b. Identity of Extensive and Intensive Magnitude

Nothing external to itself lies within degree. Still, degree is not on that account the indeterminate one, the principle of number generally, whose sole amount is the negative one of being no amount whatever. To begin with, what intensive magnitude comes to is a simple one of the several; degrees there are, several of them; determinate, however, they are neither qua simple one nor qua several, but solely within the reference of this being-outside-itself, that is, within the identity of one and multiplicity. Granted that the several as such fall outside the simple degree, still it is in the degree's reference to them that its determinateness consists; the amount is accordingly harbored within it. Just as twenty, taken extensively, keeps the twenty ones within itself in discrete fashion, so the determinate degree keeps them as continuity, being in simple fashion this determinate multiplicity; such a degree is the twentieth; and the twentieth degree it is only by mediation of that amount, which qua amount falls outside it.

Two sides, then, call for consideration in the determinateness of intensive magnitude. Other intensive quanta determine it, and with its otherness it stands in continuity, its determinateness consisting precisely in that reference to the otherness. Now insofar as, first, it is the simple determinateness, its determination is one against other degrees; those it shuts out of itself, and in this shutting out its determinateness lies. Secondly, however, determinate it is in its own self; such it is in the amount, taken as its own amount, not in an amount excluded from it, nor in the amount belonging to other degrees. Within its own self the twentieth degree keeps the twenty; being marked off from the nineteenth, the twenty-first etc. is not all that determines it, for its amount is what its determinateness consists in. Once the amount counts as belonging to the degree, though, and the determinateness has essentially the shape of amount, extensive quantum is what the degree turns out to be.

One and the same determinateness of the quantum, then, is what extensive and intensive magnitude both are; nothing sets them apart save this, that in the former the amount lies within, in the latter that very amount lies without. Because its many, in and for itself, falls together into the unit, and the many then steps outside that unit, extensive magnitude passes over into the intensive. Conversely, though, what this simple thing owes its determinateness to is nothing but the amount, and indeed the amount as its own; indifferent as it is toward intensities determined otherwise, it bears the externality of amount upon its own self; and so intensive magnitude proves no less essentially to be extensive.

Along with this identity the qualitative something makes its entrance; for the identity is a unity referring itself to itself by way of the negation of its differences, while those differences are what the existent determinateness of magnitude consists in; hence this negative identity is something, and indeed a something to which its quantitative determinateness is a matter of indifference. A quantum is what something is, yet by now qualitative existence, such as it is in itself, stands posited as indifferent toward that quantum. Talk of quantum, of number as such etc. had been possible with no something to serve them as substrate. By now, however, something steps over against these determinations of its own, mediated with itself through their negation, as existing for itself, and, in possessing a quantum, as one and the same possessor of an extensive and of an intensive one. That one determinateness which something has qua quantum stands posited in the differentiated moments of unit and of amount; nor is that determinateness merely in itself one and the same, for its being posited in these differences, as extensive and as intensive quantum, amounts to the return into this unity, a unity which, being negative, is the something posited as indifferent toward it.

Remark 1

In the ordinary way of representing things, extensive and intensive quantum are apt to get distinguished as kinds of magnitude, as though certain objects possessed intensive magnitude only, others extensive only. To this there has been added the representation belonging to a philosophical natural science that converted the plural, the extensive, in the fundamental determination of matter, say, of filling a space, and likewise in other concepts, into an intensive, in the sense that the intensive, qua dynamic, is the genuine determination, and that density or specific filling of space, for example, must essentially be grasped not as a certain multitude and amount of material parts occupying a quantum of space, but rather as a certain degree of matter's space-filling force.

Determinations of two sorts require distinguishing here. Where one speaks of the conversion of the mechanical manner of consideration into the dynamic, there occur the concept of self-subsistent parts subsisting outside one another, joined into a whole only externally, and, distinct from it, the concept of force. What in the filling of space counts on the one side as nothing but a multitude of mutually external atoms is on the other side taken for the expression of a simple force lying at the ground. – Now the relation of whole to parts, and that of force to its expression, which here come to confront each other, belong not yet in this place but will be considered further on. This much may be recalled at once: that the relation of force to its expression, corresponding as it does to the intensive, is at first indeed the truer of the two as against the relation of whole and parts; but that force is on that account no less one-sided than the intensive, and the expression, the externality of the extensive, is from force just as inseparable, so that in both forms alike, the intensive and the extensive, one and the same content is at hand.

The other determinateness cropping up here is the quantitative as such, sublated as extensive quantum and converted into degree, this being taken for the determination that alone should truly be; though it has already been shown that this degree contains the former quite as much, so that each form is essential to the other, and hence that every existence exhibits its determination of magnitude no less as extensive than as intensive quantum. Anything whatever therefore serves as an example of this, insofar as it makes its appearance in a determination of magnitude. Number itself necessarily bears this doubled form immediately upon it. An amount it is, and to that extent extensive magnitude; but it is also one, a ten, a hundred; to that extent it stands upon the transition to intensive magnitude, since within this unit the manifold goes together into what is simple. One is extensive magnitude in itself, being representable as any amount of parts one pleases. Thus the tenth, the hundredth is this simple, intensive thing, whose determinateness lies at the several that fall outside it, i.e. at the extensive. Ten, a hundred is what number is, and simultaneously the tenth, the hundredth within the number system; the determinateness is in both cases the same.

Within the circle the one goes by the name of degree, because what the part of the circle owes its determinateness to is essentially the several lying outside it, the part being determined as one out of a closed amount of such ones. Regarded as mere magnitude of space, the circle's degree is nothing but an ordinary number; regarded as degree, it is intensive magnitude, having sense only insofar as it is determined by the amount of degrees the circle is divided into, much as number in general has its sense only within the number series.

A more concrete object's magnitude exhibits its doubled side, that of being extensive and intensive, at the doubled determinations of the object's existence, in the one of which the object appears as something external, in the other as something internal. A mass, for instance, is as weight something extensively large insofar as it makes up an amount of pounds, hundredweights etc.; something intensively large insofar as it exerts a certain pressure; and the magnitude of that pressure is a simple thing, a degree whose determinateness lies at a scale of degrees of pressure. In exerting pressure the mass shows itself as a being-within-itself, as a subject to which the intensive difference of magnitude accrues. – Conversely, whatever exerts this degree of pressure is capable of shifting a certain amount of pounds etc. from their place, and measures its own magnitude thereby.

Or heat has a degree; the degree of warmth, whether the 10th, the 20th etc. is a simple sensation, subjective in kind. Yet just as much is this degree at hand as extensive magnitude, namely as the expansion of a fluid, of the mercury in the thermometer, of air or of clay etc. As a taller column of mercury, or as a narrower cylinder of clay, a higher degree of temperature expresses itself; a larger space it warms in the same way in which a lesser degree warms only the smaller space.

Being the more intensive one, the higher tone is at the same time a greater multitude of vibrations, or a louder tone, one to which a higher degree gets ascribed, makes itself audible across a larger space. – A larger surface can be coloured in like manner with the more intensive color than with a weaker one; or what is brighter, another sort of intensity, is visible from farther off than what is less bright etc.

Just so within the realm of spirit: the high intensity of character, of talent, of genius goes together with an existence equally far-reaching, an effect equally extended, a contact equally many-sided. The deepest concept has the most universal significance and application.

Remark 2

A peculiar use has Kant made of the application of the determinateness of intensive quantum to a metaphysical determination of the soul. Where he criticizes the metaphysical propositions concerning the soul – paralogisms of pure reason, as he names them – he comes upon the consideration of the inference drawn from the soul's simplicity to its permanence. Against that inference he sets (Critique of Pure Reason, p. 414) »that, even granting the soul this simple nature, seeing that it contains no manifold outside one another and therefore no extensive magnitude, one could nonetheless as little deny it, as one could any existing thing, intensive magnitude, i.e. a degree of reality as regards the whole of its faculties, indeed the whole of what constitutes existence, a degree that may decrease through all the infinitely many smaller degrees, whereby the supposed substance might be turned into nothing, though not through partition, yet through a gradual slackening (remissio) of its powers; for even consciousness possesses at all times a degree open to still further lessening, and consequently so does the faculty of being conscious of oneself, and so do all the remaining faculties.« – Within rational psychology, such as that abstract metaphysics was, the soul counts not as spirit but as something that merely immediately is, as a soul-thing. Kant is thus within his right in applying to it the category of quantum, »as to anything existing whatever«, and, insofar as this immediate being stands determined as simple, that of intensive quantum. To spirit, admittedly, being does belong, though of a wholly other intensity than that of intensive quantum – rather of an intensity wherein the form of the merely immediate being, and every category thereof, count as sublated. It was not only the removal of the category of extensive quantum that had to be conceded; that of quantum in general had to be removed. Yet something further still awaits recognition: how, in the eternal nature of spirit, existence, consciousness, finitude are, and how they proceed from it without spirit thereby turning into a thing.

c. The Alteration of the Quantum

Towards the determinateness of quantum as such, the difference between extensive and intensive quantum is a matter of indifference. Quantum in general, however, just is determinateness in the shape of being posited as sublated – limit grown indifferent, determinateness that is no less its own negation. Within extensive magnitude this difference stands developed, while intensive magnitude constitutes the existence of that externality which quantum inwardly is. Posited within itself as quantum's contradiction, the difference amounts to this: to be the simple determinateness relating itself to itself which negates its own self, and to hold its determinateness not at it but in a further quantum.

A quantum accordingly stands posited, by its quality, in absolute continuity with what is external to it, with its otherness. Every determinateness of magnitude, therefore, not merely can be overstepped, not merely can be altered; it stands posited that alter itself it must. The determination of magnitude continues itself into its otherness in such fashion that only within this continuity with an other does it possess its being; a limit that is it therefore is not, but one that becomes.

Infinite is the one, or the negation that relates to itself, and hence repulsion of it away from its own self. Quantum too is infinite, posited as negativity relating to itself; away from itself it repels itself. Yet a determinate one it is, the one gone over into existence and into limit, and so repulsion of determinateness away from its own self – no generating of what is self-same, such as the repulsion of the one gives, but generating of its otherness; at it itself there now stands posited the task to send [itself] out beyond itself and to turn into an other. Increasing or diminishing itself is what it consists in; it is the externality of determinateness at it itself.

Beyond its own self, then, quantum sends itself; this other, into which it becomes, is at first a quantum likewise, yet no less a limit that, rather than merely being, drives itself out past itself. Hence the limit arisen anew in this going-out is utterly nothing else than one which sublates itself once more and passes itself along to a further one, and so on into infinity.

C. Quantitative Infinity

a. Its Concept

Quantum alters and turns into another quantum; that this alteration presses on into infinity – its further determination – has its ground in quantum's being set down as contradicting itself at it itself. – An other is what quantum becomes; but into its otherness it continues itself; a quantum, then, is what the other is too. This, however, is the other not of some quantum merely, but of the quantum itself, the negative of quantum as something limited, and thus quantum's unlimitedness, its infinity. Quantum is an ought; what it holds within it is to be determined for itself, and such being-determined-for-itself is rather a being determined in an other; conversely again it is that being-determined-in-an-other sublated, an indifferent subsisting-for-itself.

Thereby finitude and infinity each acquire at once, at it itself, a doubled and indeed opposed signification. Finite the quantum is, first as something limited in general, second as the sending of itself out past itself, as the being determined in an other. Its infinity, on the other hand, is first its not-being-limited, second its having-returned-into-itself, that indifferent being-for-itself. Set these moments side by side at once, and it emerges that quantum's determination of finitude – the sending of itself past itself to an other wherein its determination is supposed to lie – counts equally as determination of the infinite; negating the limit is that very same passing out beyond determinateness, so that in this negation, in the infinite, quantum would hold its last determinateness. Infinity's other moment is the being-for-itself indifferent towards the limit; quantum itself, however, is limited in just this way, that towards its limit, and thereby towards other quanta and towards its own going-beyond, it is what is for itself indifferent. In the case of quantum, finitude and the (bad) infinity supposed to stand apart from it each already carry the other's moment at it.

Qualitative and quantitative infinite part company in this, that with the former the opposition of finite and infinite is qualitative, and that the passing of the finite into the infinite, or the relating of the two to one another, lies only in the in-itself, in their concept. As immediate is qualitative determinateness, and it relates to otherness essentially as to a being other than itself; that it has its negation, its other, at it itself is not posited of it. Magnitude, by contrast, is as such determinateness sublated; posited it is as unequal with itself, indifferent towards its own self, and therefore as the alterable. Absolutely, i.e. abstractly, do the qualitative finite and infinite therefore stand over against each other; their unity is the inward relation lying at the ground; only in itself, then, and not at it, does the finite continue itself into its other. The quantitative finite, conversely, relates itself at it itself into its infinite, wherein its absolute determinateness is supposed to reside. What first exhibits this relation of theirs is the quantitatively infinite process.

b. The Quantitative Infinite Progress

Contradiction generally is what the progress into infinity expresses – here the contradiction contained in the quantitatively finite, in quantum as such. It is that reciprocal determining of finite and infinite already considered within the qualitative sphere, save for this difference, just recalled, that in the quantitative the limit sends and continues itself at it itself into its beyond, so that conversely the quantitatively infinite too stands posited as having quantum at it itself; for quantum in its being-outside-itself is at once its own self, its externality belonging to its determination.

Merely the expression of this contradiction is the infinite progress, not its resolution; yet because the one determinateness is continuous into its other, it brings about an apparent resolution in a unification of the two. In its first positing it is the task of the infinite, not the reaching of it – the perennial generating of the infinite, without ever getting past quantum itself, and without the infinite becoming anything positive and present. It lies in quantum's concept to have a beyond of itself. Such a beyond is, first, the abstract moment of quantum's non-being; quantum dissolves in its own self; so does it relate to its beyond as to its infinity, following the qualitative moment of the opposition. Second, however, quantum stands in continuity with that beyond; for quantum consists precisely in being the other of its own self, external to itself; this external, accordingly, is just as much not an other than quantum; hence the beyond, the infinite, is itself a quantum. Called back thereby out of its flight is the beyond, and the infinite is reached. But since what has come over to this side is a quantum again, nothing has been posited except a new limit once more; and this limit, being quantum, has fled from its own self again, is as such out past itself, and has repelled itself away into its non-being, into that beyond of its own self, which becomes quantum just as perennially as quantum thrusts itself away from itself towards the beyond.

Quantum's continuity into its other yields the linking of the two in the expression of an infinitely great or an infinitely small. Because both still bear the determination of quantum at them, alterable they remain, and the absolute determinateness that would amount to a being-for-itself is thus not reached. In the doubled infinite, opposed to itself according to more and less, in the infinitely great and the infinitely small, this being-outside-itself of the determination stands posited. At each of them quantum stays preserved in perennial opposition against its beyond. Widen the great as far as you will, it collapses into insignificance; in relating to the infinite as to its own non-being it renders the opposition qualitative; nothing, then, has the widened quantum wrested from the infinite, which stays, now as before, its non-being. Or again, magnifying the quantum brings no approach to the infinite, since the difference between quantum and its infinity carries essentially also the moment of not being a quantitative difference. Only the contradiction driven into narrower compass is thereby expressed: a great it is supposed to be, i.e. a quantum, and infinite, i.e. no quantum. – Likewise the infinitely small, being small, is a quantum, and stays therefore absolutely, i.e. qualitatively, too great for the infinite, standing opposed to it. Preserved in both is the contradiction of the infinite progress that was supposed to have found its goal in them.

Such infinity, persistently determined as the finite's beyond, deserves the name of the bad quantitative infinity. Like the qualitative bad infinity it is the perennial passing across and back from one member of the persisting contradiction over to the other, from limit to limit's non-being, and from that back once more to the very same, to the limit. In the quantitative progress, what is advanced to is indeed no abstract other in general, but a quantum posited as diverse; still, it stands in opposition to its negation just the same. Not an advancing and getting further on, therefore, is this progress, but a repeating of one and the very same – positing, sublating, positing again and sublating again; an impotence of the negative, which through its own sublating sees what it sublates return as something continuous. Two are so knotted together that they flee one another utterly; and in fleeing one another they cannot part, but are tied together in their mutual flight.

Remark 1

Chiefly in the form of the progress of the quantitative into infinity – this ongoing overflying of the limit, which is the impotence to sublate it, and the perennial relapse into it – is the bad infinity apt to be held for something sublime and for a kind of divine service, just as in philosophy that progress has been looked upon as an ultimate. Many a tirade has this progress served, tirades that have won admiration as sublime productions. What this modern sublimity in fact magnifies, however, is not the object, which rather flees away, but only the subject, which gulps down such great quantities into itself. The paltriness of this elevation, remaining as it does subjective and climbing the ladder of the quantitative, announces itself in the very fact that amid futile labor it confesses not to come nearer the infinite goal, a goal that, if it is to be reached, must of course be tackled quite otherwise.

In the following tirades of this kind there is expressed at the same time what such elevation passes over into and where it ceases. Kant for instance presents as sublime (Kr. d. pract. V. Schl.) »the case where the subject in thought lifts itself above the place it occupies in the world of sense and widens the linkage into the infinitely great, a linkage with stars above stars, with worlds above worlds, systems above systems, and beyond that into boundless times of their periodic motion, of its beginning and its endurance. – Representing succumbs to this advance into the immeasurably remote, where the remotest world still has one remoter yet, the past traced back so far has yet a further one behind it, the future carried out ever so far has still always another before it; thought succumbs to this representation of the immeasurable; as a dream in which one walks a long passage ever further and unforeseeably further, without descrying an end, closes with a fall or with dizziness.«

Besides compressing the content of quantitative elevation into a richly furnished picture, this presentation deserves praise above all for the truthfulness with which it states how this elevation fares at the end: thought succumbs, the end is falling and dizziness. What makes thought succumb, and brings forth its falling and the dizziness, is nothing other than the tedium of the repetition, which lets a limit vanish and appear again and vanish again – so always the one for the other, and each in the other, in the beyond the this-side, in the this-side the beyond, perennially coming to be and ceasing to be – and which yields no more than the feeling of impotence in this infinite or this ought, which wants to become master over the finite and cannot.

Haller's horrifying description of eternity, as Kant called it, is likewise apt to be particularly admired, though often precisely not on account of that side which constitutes its genuine merit:

»Monstrous numbers I heap up, mountains of millions I raise, Time upon time I set, and world on world I lay, And when from that grim height With reeling eyes I look back to you, All the power of number, multiplied a thousandfold, Is not yet one part of you.«

»I take them away, and you lie wholly before me.« When value is set upon that heaping and towering of numbers and worlds as upon a description of eternity, then it goes unnoticed that the poet himself pronounces this so-called horrifying going-beyond futile and hollow, and that he closes by saying that only through the abandoning of this empty infinite progress does the genuine infinite itself come into presence before him.

There have been astronomers who liked to pride themselves greatly on the sublimity of their science, and this because it has to do with an immeasurable multitude of stars, with spaces and times so immeasurable that within them distances and periods already so great in themselves serve as units which, taken ever so many times over, shrink back once more into insignificance. The stale astonishment to which they abandon themselves in this, the insipid hopes of travelling one day in that other life from one star to another and of acquiring the like new knowledge onward into the immeasurable – these they passed off as a chief moment in their science's excellence, a science admirable not on account of any such quantitative infinity, but on the contrary on account of the ratios of measure and the laws that reason cognizes in these objects, laws which are the rational infinite as against that irrational infinity.

