of the solvent, so that with a uniform mixture of the liquid there is produced a symmetry in the arrangement of the atoms such that all the atoms of the individual substances are uniformly arranged in relation to the atoms of the other bodies; it could therefore be said that the solution is characterised by symmetry in the arrangement of the atoms, and the combination by the fixed proportions.' This is then illustrated by an example of compounds formed when sulphuric acid is added to a solution of copper chloride; but the example certainly does not demonstrate that atoms exist; neither does it show that a number of atoms of the dissolved substances surround atoms of the fluid, nor that free atoms of the two acids arrange themselves around the atoms which remain combined (with the copper oxide), nor that there exists a symmetry in their position and arrangement, nor that interstices between the atoms exist — and least of all that the dissolved substances share among themselves the interstices between the atoms of the solvent. This would mean that the atoms of the dissolved substances take up their positions where the solvent is not — for the interstices between the atoms of the solvent are spaces empty of it — and hence that the dissolved substances are not present in the solvent, but rather are outside it, even if the solvent is arranged around them or they around it, and it is also certain therefore that they are not dissolved by it. One fails therefore to see why it is necessary to form such conceptions which are not empirically demonstrated, are in essence directly self-contradictory and are not corroborated in any other way. Corroboration could be provided only by a consideration of these conceptions themselves, i.e. by metaphysics, which is logic; but this cannot confirm them any more than experience can — on the contrary! For the rest, Berzelius admits what was also said above, that Berthollet's propositions are not opposed to the theory of fixed proportions, although he does add that they are also not opposed to the views of the corpuscular theory, i.e. the ideas mentioned above of atoms, of the filling of the interstices of the solvent by the atoms of the solid substances, and so on; this latter baseless metaphysics however, has essentially nothing to do with the proportions of saturation as such.
Hence, what finds specific expression in the laws of saturation concerns only the amount of units themselves quantitative (not atoms), of a substance with which the quantitative unit (equally not an atom) of another substance chemically distinct from the first, is neutralised; the difference between them consists solely in these different proportions. When Berzelius, then, notwithstanding the fact that his theory of proportions is wholly and solely a determination of amounts, nevertheless also speaks of degrees of affinity' in explaining Berthollet's chemical mass as the sum of the degree of affinity, from the given quantity of the active substance — although Berthollet is more consistent for he uses the expression capacite de saturation — then he himself lapses into the form of intensive magnitude; but it is this form which is the characteristic feature of the so-called dynamic philosophy which earlier on he calls 'the speculative philosophy of certain German schools',' and emphatically rejects in favour of the excellent 'corpuscular theory'. He there states of this dynamic philosophy that it assumes that the elements interpenetrate one another in their chemical combination, and that neutralization consists in this mutual interpenetration; this means nothing else than that the chemically different particles, which are interrelated as a plurality, collapse into the simplicity of an intensive magnitude, a fact which also finds expression as a diminution of volume. In the corpuscular theory, on the other hand, even the atoms which are chemically combined are supposed to be preserved in the interstices, i.e. outside one another (juxtaposition); in such a relationship which is one of merely extensive magnitude, of a perpetuated amount, a degree of affinity has no meaning. In the same place it is stated that for the dynamic view the phenomena of specific proportions came as something quite unforeseen; but this would be only an external, historical circumstance, apart from the fact that Richter's stoechiometric series in Fischer's compilation of them were already known to Berthollet and are quoted in the first edition of this Logic which proves the nullity of the categories on which the old, like the would-be new, corpuscular theory is based. But Berzelius is in error when he judges that if 'the dynamic view' had prevailed, the phenomenaof specific proportions would have remained 'for ever' unknown-meaning that this view is incompatible with the determinateness of proportions. This is, in any case, only a determinateness of quantity, no matter whether in an extensive or an intensive form; so that even Berzelius, much as he adheres to the first of these forms, that of aggregate or amount [Menge], himself makes use of the conception of degrees of affinity.
Since in this way affinity is reduced to a quantitative difference, it is sublated as elective affinity; but the exclusive factor which occurs in it is ascribed to circumstances, i.e. to determinations which appear as something external to the affinity, to cohesion, insolubility of the compounds formed, and so on. A partial comparison may be made between this conception and the manner of considering the effect of gravity on a moving pendulum. Through gravity the pendulum necessarily passes into a state of rest; but this intrinsic effect of gravity itself is treated as a merely concomitant circumstance of the external resistance of the air, the thread and so on, and it is ascribed solely to friction instead of to gravity. Here it makes no difference to the nature of the qualitative element present in elective affinity whether this is manifested in the form of these circumstances taken as its conditions, and is so interpreted. With the qualitative aspect as such there begins a new order, the specifying of which is no longer only a matter of quantitative difference.
