Remark 2

It was recalled above that the Kantian antinomies are presentations, in a more concrete shape, of the opposition between finite and infinite, brought to bear upon more special substrates of representation . The antinomy considered above contained rather the opposition of qualitative finitude and infinity. In another, the first among the four cosmological antinomies, what is considered in its conflict is rather the quantitative limit. I shall therefore undertake the examination of this antinomy here.

It concerns, namely, the limitedness or unlimitedness of the world in time and space. — This opposition could with equal right have been considered in respect of time and of space themselves, for nothing in the antinomic character of limitedness or unlimitedness is altered by whether time and space are relations of the things themselves, or are merely forms of intuition.

The closer unfolding of this antinomy will likewise show that the two propositions, and equally their proofs — which, as with the antinomy considered above, are conducted apagogically — run out into nothing but the two simple, mutually opposed assertions: a limit is, and there must be a going out beyond the limit.

The thesis reads:

“The world has a beginning in time, and as regards space it too is enclosed within limits.”

The one part of the proof, the one concerning time, assumes the contrary,

“that as regards time the world has no beginning; then up to every given point in time an eternity has run out, and consequently an infinite series of states of things in the world following upon one another has ela psed. But the infinity of a series consists precisely in this, that through successive synthesis it can never be completed. An infinite elapsed world-series is therefore impossible, and a beginning of the world is accordingly a necessary condition for its existence; which was the thing to be shown.”

The other part of the proof, the one that concerns space, is led back to time. The gathering together of the parts of a world infinite in space would require an infinite time, which would have to count as run out, insofar as the world in space is to be taken not as something becoming but as something completed and given. But of time it was shown in the first part of the proof that to assume an infinite time as run out is impossible.

One sees at once, however, that it was unnecessary to make the proof apagogic, or to conduct a proof at all, since what lies immediately at its own base is the assertion of the very thing that was supposed to be proved. For some one or any given point in time is assumed, up to which an eternity (-- eternity here has only the meagre sense of a badly infinite time) is supposed to have run out. A given point in time signifies nothing other than a determinate limit within time. Within the proof, accordingly, a limit of time is presupposed as something actual; yet this is precisely that which was to be proved. For the thesis consists in this, that the world has a beginning in time.

The only difference that occurs is that the assumed temporal limit is a now as the end of the time previously elapsed, whereas the one to be proved is a now as the beginning of a future. Yet this difference is inessential. Now is assumed as that point wherein an infinite series of states of things in the world, one following upon another, is supposed to have elapsed, hence as an end, as a qualitative limit. Were this now regarded merely as a quantitative limit, one that is to be gone out beyond and that flows, then the infinite series of time would not stand elapsed within it but would go on flowing, and the reasoning of the proof would fall away. But this point in time, assumed as a qualitative limit for the past, is at the same time the beginning for the future, — for in itself every point in time is the reference of past and future, — and indeed it is an absolute beginning for that future. For it makes no difference to the matter that before this future of its and before the beginning of that future a past already is; since this point in time is a qualitative limit, — and to assume it as qualitative lies in the determination of the completed, of what has run out, hence of what does not continue itself, — time is thereby broken off in it, and the past that is being spoken of stands without reference to the time which could be called future only in regard to this past, and which is therefore only time as such, time having an absolute beginning. But did it stand, — (as it indeed does —) through the now, the given point in time, in a reference to the past, were it in fact future, then from the other side this point in time too would be no limit, the infinite series of time would carry itself on into that which was called future, and would not, as was assumed, be completed.

In truth time is pure quantity; the point in time employed in the proof, at which time was supposed to be interrupted, is rather only the self-sub lating being-for-itself of the now. The proof achieves nothing beyond setting before representation, as a given point in time, that absolute limit of time which the thesis asserts, and flatly assuming it, a popular determination which sensuous representing readily lets pass as a limit, thereby letting hold good within the proof, as an assumption, that which had earlier been put forward as what was to be proved.

The antithesis states:

“The world has neither a beginning nor limits in space, but is infinite in view of time as well as of space.”

The proof posits the contrary:

“Let the world have a beginning. Since the beginning is an existence before which there goes a time wherein the thing is not, then a time wherein the world was not must have gone before, that is, an empty time. But in an empty time no coming-to-be of any thing whatever is possible; because no part of such a time has in itself, ahead of another, any distinguishing condition of existence over that of non-existence. Hence many a series of things can indeed begin within the world, but the world itself can take no beginning, and in respect of elapsed time it is infinite.”

This apagogic proof contains, like the others, nothing beyond the direct and unproven assertion of the very thing it was meant to prove. For it first assumes a beyond of worldly existence, an empty time; but then it just as much continues worldly existence out beyond itself into this empty time, thereby sublates that time, and thus carries existence on into infinity. The world is an existence; and the proof presupposes of this existence that it comes to be, and that coming-to-be has a preceding condition in time. But in this precisely the antithesis itself consists, that no existence is unconditioned and no limit absolute, and that essential existence always demands a preceding condition. This condition is at the same time itself conditioned; it is sought in empty time, which amounts to saying that it is itself assumed as temporal and hence as existence, and as something limited. Quite generally, then, the assumption has been made that the world as existence presupposes another existence, and so on into infinity.

The proof with regard to the world's infinity in space is just the same. In apagogic fashion the spatial finitude of the world is assumed; “it would thus find itself in an empty unlimited space and would have a relation to it; yet a relation of this sort, of the world to no object, is nothing.”

What was supposed to be proved is here likewise directly presupposed in the proof. For it is directly assumed that the limited spatial world should find itself in an empty space and have a relation to it, which is to say that it must be gone out beyond, on the one side into the void, into the world's beyond and non-being, while on the other side it thereby stands in relation to that void, hence continues itself into it, and that the beyond is to be represented as filled with worldly existence. What the antithesis asserts, the infinity of the world in space, is nothing other than, on the one part, empty space and, on the other part, the relation of the world to it, which is to say the continuity of the world within it, or the filling of it; and this contradiction, space at once as empty and at once as filled, is the infinite progress of existence within space. But this contradiction itself, the world's relation to empty space, is what is directly assumed in the proof.

Thesis and antithesis and their proofs therefore present nothing but the opposed assertions that a limit is, and that this same limit is no less merely a sublated one; namely, that the limit has a beyond with which it stands in reference, towards which one is to go out beyond it, but wherein there arises once more such a limit as is none.

The resolution of these antinomies is, like that of the one above, transcendental, which is to say that it lies in asserting the ideality of space and of time, as forms of intuition, in this sense: that the world in its own self is not in contradiction with itself, is not something self-sublating, but that consciousness, in its intuiting and in the reference of intuition to understanding and reason, is an essence contradicting itself.

3. Infinity of the Quantum

1. The infinite quantum, as infinitely great or infinitely small, is itself the infinite progress; it is quantum as a great or a small, and is non-being of quantum as infinite. The infinitely great and the infinitely small are therefore images of representation which on closer consideration show themselves to be null mist and shadow. But the infinite progress expresses nothing other than the nature of quantum, which as intensive magnitude has attained its reality.

Quantum, returned into itself, is simple, referred to itself and as determined in itself. But since through this simplicity otherness and determinateness are sublated at its own self, determinateness is external to it; it has its absolute determinateness rather outside it. This being-outside-itself of quantum is at first the abstract non-being of quantum as such, bad infinity. But further it is also a great, quantum continues itself into its non-being, for it has its determinateness precisely in its externality; this externality of quantum is therefore just as much itself quantum, only another quantum, which however sublates itself once more as the first did.

Quantum is therefore something determined in itself; but this determinateness of its it has outside itself, so that it sublates itself; conversely, in its being-outside-itself it has returned into itself, its being-outside-itself is just as much sublated.

This circle is the genuine element that is posited in the infinite progress. There is present quantum and its beyond. First, quantum sublates itself, it is at its own self the going out beyond its limit; the beyond is infinity, but it is bad infinity, for second, quantum continues itself into it. This beyond, the non-being of quantum, infinity itself is limited, and a quantum is posited anew, which is to say, this beyond is itself sub lated. Quantum is precisely itself through its being-external; this is exactly what makes up the determinateness of quantum, or what quantum is. In the infinite progress, therefore, there is the concept of quantum as it is in itself; and there is present in the progress the sublating of quantum, but just as much of its beyond; or the negation of quantum as well as the negation of this negation.

