A. The Direct Ratio

1. In the ratio the determinateness of the one quantum is that of the other. The two have but one determinateness, or limit. Of two quanta standing in no ratio, each has its own determinateness, indifferent towards that of the other. But the quanta of the ratio have only one common determinateness, the exponent of the ratio.

In the immediate ratio this exponent is itself an immediate quantitative determination, or some quantum or other. It makes up the one quantum that alone is in the ratio as such. The quanta which make up the sides of the ratio are the quanta posited as sublated; they are not indifferent quanta, hence not two; rather each has its determinateness at the other; they therefore make up only one, the simple exponent, and they themselves are posited in this unity as indifferent ones.

2. The exponent is the simple determinateness of the ratio. But as such it is not the qualitative determinateness, but some quantum or other. As the qualitatively determined quantum it is the quantum that has the difference of itself, its beyond and otherness, at its own self. This is not the external difference of quantum whereby it is greater or smaller than another, but its qualitative determinateness, its own difference at its own self. But the difference of quantum at its own self is the difference of unit and amount. The unit is itself the simple, absolute being-determined; the amount, by contrast, is the indifferent moving to and fro upon determinateness, the indifference that quantum has as external. At the outset unit and amount were the moments belonging to quantum; now, at the same time, each of these moments appears as a quantum of its own; they are the determinations of its existence, the delimitations within which the otherwise merely external, indifferent magnitudes are posited over against one another.

These two moments of quantum make up the moments of the quantitative ratio itself; for it is the unity of the qualitative and the quantitative determinateness. Thus in it quantum is partly as determined in itself, partly as indifferent and external. It is therefore the immanent determinateness of quantum itself, or its quality, that the sides of the ratio have towards one another. The one quantum of the ratio is not merely amount; rather this amount is unit over against the amount, and essentially has this value and significance, to count as unit; and the other is likewise not merely amount in general, as an indifferent quantum in general, but is amount as over against the other quantum, insofar as this latter is the unit.

The exponent is this difference as simple determinateness; it is first quantum; as such it is the side of the amount. When that side of the ratio which is taken for the unit gets expressed as the numerical one, the other side, the amount, is then the exponent's own quantum; second, the exponent is the simple unity, the qualitative element of the quanta which are sides of the ratio; they are moments in this unity. If the one is determined, then the other too is determined through the exponent, and it is wholly indifferent how the first is determined; as something determined for itself, as an indifferent quantum, it no longer has any significance, but can just as well be any other, without altering the determinateness of the ratio, which rests upon the exponent alone.

3. Inasmuch as the sides of the ratio are determined over against one another through the moments of quantum, they therein properly make up only one quantum. Conversely they are not qualitatively determined over against one another, insofar as they are diverse quanta. — As regards the first respect, the one quantum has only the value of the unit, not of an amount; the other only that of the amount; according to their determinateness, therefore, they are not complete quanta. If the one is altered, then the other is increased or diminished by just as much; that is to say, simply only the one, the unit, is altered, and the other determinate side, the amount, always remains the same quantum. Accordingly, as quanta they are not qualitatively determined over against one another; or this alteration, in which the two behave as quanta, is no negative determination at the ratio as such. — The exponent for its part is only the amount of the ratio, and has no negative determination at its own self. — Insofar as the other side, that of the unit, is a quantum, there are present two indifferent quanta, the exponent or the amount as such, and that quantum, and they are present as indifferent ones, not determined through the ratio. But insofar as the other side counts as unit, the other, the amount or the exponent, is not determined through it, but is an indifferent quantum in general.

But quantum is posited in the ratio as infinite only insofar as it has in the other quantum its beyond, its non-being. The two sides of the ratio are not only determined as unit and amount; according to these moments they make up only one quantum; but they are both quanta, and inasmuch as they are this in the simple unity of the ratio, their negativity is essentially posited therein. — They are qualitatively referred to one another; but quality is essentially negation, and the one side in fact relates itself to the other only as an other, insofar as it is as a non-being, as a sublating of that other. The simple determinateness, the exponent, is in this way genuine determinateness insofar as it is not only immediate quantum that is, but at the same time quantum that is not, and in its simplicity is not something determined over against an other but something determined in itself, and thus the negative of its own self.

B. The Inverse Ratio

1. The ratio has now determined itself in such a way that the positedness of a quantum is at the same time as the non-being of this quantum. In the inverse ratio there is present this, that the same quantum is posited as one that is and as one that is not. The one side of it relates itself to the other in such a way that, however great the one is, by so much is the other lacking. By as much as the one increases, by so much does the other decrease.

The one of the magnitudes standing in this ratio therefore does not continue itself into the other in such a way that it would remain the unit of its other, the amount; rather it continues itself negatively into it; it sublates in it as much as it itself is. Each is, as amount, the negative of the other; each is as great as what is wanting to the other. Each in this way contains the other and is measured by it; for each is only the quantum that the other is not. — The continuity of each in the other makes up the moment of simplicity in this ratio. The one quantum is the non-being of the other; hence neither is an indifferent one; rather it is, first, as its own self, and second it is as negated, and so it is the other. Insofar as it is as its own self, it is the negation of the other quantum.

These two sides, which each of the two magnitudes standing in the ratio has, do not fall apart, or into an external reflection which would merely compare them and find that the one quantum is less than the other, that in the one there is a being which in the other is a non-being. What a quantum is not for this external reflection, or in the comparison, is no concern of that quantum. What in that comparison the smaller quantum is, or its positive magnitude, is not a non-being, a lack of the other, of the greater; rather the greater contains the smaller within itself.

But in the negative reference of the quanta, in which they stand in the inverse ratio, otherness is their own restricting, and non-being is a lack and an ought of their own comparison with themselves. Therein quantum has another quantum over against it in such a way that it is at its own self this other quantum, but at the same time as its non-being. The alteration of the one is therefore this doubled thing: first, that it is an alteration of the one as a quantum that is, and at the same time of its other side, namely of its non-being or of the other quantum; second, however, that what thereby becomes being of the one is non-being of the other; and so it is not, as in the direct ratio, properly only the one side, the unit, that alters.

2. Quantum in the inverse ratio therefore goes out beyond itself in such a way that it has its determinateness in that to which it is referred; it has it therein as in its non-being, and precisely thereby, because its non-being makes it into what it is, this non-being of its is its own self.

The one quantum in this way makes up with its other one sphere; each of the two quanta is itself this whole. This whole is thus here the exponent. It is the limit and the simple determinateness of this ratio. It is, first, the simple determinateness of that ratio as immediate quantum. As such it is some indifferent magnitude or other; the whole as quantum that is. For the quantitative ratio has quantum in general for its foundation. — In this immediate determinateness it is the limit of the sides of its ratio, within which they increase and decrease over against one another, but which they cannot overstep. It makes up their limit, their non-being, in that it is the whole that is, whereas the sides are the whole only in such a way that according to one part they are and according to the other they are not. It is thus their beyond, to which they approach infinitely, but which they cannot reach. This infinity, in which they approach it, is the bad infinity of the infinite progress; it is itself finite, limited by its opposite, and therefore only an approximation; for neither of the quanta can overcome the other and reach the whole, but remains affected by this its negation, its other. But the bad infinity is here posited as what it is in truth, namely only as a moment of the whole, of the exponent. It is at the same time sublated, the beyond is reached; for the sphere is the unity of the beyond and the this-side of each of the two magnitudes; the beyond of each is the other, and each is in itself its other, each is in itself this whole.

3. Of the two magnitudes of the negative ratio the one increases as the other decreases, and conversely; the being of the one is essentially the non-being of the other. But this makes up no difference between them; for the same is the case with the one as with the other. Their quantitative difference — which is the greater or the smaller, or whether they are equal — is in any case their indifferent difference; and indeed in the ratio it is posited as unessential; they count only as such as can increase or decrease. It is therefore not to one of the sides over against the other, but to the two of them together, the whole sphere, that the difference belongs.

The whole now, or the exponent, is, as it has resulted, an immediate quantum that makes up the limit for the quanta contained under it. It is not only immediate quantum, but is being-differentiated at its own self, at first into two sides, each of which is in itself the whole sphere, is its own self, and essentially also has the other at it as its negation. Thereby the whole itself is posited in a doubled way. — First, it is the sum of the two sides, insofar as they are quanta that are, the whole quantum that is. But second, this whole is also as a negative one. For each of the two sides is the lack, or is as the being-negated of the other; each is as great as what is missing to the other. Hence the whole too is at the same time posited as an ought, as something negated. As has been recalled, each is not merely a non-being of the other in an external reflection; rather this is here its value, that the non-being of each is the other; both are thus, and thereby the whole, posited as a non-being.

