A. The Direct Ratio

1. Ratio in its immediacy is the direct ratio, and there the determinateness of the one quantum lies, reciprocally, within the determinateness of the other. Only one determinateness or limit belongs to both — a determinateness itself quantum, namely the ratio's exponent.

2. Some quantum or other is what the exponent is; but a quantum qualitatively determined, one that within its externality stands at its own self in reference of itself to itself, it is only so far as it bears at its own self the difference of itself, its beyond and its otherness. Now the difference of quantum at its own self is the difference between unit and amount: unit, that is being-determined-for-itself; amount, that is the indifferent to-and-fro at the determinateness, the external indifference belonging to quantum. Moments of quantum was what unit and amount were at the outset; within ratio, the quantum realized to that degree, each of quantum's moments now shows itself as a quantum of its own, and as determinations of quantum's existence, as bounds drawn against a determinateness of magnitude that is otherwise merely external and indifferent.

This difference taken as simple determinateness is the exponent, which is to say that immediately at its own self it bears the significance of both determinations. First it is quantum, and taken so it is the amount; let the one side of the ratio, the side taken for unit, be expressed as a numerical one, and let it count for nothing but that, and then the other side, the amount, will be the exponent's own quantum. Secondly it is simple determinateness in the shape of what is qualitative about the ratio's sides; determine the quantum of the one, and through the exponent the other stands determined too, while how the first comes to be determined is utterly a matter of indifference; as quantum determined for itself the first bears significance no longer, but might equally be any other whatever, and the ratio's determinateness, hanging as it does upon the exponent alone, would be unaltered. However great the one taken for unit may grow, unit is all it ever remains, and however great the other may grow along with it, it has to persist as the same amount of that unit.

3. What the two really make up, then, is but one quantum: against the other the one counts merely for the value of unit and not of an amount, while the other counts merely for that of amount; hence by their conceptual determinateness they are themselves not complete quanta. Yet an incompleteness of this kind is a negation at them — a negation arising not from their variability at large, whereby the one (and either is one of the two) may assume any magnitude whatever, but from the determination that alteration of the one brings with it increase or diminution of the other by precisely as much; which is to say, as was shown, that the one, the unit, is alone altered as quantum, the other side, the amount, persisting as the same quantum of units, while even that first side keeps counting for nothing but unit, be its alteration as quantum what it may. So each side is no more than one moment out of the two belonging to quantum, and self-subsistence, which belongs to what is peculiar about quantum, stands in itself negated; within such a qualitative connection the two are to be posited as negative towards one another.

Since in it the determination of both sides runs together, the exponent ought to be the quantum in its completeness; in fact, though, being a quotient it likewise bears no more than the value of amount, or of unit. Nothing at hand determines which of the ratio's sides has to be taken for unit and which for amount: measure the one, quantum B, against quantum A serving as unit, and quotient C gives the amount of units of that kind; take A itself for amount, however, and quotient C gives the unit which the amount A requires if quantum B is to result; thus as exponent this quotient is not posited as what it ought to be, – as that which determines the ratio, or as the ratio's qualitative unity. Only insofar as it bears the value of being the unity of the two moments, of unit and of amount, is it posited as such. On hand these sides are, to be sure, as quanta, as within the explicit quantum, the ratio, they ought to be, yet on hand at the same time only with that value which as its sides they ought to have — namely to be incomplete quanta and to count for one alone among those qualitative moments; and so they have to be posited together with this negation of theirs, out of which arises a ratio more real, more answerable to its determination, wherein the exponent bears the significance of their product; by such determinateness it is the inverse ratio.

B. The Inverse Ratio

1. Ratio as it has now come out is direct ratio sublated; direct ratio was the immediate one and therefore not yet determined in truth; by now determinateness has been added, such that the exponent holds good as product, as unity of unit and amount. In point of its immediacy the exponent admitted of being taken indifferently for unit no less than for amount, as was shown a moment since; whereby it also was merely quantum at large, and thus amount by preference; the unit was one of the sides, to be taken as a one, and to it the other stood as fixed amount, which is at the same time the exponent; the quality of that exponent accordingly amounted to nothing beyond this, that such a quantum gets taken as something fixed, or rather that the fixed carries merely the sense of quantum.

Now in the inverse ratio the exponent likewise, qua quantum, is something immediate, some quantum or other assumed to be fixed. This quantum, however, is no fixed amount answering to the one of the other quantum within the ratio; that ratio, in what went before a fixed one, is now posited rather as variable; let another quantum be taken for the one of the single side, and the other no longer stays the same amount of units of the first. In direct ratio such a unit is nothing but what the two sides hold in common; as such it carries itself on into the other side, into the amount; while amount by itself, that is to say the exponent, stands indifferent towards the unit.

As the ratio's determinateness now stands, however, amount as such undergoes alteration over against the one to which it makes up the ratio's other side; take another quantum for the one, and amount becomes another. Hence the exponent too is admittedly no more than an immediate quantum, assumed to be fixed at mere pleasure, yet as such it fails to hold itself fast within the side of the ratio; that side is variable instead, and with it the direct ratio of the sides. With this the exponent — the quantum that does the determining — stands posited, in the ratio as it now is, negatively towards itself qua quantum of the ratio, posited therefore as qualitative, as limit, whereby the qualitative steps forth for itself in distinction from the quantitative. – Within direct ratio, whatever alteration befalls the two sides amounts to a single alteration of the quantum taken for the unit, the unit being the common element; by however much, then, the one side grows or shrinks, by just so much does the other; and the ratio itself stays indifferent towards such alteration, which is external to it. In indirect ratio, by contrast, the alteration, arbitrary though it also is so far as the indifferent quantitative moment goes, is kept within the ratio, and even this arbitrary quantitative going out beyond finds itself restricted, as by a limit, through the negative determinateness of the exponent.

2. Closer consideration is owed to this qualitative nature of indirect ratio, namely consideration of it in its realization, and the involvement of the affirmative with the negative contained therein must be laid out. – Quantum stands posited qualitatively as quantum, that is, as determining its own self, as exhibiting at itself the limit of itself. So it is first an immediate magnitude in the shape of simple determinateness, the whole as an affirmative quantum that is. Secondly, though, such immediate determinateness is at once limit; on that account quantum falls apart into two quanta, others to each other in the first instance; but as their qualitative determinateness, and that in complete form, quantum is unity of unit and amount, a product whose factors those two make. Thus the exponent of their ratio is, for one part, self-identical in them and the affirmative in them by virtue of which they are quanta; for the other part, being the negation posited at them, it is the unit at them, whereby each — immediate, a bounded quantum at large to begin with — is at the same time bounded so as to be identical with its other only in itself. Thirdly, being simple determinateness, it is the negative unity of its own thus differentiating into two quanta, and the limit within which they limit one another.

Following these determinations the two moments limit one another inside the exponent, and each is the negative of the other, since the exponent is their determinate unity; by as many times as the other grows greater, the one grows smaller, and each possesses its magnitude just so far as it possesses at itself the magnitude of the other, so far as that magnitude is wanting to the other. Negatively, in this fashion, each carries itself on into the other; whatever amount it comes to, that it sublates as amount at the other, and only through the negation or the limit which the other posits at it is it what it is. In this fashion each contains the other as well and is measured at it, for what each ought to be is nothing but the quantum the other fails to be; the magnitude of the other is indispensable to the value of each and thus not to be severed from it.

This continuity of each within the other makes up that moment of unity whereby the two stand in ratio; the moment, that is, of a single determinateness — the exponent being just this simple limit. Such unity, the whole, makes up the being-in-itself of either, from which the magnitude either has present is distinct, a magnitude by which each only is so far as it draws off from the other something of their common being-in-itself, the whole. Draw off from the other it can, however, only so much as it makes equal to that being-in-itself; its maximum it has in the exponent, which by the second determination stated above is the limit within which they limit each other. And because each counts as moment of the ratio only so far as it limits the other and thereby gets limited by the other, it forfeits this determination of its own the moment it makes itself equal to its being-in-itself; then not merely does the other magnitude drop to zero — the first vanishes too, since it is meant to be no bare quantum but, in what it is as such, nothing except such a moment of ratio. Each side is thus the contradiction between the determination that is its being-in-itself, that is to say the unity of the whole which the exponent is, and the determination that is its being a moment of ratio; and once more this contradiction is infinity, under a new and peculiar shape.

Limit of the sides of its ratio is what the exponent is, a limit inside which those sides wax and wane against each other and which, in respect of the affirmative determinateness it has as quantum, they can never come to equal. Taken thus, as the limit within which they limit one another, it is α) their beyond, one they approach infinitely without power to reach it. Such infinity, as the infinity in which they draw near to it, is the bad infinity belonging to infinite progress; finite itself, it finds its restriction in its own opposite, in the finitude of either side and of the exponent as well, and amounts therefore to mere approximation. But β) the bad infinity is here at once posited as what in truth it is, namely as nothing but the negative moment at large, whereby the exponent, set against the differentiated quanta of the ratio, is the simple limit qua being-in-itself, to which their finitude, as the utterly variable, gets referred, while it remains utterly diverse from them, their negation. This infinite, which they can do no more than approach, is then likewise on hand and present as an affirmative this-side; the plain quantum of the exponent. Therein is reached that beyond which weighs upon the ratio's sides; in itself the exponent is unity of the two, and thereby in itself the other side of either; for the value each holds is precisely what the other does not, its whole determinateness lying accordingly in the other, and this being-in-itself of theirs is, as affirmative infinity, simply the exponent.

3. But herewith the inverse ratio's transition has resulted — a transition into some determination differing from the one it bore at the outset. That first determination consisted in this, that a quantum, immediate as it is, stands at the same time in the reference to another of being greater by just so much as that other is smaller, of being what it is through negative comportment towards the other; and likewise that a third magnitude is the common restriction upon this growing greater of theirs. Peculiar to them here, in contrast with the qualitative as fixed limit, is this alteration; theirs is the determination of variable magnitudes, for which that fixed element counts as an infinite beyond.