Now although the chemical affinity in a series of quantitative ratios has thus been accurately distinguished from elective affinity which occurs as a qualitative determinateness whose behaviour in no way coincides with the order of that series, this distinction in turn gets utterly confused by the way in which electrical action has recently been coupled with chemical action; and the hope that this supposedly profounder principle would throw light on the most important relation, that of measure, has met with complete disappointment. We need not here consider more closely this theory in which the phenomena of electricity and chemistry are completely identified, since it concerns the physical nature of substances and not merely their measure relations and it calls for mention only in so far as the distinctive character of the determinations of measure is confused by it. The theory as such must be dubbed shallow, for shallowness consists in omitting the difference between distinct terms and then treating them as identical. As for affinity, chemical processes being thus identified with electrical, and also with the phenomena of fire and light, this is reduced 'to neutralisation of opposite electricities'. It is almost comical to find the identification of electricity and chemical action expounded in the following manner: 'it is true that electrical phenomena explain the action of bodies at a greater or lesser distance, their attraction before combination (i.e. a behaviour which is not yet chemical) and the fire(?) caused by this combination, but they throw no light on the cause of the combination which persists with such strength after the opposite electrical condition has been destroyed;' that is to say, the theory tells us that electricity is the cause of the chemical action, but about the specifically chemical nature of the chemical process electricity tells us nothing. Chemical difference as such being thus reduced to the opposition of positive and negative electricity, the different affinities of the agents on either side are determined as the order of two series of electro-posit' e and electro-negative substances. In identifying electricity and chemical action it is overlooked that the former generally (and its neutralisation) is transient and remains external to the quality of substances, whereas chemical action, especially in the process of neutralisation, embraces and alters the entire qualitative nature of substances. Equally transient within electricity is its opposition of positive and negative; this is so unstable that it is dependent on the most trivial outer circumstances and cannot be compared with the definiteness and fixity of the opposition between acids, for example, and metals, and so on.
The alterations which can be produced by chemical action under extremely powerful influences, e.g., of a raised temperature are not comparable with the superficial nature of the opposition in electricity. Also the further distinction within the series of each of the two sides, between a more or less positive-electrical or more or less negative-electrical disposition is quite uncertain and entirely unconfirmed. But it is on the basis of the 'electrical dispositions' of these series of substances that 'the electro-chemical system is to be set up, which above all would be best fitted to provide an idea of chemistry': these series are now quoted; but what their nature really is, is indicated in the remark 'that this is approximately the order of these substances, but so little investigation has been made into this matter, that as yet nothing really certain can be ascertained about this relative order. Both the numerical ratios of the series of affinities (first made by Richter) as well as the extremely interesting reduction by Berzelius of the combinations of two substances to the simplicity of a few quantitative ratios, are absolutely independent of that hypothetical electro-chemical hotchpotch. If the experimental method has been the correct guiding star in the theory of proportions and its universal expansion since Richter, then the mixing of these great discoveries with the so-called corpuscular theory, a desert lying away from the path of experience, forms all the greater contrast with it; only this beginning, the abandoning of the principle of experience, could be the reason for taking up again and developing that idea introduced earlier, especially by Ritter, namely, the setting up of fixed classifications of electro-positive and electronegative substances, which classifications were also supposed to have a chemical significance.
Even if the opposition of electro-positive and electro-negative substances were more in keeping with the facts than it is, then the nullity of this assumed basis of chemical affinity soon shows itself even experimentally and this again leads to further inconsistencies. It is admitted that two so-called electro-negative substances such as sulphur and oxygen combine in a much more intimate way than, e.g. oxygen and copper, although the latter is electro-positive.2 Here therefore, the basis for the affinity founded on the general opposition of positive and negative electricity must give place to a mere more or less within one and the same series of electrical quality. From this it is now inferred that the degree of affinity of the substances depends therefore not only on their specific unipolarity (with what hypothesis this determination is connected is irrelevant here; it is significant here only for the 'either' of the positive and the 'or' of the negative); the degree of affinity must be derived mainly from the intensity of their polarity generally. At this point then, the consideration of affinity passes on to the relationship of elective affinity which is our chief concern; let us see then what the result now is for this subject. It is at once admitted that the degree of this polarity, if this does not exist merely in our imagination, does not seem to be a constant quality but to depend very much on temperatures so that, after all, the result turns out to be not only that every chemical action is therefore at bottom an electrical phenomenon, but also that what seems to be an effect of so-called elective affinity is brought about only by an electrical polarity which in certain substances is present in greater strength than in others. The conclusion then after all this meandering in hypothetical conceptions, is that we are left with the category of greater intensity, which is the same formal determination as elective affinity in general; and since the latter is made to depend on a greater intensity of electrical polarity, it is not a whit nearer to being put on a physical basis than it was before. But even what is here supposed to be determined as a greater specific intensity is subsequently reduced to the modifications demonstrated by Berthollet which have already been cited.
The merit and fame which Berzelius has earned by his theory of proportions, which has been extended to all chemical relations, ought not as such to be made a reason for not setting forth the weaknesses of this theory; but a more particular reason for doing so must be the circumstance that such merit in one aspect of a science, as with Newton, tends to become an authority for a baseless structure of spurious categories which is attached to it and that it is just this kind of metaphysics which is proclaimed and echoed too with the greatest pretension.
Apart from the forms of measure relation connected with chemical and elective affinity, others too can be considered with respect to quantities which are specified into a system. Chemical substances form a system of relations with respect to saturation; saturation itself rests on the specific proportion in which the reciprocal amounts of two substances, each of which has a particular material existence, combine with each other. But there are also measure relations the moments of which are inseparable and cannot be displayed in a separate and distinct existence of their own. These are what we called earlier on, immediate self-subsistent measures and which are displayed in the specific gravity of substances. They are a ratio within the substances of weight to volume; the exponent of the ratio, which is the expression of the difference between one specific gravity and another, is a definite quantum only as a result of comparison. This is a relationship external to the substances in an external reflection, and is not founded on the one substance's own qualitative behaviour towards another contrasted substance. The problem would be to recognize the exponents of the ratios of the series of specific gravities as a system based on a rule which would specify a merely arithmetical plurality into a series of harmonic nodes. The same demand would apply to our knowledge of the series of chemical affinities already mentioned. But the accomplishment of this task still lies a long way ahead, as far ahead as the problem of grasping the numbers of the relative distances of the planets from the sun as elements in a system of measure.
