I.3.1. The Specific Quantity

Qualitative Quantity is, firstly, an immediate specific Quantum; this, secondly, is in a relation to another, and therefore becomes a quantitative specifying, a transcendence of indifferent Quantum. This Measure thus is a Rule, and contains, unsynthetized, the two moments of Measure, namely, self-existent quantitative determinateness and external Quantum. But being thus held apart the two sides become Qualities, and the Rule becomes a relation between them; hence Measure manifests itself, thirdly, as a Relation of Qualities, which have first One Measure; later on, this specifies itself as an internal difference of Measure.

1. Measure is the simple self-relation of Quantum, its own determinateness determined by itself: thus Quantum exists qualitatively. And first, as immediate Measure, it is an immediate Quantum, determined accordingly as being just any Quantum; and the Quality belonging to it is similarly immediate and is determined similarly.—Thus Quantum, as being this no longer indifferent limit, but a self-relating externality, is itself Quality; and being differentiated from Quality it does not pass beyond it, nor does Quality pass beyond Quantum. It is thus determinateness which has returned into simple self-identity; it is one with Determinate Being as Determinate Being is one with its Quantum.

If the determination which has been reached is to be formulated, it may be expressed in the proposition that everything which exists has a Measure. All Determinate Being has a magnitude, and this magnitude belongs to the nature of Something itself; it constitutes its determinate nature and Being-in-Self. Something is not indifferent to this magnitude, nor does it remain unchanged when the latter changes: a change in the magnitude would change its quality. As Measure, Quantum has ceased to be a limit which is no limit; it now is a determination of the Thing in such a manner that an increase or decrease in this Quantum would destroy it.—

A Measure as standard in the ordinary sense is a Quantum which is arbitrarily taken as Unit, determinate by itself, against an external Amount. Such a unit can, of course, also be a unit determinate by itself in fact, like a foot and other original measures; but, in so far as it is also used as standard for other things, it is for them an external and not their original Measure.—In this way the diameter of the earth or the length of the pendulum may be taken as specific Quantum for itself. But what fraction of the diameter of the earth or length of the pendulum is taken, and the degree of latitude under which the latter is taken when it is to be used as standard,—this is arbitrary. And for other things such a standard is still more something external. These have specified the general specific Quantum in yet another particular manner, and have thereby become particular things. It is, therefore, foolish to speak of a natural standard of things. Moreover, a general standard is designed to serve only for external comparison; and in this most superficial meaning, where it is taken as General Measure, it is quite indifferent what is used as Measure. It is not meant to be a fundamental Measure, which would mean that in it the natural Measures of particular things would be represented and would hence, according to a Rule, be recognized as specifications of a universal Measure, the Measure of their universal body. But without this meaning an absolute standard is interesting and significant only as being common to all; and such a common element is universal not in itself, but only by convention.

This immediate Measure is a simple magnitudinal determination, like the magnitude (for example) of organic beings, of their members, etc. But everything that exists has a magnitude which makes it what it is and allows it to have Determinate Being.—As Quantum it is an indifferent magnitude, open to external determination and capable of oscillation along the scale of more and less. But, as Measure, it is also distinct from itself as Quantum (which is such an indifferent determination), and is a limitation of this indifferent oscillation along a limit.

In Determinate Being quantitative determinateness thus is double; first, Quality is bound to it, but, secondly, oscillation may take place along it and Quality yet survive. It thus comes about that whatever has a Measure may perish when its Quantum is varied. In a manner this destruction appears unexpected in so far as changes can be made in Quantum while Measure and Quality are unchanged; and yet it can be made quite intelligible—for it is gradual. Recourse is so readily made to this category in order to render intelligible to the eye or to the mind the disappearance of a Quality or of something; for thus the illusion is created that one can Almost be eye-witness of disappearance; for, Quantum being posited as limit external and variable by its very nature, change (as a change of Quantum only) needs no explanation. But in fact nothing is thereby explained; the change is also essentially the transition of one Quality into another, or (a more abstract transition) of one existence into a non-existence; and this contains a determination different from that of “gradual,” which is only a decrease or increase and a one-sided retention of magnitude.