The determinations, though, that have come out and that we must now draw together are these: not merely that such an infinite beyond is at once something present and some finite quantum, but that its fixity — that whereby, over against the quantitative, it is an infinite beyond of this sort, and which is the qualitative of being taken merely as abstract self-reference — has developed into a mediation of itself with itself within its other, within the finites of the ratio. The universal aspect of this lies herein, that the whole at large, as exponent, is posited as the limit within which the two members limit each other, hence as the negation of the negation, hence as infinity, as affirmative comportment towards itself. More determinately: in itself the exponent is already, as product, unity of unit and amount, whereas either member is only the one of these two moments, so that the exponent shuts them up within itself and in itself refers to itself within them. In the inverse ratio, though, difference has been developed out into the externality that belongs to quantitative being, and what is qualitative is no longer merely the fixed element, nor does it merely shut the moments up immediately within itself; on hand it is, rather, as something closing together with itself within the self-external otherness. It is this determination that lifts itself out as result within the moments that have shown themselves. The exponent, namely, yields itself as the being-in-itself whose moments stand realized in quantis and in the variability of these at large; the indifference of their magnitudes throughout their alteration exhibits itself as infinite progress; and what lies at the ground of this is that, indifferent though they are, their determinateness consists in holding their value within the value of the other, hence α) in being in themselves, on the affirmative side of their quantum, the whole of the exponent. Likewise β) it is the magnitude of the exponent that serves them for their negative moment, for their limiting of each other; the limit belonging to them is the exponent's own. That no other immanent limit, no fixed immediacy, belongs to them any longer stands posited in the infinite progress of their existence and of their being limited, in the negating of every particular value. Such negating is accordingly the negation of that being-outside-itself of the exponent which is displayed in them; and the exponent — at once itself a quantum at large and laid out into quanta besides — stands thereby posited as what preserves itself in the negation of their indifferent subsistence, what goes together with itself, and hence as what determines a going out beyond self of that kind.

Ratio is herewith determined into the ratio of powers.

C. The Ratio of Powers

1. Positing itself as identical with itself in its otherness, determining its own going out beyond itself, quantum has arrived at being-for-itself. Qualitative totality of that kind, in positing itself as developed, takes for its moments those conceptual determinations belonging to number, namely unit and amount; within the inverse ratio the latter is still a multitude determined not through the former as such but from elsewhere, through a third; now it stands posited as determined through the former alone. So it stands in the ratio of powers: there the unit, being amount at its own self, is at the same time the amount over against itself as unit. Otherness, the amount of units, is the unit itself. A multitude of units is what the power is, every one of them being this multitude itself. Quantum as indifferent determinateness undergoes alteration; but insofar as such alteration is a raising into the power, this otherness of quantum's is bounded purely through quantum itself. – In the power, then, quantum stands posited as having gone back into its own self; immediately it is itself and its otherness besides.

No longer an immediate quantum, as in direct and likewise in inverse ratio, is the exponent of this ratio. In the ratio of powers it is of wholly qualitative nature, this simple determinateness, that amount is the unit itself, that quantum in its otherness is identical with its own self. Therein lies at the same time the side of its quantitative nature, that limit or negation is posited not as something immediately that is, but existence is posited as carried on into its otherness; for the truth of quality is precisely this, to be quantity, to be immediate determinateness as sublated.

2. At first the ratio of powers looks like an external alteration into which some quantum or other gets put; yet it bears the closer reference to the concept of quantum, namely that within the existence to which it has been carried forward in that ratio quantum reaches that concept, has realized it in complete fashion; this ratio is the exhibition of what quantum in itself is and expresses that determinateness or quality of quantum whereby it marks itself off from other. Quantum is the indifferent determinateness, determinateness posited as sublated, which is to say determinateness as a limit that quite as much is none, that carries itself on into its otherness and therein accordingly stays identical with itself; so it stands posited in the ratio of powers, its otherness, its going out beyond itself into another quantum, being determined through quantum itself.

Compare the advance of this realization through the ratios so far treated, and the quality of quantum — its being posited as difference of itself from itself — proves in general to be this, to be ratio. As direct ratio, quantum is such posited difference only in general or immediately, its self-reference — that reference which it holds as exponent over against its differences — counting merely as the fixedness of some amount of the unit. In inverse ratio quantum, under negative determination, is a comporting of itself towards itself, – towards itself as its own negation, wherein nevertheless it holds its value; as affirmative self-reference it is an exponent that, being quantum, is only in itself what determines its moments. In the ratio of powers, however, quantum is on hand within the difference as the difference of itself from itself. Quantum's quality is the externality of determinateness, and conformably to quantum's concept that externality now stands posited as quantum's own determining, as its self-reference, its quality.

3. But in that quantum stands posited as it is conformably to its concept, it has passed over into another determination; or, as the point may also be put, its determination now is the determinateness too, its being-in-itself is existence too. It is quantum insofar as the externality or indifference of being-determined (– that it is, as people say, what can be enlarged or diminished) counts and stands posited only simply or immediately; it has become its other, quality, insofar as that externality is now posited as mediated through quantum itself, posited as a moment in such fashion that precisely in it quantum refers to its own self, is being as quality.

To begin with, then, quantity as such makes its appearance over against quality; yet a quality is what quantity itself is, determinateness that refers to itself at large, marked off from the determinateness other to it, from quality as such. Only, quantity is not merely a quality; rather the truth of quality itself is quantity; quality has shown itself as passing over into quantity. Quantity, by contrast, in its truth is externality gone back into its own self, externality that is not indifferent. So it is quality itself, so much so that outside this determination quality as such would not be anything further. – Totality's being posited demands the doubled transition: not merely the passing of the one determinateness over into its other, but just as much the passing of that other back, its return, into the first. By way of the first, their identity is on hand only in itself; – quality is contained within quantity, which thereby, though, remains still a one-sided determinateness. That conversely quantity is contained just as much within the first, is there just as much only as sublated, comes out in the second transition, – the return into the first; and for the whole of scientific method this remark upon how the double transition is necessary carries great importance.

Quantum now being an indifferent or external determination, sublated equally as such, and being quality, being that through which something is the very thing it is — herein lies the truth of quantum, to be measure.

Remark

Above, in the Remarks on the quantitatively infinite, it was set out that this infinite, and likewise the difficulties gathering around it, take their origin in the qualitative moment that comes forward within the quantitative, and how the qualitative side of the ratio of powers in particular issues into those manifold developments and entanglements; what was exhibited as the root defect blocking any grasp of the concept is that, where the infinite is concerned, one halts at the negative determination alone, that of being the negation of quantum, and never presses on to the simple determination, the affirmative one, namely that this infinite is the qualitative. – Here nothing further remains than to add one observation about the intrusion, carried out within philosophy, of quantitative forms into the pure qualitative forms of thinking. It is above all the ratio of powers that has of late been applied to determinations of the concept. The concept in its immediacy was called the first power, in its otherness or difference, the existence of its moments, the second, and in its return into itself or as totality the third power. – Against this it strikes one at once that power, so employed, is a category belonging essentially to quantum; – with these powers there is no thought of Aristotle's potentia, δύναμις. The ratio of powers accordingly expresses determinateness in the manner in which determinateness, as the difference obtaining in the particular concept of quantum, arrives at its truth, yet not in the manner in which it obtains at the concept as such. Quantum has not yet got the negativity proper to the nature of the concept posited in the concept's own peculiar determination; differences accruing to quantum are surface determinations for the concept itself; they fall far short of being determined as they are within the concept. It belongs to the childhood of philosophizing that numbers – and first, second power and so on have in this respect no advantage over numbers – were used, as by Pythagoras, for designating universal, essential differences. It was a stage preliminary to pure conceptual apprehension; the determinations of thought themselves came to be discovered only in the wake of Pythagoras, that is, were raised for themselves into consciousness. But going back from such determinations to determinations of number belongs to a thinking that feels its own incapacity, one which now, pitted against an existing philosophical culture at home in determinations of thought, adds the ridiculous to it besides by wanting to press that weakness as something new, distinguished, and as an advance.

Insofar as the expression of powers is employed merely as a symbol, as little is to be said against it as against numbers or symbols of another sort standing 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 stands in no need of such assistance, neither from the sensible world nor from the representing imagination, nor from spheres of its own proper ground that are subordinate and whose determinations therefore do not suit higher circles and the whole. This last is what occurs each time finite categories get carried over to the infinite at all; familiar determinations such as force, or substantiality, cause and effect, and the rest, are likewise no more than symbols for expressing, say, vital or spiritual relations, hence untrue determinations of these, while the powers of quantum together with numbered powers are so in a still higher degree, both for relations of that sort and for speculative relations at large. – Suppose numbers, powers, the mathematically infinite and their like were meant to serve, in place of the role of symbol, as the very forms in which philosophical determinations are cast, and so as philosophical forms themselves: then before all else their philosophical import, their conceptual determinateness, would have to be exhibited. Should that be done, they are themselves superfluous designations; conceptual determinateness designates itself, and its own designation is alone the correct and the fitting one. Recourse to those forms is therefore nothing further than a convenient device for sparing oneself the labour of grasping, stating and justifying the determinations of the concept.

Third Section. Measure

In measure, expressed abstractly, quality and quantity are joined together. Taken as such, being amounts to determinateness immediately equal to its own self. Such immediacy on the part of determinateness has by now sublated itself. Quantity, for its part, is being gone back into itself in such fashion as to be simple equality with itself, namely indifference towards determinateness. But this indifference is nothing else than the externality of having determinateness not at its own self but in an other. The third is now externality relating itself to its own self; as relation to self it is at the same time sublated externality, and it possesses at itself the difference from itself – a difference that counts as the quantitative moment insofar as it is externality, and, once withdrawn into [itself], as the qualitative one.

Since modality, among the categories of transcendental idealism, is listed after quantity and quality, with relation slipped in between, it may be mentioned here. There this category carries the sense of being the relation of the object to thinking. In the sense of that idealism, thinking as such is essentially external to the thing-in-itself. Insofar as the other categories bear only the transcendental determination of belonging to consciousness, yet as what is objective in it, modality, being the category of relation to the subject, comparatively contains within it the determination of reflection; that is, the objectivity said to accrue to the other categories is wanting in those of modality; these, in Kant's phrase, do not in the least enlarge the concept as determination of the object, but express only the relation to the faculty of cognition, (Crit. of Pure Reason, 2nd ed., see pp. 99, 266). – Possibility, actuality and necessity, the categories Kant assembles under modality, will each turn up further on at their proper place; Kant applied the infinitely important form of triplicity, however much it appeared with him at first merely as a formal spark of light, not to the genera of his categories (quantity, quality and so forth), any more than he applied this name to them, but only to their species; hence he could not arrive at the third of quality and quantity.

With Spinoza, mode, coming after substance and attribute, is likewise the third; he declares it to be the affections of substance, or that which is in an other through which it is also conceived. According to this conception the third is nothing but externality as such; as has otherwise been recalled, with Spinoza the rigid substantiality throughout lacks the return into its own self.