Although at first specific gravities do not seem to have any qualitative relationship to one another, yet they likewise enter into a qualitative relation. When substances are chemically combined, or even form only amalgams or synsomates, there occurs also a neutralisation of the specific gravities. We mentioned earlier on the fact that the volume of a mixture, even a mixture of substances remaining really indifferent chemically to each other, is not of the same magnitude as the sum of the volumes of the substances before mixing. There is a reciprocal modification in the mixture of the quantum of specific gravity with which the substances enter into the relation and in this way they indicate their qualitative behaviour towards each other. Here the quantum of specific gravity is expressed not merely as a fixed comparative number, but as a numerical ratio which can be varied; and the exponents of the mixtures give series of measures the specifying principle of which is other than the numerical ratios of the specific gravities of the substances in combination. The exponents of these ratios are not exclusive determinations of measure; their progress is continuous but it contains an immanent specifying law which is distinct from the formally progressive ratios in which the amounts are combined and makes the former progress incommensurable with the latter.
B. NODAL LINE OF MEASURE-RELATIONS
The last determination of the measure relation was that being specific it is exclusive; the neutrality is exclusive because it is a negative unity of the distinct moments. For the relation of this self-subsistent unity, of the elective affinity to the other neutralities, no further principle of specification has offered itself; the specification resides only in the quantitative determination of affinity in general, according to which the amounts which neutralise themselves are specific and therefore stand opposed to other relative elective affinities of their moments. But further, because the fundamental determination is quantitative, the exclusive elective affinity also continues itself into the opposed neutralities; and this continuity is not only an external relation of the different ratios of the neutralities in the form of a comparison, but the neutrality is, as such, separable into the moments which united to produce it, since it is as self-subsistent somethings that these enter into relation indifferently with one or the other of the opposite series, although combining in different, specifically determined amounts. This measure, based on such a relation, is thus infected with its own indifference; it is in its own self something external and alterable in its relation to itself.
The relation to itself of the measure relation is distinct from its externality and alterableness which represent its quantitative aspect. As related to itself in contrast to these, it is an affirmatively present [seiende], qualitative foundation — a permanent, material substrate which, as also the continuity of the measure with itself in its externality, must contain in its quality the principle of the specification of this externality referred to above.
Now the exclusive measure as thus more precisely determined is external to itself in its being-for-self and hence repels itself from itself, positing itself both as another measure relation and also as another, merely quantitative, relation; it is determined as in itself a specifying unity which produces measure relations within itself. These relations differ from the affinities of the kind above-mentioned in which a self-subsistent measure relates itself to self-subsistent measures of a different quality and to a series of such. They take place in one and the same substrate within the same moments of the neutrality; the self-repelling measure develops other, merely quantitatively different relations which likewise form affinities and measures, alternating with those which remain only quantitatively different. They form in this way a nodal line of measures on a scale of more and less.
Here we have a measure relation, a self-subsistent reality which is qualitatively distinguished from others. Such a being-for-self, because it is at the same time essentially a relation of quanta, is open to externality and to quantitative alteration; it has a range within which it remains indifferent to this alteration and does not change its quality. But there enters a point in this quantitative alteration at which the quality is changed and the quantum shows itself as specifying, so that the altered quantitative relation is converted into a measure, and thus into a new quality, a new something. The relation which has taken the place of the first is determined by this, partly according to the qualitative identity of the moments which are in affinity, and partly according to the quantitative continuity. But because the difference falls into this quantitative aspect, the relation between the new something and its predecessor is one of indifference; their difference is the external one of quantum. The new something has therefore not emerged from or developed out of its predecessor but directly from itself, that is, from the inner specifying unity which has not yet entered into existence. The new quality or new something is subjected to the same progressive alteration, and so on to infinity.
Since the progress from one quality [to another] is in an uninterrupted continuity of the quantity, the ratios which approach a specifying point are, quantitatively considered, only distinguished by a more and a less. From this side, the alteration is gradual. But the gradualness concerns merely the external side of the alteration, not its qualitative aspect; the preceding quantitative relation which is infinitely near the following one is still a different qualitative existence. On the qualitative side, therefore, the gradual, merely quantitative progress which is not in itself a limit, is absolutely interrupted; the new quality in its merely quantitative relationship is, relatively to the vanishing quality, an indifferent, indeterminate other, and the transition is therefore a leap; both are posited as completely external to each other. People fondly try to make an alteration comprehensible by means of the gradualness of the transition; but the truth is that gradualness is an alteration which is merely indifferent, the opposite of qualitative change. In gradualness too, the connection of the two realities, whether these are taken to be states or self-subsistent things, is eliminated; gradualness necessarily implies that neither of the two is the limit of the other but that each is completely external to the other. In this way there is eliminated the very factor which is necessary for an understanding of change, although little enough is required for that purpose.
