I.2.2. Quantum: The Mathematical Infinite

The mathematical infinite is interesting, first, because its introduction has widened the scope of mathematics and has led to important results therein; next it is remarkable because this science has not yet succeeded in vindicating this use of it conceptually (“concept” being here taken in its proper meaning). Ultimately, such vindications are based on the correctness [demonstrated on other grounds) of the results achieved by means of this determination, and not on the clarity of the object and of the process which produces the results: indeed, the process itself is admitted to be faulty.

This in itself is a flaw: such a procedure is unscientific. But it also involves the drawback that mathematics, not knowing the nature of this its instrument (and this, because it cannot cope with the metaphysics and critique of it), could not determine the sphere in which it must be applied, and could not safeguard itself against misapplication.

In its philosophical aspect, however, the mathematical infinite is important because the concept of the true infinite is, in fact, its basis: it stands on a much higher level than the so-called metaphysical infinite, whence objections against the former are launched. The science of mathematics often has no refuge from these objections except in denying the competence of metaphysics, asserting that it has nothing to do with this science and need not trouble about its concepts if only it acts consistently within its own sphere. It need not consider truth in itself, but only what is true in its own domain. And metaphysics, though it opposes them, cannot deny nor invalidate the brilliant results of the employment of the mathematical infinite; while mathematics cannot achieve clearness about the metaphysics of its own concept, nor, therefore, about the derivation of the methods which the employment of the infinite necessitates.

If it were this single difficulty of the concept in general which oppressed mathematics, it could without ceremony neglect this, at least in so far as the concept is more than an expression of the essential determinations of a thing, that is, those which determine it for the mere Understanding: and mathematics has not allowed such determinateness to be lacking in precision; for it is not a science which is concerned with the concepts of its objects or has to create its content by the development (even if this were only through a process of argumentation) of the concept. But in the method of the mathematical infinite mathematics finds a radical contradiction to that very method which is characteristic of itself, and on which it rests as a science. For the calculation of the infinite admits of, and demands, modes of procedure which mathematics, when it operates with finite magnitudes, must altogether reject, and at the same time it treats these infinite magnitudes as finite Quanta, seeking to apply to the former those same methods which are valid for the latter. The most important step in the development of this science is to have imposed on transcendent determinations and the treatment of these the form of ordinary calculation.

In spite of this conflict in its operations, mathematics shows results which agree exactly with those found by the true mathematical method, the geometrical and analytic method. However, this does not apply to all results; and the end for which the infinite is introduced is not to shorten the beaten road, but to reach results to which it cannot lead. And also the success does not justify the manner of road. Now, this manner of calculating the infinite is plainly oppressed by the apparent inexactitude in which it involves itself when, first, it increases finite magnitudes by an infinitely small amount, and then partly retains these during the further processes, but partly also neglects. This procedure has this strange quality, that in spite of the admitted inexactitude a result is achieved which is exact,—not merely tolerably or to such a degree that the difference might be neglected, but absolutely. But in the process itself which precedes the result we cannot do without the suggestion that a certain quantity is not equal to nil but is so inconsiderable that it may be neglected. But where mathematical exactitude is meant, the distinction between greater and less accuracy quite disappears, just as in philosophy there can be no question of greater or less probability, but of truth alone. If the method and employment of the infinite are justified by success, it is not as superfluous to demand notwithstanding this a justification as to ask in the case of the nose for a demonstration of the right to use it. For mathematical cognition is a scientific cognition, and therefore the proof is essential; and as regards the results, too, it is the fact that a strictly mathematical method does not stamp all with the mark of success,—which mark in any case is only external.

It is worth while more closely to consider the mathematical concept of the infinite, together with the more remarkable among the attempts designed to justify its employment and to remove the difficulty by which the method feels itself oppressed. The consideration of these justifications and determinations of the mathematical infinite, which I shall undertake at some length in this Observation, will also throw the clearest light on the nature of the true concept itself, and will show how a confused notion of this latter was present as a foundation for those processes.

The mathematical infinite is commonly determined as a magnitude than which there is no greater (when it is determined as the infinitely great) or no smaller (when it is determined as the infinitely small); or as one which is greater (in the former) or smaller (in the latter instance) than any given magnitude.—It is true that in this definition the true concept is not expressed, but only, as has already been remarked, the same contradiction as is contained in the infinite progress; however, let us see what it contains in itself. A magnitude is defined in mathematics as something which can be increased and decreased: that is, then, as an indifferent limit. Now since the infinitely great or small is such as cannot be increased nor decreased, it is in fact no longer a Quantum as such.