2. But already the ancients were aware of the connexion by which a change appearing as merely quantitative turns into one which is qualitative, and they illustrated the confusions which arise from ignorance of this connexion by popular examples; the well-known ἔλεγχοι called those of the Bald and of the Heap belong here; these (according to Aristotle) are devices to compel the assertion of the contradictory of a first assertion. The question was, whether (for example) the removal of one hair from a head, or from a horse’s tail, produced baldness; or whether a heap ceased to be a heap after the removal of one grain. This could be denied without hesitation, for such a removal constitutes a wholly negligible quantitative difference; and thus one hair or grain is removed, and this is repeated, one only being removed after each acquiescence: there appears at last the qualitative change,—the head (or tail) is bare, the heap has vanished. At each acquiescence not only the repetition was forgotten, but also the fact that the quantities negligible in themselves (like disbursements negligible in themselves from a capital) add up, and that the sum forms the qualitative whole, which, therefore, at the end has vanished: the head is bald, and the purse empty.

The embarrassment and contradiction which result are no sophism in the ordinary meaning of the term: this contradiction is not vicious nor illusory. The error is committed by the Other which was assumed, namely by ordinary consciousness, when it takes a quantity as a merely indifferent limit, that is, precisely, in the definite meaning of a Quantity. This assumption is upset by the truth to which it is led, namely, that it is a moment of Measure and is connected with Quality: what is refuted is the one-sided clinging to abstract determinateness of Quantum.—And therefore these divagations are no idle and pedantic joke; they are in themselves correct and the product of a consciousness which takes an interest in the phenomena which occur in thought.

Quantum when it is taken as indifferent limit is that side from which a Determinate Being can unsuspectedly be attacked and destroyed. It is the cunning of the Notion to seize it from this side, where its Quality does not appear to come into play;—and this so much so that the aggrandizement of a State or of a property, and so on, which leads in the end to disaster for the State or the owner of the property, may at first actually appear as their good fortune.

3. Measure in its immediacy is an ordinary Quality of a determinate magnitude belonging to it. Now that side according to which Quantum is indifferent limit along which oscillation can take place while the Quality remains unchanged, is also different from its other side, according to which it is qualitative or specific. Both are magnitudinal determinations of one and the same thing; but, further, this difference must be taken as immediate in accordance with the immediacy in which Measure first is found: hence the two sides have different existences. The existence of Measure, then, which is magnitude determinate in itself, is, in its attitude to the existence of the variable and external side, a transcendence of its indifference; it is a specifying of Measure.

This is firstly, a Rule, a Measure external to mere Quantum;

secondly, Specific Quantity, determining external Quantum;

thirdly, both sides, as Qualities of specific quantitative determinateness, are related to one another as One Measure.

Rule, or Standard (which has already been spoken of), is first a magnitude determinate in itself; it is Unit with reference to a Quantum, which is a particular existence, existing in a Something other than the Something of the Rule, and measured by the Rule, that is, determined as Amount of that Unit which is the Rule. This comparison is an external activity, and the Unit is an arbitrary magnitude, which in turn can be posited as Amount (the foot as an Amount of inches). But Measure is not only external Rule, but being specific it must also in itself be related to an Other which is a Quantum.

Measure is a specific determining of external magnitude, that is, of the indifferent magnitude which is posited by some other existence in general in the Something of Measure; Measure is itself a Quantum, but, as distinguished from this, it is the qualitative element determining merely indifferent and external Quantum. Something has in it that side of Being-for-Other to which indifferent increase and decrease is proper. This immanent mensor is a Quality of Something, to which the same Quality in another Something is opposed; but in the latter the Quantum is relatively measureless as opposed to the former Quality which is determined as mensor.

In so far as Something is a Measure in itself, a change in the magnitude of its Quality is external, and does not make of it an arithmetical plurality. But its Measure reacts against it, is in the relation of an intensive to this plurality, and assimilates it in a peculiar manner; it changes the change which is externally posited, makes this Quantum another, and, by means of this Specification, manifests itself in this externality as Being-for-Self.—The plurality which is specifically assimilated is itself a Quantum, dependent upon the other plurality, which also remains merely external to it. The specified plurality is consequently also variable; but it is not, therefore, a Quantum as such, but is external Quantum specified in a constant manner. Thus Measure has its determinate being as Ratio, and its specific element is in general the exponent of this Ratio.