The going out beyond quantum is the negation of quantum, the infinite; but a new quantum is posited, and this is the negation of the infinite, of this bad infinite which counts for representation as an absolute, as a last term that does not sublate itself again and out beyond which nothing further could be gone. The truth of the infinite progress is therefore that quantum and its beyond are posited, but that they are posited as sublated. Its truth is accordingly their unity, wherein they are, but as moments.

This is thus the true resolution of the contradiction whose expression the infinite progress is. It consists in nothing other than the restoration of the concept of magnitude, namely that magnitude is an indifferent or external limit. In the infinite progress as such, reflection is usually directed only upon this, that every quantum, be it ever so great or small, vanishes, that it must be possible to go out beyond it; but not upon this, that this sublating of quantum, the beyond, the bad infinite, itself vanishes too. This happens, however, in that quantum continues itself into its negation, in that out beyond every quantum, into its sublating, a new quantum is posited. The first sublating is indeed in itself the sublating of negation, — for quantum is sublated limit, — but it is at the same time only in itself this; for this infinite is fixed as the beyond of quantum, which still remains subsisting as a this-side; or quantum is taken only as something immediate, and the infinite only as the first negation. But in the infinite progress there is more present, — than merely the sublating of immediate quantum, or than merely a first sublating; therein this bad infinite too is, — through the new limiting, sublated; there is therefore present in it the negation of negation, or that which the infinite in truth is. — But the concept of quantum is not merely restored, it has also received its closer determination; there has arisen the quantum determined by its concept, which is distinct from immediate quantum.

2. The beyond of quantum has, namely, a more determinate, positive significance than merely that of the non-being of quantum; and so likewise does the sublating of this beyond, and the uniting of it with quantum itself.

Quantum as indifferent limit is determined at its own self; this self-referring being-determined is the having-vanished of its externality, which it has at its own self; this externality thereby steps outside quantum; its going out beyond itself is its essential moment, it refers itself through its own self to its externality; but this externality makes up its being determined in itself, and the nature of its being-determined-in-itself consists in this externality. The beyond of quantum is therefore not the mere non-being, the empty, indeterminate negation of quantum. Rather quantum goes out beyond itself just insofar as it is indifferent limit; it sublates this indifference and posits the being-in-itself of that indifference as an infinite beyond, as that wherein it is negated, wherein it is not its own self but the externality of its own self. But this externality is rather the opposite of itself; it is an absolute moment of magnitude itself; for quantum is not itself in its immediacy, but is essentially a going out beyond itself; this going out beyond itself, this externality of its, therefore belongs to quantum itself.

But its going out beyond itself is the sublating of its indifference towards the external, which is its determinateness, and it thereby posits this determinateness as its own self. It sublates its beyond, its negation, which is to say, it sublates the externality of its being-determined; unites it with itself and thereby makes itself determined in itself.

Each of the moments of the movement of the infinite progress is the opposite of its own self; for the infinite progress is the posited contradiction. First, quantum goes out beyond itself; this therefore means 1) it sublates itself, posits its negation, its beyond, and 2) it thereby posits rather its absolute being-determined, that which it is in itself. Second, this infinite is determined again, a new limit is posited; this accordingly means 1) the being-in-itself of quantum is sublated, it only withdraws an indifferent quantum once more, 2) the negation of quantum, the beyond of quantum, is sublated, its going out beyond itself is thus taken back into quantum itself. Both sides express this, that quantum is negated and that the negation of quantum is negated; there is therefore posited its infinite reference to its own self, or its being-determined-in-itself. Infinity, which was only the bad one and a beyond of quantum, belongs to quantum, quantum is itself infinite.

In this restoration of quantum, quantum is sublated as indifferent limit, as this perennial going out beyond itself. The indifference and externality of quantum therefore vanishes only insofar as the beyond of quantum is sublated. Quantum no longer has infinity, being-determined-in-itself, outside itself. The limit is therefore sublated as indifferent or as sublated. It has thus become qualitative once more.

The infinite, then, which in the infinite progress has only the empty significance of a non-being, of a beyond, is in fact nothing other than quality. Quantum is indifferent limit; it goes out beyond itself into infinity; in doing so it seeks nothing other than being-determined-in-itself, the qualitative moment. But this qualitative moment is not a beyond of quantum, it lies within quantum itself. For precisely this going out beyond itself, or the beyond, the negation of quantum, is what makes quantum into quantum; this is its determinateness in itself; its very indifference is its determination itself.

Or, quantum is sublated quality; but quantum is infinite, goes out beyond itself, it is the negation of itself. It is therefore the negation of negated quality, or it is the restoration of that quality.

But the quantum that is sublated as indifferent limit and qualitatively determined is the quantitative ratio. In the ratio quantum is external to itself, distinct from its own self; but this externality of its, the reference to the other quantum, at the same time makes up its determinateness; therein it has not an indifferent but a qualitative determination; in its externality it has returned into itself.

Remark

On the one side the mathematical infinite holds interest through the expansion of mathematics and through the great results that its introduction into that science has brought forth; on the other side, however, it is remarkable in that this science has not yet succeeded in justifying its use of it by way of the concept. The justifications rest on the correctness of the results that come out with its help, a correctness demonstrated on other grounds; they do not rest on the clarity of the object and of the operation through which the results are brought out, so little indeed that this operation is rather admitted to be incorrect.

This is in and for itself already a defect, for a procedure of that sort is unscientific. But it also carries a disadvantage with it, namely that mathematics, in not knowing the nature of this instrument of its own, because it has not come to terms with the metaphysics or critique of it, can neither determine the range of its application nor secure itself against misuses of it.

In a philosophical respect, however, the mathematical infinite is important because the concept of the genuine infinite does in fact lie at its base, and because it stands far higher than what is ordinarily called the metaphysical infinite, from which the objections against the former are made. Against these objections the science of mathematics ordinarily knows how to save itself only by rejecting the competence of metaphysics, maintaining that it has nothing to settle with that science and need give no thought to its concept, provided only that it proceeds consistently on its own ground. Its business is not with what is true in itself, but with what holds as true upon its own field. Metaphysics cannot manage to deny or to overturn the brilliant results of the use of the mathematical infinite, and mathematics cannot manage to get clear about the metaphysics of its own concept and hence also about the derivation of the procedures which the use of the infinite makes necessary.

If the difficulty of the concept as such were the only one pressing upon mathematics, it might let this lie aside without further ado, insofar, that is, as the concept is more than the mere specification of the essential determinateness of a matter; for it is not a science that has to do with the concepts of its objects and has to generate its content through the development of the concept, even if only by way of ratiocination. In the method of its infinite, however, it encounters the principal contradiction at the very distinctive method upon which, as a science, it rests altogether. For the calculus of the infinite permits and requires procedures that mathematics must otherwise utterly repudiate in operations with finite magnitudes, and at the same time it treats its infinite magnitudes like finite quanta and would apply to the former the very procedures that hold good for the latter.

In its use of the infinite, and in the operations directly at odds with the mathematical way of proceeding that this use makes necessary, mathematics shows that the results it thereby finds agree entirely with those found by the properly mathematical route, the geometrical and the analytic. But for one thing this does not concern all results, and the purpose of introducing the infinite is not merely to abbreviate the ordinary path but to arrive at results that cannot be achieved along it. For another, success does not justify the manner of the path in and for itself. This manner of calculating the infinite, however, is always burdened with the semblance of inexactitude that it gives itself when on the one occasion it augments finite magnitudes by an infinitely small magnitude, retains part of that magnitude in the further operation, and yet also neglects a part of it. This procedure displays the oddity that, the admitted inexactitude notwithstanding, a result comes out which is not merely tolerable and so close that the difference could be left out of account, but perfectly exact. In the operation itself, though, which precedes the result, the representation cannot be dispensed with that something is not equal to zero yet is so inconsiderable as to admit of being left out of account. Yet with what is to be understood by mathematical determinateness, all difference of a greater or a lesser exactitude drops away entirely, just as in philosophy there can be no talk of greater or lesser probability but only of truth. If the method and the use of the infinite are justified by success, and even this only in part, it is nevertheless not so superfluous to demand a justification of them as it seems superfluous, in the case of one's nose, to ask for proof of the right to make use of it. For with mathematical cognition, as a scientific cognition, everything essentially turns on proof, and with regard to the results too it is the case that the strictly mathematical method does not supply for all of them the corroboration of success, which is in any event only an external corroboration.