But herewith, third, this being and non-being are one and the same. The whole sphere is at first immediate quantum; then it is posited as a non-being; but precisely this its non-being is itself only the whole sphere that is. For each of the two sides, insofar as it is the negation of the other, has existence; what vanishes from the other accrues to it; the non-being of each therefore makes up what the other is, and sublatedness is in this mutuality the existence of that which is sublated.

What is present, therefore, consists in this, that the exponent of the ratio, an immediate quantum, is as its non-being, as an other, but that this otherness is its own self. Quantum continues itself into its otherness, and the negation is only an otherness in which it preserves itself as the sphere lying at the ground, and remains the unity in this otherness.

Thus the inverse ratio, as it appears according to its determination, is sublated. It consists in this, that quantum was therein supposed to refer itself to its other in such a way that this other should be only its non-being, that the positive of its beyond should be a quantum diverse from it. But the nature of quantum is to be an indifferent limit, and hence to have sublated this non-being, the absolute limit, and to preserve itself within it.

The inverse ratio is therefore such an ought as has sublated its restriction, its otherness; an infinity which as beyond has at the same time vanished and has returned into unity with its this-side.

Inasmuch as quantum continues itself in this way into its otherness, it is the unity of itself and of its otherness. It lies at the ground of its otherness; it is the unity of that otherness. Thus the direct ratio has restored itself again. But at the same time in such a way that the other is not an immediate quantum, but simply has its determinateness, its otherness, only in the unity itself.

The ratio has passed over into the ratio of powers.

C. The Ratio of Powers

1. The ratio of powers has, according to what has resulted, sublated on the one hand the externality with which the direct ratio is afflicted, namely the indifference of the determination of the quantum which is unit towards the other quantum which is amount or exponent, — and the opposed non-being, the abstract qualitative determinateness of the inverse ratio.

Otherness, or the difference of quantum, is at first multiplicity, but qualitatively determined, in such a way that it relates itself to another quantum as amount to its unit. Now, in the ratio of powers, the unit, which is amount at its own self, is at the same time the amount over against itself as unit. Or the otherness, the amount of the unit, is the unit itself.

Quantum raises itself into its power insofar as it becomes an other to itself; but this otherness of its is at the same time bounded purely through its own self. Insofar as in the direct ratio it is unit, it is also the unit of the amount; the side which is amount as such has the difference of quantum at it, it is an amount of units, these are amount, and indeed the amount which is the first side. But from this amount, which is the unit, the amount of the second side is distinguished; it is the exponent or an immediate quantum. In the power, however, the otherness, the side which in the ratio is as amount, is not distinguished from the amount insofar as this is its unit; or conversely, the power is a multitude of which each is this multitude itself. Thereby it at the same time contains the moment of the inverse ratio; the otherness, the amount as such, is determined through its first quantum. — Quantum is therefore, in the power, returned into its own self; it is immediately its own self and also its otherness.

The exponent of this ratio is now no longer an immediate quantum, as in the direct one. In the inverse ratio too it is, considered as the sum, indeed something mediated, but at the same time only an indifferent quantum; or, taken as the reference of the sum to one of the simply alterable sides of that sum, it is only this simply alterable quantum. — In the ratio of powers, however, the exponent is of a wholly qualitative nature, the simple determinateness that the amount is the unit itself, the identity of quantum with its own self in its otherness. Therein lies also its quantitative nature, that otherness, the limit or negation, is simply only as something sublated, that existence is continued into its otherness; for the truth of quality is precisely this, to be quantity.

2. The ratio of powers appears as an external alteration into which some quantum or other is transposed, and as if it could just as well be transposed into any other alteration. But this ratio has a closer reference to the concept of quantum; according to what has gone before, quantum has itself passed over into this alteration, and in this existence has reached its concept, or has realized itself therein in a complete way. This ratio is the exposition of what quantum is at its own self; it expresses that determinateness of quantum whereby it distinguishes itself from other. For quantum is the indifferent, sublated determinateness, that is to say, the determinateness which continues itself into its otherness and is therein equal to its own self. But such is quantum as ratio of powers; for its otherness is therein its own self. — In the direct ratio this quality of quantum, to be the difference of itself from its own self, is as yet posited only in general or immediately, so that there is still present the indifference of the two sides of the difference, not the difference of quantum from itself but from something external. In the inverse ratio quantum is the difference of itself from itself as from its non-being, the relating to itself as to its negation. In the ratio of powers, finally, it is the difference of itself as from its own self; its otherness determined through itself, or simply continued into it.

Quantum has thereby exhibited itself not merely with a qualitative determinateness, but as quality. But to that extent it has at the same time passed over into another determination. For it has sublated the moment of its externality or indifference, which was its determination, and has become its other, quality. That quantum steps into the ratio, and more determinately into the ratio of powers, appears at first as a mere constitution, as an externality of quantum. But in this externality the determination of quantum, which is itself externality, is sublated; this externality becomes external to its own self; — inasmuch as it thereby sublates itself, it just as much finds itself therein, or returns therein into itself, for externality is the determination of quantum itself.

Quantum is thus now the unity of its determination and of its becoming-other or its constitution; it is quality.

At first quantity as such appears over against quality; but quantity is itself a quality; a self-referring determinateness, distinguished from the determinateness that is other to it, from quality as such. But therewith it is itself a quality.

Yet it is not only a quality; rather the truth of quality itself is quantity; the former has passed over into the latter. But quantity, on the contrary, is in its truth the externality that has returned into its own self, the externality that is not indifferent. Thus it is quality itself, so that apart from this determination quality as such would not be anything further.

Quantity, which at first is determinateness in general, quantum, or rather quantum, is now no longer an indifferent or external determination, but that whereby something is what it is. The truth of quantum is to be measure.

Remark

In recent times the ratio of powers has been applied to determinations of the concept. The concept in its immediacy has been named the first power, in its otherness or in the difference, the existence of its moments, the second, and in its return into itself or as totality the third power. — The more precise significance of the particular powers does not, however, belong here; the power itself becomes once again a formal numerical ratio insofar as one proceeds to the second, the third, the fourth and so forth into infinity. Their significance as second, third, and so forth into infinity would depend on a conceptual worth of numbers in general, something already spoken of above.

As regards the application of the determination of power itself, in order to designate moments of the concept, it is evident that power belongs essentially to quantum. It is a becoming-other of quantum in which quantum itself remains. The difference is a difference of unit and multitude or amount, strictly nothing but an otherness of the quantum. It is quantum's own difference, in which quantum expresses itself as quality, or as that determinateness which it essentially is. The ratio of powers is therefore only the genuine difference of the particular concept of the quantum, not the difference of the concept itself. Yet quantum stands far below the concept; it does not contain the negativity that belongs to the nature of the concept in that negativity's own proper determination; differences that accrue to quantum are for this reason very superficial determinations for the concept itself.

Insofar as the expression of powers is used only as a symbol, there is as little to object against it as against symbols of another kind for concepts; but at the same time just as much as against all symbolism whatever, in which pure conceptual or philosophical determinations are supposed to be exhibited. Philosophy has no need of such help, neither from the sensible world, nor from representing imagination, nor from spheres of its own proper ground which are subordinate and whose determinations therefore do not suit higher circles and the whole. This is the same thing as when categories of the finite are applied to the infinite at all. As the current determinations of force, or of substantiality, cause and effect and so forth are unsuitable symbols for expressing, for instance, living or spiritual relations, so still more are the powers of quantum and numbered powers unsuitable for relations of that kind and for speculative relations generally.

Third Section. Measure

In measure quality and quantity are united. Being as such is immediate equality with its own self. This immediacy has sublated itself. Quantity is being that has gone back into itself; simple equality with itself as indifference towards determinateness. But this indifference shows itself to be pure externality, to have its determination not at its own self but in an other. The third is now externality referring itself to its own self; for the sake of the reference to itself it is at the same time sublated externality, indifference towards being-determined, in that it has at its own self its difference from itself.

Were the third taken as mere externality, it would be mode. — In this sense the third is not a return into itself; rather, since the second is the incipient reference to externality, a stepping outwards that still stands in reference to the original being, the third is the completed falling away. — Modality, among the categories of transcendental idealism, carries the significance of being the reference of the object to thinking. In this, from one side, only pure externality is contained; for the reference to thinking, which might be the moment of reflection into itself, is here rather externality itself; in the sense of transcendental idealism, that is, thinking is essentially external to the thing-in-itself. But insofar as the other categories too have only the transcendental determination of belonging to consciousness, modality, as the category of the reference to the subject, to that extent relatively contains the determination of reflection into itself. — With Spinoza, following upon substance and attribute, mode is likewise the third; his account makes of it the affections of substance, or whatever has its being in something else by means of which it gets grasped as well. This third is, according to this concept, only externality; as has been recalled elsewhere, in Spinoza rigid substantiality altogether lacks the return into its own self.