It was seen, when these determinations were being considered, that in Intensive and Extensive Quantum it is one and the same Quantum which we find, in one case in the form of intensity, and in the other in the form of extensity. In this difference the basic Quantum undergoes no change: the difference is only an external form. On the other hand, in Specifying Measure, Quantum in the first case exists in its immediate magnitude, but in the second is taken (by virtue of the exponent of the Ratio) in another Amount.

The exponent, which constitutes the specific element, might at first appear to be a fixed Quantum, as being the quotient of the ratio between the external and the qualitatively determinate term. But then it would be nothing but an external Quantum; by exponent nothing must here be understood but the qualitative moment itself which specifies the Quantum as such. The properly immanent qualitative element of Quantum is (as was seen above) only the Determination of Power. This it must be which constitutes the ratio, and which here, as the self-existent determination, has opposed Quantum as being external modification. The principle of Quantum is the numerical One, which constitutes its determinateness-in-itself; and the relation of the numerical One is external; and the change (which is determined only by the nature of immediate Quantum as such) consists in itself in the addition of such a numerical One, of yet another, and so on. Thus external Quantum changes in arithmetical progression; and thus the specifying reaction of the qualitative nature of Measure produces another series, which is related to the first and increases and decreases with it, but in a ratio which is not determined by a numerical exponent, but is incommensurable with a number, according to a determination of powers.

Temperature—to cite an example—is a Quality where these two sides, of being external and specified Quantum, are distinguished. As Quantum it is external temperature (even that of a body as the general medium), of which it is assumed that its change moves along the scale of arithmetical progression, and that it increases or decreases uniformly; whereas it is absorbed in a different manner by the different individual bodies which it includes, since these, by virtue of their immanent Measure, determine the temperature which they receive from without, and the change in the temperature of any one does not correspond directly with that of the medium or of any other. Different bodies, compared under the same temperature, give relative numbers of their specific heats, of their capacities for heat. But these capacities of bodies vary under different temperatures, and thus a change in the specific configuration is introduced. Hence a particular specification manifests itself in the increase or decrease of temperature. The relation of the temperature which is imagined as external, to the temperature of a given body, which also depends upon the former, has no fixed exponent of ratio; the increase or decrease of this heat does not progress uniformly with the increase or decrease of external heat.—A temperature is here assumed which is altogether external, and changes quite externally or purely quantitatively. But the temperature is itself the temperature of air or some other specific temperature. More closely considered, therefore, the ratio would really have to be taken not as the relation between a merely quantitative and a qualifying Quantum, but between two specific Quanta. And indeed Specifying Ratio will immediately proceed to determine itself in such a manner that the moments of Measure do not consist only of two sides of one and the same Quality, a quantitative side and a side qualifying the Quantum, but of the relation between two Qualities which in themselves are Measures.

1. The qualitative and self-determinate side of Quantum exists only as relation to the externally quantitative; as its specification it is the transcendence of its externality, by virtue of which Quantum exists as such; thus it logically depends on, and begins from, Quantum. But Quantum differs from Quality itself also in a qualitative manner; and this difference between the two must be posited in the immediacy of Being in general, which as yet is still the sphere of Measure: the two sides are thus qualitatively related to each other, and each for itself is such a Determinate Being; and the one Quantum (as yet formal and not determinate in itself) is the Quantum of a Something and of its Quality, and also is the specific magnitude of these qualities, since it is the case that their mutual relation has proceeded to the determination of Measure in general. These Qualities are related to each other according to the determination of Measure, and this determination is their exponent; but they are so related in themselves already in the Being-for-Self of Measure: in its double existence Quantum is both external and specific, so that each of the different Quantities has this double determination in itself and also is absolutely interlocked with the other: it is just in this alone that the Qualities are determinate. Thus they are not only determinate existences existing for each other, but they are posited as inseparable, and the magnitudinal determinateness connected with them is a qualitative unity—one determination of Measure in which, according to their concept, they cohere. Measure is thus the immanent quantitative mutual attitude of two Qualities.