It is worth the trouble to look more closely at the mathematical concept of the infinite and at some of the most remarkable attempts whose aim is to justify its use and to remove the difficulty by which the method feels itself pressed. Consideration of these justifications and determinations of the mathematical infinite, which I mean to carry out at greater length in this Remark, will at the same time cast the best light on the nature of the true concept itself, and will show how it has hovered before them and lain at their ground.

The customary determination of the mathematical infinite is that it is a magnitude beyond which there is no greater or smaller one any more. — In this definition the true concept is indeed not yet immediately expressed, but it is contained in it once the definition is looked at more closely. For a magnitude is defined in mathematics as something that admits of increase and decrease; hence, quite generally, as an indifferent limit. And since the infinitely great or small is such as admits of no further increase or decrease, it is in fact no quantum as such any longer.

This consequence is necessary and immediate. But the reflection that the quantum — and in this Remark I call the finite quantum simply quantum in general — is sublated is the one that is not commonly made, and it is this that makes the difficulty for ordinary comprehension, since the quantum, in being infinite, is required to be thought as something sublated, as one that is at the same time not a quantum.

To cite how Kant appraises that concept ⁴, he finds it out of agreement with what is understood by an infinite whole. „On the ordinary concept, a magnitude is infinite beyond which none greater (that is, beyond the multitude, contained in it, of a given unit) is possible. — By an infinite whole, he says, there is no representation of how great it is; its concept, accordingly, is not the concept of a maximum (or minimum), but by it there is thought only its ratio to a unit that may be assumed at will, in respect of which the whole is greater than any number. According as one assumed this unit greater or smaller, so would the infinite be greater or smaller; infinity, however, consisting as it does merely in the ratio to this given unit, would always remain the same, though of course the absolute magnitude of the whole would thereby not be cognized at all.“

What Kant censures, then, is that infinite wholes should be viewed as a maximum, as a completed multitude of a given unit. For the maximum or minimum is itself a quantum, a multitude, not merely a ratio. The ordinary representation, to which the infinitely great or small appears as a something that is a quantum, cannot fend off the consequence adduced by Kant, which leads to a greater or a smaller infinite according as the unit lying at the base, being a variable one, were assumed greater or smaller. Or, quite generally, so long as the infinite is represented as quantum, the difference of a greater and a smaller still holds good for it. This criticism, however, does not strike the concept of the genuine mathematical infinite, of the infinite difference, for that difference is no longer a finite quantum.

Kant's own concept, by contrast, the one he calls the true transcendental concept, is „that the successive synthesis of the unit in the measuring-through of a quantum can never be completed.“ On the one side a quantum is here indeed presupposed as given; but this is first to be synthesized, and this synthesizing, through which it would be made into an amount and a quantum, is never to be completed. With this, as is evident, nothing but the progress into infinity is enunciated, only transcendentally, or properly speaking subjectively and psychologically represented. In itself the quantum is indeed to be completed, but transcendentally, namely within the subject, there arises only such a quantum as is uncompleted and simply afflicted with a beyond. Here, then, one comes to a halt quite generally at the contradiction that magnitude contains, but the contradiction is parcelled out to object and subject, so that to the former falls limitedness, to the latter the going out beyond it, the bad infinite.

The genuine infinite quantum, however, is infinite in its own self; it is this, as emerged above, as that in which the finite quantum, or quantum in general, and its beyond, the bad infinite, are sublated in the same way. The sublated quantum, however, has gone back into simplicity and into reference to itself, — not merely in the manner of the extensive, which in passing over into intensive quantum has its determinateness only in itself at an external manifoldness, toward which it is nevertheless indifferent and from which it is supposed to be distinct. The infinite quantum, by contrast, contains externality and the negation of itself at its own self; thus it is no longer some finite quantum, not a determinateness of magnitude that would have an existence as quantum, but it is simple, and so only as moment; it is only the concept of its being-determined, or a determinateness of magnitude in qualitative form. As moment it stands in essential unity with its other, only as determined through this other of its. Or it has meaning only in reference to something standing in ratio with it. Outside this ratio it is zero; — whereas quantum as such is precisely supposed to be indifferent toward the ratio and to need no other for its determination. In the ratio, however, it is just as much no quantum, and precisely for this reason, that it is only moment, only something within the ratio, not something indifferent for itself.

Since quantum thus is, according to its truth, only as determination of magnitude, it has qualitative nature and is infinite; for, first, this contains its negation, — it has ceased, namely, to be what according to its determination it was supposed to be, something indifferent. Second, it has being-determined-in-itself at it, for it no longer has it as a beyond outside it.

This concept will show itself to lie at the base of the mathematical infinite, and it will grow more distinct as we consider the various stages of the expression of quantum as a moment of ratio, beginning from the lowest, where it is at the same time quantum as such, up to the higher, where it has the meaning and expression of infinite magnitude proper.

Let us take first the quantum in the ratio, in the shape in which it is a fractional number. The fraction 2/7, for instance, is not a quantum like 1, 2, 3 and so forth; an ordinary finite number it certainly is, yet not an immediate one like the whole numbers, but as fraction it is determined mediately through two numbers which are amount and unit relative to each other, so that the unit is itself a determinate amount. But if we abstract from this closer qualitative determination of them relative to each other and look at them merely according to what befalls them here as quantum, then 2 and 7 are otherwise indifferent quanta, though here they come on the scene only as moments of another. For this reason 2 and 7 are here to count forthwith not as 2 and 7 but as their determination relative to each other. In their stead, therefore, 4 and 14, or 6 and 21 and so forth, can just as well be posited. With this they begin to have a qualitative character. Were they to count as mere quanta, then 2 and 7 are simply only 2 and 7; 4 and 14, 6 and 21 and so forth are simply something else and cannot be set in the place of those numbers. Insofar as 2 and 7 do not count according to this determinateness, their indifferent limit has been sublated; they thus have, though still imperfectly, the moment of infinity at them, since they at the same time not merely are no longer just they themselves, but their determinateness also remains, as a qualitative one that is in itself, — namely according to what they count for in the ratio. Infinitely many others can be set in their place, so that at the same time the value of the fraction, the determinateness which the sides of the ratio have, does not alter.

The exhibition which infinity has at a numerical fraction is still imperfect, however, for this reason, that the two sides of the fraction, 2 and 7, are ordinary indifferent quanta once they are taken out of the ratio; their reference, that of being in a ratio and being moments, is to them something external and indifferent.

The letters operated with in general arithmetic do not have the property of possessing a determinate numerical value; they are universal signs, and indeterminate possibilities of every determinate value. The fraction therefore seems, on account of its elements, to be a more fitting expression of the infinite, because a and b, taken out of their reference to each other, remain indeterminate and separated have no particular value of their own either — yet although these letters are indeed indeterminate magnitudes, their sense is that they be some finite quantum. Since they are thus only the universal representation, but the representation of the determinate number, it is likewise indifferent to them to stand in the ratio, and outside it they keep this value.

The two sides which the magnitudes have in the fraction consisted in this, that they are finite magnitudes, quanta, and at the same time infinite, no quanta. The ratio itself as such is first a quantum; second, however, not an immediate one but one that has the qualitative opposition in it: to be something not indifferent toward the other but determined through it, something returned into itself in its otherness and thus infinite. These two sides exhibit themselves in the following manner.

The fraction 2/7 can be expressed as 0,285714 ... as also as 1 + a + a2 + a3 and so forth. So taken it is exhibited as an infinite series, and the fraction itself goes by the name of the sum, or the finite expression, of that series. If we compare these two expressions, the infinite series exhibits the fraction no longer as a ratio but only on the side that it is a quantum, as a multitude of such quanta accruing to one another, as an amount, or at least it has the determination of exhibiting it so. — That the magnitudes which are to make it up as amount consist in turn of decimal fractions, and hence themselves of ratios, is here beside the point; for that circumstance concerns their unit, not them insofar as they constitute the amount; just as a whole number of the decimal system consisting of several digits counts essentially as an amount, and no notice is taken of its consisting of products of a number and the number ten and the powers of ten. Just as little does it matter here that there are fractions other than the 2/7 taken as example which, turned into decimal fractions, do not yield an infinite series; the point is only that every one of them can be expressed as such a series.