Remark: Examples of Such Nodal Lines; the Maxim, ‘Nature Does Not Make Leaps’
The system of natural numbers already shows a nodal line of qualitative moments which emerge in a merely external succession. It is on the one hand a merely quantitative progress and regress, a perpetual adding or subtracting, so that each number has the same arithmetical relation to the one before it and after it, as these have to their predecessors and successors, and so on. But the numbers so formed also have a specific relation to other numbers preceding and following them, being either an integral multiple of one of them or else a power or a root. In the musical scale which is built up on quantitative differences, a quantum gives rise to an harmonious relation without its own relation to those on either side of it in the scale differing from the relation between these again and their predecessors and successors. While successive notes seem to be at an ever-increasing distance from the keynote, or numbers in succeeding each other arithmetically seem only to become other numbers, the fact is that there suddenly emerges a return, a surprising accord, of which no hint was given by the quality of what immediately preceded it, but which appears as an actio in distans, as a connection with something far removed. There is a sudden interruption of the succession of merely indifferent relations which do not alter the preceding specific reality or do not even form any such, and although the succession is continued quantitatively in the same manner, a specific relation breaks in per saltum.
Such qualitative nodes and leaps occur in chemical combinations when the mixture proportions are progressively altered; at certain points in the scale of mixtures, two substances form products exhibiting particular qualities. These products are distinguished from one another not merely by a more or less, and they are not already present, or only perhaps in a weaker degree, in the proportions close to the nodal proportions, but are bound up with these nodes themselves. For example, different oxides of nitrogen and nitric acids having essentially different qualities are formed only when oxygen and nitrogen are combined in certain specific proportions, and no such specific compounds are formed by the intermediate proportions. Metal oxides, e.g. the lead oxides, are formed at certain quantitative points of oxidation and are distinguished by colours and other qualities. They do not pass gradually into one another; the proportions lying in between these nodes do not produce a neutral or a specific substance. Without having passed through the intervening stages, a specific compound appears which is based on a measure relation and possesses characteristic qualities. Again, water when its temperature is altered does not merely get more or less hot but passes through from the liquid into either the solid or gaseous states; these states do not appear gradually; on the contrary, each new state appears as a leap, suddenly interrupting and checking the gradual succession of temperature changes at these points. Every birth and death, far from being a progressive gradualness, is an interruption of it and is the leap from a quantitative into a qualitative alteration.
It is said, natura non facit saltum [there are no leaps in nature]; and ordinary thinking when it has to grasp a coming-to-be or a ceasing-to-be, fancies it has done so by representing it as a gradual emergence or disappearance. But we have seen that the alterations of being in general are not only the transition of one magnitude into another, but a transition from quality into quantity and vice versa, a becoming-other which is an interruption of gradualness and the production of something qualitatively different from the reality which preceded it. Water, in cooling, does not gradually harden as if it thickened like porridge, gradually solidifying until it reached the consistency of ice; it suddenly solidifies, all at once. It can remain quite fluid even at freezing point if it is standing undisturbed, and then a slight shock will bring it into the solid state.
In thinking about the gradualness of the coming-to-be of something, it is ordinarily assumed that what comes to be is already sensibly or actually in existence; it is not yet perceptible only because of its smallness. Similarly with the gradual disappearance of something, the non-being or other which takes its place is likewise assumed to be really there, only not observable, and there, too, not in the sense of being implicitly or ideally contained in the first something, but really there, only not observable. In this way, the form of the in-itself, the inner being of something before it actually exists, is transformed into a smallness of an outer existence, and the essential difference, that of the Notion, is converted into an external difference of mere magnitude. The attempt to explain coming-to-be or ceasing-to-be on the basis of gradualness of the alteration is tedious like any tautology; what comes to be or ceases to be is assumed as already complete and in existence beforehand and the alteration is turned into a mere change of an external difference, with the result that the explanation is in fact a mere tautology. The intellectual difficulty attendant on such an attempted explanation comes from the qualitative transition from something into its other in general, and then into its opposite; but the identity and the alteration are misrepresented as the indifferent, external determinations of the quantitative sphere.
In the moral sphere, in so far as it is considered under the categories of being, there occurs the same transition from quantity into quality and different qualities appear to be based in a difference of magnitude.
It is through a more or less that the measure of frivolity or thoughtlessness is exceeded and something quite different comes about, namely crime, and thus right becomes wrong and virtue vice. Thus states, too, acquire through their quantitative difference, other things being assumed equal, a distinct qualitative character. With the expansion of the state and an increased number of citizens, the laws and the constitution acquire a different significance. The state has its own measure of magnitude and when this is exceeded this mere change of size renders it liable to instability and disruption under that same constitution which was its good fortune and its strength before its expansion.
C. THE MEASURELESS
The exclusive measure, even in its realised being-for-self, remains burdened with the moment of quantitative determinate being and is therefore open to movement up and down a scale of fluctuating ratios. Something, or a quality, based on such a ratio is impelled beyond itself into the measureless and is destroyed by the mere alteration of its magnitude. Magnitude is that side of determinate being through which it can be caught up in a seemingly harmless entanglement which can destroy it.