2. In Measure the essential determination of variable magnitude appears, for Measure is Quantum as transcended, that is, no longer as that which it is supposed to be in order to be Quantum, but as Quantum and also something else; this Other is the Qualitative, and, as was determined, nothing else than its Ratio of Powers. This change is not yet posited in immediate Measure: any Quantum (and in fact one individual Quantum) is there connected with the one Quality. But in the Specifying of Measure (the preceding determination) as a changing of merely external Quantum by means of the Qualitative, a difference is posited between the two magnitudinal determinatenesses, and thus a plurality of Measures in a common external Quantum; and Quantum shows itself as existing Measure only here where it is differentiated from itself, for here it is manifested both as one and the same (e.g. the same temperature of the medium) and also as different (quantitative) Determinate Being (in the different temperatures of the bodies contained in the medium). This differentiation of Quantum in the different Qualities (the different bodies) gives another form of Measure, namely the one where both sides are mutually related as Quanta qualitatively determined: this might be called Realized Measure.

Magnitude as magnitude in general is variable, for its determinateness is a limit which also is no limit; variation then affects only a particular Quantum, for which another is substituted. But the true change is the change of Quantum as such; this leads to the determination, interesting if taken in this way, of the variable magnitude of higher mathematics: here no halt must be made at the mere form of variability in general, nor must any other determination be introduced than the simple determination of the concept, according to which the Other of Quantum is only the Qualitative. Thus the true determination of real variable magnitude is that it is magnitude qualitatively determined, that is (as has sufficiently been shown) determined by a Ratio of Powers: in this variable magnitude it is posited that Quantum is counted not as such, but according to its other, or qualitative, determination.

ThE two sides between which this attitude subsists are, in the abstract, Qualities in general, and as such have some particular meaning, such as Space and Time. Taken in their general relation of Measure as magnitudinal determinations they are, the one Amount oscillating in external arithmetical progression, and the other an Amount which is specifically determined by the first, which for it is Unit. If each were only any particular quality in general, they would contain no difference that could determine which should, in view of its magnitudinal determination, be taken as merely externally quantitative, and which as varying in quantitative specification. If, for instance, they are related as root and square, it is indifferent in which the increase or decrease is considered to be merely external and moving in arithmetical progression, and which on the other hand as specifically determining itself in this Quantum.

But in fact the Qualities are not indeterminately different from each other, for being moments of Measure they are to contain its qualification. The next determinateness of the Qualities themselves is, for the one, that it is the Extensive, or externality, in itself, and, for the other,’ that it is the Intensive, that which is in itself and is negative to the former. Thus of the quantitative moments Amount belongs to the former and Unit to the latter; in simple direct ratio the former must be taken as dividend and the latter as divisor, in specifying ratio the former as the power or other-becoming and the latter as root. In so far as counting here still takes place, and external Quantum (which thus is a magnitudinal determinateness wholly contingent and empirically hit on) is reflected upon, and consequently change too is taken as proceeding in external arithmetical progression, this falls within the side of Unit or intensive Quality; while the external or extensive side must be represented as varying in the specified series. Direct relation, however (like velocity in general,), is here reduced to a formal and not existing determination, belonging only to abstracting reflection; even in the relation of root and square (as in s = at²) the root must be taken as empirical Quantum and proceeding in arithmetical progression, while the other side must be taken as specified: but the higher realization of the qualification of the quantitative, more truly corresponding with the concept, is this, that both sides are related in higher determinations of powers (as is the case in s³ = at²).

What has here been discussed with respect to the connexion of the qualitative nature of a Determinate Being with its quantitative determination in Measure, has an application in the example (already indicated) of movement: here, in velocity as the direct ratio of space passed through and time elapsed, the magnitude of time is taken as denominator, and the magnitude of space as numerator. If velocity is altogether only a ratio of the space and time of a movement, it is indifferent which of the two moments is to be considered as Amount or as Unit. But space (like weight in specific gravity) is external and real Whole as such, and hence Amount: time, on the other hand (like volume), is that which is of ideal nature, the negative, the side of Unit.—Essentially, too, this is’ the proper place of a more important ratio, that of free movement;—movement first conditional, as in gravitation (where, of temporal and spatial Quantity, the former is root and the latter square), or next absolutely free, as of the celestial bodies, where, of the period of revolution and the distance, the former is by one power lower than the latter, since the two are related to each other as square and cube. Such fundamental ratios are based upon the nature of the related qualities of space and time, and upon the class of relation in which they stand, which is either mechanical movement—that is, not free and not determined by the concept of the moments,—or gravitational—that is, movement conditionally free,—or, thirdly, absolutely free, as celestial movement. These classes of movement and their laws are based upon the development of the concept of their moments, space and time; for these Qualities prove as such and in themselves, that is, in their concept, to be inseparable, and their quantitative ratio is the Being-for-Self of Measure, it is only one determination of Measure.