In the infinite series, which is to exhibit the fraction essentially as amount, the side of its being a ratio therefore vanishes; and even when it is expressed as a sum of ratios, then, since these are taken as terms of a sum, abstraction is made from their being ratios. With the ratio, then, there vanishes also the side on which the fraction had infinity at it. This latter, however, has come in in another manner; the series, namely, is itself infinite.

Of what sort the infinity of the series is, is evident of itself; it is the bad infinity of the progress. For the series contains the contradiction of exhibiting something that is a ratio and of qualitative nature as something ratioless, as a mere quantum, as amount. At the amount expressed in the series something is always lacking, so that one must always go out beyond what is posited in order to attain the required determinateness. The law of the advance is known; it lies in that determination of quantum which the fraction contains, and in the nature of the form in which this determination is to be expressed. By continuing the series it can be made as exact as one requires; but the exhibition through it always remains a mere ought; it is afflicted with a beyond that cannot be sublated, because to express something qualitative as amount is the abiding contradiction.

In this infinite series that inexactitude is actually present of which, at the genuine mathematical infinite, only the semblance occurs. One must no more confuse these two kinds of mathematical infinite than the two kinds of philosophical infinite. In exhibiting the genuine mathematical infinite, the form of series was used at the start, and has of late been called up again. But that form is not essential to it; quite the reverse, what is infinite about the infinite series differs essentially from it, as the sequel is to show; it even ranks below the expression of the fraction.

The infinite series, namely, harbours the bad infinity for this reason, that what it is meant to express remains an ought; and what it does express is afflicted with a beyond that does not vanish, and is different from what is to be expressed. It is infinite not on account of the terms that are posited, but for this reason, that they are incomplete, that the other which belongs to them essentially lies beyond them; what is there in it, be the posited terms as many as one pleases, is only a finite, and indeed posited as finite, as one that is not what it ought to be. By contrast, that which bears the name finite expression, or sum, of such a series has no defect; it rather contains in full what the series only seeks; the beyond has been summoned back out of its flight; what it is, and what it ought to be, stand not apart but are one and the same. It therefore contains no finitude, nothing beyond which one must look.

This can also be looked at thus, that within the infinite series the negative falls outside its terms, which have presence in that they count only as parts of the amount. In the finite expression, by contrast, which is a ratio, the negative is immanent, as the being-determined of the sides of the ratio through one another.

In fact, then, what is usually called the sum, the 2/7 or , is a ratio; and the so-called finite expression is the genuinely infinite expression. The infinite series, however, is in truth the sum; its purpose is to exhibit in the form of a sum what in itself is ratio, and the terms present in the series are terms not of a ratio but of an aggregate. It is, further, rather the finite expression; for it is the imperfect aggregate, and remains essentially something defective. — If the fraction is called the finite expression insofar as it is a determinate quantum, then the infinite series is, first, according to what is there in it, likewise a determinate quantum, yet at the same time a lesser one than it ought to be; then too what it lacks is a determinate quantum; and what is there in it, taken together with what it lacks, is just such a quantum, the same as the fraction is. Insofar, then, as the fraction is a finite, that is, a determinate quantum, the series is one likewise, and still more so than the fraction. But insofar as the fraction is infinite, and indeed infinite in the genuine sense at its own self, because it has the negative beyond at its own self, the series is defective, and has the infinite only as a beyond outside it.

With infinite series that are not summable the case stands otherwise; mathematics, however, comes to a halt at this difference, as at an external and contingent circumstance, whether they can be summed or not. For they contain a higher kind of infinity than the summable ones; an incommensurability, or the impossibility of exhibiting the quantitative ratio contained in them as a quantum — be it even as a fraction; the form of series, however, which they do have, is the same bad infinity that is present in the summable series.

The same inversion noted here at the fraction and at its series takes place insofar as the mathematical infinite, namely the genuine one, has been called the relative infinite, whereas the ordinary metaphysical one has been called the absolute infinite. In fact it is rather the metaphysical that is only the relative one, because the negation it expresses stands only over against a limit that does not get sublated by it; the mathematical infinite, by contrast, has genuinely sublated the finite limit within itself, because the beyond of that limit is united with it.

It is above all in the sense I have pointed out, namely that what is termed the sum, or the finite expression, of a series that is infinite counts rather as the infinite one, that Spinoza sets up the concept of true infinity against that of the bad, and elucidates it by examples. His concept gains most in light if I attach what he says on this head to the present development.

He defines the infinite, to begin with, as the absolute affirmation of the concrete existence of some nature, the finite on the contrary as determinateness, as negation. The absolute affirmation of a concrete existence is, namely, to be taken as its reference to itself, as its not being by virtue of an other's being; the finite, by contrast, is negation, a ceasing, insofar as an other begins outside it. The absolute affirmation of a concrete existence does not, indeed, exhaust the concept of infinity; this concept implies that infinity is affirmation not as immediate affirmation, but only as one that has been reinstated, by way of the other's reflection into itself; or as negation of what is negative. With Spinoza, however, substance and its absolute unity have the form of an unmoved unity, of a rigidity in which the concept of the negative unity of the self, subjectivity, is not yet to be found.

His mathematical example of the true infinite is, as is well known, a space between two unequal circles, one of which falls inside the other without touching it, and which are not concentric. He made so much, it seems, of this figure and of the concept whose example he used it to be, that he set it as the motto of his Ethics. — „The mathematicians, he says, conclude that the inequalities possible within such a space are infinite, not from the infinite multitude of the parts, for its magnitude is determinate and bounded, and I can posit greater and smaller spaces of the sort, but because the nature of the matter surpasses every determinateness.“ — One sees that Spinoza rejects that representation of the infinite according to which it is represented as a multitude or as a series that is not completed, and recalls that here, at the space of the example, the infinite is not beyond but present and complete; this space is an infinite one for this reason, „because the nature of the matter exceeds every determinateness,“ because the determination of magnitude contained in it is at the same time not a quantum. That infinite of a series Spinoza calls the infinite of the imagination; the infinite, by contrast, as reference to itself, the infinite of thinking, or infinitum actu. It is, namely, actu, it is actually infinite, because it is completed within itself and present. Thus the series 0,285714 ... or 1 + a + a2 + a3 ... is the infinite merely of imagining, or of supposing; for it has no actuality, something is simply lacking to it; whereas 2/7 or is that actually, not only what the series is in the terms present in it, but in addition what it lacks, what it only ought to be. The 2/7 or is likewise a determinate magnitude, like Spinoza's space enclosed between the two circles and its inequalities; and like this space it can be made greater or smaller. But there does not thereby come out the absurdity of a greater or a smaller infinite; for this quantum of the whole is no concern of the ratio of its moments, of the nature of the matter, that is, of the qualitative determination of magnitude. Imagining, by contrast, comes to a halt at quantum as such, and does not reflect on the qualitative reference which makes up the ground of the incommensurability present.

This incommensurability in the more general sense is already present at the 2/7 as well, insofar as 2 and 7 are prime to each other, so that the quantum 2/7 cannot be expressed as a whole number, or as an immediate, ratioless quantum. The higher, proper incommensurability, however, takes in Spinoza's example and the functions of curved lines generally. It leads us on more nearly to the infinite which mathematics needs in the case of such functions, in the case of the functions of variable magnitudes generally, and which is the genuine mathematical infinite, the absolute quantitative infinite generally, that Spinoza too had in mind.

The concept of the magnitudes whose reference these functions express, namely of the variable magnitudes, is to be grasped more precisely, however, than commonly happens. They are variable, namely, not in the sense in which in the fraction 2/7 the two numbers 2 and 7 are variable, in that other numbers, 4 and 14, 6 and 21 and so on into infinity, can just as well be set in their place without altering the determination of magnitude posited in the fraction. So too in any number one pleases can be set in the place of a and b, without altering what is meant to express. In the sense that any number one pleases can be set in the place of the x and y of a function, a and b are variable magnitude just as much, or are so still more, insofar as the function encloses the x and y within a limit generally, or at least within reference to each other. The expression variable magnitudes is on that account superficial and clumsy for determining what distinguishes the magnitudes of a function.