To the infinity that relates to outer sensuous intuition Kant opposes the other infinity, when

»the individual goes back into its invisible I, and sets the absolute freedom of its will, as a pure I, against all the terrors of fate and of tyranny, beginning with its nearest surroundings, lets these vanish for it, lets likewise what appears as enduring, worlds upon worlds, collapse into rubble, and, alone, cognizes itself as equal to itself.«

The I in this solitude with itself is indeed the beyond attained; it has come to its own self, is with itself, on this side; in pure self-consciousness the absolute negativity is brought to affirmation and presence, the very negativity which in that advance beyond the sensuous quantum does nothing but flee. But in fixing itself in its abstraction and lack of content, this pure I finds existence in general, the whole fullness of the natural and of the spiritual universe, standing over against it as a beyond. There presents itself the same contradiction that lies at the ground of the infinite progress: namely a having-returned-into-itself which is immediately at the same time a being-outside-itself, a relating to its other as to its own non-being; and such relating stays a longing, because the I has fixed for itself its own contentless, untenable emptiness on the one hand, and on the other, as its beyond, that fullness which in the negation nonetheless stays present.

To these two sublimities Kant appends the remark »that admiration (for the first, the outer) and respect (for the second, the inner) sublimity do indeed stimulate inquiry, but cannot make good the want of it«. – He thereby declares those elevations unsatisfying for reason, which cannot come to rest with them and with the sentiments bound up with them, nor let the beyond and the empty pass for the ultimate.

As an ultimate, however, the infinite progress has been taken above all in its application to morality. The second opposition of finite and infinite just adduced, that of the manifold world and of the I raised into its freedom, is at first qualitative. The self-determining of the I aims at once at determining nature and at freeing itself from nature; thus through its own self it relates to its other, which as outer existence is something manifold and also something quantitative. The relation to something quantitative itself becomes quantitative; the negative relation of the I to it, the power of the I over the not-I, over sensibility and outer nature, is therefore represented in such a way that morality can and ought to become ever greater, the power of sensibility ever smaller. As for the will's complete adequacy to the moral law, that gets displaced into the progress running on into infinity, i.e. represented as an absolute unattainable beyond, and just this unattainability is supposed to yield the true anchor and the right consolation; for morality is supposed to be as struggle, and struggle there is only where the will fails to match the law, the law thereby being utterly a beyond for it.

Within this opposition, I and not-I – or the pure will and the moral law, and again the will's nature and sensibility – count as presupposed, each perfectly self-subsistent and indifferent towards the other. The pure will has its own peculiar law standing in essential relation to sensibility; and nature and sensibility have for their part laws that are neither drawn from the will nor answerable to it, nor would even, though different from it, have in themselves an essential relation to it, but are determined altogether for themselves, complete and closed within themselves. Both, however, are at once moments of one and the same simple essence, namely of the I; over against nature the will stands determined as the negative, so that it only is insofar as something distinct from it is there for sublating by it, something by which it is in this very act touched and itself affected. To nature, and to nature as the sensibility of the human being, taken as a self-subsistent system of laws, the being limited by an other is a matter of indifference; it maintains itself in this being limited, enters self-subsistently into the relation, and sets a limit to the will of the law no less than that will sets one to it. – It is one act, that the will determines itself and sublates the otherness of a nature, and that this otherness is posited as existent, continues itself into its being sublated, and is not sublated. The contradiction lying in this is not resolved in the infinite progress but on the contrary is set forth and asserted as unresolved and unresolvable; the struggle of morality and sensibility is represented as the absolute relation that is in and for itself.

The impotence to master the qualitative opposition of the finite and the infinite, and to grasp the idea of the genuine will, of substantial freedom, takes refuge in magnitude, so as to employ it as the mediatrix, magnitude being the qualitative sublated, difference grown indifferent. Yet since both members of the opposition remain lying at the ground as qualitatively diverse, the very fact that in their mutual relation they behave as quanta rather posits each of them at once as indifferent towards this alteration. Nature is determined by the I, sensibility by the will of the good; the alteration brought about at nature by that will is only a quantitative difference, one that lets nature subsist as what it is.

Within the more abstract presentation of the Kantian philosophy, or at any rate of its principles – Fichte's Wissenschaftslehre – the infinite progress makes up, in the same way, both foundation and ultimate. Upon the first principle of that presentation, I=I, there follows a second independent of it, the opposing of the not-I; the relation of the two is at once likewise assumed as a quantitative difference, in that the not-I is in part determined by the I and in part not. In this manner the not-I continues itself into its non-being so as to stay opposed to that non-being, as something unsublated. After the contradictions lying therein have accordingly been developed within the system, the final result is that very relation which was the beginning; the not-I remains an infinite check, an absolute other; the last relation of it and the I to one another is the infinite progress, longing and striving – the very contradiction with which the start was made.

Since the quantitative is determinateness in the shape of being posited as sublated, people believed that much – rather, that everything – had been gained for the absolute's unity, for the One substantiality, once opposition as such was demoted to a difference merely quantitative. All opposition is merely quantitative was for a time a chief proposition of the more recent philosophy; one essence, one content belongs to the opposed determinations, which are real sides of the opposition insofar as each of them carries both its determinations, both factors, in it, save that on the one side the one factor and on the other the other is preponderant, in the one side the one factor, a matter or an activity, being at hand in greater amount or in stronger degree than in the other. Wherever different materials or activities are presupposed, what the quantitative difference does is rather to confirm and complete their externality, their indifference towards each other and towards the unity they have. Merely quantitative, so it is said, is the difference of the absolute unity; the quantitative is indeed immediate determinateness sublated, yet only imperfectly so, being as yet the first negation only, not the infinite one, not negation of the negation. – In being represented as quantitative determinations of absolute substance, being and thinking too become, as quanta, perfectly external and relationless to each other, as carbon, nitrogen and so forth do in a subordinate sphere. It is a third thing, an external reflection, that abstracts from their difference and cognizes their inner unity, a unity merely in-itself and not equally for-itself. Thus in fact this unity gets represented merely as a first immediate one, or merely as being that in its quantitative difference stays equal to itself without positing itself equal through its own self; grasped as negation of the negation, as infinite unity, it accordingly is not. Only in qualitative opposition does the posited infinity, being-for-itself, come forth, and the quantitative determination itself passes over, as will shortly appear more closely, into the qualitative.

Remark 2

Recalled above already was the fact that the Kantian antinomies present the opposition of finite and infinite in a more concrete shape, brought to bear on more special substrates of representation. Qualitative finitude and infinity in their opposition — that was the content of the antinomy dealt with there. Another of them, the first among the four cosmological antinomies, has rather the quantitative limit for what it examines in its conflict. For that reason I take up the investigation of this antinomy here.

At issue in it is the limitedness or unlimitedness of the world in time and space. – One might with equal right have examined this opposition in regard to time and space taken by themselves, since nothing about the antinomic character of limitedness or unlimitedness in them turns on whether time and space are relations belonging to things themselves or nothing but forms of intuition.

That the two propositions, along with the proofs of them — apagogically conducted, as were those of the antinomy just examined — issue in nothing beyond two plain and mutually opposed claims will emerge equally from a closer unfolding of this antinomy: there is a limit, and: the limit must be gone beyond.

The thesis runs:

»The world has a beginning in time, and in respect of space it is likewise shut in within limits.«

The one part of the proof, the one bearing on time, supposes the contrary,

»that as regards time the world has no beginning; then up to any given instant of time an eternity has run out, and consequently an infinite series of states of things in the world, one following upon another, has elapsed. Now the very infinity of a series lies in this, that by successive synthesis it can never reach completion. Thus an infinite elapsed world-series is impossible, and a beginning of the world accordingly a necessary condition for its existence; which is what had to be demonstrated.«

The other part of the proof, the one dealing with space, gets traced back to time. An infinite time would be required for gathering together the parts of a world infinite in space, and that time would have to count as run out, given that a world in space is to count not as something in the making but as something finished and given. Yet with respect to time the first part of the proof established the impossibility of supposing an infinite time to have run out.

One notices straightaway, however, that dressing the proof up apagogically, or mounting any proof whatever, was quite unnecessary, seeing that what the proof itself rests on immediately is the very claim that was to be established. Some particular or any given instant of time is posited, namely, up to which an eternity (–eternity carries here only the paltry sense of a badly infinite time) is supposed to have run out. But a given instant of time signifies nothing other than a determinate limit within time. A limit of time is thus presupposed in the proof as something actual; and this is exactly what was to be established. For what the thesis maintains is that the world has a beginning in time.

The sole difference to be found is that the temporal limit assumed is a now serving as end of the time already run out, while the one to be established is a now serving as beginning of a future. This difference, though, carries no weight. As the point at which an infinite series of states of things in time, one succeeding another, is supposed to have elapsed, the now gets posited — hence as an end, as qualitative limit. Suppose this now were taken merely for a quantitative limit, one that flows and that is not only to be surpassed but is rather nothing but this, to surpass itself: then within it the infinite series of time would not stand elapsed but would go on flowing, and the reasoning of the proof would collapse. The instant of time is instead supposed as qualitative limit for the past, yet just so it is at once beginning for the future, – since in itself every instant of time is the connecting of past with future, – and it is moreover an absolute, which is to say abstract, beginning for that future, hence exactly what was to be established. That a past already is, lying before this future of its and before that beginning of the future, has no bearing on the matter; for inasmuch as this instant of time is qualitative limit – and taking it as qualitative is what the determination of the completed, the run-out, hence non-self-continuing, involves – time stands broken off in it, while that past, bearing no connection to the time that could be styled future only in respect of this past, and thus, lacking such connection, being simply time as such, possesses an absolute beginning. Were that time, though, – (as in fact it does –) to stand by way of the now, the given instant of time, in a connection with the past, and were it thereby determined as future, then from the other side this instant too would be no limit, the infinite series of time would carry itself on in what went by the name of future, and would not, as was supposed, be completed.

Time is in truth pure quantity; and the instant of time deployed in the proof, at which time was to suffer interruption, is rather only the self-sublating being-for-itself of the now. All the proof achieves is to set before representation, as a given instant of time, the absolute limit of time that the thesis maintains, and then flatly to suppose it a finished, that is abstract, point, – a popular determination that sensuous representing readily waves through as a limit, thereby letting stand within the proof as supposition what had earlier been put forward as the thing to be established.

The antithesis states:

»The world has no beginning and no limits in space; rather, in respect of time as of space alike, it is infinite.«

Its proof likewise posits the contrary:

»Let the world have a beginning. A beginning being an existence ahead of which there goes a time wherein the thing is not, some time must have gone before in which the world was not — an empty time, namely. But within an empty time no coming-to-be of anything at all is possible; for one part of such a time possesses in itself, over against another, no distinguishing condition favouring existence rather than non-existence. So while many a series of things may well begin within the world, the world itself can take no beginning, and with respect to elapsed time it is infinite.«

Like the others, this apagogic proof harbours the direct, unproved affirmation of what it was meant to establish. For it first supposes a beyond of worldly existence, an empty time; but then it equally continues worldly existence out beyond itself into this empty time, sublating the latter thereby, and so carries existence on into infinity. An existence is what the world is; and that this existence comes to be, its coming-to-be having a preceding condition in time, is what the proof presupposes. Yet just in this the antithesis itself consists, that no existence is unconditioned and no limit absolute, worldly existence always calling instead for a preceding condition. So the thing to be demonstrated turns up inside the proof as supposition. – The condition is then looked for, further, in empty time, which comes to saying that it is supposed as temporal and hence as existence, as something limited. Quite generally, then, the supposition has been made that the world, as existence, presupposes some other conditioned existence in time, and this onward into infinity.

With the infinity of the world in space the proof runs just the same. Apagogically the spatial finitude of the world gets posited; »the world would then be situated in an empty unlimited space and would bear a relation to it; but a relation of the world such as this, to no object, is nothing.«

Here too what should have been proved is presupposed outright within the proof. Outright supposed it is that a limited spatial world be situated in an empty space and bear a relation to it, which means that the world must be passed beyond, – towards the void on the one hand, towards the beyond and non-being of the world, but on the other hand that it stands therewith in relation to that void, that is, continues itself into it, so that the beyond has to be represented as filled with worldly existence. What the antithesis maintains as the world's infinity in space amounts to nothing but empty space on the one side and, on the other, the world's relation to that space, which is to say its continuity within it, or its filling of it; and this contradiction — space at once empty and at once filled — is what the infinite progress of existence in space is. This very contradiction, the world's relation to empty space, has been laid down directly as the basis in the proof.

Nothing else, therefore, do thesis and antithesis with their proofs set forth than these mutually opposed claims: that a limit is, and that this same limit is no less a merely sublated one; that the limit has a beyond standing nonetheless in connection with it, towards which one is to pass out beyond the limit, but wherein a limit of that same kind arises once more, a limit that is none.

Transcendental, like the resolution of the previous antinomy, is the resolution of these — which is to say that it lies in maintaining that space and time, qua forms of intuition, are ideal, in the sense that the world in its own self stands in no contradiction with itself and is nothing self-sublating, and that only consciousness, in its intuiting and in the bearing of intuition on understanding and reason, is an essence contradicting itself. To banish contradiction from the world and shift it instead into spirit, into reason, leaving it there to subsist unresolved, is an excess of tenderness towards the world. Spirit is in fact what has the strength to bear contradiction, but it is likewise spirit that understands how to resolve it. The so-called world, however (call it the objective, real world, or, on transcendental idealism's showing, subjective intuiting plus a sensibility determined through the category of understanding), does not on that account lack contradiction anywhere, but is incapable of bearing it and is for that reason given over to coming-to-be and ceasing-to-be.

c. The Infinity of the Quantum

As infinitely great or infinitely small, the infinite quantum is itself in itself the infinite progress; a great or a small it is, quantum therefore, and simultaneously non-being of quantum. Pictures of representation, accordingly, are what the infinitely great and the infinitely small come to, pictures that on closer inspection prove to be null mist and shadow. Within the infinite progress, by contrast, that contradiction stands explicitly at hand, and with it whatever belongs to the nature of quantum — of quantum which, as intensive magnitude, has reached its reality and stands posited now in its existence just as it stands in its concept. It is this identity that calls for consideration.

Simple is quantum as degree, self-referred, and determined as at its own self. Because through such simplicity otherness and determinateness stand sublated at it, determinateness falls outside it; beyond itself quantum has its determinateness. To begin with, this being-outside-itself of quantum is the abstract non-being of quantum as such, bad infinity. Further, though, that non-being is a great as well, quantum carries itself on into its non-being, having its determinateness precisely in its externality; and so this externality of quantum's is itself quite as much quantum; hence a limit falls to that non-being of quantum, to the infinity, which means that the beyond gets sublated, since the beyond is determined as quantum itself, quantum being thereby with itself in its own negation.

But that is what quantum as such is in itself. For through being external is precisely how it is it itself; what makes it quantum, with itself, is constituted by externality. Within the infinite progress, then, quantum's concept stands posited.

Taking it first in the abstract determinations it comes with, then what lies in it is the sublating of quantum, but quite as much of its beyond, hence both the negation of quantum and the negation of that negation. Their unity is its truth, and in that unity they are, though as moments. – The unity dissolves the contradiction whose expression the progress is, and its nearest sense is accordingly the restoration of the concept of magnitude, magnitude being an indifferent or external limit. Where the infinite progress as such is concerned, reflection habitually settles only on this, that any quantum, be it ever so great or small, vanishes, that going out beyond it must be possible; not, however, on this, that the sublating of quantum, the beyond, the bad infinite itself, vanishes too.

Even the first sublating, the negation of quality as such whereby quantum gets posited, is in itself a sublating of negation, – quantum being a sublated qualitative limit and hence a sublated negation, – yet only in itself is it this; posited, it is an existence, and then its negation stands fixed as the infinite, as quantum's beyond, quantum itself standing over here as a this-side, as something immediate; the infinite is on that footing determined merely as first negation, and in the infinite progress that is how it shows up. More than this, however, has been shown to lie within the progress, namely negation of negation, or what the infinite truly is. Just now this was viewed as amounting to the concept of quantum being restored thereby; and such restoration means first of all that quantum's existence has acquired its closer determination; there has arisen, that is, the quantum determined after its own concept, distinct from immediate quantum, while externality has now become the opposite of itself, posited as a moment belonging to magnitude itself, – quantum posited so that through its non-being, through infinity, its determinateness lies in another quantum, so that qualitatively it is what it is. Holding quantum's concept against its existence belongs rather to our reflection, though, to a relation not yet on hand at this stage. Nearest to hand as determination is that quantum, having gone back to quality, is henceforth qualitatively determined. For what is peculiar to it, its quality, is externality, the indifference of determinateness; and it now stands posited as being, rather, itself within its externality, as referring itself therein to itself, in simple unity with itself, hence as qualitatively determined. – Closer still is this qualitative element determined, namely as being-for-itself; for the self-reference at which quantum has arrived issued out of mediation, out of negation of negation. Infinity, being-determined-for-itself, quantum no longer has outside it but at its own self.

In the infinite progress the infinite carries only the empty significance of a non-being, of a beyond sought yet never reached, whereas it is in fact quality and nothing else. As indifferent limit quantum passes out beyond itself into infinity; what it seeks in doing so is simply being-determined-for-itself, the qualitative moment — which, taken so, remains a mere ought. Its indifference towards the limit, and with that its want of a determinateness that would be for itself, together with its passing out beyond itself, is what makes quantum quantum; and that passing beyond is to be negated, quantum finding its absolute determinateness in the infinite.

Quite generally: sublated quality is what quantum is; but quantum is infinite, passes out beyond itself, is negation of itself; so this passing beyond is in itself negation of the negated quality, restoring that quality; and what stands posited is this, that the externality which showed up as beyond is determined as quantum's own moment.

Posited hereby as repelled from itself, quantum yields two quanta, which nonetheless, being sublated, are only moments of one unity, and the determinateness of quantum is this unity. – Quantum thus referred to itself within its externality as indifferent limit, and hereby posited qualitatively, is quantitative ratio. – Within ratio quantum stands external to itself, distinct from itself; that externality of its is one quantum's reference to another quantum, each counting only in this its reference to its other; and the determinateness of quantum, which is as such a unity, is constituted by that reference. Therein it holds a determination not indifferent but qualitative; within this externality of its it has gone back into itself, and in that same externality it is what it is.

Remark 1. The Conceptual Determinateness of the Mathematical Infinite

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 (concept taken in its proper sense). In the end the justifications rest on the correctness of the results that come out with the help of that determination, which is demonstrated on other grounds; they do not rest on any clarity in the object and in the operation whereby those results get brought out – so little, indeed, that the operation is rather admitted to be itself incorrect.

In and for itself this is already a defect; a procedure of that sort is unscientific. Yet it carries a further disadvantage with it, namely that mathematics, not knowing the nature of this instrument of its own – for it has not come to terms with the metaphysics and the critique of that instrument – could neither determine the range of its application nor secure itself against misuses of it.

In a philosophical respect, however, the mathematical infinite is important precisely 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 launched. Against these objections the science of mathematics frequently knows no other rescue than to reject the competence of metaphysics, maintaining that it has nothing to settle with that science and need give no thought to its concepts, provided only that it proceeds consistently upon its own ground. Not what is true in itself, so it says, but what is true on its own field is what it has to consider. Metaphysics, contradicting the mathematical infinite as it does, cannot manage to deny or overturn the brilliant results yielded by its use, while mathematics cannot manage to get clear about the metaphysics of its own concept and hence about the derivation of the procedures which the use of the infinite makes necessary.

If the sole difficulty pressing upon mathematics were the difficulty of the concept as such, it might let this lie aside without further ado – insofar, that is, as the concept is more than the mere specification of the essential determinatenesses, i.e. of the determinations of the understanding, of a matter, and in the sharpness of these determinatenesses it has let nothing be wanting; for it is not a science that has to do with the concepts of its objects and would have 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 is bound utterly to 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; a principal aspect of the elaboration of this science consists in having won, for the transcendent determinations and their treatment, the form of the ordinary calculus.