According to the foregoing, mode here has its determinate significance as measure. Measure is not yet the absolute return of being into itself, but rather its return into itself within its own sphere. It is the externality of quantum reflected into itself; through its reflection its worth has determined itself, namely to count for this, that it is being-in-itself. Quantum is the quality. The being reflected into itself, the being that holds good, consists therefore in the manner and way, in the more or the less, in the measure, in which something is. — This is the truth to which being has now determined itself, to be the equality of externality with its own self.

Quantum has in its return into itself sublated its externality and with that itself as quantum. But this sublating has at first the quan tum for its foundation; and the form of quantum which it has attained, that of being an indifference referring itself to itself, constitutes being-in-itself. Measure is the unity of quality and of quantity, of being-determined in itself and of being-determined externally, but the immediate unity of these; this immediate unity, however, is thereby qualitative determinateness over against the mediation and externality of quantum; the simplicity of its being-gone-back-into-itself stands over against the latter. Measure is therefore a reference of the qualitative and the quantitative in which these are still distinguished ones. In the movement, then, in which measure realizes itself, they compare themselves with one another in the determinate significance which they have over against one another; but they thereby posit themselves into the negative identity in which the determination of the immediacy of being absolutely vanishes and becomes essence.

The idea of essence already lies at hand in measure, namely that of being identical with itself in the immediacy of being-determined; or reflection, whose determinations subsist self-subsistently, yet in this self-subsistence are strictly only moments of their negative unity. In measure the qualitative is quantitative; it has an indifferent subsistence, the difference is indifferent to it; with that it is a difference which is none; it is sublated; this quantitativity is the return into itself, the being-in-and-for-itself which is essence. But in measure the qualitative and the quantitative, as has been recalled, at first still have their determinateness over against one another; it is the first negation of the externality of quantum; or the identity of the qualitative and the quantitative, the concept of essence, which in measure has already come to be, is not yet realized in its moments and thereby not yet posited.

Measure is, to begin with, immediate unity of what is qualitative and what is quantitative, so that

first, it is a quantum that has qualitative significance and stands as measure. — Measure, however, determines itself further to be that which is determined in itself, insofar as at its own self there is the difference of its moments, of qualitative and quantitative being-determined. These moments determine themselves further into wholes of measure, into the one immediately determined in itself and into the ratio specifying another; measure as the unity of them is something self-subsistent. — Measure thereby becomes

second, a ratio of specific quanta, as self-subsistent measures. But since their self-subsistence rests only on the quantitative ratio and on the difference of magnitude, they are in themselves the same, and are the passing over into one another. Considered more closely, measure thereby founders in the measureless. — This beyond of measure is the negativity of measure only at its own self; it is thereby

third, measure posited as an inverse ratio of measures. In this ratio the qualitative difference of the self-subsistent ones becomes their identical reference, and their indifferent immediacy consists in the reflection into this their negative immediacy and unity, which is essence. The indifference and immediacy of the self-subsistent sides themselves make up their negative immediacy, which is essence.

Chapter 1. Specific Quantity

Qualitative quantity is at first a specific quantum. But it becomes

second a rule, which is not itself a quantum but rather a specifying of the quantitative, a sublation of quantum in its indifference. What the rule holds within it are the two moments of measure as distinguished ones, namely quantitative determinateness that is in itself, along with external quantum. By virtue of this difference each of the two sides turns into a quality, while the rule turns into a ratio; measure accordingly exhibits itself

third as a ratio of qualities which at first have One measure; but which further also specify themselves into peculiar measures over against one another.

A. The Specific Quantum

Measure is the simple reference of quantum to itself, quantum's own determinateness at its own self; so taken, quantum is qualitative. In this immediate unity with itself it is a quantum which makes up the quality of something; an immediate measure. It is a quantum, yet this limit, indifferent in itself, carries the determination of being not an indifferent externality but an externality referring itself to itself, one that does not go out beyond itself; thus it is determinateness gone back into simple equality with itself, determinateness that is one with its being, an immediate determinateness, a quality.

Insofar as one wishes, with this immediacy, to let the forms of existence come back and to make a proposition out of the determination obtained, one may put it thus: everything that is has a measure. This magnitude belongs to the nature of something itself, or rather it alone makes up something's determinate nature and its being-within-itself. Something is not indifferent towards this magnitude, as though, were the magnitude altered, it would stay what it is; on the contrary, altering the magnitude would alter its quality. In the shape of measure, quantum no longer stands as a limit that is none; what it now is is the determination of the matter itself, such that the matter would founder were it augmented or lessened beyond this measure. — A measure, taken as a standard in the ordinary sense, is a quantum which is taken as the unit determined in itself over against an external amount, though for itself it is arbitrary. Such a unit may indeed also in fact be a unit determined in itself, like the foot and similar original measures; insofar as it is at the same time used as a standard for other things, however, it is for these only an external measure, not their original one. — Thus the earth's diameter, or the length of the pendulum, may be taken for itself as a specific quantum. But which fraction of the earth's diameter or of the pendulum's length one wishes to take, and the latter under which degree of latitude, in order to use it as a standard, is arbitrary. Still more, however, is a standard of this kind something external for other things. These have specified the universal specific quantum once again in a particular way, and have thereby made themselves into particular things. Besides, a universal standard has anyway no other office than that of the comparison carried out from outside; on this shallowest reading of it, where it counts as universal measure, whatever serves the purpose is entirely a matter of indifference. It is not supposed to be a fundamental measure in the sense that the natural measures of particular things would be exhibited by reference to it and known from it according to a rule, as specifications of One universal measure, the measure of their universal body. Lacking this sense, however, an absolute standard loses its significance and its interest. —

The immediate measure is a simple determination of magnitude; as, for instance, the specific gravity of metals, the magnitude of organic beings, of their limbs and so forth. — But existing in this way as quantum, it is indifferent magnitude, open to external determination and capable of running up and down in the more and the less. Yet as measure it is at the same time determinateness in itself, and to that extent is distinct from itself as quantum, as a wholly indifferent determination, and is rather the negative of this indifferent immediacy. Measure is what quantum is in itself; it therefore has throughout the doubled side of being quantum as something that is in itself, and quantum as something external or immediate. As the latter it is the indifferent limit; measure itself, however, is simple, inner determinateness of quantity, which sublates the alteration of the external quantum and thereby proves and preserves itself as determinateness that is in itself.

It is essentially not itself a fixed quantum, but a rule of quantum.

B. The Rule

The rule has

first the qualitative and quantitative determinateness of magnitude for its moments;

second, these moments separate into the difference of quality and its quantitative determination;

third, these two sides determine themselves into qualities over against one another.

1. The Qualitative and Quantitative Determinateness of Magnitude

The rule is at first the specific determining of external magnitude. It contains the two determinations of the qualitative and the quantitative. In their difference these are at the same time within the unity of the rule. In this unity they are moments, each in essential reference to the other. The rule is thus measure as this reflected unity of its self-differentiating moments.

It is therefore first the magnitude determined in itself, or rather determinateness of magnitude; this moment is not itself quantum, but the qualitative as determining the quantum. Second, it has quantum as the side of externality, of being-for-other; this side moves to and fro in indifferent increase and decrease; but its reference to the first moment is its essential being, namely to be sublated in respect of its indifference.

To something, insofar as it is a measure, an alteration of its magnitude comes from outside; it does not take on from this the arithmetical multitude. Its measure reacts against that, comports itself as something intensive over against the multitude, and takes the multitude up in a peculiar way. It alters the externally posited alteration, makes something other out of this quantum, and shows itself through this specification to be being-for-itself within this externality.

Two quanta arise in such comportment; the one is external multitude; the other the specifically taken-up one. — The latter is itself a quantum, and dependent upon the former. It is therefore alterable as well; but on that account it is not a quantum as such, but the outer quantum as specified in a constant manner. Measure has its existence, then, as a ratio, and the specific element in it is throughout the exponent of that ratio.

In intensive and extensive quantum it is, as emerged above in connection with these determinations, the same quantum which is present the one time in the form of intensity, the other time in the form of extensity. The underlying quantum suffers no alteration in this difference, which is only an outer form. In the rule, by contrast, quantum stands the one time in its immediate magnitude, whereas the other time it is taken, by way of the exponent of the ratio, in another amount.