This consequence follows necessarily and immediately. But it is just the reflection that Quantum has been transcended which is generally not made (and in this Observation I designate as “finite Quantum,” Quantum in general, as we find it): and it is just this which constitutes the difficulty for ordinary thought; for it is demanded that Quantum, in so far as it is infinite, be thought of as transcended and as something which is not a Quantum, although its quantitative determinateness remains.

We may here quote Kant’s judgment about this determination: he finds that it does not agree with the common notion of an “infinite whole.” “According to the ordinary concept, that magnitude is infinite beyond which there can be no greater magnitude (that is, beyond the multitude of a given unit which it contains). But no multitude is the greatest, because it is always possible to add one or more units.—But the notion of an infinite whole does not imply any magnitude, and therefore the concept of an infinite whole is not the concept of a maximum (or minimum): this concept rather expresses the relation of the whole to any given unit, with respect to which it is greater than any number. The infinite would be greater or less according as this unit was taken to be greater or smaller; but infinity, which consists only in the relation to this given unit, will always remain the same, although of course the absolute magnitude of the whole cannot be known through it.”

Kant disapproves when infinite wholes are regarded as a maximum, as the entire multitude of a given unit. Maximum or minimum as such still appears as a Quantum, or a multitude. And this idea cannot avert the conclusion, mentioned by Kant, which leads to a greater or lesser infinite. And generally, so long as the infinite is imagined as Quantum, the distinction between greater and less applies to it. But this criticism has no force against the concept of the true mathematical infinite, the infinite difference, for this no longer is a finite Quantum.

Opposed to this is Kant’s concept of infinity, which he calls the true and transcendental,—namely, “that the successive synthesis of the unit, in measuring a Quantum, can never be completed.” A Quantum in general is supposed to be given: it is demanded that, by synthetizing the unit, this be made into an amount, into a Quantum which can definitely be indicated; and it is asserted that this synthetizing can never be completed. Clearly, it is only the progress to infinity which has here been posited; only it is figured in a transcendental, that is really in a subjective and psychological, manner. It is demanded that in itself Quantum be complete; but (it is said) in the transcendental manner, that is, in the subject, which gives it a relation to a unit, the only determination of the Quantum which arises is such as is incomplete and altogether infected with a Beyond. The contradiction which magnitude contains still remains the limit of our progress; but it is distributed between object and subject, limitedness being attributed to the former, and to the latter the passing beyond every determinate measure which it may have reached at any stage into the bad infinite.

On the other hand it was stated above that the determination of the mathematical infinite, as it is used in the higher analysis, corresponds to the concept of the true infinite: the comparison of the two determinations will now be undertaken in its more detailed development.—And, first, as regards true infinite Quantum, it determined itself as intrinsically infinite; and it is so because, as was seen, finite Quantum (or Quantum in general) and its Beyond, the bad infinite, are alike transcended. Quantum transcended has thus returned into simplicity and self-relation, and this not merely as did Extensive Quantum when it passed over into Intensive Quantum; for this latter has its determinateness only in itself in an external multiplicity, towards which it is required however to be indifferent and from which it must be distinct. But Infinite Quantum contains in itself, first externality, and then the negation of this; and thus it no longer is a finite Quantum or determinateness of magnitude having a determinate existence as Quantum, but is simple, and therefore exists only as moment: it is determinateness of magnitude in a qualitative form, and its infinite nature consists in the fact that it is qualitative determinateness.—As moment, then, it is essentially at one with its Other only as determined through this its Other, that is, it has significance only with reference to something which stands in a relation towards it. Apart from this relation it is nil,—just because the Quantum as such has to be indifferent towards the relation, whilst in the relation it is an immediate constant value. In the relation, as a mere moment, it is not indifferent for itself; since it is at the same time a quantitative determinateness, it exists in infinity (regarded as Being-for-Self) only as a “For-One.”

It will appear that the concept of the infinite as here abstractly expounded is the basis of the mathematical infinite: the concept itself will become clearer if we consider the several steps of the expression of Quantum, as a moment of relation, beginning with the lowest step where it is still also Quantum as such, and proceeding to the higher where it receives the meaning and expression of real infinite magnitude.