Their genuine concept lies in the following. In 2/7 or , 2 and 7 are, each for itself, determinate quanta, and the reference is not essential to them; a and b are likewise supposed to represent such quanta as remain outside the ratio too what they are. Further, 2/7 and are a fixed quantum, a quotient; the ratio is an amount whose unit the denominator, and the amount of these units the numerator, expresses — or the reverse; even if 4 and 14 and so forth step into the place of 2 and 7, the ratio remains, as quantum too, the same. In the function = p, for instance, by contrast, x and y do have the sense of being able to be determinate quanta; but it is not x and y, only x and y2, that have a determinate quotient. Thereby these sides of the ratio are, first, not merely no determinate quanta, but, second, their ratio is not a fixed but a variable quantum. Nor are they merely universal quanta, with which, as with their ratio, a determinate quantum was supposed to be meant. Rather, their ratio itself is as quantum variable in and for itself. This, however, is contained in the fact that x stands in a ratio not to y but to the square of y, because the ratio of a magnitude to the power is not a quantum but a ratio of the concept. The power-ratio is not an external bounding but one determined through itself; hence an essentially qualitative ratio; there will be more to say of it below. If a determinate value is given to x, then y too receives a determinate value through the function; but if x receives another value, the previous ratio does not remain as quantum but is altered. In the function of the straight line y = a x, = a is an ordinary fraction and quotient; this function is therefore only formally a function of variable magnitudes, or x and y are here what a and b are in , not truly what the variable magnitudes in the proper functions are. — On account of the particular nature of the variable magnitudes in the proper functions, it would surely have been serviceable to introduce for them designations other than the usual ones of the unknown magnitudes in every finite equation, determinate or indeterminate, since they are also essentially different from such merely unknown magnitudes, which are in themselves perfectly determinate quanta, or a determinate range of determinate quanta.

In functions of genuinely variable magnitudes, then, the ratio as quantum is a variable one. What is constant in the ratio of these magnitudes — for the parameter or the constant does not express an immediate ratio of them, but only insofar as they are, as was said, still determined against each other through a power-ratio — is not to be expressed by a number or a numerical fraction, nor to be brought back to the function of a straight line, but is a ratio of quantity that is only of qualitative nature.

The sides x and y of such a function can, however, still signify quanta; only their determination relative to each other is of qualitative nature, and their being-determined through the ratio makes up their essential magnitude. They are not supposed to have the quantitative determinateness that belongs to them already immediately for themselves outside the ratio, nor is the reference to them, as to the 2 and 7 in , to be merely external. If 2 is assumed as numerator of a fraction, the denominator is thereby not yet determined. Combined into a function, however, if the one magnitude is determined, the other is thereby determined likewise; and indeed not according to a constant quotient. The quantitative determinateness, the exponent of the ratio of the variable magnitude, is thus of qualitative nature. In this the variable magnitudes, as the sides of the ratio, still have the signification of quanta, though the exponent has it no longer.

This signification, however, is lost altogether and completely in the infinitely small differences. d x, d y are quanta no longer, nor are they supposed to signify such, but have a signification solely in their reference, a sense merely as moments. They are no longer something, that something taken as quantum, not finite differences; but also not nothing, not the determinationless zero. Outside their ratio they are pure zeros, but they are to be taken only as moments of the ratio, as determinations of the differential coefficient.

Within this concept of the infinite, quantum has been genuinely brought to completion as something qualitative; it has been rendered actually infinite; what is sublated in it is not this or that quantum alone, but quantum as such. There remains, however, determinateness of quantity, element of quanta, principle, or that determinateness in its first concept.

It is against this concept of the infinite that every attack has been directed which has been made upon the mathematics of the genuinely infinite, the differential and integral calculus. Incorrect representations on the part of the mathematicians themselves occasioned at times its going unrecognized; chiefly, however, the blame for these contestations lies with the incapacity to exhibit the subject matter as concept. Yet mathematics, as was already recalled above, cannot here get around the concept; for being the mathematics of the infinite, it does not hold itself within the finite determinateness that belongs to its objects — as in pure mathematics space and number, with their determinations, are considered and related to one another only according to their finitude —; it rather posits a determination in identity with its opposite. The operations which it allows itself as differential and integral calculus therefore contradict entirely the nature of merely finite determinations and their references, and have on that account their justification in the concept alone.

If the mathematics of the infinite held fast to this, that those determinations of quantity are vanishing magnitudes, that is, such as are no longer any quantum whatever, yet are not nothing either, but are still a determinateness over against another, then nothing seemed clearer than that there is no such intermediate state, as it was called, between being and nothing. — What is to be made of this objection and of the so-called intermediate state has already been shown above. Certainly the unity of being and nothing is no state; a state would be some determination belonging to being and to nothing, one into which these moments were supposed to have fallen merely by chance, as though into a sickness or an external affection; this middle and unity, the vanishing or equally the becoming, is rather their truth and theirs alone.

What is infinite, it has further been said, is not comparable as a greater or a smaller; there can therefore be no ratio of infinites to infinites, nor orders or dignities of the infinite, such as the differences of the infinite differences that occur in the science of them. — Underlying these objections there is always the representation that what is here to be spoken of are quanta compared as quanta; that determinations which are quanta no longer have no ratio to each other any more Rather, what is only in the ratio is no quantum; for quantum is a determination of the kind that is supposed to have a perfectly indifferent existence outside its ratio, one to which its difference from another is indifferent, whereas the qualitative is only what it is in its difference from an other. Those infinite magnitudes are therefore not merely comparable, but are only moments of the comparison or of the ratio.

I adduce here the most important determinations that have been given by mathematicians concerning this infinite. It will be evident from them that the thought of the matter, in agreement with the concept developed here, lies at the base of these determinations of theirs, but that they did not fathom it as concept and for that reason again stood in need, in applying it, of expedients contradicting their better cause.

The thought cannot be determined more correctly than Newton has given it. In doing so I set apart the determinations belonging to the representation of motion and of velocity (from which he chiefly took the name fluxions), because in these the thought appears not in the requisite abstraction but concretely, mixed with inessential concepts. — These fluxions Newton explains more closely (Princ. mathem. phil. nat. L. 1. Lemma XI. Schol.) to this effect, that by them he understands not indivisibles — a form employed by earlier mathematicians, Cavalleri and others, and one that contains the concept of a quantum determinate in itself, — but vanishing divisibles. Further, not sums and ratios of parts that are determinate, but the limits (limites) of the sums and of the ratios. The objection is raised that vanishing magnitudes have no last ratio, because before they have vanished it is not the last, and when they have vanished there is none at all. But by the ratio of vanishing magnitudes is to be understood the ratio not before they vanish, and not afterwards, but with which they vanish (quacum evanescunt). In the same way the first ratio of becoming magnitudes is that with which they become.

Given the state of scientific method at that time, all that was explained was what is to be understood by an expression; that this or that is now to be understood by it is properly a subjective demand, or else a historical requirement, where it goes unshown that such a concept is in and for itself necessary and has inner truth. But what has been adduced shows that the concept set up by Newton answers to the way in which infinite magnitude issued, in the presentation given above, from the reflection of quantum into itself. Magnitudes are what is meant, in their vanishing, that is, such as are quanta no longer; further, not ratios of determinate parts, but the limits of the ratio. For the immediate ratio too, insofar as it has an exponent, is a quantum; both the quanta for themselves, then, the sides of the ratio, and with them the ratio too, insofar as it would be a quantum, are to vanish; the limit of the ratio of magnitudes is that in which it is and is not; more precisely, this means that in which the quantum has vanished and with it the ratio is preserved only as qualitative ratio of quantity. — Newton adds that from there being last ratios of the vanishing magnitudes one is not to conclude that there are last magnitudes, indivisibles. That would be, namely, once more a leap from the ratio as such over to its sides, which would be supposed to have a value for themselves outside their reference, as indivisibles, as something that would not be a relative. — He holds fast to divisibility in order still to preserve the quantitative, because the indivisible, or atoms, the one, would be something ratioless.

Against that misunderstanding he further recalls that the last ratios are not ratios of last magnitudes, but limits which the ratios of magnitudes decreasing without limit approach more nearly than any given, that is, finite difference, yet which limit they do not overstep, so that they would become nothing. — By last magnitudes, namely, as was said, indivisibles or ones could have been understood. In the determination of the last ratio, however, the representation both of the indifferent one, the ratioless one, and of the finite quantum is removed. Yet neither the decreasing without limit, into which Newton transposes the quantum and which expresses only the progress into infinity, nor the determination of divisibility, which here has no immediate significance any more, would have been called for, had the concept demanded developed itself further into the concept of a determination of magnitude that is purely and only moment of the ratio.

