I.2.3. The Quantitative Ratio

The Infinity of Quantum has proved itself to be the negative Beyond of Quantum; but it belongs to Quantum. This Beyond is the Qualitative in general. Infinite Quantum, as the union of the two moments (quantitative and qualitative determinateness), is Ratio.

In Ratio the determinateness of Quantum no longer is indifferent: it is qualitatively determined as absolutely related to its Beyond. It continues itself into its Beyond; and this, for the present, is simply any other Quantum. But essentially they are related to one another not as external Quanta; each has its determinateness in this relation to the other. In this their otherness they have thus returned upon themselves; what each is, it is in the other; and the other constitutes the determinateness of each.—Thus the passing of Quantum beyond itself now has this meaning, that it did not merely change into an Other, nor into its abstract Other (its negative Beyond), but that it there reached its determinateness: it finds itself in its Beyond, which is another Quantum. The quality, or conceptual determinateness, of Quantum is its externality in general. Now in Ratio it is posited as having its determinateness in its externality and in another Quantum, and as finding its true nature in its Beyond.

The entities that have the relation which we have found, are Quanta. This relation is itself a magnitude; Quantum is posited, not only as being in a Ratio, but as being itself a Ratio; it is a Quantum in general which contains this qualitative determinateness within itself. Taken thus as Ratio, it expresses its existence as self-contained totality and its indifference to limit, through the fact that it has within itself the externality of its determinateness and in this externality is related only to itself, that is, is in itself infinite.

Ratio is divided into:—

(1) Direct Ratio. Here the qualitative element does not show forth for itself as such: it is in a purely quantitative manner that Quantum is posited as having its determinateness in its externality.—In itself Quantitative Ratio is the contradiction of externality and of self-relation, of the existence of Quanta and of their negation;—it transcends itself when next

(2) in Indirect Ratio there is posited the negation as such of one Quantum together with the alteration of the other, and the alterability of Direct Ratio itself;

(3) in Ratio of Powers, however, unity, which in its difference is self-related, asserts itself as simple self-production of Quantum; this qualitative element being finally posited in simple determination and as identical with Quantum, becomes Measure.

—In the preceding Observations, which deal with the infinite of Quantity (that is, with the qualitative moment in Quantity), much that is true of the nature of the following Ratios has been anticipated. All that remains, therefore, is to explain the abstract concept of these Ratios.

1. In that Ratio which is immediate and, therefore, direct, the determinateness of one Quantum is contained reciprocally in the determinateness of the other. There is only one determinateness or limit of the two, which is also itself a Quantum, and this is the exponent of the Ratio.

2. The exponent may be any Quantum; but it is a qualitatively determinate Quantum self-related in its externality only in so far as it has in itself its difference, its Beyond and otherness. But this difference of the Quantum in itself is the difference of Unit and Amount: Unit is self-determinateness; Amount, the indifferent oscillation about determinateness, the external indifference of Quantum. At first Unit and Amount were moments of Quantum; now, in Ratio (which to this extent is Quantum realized) each of its moments appears as an independent Quantum, and as determinations of its existence, as limitations of its otherwise purely external and indifferent magnitudinal determinateness.

The exponent is this difference as simple determinateness, that is, it immediately contains in itself the meaning of both determinations. It is, first, Quantum, and thus is Amount; if one side of the Ratio, which is taken as Unit, is expressed as numerical One—and as such only it is counted,—then the other, Amount, is the Quantum of the exponent itself. Secondly, it is simple determinateness as the qualitative element of the sides of the Ratio: if the Quantum of one is determined, the Quantum of the other too is determined by the exponent, and it is quite indifferent how the first is determined: as Quantum determinate for itself it no longer has a meaning, but can equally well be any other without changing the determinateness of the Ratio, which rests wholly upon the exponent. The one, which is taken as Unit, always remains Unit, however great it becomes; and the other, however great it likewise becomes in the process, must always remain the same Amount of that Unit.