In this conflict of its operations mathematics shows that results found by that route agree entirely with those found by the properly mathematical route, the geometrical and 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 reach results that cannot be achieved along it. For another, success does not of itself justify the manner of the path. This manner of calculating the infinite, moreover, shows itself burdened with the semblance of inexactitude that it gives itself when it augments finite magnitudes by an infinitely small magnitude on the one occasion, retains part of that increment in the further operation, and yet also neglects a part of it. There lies in this procedure the oddity that, the admitted inexactitude notwithstanding, a result emerges that 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 or other is not equal to zero yet is so inconsiderable as to admit of being left out of account. All difference of greater or lesser exactitude drops away entirely, however, once we ask what is to be understood by mathematical determinateness, just as in philosophy there can be no talk of greater or lesser probability but only of truth. Granted that the method and the use of the infinite are justified by success, it is nonetheless 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, being a scientific cognition, everything essentially turns on proof, and even with regard to the results 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 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 clearest light upon what the true concept itself is by nature, 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, – if it is determined as the infinitely great – there is no greater, or, – if it is determined as the infinitely small – no smaller one, or which, in the former case, is greater and, in the latter case, smaller than any magnitude one pleases. – The true concept, to be sure, finds no expression in this definition; rather, as already remarked, only that same contradiction which the infinite progress harbours; but let us see what it contains in itself. A magnitude is defined in mathematics as being something that admits of increase and decrease; hence, quite generally, as a limit that is indifferent. And since the infinitely great or the infinitely small is such as admits of no further increase or decrease, it is in fact no quantum any longer as such.

This consequence is necessary and immediate. But the reflection that the quantum – and by quantum in this Remark I mean quantum in general, as it is, the finite quantum – is sublated is the one that people are not in the habit of making, 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 not a quantum and whose quantitative determinateness nevertheless remains.

To cite how Kant appraises that determination⁹, 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 (i.e. beyond the multitude, contained in it, of a given unit) is possible; but no multitude is the greatest, since one or more units can always still be added. – By an infinite whole, on the contrary, 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 this unit were assumed greater or smaller, the infinite would be greater or smaller; yet infinity, 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 is that infinite wholes should be viewed as a maximum, as a completed multitude of a given unit. Maximum or minimum as such still appears as a quantum, a multitude. A representation of that sort cannot fend off the consequence Kant adduces, which leads to a greater or a smaller infinite. 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 of infinity, 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.« A quantum in general is presupposed as given; through the synthesizing of the unit it is to be made into an amount, into a quantum that can be definitely specified, but this synthesizing can never be brought to completion. With this, as is evident, nothing but the progress into infinity is enunciated, only transcendentally, i.e. represented in a way that is properly subjective and psychological. In itself, to be sure, the quantum is to be complete; transcendentally, however – namely within the subject, which confers on it a ratio to a unit – there arises only such a determination of the quantum as is uncompleted and simply afflicted with a beyond. Here, then, one comes to a halt 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 every determinateness it has apprehended, into the bad infinite.

Against this it was said earlier that the determination of the mathematical infinite, and specifically as it is used in the higher analysis, answers to the concept of the genuine infinite; the collation of the two determinations shall now be undertaken in a fuller development. – Taking first the genuine infinite quantum, it determined itself as infinite in its own self; it is this because, as has emerged, the finite quantum, or quantum in general, and its beyond, the bad infinite, are sublated in the same way. The sublated quantum has thereby gone back into simplicity and into reference to itself, yet 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. Rather, the infinite quantum contains, first, externality and, second, the negation of that externality at its own self; hence it is no longer some finite quantum, not a determinateness of magnitude that would have an existence as quantum, but is simple, and therefore only as moment; it is a determinateness of magnitude in qualitative form; its infinity consists in being a qualitative determinateness. – Being moment in this way, it stands in essential unity with its other, only as determined through this other of its, i.e. 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, within it, still an immediate determination at rest. In the ratio, as only moment, it is nothing indifferent for itself; in infinity as being-for-itself, being at the same time a quantitative determinateness, it is only as a for-one.

The concept of the infinite, as it has exposed itself here in abstract fashion, will show itself [to] lie at the base of the mathematical infinite, and it will itself grow more distinct as we run through the various stages of the expression of quantum as a moment of ratio, beginning from the lowest, where it is still at the same time quantum as such, up to the higher, where it takes on the meaning and expression of infinite magnitude proper.

Let us take, then, first the quantum in the ratio, in the shape in which it is a fractional number. A fraction of this sort, 2/7 for instance, is not a quantum like 1, 2, 3, etc.; an ordinary finite number it certainly is, yet not an immediate one like the whole numbers, but as fraction is determined mediately through two other numbers which are amount and unit relative to each other, where the unit too is a determinate amount. But if we abstract from this closer determination of them relative to each other and look at them merely with respect to what befalls them as quanta in the qualitative reference in which they here stand, then 2 and 7 are otherwise indifferent quanta; since, however, they come on the scene here only as moments, each of the other and thereby of a third (of the quantum that is called the exponent), they count forthwith not as 2 and 7 but solely according to their determinateness relative to each other. For that reason 4 and 14, or 6 and 21, etc. into infinity, can just as well be posited in their stead. With this, accordingly, they begin to have a qualitative character. Were they to count as mere quanta, then of 2 and 7 the one is simply only 2 and the other simply only 7; 4, 14, 6, 21, etc. are simply something else than those numbers, and insofar as they were merely immediate quanta the one set could not be set in the place of the other. But insofar as 2 and 7 do not count according to the determinateness of being such quanta, their indifferent limit has been sublated; from this side, accordingly, they carry the moment of infinity at them, since they not merely are no longer just they themselves, but their quantitative determinateness remains, though 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 without the value of the fraction, the determinateness which the ratio has, thereby altering.

Yet the exhibition that infinity has at a numerical fraction is still imperfect, and imperfect for this reason, that the fraction's two sides, 2 and 7, admit of being taken out of the ratio and are ordinary indifferent quanta; their reference, that of being in a ratio and being moments, is to them something external and indifferent. Their reference itself is, in like manner, an ordinary quantum: the exponent of the ratio.

The letters operated with in general arithmetic, the nearest universality into which numbers are raised, do not have the property of being of a determinate numerical value; they are only universal signs and indeterminate possibilities of every determinate value. The fraction a/b therefore seems a more fitting expression for the infinite, seeing that a and b, once 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 posited as indeterminate magnitudes, their sense is that they be some finite quantum. Being thus the universal representation, but only of the determinate number, it is likewise indifferent to them to stand in the ratio, and outside it they keep this value.

If we look still more closely at what is present in the ratio, it has the two determinations at it: first, of being a quantum, but this is, second, not as an immediate quantum but as one having the qualitative opposition at it; within the ratio that determinate, indifferent quantum at the same time abides through the fact that, having returned into itself out of its otherness, out of the opposition, it is thereby also an infinite. In the following familiar form these two determinations exhibit themselves developed in their difference from each other.

As 0,285714 … the fraction 2/7 admits of being expressed, and 1/1−a as 1 + a + a² + a³ etc. So taken, it is an infinite series; the fraction itself bears the name sum, or finite expression, of that series. Set the two expressions side by side, and the one, the infinite series, exhibits it no longer as ratio but on the side that it is a quantum as a multitude of such quanta accruing to one another, as an amount. – That the magnitudes which are to constitute it as amount consist in turn of decimal fractions and hence themselves of ratios is here beside the point; for that circumstance bears on the particular sort of unit these magnitudes have, not on them insofar as they make up 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; every one of them, however, can be expressed as such a series for a number system of another unit.

Now since in the infinite series, which is to exhibit the fraction as amount, the side of its being a ratio vanishes, there vanishes with it the side on which, as was shown just now, it 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. The series contains and exhibits the contradiction of exhibiting something that is a ratio and has qualitative nature in it as something ratioless, as a mere quantum, as amount. The upshot is that 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 wherein this determination is to be expressed. By continuing the series the amount can indeed 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 as amount something resting on qualitative determinateness is the abiding contradiction.

In this infinite series that inexactitude is actually present of which, at the genuine mathematical infinite, only the semblance occurs. Confounding these two kinds of mathematical infinite is as little permissible as confounding 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. Necessary for it, however, that form is not; quite the reverse, the infinite belonging to the infinite series differs essentially from the former, as the sequel is to show. This latter infinite even ranks below the expression of the fraction.

The infinite series, namely, harbours the bad infinity, since what the series is meant to express remains an ought, while what it does express is afflicted with a beyond that does not vanish and is different from what is to be expressed. Infinite it is not on account of the terms that are posited, but rather because they are incomplete, because the other that essentially belongs to them lies beyond them; what is there in it, be the posited terms as many as one pleases, is only a finite, in the proper sense, posited as finite, i.e. as one that is not what it ought to be. By contrast, what one calls the finite expression, or sum, of such a series is without defect; it contains in full the value that the series only seeks; the beyond has been called back from flight; there is no sundering here of what it is from what it ought to be – the two are the same.

What distinguishes the two lies more nearly to hand in this: within the infinite series the negative falls outside the terms, and these have presence in that they count only as parts of the amount. In the finite expression, on the other hand, which is a ratio, the negative is immanent, being the determinedness of the sides of the ratio through one another, a having-returned-into-self, a unity referring itself to itself, as negation of negation (both sides of the ratio are only as moments), and hence has the determination of infinity within itself. – In fact, then, what is usually called the sum, the 2/7 or the 1/1−a, is a ratio; and it is this so-called finite expression that is the genuinely infinite expression. The infinite series, conversely, is in truth sum; its purpose is to exhibit in the form of a sum what in itself is ratio, and the terms on hand 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. According to what is there in it, it is a determinate quantum, yet at the same time a lesser one than it ought to be; and then what it lacks is likewise a determinate quantum; this lacking part is in fact what is called the infinite at the series, on the merely formal side that it is something lacking, a non-being; in point of content it is a finite quantum. What is there in the series, taken together with what it lacks, is what first makes up what the fraction is, the determinate quantum that the series equally ought to be but is incapable of being. – The word infinite, even in the infinite series, is wont to pass for something lofty and exalted; this is a kind of superstition, the understanding's superstition; we have seen that it reduces rather to the determination of defectiveness.

That there are, it may still be remarked, infinite series which are not summable is, with respect to the form of series in general, an external and contingent circumstance. They contain a higher kind of infinity than the summable ones, namely 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 as such, however, which they do have, contains the same determination of bad infinity that is present in the summable series.

The inversion just noted at the fraction and at its series, with respect to the expression, also takes place insofar as the mathematical infinite – namely not the one just named but the genuine one – has been called the relative infinite, whereas the ordinary metaphysical one, by which the abstract, bad infinite is understood, has been called the absolute. In fact it is rather this metaphysical infinite that is only the relative one, because the negation it expresses stands over against a limit only in such a way that this limit goes on subsisting apart from it and 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 preeminently in the sense established above – that the so-called sum, or finite expression, of an infinite series should rather count 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 to the present development what he says on this head.

He defines the infinite, to begin with, as the absolute affirmation of the concrete existence of some nature, and 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 as reference to an other that begins outside it. Now the absolute affirmation of a concrete existence does not indeed exhaust the concept of infinity; that concept implies that infinity is affirmation not in immediate fashion but only as reinstated by way of the other's reflection into itself, that is, as negation of the negative. With Spinoza, however, substance and its absolute unity have the form of an unmoved unity, i.e. of a unity not mediating itself with itself, of a rigidity in which the concept of the negative unity of the self, subjectivity, is not yet to be found.

The mathematical example by which he elucidates the true infinite (Epist. XXIX.) is a space between two unequal circles, one of which falls inside the other without touching it, and which are not concentric. So much, it seems, did he make of this figure and of the concept whose example he took it to be, that he set it as the motto of his Ethics. – »The mathematicians, he says, conclude that the inequalities which are possible within such a space are infinite, not on account of an infinite multitude of 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 something bounded, yet for that very reason an infinite, »because the nature of the matter exceeds every determinateness,« because the determination of magnitude contained in it is at the same time not exhibitable as a quantum, or, in the Kantian expression cited above, because the synthesizing cannot be carried to completion into a – discrete – quantum. – In what way the opposition of continuous and discrete quantum leads over to the infinite generally is to be set forth in a later Remark. – That infinite of a series Spinoza calls the infinite of the imagination; the infinite as reference to itself, by contrast, 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 1/1−a is that actually, being not only what the series is in its available terms but also, beyond that, what is wanting to it, what it merely ought to be. Like Spinoza's space enclosed between the two circles together with its inequalities, the 2/7 or 1/1−a is likewise a finite magnitude; and, like that space, it admits of being made greater or smaller. But no absurdity of a greater or a smaller infinite arises from this; for this quantum of the whole is no concern of the ratio of its moments, of the nature of the matter, i.e. of the qualitative determination of magnitude; what is there in the infinite series is equally a finite quantum, but is besides something defective. – Imagining, on the other hand, halts at quantum as such and gives no thought to the qualitative reference which makes up the ground of the incommensurability present.

The incommensurability lying in Spinoza's example takes in the functions of curved lines generally, and leads on more nearly to the infinite that mathematics has introduced in connection with such functions, in connection generally with the functions of variable magnitudes, and which is the genuine mathematical, quantitative infinite that Spinoza too had in mind. This determination shall now be discussed here more closely.

As regards first of all the category of variability, held to be so important, under which the magnitudes related in those functions are brought, they are for a start supposed to be variable not in the sense in which, in the fraction 2/7, the two numbers 2 and 7 are so, 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 value posited in the fraction. Still more can any number one pleases be set in a/b in the place of a and b, without altering what a/b is meant to express. Only in the sense that in the place of x and y of a function too an infinite, i.e. inexhaustible, multitude of numbers can be set, are a and b variable magnitude just as much as those, x and y. The expression variable magnitudes is on that account very vague, and unhappily chosen for determinations of magnitude that have their interest and their mode of treatment in something quite other than their bare variability.

In order to make plain wherein the genuine determination of the moments of a function lies, the moments with which the interest of the higher analysis is occupied, we must run once more through the stages already brought to notice. Within 2/7 or a/b, 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 outside the ratio too remain what they are. Further, 2/7 and a/b are also a fixed quantum, a quotient; the ratio makes up an amount whose unit the denominator, and the amount of these units the numerator, expresses – or the reverse; even if 4 and 14, etc. step into the place of 2 and 7, the ratio remains, as quantum too, the same. This alters essentially, however, in the function y²/x = p, for instance; here 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, x and y, are, first, not merely no determinate quanta, but, second, their ratio is not a fixed quantum (nor is any such thing here meant, as it was with a and b), not a fixed quotient, but is as quantum simply variable. This, however, is contained solely in the fact that x stands in a ratio not to y but to the square of y. The ratio of a magnitude to the power is not a quantum but essentially a qualitative ratio; the power-ratio is the circumstance that is to be regarded as the fundamental determination. – In the function of the straight line y = a x, however, y/x = a is an ordinary fraction, an ordinary quotient; such a function is thus one of variable magnitudes only formally; here x and y stand as a and b do in a/b, and they do not occupy the determination under which the differential and integral calculus regards them. – Owing to the particular nature of the variable magnitudes in this way of considering them, it would have been to the purpose to introduce for them a particular name as well as designations other than the usual ones for the unknown magnitudes in any finite equation, determinate or indeterminate; and this for the sake of their essential diversity from such merely unknown magnitudes, which in themselves are perfectly determinate quanta, or a determinate range of determinate quanta. – Again, it is only for want of consciousness regarding the peculiarity of what makes up the interest of the higher analysis and has brought on both the need for the differential calculus and its invention, that functions of the first degree, such as the equation of the straight line, get drawn for their own sake into the treatment of this calculus; a further share in such formalism falls to the misunderstanding which supposes that the demand, correct in itself, for the generalization of a method is met by having the specific determinateness on which the need is founded left out, so that it passes for settled that in this field only variable magnitudes in general are at issue. Much formalism in the consideration of these objects, and in their treatment as well, would surely have been spared had it been seen that what is at stake is not variable magnitudes as such but determinations of powers.

But there is yet a further stage at which the mathematical infinite steps forth in its peculiarity. In an equation in which x and y are posited as determined in the first instance through a power-ratio, x and y as such are still to mean quanta; this meaning, however, is now lost altogether and completely in the so-called infinitely small differences. dx, dy are quanta no longer, nor are they supposed to signify such, but have a signification solely in their reference, a sense merely as moments. No longer are they something – something taken as quantum – nor finite differences; but they are also not nothing, not the determinationless zero. Outside their ratio they are pure zeros, yet they are to be taken only as moments of the ratio, as determinations of the differential coefficient dx/dy.

In this concept of the infinite, quantum has been genuinely completed into a qualitative existence; posited it is as actually infinite; sublated it is not merely as this or that quantum, but as quantum in general. There remains, however, the determinateness of quantity as element of quanta, as principle, or, as it has also been put, in its first concept.

It is against this concept that every attack has been directed which has been made upon the fundamental determination of the mathematics of this infinite, upon the differential and integral calculus. Incorrect representations on the part of the mathematicians themselves occasioned its going unrecognized; chiefly, however, the blame for these contestations lies with the incapacity to justify the subject matter as concept. Yet mathematics, as was recalled above, cannot here get around the concept; for being mathematics of the infinite, it does not restrict itself to that finite determinateness which its objects have – whereas in pure mathematics space and number, with their determinations, get considered and related to one another only according to their finitude –; it rather transposes a determination taken up from that quarter and treated by it into identity with its opposite, as when it turns a curved line into a straight, a circle into a polygon, and so on. The operations which it allows itself as differential and integral calculus therefore contradict outright the nature of merely finite determinations and their references, and would for that reason have their justification in the concept alone.

If the mathematics of the infinite held fast to this, that those determinations of quantity are vanishing magnitudes, i.e. 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, at Remark 4 to the category of becoming. Certainly the unity of being and nothing is no state; a state would be some determination of being and of nothing, one into which these moments were supposed to have slipped merely by accident, as into a sickness or an external affection, through an erroneous thinking; this middle and unity – the vanishing, or equally the becoming – is rather their truth and theirs alone.

It has further been said that what is infinite is not comparable as a greater or a smaller; that there can therefore be no ratio of infinites to infinites, nor orders or dignities of the infinite, whereas differences of the infinite differences do occur in the science of them. – Underlying this already mentioned objection 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 one another any more. On the contrary, what is only in the ratio is no quantum; quantum is a determination of the kind that outside its ratio is supposed to have a perfectly indifferent existence, 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 accordingly not merely comparable; they are only as moments of the comparison, of the ratio.

I adduce the most important determinations that have been given in mathematics concerning this infinite; from them it will be evident that the thought of the matter lies at their base, in agreement with the concept developed here, but that their authors did not fathom it as concept and, in applying it, again stood in need 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 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 forms. Newton explains these fluxions (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 among them, and one that contains the concept of a quantum determinate in itself – but vanishing divisibles. Further: not sums and ratios of determinate parts, but rather the limits (limites) of sums and of ratios. The objection is raised, he says, 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 the ratio of vanishing magnitudes is to be taken as the ratio neither before they vanish nor afterwards, but that 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; but that this or that is 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 possesses inner truth. Yet what has been adduced shows that the concept Newton set up 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, i.e. such as are quanta no longer; further, no ratios of determinate parts, but rather the limits of the ratio. 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, and its sides likewise as qualitative moments of quantity. – Newton adds that from this, that there are 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 abstract ratio over to such of its sides as would be supposed to have a value for themselves outside their reference, as indivisibles, as something that would be a one, a ratioless thing.

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, i.e. 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 set aside. Yet neither the decreasing without limit, into which Newton transposes 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 determination demanded developed itself further into the concept of a determination of magnitude that is purely and only moment of the ratio.