The observation made here extends more generally over those systems of pantheism which thought has to some degree worked out. The first is being, the one, substance, the infinite, essence; over against this abstractum the second, all determinateness, can be lumped together just as abstractly as the merely finite, the merely accidental, transitory, extra-essential and inessential and so forth, in the way this usually and initially happens in wholly formal thinking. But the connectedness of this second with the first presses forward too insistently for it not to be grasped at the same time in a unity with the first, as the attribute with Spinoza is the whole substance, though grasped by the understanding, itself a limitation or mode; mode, however, the non-substantial in general, which can be grasped only out of an other, thus makes up the extreme opposite to substance, the third as such. Indian pantheism too, in its monstrous fantasy, has received this same elaboration, taken abstractly, one that runs through what is measureless in it as a moderating thread of some interest, namely that Brahma, the one of abstract thinking, advances, by way of its shaping in Vishnu and above all in Krishna's form, to Shiva as the third. What determines this third is mode, is alteration, coming-to-be along with ceasing-to-be, the whole field of externality. If this Indian threefoldness has tempted people into a comparison with the Christian one, a common element of conceptual determination is indeed to be recognized in them, but concerning the difference a more determinate consciousness has essentially to be reached; the difference is not merely infinite, rather genuine infinity itself makes up the difference. That third principle is by its determination the flying apart of substantial unity into its opposite, not the return of that unity into itself – what is spiritless, rather, not spirit. In the genuine threefoldness there is brought about not merely unity but oneness, the syllogism issuing in the contentful and actual unity that in its wholly concrete determination is spirit. That principle of mode and alteration does not, to be sure, exclude unity altogether; just as within Spinozism mode as such is precisely the untrue and substance alone the genuine, everything being supposed to be led back to substance, which then amounts to a submersion of all content in emptiness, in a merely formal, contentless unity, so too Shiva is in turn the great whole, not distinguished from Brahma, is Brahma himself; that is, difference and determinateness merely vanish once more instead of being preserved, instead of being sublated, and unity does not become concrete unity, nor is disruption led back to reconciliation. For the human being displaced into that sphere where things come to be and cease to be, the sphere of modality at large, the highest goal is to sink into unconsciousness, into unity with Brahma, into annihilation; the Buddhist nirvana, nieban and the rest are the same thing.

Now if mode as such is the abstract externality, the indifference towards qualitative as much as towards quantitative determinations, and if in the essence what is external and inessential is supposed not to matter, then in many things it is conversely conceded that everything turns on the manner and way; mode is thereby itself declared to belong essentially to the substantial side of a matter; in which very indeterminate connection there lies at least this, that this external element is not so abstractly the external.

Here mode has the determinate significance of being measure. Both the Spinozistic mode and the Indian principle of alteration are the measureless. The Greek consciousness, still itself indeterminate, that everything has a measure, so that even Parmenides introduced, after abstract being, necessity as the ancient limit set to all things, is the beginning of a far higher concept than substance and the difference of mode from substance contain. –

Measure in a more developed, more reflected shape is necessity; fate, nemesis, confines itself broadly to the determinateness of measure, in that whatever mismeasures itself, pitching itself too large and too lofty, gets driven to the counter-extreme of debasement into nullity, whereby the middle of measure, mediocrity, is re-established. – To say that the absolute, God, is the measure of all things is no more pantheistic than the definition which says that the absolute, God, is being, though it is infinitely more true. – Measure is, to be sure, an external manner and way, a more or a less, and yet at the same time a determinateness reflected into itself just as much, not merely indifferent and external but one that is in itself; hence what measure is is the concrete truth of being; and for this reason peoples have venerated in measure something untouchable, something holy.

Measure already harbours the idea of essence, namely that of being self-identical within the immediacy of determinedness, such that immediacy of this sort is degraded by that identity-with-self into something mediated, while the identity in turn, mediated as it is solely through this externality, nonetheless is mediation with itself; – reflection, that is, whose determinations are, though within such being they stand strictly only as moments of its negative unity. In measure the qualitative is quantitative; determinateness or difference is there as something indifferent, and with that it is a difference which is none; it is sublated; this quantitativity, as the return into itself in which it has the standing of the qualitative, is what constitutes being-in-and-for-itself, and that is essence. Yet measure is essence only in itself or according to its concept; this concept of measure has not yet been posited. Taken still as such, measure is itself the unity of qualitative and quantitative that is; its moments have the standing of an existence, a quality along with quanta of that quality, inseparable as yet only in themselves and lacking so far the sense carried by this reflected determination. What the development of measure contains is the distinguishing of these moments, yet with it their connection as well, so that the identity which they are in themselves arises as their relation to each other, that is, becomes posited. This development means the realization of measure: in it measure places itself into a ratio to its own self and so posits itself at the same time as a moment; by such mediation it is determined as something sublated; its immediacy vanishes, as does that of its moments, which now are as reflected; having stepped forth in this way as what it is according to its concept, measure has crossed over into essence.

To begin with, measure is that unity of qualitative and quantitative which is immediate, so that first, a quantum is there carrying qualitative significance and standing as measure. Its further determination consists in this, that at it, the thing determined in itself, – the difference of its moments, of qualitative and quantitative determinedness, steps forth. These moments determine themselves further into wholes of measure, which are to that extent self-subsistent; since they relate essentially to one another, measure becomes

second, a ratio of specific quanta, as self-subsistent measures. Their self-subsistence, however, rests essentially at the same time on the quantitative ratio and on the difference of magnitude; thus their self-subsistence becomes a passing over into one another. Measure thereby founders in the measureless. – This beyond of measure is, however, the negativity of measure only in itself; through this it is

third, the indifference of the determinations of measure, measure now posited as real together with the negativity such indifference contains, that is, as an inverse ratio of measures that are self-subsistent qualities resting essentially on nothing but their quantity and their mutually negative relation, and so proving to be no more than moments of the unity which is truly self-subsistent, the unity that is their reflection-into-self and the positing thereof, namely essence.

The development of measure attempted in what follows is one of the most difficult matters; beginning as it does from immediate, external measure, it would have on the one hand to proceed to the abstract further determination of the quantitative (a mathematics of nature), on the other hand to indicate the connection of this determination of measure with the qualities of natural things, at least in general terms; for it falls to the particular science of the concrete to demonstrate determinately the connection of qualitative and quantitative as this issues from the concept of the concrete object; for examples one may consult the Encyclopaedia of the Philosophical Sciences, 3rd ed., §§ 267 and 270, Remark, concerning the law of fall and that of free celestial motion. Here it may be noted quite generally that the various forms in which measure realizes itself likewise belong to different spheres of natural reality. Only within the sphere of mechanism can developed measure, that is, can its laws, hold with complete, abstract validity, since there the concrete corporeal is nothing but matter which is itself abstract; the qualitative differences of matter have the quantitative essentially for their determinateness; space and time are pure externalities themselves, while the multitude of the matters, the masses, the intensity of weight count equally as external determinations that possess their peculiar determinateness in the quantitative. Such determinateness of magnitude in what is abstractly material is on the other hand already disturbed, in the physical realm and still more in the organic, by multiplicity and thus by a conflict of qualities. But it is not merely the conflict of qualities as such that sets in here; measure is rather subordinated to higher relations, so that the immanent development of measure gets reduced instead to immediate measure in its simple form. A measure belongs to the limbs of the animal organism which, as a simple quantum, stands in a ratio to the quanta of the other limbs; the human body's proportions are ratios of that kind, fixed among such quanta; and natural science has still a long way to go before it grasps anything of how such magnitudes hang together with the organic functions on which they wholly depend. Motion, however, offers the nearest example of how an immanent measure is brought down to a magnitude determined merely from outside. In the heavenly bodies it is free motion determined only by the concept, whose magnitudes accordingly likewise depend only on the concept (see above), whereas by the organic it is brought down to arbitrary or mechanically regular, that is, to abstract formal motion generally.

Still less, however, does a peculiar, free development of measure take place in the realm of spirit. One may well see, for instance, that a republican constitution such as the Athenian, or an aristocratic one shot through with democracy, can have its place only given a certain magnitude of the state; that in developed civil society the multitudes of individuals belonging to the various trades stand in a ratio with one another; but this yields neither laws of measures nor peculiar forms of measure. In what is spiritual as such there occur differences of intensity of character, of strength of imagination, of sensations, of representations and so forth; but beyond this indeterminacy of strength or weakness the determination does not reach. Anyone who looks into the psychologies that busy themselves with such matters becomes aware how feeble, how wholly vacuous, are the so-called laws laid down about the relation of strength and weakness in sensations, representations and the rest.

Chapter 1. Specific Quantity

Qualitative quantity is initially an immediate specific quantum, which

second, comporting itself towards another, turns into a quantitative specifying, into the indifferent quantum's getting sublated. Measure of this sort is to that extent a rule, and it holds the two moments of measure apart as distinguished, namely the quantitative determinateness that is in itself, and the external quantum. Within this difference, however, both sides turn into qualities, and the rule into a ratio between them; measure accordingly exhibits itself

third, as a ratio of qualities that at first have One measure, though it further specifies itself within itself into a difference of measures.

A. The Specific Quantum

1. Measure is the simple relation of quantum to itself, its own determinateness at its own self; quantum is thus qualitative. To begin with, being immediate measure, it is an immediate and hence some determinate quantum; equally immediate is the quality belonging to it, which is some determinate quality. – Quantum, taken as this limit that is no longer indifferent but rather externality relating [itself] to itself, is on that account the quality itself, and though distinguished from the quality it does not reach out past it, any more than the quality reaches out past quantum. It is determinateness returned in this way into simple equality with itself; one with determinate existence, as this existence is one with its quantum.

Should one wish to make a proposition out of the determination obtained, one can put it thus: everything that is there has a measure. Every existence has its magnitude, and such magnitude falls within the nature of the something itself; what it constitutes is that something's determinate nature and its being-within-itself. Something is not indifferent towards this magnitude, as though it would stay what it is were the magnitude changed; on the contrary, altering the magnitude would alter its quality. As measure, quantum has ceased to be a limit that is none; it is by now the determination of the matter, so that the matter, increased or diminished past this quantum, would founder. –

A measure, taken as a standard in the ordinary sense, is a quantum arbitrarily assumed as the unit determined in itself over against an external amount. Such a unit can indeed also be in fact a unit determined in itself, like the foot and similar original measures; insofar as it is at the same time employed as a standard for other things, however, it is for these only an external measure, not their original one. – The earth's diameter, or the length of the pendulum, may thus be taken for itself as a specific quantum. But it is arbitrary which portion of that diameter or of that pendulum length one chooses, and under which degree of latitude, when the thing is to serve as a standard. Still more is a standard of this kind something external for other things. These have specified the universal specific quantum once again in their own particular way, and are made into particular things through that. To speak of some natural standard for things is therefore a folly. In any case a universal standard is supposed to serve only the external sort of comparison; where it is taken in that most superficial sense, as a universal measure, whatever gets used for it makes no difference at all. It is not supposed to be a basic measure in the sense that the natural measures of particular things would be exhibited by reference to it, and known from it by a rule, as specifications of One universal measure, the measure of their universal body. Lacking this sense, however, an absolute standard has only the interest and the significance of a common one, and such a thing is a universal not in itself but by convention.

A simple determination of magnitude is what immediate measure amounts to, as with the magnitude of organic beings, of their limbs, and so on. But everything that concretely exists has a magnitude in order to be what it is, and in general in order to have existence. – 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 once other than itself as quantum, other than determination of that indifferent sort, and a limitation, at a limit, of the indifferent to-and-fro.