3. Thus really both only constitute one Quantum; one relatively to the other has the value only of Unit and not of an Amount, and the other has only that of an Amount; and thus, according to their conceptual determinateness, they are not themselves complete Quanta. But this incompleteness is a negation in them,—not a negation which follows from their general alterability according to which one (and each is one of the two) can assume any magnitude, but according to the determination that when one is altered the other increases or decreases proportionally; this means (as has been shown) that only one (Unit) changes as Quantum; the other side (Amount) remains the same Quantum of units, and even the former has meaning only as Unit, however much it may change as Quantum. Thus each of the two sides is only one of the two moments of Quantum, and that independence which is its peculiarity, is negated in itself: in this qualitative connexion they must be posited as negative to each other.

The exponent is supposed to be the complete Quantum, for in it the determination of the two sides coincides; but in fact, as quotient, it only has the value of the Amount, or of the Unit. There is here no determination showing which of the sides of the ratio must be taken as Unit and which as Amount; if one (the Quantum B) be measured against the Quantum A as Unit, then the quotient C is the Amount of such Units; but if A itself be taken as Amount, then the quotient C is the Unit which the Amount A demands for the Quantum B; thus this quotient, as exponent, is not posited as what it is supposed to be, namely, the determinant of the ratio or its qualitative unity. It is posited as such only in so far as it has the meaning of being the union of both moments, of Unit and Amount. These sides are present as Quanta (such as they are supposed to be in explicit Quantum, which is Ratio), but also only in the value which they are supposed to have as being sides of the Ratio,—which means that they are incomplete Quanta and count only as one of these qualitative moments; and, therefore, they must be posited as having this their negation. Thus there arises a Ratio more real and better corresponding to its determination; the exponent here has the meaning of being their product; Ratio according to this determinateness is Inverse Ratio.

1. The Ratio which has now been reached is transcended Direct Ratio; the first stage was immediate and, therefore, not truly determinate Ratio; here determinateness has been added, in such a manner that the exponent counts as product, as the unity of Unit and Amount. With regard to immediacy, it could indifferently be taken as Unit and as Amount, as was shown above, and so it also existed as Quantum only in general, and hence, by preference, as Amount; one side had to be taken as Unit and as One, to which the other stood in the relation of fixed Amount, which also was the exponent; its quality, therefore, was only this, that this Quantum is taken as fixed, or rather that what is fixed has the meaning only of Quantum.

Now in Inverse Ratio, too, the exponent, as Quantum, is taken as immediate; and any Quantum can be taken as fixed. But this Quantum is not related as fixed Amount to the One of the other Quantum in the ratio; this ratio, which before was fixed, is now posited as variable; if another Quantum is taken as the One of the one side, then the other no longer remains the same amount of units of the first. In Direct Ratio this unit is only the common part of both sides; as such it continues itself into the other side, which is Amount; and Amount itself for itself, or the exponent, is indifferent to the Unit.

But under the present determinateness of Ratio, Amount as such is varied relatively to the One to which it is related as the other side of the Ratio. It becomes another when another One is taken as Quantum. Consequently, though the exponent is only immediate and a Quantum arbitrarily taken as fixed, yet it does not preserve itself as such in the side of the Ratio: it, and with it the direct ratio of the sides, is variable. Thus in the present Ratio the exponent as determining Quantum is posited as negating itself as Quantum of the Ratio: hence it is posited as qualitative, as Limit, so that the qualitative element emphasizes itself in opposition to the quantitative.—In Direct Ratio the variation of both sides is the single alteration only of the Quantum which the unit (the common element) is supposed to be; one side is increased or decreased in the same proportion as is the other; the Ratio itself is indifferent to this variation, which remains external. But in Inverse Ratio the variation, although likewise arbitrary according to the indifferent quantitative moment, is contained within the Ratio; and even this arbitrary quantitative overpassing is circumscribed by a limit, namely, the negative determinateness of the exponent.

2. This qualitative nature of Indirect Ratio must be further considered—in its realization; and the complication of the affirmative with the negative which it contains must be analysed.—Quantum is posited as qualitatively determining Quantum, that is, itself, as representing itself in itself as its own limit. Thus, first, it is an immediate magnitude as simple determinateness; it is the whole as existent and affirmative Quantum. Secondly, however, this immediate determinateness is also limit; and hence it is differentiated into two Quanta, which stand primarily in the relation of Other to each other;—but, since it is their qualitative and their complete determinateness, it is the unity of Unit and Amount; it is product, and they are its factors. Thus partly the exponent of their ratio is identical with itself in them and is their affirmative, and thus they are Quanta; and partly, as negation posited in them, it is the element of unity in them, according to which each is an immediate and limited Quantum, and limited in such a way that it is identical with its Other only in itself. Thirdly, as simple determinateness, it is the negative unity of this its differentiation into two Quanta and the limit of their reciprocal limitation.