Equally interesting is the other form of the Newtonian presentation of these magnitudes, namely as generated magnitudes. A generated magnitude (genita) is a product or quotient, roots, rectangles, squares, also sides of rectangles and squares; — in general a finite magnitude. — „Considering it as variable, as it increases or decreases in continuing motion and flux, he understands its momentary increments or decrements under the name of moments. These, however, are not to be taken for particles of determinate magnitude (particulae finitae). Such are not themselves moments, but magnitudes generated from moments; what is to be understood are rather the becoming principles or beginnings of finite magnitudes.“ — The quantum is here distinguished from itself, as it is as a product, or something existent, and as it is in its becom ing, in its beginning and principle, that is, as it is in its concept, or what here amounts to the same, in its qualitative determination; within the latter, the differences of quantity, the infinite increments or decrements, count merely as moments; it is only what has become that has passed over into the indifference of existence and into the externality wherein it is quantum. — The increments and decrements do indeed fall within the sensuous representation of quantum; but the other determinations adduced must be acknowledged by the philosophy of the concept of the genuinely mathematical infinite.

Far behind the determinations considered stands the ordinary representation of infinitely small magnitudes. On this representation they are supposed to be of such a constitution that not only they themselves as against finite magnitudes, but also their higher orders as against the lower, or the products of several as against a single one, are to be neglected. — Leibnitz, like the earlier inventors of methods bearing on this magnitude, held to this representation; it is chiefly this that gives to that calculus, along with the gain in convenience, the semblance of inexactitude in the path of its operations. — Wolf sought to make it intelligible in his own way of making things popular, that is, of defiling the concept and setting incorrect sensuous representations in its place. He compares, namely, the neglecting of infinite differences belonging to higher orders over against lower ones with how a surveyor proceeds who, while gauging how high a mountain is, would have been no less exact had the wind meanwhile blown a grain of sand off its summit.

If the fair-mindedness of common human understanding permits such an inexactitude, all geometers, on the contrary, have rejected this representation. — It presses itself upon one of its own accord that in the science of mathematics there is no question whatever of such empirical exactitude, that mathematical measuring by operations of the calculus, or by constructions and proofs of geometry, is entirely distinct from land-surveying, from the measuring of empirical lines, figures and so forth. In any case the analysts show, as was adduced above, through the comparison of the result as it is obtained on the strictly geometrical route and as it is obtained by the method of infinite differences, that the one is the same as the other, and that a more or less of exactitude has no place whatever. And it goes without saying that an absolutely exact result could not come from a procedure that was inexact. Yet on the other side, again, the procedure itself cannot dispense with that neglect on the ground of insignificance. And this is the difficulty around which the efforts of the analysts turn, of making comprehensible to themselves the absurdity lying in this.

Euler, taking the general Newtonian definition for his base, insists above all that what the differential calculus has under consideration are the ratios of the increments belonging to a magnitude, whereas the infinite difference taken by itself must count wholly as zero. — How this is to be understood has been sufficiently elucidated; the infinite difference is zero only of the quantum, not a qualitative zero, but as zero of the quantum it is rather pure moment only of the ratio. It is not a difference by a magnitude, as when one quantum is subtracted from another, where their difference is itself also a quantum, to which it is indifferent whether it is regarded as a difference or as a sum, product and so forth, and which therefore does not have merely the sense of a difference. Since the ratios of the infinite differences are derived from the ratios of magnitudes that are variable but considered as finite, those ratios contain as results that within themselves as moment which the latter express as existent, or in finite determination, — or rather only in finite determinability, for the finite magnitudes that have such increments as are here considered are variable ones which do not themselves have a determinate quantum, though they can have one. On the one side it is, as was recalled, altogether awry, and belongs to the sensuous representation, to enunciate the infinitely small magnitudes as increments or decrements and as differences. For underlying this presentation is the idea that to the finite magnitude present at the start something is added or from it something is subtracted, that a subtraction or an addition, an arithmetical, external operation, takes place; the transition from the variable magnitude into its infinite difference, or of the function into its differential, is rather of a quite other nature; it is to be regarded as the leading back of that magnitude to the qualitative ratio of its determinations of quantity. — On the other side it has an awry aspect for this reason, when it is said that the increments are for themselves zeros, that only their ratios are considered. For a zero has no determinateness left at all. This representation does indeed reach as far as the negative of quantum, and enunciates it determinately, but it does not at the same time grasp this negative in its positive meaning, which, as was shown, consists in this, that the variable magnitudes, in that their ratio goes back into its qualitative determinateness, are no quanta, but are also not determinationless zeros, but moments; it is a ratio of determinations of quantity which, were they to be torn out of the ratio and taken as quanta, would be nothing but zeros. — Lagrange judges of the method which lays the representation of limits or last ratios at the base, — which L' Huillier in particular elaborated, — that although one may very well represent to oneself how two magnitudes stand in ratio so long as they remain finite, such a ratio yields the understanding no distinct and determinate concept as soon as its terms become zero together. — In fact the understanding must go out beyond this merely negative side, that the terms of the ratio are zeros as quanta, and apprehend them positively, as qualitative moments.

As concerns how the ratio keeps itself in the vanishing of the quanta, one meets, for instance in Carnot, the expression that by virtue of the law of continuity magnitudes which vanish hold on still to that ratio out of which they have come, before ever they vanish. — This representation expresses the true nature of the matter, provided that what is understood is not that continuity of quantum which it has in the infinite progress, where it continues itself into its vanishing, namely where in its beyond there arises once more only a finite quantum, a new term of the series, or the sum of that term with the preceding ones. In that negation, by contrast, which is the genuine infinite, the quanta vanish as indifferent, external determinations, and become only moments of the ratio. The ratio is therefore so continuous in this transition, and so preserves itself, that the transition rather consists solely in lifting the ratio out purely and making the ratioless side vanish. This purification of the quantitative ratio is nothing other than what happens when an empirical existence is comprehended. This existence is thereby raised above itself in such fashion that its concept contains the same determinations as it does itself, but grasped in their essentiality and into the unity of the concept, in which they have lost their indifferent, conceptless subsistence.

I refrain from multiplying the citations, since the determinations considered have shown well enough that the genuine concept of the quantitative infinite lies at their base, even though it has not been lifted out and grasped in its determinateness. For this very reason, however, it comes about that the concept does not maintain itself in its application and that the operation becomes unfaithful to it. The operation is grounded chiefly on the representation of a merely relatively small. The calculus makes it necessary to subject the infinite magnitudes to the ordinary arithmetical operations of adding and so forth, which are grounded on the nature of finite magnitudes, and thus to let them count for a moment as finite magnitudes and to treat them as such. The calculus would have to justify itself, on the one side, for dragging them down on the one occasion into this sphere, and on the other side for now and again dropping them and neglecting them as quanta, right after having applied to them the laws of finite magnitudes.

I adduce a few more things concerning the attempts of the geometers to remove the difficulty which gives the method the semblance of inexactitude.

The older analysts made fewer scruples about this; but the efforts of the moderns went chiefly to bringing the calculus of the infinite back to the evidentness of the properly geometrical method and to attaining in it the rigour of the proofs of the ancients in mathematics. Yet since the principle of the analysis of the infinite is of higher nature than the principle of the mathematics of finite magnitudes, the former must necessarily renounce the lesser merit of evidentness, which the latter owes chiefly to the conceptlessness of its content and of its method, just as philosophy too can lay no claim to that distinctness which the sciences of the sensuous, natural history for instance, possess, and just as eating and drinking pass for a more intelligible business than thinking and comprehending.