Let us first take Quantum in a ratio, such as is a fraction. Such a fraction, for instance 2/7, is not a Quantum like 1, 2, 3 and so on; it is an ordinary finite number, but not absolute, like integers: as fraction it is determined mediately by two other numbers which stand in the relation of Amount and Unit, the Unit itself being a determinate Amount. However, if we abstract from this closer mutual determination of these, and consider them only according to their fate as Quanta, in the qualitative relation in which they here are, 2 and 7 are Quanta otherwise indifferent: here, however, they act only as moments, one of the other, and, therefore, of a third (namely, that Quantum which is called the Exponent): here it is not their status as 2 and 7, but their mutual determinateness, which matters. In their place 4 and 14, 6 and 21, and so on to infinity, might equally well be put. They thus herewith assume a qualitative character. If they counted as mere Quanta, then 2 and 7 are, the one just 2 and the other just 7; and 4,14, 6, 21 and so on are simply other than those numbers, and, in so far as they would be absolute Quanta, they could not be substituted for one another. But, since 2 and 7 are not counted according to their determinateness as such Quanta, their indifferent limit is transcended; and thus in this aspect they contain the moment of infinity; for not only is each number not itself, but also their quantitative determinateness remains: it remains, however, as qualitative in itself, and depends on their value as related. An infinite number of others could be put in their place, and yet the value of the fraction, owing to the determinateness possessed by the relation, is not varied.

But infinity is still not adequately represented by a fraction, and this because the two sides of the fraction, 2 and 7, can be taken apart from the ratio, and are ordinary indifferent Quanta: the relation which subsists between them (namely, that of being in ratio, and of being moments) is external and indifferent to them. And their relation too is an ordinary Quantum, the exponent of the ratio.

The letters with which general arithmetic operates form the next universality into which numbers rise: they have not the peculiarity of bearing a determinate numerical value; they are only general symbols and indeterminate potentialities of any determinate value. And therefore the fraction a/b seems to be a more fitting expression of the infinite, since a and b, taken apart from their relation to one another, remain indeterminate, and severally too have no special and peculiar value.—But, though these letters are posited as indeterminate magnitudes, their significance is that they are some finite Quantum. Thus they are the general image, but the image only of determinate number, and therefore with them too it is indifferent that they should be in a ratio, and they retain this value also when they are no longer in such ratio.

If we consider more closely what is present in Ratio, we find therein these two determinations, first that it is a Quantum, and secondly that this Quantum is not immediate, but contains the qualitative opposition; and also there remains in it that determinate and indifferent Quantum, because it returns upon itself from its otherness (the opposition), and therefore is infinite itself. These two determinations are seen clearly developed in their distinctness from each other in the following well-known form.

The fraction 2/7 may be expressed as 0·285714 . . ., 1/(1 − a) as 1 + a + a² + a³ + . . . It thus appears as an infinite series; the fraction itself is called the sum or finite expression of it. If these two expressions be compared, then one, the infinite series, no longer represents it as a ratio, but in its aspect as Quantum (since it is a multitude of added units) or as Amount.—It is here irrelevant that the magnitudes of which it is supposed, as Amount, to consist are in turn themselves made up of decimal fractions, that is, of ratios; for this fact applies only to the special kind of unit of which these magnitudes are composed, and not to the magnitudes as constituting an Amount; and, similarly, a multi-figured integer of the decimal system is counted essentially as Amount, and the fact is neglected that it consists of the products of a number and of the number ten and the powers of ten. And, similarly, it is here irrelevant that there are other fractions besides the one taken of-, which, if converted into decimals, do not yield an infinite series; every fraction can be so expressed in a numerical system having a different unit.

Thus, in the infinite series which is to represent the fraction as Amount, its aspect of ratio vanishes; and simultaneously there vanishes that aspect which—as was shown above—gave to it infinity. Infinity, however, has now been introduced in another manner; for the series itself is infinite.

Now it is self-evident what is the nature of the infinity of the series: it is the bad infinity of the progression. The series contains and illustrates the contradiction which would represent as a mere Quantum or Amount, or as non-relational, that which is a ratio and includes a qualitative nature. The consequence of this is that the Amount expressed in the series is ever incomplete, so that what is posited must still be surpassed if the determinateness demanded would be reached. The law of progress is known, and is implicit in the determination of Quantum which is contained in the fraction, and in the nature of the form in which this determination has to be expressed. The Amount can indeed be made as exact as is necessary by a continuation of the series; but the representation through the series still remains an Ought; it is infected with a Beyond which cannot be transcended, because to express as Amount that which is based on qualitative determinateness, is an abiding contradiction.