According to these determinations, the two moments limit each other in the exponent; and since the exponent is their determinate unity, each is the negative of the other; one increases in the same ratio in which the other decreases; each has its magnitude in so far as it is in contact with the magnitude of the other and is what the other lacks. Thus each negatively continues itself into the other; it cancels its own Amount in the other, and is what it is only by virtue of the negation or limit which the other posits in it. Thus each contains and has its measure in the other, for each is to be only that Quantum which the other is not; the magnitude of the other is indispensable to the value of each, and therefore inseparable from it.

This continuity of one in the other constitutes the moment of unity by which they form a Ratio,—or the one determinateness and simple limit which is the exponent. This unity, the whole, constitutes the Being-in-Self of each, which is distinct from their magnitude as it is found: according to the latter, each exists only in so far as it withdraws from the other part of their common Being-in-Self or whole. But it can withdraw from the other only as much as makes it equal to this Being-in-Self; it has its maximum in the exponent, which, according to the second determination which we indicated, is the limit of their reciprocal limitation. And since each is moment of the Ratio only in so far as it limits the other and therefore is limited by it, it loses this its determination by equating itself with its Being-in-Self; in it not only the other magnitude becomes nought, but it vanishes itself, since it is not mere Quantum, but is to be what it is as such only as being such a moment of the Ratio. Thus each side is the contradiction of the determination as their Being-in-Self, that is, of the unity of the whole (which is the exponent), and of the determination as moment of the Ratio: and this contradiction is once more infinity in a new and peculiar form.

The exponent is limit of the sides of its Ratio, within which they increase and decrease relatively to each other; and according to the affirmative determinateness which the exponent (as Quantum) is, neither can become equal to it. As limit thus of their reciprocal limitation, it is (α) their Beyond, to which they can approximate infinitely; but they cannot reach it. This infinity in which they approximate to it is the bad infinity of the infinite progress; this itself is finite, and has its barrier in its opposite, in the finitude of the two sides and of the exponent; it, therefore, is only approximation. But (β) bad infinity is here also posited as what in truth it is, namely as the merely negative moment in general, according to which the exponent is simple limit as Being-in-Self in relation to the distinct Quanta of the Ratio; their finitude is referred to this as to a pure variable, although it remains quite distinct from them as their negation. This infinite, to which they can but approximate, further is found and is present as affirmative Hither: this is the simple Quantum of the exponent. Here the Beyond is reached with which the sides of the Ratio are tainted: it is in itself the unity of both and, therefore, in itself the other side of each; for each only counts for as much as the other does not, and thus the whole determinateness of each resides in the other; and this their Being-in-Self is, as affirmative infinity, simply the exponent.

3. But now the result is that Inverse Ratio has passed over into a determination different from that which it had at first. The determination was, that a Quantum (as immediate) is related also to another Quantum in such a manner that it is greater in the proportion in which the other is smaller, that it is what it is by virtue of its negative attitude to the other; and similarly a third magnitude is the common barrier of this their increase. Here this variation (as opposed to the Qualitative as fixed limit) is their peculiarity: they have the determination of variable magnitudes, having this fixed limit as an infinite Beyond.