Since the determinateness of quantity at existence is thus a doubled one, the one time that to which quality is bound, the other time that at which one may move to and fro without prejudice to quality, the perishing of a something that has a measure takes place in this, that its quantum is altered. Such perishing appears in one respect as unexpected, insofar as change can be made at the quantum without altering measure and quality, while in another respect it is turned into something wholly comprehensible, namely through gradualness. This category is seized upon so readily in order to make the ceasing of a quality or of a something representable or to explain it, since one then seems almost able to watch the vanishing with one's own eyes, because quantum is posited as the limit which is external and by its nature alterable, so that alteration, as alteration merely of quantum, goes without saying. In fact, however, nothing gets explained in this way; essentially, and at the same time, the alteration is a quality's passing over into another quality, or, more abstractly, an existence's passing over into a non-existence; there lies in that a determination other than the one lying in gradualness, which is merely a diminishing or an increasing, the one-sided clinging to magnitude.

2. That an alteration appearing as merely quantitative also strikes over into a qualitative one, of this connection the ancients were already aware, and they represented in popular examples the collisions that arise from ignorance of it; under the names of the bald man and of the heap, elenchi belonging here are familiar, that is, according to Aristotle's explanation, ways by which one is forced to say the opposite of what one had previously maintained. The question was put: does plucking out one hair from the head or from a horse's tail make it bald, or does a heap cease to be a heap when one grain is taken away. This one can concede without hesitation, since such removal makes only one quantitative difference, and indeed one that is itself utterly insignificant; so one hair, one grain is taken away, and this is repeated in such a way that each time, according to what has been conceded, only one is removed; at the end the qualitative alteration shows itself, that the head, the tail is bald, the heap has gone. In conceding this one forgot not only the repetition, but also that quantities insignificant in themselves (like outlays insignificant in themselves from a fortune) add up, and that the sum makes up the whole qualitatively, so that finally the whole is gone, the head bald, the purse empty.

The embarrassment, the contradiction that comes out as the result, is not something sophistical in the customary sense of the word, as though such contradiction were a false pretence. What is false is what the assumed other party, that is, our ordinary consciousness, commits: taking a quantity only for an indifferent limit, that is, taking it precisely in the determinate sense of a quantity. The truth to which this assumption gets led, namely that of being a moment of measure and hanging together with quality, confounds it; what gets refuted is the one-sided clinging to the abstract determinateness of quantum. – Those turns of argument are therefore no empty or pedantic joke either, but correct in themselves and products of a consciousness that takes an interest in the appearances occurring within thinking.

Quantum, in being taken as an indifferent limit, is the side at which an existence is unsuspectingly attacked and brought to ruin. It is the cunning of the concept to grasp an existence at just that side where its quality seems to stay out of the game – to such a degree, indeed, that the enlargement of a state, of a fortune and so forth, which brings on the ruin of the state, of the owner, even looks at first like its good fortune.

3. In its immediacy, measure is an ordinary quality having a determinate magnitude that belongs to it. Now from that side on which quantum is the indifferent limit, where one may move to and fro without any change of quality, its other side is distinguished too, the side on which it is qualitative, specific. Both are determinations of magnitude of One and the same; but in keeping with the immediacy in which measure first is, this difference is further to be taken as an immediate one, and accordingly a different concrete existence belongs to each of the two sides. That concrete existence of measure which is the magnitude determined in itself then stands, in its bearing towards the concrete existence of the alterable, external side, as a sublating of the latter's indifference, a specifying of it.

B. Specifying Measure

The same is

firstly a rule, a measure external over against mere quantum;

secondly specific quantity, which determines the external quantum;

thirdly both sides comport themselves towards each other as qualities of specific quantitative determinateness, as One measure.

a. The Rule

The rule, or measuring-stick, already spoken of, counts first of all as a magnitude determinate in itself; it serves as unit against a quantum which is a particular concrete existence, one that exists in a something other than the something the rule is – and it is by the rule that such a quantum gets measured, that is, gets determined as an amount of that unit. Such comparing is an external business, and the unit is itself a magnitude arbitrarily chosen, one that may in its turn be posited as an amount (a foot, say, counted out in inches). But measure is not just an external rule: being specific, what it amounts to is this, that within its very own self it comports itself towards its other, an other which is a quantum.

b. The Specifying Measure

Measure is the specific determining of external magnitude, that is, of the indifferent magnitude which is now posited by some other concrete existence in the something belonging to measure, a something which, though itself a quantum, is nevertheless, as distinguished from quantum of that kind, the qualitative factor, the factor determining the merely indifferent, external quantum. The something carries this side of being-for-other about it, the side to which indifferent increase and decrease accrues. That immanent measuring factor is a quality of the something, and over against it stands the same quality in a different something; only that in this latter the quality stands at first with a quantum relatively measureless, over against the quality determined as the measuring one.

To a something, so far as it is a measure within itself, an alteration in the magnitude of its quality comes from outside; the something does not, however, take in the arithmetical multitude of that alteration. Its measure reacts against it, comports itself towards the multitude as something intensive, and receives it in a manner peculiar to itself; it alters the alteration posited from without, turns this quantum into an other, and by such specification manifests itself as being-for-itself within that externality. – This specifically-received multitude is itself a quantum, dependent also on the other, or on the multitude taken as merely external multitude. Thus the specified multitude is alterable too, yet for that reason it is not a quantum as such, but rather the external quantum specified in a constant fashion. Measure accordingly has its existence as a ratio, and what is specific about it is, quite generally, the exponent of that ratio.

With the intensive and the extensive quantum it is, as those determinations brought to light, one and the same quantum which is at hand now in the form of intensity, now in the form of extensity. The quantum lying at their base undergoes no alteration through this difference, the difference being only an outward form. In the specifying measure, by contrast, the quantum stands at one time in its immediate magnitude, whereas at another it is taken, by way of the exponent of the ratio, in a different amount.

The exponent which makes up what is specific may at first look like a fixed quantum, being the quotient of the ratio between the external side and the qualitatively determined one. So taken, however, it would be nothing beyond an external quantum; the exponent must here be understood as nothing other than the moment of the qualitative itself, the moment which specifies the quantum as such. What is properly immanent and qualitative in quantum is, as came out earlier, solely the power-determination. A determination of that kind it must be which constitutes the ratio, and which here, as the determination that is in itself, has come to confront the quantum as the external constitution . This quantum has for its principle the numerical one, wherein its being-determined-in-itself consists; and the connecting of the numerical one is the external sort, while the alteration determined solely by the nature of immediate quantum as such consists on its own in one such numerical one joining on, and then another such, and so on without end. If, then, the external quantum alters in arithmetical progression, the specifying reaction of the qualitative nature of measure yields another series, one bound up with the first, waxing and waning along with it, yet in a ratio not fixed by any numerical exponent but incommensurable with number, a ratio following a determination by powers.

Remark

To adduce an example: temperature is a quality at which these two sides, that of being external quantum and that of being specified quantum, come apart. As quantum it is external temperature, temperature moreover of a body serving as universal medium, of which it is assumed that its alteration proceeds along the scale of arithmetical progression and that it waxes or wanes uniformly, whereas it is taken up differently by the various particular bodies situated within it, inasmuch as these, through their immanent measure, determine the temperature received from without, the temperature-alteration of these bodies answering neither to that of the medium nor to one another's in direct ratio. Various bodies compared at one and the same temperature yield the ratio-numbers of their specific heats, of their heat capacities. But these capacities of the bodies shift at different temperatures, and bound up with that shift is the onset of an alteration of the specific shape. With the increase or diminution of temperature, accordingly, a particular specification makes itself apparent. Between the temperature represented as external and the temperature of a determinate body, itself at the same time dependent upon the former, the relation has no fixed exponent of ratio; the increase or decrease of this warmth does not keep uniform pace with the waxing and waning of the external one. – A temperature is here assumed as external in general, one whose alteration is supposed to be merely external, that is, purely quantitative. It is, however, itself the temperature of air, or else some other specific temperature. Regarded more closely, therefore, the ratio would properly have to be taken not as one between a merely quantitative and a qualifying factor, but as one between two specific quanta. As the specifying ratio will presently determine itself further, the moments of measure consist not merely in a quantitative side together with a side that qualifies the quantum, both belonging to one and the same quality, but rather in the relation of two qualities, each of which is in its own self a measure.

c. Relation of Both Sides as Qualities

1. The qualitative side of quantum, the side determinate in itself, exists only as connection to the externally quantitative; as the specifying of that quantum it is the sublating of the externality through which quantum as such is at all, and so it has that quantum for its presupposition and makes its start from it. This quantum, however, is also qualitatively distinct from the quality itself; and since the difference of the two has to be posited within the immediacy of being at large, wherein measure still stands, the two sides are qualitative over against each other, each of them on its own such an existence; and the one of them, at first only a formal quantum undetermined in its own self, is the quantum of a something and of that something's quality, and, now that the bearing of these upon one another has determined itself into measure generally, it is likewise the specific magnitude of these qualities. These qualities stand, according to the measure-determination, in relation to one another; that determination is their exponent, yet in themselves they are already referred to each other within the being-for-itself of measure, quantum being there in its double being as external and as specific, so that each of the distinguished quantities carries this twofold determination about it and is at the same time utterly interlaced with the other; in just this, and in nothing else, are the qualities determinate. They are in this way not merely an existence which is there for one another at large, but are posited as inseparable; and the determinateness of magnitude knit to them is a qualitative unity, – One measure-determination in which, conformably to their concept, they hang together in themselves. Measure is thus the immanent quantitative comportment of two qualities towards one another.

2. In measure the variable magnitude enters as an essential determination, since measure is quantum in sublated shape – no longer, therefore, what a quantum has to be in order to count as one, but quantum and at the same time something other besides; that other is the qualitative, and, as was settled above, nothing else than the quantum's ratio of powers. In immediate measure such alteration is not yet posited; there is only some single quantum or other at large, to which a quality is knit. In the specifying of measure – the determination just preceding – taken as an alteration which the qualitative works upon the merely external quantum, there is posited a differentiatedness of the two determinatenesses of magnitude, and with it, quite generally, a multiplicity of measures at one common external quantum; quantum first shows itself as existent measure in such differentiatedness of itself from itself, in that it, one and the same (e.g. the same temperature of the medium), steps forth at once as a diverse and indeed quantitative existence (– in the differing temperatures of the bodies lodged in that medium). This differentiatedness of quantum in the differing qualities – in the differing bodies – yields a further form of measure, the form wherein both sides comport themselves towards each other as qualitatively determined quanta, and this may be called the realized measure.

Magnitude, being magnitude at large, is alterable, for its determinateness is as a limit which is at the same time none; the alteration to that extent touches only a particular quantum, in whose place another gets posited; the genuine alteration, however, is that of quantum as such; and this yields what, so grasped, is the interesting determination of variable magnitude in higher mathematics; where one must neither halt at the formal aspect of variability at large, nor fetch in anything beyond the simple determination of the concept, according to which the other of quantum is only the qualitative. The genuine determination, then, of real variable magnitude is that it is the qualitatively determined one, and hence, as has been sufficiently shown , the one determined through a ratio of powers; in such variable magnitude it is posited that quantum does not hold good as such, but according to the determination that is other to it, the qualitative determination.