In view of the preservation of the ratio amid the vanishing of the quanta one meets (elsewhere, as in Carnot, Reflexions sur la Métaphysique du Calcul Infinitésimal.) with the statement that by virtue of the law of continuity the vanishing magnitudes retain, before they vanish, the ratio out of which they come. – This representation expresses the true nature of the matter, provided that what is meant is not the continuity that quantum has in the infinite progress, of continuing itself into its vanishing in such a way that in the beyond of it there arises once again only a finite quantum, a new term of the series; a continuous advance, however, is always so represented that the values run through are ones which are still finite quanta. In that transition, by contrast, which is made into the genuine infinite, it is the ratio that is the continuous; so thoroughly continuous and self-preserving is it, that the transition consists rather solely in this, in lifting the ratio out purely and making the ratioless determination vanish, i.e. the determination that a quantum which is a side of the ratio, posited also outside this reference, is still quantum. – This purification of the quantitative ratio is to that extent nothing other than what occurs when an empirical existence is comprehended. By this it is so raised above itself that its concept contains the same determinations as it does itself, but grasped in their essentiality and taken up into the unity of the concept, wherein they have lost their indifferent, conceptless subsistence.

Equally interesting is the other form of the Newtonian presentation of the magnitudes in question, namely as generating magnitudes or principles. A magnitude that has been generated (genita) is a product or a quotient, roots, rectangles, squares, and also sides of rectangles and squares; – in general a finite magnitude. – »Considering it as variable, as increasing or decreasing in continual motion and flux, he understands its momentaneous 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 out of moments; what are to be understood are rather the becoming principles or beginnings of finite magnitudes.« – Quantum is here distinguished from itself, as it is qua product or existent thing, and as it is in its becoming, 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 quantitative differences – the infinite increments or decrements – count as moments merely; only that which has become is the quantum, the thing gone over into the indifference of existence and into externality. – But if these determinations of the infinite, adduced with respect to increments or decrements, must be acknowledged by the philosophy of the genuine concept, it is at once to be remarked as well that the forms of increments and the like themselves come within the category of the immediate quantum and of that continuous advance mentioned, and that the representations of increment, accretion, increase of x by dx or i and so forth are rather to be regarded as the fundamental evil present in the methods; – as the abiding obstacle to lifting purely out of the representation of ordinary quantum the determination of the qualitative moment of quantity.

Far behind the determinations given stands the representation of infinitely small magnitudes, a representation lurking in the increment or decrement itself as well. According to it they are supposed to be of such a constitution that not only they by comparison with finite magnitudes, but also their higher orders by comparison with the lower, or again the products of several by comparison with a single one, are to be neglected. – In Leibnitz the demand for this neglect, which the earlier inventors of methods bearing on this magnitude likewise let come into play, stands out more strikingly. It is chiefly this demand that, along with the gain in convenience, lends this calculus the semblance of inexactitude and of express incorrectness in the course of its operation. – Wolf sought to make it intelligible in his own way of popularizing things, i.e. of adulterating the concept and putting incorrect sensuous representations in its place. He compares, namely, the neglect of the infinite differences of higher orders by comparison with the lower to how a surveyor proceeds who, when he measures the height of a mountain, would have been no less exact if the wind had meanwhile blown a grain of sand from the summit, or to the neglect of the heights of houses and towers when lunar eclipses are computed (Element. Mathes. univ. Tom. I. El. Analys. math. P. II. C. I. s. Schol.).

Though the fair-mindedness of common human understanding permits an inexactitude of this sort, all geometers have by contrast rejected the 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 wholly distinct from land-surveying, from the measuring of empirical lines, figures, and so forth. Besides, as was cited above, the analysts show by comparing the result as obtained on the strictly geometrical path with the result obtained by the method of infinite differences that the one comes out the same as the other, and that a more or a less of exactitude has no place at all. And it goes without saying that an absolutely exact result could not come from a procedure that were inexact. Yet again, on the other side, the procedure itself cannot do without that neglect on the ground of insignificance, protests against the manner of justification adduced notwithstanding. And this is the difficulty around which the efforts of the analysts revolve, the effort to make comprehensible the absurdity lying herein, and to remove it.

In this respect it is above all Euler's representation that is to be adduced. Laying the general Newtonian definition at the base, he presses the point that what the differential calculus considers are the ratios of the increments of a magnitude, while the infinite difference as such must be viewed wholly as zero (Institut. Calc. different. P. I. C. III.). – How this is to be understood lies in what precedes; the infinite difference is zero only of the quantum, not a qualitative zero, but, as zero of the quantum, rather pure moment only of the ratio. It is not a difference by a magnitude; yet for that very reason it is, on the one hand, altogether awry to enunciate those moments which are called infinitely small magnitudes also as increments or decrements and as differences. Underlying this determination is the idea that to the finite magnitude present at the start something is added or from it something subtracted, that a subtraction or an addition takes place, an arithmetical, external operation. The transition from the function of the variable magnitude into its differential, however, is to be viewed as being of a quite other nature, namely, as was discussed, as a leading back of the finite function to the qualitative ratio of its determinations of quantity. – On the other hand, the awry side strikes one of itself 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 therefore 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, that of qualitative determinations of quantity which, were they to be torn out of the ratio and taken as quanta, would be nothing but zeros. – Lagrange (Théorie des fonct. analyt. Introd.) judges of the representation of limits or last ratios that, however well one may represent to oneself two magnitudes' ratio while 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. – What Euler (at the place cited §. 84. ff.) further adds concerning the determination given, in order to show that two so-called infinitely small magnitudes, which are supposed to be nothing else than zeros, nevertheless have a ratio to each other and that for this reason not the sign of zero but other signs are in use for them, cannot be regarded as sufficient. He would ground this on the difference between arithmetical and geometrical ratio; in the former we look to the difference, in the latter to the quotient, and though the former, between two zeros, is equal, the geometrical is not equal on that account; if 2 : 1 = 0 : 0, then, by the nature of the proportion, since the first member is twice the second, the third member too must be twice the fourth; 0 : 0 is accordingly to be taken, on the strength of the proportion, as the ratio of 2 : 1 itself. – In common arithmetic too n · 0 = 0; hence n : 1 = 0 : 0. – Yet precisely because 2 : 1 or n : 1 is a ratio of quanta, there corresponds to it neither a ratio nor a designation of 0 : 0.

I refrain from multiplying the citations, since those considered have shown well enough that the genuine concept of the infinite does indeed lie in them, but that it has not been lifted out and grasped in its determinateness. Hence, when the advance is made to the operation itself, it cannot come about that the genuine conceptual determination should make itself count in it; the finite determinateness of quantity rather returns, and the operation cannot do without the representation of a merely relatively small. The calculus renders it necessary to subject the so-called infinite magnitudes to the ordinary arithmetical operations of adding and the like, 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 for dragging them down on the one occasion into this sphere and treating them as increments or differences, and for neglecting them, on the other side, as quanta, after having just applied to them forms and laws of finite magnitudes.

Of the attempts of the geometers to remove these difficulties I adduce still the principal ones.

The older analysts made fewer scruples over the matter; 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 (– expressions of Lagrange –) in mathematics. Yet since the principle of the analysis of the infinite is of higher nature than that on which the mathematics of finite magnitudes rests, the former had at once, of itself, to renounce that sort of evidentness, just as philosophy too can lay no claim to the kind of distinctness that the sciences of the sensuous possess, natural history for example, and just as eating and drinking pass for a more intelligible business than thinking and comprehending. What is at issue, accordingly, will only be the effort to attain the rigour of the proofs of the ancients.

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 speaks, for example, 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. – This is, to be sure, a procedure having something answering to that from which Descartes' method of tangents starts out, a method still to be mentioned more closely further on. This much, it may be remarked here, is at once evident in general: that the procedure of assuming different values of the variable magnitudes and afterwards setting them equal belongs to another circle of mathematical treatment than does the method of the differential calculus itself, and that the peculiarity of the simple ratio, to be discussed more closely later on – the ratio to which its actual concrete determination leads back, namely that of the derived function to the original – is not lifted out.

The elder among the moderns, such as Fermat, 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, Euler among them, have always openly believed themselves entitled to drop the products of infinite differences, and likewise their higher powers, on no other ground than that these vanish relatively over against the lower order. On this alone rests, with them, the fundamental theorem, namely the determination of what the differential of a product or of a power is, for to this the whole theoretical doctrine reduces itself. The rest is partly mechanism of development, partly application, into which, however, as is to be considered further on, the higher or rather the sole interest in fact falls. – With regard to the present matter, only the elementary point is to be adduced here, that on the same ground of insignificance the main proposition concerning curves is assumed, namely that the elements of the curves, that is to say the increments of the abscissa and of the ordinate, have to each other the ratio of the subtangent and the ordinate; with a view to obtaining similar triangles, the arc, which makes up a triangle's third side alongside the two increments – the triangle formerly and with good right called the characteristic one – counts as a straight line, as a piece of the tangent, and one of the increments accordingly as reaching right up to the tangent. These assumptions raise those determinations, on the one hand, above the nature of finite magnitudes; on the other hand, however, a procedure is applied to the moments now called infinite that holds only of finite magnitudes and in which nothing may be neglected out of regard for insignificance. With a manner of proceeding of this sort the difficulty pressing upon the method remains in its full force.

A remarkable procedure of Newton's is to be adduced here; (Princ. Math. phil. nat. Lib. II. Lemma II. Propos. VII.) – the invention of an ingenious artifice for getting 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 – whence the differentials of quotients, powers and so forth then follow easily – he finds in the following manner. The product, if x and y are each taken smaller by half of its infinite difference, passes over into xy – xdy/2 – ydx/2 + dxdy/4; but if one lets x and y increase by just as much, into xy + xdy/2 + ydx/2 + dxdy/4. Now with the first product subtracted from this second one, ydx + xdy remains as excess, and this, he says, is the excess of the growth by a whole dx and dy, for it is by this growth that the two products are distinguished; it is therefore the differential of xy. – One sees that in this procedure the term making up the chief difficulty, the product of the two infinite differences, dxdy, falls away of itself. But the Newtonian name notwithstanding, it must be permitted to say that such an operation, elementary though it is, is incorrect; it is incorrect that (x + dx/2) (y + dy/2) – (x – dx/2) (y – dy/2) = (x + dx) (y + dy) – xy. Only the need to give the calculus of fluxions, in view of its importance, a foundation could have brought a Newton to practise upon himself the deception of such a proof.

Other forms which Newton employs in deriving the differential are tied to concrete significations of the elements and their powers, significations bearing on motion. – In his use of the series-form, which otherwise distinguishes his method, it lies too near at hand to say that it always stands in one's power, by adding further terms, to take the magnitude as exactly as one requires, and that the terms dropped are relatively insignificant, the result on the whole being only an approximation, for him not to have contented himself with this ground here as well, just as in his method for solving equations of higher degrees by approximation he drops, on the crude ground of their smallness, the higher powers arising when each value found, still inexact, is substituted into the given equation; see Lagrange Equations Numériques p. 125.

That error into which Newton lapsed when solving a problem, through dropping essential higher powers, an error that gave his opponents the occasion for a triumph of their method over his, and whose true origin Lagrange has exhibited in his recent investigation of it (Théorie des fonct. analyt. 3me P. Ch. IV.), proves the formalism and the uncertainty still present in the use of that instrument. Lagrange shows that Newton slipped into the error because he neglected the term of the series containing the power upon which everything turned in the determinate problem. Newton had held to that formal, superficial principle of dropping terms on account of their relative smallness. – It is well known, namely, that in mechanics the terms of the series developing the function of a motion are given a determinate signification, such that the first term, or first function, has reference to the moment of velocity, the second to accelerating force, the third to resistance of forces. Here, accordingly, the terms of the series are to be viewed not merely as parts belonging to 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.¹⁰ The Newtonian solution contained that error, not because terms of the series were in it dropped merely qua parts of a sum, but because no heed was paid to the term containing the qualitative determination upon which everything turned.

In this example it is the qualitative sense on which the procedure has been made to depend. In connection with this the general assertion may at once be advanced that the entire difficulty of the principle would fall away if, in place of the formalism of putting the determination of the differential merely into the task that gives it its name – that of [specifying] the difference in general of a function from its alteration after its variable magnitude has received an increment – the qualitative meaning of the principle were specified and the operation were made dependent upon this. In this sense the differential of xⁿ shows itself entirely exhausted by the first term of the series that results from the development of (x + dx)ⁿ. That the remaining terms go unheeded does not, on this view, come from their relative smallness; – no inexactitude, no fault or error is presupposed here that would be compensated and corrected by another error, a view from which Carnot above all justifies the ordinary method of the infinitesimal calculus. Since it is not a sum but a ratio that is at issue, the differential is found completely through the first term; and where there is need of further terms, of differentials of higher orders, their determination involves not the continuation of a series as sum but the repetition of one and the same ratio – the ratio alone being wanted, and it stands already perfectly determined in the first term. Whatever need there is for the form of a series, for summing it, and whatever hangs together with that, must then be held entirely apart from that interest of the ratio.

The elucidations which Carnot gives concerning the method of infinite magnitudes contain in the most refined form, and exposed with the utmost clarity, what occurred in the representations cited above. But at the transition to the operation itself the ordinary representations, of the infinite smallness of the dropped terms relative to 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 that accrues, for simplification and abbreviation of the calculus, from introducing imperfect equations, as he calls them, i.e. such as have had an arithmetically incorrect omission made in them, 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 carries with it, as well as of those carried by the method of first and last ratios and limits. Of his calculus of functions, whose other merits with respect to precision, abstraction and universality are recognized well enough, only this is to be adduced as belonging here, that it rests on the fundamental theorem that the difference, though it does not become zero, can be assumed so small that every term of the series exceeds in magnitude the sum of all those following. – Here too the start is made from the categories of increment and of difference of the function whose variable magnitude receives the increment, whereby the burdensome series of the original function comes in; just as in the sequel the terms of the series to be dropped come into consideration only in the respect that they constitute a sum, and the ground for dropping them is placed in the relativity of their quantum. Here too, then, the omission is not for the general case led back to the point of view which occurs partly in some applications where, as was recalled earlier, the series' terms are supposed to carry a determinate qualitative signification and terms are left out of account, not on the score of being insignificant in magnitude, but on the score of being insignificant in point of quality; while partly the omission itself then falls away in the essential point of view which, for the so-called differential coefficient, first emerges determinately in Lagrange's so-called application of the calculus, as the Remark that follows will set forth at greater length.

The qualitative character in general, which has here been shown at the form of magnitude in question in what is thereby called the infinitely small, is met with most immediately in the category of the limit of the ratio adduced above, whose implementation in the calculus has been stamped as a method of its own. Of what Lagrange judges of this method, that it wants ease in application and that the expression limit offers no determinate idea, we shall take up the second point here and look more closely at what is set up regarding its analytic meaning. In the representation of the limit there does indeed lie the genuine category, indicated above, of the qualitative determination of ratio of the variable magnitudes, for the forms in which these come in, dx and dy, are to be taken purely and only as moments of dy/dx, while dx/dy itself is to be regarded as a single indivisible sign. That there is thereby lost, for the mechanism of the calculus and especially in its application, the advantage which it draws from having the sides of the differential coefficient separated off from each other, is to be set aside here. That limit is now supposed to be limit of a given function; – it is to specify a certain value in relation to that function, a value determined by the manner of the derivation. With the bare category of limit, however, we should be no further than with what has been at issue in this Remark, namely to show that the infinitely small occurring in the differential calculus as dx and dy carries not merely the negative, empty sense of a magnitude that is not finite, not given – as when one says an infinite multitude, on into infinity and the like – but the determinate sense of the qualitative determinateness of the quantitative, of a moment of ratio as such. This category, however, taken so, still has no relation to what a given function is, and does not of itself reach into the treatment of such a function or into a use to be made of that determination at it; thus the representation of limit too, held back within this determinateness demonstrated of it, would lead to nothing. But the expression limit itself already contains this, that it is limit of something, i.e. that it gives voice to a certain value lying in the function of variable magnitude; and it is to be seen how this concrete dealing with it is constituted. – It is to be the limit of that ratio which the two increments bear to each other, the increments by which the two variable magnitudes, joined in an equation and the one of them regarded as function of the other, have been assumed to be increasing; – the increment is here taken indeterminately in general, and to that extent no use is made of the infinitely small. But in the first place the path to finding this limit brings on the same inconsistencies as lie in the other methods. This path, namely, is the following. If y = fx, then fx, when y passes over into y + k, is supposed to alter into fx + ph + qh² + rh³ etc.; hence k = ph + qh² etc. and k/h = p + qh + rh² etc. Now when k and h vanish, the second term vanishes apart from p, and this p is now supposed to be the limit of the ratio of the two increments. One sees that h as quantum is posited = 0, but that k/h is on that account not supposed to be at the same time = o/o but to remain still a ratio. The advantage of warding off the inconsistency lying herein is now supposed to be afforded by the representation of the limit; p is supposed at the same time not to be the actual ratio, which would be = o/o, but only the determinate value which the ratio can approach infinitely, i.e. so approach that the difference can become smaller than any given one. The more determinate sense of approximation with regard to what is properly supposed to approach what will be considered below. – But that a quantitative difference which has the determination not merely of being able to be smaller than any given one, but of being obliged to be so, is a quantitative difference no longer, this is clear of itself, as evident as anything in mathematics can be evident; with this, however, nothing beyond dy/dx = o/o has been reached. If, on the contrary, dy/dx = p is assumed, i.e. as a determinate quantitative ratio, as is in fact the case, then conversely the presupposition which has posited h = 0 lands in embarrassment, a presupposition through which alone k/h = p is found. But if one grants that k/h = 0, and with h = 0 in fact k = 0 of itself as well – for the increment k to y takes place only on the condition that the increment h is there – then it would be for one to say what p is then supposed to be, p being an altogether determinate quantitative value. On this the simple, dry answer offers itself at once, that it is a coefficient, and from what derivation it arises, – the first function of an original function, derived in a certain determinate manner. Were one to content oneself with that, as Lagrange has in fact contented himself with it so far as the matter goes, then the general part of this science of the differential calculus – and immediately this form of it which goes by the name theory of limits – would be freed from the increments, then from their infinite or arbitrary smallness, from the difficulty of getting rid again – apart from the first term, or rather only the coefficient of the first term – of the further terms of a series, terms which through the introduction of those increments unavoidably present themselves; and purified besides of what further hangs together with this, of the formal categories above all of the infinite, of infinite approximation, and of the further categories, here equally empty, of continuous magnitude¹¹ and of such as one otherwise deems necessary, like striving, becoming, occasion of an alteration. But then it would be required to show what meaning and worth p has, apart from the dry determination, quite sufficient for the theory, that it is nothing further than a function derived from the development of a binomial, i.e. what connection and use it has for further mathematical need; and of this the second Remark is to treat. – What follows here first, however, is the exposition of the confusion which has been brought into the apprehending of the proper, qualitative determinateness of the ratio, the thing at issue in the first place, through the cited use, so current in the presentations, of the representation of approximation.

It has been shown that what the so-called infinite differences express is the vanishing of the ratio's sides qua quanta, and that what stays behind is their ratio of quantity, purely insofar as this is determined in qualitative fashion; so little is the qualitative ratio lost in this that it is rather what results precisely through the conversion of finite magnitudes into infinite ones. In this, as we have seen, consists the entire nature of the matter. – Thus in the last ratio the quanta of the abscissa and the ordinate, for example, vanish; but the sides of this ratio remain essentially the one element [of the] ordinate, the other element of the abscissa. Where the manner of representation is used according to which one lets the one ordinate approach the other infinitely, then the ordinate previously distinguished passes over into the other ordinate, the abscissa previously distinguished into the other abscissa; essentially, however, no passing over of ordinate into abscissa, or of abscissa into ordinate, takes place. The element of the ordinate – to stay with this example of variable magnitudes – is to be taken not as the difference of one ordinate from another ordinate; it is rather the difference, 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.