We have found and must coordinate the determinations that this infinite Beyond exists as a present and finite (but optional) Quantum; and, further, that its fixity (by virtue of which it is thus related to the quantitative as infinite Beyond) which is the qualitative element of Being only as abstract self-relation, has developed itself with itself as mediation of itself in its Other, namely the finite moments of the Ratio. The universal part of this is contained in the fact that the whole as exponent is the limit of the reciprocal limitation of the two members; thus negation of negation and, therefore, infinity (an affirmative attitude to itself) is posited. A closer determination is, that in itself the exponent is product and, therefore, unity of Unit and Amount, whereas each of the two members is only one of these moments; it, therefore, includes them, and in them (as containing them) relates itself to itself. But in Inverse Ratio the difference develops into the externality of quantitative Being; and the Qualitative is not merely what is fixed or immediately includes the moments, but is found coalescing with itself in self-external otherness. It is this determination which manifests itself as result in the moments which we discovered. For the exponent is found to be Being-in-Self, the moments of which are realized in Quanta and in their variability in general; the indifference of their magnitudes in their variation manifests itself as infinite progress, and this is based upon the fact that in their indifference it is their determinateness that they have their value in the value of an other. Thus (α) in the affirmative aspect of their Quantum they are, in themselves, the whole of the exponent. Equally, (β) their negative moment, for their reciprocal limitation, is the magnitude of the exponent; their limit is that of the exponent. The infinite progress of their determinate existence and of their limitation, and the negation of any particular value, implies that they have no other immanent limit or fixed immediacy. This negation, therefore, is the negation of the self-externality of the exponent which is represented in them; and the exponent (taken as Quantum in general and as analysed into Quanta) is posited as that which preserves itself amid the negation of their indifferent persistence and collapses into itself, and, therefore, as the determinant of any such transition beyond itself.

Hence Ratio has come to be determined as the Ratio of Powers.

1. Quantum, which in its other-being identifies itself with itself and determines its self-overpassing, has reached Being-for-Self. It is thus qualitative totality, and, positing itself as developed, it has for moments the conceptual determinations of number—Unit and Amount. Even in Inverse Ratio the latter is a multitude determined, not by the former as such, but from without by a third element: now it is posited as determined only by the former. This happens in the Ratio of Powers, where Unit, which in itself is Amount, is also Amount relatively to itself as Unit. Other-being, the Amount of Units, is Unit itself. Power is a multitude of Units each of which itself is this multitude. Quantum as indifferent determinateness varies; but where this variation means raising to a power, this its other-being is limited purely by itself.—Thus in Power Quantum is posited as having returned upon itself: it is immediately itself and also its other-being.

The exponent of this Ratio no longer is an immediate Quantum as in Direct and also in Inverse Ratio. In the Ratio of Powers it is of a wholly qualitative nature; it is the simple determinateness that Amount here is Unit and that Quantum in its other-being is self-identical. This also contains the aspect of its quantitative nature, which is, that limit or negation is not posited as existing immediately, but that Determinate Being is posited as continued into its otherness; for the true nature of quality is this, that it is quantity (or immediate determinateness in so far as it is transcended).

2. Ratio of Powers first appears as an external variation applied to any Quantum; but it has this closer connexion with the concept of Quantum, that Quantum, in that Determinate Existence into which it has developed in this Ratio, reaches and completely realizes this concept: this Ratio represents what Quantum is in itself, and expresses its determinateness or quality by which it is distinct from anything else. Quantum is determinateness indifferent and posited as transcended; that is, determinateness as limit which also is no limit, and continues itself into its otherness, and consequently remains self-identical. It is thus posited in the Ratio of Powers, and its other-being or self-overpassing into another Quantum is determined by itself.

If we compare the progress of this realization in the Ratios with which we have dealt hitherto, then the quality of Quantum to be its own posited self-difference, is just this, that it is Ratio. As Direct Ratio it is, as such posited difference, only general or immediate, so that its self-relation, which, as exponent, it possesses in relation to its differences, counts only as the fixity of an Amount of Units. In Inverse Ratio, Quantum in its negative determination is an attitude of itself to itself,—to itself as its own negation, wherein, however, it has its value; as affirmative self-relation it is an exponent, which, as Quantum, is the determinant of its moments only in itself. But in the Ratio of Powers it is present in the difference, because the difference is a self-difference. Externality of determinateness is the quality of Quantum; and according to its concept this externality is posited as its self-determination, as its self-relation and its Quality.

3. But Quantum, being now posited as it is in accordance with its concept, has passed over into another determination. This might also be expressed thus—that its determination is its determinateness, and its Being-in-Self its Determinate Being. It is Quantum in so far as the externality or indifference of determinateness (which means that it is that, as the phrase goes, which can be increased or diminished) counts and is posited only simply or immediately; and it has become its Other (namely Quality) in so far as this externality is now posited as mediated by Quantum and as being such a moment as refers itself in it to itself and is Being as Quality.