The sides of this comportment have, on their abstract side, as qualities at large, some particular signification or other, space and time for instance. Taken at first quite generally in their measure-relation as determinatenesses of magnitude, one of them is an amount that rises and falls in external, arithmetical progression, the other an amount that is specifically determined by the first, which is unit for it. So far as each would equally be only some particular quality at large, there would lie in them no difference as to which of the two is to be taken, in respect of its determination of magnitude, as the merely externally quantitative one, and which as the one that alters in quantitative specification. Should they comport themselves, say, as root and square, it is all one at which of them the increase or diminution is viewed as merely external, proceeding in arithmetical progression, and which by contrast is viewed as determining itself specifically at that quantum.

But the qualities are not indeterminately diverse over against each other, for in them, as moments of measure, the qualification of measure is supposed to lie. The nearest determinateness of the qualities themselves is, for the one, to be the extensive, externality in its own self, and for the other, the intensive, what is within itself or is negative over against the first. Among the quantitative moments, then, amount belongs to the extensive quality and unit to the intensive; where the ratio is simply direct, the first counts as dividend and the second as divisor, while in the specifying ratio the first must be read as power, or as the becoming-other, and the second as root. So far as counting still goes on here, that is, so far as reflection is still cast upon the external quantum (which is thus the wholly contingent determinateness of magnitude, the one called empirical), and the alteration is accordingly likewise taken as proceeding in external, arithmetical progression, this falls to the side of the unit, of the intensive quality, whereas the external, extensive side is to be exhibited as altering within the specified series. But the direct ratio (like velocity at large, s/t) is here brought down to a formal determination, one that does not exist but pertains merely to abstracting reflection; and if root and square still stand so related (as in s= at²) that the root counts as empirical quantum advancing in arithmetical progression while the other side counts as specified, then the qualification of the quantitative reaches its higher realization, the one closer to the concept, when the two sides comport themselves in powers of a loftier determination (as happens in s³= at²).

Remark

What has been discussed here regarding the connection between the qualitative nature of an existence and its determination of quantity within measure finds its application in the example of motion already hinted at, first of all 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. Were velocity at large nothing but a ratio between the space and the time of a motion, it would be a matter of indifference which of the two moments should count as the amount and which as the unit. But space, like weight in the case of specific gravity, is an external, real whole at large, hence amount, whereas time, like volume, is the ideal element, the negative, the side of unit. – What belongs here essentially, however, is the weightier relation, namely that in free motion – at first the still conditioned sort, that of fall – the quantity of time and the quantity of space stand determined against each other, the former as root, the latter as square, – or that in the absolutely free motion of the heavenly bodies the period of revolution and the distance stand so, the former a power lower than the latter, – the former as square, the latter as cube. Fundamental relations of this kind rest upon the nature of the qualities that stand in relation, space and time, and upon the manner of connection wherein they stand, whether as mechanical motion, that is, as unfree motion undetermined by the concept of its moments, or as fall, that is, conditionally free motion, or as absolutely free celestial motion; – and these kinds of motion, quite as much as their laws, rest upon the development of the concept of their moments, of space and time, inasmuch as these qualities as such prove themselves in themselves, that is, in the concept, to be inseparable, and their quantitative relation is the being-for-itself of measure, only One measure-determination.

Concerning the absolute measure-relations it may well be recalled that the mathematics of nature, if it means to deserve the name of a science, must essentially be the science of measures, – a science for which a great deal has indeed been done empirically, but as yet little in a properly scientific, that is, philosophical, way. Mathematical principles of natural philosophy – such being the title Newton gave his work – would have had to contain matters of a wholly different order, were they to satisfy that vocation in a deeper sense than he and the entire Baconian lineage of philosophy and science conceived it, so as to throw light into these regions, dark still yet in the highest degree deserving of consideration. – It is a great merit to come to know the empirical numbers of nature, the distances of the planets from one another for instance; but an infinitely greater one to make the empirical quanta vanish and to raise them to a universal form in which quantity stands determined, whereby they turn into moments of a law or a measure; – such are the immortal merits earned, for instance, by Galileo with respect to fall and by Kepler with respect to the motion of the heavenly bodies. They established the laws they had found in this manner, that they showed the range of the singularities of perception to answer to them. A higher proving of these laws must nevertheless still be demanded; nothing else, namely, than that their determinations of quantity be cognized out of the qualities, or determinate concepts, which are brought into relation (such as time and space). Of this manner of proving not a trace is yet to be met with in those mathematical principles of natural philosophy, nor in the later labours of the same kind. It was noted above, on the occasion of that semblance of mathematical proofs of natural relations which rests on the misuse of the infinitely small, that the attempt to carry such proofs through properly mathematically, that is, neither out of experience nor out of the concept, is a preposterous undertaking. Such proofs presuppose their theorems, precisely those laws, from experience; what they achieve consists in bringing them to abstract expressions and handy formulas. The whole real merit ascribed to Newton in preference to Kepler with regard to these very same objects will one day, the sham scaffolding of proofs once deducted, – and doubtless upon a purer reflection on what mathematics is capable of achieving and on what it has achieved, – come to be restricted, with distinct knowledge, to that transformation of the expression¹⁴ and to the analytic treatment introduced so far as its beginnings go.

C. Being-for-itself in Measure

1. Within the shape of specified measure considered a moment ago, what is quantitative on either side stands qualitatively determined (both being in the ratio of powers); so the two count as moments of One measure-determinateness having qualitative nature. In all this, however, the qualities are as yet posited only as immediate ones, as merely diverse, which do not themselves stand in the relation wherein their determinatenesses of magnitude stand, namely that of having neither sense nor existence outside such a relation, this being what the power-determinateness of magnitude carries with it. The qualitative thus keeps itself veiled, since it specifies not itself but the determinateness of magnitude; only at that determinateness is it posited, while on its own it is immediate quality as such, quality which, apart from the setting of magnitude in difference by it and apart from its connection to its other, would still have an existence subsisting on its own. So space and time both hold good, apart from the specification which their determinateness of magnitude receives in the motion of fall or in absolutely free motion, as space at large and time at large, space subsisting on its own outside and without time, as enduring, and time as flowing on its own independently of space.

This immediacy of the qualitative over against its specific measure-connection is, however, just as much knit together with a quantitative immediacy and with the indifference which a quantitative factor in it shows towards this relation of its own; the immediate quality possesses likewise a merely immediate quantum. Hence the specific measure also has a side of at first external alteration, an alteration whose advance is merely arithmetical, is not disturbed by the specifying factor, and into which the external, and therefore merely empirical, determinateness of magnitude falls. Quality and quantum, cropping up in this fashion outside the specific measure as well, stand at the same time in connection with it; the immediacy is a moment of such determinations as themselves belong to measure. The immediate qualities are thus appurtenant to measure too, are likewise in connection, and stand, as regards determinateness of magnitude, in a relation which, lying outside the specified relation, outside the power-determination, is itself only the direct ratio and immediate measure. This consequence, and how it hangs together, is now to be stated more closely.

2. The immediately determined quantum as such, even where as a moment of measure it is otherwise grounded in itself within a nexus of the concept, is, in its bearing upon the specific measure, something given from without. The immediacy thereby posited is, however, the negation of the qualitative measure-determination; that negation was pointed out just above at the sides of this measure-determination, sides which on that account came to look like self-subsistent qualities. Such negation, and the return to immediate determinateness of quantity, lie in the qualitatively determined relation insofar as the relation of things distinguished contains at large their connection as One determinateness, which accordingly, here in the quantitative and as distinguished from the relation-determination, is a quantum. Being the negation of the distinguished sides that are qualitatively determined, this exponent amounts to a being-for-itself, to being-determined-outright; yet a being-for-itself of that kind is only in itself; qua existence it is a simple and immediate quantum, quotient or exponent of what counts as a ratio between the sides of measure, that ratio being taken as direct; but it is, in general terms, the unit appearing as empirical within the quantitative aspect of measure. – In the fall of bodies the spaces traversed stand in the ratio of the square of the times elapsed; s = at²; – this is what stands specifically determined, a ratio of powers holding between space and time; the other relation, the direct ratio, would fall to space and to time taken as qualities indifferent to each other; it is meant to be that of space to the first temporal moment; one and the same coefficient, a, persists through every later point of time; – the unit, an ordinary quantum, for the amount otherwise determined by the specifying measure. This unit passes at the same time for the exponent belonging to that direct ratio which attaches to represented bad velocity, that is, to formal velocity, not specified through the concept . Velocity of that sort does not exist here, no more than does the one mentioned earlier, which was to belong to the body at the end of a moment of time. The former is attributed to the first moment of time in the fall, but this so-called moment of time is itself only an assumed unit and, as such an atomic point, has no existence; the beginning of the motion – the smallness alleged on its behalf could make no difference – is straightway a magnitude, and indeed a magnitude specified by the law of fall. That empirical quantum is attributed to the force of gravity, in such wise that this force itself is supposed to bear no relation to the specification at hand (the power-determinateness), to what is peculiar to the measure-determination. The immediate moment – that in the motion of fall there answers to one unit of time (– a second, and indeed the so-called first one –) an amount of roughly fifteen units of space reckoned as feet – is an immediate measure, on a par with the measure-magnitude of human limbs, or the distances and diameters of the planets, and the like. Where such a measure gets determined lies elsewhere than within the qualitative measure-determination, here that of the law of fall itself; but upon what such numbers hang, the merely immediate and therefore empirically appearing element of a measure – on that head the concrete sciences have so far told us nothing. Here we have to do only with this determinateness of the concept; it is this, that the empirical coefficient makes up the being-for-itself within the measure-determination, but only the moment of being-for-itself, so far as this is in itself and therefore something immediate. The other side is what this being-for-itself has developed into, the specific measure-determinateness belonging to the two sides. – Gravity, in the relation of falling, a motion admittedly still half conditioned and only half free, is by this second moment to be looked upon as a force of nature, so that its relation is determined through the nature of time and of space, and hence that specification, the ratio of powers, falls within gravity; the former, the simple direct ratio, expresses only a mechanical comportment of time and space, the formal velocity, externally brought forth and settled.