The exponent, in which what is specific consists, may look at first like a fixed quantum, namely the quotient yielded by the ratio of the outer quantum to the one qualitatively determined. But so taken it would be nothing but an external quantum; by the exponent nothing else is to be understood here than the moment of the qualitative itself, which specifies quantum as such. For what stands in reference here is quantum and the qualitative; not two immediate quanta. — But the properly immanent qualitative element of quantum is, as has emerged, only the determination of power. It showed itself to be the determinateness of quantum itself that is in itself, so that quantum through its nature or concept is that which produces itself and raises itself into the power. Here this concept, as the determination that is in itself, has stepped over against quantum as the external constitution. For since, as emerged above, measure is immediate unity of quantum and quality, this unity is itself the qualitative and stands over against quantum as such. — Insofar as the specified as much as the external appears as quantum, they display the difference of their nature in their alteration. The external quantum has for its principle the numerical one; this makes up its being-determined-in-itself, and the reference of the numerical one is the external reference. The alteration of the immediate quantum which is determined by the nature of such quantum as such consists therefore in the accession of one such numerical one and again of one such and so forth. Should the external quantum therefore alter itself in arithmetical progression, then the specifying reaction of measure's qualitative nature calls forth a second series, one referring to the first and rising and falling with it, though in a ratio that no numerical exponent determines, one that is rather incommensurable with any number.

2. Quality and Quantum

The rule contains quantum in the doubled determination, as immediate and as specified, and the two are different quanta. The qualitative as specifying, the exponent of the ratio, is the negative reference to the immediate quantum, it has its existence as the specified quantum, and is the self-identical moment of this second quantum; the qualitative over against the immediacy of the first. Both sides are quanta, go out beyond themselves and have their beyond in the other; the qualified side is not itself indifferent towards quantum, but rather is strictly referred to it, and just thereby is itself quantum. Because both sides are quanta, external differences, their reference is what is determined in itself, the moment of the exponent insofar as it is simple unity with itself. In this reference the immediate and the specified quantum are themselves moments; the reference is the continuity in which both quanta are as the indifferent determinations. As the external quantum is immediate externality, so this reference is immediate being-determined-in-itself. It is a quality.

This quality and quantum make up two extremes over against one another, which mediate themselves through the specified quantum, which holds both moments, the qualitative and the quantitative, united within it. The qualitative separates itself out into abstract quality insofar as quantum, in its otherness, namely in its specification, attains equality with itself, and this equality with itself makes up its being-in-itself, which is indifferent towards quantum. This being-in-itself has the character of immediacy as of being, in opposition to the self-sublating and mediating immediacy of quantum. It is therefore a being, and indeed one negative towards this mediation, a determinate being; its determinateness, further, does not go out beyond its being; rather, since quantum goes out beyond itself, and the specified quantum itself only is in ratio to the first, that determinateness is the negative moment of both, the self-equal exponent as the simple reference of the two. It is therefore determinate being as quality.

This quality is thus immediate being, it has an existence, and this its existence is the quantitative, which is external quantum, and which is then determined through the quality of being, the quality of being immediately determined in itself. — It is therefore only here that quality has arisen, as that which has a quantum; it is pure quantity at which determinateness is as something indifferent. Insofar as it is first immediate determinateness, it is some one quality; but it is second posited as determined in reference to quantum, and so it is pure quantity; the externally determinable, which is indifferent towards this. But since quantum, as something sublated at quality, in that quality is quality, is through the return of quantum into its own self, quality is the negative unity of its own first immediacy and of quantum; it is something sublating, a reacting negation of its external being-determined. There is present a being-within-itself over against this its limit, and a determining being-for-itself over against this its existence.

This something that is for itself has a quality, a determinateness; this is constitution, and indeed this constitution is the quantum. Quality, however, no longer passes over into this its constitution, but preserves itself within it; for the constitution is the quantum that sublates itself and goes back into quality. Quality itself is properly only this determinateness, that of sublating the immediacy of quantum and specifying it. A further significance which quality has as some other determinateness is here inessential; a significance of that kind belongs only to that abstract moment according to which the qualitative exponent is quality, immediate, unreflected being-determined-in-itself in general, or according to which it is not exponent. But the qualitative, as it essentially is, namely as exponent, is what is for itself, and thus has its determination, whereby it distinguishes itself from others, solely in this, that it announces itself as something measure-determining; its nature consists in this rule which it is, and its existence in this negative comportment towards external immediacy. This comportment itself, however, consists more precisely, as emerged just before, in the qualifying, that is, in the raising into power of the external quantum.

— Thus, to adduce an example, temperature is a quality at which these two sides, that of being external and that of being specified quantum, are distinguished. As quantum it is an external temperature which, proceeding along the scale of arithmetical progression and considered as uniformly increasing or decreasing , is by contrast taken up differently by the various bodies situated within it, in that these bodies determine, through their immanent measure, the temperature received from outside. Insofar as different bodies are compared at one and the same temperature, the ratio-numbers of the comparison yield their specific heats, or their capacities. But the capacities of bodies alter at different temperatures. In the increase or decrease of temperature a particular specification shows itself. The ratio of the temperature that is represented as external to the temperature of a determinate body has no fixed exponent of ratio; the increase or decrease of the heat existing at the body does not proceed uniformly with the increase and decrease of the external heat. If therefore the outer temperature were represented as an abscissa and the other as an ordinate, then, as the former grew uniformly, a curved line would be described by the corresponding alteration of the latter. — A temperature is in this connection assumed as external in general, whose alteration is supposed to be merely external or purely quantitative. But it is the temperature of the air or else some other specific temperature, and looked at more closely the ratio ought therefore, strictly speaking, to be understood not as one holding between something merely quantitative and something qualifying, but as one holding between two specific quanta. As the specifying ratio will directly determine itself further, the moments of measure consist not merely in a quantitative side and a side qualifying the quantum of one and the same quality, but in the ratio of two qualities which at their own selves are measures.

3. Distinction of Both Sides as Qualities

What is for itself has its determination in its specific bearing towards quantum. It has two sides, the one qualitative, the being-determined-in-itself; the other the externally quantitative. But the former is only as reference to the latter; it is the sublated quantum; it therefore has that quantum for its presupposition and begins from it. Quantum accordingly is indeed only as sublated immediacy; but with this it has itself an immediacy over against its sublatedness, the qualitative. The qualitative and the quantitative are at large qualitatively distinguished from each other; quantity is itself a quality over against quality as such. Here in measure the quantitative itself bears itself as something qualitative; insofar as it is merely quantum, it bears itself only towards another quantum; here, however, it bears itself towards the qualitative. — Or, the quantitative side considered for itself, it is itself determined in itself. Quality, namely, as such is the moment of the simple determinateness of the exponent; over against this stands the other side, the reference of the external quantum to the specified one. This side is the quantitative as such; it does not contain one immediate quantum, but that quantum as ratio and as quantitative exponent; it is thus the quantitative itself as quality at large.

But these two qualities are furthermore still comprehended within measure; they have it for their foundation and make up One measure. For first, according to the first consideration, insofar as the two proper sides of measure, the specified and the external quantum, determine themselves into qualities, these two quantitative sides make up the determinateness which their qualities have over against each other. Second, according to the other consideration, the one quality is indeed the immediate being-determined-in-itself, and the whole difference of the quantities falls upon the other side, and this side is itself ratio and quality only insofar as it has the whole difference of quantum at it. Yet the former is now no longer pure quantity, at which the difference is indifferent; rather, since this difference, as a difference referring itself to itself, is itself the being-determined-in-itself, only thereby is the former a genuine quality and determined over against another. This determinateness, however, or the limit in which they refer themselves to each other, is the quantitative at large; they have it for their foundation; the qualitative has here no other significance whatever than this, to be the reference of quantum to itself.

They are accordingly now qualities which stand in the reference of measure to one another. On their abstract side, as quality at large, they have some particular signification or other, (for instance space and time). But furthermore they enter into the measure-relation as determinatenesses of magnitude, and of the determinatenesses of magnitude of measure the one is that amount which goes up and down in an external, arithmetical progression, the other an amount that is specifically determined through measure.

As concerns the difference of the sides in the comparison of their qualitative determination with their quantitative one, each is at first a particular quality at large. To that extent there lies in them no difference as to which of the two qualities, in respect of the determination of quantity, is to be taken as the merely externally quantitative one, and which as the one that alters in quantitative specification . If the one side, which is regarded only as quantum, stands to the other, say, as root to square, then it is all one at which of them the increase or decrease is regarded as merely external, proceeding in arithmetical progression, and which by contrast is regarded as determining itself specifically at this quantum. If one lets the side of the root proceed in arithmetical progression, then the other contains the corresponding squares, which make up the series that does not progress arithmetically; if by contrast one lets the side of the square alter in the arithmetical progression, then the other side contains the corresponding roots, and exhibits its alteration as not lying in an external progression but as specifically determined.