With respect to the absolute relations of Measure, it may be proper to recall that the Mathematics of Nature, if it would be worthy of the name of Science, must essentially be a science of measures,—a science for which much has been done empirically, but little in a truly scientific, that is philosophic, manner. Mathematical principles of Natural Philosophy—as Newton called his work—if they were to fulfil this determination in a philosophic and scientific meaning deeper than that which was reached by Newton and the whole Baconian generation, must contain quite other things in order to bring light into these regions, dark as yet, but most worthy of contemplation.—It is a great merit to become acquainted with the empirical numbers of nature (as the distances from one another of the planets), but an infinitely greater merit to cause the empirical Quanta to disappear and to raise them into a universal form of quantitative determinations, so that they become moments of a law or Measure;—immortal merits, acquired (for instance) by Galileo with regard to gravity and by Kepler with regard to the movements of the celestial bodies. They demonstrated the laws which they discovered by showing that with them the totality of details corresponds with observation. But a higher proof of these laws must be demanded, nothing less than that their quantitative determinations be known from the qualities or determinate concepts which (like time and space) are related. There is no trace of this kind of demonstration either in those mathematical principles of natural philosophy nor in further works of this kind. We observed above (in connexion with the show of a mathematical demonstration of natural relations based on an abuse of the infinitesimally small) that the attempt to conduct such demonstrations in a properly mathematical manner, that is, with neither experience nor the concept for starting-point, was a contradictory undertaking. Such proofs presuppose their theorems (namely precisely these laws) out of experience: their work is to reduce them to abstract expressions and convenient formulae. No doubt but that the whole real merit which is ascribed to Newton over Kepler in relation to the same objects will be reduced, with full knowledge (the false fabric of proofs being banished) and after clearer reflection upon what mathematics can do and has done, to this transformation of the expression, and to the analytic treatment introduced in a rudimentary form.

1. In the form of specified Measure which we have just considered, the quantitative element of both sides is qualitatively determined (both in the Ratio of Powers): thus they are moments of one determinateness of Measure of qualitative nature. But the qualities are thus posited so far only as immediate and merely different; they are not themselves in the relation in which is the magnitudinal determinateness of each, that is, having neither meaning nor existence outside this ratio; for this is contained in the determinateness of powers of magnitude. Thus the qualitative veils itself, as though specifying not itself, but magnitudinal determinateness: it is posited only as applied to this, while for itself it is immediate quality as such, which, beyond the fact that magnitude is differentiated from it, and beyond its relation to its other, has a Determinate Being which subsists for itself. Thus space and time, apart from that specification which their magnitudinal determinateness contains in the movement of gravitation, or in absolutely free movement, count as space or as time in general, space persisting permanently for itself beyond and without time, and time flowing for itself independently of space.

This immediacy of the qualitative as against its specific relation of Measure is, however, equally bound up with a quantitative immediacy and with the indifference of a quantitative element in itself as opposed to this its relation; immediate Quality has a Quantum which also is only immediate. Consequently specific Measure has also a side of—at this point-external variation; it does not interfere with its progress, which is merely arithmetical and contains external and therefore only empirical magnitudinal determinateness. When Quality and Quantum thus appear outside of specific Measure, they are also related to it; immediacy is a moment of such things as themselves belong to Measure. Thus the immediate Qualities also belong to Measure, they are also related, and, apart from magnitudinal determinateness, they are in a relation which in being outside the specified relation (determination of powers) is itself only direct relation and immediate Measure. This conclusion and its corollaries must be expounded in greater detail.