3. Measure has determined itself into a specified relation of magnitudes which, qua quantitative, carries the ordinary external quantum about it; that quantum, though, is no quantum at large but essentially serves as the moment determining the relation as such; hence it is exponent, and, since it is now an immediate being-determined, an exponent that cannot alter, the exponent accordingly of that direct ratio, mentioned already, between the very same qualities, whereby their mutual relation of magnitude also gets specifically determined. This direct ratio is, in the example used of the measure of the motion of fall, anticipated as it were and assumed to be present; but, as was noted, it does not yet exist in that motion. – It makes up, however, the further determination that measure is now realized in such a way that its two sides are measures, distinguished as immediate and external and as specified within itself, and that measure is the unity of them. Measure, being this unity, holds within it the relation wherein the magnitudes stand determined and set in difference by the nature of the qualities, a relation whose determinateness is on that account entirely immanent and self-subsistent, and which has at once run together into the being-for-itself of immediate quantum, into the exponent of a direct ratio; therein measure's determining of itself is negated, seeing that its ultimate determinateness, the one that is for itself, lies in this other of its own; and inversely, the immediate measure, which ought to be qualitative in its own self, wins its qualitative determinateness in truth only at that other. Such negative unity constitutes real being-for-itself, the category of a something taken as unity of qualities that stand within the measure-relation; – a complete self-subsistence. Immediately the two, which have turned out to be two different relations, also yield a twofold existence, or more precisely, such a self-subsistent whole is, as something for itself at large, at the same time a repelling within its own self into distinct self-subsistent things, whose qualitative nature and persistence (materiality) reside in their measure-determinateness.

Chapter 2. Real Measure

Measure has been determined into a connection of measures which make up the quality of distinct self-subsistent somethings, more familiarly: things. The measure-relations just considered belong to abstract qualities such as space and time; for those to be considered in what lies ahead the examples are specific gravity and, further on, the chemical properties, which stand as determinations of material concrete existences. Space and time are moments of such measures too, but moments now subordinated to further determinations, no longer comporting themselves to one another merely according to their own conceptual determination. In sound, e.g., the time in which an amount of vibrations occurs and the spatial factor of the length and thickness of the vibrating body are among the determining moments; but those ideal moments have their magnitudes fixed from outside, they no longer show themselves against each other in a ratio of powers but in an ordinary direct one, and the harmonic reduces itself to the wholly external simplicity of numbers whose ratios are the easiest of all to take in, and which thereby afford a satisfaction falling entirely to sensation, since for spirit no representation, image of fantasy, thought or anything of the sort is present to fill it. Since the sides that now constitute the measure-relation are measures themselves, and yet at once real somethings, the measures belonging to them are immediate ones to begin with and, taken as ratios within them, direct ones. It is the relation of such ratios to one another that must now be considered in its onward determination.

Measure, as it is henceforth real, is

firstly a self-subsistent measure of a corporeality which comports itself towards others and in this comportment specifies them, and along with them self-subsistent materiality as well. Such specifying, being an external act of connecting to many others at large, is the bringing forth of further ratios, hence of further measures, and specific self-subsistence no longer stays within one direct ratio but goes over into a specific determinateness that constitutes a series of measures.

Secondly, the direct ratios arising in this way are measures determined in themselves and exclusive ones (elective affinities); yet because their mutual difference is at the same time only quantitative, there is present an onward course of ratios that runs in part merely externally quantitative, though it is broken into by qualitative relations as well, and yields a nodal line of specific self-subsistent entities.

Thirdly, however, in this onward course there enters for measure measurelessness in general, and more determinately the infinity of measure, in which the self-subsistences that exclude one another are at one with each other, and the self-subsistent steps into negative connection with its own self.

A. The Relation of Self-subsistent Measures

The measures are now called no longer merely immediate but self-subsistent, insofar as within themselves they [become] relations of measures which are specified, and so in this being-for-itself are somethings, physical, in the first instance material things. The whole, however, being itself a relation of measures of that kind, is

a. to begin with immediate itself; the two sides, determined as self-subsistent measures of this sort, accordingly subsist outside one another in particular things, and get set externally into combination;

b. the self-subsistent materialities, however, are what they qualitatively are only through the quantitative determination which they have as measures, and hence are determined through a connection to others which is itself quantitative, as different over against them (so-called affinity), and indeed as members of a series made up of quantitative comportment of that kind;

c. this indifferent manifold comportment at the same time closes itself off into an exclusive being-for-itself; – so-called elective affinity.

a. Combination of Two Measures

Within itself a something has been determined as a measure-relation of quanta to which, furthermore, qualities accrue, and the something is what connects these qualities. The first of them is its being-within-itself, whereby it is a being-for-itself, – something material – (weight, if taken intensively, or, extensively, the multitude of material parts); the second, by contrast, is the externality of that being-within-itself, (the abstract, the ideal, space.) Quantitatively determined as these qualities are, it is their ratio to each other that constitutes the qualitative nature of the material something; – weight in ratio to volume, determinate specific gravity. Volume, the ideal side, has to be assumed as unit, whereas the intensive side, appearing in quantitative determinateness and in comparison with volume as extensive magnitude, as a multitude of ones existing for themselves, has to be assumed as amount. – What has vanished here is the purely qualitative comportment of the two determinatenesses of magnitude according to a ratio of powers, and it has vanished because in the self-subsistence of being-for-itself (– of material being –) immediacy has returned, an immediacy at which the determinateness of magnitude counts as quantum as such, and at which the ratio of such a quantum to the further side is likewise fixed in the ordinary exponent of a direct ratio.

Now this exponent, though it is the specific quantum belonging to the something, is an immediate quantum, and only in comparison with the exponents of other such ratios does it, and with it the specific nature of such a something, become determined. What it constitutes is the specific being-determined-in-itself, the inner measure peculiar to a something; but because that measure of its own rests on quantum, it too is nothing more than an external, indifferent determinateness, and such a something is on that account alterable, its inner measure-determination notwithstanding. The other towards which it can comport itself as alterable is no multitude of matter, no quantum at large; against that its specific being-determined-in-itself holds firm; rather it is a quantum that is itself equally the exponent of a specific ratio of that kind. What stand in connection and enter into combination are two things of unlike inner measure; – two metals, say, of unlike specific gravity; – and what homogeneity of nature may otherwise be requisite for such a combination to be possible, so that, e.g., no metal is at issue whose combination with water would be spoken of, does not belong here for consideration. – Each of the two measures, on the one side, preserves itself in the alteration that was to befall it through the externality of quantum, and does so because it is measure, while on the other side this preserving-of-itself is itself a negative comportment towards that quantum, a specifying of it, and, the quantum being exponent of the measure-relation, an alteration of measure itself, and indeed a mutual specifying.

Taken by merely quantitative determination, the combination would be a mere summing of the two magnitudes of the one quality with the two of the other, e.g., in combining two matters of unlike specific gravity, the sum of both weights and of both volumes, so that not merely the weight of the mixture but likewise the space it occupies would stay equal to those respective sums. Yet it is the weight alone that turns out to be the sum of the weights present before the combination; what gets summed is that side which, as the side existing for itself, has become fixed existence and thereby a lasting immediate quantum, – matter's weight, or what passes for it in respect of quantitative determinateness, the multitude of material parts. Into the exponents, though, the alteration falls, since they are the expression of qualitative determinateness, of being-for-itself as measure-relations, which, while quantum as such undergoes the contingent, external alteration by an addendum that is summed, proves itself at the same time to negate this externality. This immanent determining of the quantitative, unable as it is, per the showing above, to come out at the weight, gives proof of itself at the other quality, the side of the ratio that is ideal. To sense perception it may be striking that after two specifically unlike matters have been mixed an alteration – as a rule a diminution – shows up in the summed volume; space itself constitutes the subsisting of matter that lies outside itself. But over against the negativity which being-for-itself harbours within, this subsisting is what has no being in itself, what is alterable; in this way space gets posited as what it truly is, as ideal.

Herewith, though, it is not merely one of the qualitative sides that gets posited as alterable but measure itself, and thereby the qualitative determinateness of the something grounded on measure has shown itself to be nothing fixed in its own self, having rather, like quantum at large, its determinateness in other measure-relations.

b. Measure as a Series of Measure-Relations

1. Were a something that gets united with an other, and equally this other, to be what it is solely by way of simple quality, then in such a combination the two would do nothing but sublate themselves; a something that is measure-relation within itself, however, is self-subsistent and for that very reason unitable with a something of like kind; in getting sublated within this unity it holds itself in being through its indifferent, quantitative subsisting, and comports itself at once as the moment that specifies a fresh measure-relation. Wrapped up in the quantitative, its quality is thereby just as indifferent towards the other measure, continuing itself both into that measure and on into the one newly formed; as for the exponent of this new measure, it is nothing but some quantum or other, a determinateness that is external; and the indifference exhibits itself in this, that the specifically determined something engages with further measures of like kind in just such neutralizations of the measure-relations on either side; in a single one alone, formed by it together with an other, its specific peculiarity finds no expression.

2. Combining in this way with several others, which within them are measures likewise, yields ratios that differ, and hence exponents that differ. Only in comparison with others does the self-subsistent possess the exponent of its being-determined-in-itself; its neutrality with others, however, is what constitutes the real comparing of it with them; here it is compared with them by way of its own self. Since the exponents of these ratios differ, though, the self-subsistent thereby sets out its qualitative exponent as the series of these differing amounts to which it stands as unit; – as a series of specific comportment towards others. Taken as One immediate quantum, the qualitative exponent gives voice to a single relation. What genuinely marks off the self-subsistent is the peculiar series of exponents that it, assumed as unit, builds with other self-subsistents of the kind, seeing that another among them, brought into connection with those very ones and assumed as unit, will shape a different series. – What now constitutes the qualitative side of the self-subsistent is the relation obtaining inside such a series.

Now insofar as a self-subsistent of this kind builds a series of exponents together with a series of self-subsistents, it looks at first as though what marks it off from another lying beyond that series, one with which it gets compared, were this, that the latter makes up a different series of exponents with the very same partners over against it. In this way, however, the two self-subsistents would be not comparable, insofar as each gets regarded as unit over against its own exponents while the two series springing from that connection are indeterminately other. The pair meant to be compared as self-subsistents stand at first distinguished from each other merely as quanta; determining their ratio calls for a common unit that exists for itself. Such a determinate unit is to be looked for nowhere but in that wherein the ones to be compared have, as shown, the specific existence of their measure, that is to say, in the relation borne to one another by the ratio-exponents of the series. This relation of exponents is itself a unit existing for itself, a determinate one in fact, only so far as the members of the series bear it towards both alike as a constant relation among themselves; on that condition it can be their common unit. Here alone, accordingly, does the comparability of the two self-subsistents reside, of the two that were assumed not as neutralizing themselves with each other but as indifferent towards each other. Set apart, outside the comparing, each is unit for the ratios with the members facing it, members that are amounts over against that unit and so make up the series of exponents. Conversely that series is, on the other side, the unit for the pair which, once compared, are quanta over against each other; as such the pair are themselves differing amounts of the unit just now brought out.