But the qualities are not indeterminately diverse over against each other, for they emerge out of measure and there lie at their foundation the two sides of measure, of the original ratio of quanta, which have qualitative signification, the one to be the indifferent, the other the qualitative determinateness of quantity. The qualities are therefore essentially distinguished according to the determinate character of the quantitative moments of measure. The one accordingly has, over against the other, the determinateness of being the extensive, externality at its own self; the other, however, that of being the intensive, what is within itself or negative over against the first; the former the real, indifferent, the latter the ideal, specific side. The quantitative moment of the latter is therefore also to be taken as the unit, and that of the former as the amount, the former as divisor, the latter as dividend in the simple ratio, or the former as root and the latter as the power or the becoming-other, in the specifying ratio. — Insofar now as such a ratio also has existence at the indifferent quanta of its sides, and alterations go on at the indifferent quantum, the specific side is to be exhibited as the foundation in arithmetical progression, whereas the external side is to be exhibited as altering within the specified series; for the former, as the side specific in itself, shows through its arithmetical progressing that it has quantum as something external; the external side, on the contrary, shows itself through its specified series to be one whose quantum is determined through another. — Or, insofar as the arithmetical progression is regarded as the natural rule, the side specified in itself proceeds within that progression, because it is itself what qualifies and determines; the other, however, proceeds in a series which shows itself to have its rule in an other.

Remark

What has here been set forth in respect of the connection of the qualitative nature of an existence with its determination of quantity in measure finds its application, for example, in this: that in velocity, as the direct ratio of space traversed to time elapsed, the magnitude of the time is assumed as denominator and the magnitude of the space by contrast as numerator. If velocity at large is a ratio of the space and the time of a motion, then it is indifferent which of the two moments should be regarded as the number or as the unit, as whole or as moment of the whole. But space, like the weight in specific gravity, is number, external, real whole at large, whereas time, like volume, is the ideal, the negative, the side of unit. — Further, however, upon this is founded the weightier relation, why in free motion, — at first the still conditioned one —, that of fall, the quantity of time and the quantity of space, the former as root, the latter as square, — or in the absolutely free motion of the heavenly bodies the period of revolution and the distance, the former by one power lower than the latter, — the former as square, the latter as cube, are determined over against each other. Fundamental relations of this kind rest upon the nature of the qualities of space and of time that stand in the relation, and upon the kind of reference in which they stand, either as mechanical motion, or as fall, or as free celestial motion; — namely insofar as the qualitative at large is to be laid at the foundation, not indeed as such, but as determinate concept, one that contains the determination of space and of time according to their qualitative as well as their quantitative nature. —

In respect of the absolute measure-relations it is to be recalled at large that the mathematics of nature, if it would be worthy of the name of a science, must essentially be the science of measures, — a science for which much has indeed been done empirically, but little scientifically. Mathematical principles of natural philosophy, — as Newton entitled his work, — if they were to fulfil this vocation in a deeper sense than he and the whole Baconian breed had of philosophy and science, would have to contain quite other things, in order to bring a light into these regions, still dark yet in the highest degree worth considering. — It is a great merit to come to know the empirical numbers of nature, for instance the distances of the planets from one another; but an infinitely greater one to make the empirical quanta vanish and to elevate them into a universal form of determinations of quantity, in such a way that what they come to be are moments of a law or of a measure; — immortal merits such as Galileo earned, for instance, in respect of fall, and Kepler in respect of the motion of the heavenly bodies. The higher thing, however, is to prove these laws. But this means nothing other than to cognize their determinations of quantity out of the qualities, or determinate concepts, that are brought into reference (such as time and space). But of this manner of proving there is as yet to be found no trace in those mathematical principles of the knowledge of nature, nor in the further labours of this kind. It was remarked above, on the occasion of that shine of mathematical proofs of relations of nature which is founded upon the misuse of the infinitely small, that the attempt to conduct such proofs in a properly mathematical way is a preposterous undertaking. These proofs presuppose their theorems from experience, and what they accomplish consists solely in bringing these to abstract expressions and convenient formulas. The whole real merit which is ascribed to Newton in preference to Kepler with reference to the very same objects will one day, once the sham scaffolding of proofs is deducted, — doubtless upon a purer reflection on what mathematics is capable of accomplishing and on what it has accomplished, — be restricted, with distinct knowledge, to that transformation of the expression.

C. Relation of Qualities

Measure has determined itself into a relation of qualities. The rule is at first only a qualitative bearing towards quantum as such. The qualities have at first only One measure, and are moments of it.

These qualities have the two sides: as qualities, first, to be indifferent towards their measure-reference as towards the quantitative side, and second, to stand in this reference. It has just been shown how their purely qualitative determination stands in reference to that determination which they have towards each other in the measure-relation. But their indifference towards measure has yet another side, namely the direct signification of their stepping forth out of measure. — The qualities, namely, are only through measure itself; for in measure there lies the moment of the immediacy determined in itself. But this moment, as immediacy, is the simple, unmediated quotient of measure, or it is measure sublated; for measure is the mediation, a being-determined-in-itself through the sublating of immediate quantum. Insofar therefore as they, outside measure and as sides free from its reference, are themselves only in reference to measure, they are only the negated measure, the qualitative determination of quantum sublated once again, or the restored immediate quantum. This moment belongs to the completion of the concept of quality as it is here determined; for quality resulted as the exponent of a ratio whose sides are the immediate and the specified quantum; it therefore itself contains both sides. As the two qualities were considered as qualities of One specifying measure and as the moments of the ratio of that measure, so in this determination there was present only the one of their sides, namely the qualitatively determined one, but not the side of immediacy. — Or, quality is at large the unity of being-in-itself and of being-for-other; the former is the specific, the latter the immediate quantum.

This side is accordingly their indeterminate constitution, the external quantum which accrues to them apart from the specific determination. But the side of quantum accrues to them only in reference to measure. Measure, as abstract immediate determinateness, is a determinateness as quantum, which however is measure-determinateness or exponent of an immediate direct ratio, one that has its sides at the moment of the qualities of being external quanta. The qualities are therefore immediate quanta only insofar as they are sides of this ratio; or conversely, the quanta into whose immediacy the qualitative moments of measure lower themselves have their immediacy solely in the determinateness over against an other.

Considering the quanta more closely, as they are determined in this direct ratio, it is their units whose determination of measure over against each other it is; and this determination of measure remains the same throughout all further specific determination of their amounts. (-- It is the ratio which for instance in motion expresses the space that the body traverses in the first moment of time; but it is the ratio remaining just as much in the second, third and so forth moments of time, and it expresses at large the ratio of a quantum of space that corresponds to a unit of time; that quantum of space is the unit to the further amount of space determined through the specifying measure. —) — This results more closely from the following. The specifying measure is the purely qualitative ratio, which is in and for itself insofar as in it quantum is in its essential quality; it is the form of the reference of quantum to itself in its otherness; but as this form it presupposes quantum as something immediate. The ratio of powers has some quantum or other for its foundation, a quantum which within that ratio bears itself towards itself. It is this immediacy that the specifying measure has at the first or immediate ratio. — The specifying ratio consists furthermore in specifying an external quantum; an indeterminate amount at large is altered into another, qualified quantum; there are amounts standing over against each other whose exponent, as quantum, is simply alterable; they have only a qualitatively determined one. But the amounts are amount of units; thus they are the qualities which make up the sides of measure; quantum is quality at first as reference of amount and unit; the qualification of quantum by powers, or the real qualification, is the measure-relation itself. — They are furthermore, within the ratio, sides determined over against each other; thus each of them has also its particular unit; and since these units at the same time belong to amounts which are essentially in the ratio, or since the qualities at large stand in the measure-relation, this side of them too, the units, is determined over against each other; or they have a measure. This measure of theirs is accordingly a ratio of them as units, hence not the specifying but an immediate direct ratio. — Or immediately, the specifying ratio is present only as purely qualitative; its simple reference to its own self is its immediacy. But this immediacy, as at the same time the immediacy of measure, is the exponent as quantum, and as ratio a direct ratio; it is thus that in which the specifying ratio has returned into itself.

The reference which has resulted is herewith present as follows. There is a first immediate ratio which lies at the foundation and whose exponent is not altered. Its sides alter their quantum, and indeed in such a way that the alteration of the one side proceeds as external in arithmetical progression, while that of the other side is qualitative and a series of specified quanta. The units of these two quanta, however, do not as units enter into this alteration of their amount; they remain in their first direct ratio, since they make up the immediately determined-in-itself moment of their sides, and within the purely qualitative ratio have the value of units standing in no ratio. But apart from that ratio these two units are over against each other a determinate quantum, and stand in an immediate ratio.

These two ratios, the specifying and the immediate direct one, show themselves to be the realized moments of measure. Measure, namely, contains the side of the immediacy of quantum, or of quantum as something indifferent. Since the moment is itself the whole, it is measure, and, in the determination of immediate quantum, the immediate direct ratio. — On the other side measure contains the essentially qualitative determination of quantum; thus it is the qualitative ratio over against that first direct ratio. Both sides of measure are accordingly themselves measure-relations.