Several have attempted to dispense with the concept of the infinite altogether and to accomplish without it what seemed bound up with its use. — Lagrange, for instance, speaks of the method invented by Landen, and says of it that it is purely analytic and does not employ the infinitely small differences, but first introduces different values of the variable magnitudes and subsequently sets them equal. He judges, moreover, that in this way the advantages proper to the differential calculus, simplicity of method and ease of operations, are lost. — It is evident from what has been adduced that the vanishing of quantum occurs in this method too, namely in this, that the different assumed values of variable magnitudes are set equal to each other; for to set one quantum equal to another unequal to it means nothing else than to sublate them, and here indeed in order thereby to win their universal determination of ratio. — L' Huillier's method, which was grounded on the representation of the limits of a ratio, presses chiefly the point of regarding d x and d y simply and solely as moments of the differential coefficient, and as a single indivisible sign. But although this method remains the most faithful to the philosophical concept of the quantitative infinite, in the judgment of the geometers it does not accomplish what the calculus of the infinite attains by separating the sides of the differential coefficient from each other. Besides the fact that the limit always represents the positive, here namely a quantum, on the one side, and on the other side the negative of it, separated, and does not unite the two into the simple determination of the qualitative moment of quantity, — this method seems not to accomplish the transition, necessary for the manner of calculation which makes up the advantage of the ease of the calculus of the infinite, of the moments of ratio into the shape of finite magnitudes, nor the specification of the laws required for them on this ground and for their return into their peculiarity.

The elder among the moderns, such as Fermate, Barrow and others, who first made use of the infinitely small in that application which was later elaborated into the differential and integral calculus, and then Leibnitz too and those who followed, have always believed openly that they were entitled to drop the products of infinite differences, and likewise their higher powers, on no other ground than that relatively, as against the lower order, they vanish. On this alone rests, with them, the fundamental theorem of the whole doctrine, namely what the differential of a product or of a power is. On the same ground the main proposition concerning curves is assumed, which consists in this, that the elements of the curves, namely the increments of the abscissa and of the ordinate, have to each other the ratio of the subtangent and the ordinate; for with a view to obtaining similar triangles, the arc, which makes up the third side of a triangle alongside the two increments, is regarded as a straight line, as a piece of the tangent, and thus one of the increments as reaching right up to the tangent. These assumptions raise these moments, on the one side, above the nature of finite magnitudes; on the other side, however, a procedure is applied to them that holds only of finite magnitudes, and in which nothing may be neglected out of regard for insignificance. The difficulty by which the method is pressed remains in the manner of proceeding adduced in its full strength.

Newton employed (Princ. Math. phil. nat. Lib. II. Lemma II. after Propos. VII.) an ingenious artifice in order to get rid of the arithmetically incorrect dropping of the products of infinite differences, or of their higher orders, in the finding of the differentials. The differential of the product — from which the differentials of quotients, powers and so forth are then easily derived — he finds in the following way. The product, if x, y are each taken smaller by half of its infinite difference, passes over into ; but if x and y increase by just as much, into . With the first product subtracted from this second one, y d x + x d y remains as excess, and this, he says, is the excess of the growth by a whole d x and d y, for it is by this growth that the two products are distinguished; it is therefore the differential of x y. — One sees that in this procedure the term which makes up the chief difficulty, the product of the two infinite differences, d x d y, falls away of itself. But it is incorrect that or that the excess of a product whose factors each increase by a whole increment, over the product of the original factors, — should be equal to the excess of the product when its factors each grow by half the increment, over the product insofar as its factors have decreased by this half.

Other forms which Newton employs in deriving the differential are tied to concrete significations of the elements and of their powers. — In the use of series, which distinguishes his method, the ordinary representation of series lies too near at hand, namely that it always stands in one's power, by adding further terms, to take the magnitude so exactly as one requires, and that the terms dropped are relatively insignificant, the result on the whole being only an approximation. — The error into which Newton fell in solving a problem, through dropping essential higher powers, an error that gave his opponents an occasion for the triumph of their method over his, and whose true origin Lagrange has exhibited in his recent investigation of it, — proves at least the formal character and the uncertainty still present in the use of his instrument. Lagrange (in his Theorie des Fonctions analytiques) shows that Newton fell into the error because he neglected the term of the series which contained the power upon which everything turned in the determinate problem.

It is remarkable, namely, that in mechanics the several terms of that series into which the function of a motion is developed carry each its determinate signification, so that the first term, or the first function, has reference to the moment of velocity, the second to accelerating force, and the third to the resistance of forces. The terms of the series are here, accordingly, to be viewed not merely as parts of a sum, but as qualitative moments of a whole of the concept. Thereby the dropping of the remaining terms, which belong to the badly infinite series, acquires a wholly different signification from the dropping of them on the ground of their relative smallness. They are to be dropped because, through the determinations of the concept to which the first terms belong, the whole of the object is completed as concept, and thereby also as sum, its determination of quantity in general. The Newtonian solution contained that error, not because in it terms of the series, as parts of a sum, but because a term containing a determination of the concept, which belonged to the whole, was dropped.

In this respect too it is the case that the differential of xn is wholly exhausted in the first term of that series which the development of (x + d x)n yields; — a view upon which L' Huillier especially pressed. That the remaining terms are not taken into account does not come from their relative smallness; — no inexactitude, no fault or error is here presupposed that would be compensated and corrected by another error; a view from which Carnot above all justifies the ordinary method of the infinitesimal calculus. Rather, since what is here at issue is not a sum but a ratio, the differential is completely exhausted through the first term, in that the further terms, or differentials of higher orders, develop out of their predecessors in the same way as the differential of the original function develops out of that function, so that in them there is nothing but the repetition of one and the same ratio, which alone is wanted, and which is thus already perfectly attained in the first term.

I do not adduce separately the elucidations which Carnot gives concerning the method of infinite magnitudes. They contain what is most refined in the representations adduced above. But at the transition to the operation itself the ordinary representations, of the infinite smallness of the dropped terms as against the others, come in more or less. He justifies the method rather by the fact that the results come out correct, and by the utility which the introduction of imperfect equations, that is, of such as have had an arithmetically incorrect omission made in them, has for the simplification and abbreviation of the calculus, than by the nature of the matter itself.

Lagrange, as is well known, took up again Newton's original method, the method of series, in order to be relieved of the difficulties which the representation of the infinitely small, and of those which the method of first and last ratios and limits, carries with it. Of his calculus of functions, whose other merits with respect to precision, abstraction and universality are not to be brought out further here, only this is to be adduced, that it rests on the fundamental theorem that the difference, without its becoming zero, can be assumed so small that every term of the series exceeds in magnitude the sum of all those following. — One sees that the terms of the series to be dropped here come into consideration only in the respect that they constitute a sum, and that the ground for dropping them is placed in the relativity of their quantum. The omission is thus here too not led back, for the general case, to that ground which occurs in some applications, where, namely, as was recalled earlier, the terms of the series have a determinate qualitative signification, and following terms are left out of account, not because they are insignificant in magnitude, but because they are insignificant in point of quality.

In conclusion I set this uniquely correct point of view, the qualitative nature of the infinite differences, against the misunderstanding which seems to occur especially in the older presentations, and which takes the infinite differences as wholly ratioless moments, letting the determination of ratio vanish along with the quanta.

For since the infinite differences are the vanishing of the sides of the ratio as quanta, what remains over is their ratio of quantity, purely insofar as this depends on the qualitative determination. So little is the qualitative ratio lost in this that it is rather the determining factor, and precisely what results through the conversion of finite magnitudes into infinite ones. In this, as has been shown, consists the entire nature of the matter. — Thus in the last ratio the quanta of the abscissa and of the ordinate vanish; but the sides of this ratio remain essentially the one increment or element of the ordinate, the other increment or element of the abscissa. When, according to the ordinary manner of representation, one lets the one ordinate approach the other infinitely, the ordinate previously distinguished passes over into the other ordinate, and the abscissa previously distinguished into the other abscissa; (— as, according to the above, Landen first assigns different values to the variable magnitudes and then sets these equal —) in this passing over their finite difference vanishes, and there remains only the infinite difference, as moment of this passing over, the element of the ordinate and the element of the abscissa. Essentially the ordinate does not pass over into the abscissa, nor the abscissa into the ordinate. The qualitative ratio continues itself, as was expressed above, so far into the infinitely becoming, that is, vanishing differences of quantum, that it alone is that by which the determination of quantity is still borne.