2. Immediately determinate Quantum as such is, in itself, and as moment of Measure, based upon a conceptual complex; nevertheless it is given as external in its relation to specific Measure. But the immediacy which is hereby posited is the negation of qualitative determination of Measure: this was demonstrated above of the sides of this determination of Measure, which therefore appeared as independent ties. This negation and the return to immediate quantitative determinateness are contained in qualitatively determinate ratio in so far as the relation of differentiated somethings in general contains their relation as one determinateness, which, accordingly, here in the quantitative, and distinct from the determination of ratio, is a Quantum. As negation of the differentiated and qualitatively determined sides, this exponent is a Being-for-Self, it is absolute determinedness; but it is such Being-for-Self only in itself: as Determinate Being it is a simple, immediate Quantum, it is quotient or exponent as of a ratio of the sides of Measure (this ratio being taken as direct), but, generally, the empirically appearing unit in the quantitative element of Measure.- In the fall of bodies the spaces passed through vary as the squares of the times elapsed (s = at²);—this is a ratio of powers, specifically determined, of time and space; the other or direct ratio would belong to space and time as Qualities indifferent to each other: it is supposed to be the ratio of space to the first moment of time, and in all the following points of time the same coefficient a remains the Unit as an ordinary Quantum for the Amount which, for the rest, is determined by Specifying Measure. At the same time it counts as the exponent of that direct ratio which belongs to imaginary and absolute velocity, that is velocity which is formal and not specifically determined by the concept. Such velocity does not exist here any more than that mentioned above which was supposed to belong to a body at the end of a moment of time. The former velocity is attributed to the first moment of time occupied by the fall, but this so-called moment of time is itself only an assumed unit, and thus, as an atomic point, has no Determinate Being; the beginning of the movement—and the smallness which is attributed to the beginning could make no difference—is forthwith a magnitude, and a magnitude specified by the law of gravitation. This empirical Quantum is ascribed to the force of gravity, in such a manner that this force itself is to have no relation to the given specification (the determinateness of powers) and to what is peculiar in the determination of Measure. The immediate moment—namely, that in the movement of falling one unit of time (one second—that which is called the first) goes with an amount of some fifteen spatial units (which are taken as feet)—this moment is an immediate Measure, like the measure-magnitude of human limbs, the distances and diameters of the planets, and so on. The determination of such a measure does not belong to the sphere of qualitative determination of Measure (which here is the law of gravitation itself); and the concrete sciences have not yet given us any clue to that upon which depend such numbers, being as they are that element of a Measure which appears only immediately, that is, empirically. Here we are concerned only with this conceptual determinateness, which is, that that empirical coefficient constitutes Being-for-Self in the determination of Measure, but constitutes the moment of Being-for-Self only in so far as it exists in itself and therefore immediately. The rest is the developed aspect of this Being-for-Self, the specific measure-determinateness of the sides.—In the relation of gravitation (a movement which indeed is half conditioned and half free), gravity according to this second moment must be looked upon as a force of nature, its ratio being determined by the nature of time and of space, so that this specification (the ratio of powers) falls within gravity: the former simple and direct ratio expresses only a mechanical behaviour of time and space, formal velocity externally produced and determined.

3. Measure has reached the determination that it is a specified ratio of magnitudes, which as qualitative has applied to it ordinary external Quantum; but this is not a Quantum in general, but essentially is the determining moment of Ratio as such; thus it is exponent, and, as immediate determinateness, it is now an invariable exponent, and exponent therefore of the direct ratio (already mentioned) of those same Qualities by which their magnitudinal ratio to each other is specifically determined. In the example which we used (the Measure of the movement of gravitation) this direct ratio is anticipated, so to speak, and assumed to be given; but, as was observed, it does not yet in fact exist in this movement.—But it constitutes the further determination that Measure is now realized in this manner, that its two sides are Measures (distinguished as an immediate and external Measure and a Measure specified within itself), and that it is their unity. As such a unity Measure contains the relation in which the magnitudes are determined by the nature of the Qualities and are posited as different: its determinateness is therefore quite immanent and independent and, simultaneously, has collapsed into the Being-for-Self of immediate Quantum, the exponent of a direct ratio. And here its self-determination is negated, for it has its last and self-existent determinateness in this its Other; and, conversely, immediate Measure, which in itself is supposed to be qualitative, in truth has qualitative determinateness only in the former. This negative unity is real Being-for-Self, the category of a Something as unity of Qualities related as Measures,—a complete independence. Immediately, the two (which have shown themselves to be two different ratios) yield a twofold Determinate Being; or rather, such an independent whole, since it is a Being-for-Self in general, is also a schism into different stable somethings whose qualitative nature and subsistence (materiality) lies in their mensural determinateness.