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.

A. The Specific Quantum

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

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

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

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

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

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

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

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

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

B. Specifying Measure

The same is

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

secondly specific quantity, which determines the external quantum;

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

a. The Rule

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

b. The Specifying Measure

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

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

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

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

Remark

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

c. Relation of Both Sides as Qualities

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

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

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

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

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

Remark

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

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

C. Being-for-itself in Measure

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

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

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

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

Chapter 2. Real Measure

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

Measure, as it is henceforth real, is

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

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

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

A. The Relation of Self-subsistent Measures

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

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

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

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

a. Combination of Two Measures

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

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

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

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