In accordance with this it is essential now, against the point of view which the ordinary opinion has of the infinite differences, and which above all makes it hard to grasp the correct concept of the matter, — to remark that the element of the ordinate, — to stay with this example of variable magnitudes, — is no longer the difference of one ordinate from another ordinate, for these are no longer different quanta over against each other, since they have been infinitely approximated to each other, but is rather the difference, or the qualitative determination of magnitude, over against the element of the abscissa; the principle of the one variable magnitude over against that of the other stand in ratio with each other. The difference, in ceasing to be difference of finite magnitudes, has ceased to be a manifold within itself; it has collapsed into simple intensity, into the determinateness of one qualitative moment of ratio over against the other.

The consideration of these elements as differences, or also as increments, essentially holds fast only to the difference of the quantum of one ordinate from the quantum of another ordinate. The limit is taken as the last value which another magnitude, otherwise of like kind, constantly approaches, so that it can be distinguished from it by as little as one pleases, and so that the last ratio is a ratio of equality. Thus the infinite difference is a hovering, as difference of one quantum from a quantum, and the qualitative nature, according to which d x is essentially a determination of ratio not over against x but over against d y, recedes in the representation. One lets d x2 vanish as against d x, but far more does d x vanish as against x, or it has a ratio only to d y. — It has been recalled that this side is most of all lifted out in L'Huillier's method. But it has not yet been brought to the concept of the qualitative determination of magnitude, and the geometers who hold to the representation of limits are always concerned above all to make comprehensible the approximation of a magnitude to its limit, and to keep hold of this side of the difference of quantum from quantum, of how it is no difference and yet still a difference.

But since it has come about that the increments or infinite differences were taken merely on the side of quantum and as ratioless moments, there has sprung from this the inadmissible representation which permits itself, in the last ratio, to set abscissa and ordinate, or even sine, cosine, tangent, versed sine and whatever else, equal to one another.

The arc, too, is surely incommensurable with the straight line, and its element, in the first instance, of a quality other than that belonging to the element of the straight line. It seems accordingly still more absurd and more impermissible than the confounding of abscissa, ordinate, sine, cosine and so forth, when quadrata Rotunde, when a part of the arc, infinitely small though it be, is taken for a part of the tangent, or in general as hypotenuse in a right-angled triangle in which the two legs are the elements of the abscissa and of the ordinate, and thus is treated as a straight line. — This treatment, however, is essentially to be distinguished from the confounding just censured; it has its justification in this, that within such a triangle the ratio of the element of an arc to the element of the abscissa and of the ordinate is the same as it would be were that element the element of a straight line, of the tangent; for the angles which constitute the essential ratio, namely the ratio that remains to these elements after the finite magnitudes belonging to them have vanished as quanta, are the very same. — One can also put it thus, that straight lines, as infinitely small, have passed over into curved lines, and that the ratio of them in their infinity is a ratio of curves. For if one takes the ordinary definition of the straight line, that it is the shortest path between two points, then its difference from the curved line is grounded on the determination of multitude, on the lesser multitude of what is distinguishable along this path, which is accordingly a determination of quantum. But this determination vanishes in it, once it is taken as intensive magnitude, as infinite moment, as element; and with it also its difference from the curved line, a difference that rested merely on the difference of quantum. — Or, an infinite straight line is the sublated straight line, for the infinite straight line is the one that goes back into itself, that is, a curve. Thus, as infinite, straight line and curve retain no qualitative ratio to each other any more; the former rather passes over into the latter.

Quite otherwise, however, is it constituted with the ratios of sine, tangent and so forth to one another. It is easy to see, and has also been recalled by others, that if with the general excuse that in the last ratio everything is equal, that is, that the ratio itself too is sublated, one permits oneself to put the ordinate for the abscissa, then the most absurd things can be brought out, or, as it is called, proved. By such a confounding the underlying concept, that for the variable magnitudes in their vanishing the ratio from which they come is preserved, is entirely destroyed. There arises in the proper sense a ratio of zero to zero, for which it is wholly arbitrary and contingent what qualitative and quantitative signification is given to it. With the licence of such an equating it cannot be hard to bring forth formulas yielding as result that the diameter is greater than the circumference, the hypotenuse smaller than a leg, and so forth.

There can surely be no other ground for people's having put up with proofs built upon that equating than this, that what came out was always known beforehand anyway, and that the proof, contrived so that it should come out, notwithstanding that in such a fashion the opposite could just as well be brought out, at least brought off the semblance of a scaffolding of proof; — a semblance still preferred to mere faith or to knowledge from sensuous experience. I have no hesitation in regarding this manner as nothing more than a mere sleight of hand and charlatanry of proving, and in reckoning among it even a mass of the Newtonian proofs, but especially of those on account of which Newton was exalted to the skies and above Keppler for having set out mathematically what the latter found merely through experience. So long as the mathematics of the infinite lacks the thorough concept of its object, it is unable to specify the limit up to which that equating may go, and even to the correct ones among its operations there always clings the mistrust that springs from the uncertainty, and, in the case of the confounding adduced, — from the senselessness of this procedure, — a procedure that has nothing to reproach in the often-mentioned prattle of recent philosophers, — which at the same time is wont to make up their entire philosophy, — that in the absolute all is one.

The empty scaffolding of Newtonian proofs of that sort was erected chiefly in order to prove physical laws. But mathematics is simply not in a position to prove determinations of magnitude in physics, insofar as these are laws having the qualitative nature of the moments for their ground; and this on the plain ground that such a science is no philosophy and does not set out from the concept, so that whatever is qualitative, insofar as it is not taken up lemmatically from experience, lies outside its sphere. That scaffolding will doubtless yet meet with the same justice that has recently been done to the groundless Newtonian edifice of artifice built from optical experiments and the inferring bound up with them. Applied mathematics remains full of just such a decoction of experience and reflection; yet just as, in the case of that optics, one part after another has for a good while now begun to be factually ignored, so it is a fact as well that a portion of those deceptive proofs, grounded as they are on that lawless and senseless equating of qualitative determinations under the pretext of their infinite smallness, has already, without their defect having been seen through, fallen of itself into oblivion or been supplanted by others.

Chapter 3. The Quantitative Ratio

Quantum, having become infinite, has the negative beyond at its own self. This beyond is the qualitative in general. The infinite quantum is the unity of both moments, of the quantitative and of the qualitative determinateness. It is ratio.

In the ratio, therefore, quantum no longer has an indifferent determinateness, but is qualitatively determined, as referred simply to its beyond. Quantum continues itself into its beyond; this beyond is at first another quantum in general. But essentially they are not referred to one another merely as external quanta; the one does not have its determinateness as indifferent towards that of the other, but each has it in this reference to the other. They are therefore, in this otherness of theirs, returned into themselves; for what each is, it is not immediately for itself but in the other; the other makes up the determinateness of each. — A quantum goes out beyond itself as quantum, yet not so that it merely altered into another, nor into its abstract other, into its negative beyond, but into its determinateness; it finds its own self in its beyond, which is another quantum.

This being-determined of the quanta through one another, in which each has the essential significance of being not indifferently for itself but a moment of the ratio, and of being only in the reference to the other what it is, makes up the qualitative moment of the ratio. But the ratio at the same time remains quantitative. It is quanta that lie at the ground and have towards one another the reference that has resulted; or it is one quantum in general that has the qualitative determinateness within itself. Quantum, in that it is ratio, expresses itself as a totality closed within itself, and expresses its indifference towards the limit, in that it has the externality of its being-determined within its own self and in that externality is referred only to itself. — The qualitative and the quantitative have not here yet stepped apart; the qualitative is that of quantum itself, or that whereby quantum is quantum.

A. The Direct Ratio

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

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

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

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

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

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

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

B. The Inverse Ratio

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

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

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

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

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

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

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

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

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

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

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

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

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

The ratio has passed over into the ratio of powers.

C. The Ratio of Powers

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

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

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

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

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

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

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

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

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

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

Remark

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

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

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

Third Section. Measure

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

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

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

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

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

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

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

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

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

Chapter 1. Specific Quantity

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

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

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

A. The Specific Quantum

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

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

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

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

B. The Rule

The rule has

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

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

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

1. The Qualitative and Quantitative Determinateness of Magnitude

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

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

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

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

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

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

2. Quality and Quantum

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

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

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

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

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

3. Distinction of Both Sides as Qualities

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

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

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

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

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

Remark

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

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

C. Relation of Qualities

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

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

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

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

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

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

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

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

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

Chapter 2. Relation of Self-subsistent Measures

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

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

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

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

A. The Relation of Self-subsistent Measures

1. Neutrality

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

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

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

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