Through this realization measure has returned into itself; it has become equal to itself in its other. For the qualitative side of measure referred itself at first to an external quantum; now, however, this side is itself measure. And indeed it is measure lying at the foundation. The specifying measure, in referring itself to immediate quantum and specifying it, has amount for its content; the qualified magnitude is, according to this content, indeterminate and dependent upon the external magnitude. In the direct measure-relation, by contrast, the units of the sides stand in reference; the unit is that of quantum which is determined in and for itself.

In the rule the qualitative and the quantitative are separated, and specification was that which made up measure; but measure is, according to its concept, this, that quantum is the qualitative. Here this has restored itself, that a quantum makes up the foundation of measure, but a quantum that is itself exponent and determined as ratio.

Measure is quality at large, as being-determined-in-itself. Quality is the unity of being-in-itself and of being-for-other, of determination and of constitution. These moments of it now have the closer content that being-in-itself or determination is a direct measure-relation, whereas being-for-other or constitution is the specifying measure. Since the two sides are themselves measures, and hence determination and constitution are in themselves the same, quality has become a self-subsistence.

Chapter 2. Relation of Self-subsistent Measures

In immediate measure quantum is the quality; the quantitative determination lies at the foundation. But as measure has determined itself into something self-subsistent, the qualitative determinateness is now the first; measure is a unit determined in itself which bears itself towards an amount. The self-subsistence of measure therefore rests upon an immediate ratio lying at the foundation; it is no longer the simple, merely external quantum that is supposed to be quality; rather it is quality insofar as it is at its own self a ratio. But this direct ratio is at the same time a relation to other measures, and to that extent it is a specifying one. It is accordingly

first a self-subsistent measure which bears itself towards others and in this bearing specifies them. This specification, however, is the bringing forth of other direct ratios, and hence of other measures; and the specific self-subsistence consists not in one direct ratio, but in the specific determinateness towards the series of self-subsistent measures.

Second, the direct ratios that thereby arise are measures determined in themselves and exclusive; but since their difference from one another is at the same time only quantitative, there is present a progression of ratios which is in part merely externally quantitative, but is also interrupted by qualitative ratios, and forms a nodal line of specific self-subsistent ones.

Third, however, in this progression there enters for measure the measurelessness at large, and more determinately the infinity of measure, in which the mutually exclusive self-subsistences are one with each other, and the self-subsistent steps into negative reference to its own self.

A. The Relation of Self-subsistent Measures

1. Neutrality

A something that is self-subsistent through its measure is in itself an immediate ratio, and this makes up its nature and the ground of its difference from others. It is its determination or its being-in-itself; insofar as it is a ratio, it is a quality. But then this something also refers itself to others, and yet in this reference it is self-subsistent, or preserves itself therein; thus it specifies the external quantum that comes to it. — This side is its constitution or being-for-other. As the relation of its determination to externality it is itself a quality. The something is a self-subsistent one in that it is the unity of these its qualities. It is therefore not here merely a quality that stands in reference to another quality.

The immediate ratio which the something is at its own self is now its genuine specific quantum. The exponent of this ratio is an immediate quantum, only in comparison with other ratios of the same kind; but this determination through another is no concern of the exponent; it is at its own self, in that it is ratio within itself. — Insofar as its sides alter as quanta, it preserves itself in them, like an immediate ratio at large; since the one side is the unit, the other is the amount, and what alters herein is only the unit, not the specific amount or the exponent. (-- Such a measure is the specific gravity of bodies.)

But furthermore this measure has a side of bearing towards others. This bearing concerns the amount. Through it, namely, the self-subsistent compares itself with others; it has therein externality at it, and thus sets itself into reference according to the exponent of the ratio it is in itself. Yet in this the exponent is essentially exponent, of qualitative nature; its reference to others is neither the indifferent immediacy of one quantum over against other quanta; nor a likewise external, indifferent alteration of it. Rather, since it is a quantum determined in itself, or quantitative quality, it bears itself as measure towards the external quantum and specifies it. But conversely, insofar as it is itself quantum, it is therein likewise altered. It is a reciprocal specification which sets out from immediately determined measures and is therefore not a measure determined in and for itself, but an external one. The specific bearing towards others is therefore indeed a negative directedness upon the immediate measure, for what is determined in itself steps through this bearing into externality; but the immediate measure makes up the foundation of the relation of reference that has arisen.

This reference is a neutralization of both sides; through their quantitative nature, which lies at the foundation of the reference, they continue themselves into each other, whereby their indifferent difference is posited, and since therein there lies at the same time the qualitative determination which they have, this determination too is modified. The unity of the qualitative is here not the passing over of the one quality into the other, nor is their result the merely negative of their mutual sublating; rather it is here posited that in their sublatedness they also preserve themselves; for their difference, being quantitative, is an indifferent one and one in which what is distinguished continues itself also into its otherness and preserves itself in its change. The self-subsistent therefore does not indeed remain in the neutralization what it immediately is; it exhibits its being-determined-in-itself only as a mode, as a manner and way of being-for-other; but conversely its alteration is just as much only a mode for it, and does not concern its determination in and for itself.

2. Specification of Neutrality

The fundamental measure of something self-subsistent, then, (its weight within itself, or its peculiar gravity,) is first a quantum, and in the combination which it enters into with others this quantum is altered; this quantum is second an exponent; it makes up the quality of the self-subsistent, and this quality is thereby altered; but third this alteration is only a modification, it is only as amount that it gets specified, and amount it is only in comparison with another. Now because the self-subsistent is indifferent towards this alteration in the neutralization, it enters with several into such neutral combinations. If it were of merely qualitative nature, then in the other it would have only its non-being; a quality has its determinateness only in one other. Only when wrapped up in the quantitative is the difference from an other also indifferent to it. The self-subsistent is not one quality but a negative unity of qualities, and therein it is an essential quantum. These combinations with several are now different ratios, which accordingly have different exponents. Only in the comparison with others does the self-subsistent have the exponent of its being-determined-in-itself; the neutrality with others makes up its genuine comparison with them, for through its own self it is its comparing with them. — The exponents of these ratios, however, are different, and it thereby exhibits its qualitative exponent as the series of these different amounts to which it is the unit; — as a series of specific comportment towards others. The qualitative exponent is in and for itself not an immediate quantum. The self-subsistent therefore distinguishes itself from others through the peculiar series of exponents which it, taken as unit, forms with other self-subsistents, whereas another self-subsistent, brought into reference with those very same ones and taken as unit, forms a different series.

The self-subsistent is, as has been considered, the unit for these exponents or amounts which express its comportment towards another. For it is the being-determined-in-itself over against its constitution, that is, over against itself as quantum. Its quantum as such is its externality, which gets modified; it therefore exhibits its quantitative being-determined not in one quantum, but rather exhibits this quantum as something alterable, and accordingly displays its being-determined-in-itself in a series of exponents. The ratio of this series within itself makes up the qualitative aspect of the self-subsistent, which in this manifoldness of quantitative determination is the unity with itself. — Insofar, then, as a self-subsistent forms a series of exponents with a series of self-subsistents, it is at first distinguished from a self-subsistent not of this series, but from another one with which it is compared, only by this, that the latter makes a different series of exponents with the same self-subsistents. But in this way these two self-subsistents would not be comparable, insofar as each is to be regarded as unit over against its exponents, and the two series arising in this way are indeterminately other ones. The self-subsistent, however, is not determined in itself as the unit which is a simple one, but essentially as ratio; it is indeed a one over against the number series of its exponents; a unit determined in itself it is neither as this one nor in the ratio to one of them, for then it would have its determinateness in a quantum as such; rather it has its being-determined-in-itself only in the ratio of the series, in the ratio which the series has within its own self. This series is its unit, and insofar as the other self-subsistent comparable with it is of the same kind in general, namely insofar as it likewise has, among the self-subsistents of the other sides, those with which it neutralizes itself, it too has its being-determined-in-itself in that series. But this series is determined in itself only insofar as its members have a constant ratio among themselves towards both; so it is their common unit. In this common unit alone lies the comparability of the two self-subsistents, which were assumed not as neutralizing themselves with each other but as indifferent towards one another. They are in this respect quanta over against each other, but as such they are comparable only in the common unit that has been pointed out.