Thus first Quantity as such appears as opposed to Quality. But Quantity is itself a quality; it is self-relating determinateness in general, distinct from that determinateness which is other to it, namely Quality as such. But it is not only a quality: the truth of Quality itself is Quantity, and the former has manifested itself as passing over into the latter. On the other hand, Quantity in its truth is externality which has returned to itself and is not indifferent. And thus it is Quality itself, and, outside this determination, Quality as such would be nothing.—In order that the totality may be posited, the twofold transition is required,—not only the transition of one determinateness into its other, but also the transition of the latter into the former, its regress into the first. The first transition gives us only the identity in itself of both;—Quality is contained in Quantity, but so far Quantity is a one-sided determinateness. It is the result of the second transition that, conversely, Quantity is contained in Quality, and that it too exists only as having been transcended. This is the regress into the first. This observation upon the necessity of the twofold transition is of great importance for the whole scientific method.

Now Quantum no longer is indifferent or external determination, but, as such, is transcended, and is Quality, and is that by virtue of which something is what it is: this is the truth of Quantum, and this is Measure.

It has been explained above, in the Observations upon the Quantitative Infinite, that this and the difficulties which result have their origin in the qualitative moment which manifests itself in the quantitative; and further, how especially the qualitative moment of the Ratio of Powers runs into manifold developments and complexities. It was shown that the fundamental flaw which prevented the concept from being apprehended was, that a halt was made at the negative determination of the infinite (where it is negation of Quantum), and that no progress was made to the simple affirmative determination which states that this is the qualitative.—Here it only remains to remark upon the intrusion, in philosophy, of quantitative forms into the pure qualitative forms of thought. The Ratio of Powers has especially been applied in recent times to conceptual determinations. The concept in its immediacy was called first power, in its other-being or difference (the Determinate Being of its moments) it was called second power, and in its return upon itself or totality, third power.—Here it is immediately evident that Power thus employed is a category essentially belonging to Quantum: this Power is not meant as potentia, the δύναμις of Aristotle. The Ratio of Powers thus expresses determinateness as reaching its truth in Difference, as Difference exists in the special notion of Quantum, but not as it exists in the Notion as such. Quantum does not yet by any means possess that negativity which belongs to the nature of the Notion, as posited in its peculiar determination: differences which are proper to Quantum are superficial determinations for the Notion itself, and are far from being determinate as they are determinate in the Notion. It was in the childhood of philosophic thought that numbers were used (as by Pythagoras) to designate universal and essential distinctions; and here first, second, or other Powers are in no way better than numbers. This was a rudimentary form of pure thinking comprehension; the determinations of thought themselves were not discovered—that is, brought to consciousness for themselves—till after Pythagoras. To return from these to numerical determinations is the part of thought which knows its impotence and, in opposition to what our philosophic culture—no stranger to determinations of thought—had reached, commits the further folly of trying to vindicate this weakness as something new, superior, and in the line of progress.

In so far as Powers are used only as a symbolical expression, they are unobjectionable, just as much as are numbers and other symbols of concepts,—but also they are as objectionable as all symbolism whatever which attempts to represent pure conceptual or philosophic determinations. Philosophy needs no such help either from the sensible world or from active imagination or from subordinate fields of philosophical speculation, the determinations of which, therefore, are unfitted for higher spheres and for the whole. The latter happens whenever categories of the finite are applied to the infinite; the common determinations of force, or substantiality, cause and effect, and others, are themselves too only symbols used to express other relations, like vital and spiritual relations; that is, they are untrue determinations of those relations, and still more so are Powers of Quantum and numbered powers, both for such and for speculative relations generally.—If numbers, powers, the mathematical infinite, and the like are to be used not as symbols but as forms for philosophic determinations and hence themselves as philosophic forms, then first of all their philosophic meaning, that is, their conceptual determinateness, must be demonstrated. If this is done, they are superfluous designations: the conceptual determinateness designates itself, and its own is the only correct and fitting designation. The use of these forms is, therefore, nothing but a convenient means of escaping the trouble of seizing, proclaiming, and justifying the conceptual determinations.