This constitution of the matter is, however, obscured by the fact that what has just been called element, of the ordinate for instance, gets grasped as difference or increment in such fashion that it is supposed to be nothing but the difference of the quantum of one ordinate between the quantum of another ordinate. The limit has here, accordingly, not the sense of the ratio; it counts only as the last value which another magnitude of like kind constantly so approaches 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 the hovering of a difference of quantum from quantum, while the qualitative nature – whereby dx is essentially a determination of ratio not over against x but over against dy – recedes in the representation. One lets dx² vanish over against dx, but far more does dx vanish over against x, and this truly means: it has only a ratio to dy. – In presentations of this sort the geometers are always concerned above all to make the approximation of a magnitude to its limit comprehensible, and to keep hold of this side of quantum's difference from quantum, of how it is no difference and yet a difference still. But approximation is in any case a category for itself which says nothing and makes nothing comprehensible; dx has approximation already behind it, it is neither near nor a nearer; and infinitely near itself means the negation of nearness and of approaching.

Now since it has come about that the increments or infinite differences have been considered only on the side of the quantum that vanishes in them, and only as limit of the quantum, they are grasped in this way as ratioless moments. There would follow from this the inadmissible representation that it is permitted, in the last ratio, to set abscissa and ordinate, say, or even sine, cosine, tangent, versed sine and whatever else, equal to one another. – At first this representation seems to prevail wherever an arc gets treated as tangent; for the arc too is surely incommensurable with the straight line, and its element, in the first instance, of a quality other than that of the straight line's element. More absurd still and more impermissible it seems than the confounding of abscissa, ordinate, versed sine, cosine and so forth, when quadrata rotundis, when a part of the arc, infinitely small though it be, is taken for a piece of the tangent and thus treated as a straight line. – This treatment, however, is essentially to be kept apart from the confounding just censured; its justification lies in this, that within the triangle whose sides are the element of an arc together with the elements of its abscissa and of its ordinate, the ratio is the same as it would be were that arc-element the element of a straight line, of the tangent; and the angles making up the essential ratio – i.e. the ratio left over to these elements once one abstracts from the finite magnitudes that belong to them – are one and the 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. Since by its definition the straight line is the shortest path between two points, 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 vanishes its difference from the curved line, a difference resting merely on the difference of quantum. – Thus, as infinite, straight line and arc retain no quantitative ratio, and with it, on the strength of the definition assumed, no qualitative diversity from each other either, but the former rather passes over into the latter.

Akin to the equating of heterogeneous determinations, yet at the same time distinct from it, is the assumption, for itself indeterminate and entirely indifferent, that infinitely small parts of the same whole are equal to one another; applied, however, to an object heterogeneous within itself, i.e. afflicted with an essential non-uniformity of the determination of magnitude, it brings forth that peculiar inversion contained in the proposition of higher mechanics, that in equal, and indeed infinitely small, times infinitely small parts of a curve are traversed in uniform motion, seeing that this gets asserted of a motion in which, within equal finite, i.e. existing, parts of time, finite, i.e. existing, unequal parts of the curve are traversed, i.e. therefore of a motion which as existing is non-uniform and is assumed to be so. This proposition is the expression in words of what an analytic term is supposed to signify, a term arising in the development, cited above as well, of the formula of a motion that is non-uniform though otherwise conformable to a law. Older mathematicians sought to express in words and propositions the findings of the newly invented infinitesimal calculus, which in any case always had to do with concrete objects, and to exhibit them in geometrical figures, essentially in order to use them for the theorems in the ordinary manner of proof. The terms of a mathematical formula into which analytic treatment had broken up the magnitude of the object – of motion, say – there acquired an objective signification, that of velocity, accelerating force and so forth; according to such signification they were supposed to yield correct propositions, physical laws, and, following the analytic connection, their objective linkages and ratios too were supposed to be determined, as for example precisely that in a uniformly accelerated motion a particular velocity proportional to the times exists, but that besides this an increment coming from the force of gravity is always added. Such propositions are in the modern, analytic shape of mechanics adduced throughout as findings of the calculus, without concern whether they have a real sense for themselves at them, i.e. one to which a concrete existence would correspond, and without concern for a proof of such; the difficulty of making comprehensible the connection of such determinations when they are taken in the pronounced real sense, for example the transition from that merely uniform velocity to a uniformly accelerated one, passes for being entirely removed by the analytic treatment, in which such connection is a simple consequence of the by now settled authority of the operations of the calculus. It is given out as a triumph of science to find, by mere calculus, beyond experience, laws, i.e. propositions of concrete existence which have no concrete existence. In the earlier, still naive period of the infinitesimal calculus, however, a real sense for themselves was to be specified for those determinations and propositions, represented in geometrical figures, and made plausible, and in such a sense they were to be applied to the proof of the principal theorems that were at issue (– see the Newtonian proof of his fundamental theorem of the theory of gravitation in the Princ. mathem. philosophiae naturalis lib. I. Sect. II. Prop. I. compared with Schubert's Astronomy (first ed. III. B. §. 20.), where it is conceded that matters do not stand exactly so, i.e. that at the point which is the nerve of the proof they do not stand as Newton assumes –).

It will not be deniable that in this field much has been put up with as proof, chiefly with the aid of the fog of the infinitely small, on no other ground than that what came out was always known beforehand anyway, and that the proof, contrived so that it should come out, at least brought off the semblance of a scaffolding of proof; – a semblance that was still preferred to mere faith or to knowledge from experience. For my part, however, 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 such even Newtonian proofs, in particular those belonging to the one just cited, on account of which Newton has been exalted to the skies and above Keppler for having set out mathematically what the latter found merely through experience.

It was in order to prove physical laws that the empty scaffolding of such proofs got erected. But determinations of magnitude in physics, insofar as these are laws having the qualitative nature of the moments for their ground, mathematics is simply not in a position to prove; and this on the simple ground that this science is not philosophy and does not proceed from the concept, and that the qualitative therefore, insofar as it is not taken up lemmatically from experience, lies outside its sphere. The assertion of the honour of mathematics, that all propositions occurring in it should be rigorously proved, often made it forget its limit; thus it seemed against its honour to acknowledge, for propositions of experience, simply experience as source and as sole proof; later the consciousness of this became more cultivated; but until such consciousness gets clear about the difference between what is mathematically provable and what can only be taken from elsewhere, as about what are merely terms of an analytic development and what are physical concrete existences, scientific rigour cannot form itself into a strict and pure bearing. – That scaffolding of Newtonian proving, however, will doubtless yet meet with the same justice that has been done to another groundless Newtonian edifice of artifice built from optical experiments and the inferring bound up with them. Applied mathematics remains full of a similar brew of experience and reflection; yet just as, with that optics, one part after another has for a good while now begun to be factually ignored in science – with the inconsistency, however, of still letting the rest stand although it contradicts this – so too it is a fact that part of those deceptive proofs has already fallen of itself into oblivion or been replaced by others.

Remark 2. The Purpose of the Differential Calculus Derived from Its Application

In the previous Remark there came under consideration partly the conceptual determinateness of the infinitely small that the differential calculus employs, partly the basis on which it was brought into that calculus; both are abstract determinations and for that very reason easy enough in themselves; what is called the application, by contrast, presents both the greater difficulties and the side of greater interest; and the elements belonging to this concrete side shall be what the present Remark deals with. – The whole method of the differential calculus is dispatched in the proposition that dxⁿ = nxⁿ⁻¹ dx, or (f (x + i) − fx)/i = P, which is to say, that it equals the coefficient standing at the first term when the binomial x+d, x+i is expanded by the powers of dx or i. Nothing further needs to be learned; mechanically there follows from it the derivation of the forms lying nearest to hand, that of the differential of a product, of an exponential magnitude, and onward; within little time, perhaps within half an hour – and the converse operation, the recovery of the original function out of the differentials, integration, comes along with the finding of them – the whole theory can be had in one's possession. What alone holds one up longer is the effort to see, and to make intelligible, that once the one circumstance of the task, the finding of that coefficient, has been so easily brought about in an analytic, that is, wholly arithmetical way, through the expansion of the function of the variable magnitude after this has received the form of a binomial, the other circumstance too, namely the dropping of the remaining terms of the resulting series apart from the first, is equally in order. Were it the case that one had need of that coefficient alone, then, as was said, everything touching the theory would be settled with the determination of it in less than half an hour, and the dropping of the further terms of the series would make so little difficulty that rather of them, as terms of the series (as second, third and so forth. Functions, their determination is already accomplished along with the determination of the first as well), there would be no mention at all, since they are not in the least what the business is about.

The remark may be sent on ahead that one sees at once, in the method of the differential calculus, that it was not invented and set up for its own sake; not only does it lack a grounding of its own as another mode of analytic procedure, but the violence of simply dropping terms that issue from the expansion of a function – when the whole of this expansion is nevertheless assumed to belong completely to the matter at hand, since the matter is regarded as the difference between the expanded function of a variable magnitude, once this has been given the shape of a binomial, and the original function – runs flatly counter to every mathematical principle. Both the need for such a way of proceeding and the justification it lacks in its own self point at once to the fact that origin and foundation must be located elsewhere. Elsewhere in the sciences it happens likewise that whatever gets placed at the head as elementary, whatever the propositions of a science are then meant to follow from, carries no evidence of its own, and shows itself instead to owe its occasion and its grounding to what comes later. What the history of the differential calculus records makes plain that it took its beginning chiefly in the various so-called tangential methods, with the matter standing there, as it were, in the guise of tricks; the manner of proceeding, once it had been extended to further objects as well, was afterwards brought to consciousness and into abstract formulas, which people then also tried to raise into principles.

What has been exhibited as the conceptual determinateness of the so-called infinitely small is the qualitative determinateness of quantity in magnitudes that are in the first instance posited as quanta in ratio to one another, and to this attached itself the empirical investigation whether that conceptual determinateness can be shown in the descriptions or definitions of the infinitely small that are to be met with, insofar as it is taken as infinite difference or the like. – All this happened only in the interest of abstract conceptual determinateness as such; the further question would be how the transition from it to mathematical shape and application is constituted. To that end the theoretical side, the conceptual determinateness, is first to be pursued further, and it will prove not altogether barren in its own right; then the relation of it to the application is to be considered, and in both cases it is to be shown, so far as that is feasible here, that the general consequences are at the same time adequate both to what the differential calculus is about and to the manner in which it accomplishes it.

To begin with, it is to be recalled that the form which the conceptual determinateness under discussion has in the mathematical domain has already been indicated in passing. The qualitative determinateness of the quantitative was first exhibited in the quantitative ratio in general, but already in the exposition of the distinct so-called arithmetical operations (see the Remark bearing on this) it was anticipated that it is the ratio of powers, still to be considered later at its own proper place, in which number, through the equating of its conceptual moments, unit and amount, is posited as having returned into itself, and thereby wins in it the moment of infinity, of being-for-itself, which is to say, of determinateness through its own self. The expressly qualitative determinateness of magnitude accordingly refers, as has likewise been recalled already, essentially to determinations of powers, and since the differential calculus has as its specific business to operate with qualitative forms of magnitude, its proper mathematical object must be the treatment of forms of powers, and every problem together with its solution, on whose account the differential calculus gets employed, bears witness that determinations of powers as such, and the handling of them, are where the whole interest resides.

Important as this foundation is, and much as it sets something determinate at the head straightaway where otherwise stand the merely formal categories of variable, continuous or infinite magnitudes and their kin, or indeed of functions at large, it remains too general; other operations have to do with it just as much; already the raising to a power and the extraction of roots, then the treatment of exponential magnitudes and logarithms, series, equations of higher orders, have their interest and their labour solely with ratios that rest upon powers. Doubtless these must together make up a system for the treatment of powers; but which among the various ratios into which determinations of powers can be put is the one that is the proper object and interest for the differential calculus, this is to be gathered from the calculus itself, that is, from its so-called applications. These are in fact the matter itself, the actual procedure in the mathematical solution of a certain range of problems; this procedure came earlier than the theory or general part, and the same thing came to be called application only with reference to the theory created afterwards, which meant partly to set out the general method of the procedure and partly to furnish it with principles, that is, with a justification. What a futile labour it has been to hunt up, for the received way of grasping the procedure, principles that would really resolve the contradiction which comes to light in it, instead of merely excusing or concealing that contradiction by pleading the insignificance of what the mathematical procedure requires but what here has to be dropped, or by pleading the possibility, which comes to the same thing, of an infinite or arbitrarily close approximation and the like, has been shown in the previous Remark. If the general character of the procedure were abstracted from the actual part of mathematics that is called the differential calculus otherwise than has hitherto been done, those principles and the labour spent upon them would show themselves to be dispensable as well, just as in their own selves they prove to be something skewed and left standing in contradiction.

If we track down this distinctive feature by simply taking up what is present in this part of mathematics, we find as its object α) equations in which an arbitrary amount of magnitudes (we can here keep throughout to two) are so bound together into a whole of determinateness that these magnitudes first have their determinateness in empirical magnitudes, as fixed limits, and then in the manner of their connection with these as well as of their connection with one another, as is the case in an equation generally; but since only one equation is present for both magnitudes (and correspondingly several equations for several magnitudes, yet always fewer than the amount of the magnitudes –), these equations belong among the indeterminate ones; and that secondly one side of how these magnitudes here have their determinateness lies in this, that they (at least one of them) are present in the equation in a higher power than the first.

On this a few observations are to be made, first of all that, according to the first of the determinations indicated, the magnitudes bear wholly and only the character of such variable magnitudes as occur in the problems of indeterminate analysis. Indeterminate their value is, yet so that if from some other quarter a completely determinate value, a numerical value that is, accrues to the one, the other stands determined as well, the one being thus a function of the other. The categories of variable magnitudes, functions and the like are therefore, for the specific determinateness of magnitude under discussion here, merely formal, as was said above, because they are of a generality in which the specific feature upon which the entire interest of the differential calculus bears is not yet contained, nor can it be explicated out of them by analysis; taken for themselves they are simple, unremarkable, easy determinations, which are made difficult only insofar as that is to be put into them which does not lie in them, namely the specific determination of the differential calculus, so that it may then be derived from them. – As concerns the so-called constant, it may be observed of it that in the first instance it is an indifferent empirical magnitude, determining the variable magnitudes merely with respect to their empirical quantum, as the limit of their minimum and maximum; the manner, however, in which the constant is connected with the variable magnitudes is itself one of the moments belonging to the nature of the particular function which these magnitudes are. Conversely, though, the constants are themselves functions too; insofar as a straight line has the sense, for example, of being the parameter of a parabola, this sense of it consists in its being the function y²/x; just as, when a binomial gets expanded, the constant serving as coefficient of the expansion's first term amounts to the sum of the roots, the coefficient of the second to the sum of the products of those roots taken two by two, and onward, whereby these constants are here throughout functions of the roots; and where the integral calculus determines the constant out of a given formula, it handles the constant to that extent as a function of the formula. Those coefficients we shall then go on to consider under another determination, as functions whose meaning in the concrete is that on which the whole interest bears.

But the distinctive feature by which the consideration of variable magnitudes in the differential calculus differs from their constitution in indeterminate problems is to be placed in what has been indicated, that at least one of those magnitudes, or all of them, stands in a power higher than the first, where again it makes no difference whether all of them are of the same higher power or of unequal powers; the specific indeterminateness which they have here lies solely in this, that in such a ratio of powers they are functions of one another. Thereby the alteration of the variable magnitudes is qualitatively determined and hence continuous, and this continuity, which taken for itself is again only the formal category in general of an identity, of a determinateness preserving itself unchanged throughout the alteration, has here its determinate sense, and has it solely in the ratio of powers, a ratio which has no quantum for its exponent and which constitutes the non-quantitative, abiding determinateness of the ratio of the variable magnitudes. Hence, against another formalism, this observation is called for, that only relatively to higher powers is the first power a power at all; taken for itself, x is merely some indeterminate quantum. Thus there is no sense in differentiating for themselves the equations y = ax + b, that of the straight line, or s = ct, that of merely uniform velocity; if out of y = ax, or also out of y = ax + b, a = dy/dx arises, or ds/dt = c out of s = ct, then equally a = y/x, the determination of the tangent, or s/t = c, that of the merely uniform velocity. The latter is exhibited as dy/dx within the context of what is given out for the development of uniformly accelerated motion; but that a moment of simple, merely uniform velocity, of a velocity, that is, which no higher power of any moment of the motion determines, should occur within the system of such motion is, as was observed earlier, itself an empty assumption grounded solely in the routine of the method. Since the method starts out from the representation of an increment which the variable magnitude is supposed to undergo, a magnitude that is only a function of the first power can of course undergo an increment too; but if thereupon, in order to find the differential, the difference of the second equation thus arising from the given one is to be taken, the emptiness of the operation shows itself, in that, as was observed, the equation before and after the operation is the same for the so-called increments as for the variable magnitudes themselves.

β) What has been said determines the nature of the equation to be treated, and it is now to be stated upon what interest the treatment of it is found to be directed. This consideration can yield only familiar results, such as are present as regards their form in the Lagrangian conception in particular; but I have set up the exposition in so wholly elementary a fashion in order to remove the heterogeneous determinations that are mixed in with them. – As the foundation of the treatment of an equation of the kind indicated, it emerges that the power is grasped within itself as a ratio, as a system of ratio-determinations. Above, the power was stated to be number insofar as number has come to the point where its alteration is determined through itself, its moments, unit and amount, being identical, as was shown earlier, perfectly so at first in the square, more formally, which here makes no difference, in the higher powers. Now since the power, as number – if one prefers the expression magnitude as the more general one, still the power is in itself always number, – is a multitude, and is presented also as a sum, it can in the first place be decomposed within itself into an arbitrary multitude of numbers which stand in no further determination toward one another and toward their sum than that together they are equal to that sum. Yet the power admits also of being discerned into a sum of differences such as the form of the power determines. Where the power is taken as a sum, its base number too, the root, gets grasped as a sum, and grasped at will along any of a manifold of decompositions, a manifoldness which is precisely the indifferent, empirically quantitative side of the affair. Led back to its simple determinateness, that is, to its genuine universality, the sum which the root is supposed to be turns out to be the binomial; any further multiplying of the terms merely repeats the same determination and is therefore empty.¹² What matters is solely the thereby qualitative determinateness of the terms, a determinateness that results from the exponentiation of the root assumed as a sum and that lies solely in the alteration which exponentiation is. These terms are accordingly wholly functions of exponentiation and of the power. Now that presentation of number as the sum of a multitude of such terms which are functions of exponentiation, then the interest in finding the form of such functions, and further in finding this sum out of the multitude of such terms, insofar as this finding must depend solely on that form, – this, as is well known, makes up the particular doctrine of series. But here we have essentially to distinguish the further interest, namely the ratio of the underlying magnitude itself, whose determinateness, insofar as it is a complex, that is, here an equation, encloses a power within itself, – to the functions of its exponentiation. This ratio, wholly abstracted from the previously mentioned interest in the sum, will show itself to be the standpoint that emerges out of the actual science as the sole standpoint which the differential calculus sets before itself.