But those, further, which over against the pair – or rather the many at large – compared among themselves supply the series of the exponents of their comportment, are within themselves self-subsistents no less, each of them a specific something carrying a measure-relation that belongs to it in itself. To that extent each of them too has to be taken as unit, such that in the pair first named, merely compared among themselves, or rather in the indeterminately several, they possess a series of exponents, exponents which are the comparison-numbers of the ones just named among themselves; and conversely the comparison-numbers of those now taken singly as self-subsistent likewise make up, among themselves, the series of exponents for the members of the first series. Both sides in this way are series wherein each number is firstly unit at large over against the series facing it, at which series it possesses its being-determined-for-itself as a series of exponents; secondly, it is one among the exponents for every member of the facing series; and thirdly a comparison-number to the remaining numbers of its own series, and as an amount of that sort, which belongs to it as exponent too, it possesses its for-itself-determined unit at the series facing it.

3. What has come back in this comportment is the manner and way in which quantum, as existing for itself, that is, as degree, is posited to be simple and yet to have its determinateness of magnitude at a quantum lying outside it, one that is a circle of quanta. In measure, though, what is external here amounts not simply to a quantum and a circle of quanta but to a series of ratio-numbers, whose totality is where the being-determined-for-itself of measure resides. Just as happens with quantum's being-for-itself as degree, the nature of self-subsistent measure has turned itself around into this externality of its own self. Its connection to itself stands at first as an immediate ratio, and with that its indifference towards other lies straightway in quantum alone. Its qualitative side falls therefore into this externality, and its comportment towards other turns into that which constitutes the specific determination of this self-subsistent. So that determination lies purely and simply in the quantitative manner and way of such comportment, a manner and way determined by the other as much as by the self-subsistent itself, the other in question being a series of quanta, and it reciprocally a series of the kind. But there is more contained in this connection, wherein two specific measures specify themselves into a something, into a third, the exponent, namely this, that within it the one has not passed over into the other, so that not merely one negation at large is posited but both are posited negatively there, and, each of them holding itself indifferently within it, its negation is once more negated as well. Their qualitative unity is accordingly an exclusive unity existing for itself. Only in the moment of excluding do the exponents, comparison-numbers among themselves to begin with, come to have at them their truly specific determinateness over against each other, and their difference thereby takes on a qualitative nature as well. It rests nonetheless on the quantitative; firstly, the self-subsistent comports itself towards a several of its qualitatively other side for the sole reason that in such comportment it remains at the same time indifferent; secondly, the neutral connection, by virtue of the quantitativity it holds within, is now not alteration merely but is posited as negation of negation, and as exclusive unity. By this the affinity of a self-subsistent to the several on the other side is no longer an indifferent connection but an elective affinity.

c. Elective Affinity

The expression elective affinity has been employed here, and in what went before neutrality and affinity as well, – terms drawn from the chemical relation. For within the chemical sphere it is essentially in its connection to its other that the material has its specific determinateness; only as this difference does it exist. Bound as this specific connection further is to quantity, it is at the same time a connection not to one single other but to a whole series of differents facing it; what the combinations with that series rest upon is a so-called affinity with each of its members, and yet, indifference notwithstanding, every one of them excludes the rest; and that connection of opposed determinations has still to be looked at. – Chemistry, however, is not the only field where the specific presents itself in a circle of combinations; a single tone likewise acquires its sense only in comporting itself towards, and combining with, another tone and with the series of others; harmonious or disharmonious, such a circle of combinations makes for the tone's qualitative nature, resting as that nature does at the same time on quantitative ratios which form a series of exponents, ratios of the two specific ratios that each of the tones combined is within itself. A single tone is the fundamental tone of a system, and just as much, in turn, one member within the system of every other fundamental tone. The harmonies are exclusive elective affinities, and yet their qualitative peculiarity dissolves once more, just as readily, into the externality of a merely quantitative onward course. – As to wherein the principle of a measure lies for the affinities that are elective affinities (chemical, musical or other) among and against the rest, a remark bearing on the chemical case will still be made below; a question of this higher order, though, is bound up most intimately with the specific character of what is properly qualitative, and has its place in the particular parts of concrete natural science.

Given that a member of one series has its qualitative unity in how it comports itself towards the whole of the series facing it, whose members are diverse from each other solely through the quantum by which they neutralize themselves with that member, the more special determinateness within this manifold affinity is likewise a merely quantitative one. Being an exclusive, qualitative connection, elective affinity takes the comportment out of this quantitative difference. Here the nearest determination on offer is this: that a member's elective affinity towards the members of the other series, with all of which it stands in affinity, follows the difference in multitude, hence in extensive magnitude, that holds among the members of the one side for neutralizing a member of the other. Excluding, understood as a firmer cohesion against other possibilities of combination and grounded on that difference, would then look to be converted into a correspondingly greater intensity, in keeping with the identity established earlier between the forms of extensive and intensive magnitude, forms in both of which the determinateness of magnitude is one and the same. Yet this switching of the one-sided form of extensive magnitude over into its other, the intensive, changes nothing in the nature of the underlying determination, which is that one and the same quantum; so that in fact no excluding at all would be posited thereby, and indifferently either a single combination or else a combination of indeterminately many members might take place, provided only that the portions entering from them, conformably to their mutual ratios, matched the quantum demanded.

Only, the combination we have also named neutralization is not the form of intensity alone; the exponent is in essence a determining of measure, and thereby exclusive; on this side of exclusive comportment the numbers have forfeited their continuity and their running-together with one another; a negative character attaches to the more or less, and the precedence one exponent holds over others does not stop short at determinateness of magnitude. Yet no less is that further side at hand, on which a moment is again indifferent as to receiving the neutralizing quantum from any of several moments facing it, from each in keeping with its specific determinateness over against the other; comportment that excludes and negates suffers at the same time this inroad from the quantitative quarter. – Herewith there is posited a switching of indifferent, merely quantitative comportment into a qualitative one, and inversely a passing-over of the specific determinedness into the merely external relation; – a series of relations that are now of merely quantitative nature, now specific ones and measures.

Remark

The chemical substances are the most distinctive examples of such measures, ones that are moments of measure and have what makes up their determination solely in their bearing towards others. Acids and alkalis, or bases in general, appear as things immediately determined in themselves, yet 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. Moreover, the difference by which they count as self-subsistent lies not in that immediate quality but in the quantitative way and manner of their bearing. For it is not confined to the chemical opposition between acid and alkali, or base as such, but is specified into a measure of saturation, lying in how the quantity of the mutually neutralizing substances is specifically determined. This determination of quantity with respect to saturation makes up the qualitative nature of a substance, makes it into what it is for itself, and 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 connection were to remain of a purely qualitative nature, then – as with the connection of the magnetic poles or of the electricities – the one determinateness would be merely the negative of the other, and the two sides would not also show themselves at the same time indifferent towards each other. But because the connection 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 each other in the first instance by this, that one acid, for example, requires more of an alkali in order to saturate itself with it than another does. But the self-subsistence that is for itself shows itself in this, that the affinities behave exclusively and one has preference over another, since taken by itself an acid can enter into combination with all the alkalis, and conversely. Thus what makes up the chief difference of one acid over against another is whether it has a closer affinity to a base than another has, that is, a so-called elective affinity.

Concerning the chemical affinities of acids and alkalis the law has been found that when two neutral solutions are mixed, whereby a separation occurs and from it two new combinations arise, these 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 the series of ratio numbers in which the various acids saturate one and the same alkali has been determined, taking that alkali as unit, then this series is the same for every other alkali, save only that the various alkalis are to be taken over against each other 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 was the first to bring out these series in their simplicity from Richter's works; 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. – To want to take account here of the knowledge of the ratio numbers of the mixtures of the chemical elements, so greatly developed on all sides since this was first written, would also be a digression for the reason that this empirical, though in part merely hypothetical, extension remains enclosed within the same determinations of the concept. But on the categories employed in it, further on the views of chemical elective affinity itself and of its connection to the quantitative, as well as on the attempt to ground it upon determinate physical qualities, a few remarks may yet be added.

As is well known, Berthollet modified the general representation of elective affinity by the concept of the efficacy of a chemical mass. This modification has – and this is well worth distinguishing – no influence on the quantitative ratios of the chemical laws of saturation themselves, but the qualitative moment of exclusive elective affinity as such is not merely weakened but rather sublated. If two acids act upon an alkali, and the one of which it is said that it has the 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 so-called affinity. Berthollet's investigations have specified the closer circumstances under which the efficacy of the chemical mass is sublated and one acid (the more strongly akin) appears to drive out the other (the weaker) and to exclude its action, and so to be at work in the sense of elective affinity. What he has demonstrated is that the excluding occurs under circumstances – cohesive strength, the failure of the salts produced to dissolve in water – and not through the qualitative nature of the agents as such, circumstances whose effect can in turn be sublated by other circumstances, temperature for example. With the removal of these hindrances the chemical mass comes into efficacy undiminished, and what appeared as purely qualitative excluding, as elective affinity, shows itself to reside merely in external modifications.

Berzelius would be chiefly the one to be heard further on this subject matter. But in his Lehrbuch der Chemie he puts forward nothing distinctive or more determinate on the matter. Berthollet's views have been taken up and repeated word for word, only decked out with the peculiar metaphysics of an uncritical reflection, whose categories alone therefore offer themselves for closer consideration. The theory goes beyond experience and partly invents sensuous representations such as are not themselves given in experience, partly applies determinations of thought, and in both ways makes itself an object of logical critique. We shall therefore take up what that textbook itself (Vol. III, Part I, trans. Wöhler, p. 82 ff.) sets out on the theory. There one reads that one must represent to oneself how, in a fluid of uniform mixture, every atom of the body in solution has about it an equal number of solvent atoms surrounding it; and that where several substances have been dissolved together they must apportion among themselves the interstices lying between the solvent's atoms, so that, given a uniform mixture of the fluid, there arises a symmetry in the situation of the atoms such that all the atoms of the several bodies come to occupy a uniform situation in relation to the atoms of the other bodies; one might accordingly say that solution is marked by the symmetry in the disposition of the atoms, and combination by the determinate proportions.« – This is thereupon elucidated by an example of the combinations that arise from a solution of copper chloride to which sulphuric acid is added; but in this example there is of course no demonstration that atoms exist, nor that a number of atoms of the dissolved bodies surround atoms of the fluid, that free atoms of the two acids range themselves about those that remain combined (with the copper oxide), nor that the symmetry in the disposition and situation exists, nor that interstices between the atoms exist, – least of all that the dissolved substances apportion among themselves the interstices of the atoms of the solvent. This would mean that the dissolved substances take their place where the solvent is not, – for its interstices are the spaces empty of it, – hence that the dissolved substances are not in the solvent but, even while surrounding and encompassing it, or surrounded and encompassed by it, are outside of it, and so assuredly not dissolved by it either. One thus does not see why one must make for oneself such representations, which are not demonstrated in experience, which contradict themselves at once in essentials, and are in no other way further corroborated. This could happen only through the consideration of these representations themselves, that is, through metaphysics, which is logic, but by this they are just as little confirmed as by experience, – on the contrary! – Berzelius does, incidentally, grant the point made above as well, that Berthollet's propositions run counter in no way to the theory of determinate proportions, – he adds, of course, that neither do they run counter to the views of corpuscular philosophy, that is, to the representations adduced just now concerning atoms, concerning the filling of the interstices of the dissolving fluid by atoms of solid bodies, and so forth, – yet this latter, groundless metaphysics has in essence nothing to do with the proportions of saturation themselves.