The abstract measureless is the quantum as such which lacks an inner significance and is only an indifferent determinateness which does not alter the measure. Measure in the nodal line of measures is at the same time posited as specifying and the abstract measureless raises itself into a qualitative determinateness; the new measure relation into which the original one passes is, with respect to this, measureless, but in its own self it is equally a quality on its own account. Thus there is posited the alternation of specific existences with one another and of these equally with relations remaining merely quantitative — and so on ad infinitum. What therefore is present in this transition is both the negation of the specific relations and the negation of the quantitative progress itself — the infinite which is for itself. The qualitative infinite, as simply a determinate being, was the eruption of the infinite in the finite as an immediate transition and vanishing of the latter in its beyond. The quantitative infinite on the other hand is, simply by virtue of its determinateness, the continuity of the quantum, a continuity of it into its beyond. The qualitative finite becomes the infinite; the quantitative finite is in its own self its beyond and points beyond itself. But this infinity of the specification of measure posits both the qualitative and the quantitative as sublating themselves in each other, and hence posits their first, immediate unity, which is measure as such, as returned into itself and therefore as itself posited. The transition of the qualitative, of one specific existence, into another, is such that all that occurs is an alteration of the specific magnitude of a ratio. Hence the alteration of the qualitative itself into the qualitative is posited as an external and indifferent change, as a coming together with itself; moreover, the quantitative, in being converted into the qualitative, into that which is determined in and for itself, sublates itself. This unity which thus continues itself into itself in its alternating measures is the truly persisting, self-subsistent material substance or thing.
What therefore is present here is [a] one and the same thing which is posited as the perennial substrate of its differentiations. This severance of being from its determinateness begins already in quantum as such; under the category of magnitude, a thing is indifferent to its affirmative determinateness. In measure, the thing itself is already in itself the unity of its qualitative and quantitative moments, the two moments which constitute the element of difference within the general sphere of being and of which one is the beyond of the other; in this way the perennial substrate has directly in its own self the determination of affirmative infinity. [b] This self-sameness of the substrate is posited in the fact that the qualitative self-subsistent measures into which the specifying unity is dispersed consist only of quantitative differences, so that the substrate continues itself into this differentiation of itself; [c] in the infinite progress of the nodal series there is posited the continuation of the qualitative moment into the quantitative progress as into an indifferent alteration, but equally too, there is posited the negation of the qualitative moment contained therein and hence of the merely quantitative externality too. The quantitative reference beyond itself to an other which is itself quantitative perishes in the emergence of a measure relation, of a quality; and the qualitative transition is sublated in the very fact that the new quality is itself only a quantitative relation. This transition of the qualitative and the quantitative into each other proceeds on the basis of their unity, and the meaning of this process is only to show or to posit the determinate being of such a substrate underlying the process, a substrate which is their unity.
In the series of self-subsistent measure relations the one-sided members of the series are immediately qualitative somethings (specific gravities or chemical substances, bases, alkalis, or acids for example), and then their neutralisations (by which must also be understood here the compounds of substances of different specific gravity) are self-subsistent and even exclusive measure relations, self-determined and mutually indifferent totalities of determinate being. Now such relations are determined only as nodal points of one and the same substrate. Consequently, the measures and the self-subsistent things posited with them are reduced to states. The alteration is only change of a state, and the subject of the transition is posited as remaining the same in the process.
Surveying the progressive determinations which measure has passed through we can summarise them as follows. Measure is, in the first instance, only the immediate unity of quality and quantity as an ordinary quantum which is, however, specific. As thus a specific quantity which is related not to another but to itself, it is essentially a ratio. It therefore also contains its moments as sublated and undivided within itself; as is always the case in a Notion, the difference in the ratio is present in such a manner that each of its moments is itself a unity of quality and quantity. The difference therefore is real and yields a number of measure relations which, as formal totalities in themselves, are self-subsistent. The two series formed by the sides of these ratios are the same constant arrangement for each individual member which, as belonging to one side, enters into relationship with all the members of the opposite series.
This unity as a mere arrangement is still quite external, and although it shows itself to be an immanent specifying unity of a self-subsistent measure distinguished from its specifications, it is not yet the free Notion which alone gives its differences an immanent determination: it is as yet only a substrate, a material, and for its differentiation into totalities, i.e., into differences embodying the nature of the unchanged substrate, it is dependent solely on the external, quantitative determination which shows itself at the same time as a difference of quality. In this unity of the substrate with itself the measure determination is sublated and its quality is an external state determined by the quantum. This process is equally the progressive determination of measure in its realisation and also the reduction of measure to the status of a moment.
Chapter 3 The Becoming of Essence
A. ABSOLUTE INDIFFERENCE
Being is the abstract equivalence — for which, since it is to be thought of by itself as being, the expression indifference has been employed — in which there is supposed to be as yet no determinateness of any kind; pure quantity is indifference as open to all determinations provided that these are external to it and that quantity has no immanent connection with them; but the indifference which can be called absolute is the indifference which, through the negation of every determinateness of being, i.e., of quality, quantity, and their at first immediate unity, measure, is a process of self-mediation resulting in a simple unity. Any determinateness it still possesses is only a state, i.e. something qualitative and external which has the indifference for a substrate.
But what has thus been determined as qualitative and external is only a vanishing determinateness; quality as thus external to being is the opposite of itself and as such is only the sublation of itself. In this manner, the determinateness is still only posited in the substrate as an empty differentiation. But it is just this empty differentiation which is indifference itself as a result; and indifference is thus concrete, a mediation-with-self through the negation of every determination of being. As this mediation it contains negation and relation, and what was called state is its immanent, self-related differentiation; it is precisely externality and its vanishing which make the unity of being into indifference and they are therefore within this indifference, which therewith ceases to be only a substrate and in its own self only abstract.
B. INDIFFERENCE AS INVERSE RATIO OF ITS FACTORS
We have now to see how this determination of indifference is posited within the indifference itself and how the latter is therewith posited as being for itself.