Beforehand, however, one determination is still to be added to what has been said, or rather one that lies in it is to be removed. It was said, namely, that the variable magnitude into whose determination the power enters is regarded within itself as a sum, and indeed as a system of terms insofar as these are functions of exponentiation, whereby the root too is considered as a sum, and in the simply determined form as a binomial; xⁿ = (y + z)ⁿ = (y + nyⁿ⁻¹ z + . . .) For the expansion of the power, that is, for obtaining its functions of exponentiation, this presentation started out from the sum as such; here, however, it is neither a sum as such nor the series springing from it that is at issue, but rather only the connection is to be taken up out of the sum. What remains over on the one side, once one abstracts from the plus belonging to a sum as such, and what on the other side is required if the expansion-functions of the power are to be found, is precisely the connection as such of the magnitudes. Such a connection, however, is already determined in this, that the object here is an equation, yᵐ = axⁿ, hence already a complex of several (variable) magnitudes containing a determination of power in them. Within this complex every one of these magnitudes is posited simply as standing in connection with the other, bearing, one could say, the meaning of a plus in its own self, – as a function of the other magnitudes; it is their character as functions of each other that lends them this determination of a plus, though precisely for that reason a wholly indeterminate plus, not an increase, an increment or anything of that sort. Yet this abstract standpoint too we could leave aside; one can quite simply stop at this, that once the variable magnitudes are given in the equation as functions of one another, such that this determinateness contains a ratio of powers, the functions of the exponentiation of each are then also compared with one another, – which second functions are determined by nothing whatever other than exponentiation itself. It can at first be given out as an arbitrary choice or a possibility to put an equation of the powers of its variable magnitudes into a ratio of its expansion-functions; only a further purpose, benefit, use has to state what is serviceable in such a transformation; it was solely through the usefulness of that rearrangement that it was occasioned. If earlier the start was made from the presentation of these determinations of exponentiation on a magnitude taken as a sum differentiated within itself, this served partly only to state of what sort such functions are, and partly it holds the way to find them.

We stand herewith at the ordinary analytic expansion, which for the purpose of the differential calculus is so conceived that an increment, dx, i, is given to the variable magnitude and the power of the binomial is then explicated through the row of terms belonging to it. The so-called increment, however, is supposed to be not a quantum but only a form whose entire worth consists in being of help to the expansion; what is wanted, avowedly so, most determinately by Euler and Lagrange and in the previously mentioned representation of the limit, is only the resulting determinations of powers of the variable magnitudes, the so-called coefficients, to be sure, of the increment and of the powers of the increment, according to which the series arranges itself and to which the distinct coefficients belong. It may be observed on this head that, since an increment is assumed only for the sake of the expansion and is supposed to be without a quantum, it would have been the most adroit thing to take 1 (the one) for it, since in the expansion the increment always occurs only as a factor, and precisely the factor one fulfils the purpose that no quantitative determinateness and alteration is to be posited by the increment; whereas dx, encumbered with the false representation of a quantitative difference, and other signs like i, encumbered with the here useless semblance of generality, always have the look and the pretension of a quantum and of its powers; which pretension then brings on the trouble of clearing them away and leaving them out all the same. In order to keep the form of a series expanded according to powers, the designations of the exponents could just as well be attached as indices to the one. One must in any case abstract both from the series and from any determining of the coefficients by the place they occupy in it, since all of them stand in one and the same ratio; out of the first function the second is derived exactly as the first was derived out of the original, and for whatever gets counted as the second the first derived function is in turn an original one. Essentially, however, the interest bears not on the series but wholly and solely on the determination of power resulting from the expansion, in its ratio to the magnitude that is immediate for it. Rather, therefore, than determining that determination of power as the coefficient of the expansion's first term, since a term counts as the first only by reference to the others that follow it in the series, while a power of an increment, like the series itself, has no place here, one would do better with the bare expression derived power-function, or, as was said earlier, a function of the exponentiating of the magnitude, it being presupposed as known in what manner the derivation is taken as an expansion enclosed within a power.

If, then, in this part of analysis the properly mathematical beginning amounts to nothing more than finding the function which the expansion of a power determines, then it must next be asked what one is to start with the ratio so obtained, where a ratio of this kind finds application and use, or indeed to what purpose such functions get sought at all. It is through the finding of ratios in concrete objects which can be led back to those abstract analytic ones that the differential calculus has acquired its great interest.

As regards applicability, however, the following results of itself in the first place from the nature of the matter, without inferring as yet from the cases of application themselves, by virtue of the shape of the moments of powers that has been exhibited. The expansion of magnitudes of powers, whereby the functions of their exponentiation result, contains, abstracting from closer determination, first of all and in general the lowering of the magnitude to the next lower power. The applicability of this operation therefore takes place with such objects as likewise exhibit such a difference of determinations of powers. Reflecting now upon spatial determinateness, we come upon its three dimensions, which, to mark them off from the abstract differences of height, length and breadth, may be called the concrete dimensions, namely line, surface and total space; and once these are taken in their simplest shapes and with an eye to self-determination and hence to analytic dimensions, what stands before us is the straight line, the plane surface together with the same as square, and the cube. An empirical quantum belongs to the straight line, whereas with the plane the qualitative side sets in, namely determination by powers; closer modifications, that the like holds of plane curves for instance, may be left undiscussed, since for the present only the difference in its generality is at issue. Herewith there also arises the need to pass over from a higher determination of power to a lower one and conversely, in that, say, linear determinations have to be got out of given equations of the surface and so on, or the other way about. – Motion, further, as that in which the ratio of magnitude between the space traversed and the time elapsed for it is to be considered, shows itself in the various determinations of a merely uniform, a uniformly accelerated, an alternately uniformly accelerated and uniformly retarded motion returning into itself; and since these distinct kinds of motion are expressed according to the ratio of magnitude of their moments, space and time, equations out of distinct determinations of powers result for them, and insofar as there may be need to determine one kind of motion, or also of the magnitudes of space to which a kind of motion is bound, out of another kind of the same, the operation likewise brings with it the passing over from one function of a power to a higher or a lower one. – The examples of these two objects may suffice for the purpose for which they have been adduced.

The semblance of contingency which the differential calculus presents in its applications would already be simplified by an awareness of the nature of the domains within which the application can take place, and of the peculiar need and the condition of this application. But now, further, within these domains themselves everything turns on knowing between which parts of the objects of the mathematical problem such a ratio takes place as is peculiarly posited by the differential calculus. It must be noted at once and provisionally that two sorts of ratio are here to be attended to. The operation of lowering the power of an equation, the equation being considered according to the derived functions of its variable magnitudes, yields a result which in its own self is truly no longer an equation but a ratio; this ratio is the object of the differential calculus proper. Precisely thereby there is present, secondly, the ratio of the higher determination of power (of the original equation) itself to the lower one (to the derived function). This second ratio we have here to leave aside for the present; it will show itself to be the peculiar object of the integral calculus.

Let us consider the first ratio to begin with, and, for the determination, to be gathered from the so-called application, of the moment in which the interest of the operation lies, let us take up the simplest example in the curves that are determined by an equation of the second power. As is well known, the equation gives immediately the ratio of the coordinates in a determination of power. Consequences of this fundamental determination are the determinations of the other straight lines connected with the coordinates, of the tangent, subtangent, normal and so forth. Between these lines and the coordinates, however, the equations are linear ones; and the wholes in which these lines get determined as parts are right-angled triangles composed of straight lines. The transition from the fundamental equation, which contains the determination of power, to those linear equations now contains the transition indicated above from the original function, that is, from what is an equation, to the derived function, which is a ratio, and indeed a ratio between certain lines contained in the curve. It is the connection between the ratio of these lines and the equation of the curve whose finding is the business at hand.

It is not without interest to note this much of the historical side, that the first discoverers knew how to state their find only in a wholly empirical manner, without being able to give any account of an operation that remained entirely external. Here I content myself with citing Barrow, the teacher of Newton. His lect. Opt. et Geom., where problems of higher geometry get handled by the method of indivisibles, a method which in the first instance differs from what is peculiar to the differential calculus, also sets out, »because his friends pressed him,« (lect. X.) his procedure for determining the tangent. One must read in Barrow himself how this statement is constituted, in order to form a proper representation of how the procedure is stated entirely as an external rule, – in the same style in which formerly the rule of three, or better still the so-called check by nines of the arithmetical operations, used to be set forth in the arithmetic school-books. He draws in the little lines that were afterwards called the increments in the characteristic triangle of a curve, and now gives the prescription as a bare rule, namely to throw away as superfluous those terms which, in consequence of the expansion of the equations, come to light as powers of those increments or as products, (etenim isti termini nihilum valebunt); likewise the terms which contain only magnitudes determined from the original equation are to be thrown away (– the subsequent subtraction of the original equation from the one formed with the increments) and lastly to put in place of the ordinate's increment the ordinate itself, and in place of the abscissa's increment the subtangent. One cannot, if the phrase be allowed, put the procedure more schoolmasterly than that; – that last substitution is nothing other than the assumption of the proportionality between the increments of ordinate and abscissa on the one side and the ordinate and subtangent on the other, the very assumption laid at the base of how the ordinary differential method fixes a tangent; with Barrow the assumption stands forth in wholly naive nakedness. A simple way of determining the subtangent had been found; the manners of Roberval and Fermat come to something similar, – the method of finding the greatest and smallest values, from which the latter started out, rests upon the very same foundations and proceeds in the very same way. Hunting up so-called methods, that is, rules of that stamp, was a mathematical mania of those times, and so was making a mystery of them, something not merely easy but in one respect even necessary, and necessary for just the reason that made it easy, – namely that what the inventors had hit upon was an empirical external rule and no method at all, nothing, that is, drawn from acknowledged principles. Such so-called methods Leibnitz took over from his age, and Newton likewise from that same age and more directly from his teacher; through the generalization of their form and applicability they broke new paths for the sciences, but with that they had at the same time the need to tear the procedure out of the shape of merely external rules, and sought to procure for it the requisite justification.

If we analyse the method more closely, its true course is this. First, the determinations of powers (of the variable magnitudes, be it understood) which the equation contains are lowered to their first functions. But thereby the value of the terms of the equation is altered; no equation therefore remains any longer, but only a ratio has arisen between the first function of the one variable magnitude and the first function of the other; instead of px = y² one has p : 2y, or instead of 2 ax − x² = y² one has a − x : y, which afterwards used to be designated as the ratio dy/dx. The equation is the equation of the curve; this ratio, wholly dependent on that equation and derived from it (above, by a bare rule), is by contrast a linear one with which certain lines stand in proportion; p : 2y or a − x : y are themselves ratios out of straight lines of the curve, out of the coordinates and the parameters; but with that one still knows nothing. The interest is to know of other lines occurring on the curve that that ratio belongs to them, to find the equality of two ratios. – Secondly, therefore, it must be asked what straight lines, determined as they are by the curve's nature, stand in such a ratio? – But this is just what was already known beforehand, namely that such a ratio, obtained by that route, is the ratio of the ordinate to the subtangent. The ancients had found this by an ingenious geometrical route; what the modern discoverers have discovered is the empirical procedure of so dressing up the equation of the curve that that first ratio is yielded, a ratio of which it was already known that it equals another one containing the line whose determination is the business at hand, here the subtangent. Partly, then, that dressing up of the equation has been methodically conceived and carried out, – differentiation, – but partly there were invented the imaginary increments of the coordinates, and the imaginary characteristic triangle built out of these together with a like increment of the tangent, so that the proportionality of the ratio found by lowering the power of the equation with the ratio of ordinate and subtangent might be presented not as something taken up merely empirically out of the old acquaintance, but as something demonstrated. The old acquaintance, however, proves itself in general and most unmistakably, in the cited form of rules, to be the sole occasion and respectively the sole justification of the assumption of the characteristic triangle and of that proportionality.

Lagrange, now, discarded this pretence and struck out the genuinely scientific path; to his method we owe the insight into what matters, in that it consists in separating the two transitions that have to be made for the solution of the problem and in treating and demonstrating each of these sides for itself. The one part of this solution – and for a closer account of how it runs we keep to the elementary problem of the subtangent – the theoretical or general part, that is, the winning of the first function from the given equation of the curve, gets regulated on its own; what it yields is a linear ratio, a ratio therefore of straight lines occurring within the system by which the curve is determined. The other part of the solution is now the finding of those lines on the curve which stand in that ratio. This is now accomplished directly (Théorie des Fonct. Anal. II. P. II. Chap.), that is, with no characteristic triangle, and so without positing infinitely small arcs, ordinates and abscissas, and without loading upon them the determinations dy and dx, the sides, that is, of that ratio, along with the immediate significance of its equality with ordinate and subtangent themselves. A line (as also a point) has its determination solely insofar as it makes up the side of a triangle, just as the determination of a point too lies only in a triangle. This is, to mention it in passing, the fundamental proposition of analytic geometry, the proposition that brings in the coordinates just as, what is the same thing, in mechanics it brings in the parallelogram of forces, which for that very reason stands in no need at all of the much labour spent on proving it. – The subtangent is now posited as the side of a triangle whose further sides are the ordinate and the tangent referring to it. The latter, as a straight line, has for its equation p = aq, (adding + b is useless for the determination and is added only for the sake of a favoured generality); – what determines the ratio p/q falls into a, that is, into the coefficient belonging to q, which is the respective first function of the equation, though in general it needs regarding only as a = p/q, this being, as was said, what essentially determines the straight line laid against the curve as tangent. Now if the first function of the curve's equation be taken in turn, it likewise gives the determination of a straight line; and since further p, the one coordinate of that first straight line, and y, the curve's ordinate, get taken as identical, so that the point where the first straight line assumed as tangent touches the curve is at the same time the point where the line determined through the curve's first function begins, everything turns on showing that this second straight line coincides with the first, that is, is tangent; expressed algebraically, that since y = fx and p = Fq, and now y = p, hence fx = Fq, is assumed, f′x = F′q as well. That the line laid on as tangent coincides with the line which the equation determines through its first function, and that this latter is therefore a tangent, gets shown with the help of the increment i of the abscissa and of the increment of the ordinate determined through the expansion of the function. Here, then, the notorious increment likewise comes in; but the way in which it is introduced for the purpose just stated, and the expansion of the function according to it, must be carefully distinguished from the earlier mentioned use of the increment for finding the differential equation and for the characteristic triangle. The use made of it here is justified and necessary; it falls within the compass of geometry, for the geometrical determination of a tangent as such carries with it that no further straight line falling into the same point can run between the tangent and the curve with which it shares that point. For with this determination the quality of tangent or non-tangent is led back to the difference of magnitude, and that line is the tangent upon which the greater smallness falls, simply as regards the determination that matters. Nothing empirical whatever lies in this apparently merely relative smallness, nothing, that is, that hangs upon a quantum as such; the nature of the formula posits it qualitatively, provided that what the compared magnitude hangs upon differs as a moment by a difference of powers; and since that difference amounts to i and i², while i, which at the last is after all to mean a number, must then be pictured as a fraction, i² is in and for itself the smaller of the two, so that any representation of some arbitrary size in which i might be taken is here both superfluous and out of place. Precisely thereby the proof of the greater smallness has nothing to do with an infinitely small, which accordingly has no business at all coming in here.

Even were it only for the sake of the beauty, and of the nowadays rather forgotten but well-deserved fame, that I still wish to adduce Descartes' method of tangents; it has, moreover, a bearing on the nature of equations, about which a further observation is then to be made. Descartes sets out this self-subsistent method, in which the required linear determination is likewise found out of the same derived function, in his Geometry (liv. II. p. 357 ss. Oeuvres compl. ed. Cousin Tom. V.), which became so fruitful in other respects too, since in it he taught the great foundation concerning the nature of equations and their geometrical construction, and the application, thereby so greatly extended, of analysis to geometry in general. With him the problem takes the form of the task of drawing straight lines perpendicular to arbitrary places on a curve, whereby subtangent and so forth is determined; one appreciates the satisfaction he there expresses over his discovery, a discovery which concerned an object of general scientific interest in that day and which is so thoroughly geometrical and thereby stood so high above the bare rule-methods of his rivals mentioned above: j’ose dire que c’est ceci le problème le plus utile et le plus général, non seulement que je sache, mais même que j’aie jamais désiré de savoir en géometrie. – At the base of the solution he lays the analytic equation of a right-angled triangle, one formed by the ordinate belonging to that point of the curve on which the line demanded in the problem is to stand perpendicular, then by that line itself, the normal, and thirdly by the piece of the axis which ordinate and normal cut off between them, the subnormal. Out of the known equation of a curve the value, be it of the ordinate or of the abscissa, is now substituted into that equation of the triangle, and one thus has an equation of the second degree (and Descartes shows how curves too whose equations contain higher degrees can be led back to this), in which only one of the variable magnitudes still occurs, and indeed in the square and in the first power; – a quadratic equation which at first presents itself as one of the so-called impure sort. Descartes now reflects that, should the point assumed on the curve be pictured as a point where curve and circle intersect, that circle will cut the curve at a second point as well, so that for the two unequal x's thereby arising there result a pair of equations agreeing in form and in their constants; – or else one equation only, carrying unequal values of x. But there comes to be only one equation for that one triangle whose hypotenuse stands perpendicular to the curve, is normal to it, and this is pictured by letting the two points where the circle cuts the curve fall together, so that the circle touches it. With that, however, there drops away also the circumstance that the quadratic equation's x or y has unequal roots. Now where a quadratic equation has two equal roots, the coefficient belonging to the term that carries the unknown in the first power comes to double that single root; and from this there follows an equation by which the required determinations are found. This course is to be regarded as the ingenious stroke of a genuinely analytic mind, against which the wholly assertorically assumed proportionality of subtangent and ordinate with the so-called increments of abscissa and ordinate, increments supposed to be infinitely small, falls quite short.

The final equation obtained in the manner indicated, which sets the coefficient of the second term of the quadratic equation equal to the doubled root or unknown, is the same as the equation found through the procedure of the differential calculus. x² − ax − b = o differentiated yields the new equation 2x − a = o; or x³ − px − q = o yields 3x² − p = o. But here the observation offers itself that it by no means goes without saying that such a derived equation is also correct. In the case of an equation with two variable magnitudes which, precisely because they are variable, do not lose the character of being unknown magnitudes, there comes out, as was considered above, only a ratio, and this for the simple reason indicated, namely that putting the functions of exponentiation where the powers themselves stood alters what the two terms of the equation are worth, and whether at values so altered an equation still holds between them remains of itself unknown. The equation dy/dx = P expresses nothing further than that P is a ratio, and no other real sense is to be ascribed to the dy/dx. But of this ratio = P it is equally still unknown to what other ratio it is equal; such an equation, the proportionality, is what first gives it a value and a significance. – Just as it was stated that this significance, which was called the application, was taken up from some other quarter, empirically, so with the equations here under discussion, derived by differentiation, it has to be known from some other quarter whether they have equal roots, in order to know whether the equation obtained is still correct. This circumstance, however, is not expressly brought to notice in the textbooks; it is presumably got out of the way by the fact that an equation with one unknown, brought to zero, is straightaway set = y, whereby in differentiating there does indeed come out a dy/dx, only a ratio. The calculus of functions is indeed supposed to have to do with functions of exponentiation, or the differential calculus with differentials, but from this it by no means follows of itself that the magnitudes whose differentials or functions of exponentiation are taken should themselves be merely functions of other magnitudes. In the theoretical part, the instruction for deriving the differentials, that is, the functions of exponentiation, no thought is in any case yet given to the idea that the magnitudes one is taught to treat according to such derivation should themselves be functions of other magnitudes.

Regarding the dropping of the constant in differentiating, it can further be brought to notice that this dropping has here the sense that the constant is indifferent for the determination of the roots in the case of their equality, a determination which is exhausted by the coefficient of the second term of the equation. Thus in the cited example from Descartes the constant is the square of the roots themselves, so that these can be determined out of the constant just as well as out of the coefficients; for the constant, like the coefficients, is in general a function of the roots of the equation. In the ordinary presentation, the dropping of the so-called constants, joined to the remaining terms only by + and −, comes about through the bare mechanism of the procedure, namely that in order to find the differential of a composite expression an increment is given only to the variable magnitudes and the expression thereby formed is subtracted from the original one. The sense of the constants and of their being dropped, insofar as they are themselves functions and according to this determination serve a purpose or do not, is never brought up for discussion.