Those self-subsistents, however, which neutralize themselves with the ones standing over against them that are merely compared among themselves, and which yield the series of exponents of the comportment of the latter, are in their own selves likewise self-subsistents; each of them is to that extent likewise to be taken as a unit which has the series of its exponents in the two first-named ones, merely compared among themselves, or rather in the indeterminately many, which exponents are the comparison numbers of those first-named ones among themselves; just as, conversely, the comparison numbers of the second series among themselves are likewise the series of exponents for the first series. Both sides are in this way series of numbers, in which each is first a unit over against the series standing over against it, in which series it has its quantum as a series of exponents; second, it is itself one of the exponents for the series standing over against it; and third, a comparison number in relation to the remaining numbers of its own series, and as this amount it has its being-determined-in-itself, or its unit, in the series standing over against it. — Insofar, then, as each unity-with-itself that comports itself as self-subsistent is determined in itself, it has this its unity in a series of exponents of its comportment standing over against it. Insofar as it is quantum or amount, it is a specified quantum among others, and thereby distinguishes itself from them. Its being-determined-in-itself, then, is the series standing over against it, which for the others of its own side is only the common unit; through its being-determined-in-itself it is therefore equal to the others. But it is an other over against them, or it has a comparison number and an indifferent quantum, insofar as it is specified and posited by an alien unit. Into this externality of its own self, then, the nature of the self-subsistent measure has inverted itself, insofar as it was supposed to be an immediate ratio which would be specifying over against another and would preserve itself indifferent within this specification. Its reference to itself was supposed not to suffer from its reference to another; but its reference to itself is at first an immediate ratio; its indifference towards another consists in the quantum; therefore its qualitative side is directed against its own self; its comportment towards another, as the truly qualitative, becomes that which makes up the specific determination of this self-subsistent; that determination therefore consists simply in the way and manner of comporting itself towards an other, and this way and manner is determined as much by the other as by the self-subsistent itself.

The externality into which the specific self-subsistence inverts itself is, considered more closely, the transition of the qualitative into the quantitative, and conversely of the quantitative into the qualitative, that has here set in. In measure they are in immediate unity generally; in real measure, in the specific self-subsistence, they are distinguished, but for the sake of their essential unity this distinguishing becomes a passing over of the one moment into the other. The self-subsistent measures are in themselves immediately determined; so they are quanta; but this determination overturns into a qualitative ratio towards others, into neutralization. Towards this negative unity they are indifferent, and it passes over into quantitative determination; in this reference they stand with several; these several are determined through the qualitative reference towards each other; but their difference is only the diversity of quantum. — But they are thus only several, and different quanta in general over against one another; the specific determinateness, the return of this comportment into itself, is not yet at hand.

Only, the neutralization of self-subsistents standing over against each other is their qualitative unity in such a way that the one has not passed over into the other within it, so that there is not merely one negation in general, but both are posited negatively within it; or in such a way that, since each preserves itself indifferently within it, its negation is in turn also negated. Their qualitative unity is thus an excluding unity that is for itself. The exponents, which are comparison numbers among themselves, first acquire their truly specific determinateness in the moment of their excluding one another. — Only so is their difference not merely the indifferent difference of quantum, but also of qualitative nature. At the same time, however, it is grounded, as is evident, upon the quantitative; namely, the self-subsistent comports itself towards a several of its qualitatively other side only because in this comportment it is at the same time indifferent; and through the quantitativity of the neutral reference this reference is in its nature infinite, not merely negation in general but negation of negation; an excluding unity that is for itself. Thereby the affinity of a self-subsistent towards the several of the other side is not merely an indifferent reference but an excluding one, an elective affinity.

3. Elective Affinity

In elective affinity the specifically self-subsistent has completely lost its first character, that of being immediately determined in itself; it is determined in itself only as a negative unity that is for itself. This unity has shown itself, as the passing-over of the quantitative and the qualitative that has gone back into itself, to be the absolute unity of the quantitative and of the qualitative. It is thereby so determined that, as a difference quantitative within itself, it falls apart within itself indifferently towards its own self, or, as qualitative within itself, comports itself negatively towards itself, — both are here the same, — and specifies itself in the manner shown. In this repelling the ratios in part separate into their universal qualitative sides, and in part these sides specify one another and thereby themselves, and exclude each other. Only out of this has the self-subsistent then emerged as a ratio in which what appears as an indifferent quantum is at the same time only a moment. This self-subsistent has the doubled reference of comporting itself neutralizingly towards another, and on the one hand of not passing over immediately into the other within this its self-sublating reference, but merely of modifying itself, and on the other hand of comporting itself as referring purely to itself, of excluding other ratios from this its modification and of keeping neutrality with them away from itself. In this comportment towards an other the self-subsistence consists, and indeed in such a way that it is just as much the comportment of the others towards it, or in general of all towards all. Further, every moment is just as much of a qualitative as of a quantitative nature; and so too is the last determination, the difference of those that exclude one another being a difference of quantum.

The continuity of a specific moment with its other is neutralization; it is also of a negative nature, specifying and excluding. But what is excluded from this elective affinity is at the same time one of the exponents; it is distinguished as quantum; and so what does the excluding is likewise a different quantum. In this side of excluding comportment the numbers have lost their continuity and their capacity for running together with one another; it is the more or less that has acquired this negative character, and that gives the precedence to the one exponent over others, and among these again to one over the rest. Only, since at the same time it is again merely quanta that exclude one another, a moment which can be regarded as self-subsistent puts itself, in excluding fashion, into specific neutrality with an exponent that is a more for it; but it is again also indifferent to a moment to receive this neutralizing quantum from several moments standing over against it, from each according to its specific determinateness over against the other. Although the excluding comportment of these is here what determines, this negative comportment nevertheless also suffers this inroad from the quantitative side.

Real measure therefore began from a direct ratio determined in itself, the ratio of the units, of what is simply determined in itself on the two sides, a ratio which as the immediate one was supposed to lie firmly at the base and to distinguish itself from the specifying, the qualifying ratio of the amounts. But it has shown itself that rather only this specifying ratio, which determines itself as comportment towards others, is the total ratio, and that that first immediate one passed over into this other. The units determined in themselves, which made up the sides of the direct ratio, have themselves become amounts, such as have their unit determined in itself in a series standing over against them, and as self-differentiating, excluding amounts are only members of a specified series. The immediate qualitative unit of the first ratio itself has passed over into the negative excluding unity, which is not a reference of immediate units determined in themselves, but a specifying of them. What was a neutralization of immediately present self-subsistents is a reference of quanta which have their existence solely in this qualifying negation, the same negation that also makes up their neutralization. What is herewith at hand is the negative reference of the immediate unit of the ratio and the specified unit, and thereby the qualitative difference of the quantitative and the qualitative itself. That immediate unit is thereby determined as indifferent immediacy in general, as quantum as such, and the specific as the qualitative. Since, further, the ratios now stand under these determinations and these determinations are referred simply to one another, there is in general an overturning of indifferent, merely quantitative comportment, and conversely a passing-over of the specific being-determined into the merely external ratio; — a series of ratios that are now of merely quantitative nature, now specific ones and measures.

Remark

The chemical substances are such measures, or moments of measure, as have just now resulted, ones that have what makes up their determination solely in their comportment towards others. Acids and alkalis, or bases in general, appear as things immediately determined in themselves, but at the same time rather as incomplete bodily elements, as constituents which properly do not exist for themselves, but have only this concrete existence, namely to sublate their isolated subsistence and to combine with an other. Their difference, by which they are self-subsistent over against one another, consists not in immediate qualities but in the quantitative way and manner of their comportment. This difference, moreover, does not stop at the chemical opposition between acid and alkali, or base in general, but is specified onward into a measure of saturation, and its seat is the specific determinateness belonging to the quantity of the mutually neutralizing substances. This determination of quantity with respect to saturation makes up the qualitative nature of a substance, it makes it into what it is for itself; the number that expresses this is essentially one among several exponents for a unit standing over against it. — Such a substance stands to another in so-called affinity. Insofar as this reference were to remain of a purely qualitative nature, then — as with the reference of the magnetic poles or of the electricities — the one determinateness would be merely the negative of the other, and the two would not also be at the same time indifferent towards each other. But because the reference is also of a quantitative nature, each of these substances is capable of neutralizing itself with several, and is not restricted to one standing over against it. It is not just the acid and the alkali or base that stand towards each other, but acids and alkalis or bases. They characterize themselves over against one another, within the difference of acids from acids and of alkalis from alkalis, according as their affinities comport themselves exclusively towards each other and one has the precedence over another, since taken by itself an acid can enter into a combination with all the alkalis, and conversely. It therefore makes up the chief difference of one acid over against another whether it has a closer affinity to a base than another has. And the closer affinity rests upon the difference of the multitude of it that suffices to saturate a qualitative moment standing over against it; it is therefore a ratio number through which the specific property of such a substance is expressed.