This inexactitude, of which there is only an appearance in the true mathematical infinite, is actually present in this infinite series. These two kinds of mathematical infinite are as little to be confounded as the two kinds of philosophical infinite. In representing the true mathematical infinite the form of a series was originally used, and has again lately been invoked. But this form is not necessary for it; on the contrary the infinite of the infinite series differs from it essentially, as the sequel will show. Indeed, it is inferior even to the fraction as a form of expression.

For the infinite series contains bad infinity, since that which the series is designed to express remains as Ought; and what it does express is infected with a Beyond that never vanishes, and is distinct from what it is desired to express. It is infinite, not because of the number of its terms, but because they are incomplete, because the Other, which essentially belongs to them, is beyond them; that which it really contains (let the terms posited be as many as they will) is but something finite, in the proper sense, posited as finite—that is as something which is not what it ought to be. That, on the contrary, which is called the finite expression or sum of such a series, has nothing lacking: it contains fully the value which the series only seeks after; the Beyond is recalled from its flight; what it is, and what it ought to be, are not severed, but the same.

The differentia of these two is, more precisely, this: in the infinite series the negative is outside its terms, which are present only in so far as they count as parts of the Amount. On the other hand, in the finite expression, which is a ratio, the negative is immanent as being the mutual determinateness of the sides of the ratio; and this, being a return upon self or self-relating unity as negation of negation (for both the sides of the ratio exist only as moments), contains the determination of infinity.—Thus, in fact, what is ordinarily called sum (like 2/7 or 1/(1 − a)) is a ratio; and this so-called finite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum: it is designed to set forth in the form of sum what is in itself a ratio, and terms of the series exist not as terms of a ratio but of an aggregate. And, further, it is, more properly, the finite expression; for it is an incomplete aggregate and essentially has something lacking. According to its actual content it is a determinate Quantum, but also a Quantum less than it should be; and that which is lacking is then also a determinate Quantum. This lacking part is in fact that which is called the infinite part of the series, in this merely formal meaning, that it is a something which is lacking, or a not-being; according to its content it is a finite Quantum. It is only that which the series contains, plus that which is lacking, which constitutes the fraction,—that determinate Quantum which it too ought to, but cannot, be.—In the infinite series, too, the word “infinite” generally enjoys the reputation of something lofty and venerable; this is a kind of superstition, the superstition of the understanding; we have seen that it reduces itself rather to the determination of a deficiency.

We may still remark that the existence of infinite series which cannot be summed is a circumstance altogether external and contingent with respect to the form of the series. These contain a higher kind of infinity than do the series which can be summed: they contain incommensurability,—that is to say, the quantitative ratio which they contain cannot be represented as Quantum even in the shape of a fraction. However, the form of series as such, which is theirs, contains the same determination of bad infinity that exists in the series which admits of being summed.

The terminological inversion remarked above in connexion with fractions and their equivalent series is found also in so far as the mathematical infinite—not that just mentioned, but the true one—has been called the relative infinite, and the ordinary metaphysical infinite on the other hand (by which is meant the abstract and bad infinite), the absolute. But, in fact, this metaphysical infinite is merely relative, because the negation which it expresses remains in the opposition of a limit only in such a way that the limit persists outside it and is not transcended by it; whereas the mathematical infinite has veritably in itself transcended its finite limit, because its Beyond is incorporated with it.

It is in this sense, in which it has just been demonstrated that the so-called sum or finite expression of an infinite series is in fact to be regarded as its infinite expression, that Spinoza opposes the concept of true infinity to that of bad, and elucidates it by means of examples. His concept gains the greatest clarity if I join his observations on this subject to my exposition.

He begins by defining the Infinite as the absolute affirmation of the existence of any one Nature, and the Finite on the contrary as determinateness, or negation. For the absolute affirmation of an existence must be taken as meaning its self-relation, the fact that its existence does not depend on that of an Other. The finite, on the other hand, is negation, that is, cessation as meaning relation to an Other which begins beyond it. It is true that the absolute affirmation of an existence does not exhaust the concept of infinity; for this concept also implies that infinity is affirmation not as immediate, but only as restored through the intro-Reflection of the Other, or as negation of the negative. But with Spinoza, Substance and its absolute unity have the form of an inert, that is, of a not self-mediating, unity,—of a rigidity wherein the concept of the negative unity of the self (Subjectivity) has not yet found a place.