With the dropping of the constants there hangs together a similar observation, one that can be made about the names of differentiation and integration as was earlier made about the finite and the infinite expression, namely that their determination contains rather the opposite of what the expression says. To differentiate signifies the positing of differences; through differentiating, however, an equation is rather brought down to fewer dimensions, and through the dropping of the constant a moment of determinateness is taken away; as was observed, the roots of the variable magnitude are posited in an equality, the difference of them therefore sublated. In integration, by contrast, the constant is supposed to be added back on; the equation is thereby indeed integrated, but in the sense that the previously sublated difference of the roots is restored again, that what was made equal is differentiated once more. – The customary expression helps to cast the essential nature of the matter into shadow and to set everything under the subordinate standpoint, one indeed alien to the main point, partly of the infinitely small difference, of the increment and the like, partly of the bare difference in general between the given and the derived function, without designating their specific, that is, their qualitative difference.

Another principal domain in which use is made of the differential calculus is mechanics; the significations of the distinct functions of powers which arise with the elementary equations of its object, motion, have already been mentioned in passing; I wish to take them up here directly. Merely uniform motion has for its equation, that is, for its mathematical expression, c = s/t or s = ct, where the spaces run through stand to the times gone by in a proportion set by an empirical unit c, the magnitude of the velocity; and such an equation yields no sense for differentiation, since c as coefficient is already fully determined and known, so that no further unfolding of powers can occur. – How s = at², the equation of the motion of fall, is analysed has already been recalled earlier; – the first term of the analysis, ds/dt = 2 at, is translated into language and respectively into existence in this way, that it is supposed to be a term of a sum (– a representation which we removed long ago), the one part of the motion, and that this part is supposed to belong to the force of inertia, that is, to a merely uniform velocity, in such a way that throughout the infinitely small parts of time the motion should be uniform, while throughout the finite parts, the parts that actually exist, it is non-uniform. Certainly fs = 2 at; and the signification of a and of t is known for itself, as is the fact that thereby the determination of a uniform velocity of a motion is posited; since a = s/t², 2 at = 2s/t in general; but with that one knows not the least thing further; it is only the false assumption that 2 at is one part of the motion regarded as a sum which lends the whole the false semblance of a physical proposition. As for the factor a itself, the empirical unit – a quantum as such – it gets ascribed to gravity; and if the category of the force of gravity is to be used, then it should rather be said that precisely the whole s = at² is gravity's effect, or better, gravity's law. – No better is the proposition which gets derived from ds/dt = 2 at, namely that if gravity were to stop acting, then the body, carrying the velocity reached at the end of its fall, would in a span of time equal to the duration of that fall cover double the space through which it has passed. – There lies in this too a metaphysics skewed on its own account; the end of the fall, or the end of a part of time in which the body has fallen, is always itself still a part of time; were it no part of time, then rest and hence no velocity would be assumed, for velocity can be reckoned only according to the space traversed in a part of time, not at the end of it. – And when finally, in other physical fields where motion is not present at all, as for instance in the conduct of light (leaving aside what gets called its propagation through space) and in determinations of magnitude among the colours, the differential calculus is applied all the same, and the first function of a quadratic function is here too christened velocity, then this must count as a formalism still less admissible, a mere fabrication of existence. –

The motion represented by the equation s = a t², says Lagrange, we find in the experience of the fall of bodies; the simplest motion after this one would be the motion whose equation would be s = ct³, but nature, he says, shows no motion of this kind; we would not know what the coefficient c could signify. If that is indeed so, there is on the other hand a motion having s³ = at² for its equation, – Kepler's law for how the bodies of the solar system move; – and what the first derived function 2at/3s² and so forth is meant to signify here, together with a further direct handling of this equation through differentiation, an unfolding of the laws and determinations of that absolute motion out of this point of departure, would by contrast surely present itself as an interesting task, one in which analysis might show itself at its most worthy splendour.

Applied on its own to the elementary equations of motion, then, the differential calculus offers no real interest; whatever formal interest there is comes from the calculus's general mechanism. The resolution of motion acquires another significance, however, with reference to the determination of its trajectory; should that trajectory be a curve whose equation carries higher powers, transitions become necessary from rectilinear functions, as functions of exponentiation, to the powers themselves, and since those functions are to be won out of the original equation of motion, which contains the factor of time, with elimination of the time, this factor is at the same time to be lowered to the lower expansion-functions out of which those equations of linear determinations can be obtained. This side leads on to the interest of the other part of the differential calculus.

What has gone before has had the purpose of lifting out and establishing the simple specific determination of the differential calculus and of exhibiting it in a few of the elementary examples. That determination has turned out to consist in finding, out of an equation of functions of powers, the coefficient belonging to the expansion's term, what is called the first function, and in exhibiting the ratio which this function is within moments of the concrete object, so that by the equation thereby obtained between the two ratios those moments are themselves determined. Of the principle of the integral calculus it is likewise to be considered in brief what results from its application for the specific concrete determination of that principle. The view of this calculus has already been simplified and more correctly determined by the fact that it is no longer taken as a method of summation, as it was called in contrast to differentiating, where the increment counts as the essential ingredient, and whereby it appeared to stand in essential connection with the form of the series. – The task of this calculus is at first likewise the theoretical or rather the formal one, as is that of the differential calculus, but, as is well known, the reverse of the latter; – here the start is made from a function which is considered as derived, as the coefficient of the next term arising out of the expansion of an as yet unknown equation, and out of it the original function of a power is to be found; what in the natural order of the expansion is to be regarded as original is here derived, and what was earlier considered as derived is here the given, or in general the initiating, function. The formal side of this operation, however, now seems already to have been accomplished by the differential calculus, since in it the transition and the ratio from the original function to the expansion-function is established in general. If in this connection recourse must necessarily be had in many cases to the form of the series, partly even in order to set down the function from which one is to start, partly in order to bring about the transition from it to the original function, then it is first of all to be held fast that this form as such has nothing immediately to do with the peculiar principle of integrating.

The other part of the task of the calculus, however, appears, with regard to the formal operation, as the application of that operation. This application is now itself the task, namely to know the significance, in the sense indicated above, which the original function possesses of the given function of a particular object, that given function being regarded as the first function. In itself this doctrine too could seem already to have been quite settled in the differential calculus; but a further circumstance [enters] which does not let the matter be so simple. For since it results in this calculus that through the first function of the equation of a curve the ratio, which is a linear one, has been obtained, one thereby knows as well that the integration of this ratio yields the equation of the curve in the ratio of abscissa and ordinate; or if the equation for the plane of a curve were given, then the differential calculus ought already to have taught, concerning the significance of the first function of such an equation, that this function presents the ordinate as function of the abscissa and hence presents the equation of the curve.

Now, however, everything turns on which of the determining moments of the object is given in the equation itself; for the analytic treatment can take its start only from what is given, and only from there pass over to the remaining determinations of the object. It is not, for example, the equation of a surface-area of the curve, nor perhaps of the body arising through its rotation, nor yet of an arc of it, but only the ratio of abscissa and ordinate in the equation of the curve itself, that is given. The transitions from those determinations to this equation itself cannot therefore already be treated in the differential calculus; the finding of these ratios is saved up for the integral calculus.

Further, however, it has been shown that the differentiation of the equation of several variable magnitudes yields the power of the expansion, or the differential coefficients, not as an equation but only as a ratio; the task is then, for this ratio, which is the derived function, to state a second ratio in the moments of the object that is equal to the first. The object of the integral calculus, by contrast, is the ratio itself of the original to the derived function, the latter being here supposed to be given, and the task is to state the significance of the original function that is to be found in the object of the given first function, or rather, since this significance, for example the plane of a curve, or the curve to be rectified, represented as rectilinear, and so forth, is already pronounced as the problem, to show that such a determination is found through an original function, and which moment of the object must be assumed for this purpose as the starting (the derived) function.

The ordinary method, which uses the representation of the difference as the infinitely small, makes things easy for itself; for the quadrature of curves, accordingly, it takes a rectangle that is infinitely small, the ordinate multiplied into the element, that is, into the abscissa's infinitely small, and takes this for the trapezium one of whose sides is the infinitely small arc lying over against that infinitely small of the abscissa; the product then gets integrated in the sense that the integral is to yield the sum of infinitely many trapezia, the plane whose determination is demanded, namely the finite magnitude of that element of the plane. In the same way it forms, out of the infinitely smalls of the arc and of the ordinate and abscissa belonging to it, a right-angled triangle in which that arc squared equals the two other infinitely smalls squared and added, and the integration of these yields the arc as something finite.

This procedure has for its presupposition the general discovery lying at the base of this domain of analysis, here in the form that the squared curve, the rectified arc and so forth stands, to a certain function given through the equation of the curve, in the ratio of the so-called original function to the derived one. The point is to know, when a certain part of a mathematical object (for example of a curve) is assumed as the derived function, which other part of it is expressed by the corresponding original function. It is known that when the curve's equation supplies the function of the ordinate and this gets taken as the derived function, then the relatively original function expresses the magnitude of that area of the curve which the ordinate cuts off, and that when a certain determination of the tangent counts as the derived function, its original function expresses the magnitude of the arc answering to that determination of the tangent, and so on; but the recognizing and the proving that these two ratios, the one holding between an original and a derived function, the other between the magnitudes of two parts or circumstances of the mathematical object, make up a proportion, is just what the method spares itself which works with the infinitely small and with the mechanical operation upon it. The peculiar merit of acumen consists in having found out, from results already known from elsewhere, that certain sides of a mathematical object, and which ones, stand in the ratio of original and of derived function.

Of these two functions the derived one, or, as it has been determined, the function of exponentiation, is here in this calculus the given one, relatively to the original function, which is first to be found out of the derived one by integration. Only, the derived function is not immediately given, nor is it already given of itself which part or determination of the mathematical object is to be regarded as the derived function, in order, by leading it back to the original function, to find that other part or determination whose magnitude the problem demands. The ordinary method, which, as was said, pictures certain parts of the object straightaway as infinitely small and in the shape of derived functions determinable by differentiation out of the object's originally given equation, (– as, for the rectification of a curve, the infinitely small abscissas and ordinates), takes for this purpose such parts as can be brought into a connection, established in elementary mathematics, with the object of the problem (in the example, with the arc), which is likewise represented as infinitely small, and through which, if those parts are known, that object too is determined whose magnitude has been set as the task; thus for rectification the three infinitely smalls indicated are brought into the connection of the equation of the right-angled triangle, and for quadrature the ordinate with the infinitely small abscissa is brought into the connection of a product, since a plane is in general assumed arithmetically as a product of lines. The transition from such a so-called element of the plane, of the arc and so forth, to the magnitude of the plane, of the arc and so forth itself, then counts merely as a climb from the infinite expression up to the finite one, or up to the sum made of those infinitely many elements out of which the magnitude in question is supposed to be composed.

Only superficially, therefore, can one say that integral calculus poses simply the inverted, though on the whole harder, problem of the differential calculus; what carries the real interest in it is rather, and exclusively, how original and derived function stand to one another in concrete objects.

Lagrange was just as little inclined, in this part of the calculus, to dispose of the difficulty of the problems in the smooth manner of those direct assumptions. It will contribute to the elucidation of the nature of the matter to state likewise the closer detail of his procedure from a few examples. That procedure makes it precisely its task to prove for itself that between particular determinations of a mathematical whole, for example of a curve, a ratio of the original to the derived function takes place. In this field, however, that cannot be brought about in a direct way, on account of the very nature of the ratio, which on the mathematical object brings curved lines into connection with straight ones, linear dimensions and functions of them with dimensions of plane surface and their function and so forth, hence brings qualitatively distinct things into connection; the determination lets itself be grasped in this way only as the middle between a greater and a smaller. Herewith the form of an increment with plus and minus of course enters once more of itself, and the vigorous Développons is in its place; but that the increments here have only an arithmetical, finite significance has been spoken of above. Unfolding that condition, that the magnitude to be determined exceeds the one readily determinable limit and falls short of the other, then yields, for instance, the result that the ordinate's function is the derived first function belonging to the function of the area.

The rectification of curves as it is exhibited by Lagrange, who starts out from the Archimedean principle, holds the interest of letting one see how the Archimedean method gets translated over into the principle of modern analysis, and this affords a glance into the interior and the true sense of a business carried on mechanically in the other way. Necessarily the way of proceeding is analogous to the one just indicated; the Archimedean principle, that a curve's arc exceeds its chord while falling short of the two tangents drawn at the arc's endpoints taken together, insofar as these lie between those points and their point of intersection, gives no direct equation. The carrying over of that Archimedean fundamental determination into the modern analytic form is the invention of an expression that is for itself a simple fundamental equation, whereas that form sets up only the demand to proceed to infinity between a too-great and a too-small which have in each case determined themselves, a proceeding that again always yields only a new too-great and a new too-small, though within ever narrower limits. By means of the formalism of the infinitely small the equation dz² = dx² + dy² is set down straightaway. The Lagrangian exposition, starting out from the foundation indicated, shows by contrast that the magnitude of the arc is the original function to a derived one whose peculiar term is itself a function out of the ratio of a derived function of the ordinate to the original function of it.

Because in the Archimedean procedure, as later in the Keplerian treatment of stereometric objects, the representation of the infinitely small occurs, this has so often been adduced as an authority for the use made of that representation in the differential calculus, without the peculiar and distinguishing feature having been lifted out. In the first place the infinitely small signifies negation of quantum as such, negation, that is, of what gets called a finite expression, of that completed determinateness belonging to quantum as such. In the same way, in the famous methods that followed, those of Valerius, Cavalleri and others, which are grounded on the consideration of the ratios of geometrical objects, the fundamental determination is that the quantum as such of the determinations that are at first considered only in ratio is for this purpose set aside, and that they are accordingly to be taken as a non-magnitude. But partly the affirmative element in general, which lies behind the merely negative determination, has thereby not been recognized and lifted out, the element which above resulted abstractly as the qualitative determinateness of magnitude and, more determinately, as lying in the ratio of powers; – partly, however, since this ratio again comprises within itself a multitude of more closely determined ratios, such as that of a power and its expansion-function, these too were supposed to be grounded upon and derived from the general and negative determination of that same infinitely small. In the Lagrangian exposition just extracted, the determinate affirmative element lying in the Archimedean way of developing the problem has been found, and thereby the procedure, encumbered as it is with an unbounded going-out-beyond, has been given its correct limit. The greatness of the modern invention for itself, and its capacity to solve problems previously intractable and to treat those previously solvable in a simple way, is to be placed solely in the discovery of the ratio of the original to the so-called derived functions and of the parts which stand in such a ratio on a mathematical whole.

The citations made may suffice for the purpose of lifting out the peculiar character of that ratio of magnitudes which is the object of the particular kind of calculus under discussion. These citations could restrict themselves to simple problems and to their modes of solution; and it would neither have been appropriate for the determination of the concept, which alone was at issue here, nor would it have lain within the author's power, to traverse the whole extent of what is called the application of differential and integral calculus and to round off the induction, that the principle exhibited lies at their base, by tracing every one of their problems and solutions back to it. What has been brought forward has shown sufficiently, however, that just as every particular mode of calculation has a particular determinateness or ratio of magnitude toward its object, and just as such a ratio constitutes adding, multiplying, the raising to powers and the extraction of roots, calculation with logarithms, series and so forth, so likewise does the differential and integral calculus; for what belongs to this calculus, the name of the ratio of a function of a power and the function of its expansion or exponentiation might be the most fitting, because it lies closest to insight into the nature of the matter. Only, just as the operations according to the other ratios of magnitude, such as adding and so forth, are likewise used in this calculus generally, so too are the logarithmic, circular and series ratios applied, particularly in order to make expressions more tractable for the sake of the requisite operations of deriving the original functions out of the expansion-functions. With the form of the series the differential and integral calculus does indeed have in common the closer interest of determining the expansion-functions, which in the case of series are called the coefficients of the terms; but whereas the interest of that calculus bears only on the ratio of the original function to the next coefficient of its expansion, the series aims at presenting a sum in the multitude of terms ordered according to powers furnished with those coefficients. That infinite which turns up with the infinite series, the indeterminate expression for the negative of quantum at large, shares nothing with the affirmative determination which lies in the infinite proper to that calculus. In the same way the infinitely small, as the increment by means of which the expansion falls into the form of the series, is only an external means for the expansion, and its so-called infinity is without any other significance than that of having none whatever apart from that of being such a means; the series, since it is in fact not what is demanded, brings on a too-much, the clearing away of which occasions superfluous trouble. By this trouble the method of Lagrange, who took up the form of the series again by preference, is likewise burdened; although it is his method through which, in what is called the application, the true peculiarity lifts itself out, since without forcing the forms of dx, dy and so forth into the objects it directly demonstrates that part of them to which the determinateness of the derived (– expansion –) function belongs, and it thereby shows that the form of the series is not here the thing at issue.¹³

Remark 3. Further Forms Connected with the Qualitative Determinateness of Magnitude

The infinitely small of the differential calculus has been exhibited in its affirmative sense as the qualitative determinateness of magnitude, and regarding this determinateness it has been shown more closely that in this calculus it is on hand as determinateness of power not simply in general, but as the particular such determinateness of the ratio of a power function to the power of development . But the qualitative determinateness is on hand in a further, so to speak weaker form as well, and it is this form, together with the associated use of the infinitely small and the sense the latter carries in that use, that the present Remark still has to consider.

Starting out from what precedes, the first thing to be recalled in this respect is that on the analytic side the distinct determinations of power come forward at first as merely formal and as entirely homogeneous in this, that what they mean are numerical magnitudes, and these as such lack that qualitative diversity in relation to one another. In the application to objects of space, however, the analytic relation shows itself fully in its qualitative determinateness, as the passing over from linear determinations to determinations of surface, from rectilinear to curvilinear ones, and so forth. A further consequence of this application is that objects of space, given by their very nature in the shape of continuous magnitudes, come to be grasped in discrete fashion – the surface accordingly as a multitude of lines, the line as one of points, and so forth. The one interest of such a resolution is to determine the points themselves into which the line, and the lines into which the surface and so forth, stand resolved, so as to be able to move on from that determination analytically, i.e. properly arithmetically; with a view to the determinations of magnitude still to be found, these points of departure are the elements from which there is to be derived the function and equation for the concrete, for continuous magnitude. Wherever the interest in employing this procedure chiefly declares itself, what the element is required to supply for the starting point is something determined for itself, against a course that is indirect because it can on the contrary set out only from limits between which the self-determined is supposed to lie, this last being the goal it heads for. In both methods the result then comes out the same, provided only that the law of the further onward determining lets itself be found, even where the complete, i.e. so-called finite, determination demanded remains out of reach. To Keppler the honour is ascribed of having first conceived the thought of reversing that course and of having taken the discrete for his point of departure. The way he explains his understanding of the first proposition in Archimed’s Measurement of the Circle puts this simply. That proposition of Archimed’s runs, as is well known, that the circle is equal to a right-angled triangle whose one leg equals the radius and whose other equals the circumference of the circle. Keppler, construing the sense of this proposition to mean that there are in the periphery of the circle as many parts as there are points, hence infinitely many, and that any one of them may count as the base line of an isosceles triangle, and so forth, thereby gives expression to the resolution of what is continuous into the form of what is discrete. Still far removed is the infinite that turns up here from the determination it is meant to carry in the differential calculus. – Now when a determinateness, a function, has been found for discretes of this kind, they are then to be gathered together again, essentially to be elements of what is continuous. Since, however, no line results from a sum of points and no surface from a sum of lines, the points get taken at once as linear ones, just as the lines get taken as surface-like. Yet because those linear items are at the same time still not to be lines, which is what taking them as quantum would make of them, they get represented instead as infinitely small. Of the discrete only an external gathering is possible, one in which the moments keep the sense of discrete ones; the analytic transition from them reaches no further than their sum, it is not at once the geometrical transition from the point into the line, or from the line into the surface, and so forth; the element, therefore, whose determination is that of point or of line, is at the same time endowed in the one case with linear and in the other with surface quality, so that a sum of small lines may become a line and a sum of small surfaces a surface.