Concerning the chemical affinities of acids and alkalis, Richter and Guyton have found the law that when two neutral solutions are mixed and a separation thereby arises, the products are likewise neutral. It follows from this that the multitudes of two alkaline bases which are required for the saturation of one acid are needed in the same ratio for the saturation of another; in general, once one takes an alkali as unit and settles the series of ratio numbers in which the various acids saturate that same alkali, this series turns out the same for every other alkali, save only that the various alkalis are to be taken over against one another in different amounts; — amounts which on their side again form just such a constant series of exponents for each of the acids standing over against them, since they relate to each single acid in the same ratio as to every other. — Fischer has brought out these series from Richter's works in their simplicity (see his notes to the translation of Berthollet's Abhandlung über die Gesetze der Verwandtschaft in der Chemie, p. 232, and Berthollet, Statique chimique, Part I, p. 134 ff.)

As is well known, Berthollet has further modified the general representation of elective affinity through the concept of the efficacy of a chemical mass. This modification has no influence on the quantitative ratios of the chemical laws of saturation themselves, but only on the qualitative moment of exclusive elective affinity. Because the foundation of the qualitative comportment consists in determinations of quantity, this comportment is weakened by the indifferent nature of these. If for example two acids act upon an alkali, and the one which has a greater affinity to it is also present in the quantum capable of saturating the quantum of the base, then according to the representation of elective affinity only this saturation ensues; the other acid remains wholly without effect and excluded from the neutral combination. According to that concept of the efficacy of a chemical mass, by contrast, each of the two is effective in a ratio compounded of the multitude of it present and its capacity for saturation, or affinity. Berthollet's investigations have specified the closer circumstances under which the efficacy of the chemical mass is sublated, so that an acid more strongly akin appears to drive out the other, weaker one and to exclude its action, and thus to be at work in the sense of elective affinity. He has shown that it is circumstances, for instance the strength of cohesion, the insolubility in water of the salts formed, under which that excluding takes place, and not the nature of the agents themselves, — circumstances whose effect can in turn be sublated by other circumstances, for instance temperature. Through the removal of these hindrances the chemical mass comes into efficacy, and what appeared as purely qualitative excluding, as elective affinity, shows itself to reside merely in external modifications.

What, in the presentation of the text, are the immediate self-subsistent measures, the ratios determined in themselves which distinguish themselves from their comportment towards others, is represented by the specific gravities of bodies. — Within their own selves they are a ratio of weight to volume. The ratio exponent which expresses the determinateness of one specific gravity in distinction from others is at first a determinate quantum only of comparison; something that is a ratio external to them within the referring of an external reflection, and that is not grounded upon their own qualitative comportment towards a unit standing over against them. But since these differences, as determination in general, have a specifying unity lying at their base, and since determinate qualifying is an identity with self within its differentiating, a rule, — there is the task at hand of cognizing the ratio exponents of the series of specific gravities as a system deriving from a rule which specifies an arithmetical progression into a series of harmonic nodes; to every such node an exponent would have to correspond which is the quantum of the specific gravity of an available body. In this way the simple numbers of the specific gravities, — numbers which taken by themselves have a conceptless immediacy and can therefore display no order, — would appear as the ultimate results of ratios in which the specifying rule lying at the base would be recognizable. — The same demand is at hand for the cognition of the chemical affinity series adduced above.

The specific gravities, although at first they seem to have no qualitative ratio to one another, nevertheless likewise show themselves to stand in a qualitative reference. When bodies are chemically combined, or merely amalgamated or synsomatized, (even already when only the temperature alters) the union likewise shows itself as a neutralization of the specific gravities. As is well known, the volume even of the mixture of blended fluids or bases that are chemically indifferent to one another, properly speaking, is not of the same magnitude as the sum of the volume of the mixed constituents before their mixing. Within that mixture they mutually modify the quantum of their determinateness with which they enter into the reference, and in this way make themselves known as qualitative determinations over against each other. Here, then, the quantum of specific gravity expresses itself not merely as a fixed comparison number, but as a ratio number which, being displaceable, enters with others into a particular neutrality.

B. Nodal Line of Measure-Relations

What is at hand is the measure-relation which proves itself to be exclusive and thereby self-subsistent; the preference that the side of the relation grants to one of its exponents over others rests upon the quantum of that exponent as against the others. The more or less is what excludes, the qualitative. Conversely, however, it is the specific through which such a more or less is determined. The qualitative, which in this manner makes itself into a quantitative difference, becomes something external, something transient. At hand at large is the transition of the specific into the merely quantitative relation, and of the quantitative into the specific relation. Inasmuch as the qualitative relation makes itself into a quantitative difference, it subsists on the one hand in this difference; it remains therein what it is, and the quantitative is the indifference of its subsistence; it is this unity of the two, wherein the quantitative is determined through the specific, that makes up something self-subsistent. On the other hand, however, it is thereby altered; the quantitative is its other. Conversely, the quantitative for its part just as much makes up the foundation of the specific relation. Through the quantitative moment the self-subsistent maintains itself in its otherness; on the other hand, however, the quantitative moment in its unity with the specific is likewise altered. — Each of the two moments therefore comes forward as the determining one, in which the other is only as sublated, and thereby each is also as sublated.

The final determination of the measure-relation was that, as excluding, it is specific. But this repelling exclusion is in part in and for itself a reference to what is excluded and a mutual attraction of the two; in part, however, insofar as the indifferent subsistence of what is excluded is the quantitative moment, what excludes is indifferently distinguished from the other and continues itself into it. It continues itself into it on the one part as maintaining itself; its other is something quantitative, hence an indifferent difference which does not affect the specific; on the other hand, however, it is qualitatively distinguished from it; in this its otherness it becomes another relation and thereby another measure.

The specifying unity determines, as has emerged, numerical ratios which are of a qualitative nature and are measures. But the sides, or likewise the exponents, of these are amounts at large, hence what is in itself indeterminate and external. On the one part measure remains unaltered in this difference of its quantity; on the other part it is altered, and indeed not through itself, nor in such a way that it would maintain itself in its otherness as in that to which it refers; rather the quantitative into which it passes over is constitution, what is external in itself; it has therefore merely gone under therein. Yet since the quantitative itself has at the same time just as much a qualitative nature, another quantitative ratio becomes once again a measure too, and one determined in itself, which does not come from externality and mere constitution but hangs together with the preceding measure and stands in qualitative reference to it through a rule. What is at hand, then, is this doubled character. The transition from one measure into another is external, unconnected, the one is without the other, each appears as an immediate; they are distinguished by a more and less, and this is their reference in comparison, which is external and indifferent to them. But they also have a rule lying at their foundation, and they bear themselves towards one another as qualitative differences; for quantum has its determinateness in specification.

The self-subsistent measures, then, distinguished from one another both merely quantitatively and qualitatively, both wholly external to one another and determined through a rule, form a nodal line of measures upon a scale of the more and the less. A measure-relation is at hand; this is a self-subsistent existence, a reality qualitatively distinguished from others. Such an existence, because it rests upon a ratio of quanta, stands at the same time open to externality and to quantum-alteration, and to that extent what alters it is an indeterminate other at large, contingencies, external circumstances. It has a latitude within which it remains indifferent towards this alteration and does not change its quality. But a point of this change of the quantitative ratio sets in at which the quality is changed, or at which quantum proves to be specifying, where such another quantitative ratio takes place as is itself measure and thereby a new quality and a new something. Insofar as the preceding relation refers itself qualitatively to its other, the quantitative, in which it founders, inasmuch as the qualitative and the quantitative at large bear themselves qualitatively towards each other, so too is the relation that has stepped into the place of the first determined by the first. But this new something bears itself just as indifferently towards the preceding one, for their difference is the external one of quantum; it has therefore come forth not out of the preceding one but immediately out of itself. The new quality, or the new something, is subjected to the same going-on of its alteration, and so forth into infinity.

Insofar as the going-on from one quality runs within the steady continuity of quantity, the ratios drawing near to a qualifying point are, quantitatively regarded, distinguished only by the more and the less. On this side the alteration is a gradual one. But gradualness concerns merely what is external about the alteration, not what is qualitative about it. The preceding quantitative ratio, which is infinitely close to the one that follows, is still another reality. From the qualitative side, accordingly, the merely quantitative going-on of gradualness, which is no limit at its own self, is absolutely broken off, and since the newly entering quality is, for the sake of its quantitative difference itself, one indeterminately other than the vanishing quality, an indifferent one, the transition is a leap; the vanished quality and the newly entering one are wholly external. — One likes to make an alteration comprehensible to oneself through the gradualness of the transition; but gradualness is rather precisely the merely indifferent change, precisely the opposite of the qualitative one. In gradualness the coherence of the two realities, — they are taken as states, or as self-subsistent things, — is rather sublated; it is posited that neither is the limit of the other but that the one is simply external to the other, that within the merely quantitative going-on there show themselves ratios of quanta which are qualitatively distinguished from those immediately preceding and following them , and which, over against those merely external, qualitatively indifferent ones, present themselves as specific.