1. The reduction of measure relations which at first ranked as self-subsistent measures, establishes their common substrate; this is their continuation into one another and hence the indivisible self-subsistent measure which is wholly present in its differentiations. For this difference, there are present the determinations of quality and quantity which the measure contains, and everything turns solely on how these are posited in it. This, however, is determined by the fact that the substrate is in the first place a result and only in principle [an sich] a mediation; but because this mediation is not yet posited as such in the measure itself, this latter is in the first place a substrate and with respect to determinateness, indifference.
Consequently, at first it is essentially the merely quantitative external difference which is present in it; there are two distinct quanta of one and the same substrate which in this way would be their sum and therefore would itself be determined as quantum. But the indifference is this fixed measure, the implicit, absolute limit, only in relation to those differences in such a manner that it would not be in its own self a quantum or opposed in any way, either as a sum or even as an exponent, to other quanta whether sums or indifferences. It is only the abstract determinateness which falls into the indifference; the two quanta, in order that they may be posited in it as moments, are variable, indifferent, greater or smaller relatively to one another. Bounded however by the fixed limit of their sum, they are related to each other not externally but negatively, and this now is the qualitative determination of their relationship. They are consequently inversely proportional. This relation is distinguished from the earlier formal inverted relation or inverse ratio by the fact that here the whole is a real substrate and each of the two sides is posited as having to be itself in principle [an sich] this whole.
According to the stated qualitative determinateness, the difference is present, further, as two qualities, one of which is sublated by the other; but because both are held in a unity which they together constitute, neither is separable from the other. The substrate itself, as an indifference, is likewise in itself the unity of both qualities; therefore each side of the relation, too, contains both sides within itself and is distinguished from the other side only by a more of the one quality and a less of the other, and vice versa. The one quality is through its quantum only preponderant in the one side, and so, too, the other quality in the other side.
Consequently each side is in its own self an inverted relation. As formal, this relation recurs in the two distinct sides. These sides thus continue themselves into each other also in respect of their qualitative determinations; each quality is self-related in the other and is present in each of the two sides, only in a different quantum. Their quantitative difference is that indifference in accordance with which they continue themselves into each other and this continuation as the self-sameness of the qualities is in each of the two unities. The sides, however, each as the whole of the two determinations and hence containing the indifference itself, are thus at the same time posited as self-subsistent relatively to each other.
2. As this indifference, being is now the specification of measure no longer in its immediacy, but measure as developed in the manner just indicated; it is indifference first as in itself, the whole of the determinations of being which are resolved into this unity and secondly, as a determinate being, as a totality of the posited realisation, in which the moments themselves are implicitly the totality of the indifference, borne by this latter as their unity. But because the unity is only an indifference, and hence is held fast only as an implicit unity, and the moments are not yet explicitly self-determined, i.e. are not yet determined as sublating themselves into a unity within themselves and through one another, what is here present is simply the indifference of the unity itself towards itself as a developed determinateness.
This self-subsistent measure as thus indivisible is now to be considered in more detail. It is immanent in all its determinations and in them remains in unity with itself and unaffected by them; but [a] as remaining implicitly the totality, it possesses the determinatenesses which are sublated in it, only as groundlessly emerging in it. The implicit being of the indifference and this real being of the latter are unconnected; the deterininatenesses show themselves in the indifference in an immediate manner and the indifference is wholly present in each of them. Consequently, the difference between them is posited in the first place as sublated, therefore as only quantitative, but not as the self-repulsion of the indifference, and the indifference is not posited as self-determining but as being determinate and determined only externally. [b] The two moments are in an inverted quantitative relation-a to and fro in the scale of magnitude; but this fluctuation is determined not by the indifference, which is just the indifference of this fluctuation, but is determined herewith only externally. The principle of determination resides not in the indifference, but in something lying outside it. The absolute, as indifference, has in this aspect the second defect of the quantitative form, namely that the determinateness of the difference is not determined by the absolute itself; just as it has the first defect in the fact that the differences simply emerge in it, that is to say, the absolute's positing is immediate in character, is not the mediation of the absolute with itself. [c] The quantitative determinateness of the moments which are now sides of the relation constitutes this mode of their subsistence; through this indifference their determinate being is freed from the transition of the qualitative sphere. But in distinction from this their real being, they have an implicit subsistence in the fact that they are in themselves the indifference itself, each being itself the unity of the two qualities into which the qualitative moment splits itself. The difference between the two sides is restricted to this, that one quality is posited in the one side with a more and in the other with a less, and the other quality similarly, but conversely. Hence each side is in its own self the totality of the indifference. Each of the two qualities taken singly on its own account also remains the same sum which the indifference is; it continues itself out of the one side into the other and is not bounded by the quantitative limit which is thereby posited in it. In consequence of this the determinations come into immediate opposition and this develops itself into a contradiction which we have now to consider.
3. Namely, each quality enters within each side into relation to the other, and does so in such a manner that, as has been determined, this relation too is to be only a quantitative difference. If the two qualities are self-subsistent — taken, say, as if they were sensuous things independent of each other — then the whole determinateness of indifference falls asunder; their unity and totality would be empty names. But they are at the same time expressly determined as comprised in a single unity, as inseparable, each as having meaning and reality only in this one qualitative relation to the other. But now, because their quantitativity is simply and solely of this qualitative nature, each reaches only as far as the other. If the qualities are regarded simply as distinct quanta, then the one would reach beyond the other and would have in its more an indifferent determinate being which the other would not have. But in their qualitative connectedness, each is only in so far as the other is. From this it follows that they are in equilibrium; that by as much as the one increases or decreases, the other likewise would increase or decrease and in the same proportion.