The need to hold fast to this moment of qualitative transition, and for that purpose to take refuge in the infinitely-small, must be regarded as the source of all those representations which, though meant to smooth that difficulty away, are in their own selves the greatest difficulty of all. Making this makeshift dispensable would require showing that a multiplying is in fact already contained in the analytic procedure itself, which has the look of a mere summing. In this respect, however, a new assumption comes in, one that forms the basis of the whole application of arithmetical relations to geometrical figurations, namely that for geometrical determination too arithmetical multiplying is a transition into a higher dimension, – that magnitudes which by their spatial determination are lines, multiplied arithmetically, at the same time produce the linear into a determination of surface; 3 times 4 linear feet gives 12 linear feet, but 3 linear feet times 4 linear feet gives 12 surface feet, square feet namely, the unit in both, as discrete magnitudes, being the same. That lines should be multiplied by lines strikes one at first as absurd, insofar as multiplication concerns numbers generally, i.e. is an alteration of numbers that are wholly homogeneous with what they pass over into, with the product, and that alter the magnitude only. By contrast, what would be called multiplying line as such by line – it has been named ductus lineae in lineam, as also plani in planum, and there is likewise ductus puncti in lineam – amounts to an alteration not of magnitude alone but of magnitude as qualitative determination of spatiality, as a dimension; line's passing over into surface is to be grasped as line's coming-out-of-itself, just as the point's coming-out-of-itself is the line and the surface's is a whole space. This is the same thing represented when one says that the line is the motion of the point and so forth; but motion brings the determination of time with it, and in that representation therefore looks rather like a merely contingent, external alteration of state; what has to be taken is the conceptual determinateness expressed as coming-out-of-itself, – the qualitative alteration, which arithmetically is a multiplying of the unit (as point and so forth) into the amount (into the line and so forth). – One may add here that with the surface's coming-out-of-itself, which would look like a multiplying of surface into surface, the shine of a difference between arithmetical and geometrical producing arises thus: that this coming-out-of-itself, as ductus plani in planum, would arithmetically be a multiplication of the second dimensional determination by such a determination and would thereby give a product of four dimensions, which the geometrical determination nevertheless brings down to three. If on the one side number, precisely because the one is its principle, furnishes the fixed determination for whatever is externally quantitative, then to that same degree its producing is formal; 3 · 3, taken as a determination of number and producing itself, is 3 · 3 · 3 · 3; the same magnitude producing itself as a determination of surface, however, is held back at 3 · 3 · 3, for space, represented as a going-out from the point, that merely abstract limit, has its genuine limit as concrete determinateness in the third dimension counted from the line. The difference just adduced might prove of effect in the case of free motion, where the one, the spatial side, stands under geometrical determination (in Keppler's law s³ : t²), the other, the temporal side, under the arithmetical.

How the qualitative under consideration here differs from the subject matter of the previous Remark may now be left to become clear of itself, without further comment. There the qualitative lay in determinateness of power; here it is, like the infinitely small, merely a factor arithmetically over against the product, or a point over against the line, a line over against the surface, and so forth. As for the qualitative transition to be made from the discrete, as that into which continuous magnitude is represented as resolved, to the continuous, it is carried out as a summing.

That the alleged mere summation does in fact harbour a multiplication within itself, and hence the transition from linear determination into determination of surface, is most simply apparent in the manner in which it is proved, for instance, that the area of a trapezium is equal to the sum of the two opposite parallel lines multiplied into half the height. The height here is represented merely as the amount belonging to a multitude of discrete magnitudes which are to be summed. These magnitudes are lines lying parallel between those two bounding parallels; infinitely many of them there are; for they are to make up the surface, and yet they are lines, so that, to be something surface-like, they must be posited together with negation at the same time. The difficulty that a sum of lines should yield a surface is evaded by assuming the lines straightaway as surfaces, though equally as infinitely thin ones, for their determination lies solely in the linear character of the trapezium's parallel boundaries. Parallel, and bounded by the other pair of the trapezium's rectilinear sides, these lines admit of being represented as terms of an arithmetical progression whose difference is everywhere the same though it need not be determined, and whose first and last terms are those two parallels; the sum of such a series is, as is well known, the product of those parallels into half the amount of the terms. Only quite relatively to the representation of infinitely many lines is this last quantum called an amount; it is the determinateness of magnitude in general belonging to something continuous, – to the height. Clearly, what goes by the name of sum is at once a ductus lineae in lineam, a multiplying of linear by linear, and on the determination given above an emergence of something surface-like. Now in the simplest case, a rectangle a b in general, both factors are simple magnitudes; but in the further and itself still elementary example of the trapezium only one factor, the simple half-height, is such, while the other gets determined through a progression; the latter is likewise something linear, only that its determinateness of magnitude is more involved; insofar as this admits of expression only through a series, the analytic, i.e. arithmetical, interest is said to be that of summing it; the geometrical moment in it, however, is the multiplication, the qualitative side of the transition out of the dimension of the line into surface; the one factor was taken as discrete merely on account of the arithmetical determination of the other, and taken by itself it too, like that other, is the magnitude of something linear.

The procedure of representing surfaces as sums of lines is often resorted to, however, even where no multiplication as such occurs for the sake of the result. That happens where the business is not to state the magnitude in the equation as a quantum, but rather in a proportion. One familiar procedure, for instance, shows that a circle's area stands to the area of an ellipse whose major axis is that circle's diameter as the major axis stands to the minor, each of the two areas being taken as the sum of the ordinates belonging to it; every ordinate of the ellipse stands to the corresponding one of the circle as minor axis to major, and hence, so the inference goes, the sums of the ordinates, i.e. the areas, are related [to each other] likewise. Those who in this connection wish to avoid representing the surface as a sum of lines make the ordinates into trapezia of infinitely small breadth, by the usual and wholly superfluous expedient; because the equation amounts to nothing but a proportion, comparison touches only one of the surface's two linear elements. The other, the axis of the abscissas, is assumed equal in ellipse and circle, hence, as a factor of arithmetical determination of magnitude, equal to = 1, and the proportion accordingly hangs entirely and alone on the ratio of the one determining moment. Two dimensions are needed for the representation of the surface; but the determination of magnitude that is to be stated in that proportion bears on the one moment alone; to humour representation, or to prop it up, by adding the representation of a sum to this one moment, is really to mistake what matters here for mathematical determinateness.

What has been set out here holds also as the criterion for the method of indivisibles of Cavalleri mentioned earlier, which is thereby justified in like manner and needs no recourse to the infinitely small. Lines are these indivisibles when he is considering a surface, squares or circular areas when he is considering a pyramid or cone and so forth; the base line or base surface assumed as determined he calls the rule; it is the constant, and, in relation to a series, the first or last term of that series; those indivisibles are considered parallel to it, hence in like determination as regards the figure. Cavalleri’s general principle now runs (Exerc. Geometr. VI. – the later work – Exerc. I. p. 6.) that all figures, plane as well as solid, are in the ratio of all their indivisibles, these compared with one another collectively and, where perhaps a common ratio obtains among them, distributively.« – To this end he compares, in figures constructed with equal base line and height, the ratios of the lines drawn parallel to that base and at equal distance from it; the whole content of a figure is made up by all such lines, and every one of them carries one and the same determination. In this way Cavalleri proves, for instance, the elementary proposition too that parallelograms of equal height are in the ratio of their base lines; any two lines drawn in the two figures at equal distance from the base line and parallel with it stand in the same ratio of base lines as do the whole figures. The lines do not in fact make up the content of the figure as continuous, but they do make it up insofar as it is to be determined arithmetically; the linear is its element, and its determinateness must be grasped through this alone.

We are led at this point to reflect on the difference obtaining with regard to what the determinateness of a figure falls into, namely, either it is so constituted as the height of the figure is here, or it is outer limit. Insofar as it is as outer limit, one concedes that upon the equality or the ratio of the limit there follows, so to speak, the continuity of the figure; the equality of figures that coincide, for instance, rests on the coinciding of their bounding lines. With parallelograms of equal height and base line, though, only the latter determinateness is an outer limit; the height, not parallelism in general, on which the figures' second principal determination, their ratio, rests, brings a second principle of determination in alongside the outer limits. Euclid's proof of the equality of parallelograms having equal height and base line traces them back to triangles, to continuous magnitudes externally bounded; in Cavalleri’s proof, taking first the proportionality of parallelograms, the limit is determinateness of magnitude as such in general, explicated by being taken on each pair of lines that are drawn at equal distance in the two figures. Taken collectively, these lines, equal to the base line or standing in equal ratio with it, give the figures standing in equal ratio. Representing an aggregate of lines runs counter to the figure's continuity; the consideration of the lines nevertheless exhausts to perfection the determinateness that matters. Cavalleri answers repeatedly the difficulty that the representation of the indivisibles seems to bring with it, namely that lines or planes infinite in amount would have to be compared (Geom. Lib. II. Prop. I. Schol.); he makes the right distinction, that his comparison concerns not their amount, of which we know nothing, – or rather which, as was remarked, is an empty representation called in as a crutch, – but solely the magnitude, i.e. quantitative determinateness as such, this being equal to the space those lines occupy; and because that space lies shut in between limits, so too does its magnitude lie shut in between the very same limits; the continuous is nothing other than the indivisibles themselves, so he says; were it anything outside these, comparison of it would be impossible; yet it would surely be preposterous to hold that bounded continuous magnitudes admit of no comparison with one another.

One sees that Cavalleri means to distinguish what belongs to the external concrete existence of the continuous from that into which its determinateness falls, the latter being what alone, for comparison and for the sake of theorems about the continuous, is to be brought into relief. The categories he employs in doing so, that the continuous is composed of the indivisibles or consists of them and the like, are of course inadequate, since they lay claim at the same time to the intuition of the continuous or, as was said just now, to its external concrete existence; rather than say »that the continuous is nothing other than the indivisibles themselves,« it would be more correct, and thereby at once clear on its own account, to say that the determinateness of magnitude of the continuous is no other than that of the indivisibles themselves. – Cavalleri thinks nothing of the bad inference that there are greater and lesser infinites, an inference which the School drew from the representation that the indivisibles make up the continuous, and he goes on to express (Geom. Lib. VII. Praef.) the more definite awareness that his mode of proof in no way obliges him to represent the continuous as composed out of the indivisible; continuous magnitudes merely follow the proportion of the indivisibles. He has taken the aggregates of the indivisibles, he says, not as they seem to lapse into the determination of infinity for the sake of an infinite multitude of lines or planes, but insofar as they carry in them a determinate constitution and nature of boundedness. Still, to get this stone of stumbling out of the way, he does not shrink from the labour of proving over again, in a seventh book added expressly for the purpose, the chief propositions of his geometry in a manner that keeps free of any intrusion of infinity. – This manner brings the proofs back to the ordinary form adduced above, the coinciding of figures, i.e., as was remarked, to representing determinateness as outer spatial limit.

Regarding this form of coinciding, one further remark may be made first of all, that on the whole it is a so to speak childlike aid for sensory intuition. In the elementary propositions about triangles two such triangles are represented side by side, and, three among their six parts each being assumed equal in magnitude to the corresponding three of the other triangle, it is then shown that these triangles are congruent, i.e. that each has the remaining three parts too equal in magnitude to those of the other, – because in virtue of equality with respect to the first three they coincide with each other. Taking the matter more abstractly, it is just on account of this equality of each pair of mutually corresponding parts in the two that only one triangle is at hand; three parts in it are assumed as already determined, and from these there follows the determinateness of the other three as well. The determinateness thus shows itself complete in three parts; the other three are accordingly, for determinateness as such, a superfluity, the superfluity of sensuous concrete existence, i.e. of the intuition of continuity. Put in such a form, the qualitative determinateness stands out here in distinction from what intuition has before it, the whole as something continuous within itself; coinciding never lets this difference reach consciousness.

With parallel lines and with parallelograms, as was remarked, something new comes in, partly the mere equality of angles, partly the figures' height, and from this last their outer limits, the sides of the parallelograms, are distinct. An ambiguity surfaces here as to how far, with such figures, over and above the determinateness of the one side, the base line, which is as outer limit, the second determinateness, the other outer limit, is to be taken as the parallelogram's remaining side or rather as the height. Given two such figures of one and the same base line and height, of which the one is right-angled while the other has very acute and correspondingly very obtuse opposite angles, intuition may easily find the latter the larger, insofar as it takes the long side lying before it as determining and, after Cavalleri’s manner of representation, compares the planes by a multitude of parallel lines through which they can be cut; the larger side might be seen as a possibility of more lines than the perpendicular side of the rectangle affords. Such a representation supplies no objection to Cavalleri’s method, however; for the multitude of parallel lines represented in the two parallelograms for purposes of comparison already presupposes the equality of their distance from one another or from the base line, and it follows from this that the other determining moment is the height and not the parallelogram's remaining side. Things change further, though, when the comparison is between two parallelograms of equal height and base line that do not lie in one plane and that make differing angles with some third plane; the parallel sections arising when one represents that third plane as laid through them and as travelling on parallel to itself are then no longer equidistant, and those two planes are unequal to one another. Cavalleri very carefully calls attention to this difference, determining it as a difference of transitus rectus and transitus obliquus of the indivisibles (already in Exercit. I. n. XII. ff. as also earlier in the Geometr. 1. II.), and thereby cuts off the superficial misunderstanding that might arise on this side. I recall that Barrow, in the work cited above (Lect. Geom. II. p. 21), while making use of the method of the indivisibles too, though having already adulterated and contaminated it with the assumption, passed from him to his pupil Newton and to the other mathematical contemporaries, Leibnitz among them, that a curvilinear triangle such as the so-called characteristic one may be equated with a rectilinear one insofar as both are infinitely, i.e. very, small, cited an objection of Tacquet’s tending in just this direction, Tacquet being an acute geometer of that day who was likewise at work in the new methods. The difficulty he raised likewise bears on the question which line, in calculating conical and spherical surfaces, ought to be taken as the fundamental moment of determination for a consideration that rests on applying the discrete. Tacquet's objection to the method of the indivisibles is that when the surface of a right-angled cone is to be calculated, that atomistic method represents the triangle of the cone as put together out of the straight lines running parallel to the base line at right angles to the axis, and these are at once the radii of the circles making up the cone's surface. Should this surface now be determined as the sum of the peripheries, and that sum from the amount of their radii, i.e. from the magnitude of the axis, the height of the cone, then such a result stands in contradiction with the truth Archimed otherwise taught and proved. Barrow shows in reply that what has to be taken for determining the surface is not the axis but the side of the cone's triangle, since the revolution of this line is what generates the surface, and it therefore, and not the axis, must be assumed as the determinateness of magnitude for the multitude of the peripheries.

Objections and uncertainties of this sort have their source solely in the indeterminate representation employed of an infinite multitude of points out of which the line, or of lines out of which the surface and so forth, is held to consist; that representation puts the essential determinateness of magnitude of the lines or surfaces in the shade. – The aim of these Remarks has been to point out the affirmative determinations which, in the various uses that mathematics makes of the infinitely-small, remain so to speak in the background, and to draw them out of the nebulosity with which that merely negatively held category shrouds them. With the infinite series, as in the Archimedean measurement of the circle, the infinite signifies no more than that the law of onward determination is known while the so-called finite, i.e. arithmetical, expression is not given and no reduction of the arc to the straight line can be effected; their qualitative diversity is just this incommensurability. The qualitative diversity of the discrete from the continuous in general contains likewise a negative determination, one that makes them appear incommensurable and calls in the infinite, in this sense: that what is continuous, once it has to be taken as discrete, is now supposed to have no quantum any longer in accordance with its continuous determinateness. What is continuous, taken arithmetically as a product, is thereby posited as discrete in its own self, namely broken up into the elements that are its factors; in these its determinateness of magnitude lies; and precisely as being those factors or elements, they belong to a lower dimension and, where determinateness of power enters, to a lower power than the magnitude of which they are elements or factors. Arithmetically this difference looks merely quantitative, that of root and power or of whatever determinateness of power it may be; yet where the expression bears on the quantitative as such alone, for example a : a² or da² = 2a:a² = 2:a, or, for the law of fall, t : at², what it yields are the vacuous ratios 1:a, 2:a, 1:at; against their merely quantitative determination the sides would have to be kept apart by the distinct qualitative significance, as in s:at²; whereby magnitude gets pronounced as a quality, as function of another quality's magnitude. What then stands before consciousness here is merely the quantitative determinateness, with which one operates after its own fashion without difficulty, and one sees no harm in multiplying the magnitude of one line by that of another; yet the multiplying of just these magnitudes yields at the same time the qualitative alteration of the transition from line into surface; to that extent a negative determination sets in; it is this that occasions the difficulty, a difficulty that insight into its own peculiar character and into the simple nature of the matter resolves, but that the aid of the infinite, meant to dispose of it, rather merely throws into confusion and keeps wholly unresolved.

Chapter 3. The Quantitative Ratio

Determined as the negative beyond of the quantum, though a beyond the quantum carries at its own self, is what the infinity of quantum has come to. Such a beyond is the qualitative in general. Being the unity of the two moments, of quantitative and of qualitative determinateness, the infinite quantum is first of all ratio.

Within ratio a merely indifferent determinateness is no longer what quantum has; qualitatively determined is what it is, as referred outright to its beyond. Into its beyond it carries itself on; and that beyond is, to begin with, simply some other quantum. Yet essentially they are not referred to one another as external quanta; rather, each has its determinateness in this reference to the other. So, in this their otherness, they have gone back into themselves; whatever each is, it is in the other; the determinateness of each is made up by the other. – Quantum's passing out beyond itself accordingly carries now this sense, that it did not merely alter into an other, nor into its abstract other, its negative beyond, but that therein it has come to its determinateness; itself is what it finds in its beyond, and that beyond is another quantum. Externality in general makes up the quality of quantum, makes up its conceptual determinateness, and within ratio quantum stands now posited so as to hold its determinateness in that externality of its, at another quantum, so as to be in its beyond that which it is.

Quanta they are, which stand to one another in the reference that has emerged. That reference is itself a magnitude too; not merely in a ratio does quantum stand, but it itself is posited as ratio; one quantum in general is what it is, carrying that qualitative determinateness within itself. As ratio, therefore, it expresses itself as a totality closed within itself, expresses too its indifference towards the limit, and it does so by holding the externality of its being-determined within its own self, being in that externality referred to itself alone and hence infinite at its own self.

Ratio in general is

1. the direct ratio. Therein the qualitative does not yet come forward for itself as such; present it is in no further manner than that of quantum, namely that quantum stands posited as having its determinateness in its own externality. – Quantitative ratio is in itself the contradiction of externality and of self-reference, of the subsistence of the quantorum and of their negation; – and this contradiction sublates itself, in that first of all

2. within the indirect ratio the negation of the one quantum as such gets posited along with the alteration of the other, and so does the variability of the direct ratio itself;

3. within the ratio of powers, however, the unity that refers itself to itself in its own difference makes itself good as the quantum's simple self-production; and this qualitative element, posited at last in simple determination and identical with quantum, becomes measure.

– Much about the nature of the ratios that follow has been anticipated in the preceding Remarks, those concerning the infinite of quantity, that is, the qualitative moment at it; nothing therefore remains but to lay out the abstract concept of these ratios.

A. The Direct Ratio

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

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

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

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

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

B. The Inverse Ratio

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

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

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

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

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

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

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

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

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

Ratio is herewith determined into the ratio of powers.

C. The Ratio of Powers

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

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

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

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

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

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

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

Remark

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

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

Third Section. Measure

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

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

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

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

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

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

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

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

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

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

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

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

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

Chapter 1. Specific Quantity

Qualitative quantity is initially an immediate specific quantum, which

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

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