The specific element expressed in the laws of saturation thus concerns only the multitude of themselves quantitative units (not atoms) of one body with which the quantitative unit (just as little an atom) of another body chemically different from the first neutralizes itself; the diversity consists solely in these different proportions. When Berzelius then, notwithstanding that his doctrine of proportions is wholly and solely a determination of multitudes, nevertheless also speaks of degrees of affinity, for example on p. 86, in that he explains Berthollet's chemical mass as the sum of the degree of affinity out of the available quantity of the operative body, where Berthollet, more consistently, employs the term capacité de saturation, then he is himself thereby falling into the form of intensive magnitude. Yet that form is exactly what constitutes the distinguishing mark of the so-called dynamical philosophy, which earlier, on p. 29 of the same work, he labels »the speculative philosophy of certain German schools« and repudiates emphatically for the good of the admirable »corpuscular philosophy«. Of this dynamical philosophy he states there that it assumes the elements to interpenetrate each other in their chemical union and that neutralization consists in this mutual interpenetration; this means nothing other than that the chemically different particles, which stand over against each other as multitude, go together into the simplicity of an intensive magnitude, something that also announces itself as a diminution of volume. In the corpuscular theory, by contrast, even the chemically combined atoms are supposed to maintain themselves in the interstices, that is, outside one another, (juxtaposition); degree of affinity has no sense in such a bearing, which is a merely extensive magnitude, a perduring of multitude. When it is stated in the same place that the appearances of determinate proportions came quite unforeseen for the dynamical view, this would be merely an external historical circumstance, apart from the fact that Richter's stoichiometric series were already known to Berthollet in Fischer's compilation and were cited in the first edition of this Logic, which demonstrates the nullity of the categories on which the old corpuscular theory rests just as much as the one that would be new. But Berzelius judges erroneously, as if under the dominion of »the dynamical view« the appearances of determinate proportions would have remained unknown »for ever«, – in the sense that that view could not be reconciled with the determinateness of the proportions. This is in any case only determinateness of magnitude, indifferent whether in extensive or in intensive form, – so that Berzelius too, however much he clings to the first form, that of multitude, himself makes use of the representation of degrees of affinity.

Since affinity is herewith led back to quantitative difference, it stands sublated as elective affinity; the excluding that occurs in it, however, has been referred to circumstances, that is, to determinations which show up as something outside affinity – to cohesion, to the failure of the combinations produced to dissolve, and so forth. This representation may in part be compared with the procedure in considering the effect of gravity, where what belongs in itself to gravity itself, namely that the swinging pendulum necessarily passes over into rest through it, is taken merely as the concurrently present circumstance of the external resistance of the air, of the thread, and so forth, and is ascribed to friction alone instead of to gravity. – Here, for the nature of the qualitative that resides in elective affinity, it makes no difference whether that qualitative appears and is grasped in the form of those circumstances as its conditions. With the qualitative as such a new order begins, whose specification is no longer merely quantitative difference.

If accordingly the difference of chemical affinity fixes itself precisely in a series of quantitative ratios over against elective affinity as a qualitative determinateness that supervenes, whose bearing by no means coincides with that order, then this difference is thrown back into complete confusion by the manner in which in more recent times the electrical has been brought into connection with the chemical, and the hope of obtaining from this principle, which is supposed to be deeper, an elucidation about the most important thing, the measure-ratio, is utterly disappointed. The theory that identifies outright the appearances of electricity with those of chemism is not to be examined more closely here, so far as it touches the physical and not merely the measure-ratios, and comes up for mention only to the extent that it muddles the distinctness of the determinations of measure. Taken by itself it is to be called shallow, for shallowness consists precisely in taking what is diverse as identical by leaving out the diversity. As regards affinity in this connection, since chemical processes are thus identified with electrical ones, and likewise with appearances of fire and light, it has been reduced to »the neutralization of opposed electricities«. As for the identification of electricity and chemism itself, it is almost comic to find it presented (p. 63 of the same work) in the following manner, that »the electrical phenomena do indeed account for how bodies act across a shorter or longer interval, for their attraction prior to union (which is to say, the bearing that is not yet chemical) – and for the fire (?) that arises out of this union, but yield us no elucidation as to the cause of that union of bodies which persists with such great force once the opposed electrical state has been annihilated;« that is, the theory yields the elucidation that electricity is the cause of chemical bearing, but that electricity yields no elucidation about what in the chemical process is chemical. – With chemical difference in general being traced back to the opposition of positive and negative electricity, the diversity of affinity among the agents falling on the one side and on the other is determined as the ordering of two series, of electropositive and of electronegative bodies. In identifying electricity and chemism according to their general determination, this much is already overlooked, that the former in general and its neutralization is fleeting and remains external to the quality of the bodies, whereas chemism in its action and especially in neutralization lays claim to the whole qualitative nature of the bodies and alters it. Equally fleeting within electricity is its opposition of positive and negative; it is something so unsteady that it depends on the slightest external circumstances and can be placed in no comparison with the determinateness and firmness of the opposition of acids, for instance, over against the metals, and so on. The alterability that can take place in this chemical bearing through the most violent influences, for instance an elevated temperature and so forth, bears no comparison with the superficiality of the electrical opposition. As for the further difference within the series on either side, that between a constitution more or less positively electrical and one more or less negatively electrical, this is thoroughly uncertain and unverified alike. But out of these series of bodies (Berzelius, loc. cit., p. 64 f.) »arranged according to their electrical dispositions the electrochemical system is supposed to arise, which of all is best suited to give an idea of chemistry;« these series are then given; but as to how they are in fact constituted, it is added on p. 67 »that this is approximately the order of these bodies, but that this matter has been so little investigated that as yet nothing quite certain can be settled with respect to this relative order.« – The ratio numbers of those affinity series (which Richter first drew up), no less than the exceedingly interesting reduction Berzelius has proposed, whereby combinations of two bodies come down to the simplicity of a handful of quantitative ratios, owe nothing whatever to that concoction alleged to be electrochemical. If in those proportions and in the extension of them gained on all sides since Richter the experimental path has been the right lodestar, then all the more does the mingling of these great discoveries with the barrenness of the so-called corpuscular theory, lying as it does outside the path of experience, stand in contrast with it; only this beginning of abandoning the principle of experience could motivate the further resumption of that notion, chiefly begun earlier by Ritter, of setting up fixed orderings of electropositive and electronegative bodies that were at the same time supposed to have chemical significance.

The nullity of the foundation assumed for chemical affinity in the opposition of electropositive and electronegative bodies – even if this opposition were in itself factually more correct than it is – soon shows itself on the experimental path itself, which however leads in turn to further inconsistency. It is conceded at p. 73 (loc. cit.) that two so-called electronegative bodies, such as sulphur and oxygen, combine with each other in a far more intimate manner than, for instance, oxygen and copper, although the latter is said to be electropositive. The basis for affinity resting upon the general opposition of positive and negative electricity must here accordingly give way to a bare more-or-less falling inside one single series of electrical determinateness. The degree of affinity of bodies, so it is now concluded from this, accordingly does not depend on their specific unipolarity alone (what hypothesis this determination belongs with is nothing to the purpose here; it holds only for the either of positive and the or of negative); the degree of affinity must be derived principally from the intensity of their polarity in general. Here, then, the consideration of affinity passes over more closely into the relation of elective affinity, which is what chiefly concerns us; let us see what now results for it. When it is conceded straight away (ibid., p. 73) that this polarity's degree, supposing it has any being outside our mere representation, seems to be no constant quantity but to hinge largely on temperature, the upshot given after all of this is not only that every chemical action is thus in its ground an electrical phenomenon, but further that whatever seems to be an operation of so-called elective affinity comes about solely through an electrical polarity that is stronger in some bodies than in others. As the conclusion of all this squirming about in hypothetical representations, it thus stays with the category of greater intensity, which is the same formal thing as elective affinity in general, and which, by being placed upon a greater intensity of electrical polarity, does not in the least bring that affinity onto a physical ground any further than before. But even what is here supposed to be determined as greater specific intensity is subsequently traced back only to the modifications already cited, those demonstrated by Berthollet.

The credit and renown that Berzelius enjoys for a doctrine of proportions carried through all chemical ratios could hardly of itself count as a ground for withholding an exposure of how bare the theory cited is; the nearer ground for undertaking it must rather be this, that credit of that kind on one side of a science tends, as it did with Newton, to turn into authority on behalf of a baseless structure of poor categories tacked on to it, and that metaphysics of just this sort is what gets issued with the loudest pretension and echoed as loudly again.

Besides the forms of the measure-ratio that bear on chemical affinity and elective affinity, still others may be considered with respect to quantities that qualify themselves into a system. In relation to saturation the chemical bodies form a system of ratios; saturation itself rests on the determinate proportion in which the multitudes on both sides, which have a particular material concrete existence over against each other, combine. But there are also measure-ratios whose moments are inseparable and cannot be exhibited in a concrete existence of their own, distinct from one another. These are what were earlier called the immediate self-subsistent measures, and they are represented in the specific gravities of bodies. – Inside bodies they constitute a ratio of weight against volume; the ratio's exponent, which states what determines one specific gravity as against others, is a determinate quantum belonging only to comparison, a ratio external to them within an external reflection, not founded on their own qualitative bearing towards a concrete existence set over against them. The task stands before us of cognizing the ratio exponents of the series of specific gravities as a system deriving from a rule which would make a bare arithmetical plurality specific as a series of harmonic nodes. – That very requirement would apply to the cognition of the chemical affinity series adduced above. But science has still far to go before arriving there, as far as before grasping the numbers of the distances of the planets of the solar system in a system of measure.

The specific gravities, although at first they seem to have no qualitative ratio to one another, nevertheless likewise enter into qualitative connection. When bodies are chemically combined, or merely amalgamated or synsomatized, there likewise shows itself a neutralization of the specific gravities. The appearance was adduced earlier that the volume – even where the mixture is of materials that strictly remain chemically indifferent to one another – does not equal in magnitude what their volumes added together came to before they were mixed. In it they mutually modify the quantum of determinateness with which they enter into the connection, and in this way make themselves known as bearing themselves qualitatively towards each other. Here the quantum of specific gravity expresses itself not merely as a fixed number of comparison but as a ratio number that is displaceable; and the exponents of the mixtures yield series of measures whose progression is determined by a principle other than the ratio numbers of the specific gravities that are combined with one another. Exponents of these ratios do not amount to exclusive determinations of measure; their advance is continuous, yet harbours within it a specifying law which differs from the formally advancing ratios in which the multitudes get combined, and which renders the one advance incommensurable with the other.