Therefore on the basis of their qualitative connection there can be no question of a quantitative difference or of a more of the one quality. The more by which one of the correlated moments would exceed the other would only be a baseless determination, or else this more would only be the other itse@ again; but in this their equality, both would have vanished, for their determinate being was supposed to be based solely on the inequality of their quantum. Each of these hypothetical factors vanishes, whether it is supposed to be beyond or equal to the other. This vanishing appears to quantitative conception as a disturbance of the equilibrium so that the one factor becomes greater than the other; thus there is posited the sublating of the quality of the other and its lack of any support. One factor becomes preponderant as the other diminishes with an accelerated velocity and is overpowered by the first, which therefore constitutes itself the sole self-subsistent quality; but this being so, there are no longer two specific moments and factors but only the one whole.
This unity thus posited as the totality of the process of determining, in which it is itself determined as indifference, is a contradiction in every respect; it therefore has to be posited as sublating this its contradictory nature and acquiring the character of a self-determined, self-subsistent being which has for its result and truth not the unity which is merely indifferent, but that immanently negative and absolute unity which is called essence.
Remark: Centripetal and Centrifugal Force
The relationship of a whole which is supposed to have its determinateness in the quantitative difference of two factors determined qualitatively against each other, is applied to the elliptical motion of the celestial bodies. This example exhibits primarily only two qualities in inverse relation to each other, not two sides, each of which is itself the unity of both and their inverse relation. The fact, the inverse relation, rests on a firm empirical foundation, but the theoretical explanation of it involves a consequence which is overlooked, namely the destruction of the basic fact; or if, as is proper, the fact is retained it escapes notice that the theory proves to be meaningless in face of the fact. The ignoring of this consequence allows the fact and the theory conflicting with it to exist calmly side by side. The simple fact is that in the elliptical motion of the celestial bodies their velocity is accelerated as they approach perihelion and retarded as they approach aphelion. The quantitative side of this fact has been accurately ascertained by the untiring diligence of observation, and further, it has been reduced to its simple law and formula.
Hence all that can properly be required of a theory has been accomplished; but to reflective understanding this did not appear sufficient. For the purpose of a so-called explanation of the phenomenon and its law a centripetal and a centrifugal force are assumed as qualitative moments of the curvilinear motion. Qualitatively, their difference consists in the opposed direction of the moments, and quantitatively, the moments being determined as unequal, in the fact that as the one increases the other is supposed to decrease, and vice versa; then further, that their relationship is reversed again so that after a period during which the centripetal force has been increasing and the centrifugal force decreasing, a point is reached where the former decreases and the latter increases. But this way of representing the matter is contradicted by the essentially qualitative relation between their respective determinatenesses which makes their separation from each other completely out of the question. Each has meaning only with reference to the other; consequently, in so far as the one had an excess over the other, to that extent it would have no relation to it and the other would no longer exist. If it is assumed that one of them is at one time greater than the other, being related to it as the greater to the smaller, then what was said above applies, namely, that the greater would acquire absolute preponderance and the other would vanish; the other would be posited as a vanishing moment without any support; and nothing would be altered by supposing the vanishing to take place only gradually any more than by supposing that by as much as the vanishing moment decreases in magnitude the other increases, for the latter, too, is destroyed with the former, since it is what it is, only in so far as the other is. It requires but little consideration to see that if, for example, as is alleged, the body's centripetal force increases as it approaches perihelion, while the centrifugal force is supposed to decrease proportionately, the latter would no longer be able to tear the body away from the former and to set it again at a distance from its central body; on the contrary, for once the former has gained the preponderance, the other is overpowered and the body is carried towards its central body with accelerated velocity. Just as conversely, if the centrifugal force gains the upper hand when infinitely near to aphelion, it is equally contradictory that now, in the aphelion itself, it should be overpowered by the weaker force.
Further, it is evident that it would be an alien force which effected this reversal; and this means that this alternation of accelerated and retarded velocity of the motion cannot be ascertained or, as it is said, explained from the assumed determination of the factors although these have been assumed for the express purpose of explaining this difference. The conclusion which follows from the vanishing of one or the other direction and hence of the elliptical motion altogether, is ignored and concealed because of the undeniable fact that this motion does go on and pass from the accelerated into the retarded velocity. The assumed transformation of the weakness of the centripetal force in aphelion into a preponderant strength over the centrifugal force, and conversely in perihelion, implies first the conclusion arrived at above, namely, that each side of the inverse relation is in its own self the whole inverse relation; for the side of the motion from aphelion to perihelion (when the centripetal force is supposed to be preponderant) is still supposed to contain the centrifugal force which, however, diminishes as the other increases, while on the side of the retarded motion the centrifugal force is assumed to be present in that same inverse relation in an ever-increasing preponderance over the centripetal force. Consequently, on neither side has either force vanished but only become increasingly smaller up to the point of its transformation into a preponderance over the other. All that recurs then on either side is the defect characteristic of this inverse relation, namely, either each force is credited with an independent self-subsistence, the pair being merely externally associated in a motion (as in the parallelogram of forces), in which case the unity of the Notion, the nature of the thing itself, is left out of account; or else, since each is qualitatively related to the other by the Notion, neither can attain an indifferent independent subsistence in face of the other, a subsistence supposedly imparted to it by a more. The form of intensity, the so-called dynamic factor, does not help, because this too has its determinateness in quantum and consequently can express only as much force (which is the measure of its existence) as is opposed to it by the opposite force. Secondly, this sudden change round from a preponderance into its opposite implies the alternation of the qualitative determination of positive and negative; an increase in the one means an equivalent loss in the other. In the theory, the inseparable qualitative connectedness of this qualitative opposition is distorted into a succession in time; but it fails thereby to give an explanation of this alternation and more particularly of this distortion itself. The semblance of unity which is still implied in the increasing of one side with an equivalent decreasing of the other vanishes here completely; what is alleged is a merely external succession of the sides which only contradicts what is necessarily implied in their qualitative connectedness, namely that with the preponderance of one side the other must disappear.
The same relationship has been applied to the forces of attraction and repulsion for the purpose of explaining the different densities of bodies; and the inverse ratio of sensibility and irritability has also been invoked to explain from the inequality of these factors of life the various determinations of the whole and of health and also the variety of living species. This mode of explanation was supposed to form a basis for the natural philosophy of physiology, nosology and also zoology, but the confused hotchpotch of nonsense in which it became entangled through the uncritical use of these determinations of the Notion soon led to the abandonment in these spheres of this formalism which, however, is practised without restraint in the other sciences, particularly in physical astronomy.
Since absolute indifference may seem to be the fundamental determination of Spinoza's substance, we may add that this is indeed the case in so far as in both every determination of being, like every further concrete differentiation of thought and extension and so forth, is posited as vanished. If we stop short at the abstraction [of substance] then it is a matter of complete indifference what something looked like in reality before it was swallowed up in this abyss. But when substance is conceived as indifference, it is tied up with the need for determining it and for taking this determining into consideration; it is not to remain Spinoza's substance, the sole determination of which is the negative one that everything is absorbed in it. With Spinoza, the moment of difference — attributes, thought and extension, then the modes too, the affections, and every other determination — is introduced quite empirically; it is intellect, itself a mode, which is the source of the differentiation.
The relationship of the attributes to substance and to one another is not specified further than that they express the whole of substance, and their content, the order of things as extended and as thoughts, is the same. But by the determination of substance as indifference, the difference too, comes to be reflected on; whereas with Spinoza, the difference is an external, and more precisely a quantitative difference only by implication, now it is posited as such. The indifference does indeed, like substance, remain immanent in the differentiated moments, but only abstractly so, only implicitly; the difference is not immanent in the indifference, for as quantitative it is rather the opposite of immanence, and the quantitative indifference is rather the self-externality of the unity. Thus the difference is also not grasped qualitatively; substance is not determined as self-differentiating, not as subject. The immediate consequence with respect to the category of indifference itself is that in it the difference of quantitative or qualitative determination falls apart as we found in the explication of indifference; it is the dissolution of measure, in which both moments were directly posited as one.
C.Transition into Essence
Absolute indifference is the final determination of being before it becomes essence; but it does not attain to essence. It reveals itself as still belonging to the sphere of being through the fact that, determined as indifferent, it still contains difference as an external, quantitative determination; this is its determinate being, contrasted with which absolute indifference is determined as being only implicitly the absolute, not the absolute grasped as actuality. In other words, it is external reflection which stops short at conceiving the differences in themselves or in the absolute as one and the same, thinking of them as only indifferently distinguished, not as intrinsically distinct from one another. The further step which requires to be made here is to grasp that this reflection of the differences into their unity is not merely the product of the external reflection of the subjective thinker, but that it is the very nature of the differences of this unity to sublate themselves, with the result that their unity proves to be absolute negativity, its indifference to be just as much indifferent to itself, to its own indifference, as it is indifferent to otherness.
But we are already familiar with this self-sublating of the determination of indifference; in the development of its positedness, this determination has shown itself to be from every aspect a contradiction. It is in itself the totality in which every determination of being is sublated and contained; it is thus the substrate, but at first only in the one-sided determination of the in-itself, and consequently the differences, namely, the quantitative difference and the inverse ratio of factors, are present in it only in an external manner. As thus the contradiction of itself and its determinedness, of its implicit determination and its posited determinateness, it is the negative totality whose determinatenesses have sublated themselves in themselves and in so doing have sublated this fundamental one-sidedness of theirs, their [merely] implicit being [Ansichsein]. The result is that indifference is now posited as what it in fact is, namely a simple and infinite, negative relation-to-self, its inherent incompatibility with itself, a repelling of itself from itself. The process of determining and being determined is not a transition, nor an external alteration, nor an emergence of determinations in the indifference, but is its own self-relating which is the negativity of itself, of its [merely] implicit being.
Now these repelled determinations do not possess themselves, do not emerge as self-subsistent or external determinations, but first, as moments belonging to the implicit unity, they are not expelled from it but are borne by it as the substrate and are filled solely by it; secondly, as determinations which are immanent in the explicated unity, they are only through their repulsion from themselves. The being of the determinations is no longer simply affirmative as in the entire sphere of being, but is now a sheer positedness, the determinations having the fixed character and significance of being related to their unity, each consequently being related to its other and with negation; this is the mark of their relativity.
Thus we see that being in general and the being or immediacy of the distinct determinatenesses, no less than the implicit being, has vanished and the unity is being, an immediate presupposed totality such that it is this simple self-relation only as a result of the sublating of this presupposition, and this presupposedness and immediate being is itself only a moment of its repelling, the original self-subsistence and self-Identity is only as the resulting coming together with itself. Being, in its determining, has thus determined itself to essence, a being which, through the sublating of being, is a simple being-with-itself.