Spinoza explains the true infinite by means of a mathematical example (Epist. XXIX), namely, a space contained between two unequal circles which are not concentric, while one lies within the other without touching it. It seems that he thought much of this figure and of the concept for which he used it as example: he made it the motto of his Ethics.—“Mathematicians,” he says, “conclude that the inequalities which are possible in such a space are infinite, not from the infinite multitude of parts (for its magnitude is determinate and limited, and I can posit similar spaces both greater and smaller), but because the nature of the matter surpasses any determinateness.”—Evidently Spinoza rejects that idea of the infinite according to which it is imagined as multitude or as incomplete series, reminding us that here in the space of the example the infinite is not beyond, but is present and complete; this space is limited, and is infinite, “because the nature of the matter surpasses all determinateness,” because, that is, the magnitudinal determination which it contains cannot also be represented as a Quantum, or (to use the expression of Kant quoted above) because the synthesis cannot be completed into a (discrete) Quantum.—How in general the opposition of continuous and discrete Quantum leads to the infinite, must be explained in a later Observation.—Spinoza calls this serial infinite “the infinite of imagination,” but the infinite as self-relation “the infinite of thought,” or infinitum actu. For it is actually (actu) infinite, because it is complete and present in itself. Thus the series 0·285714 . . . or 1 + a + a² + a³ + . . . is merely the infinite of fancy or opinion, for it has no actuality and has something altogether lacking; while 2/7 or 1/(1 − a) is the actual infinite, and not only is what the series is in those of its terms which are given, but has also that which the series lacks and is what the latter only ought to be. The fraction 2/7 or 1/(1 − a) is also a finite magnitude, like Spinoza’s space enclosed between two circles, with its inequalities; and, like that space, it may be increased or decreased. But this does not result in the absurdity of an infinite which may be greater or less; for this Quantum of the whole does not touch the ratio between its moments, the “nature of the matter,” that is, the qualitative determination of magnitude. That which the infinite series presents to us is also a finite Quantum; but, further, it is defective.—But fancy stands still at Quantum as such, and does not reflect upon the qualitative relation that is the basis of the incommensurability which faces it.

That incommensurability which is implied in Spinoza’s example includes in a general way the functions of curves, and, more exactly, leads to the infinite which mathematics has introduced with such functions and, generally, with the functions of variable magnitudes; this is the true mathematical or qualitative infinite, which also Spinoza had in mind. We will now discuss this determination in greater detail.

First, as regards that important category of variability, which comprises the magnitudes which are related in these functions, these are supposed to be variable, but not with that meaning in which in the fraction 2/7 the two numbers 2 and 7 are variable, since 4 and 14, 6 and 21, and other numbers to infinity, may be put in their place without altering the value denoted in the fraction. Similarly, in a/b any number may be put in place of a or b without altering what a/b is intended to express. It is only in this sense (that an infinite, that is, an inexhaustible, multitude of numbers may be put in the place of the x and y of a function) that a and b are variable magnitudes as much as that X and that y. And hence the expression “variable magnitudes” is very vague and ill-chosen for determinations of magnitude whose interest and manner of treatment lie in something quite different from their mere variability.

In order to make clear where lies the true determination of the moments of a function to which higher analysis devotes its attention, we must once more run through the stages to which attention has already been drawn. In 2/7 or a/b, 2 and 7 are each a determinate Quantum, and their relation is not essential to them; and a and b are likewise meant to represent such Quanta as remain what they are even outside the ratio.

2 a And, further, - and - are each a fixed Quantum, a quotient; the ratio constitutes a number of which the unit is expressed by the denominator and the Amount of these units by the numerator, or conversely; and even when 4 and 14, or other numbers, are substituted for 2 and 7, the ratio, as a Quantum, remains the same. But in the function y²/x = p, for example, this is essentially changed; for, though here x and y have the meaning that they can be determinate Quanta, it is not x and y but only x and y² which here have a determinate quotient. Hence these sides of the ratio, x and y, first are not determinate Quanta, and secondly the ratio subsisting between them is not a fixed quotient or Quantum (nor is such a Quantum intended as with a and b); but this quotient, simply as a Quantum, is itself variable. But all this is implied in the fact that X is related, not to y, but to the square of y. The relation of a magnitude to a power is not a Quantum, but is essentially qualitative: the power-relation is the circumstance which must be regarded as the fundamental determination.—But in the function of the straight line, y = ax, y/x = a is an ordinary fraction and quotient; hence this function is only formally a function of variable magnitudes, and here x and y are what a and b are in a/b; that is, they are not in that determination in which the differential and the integral calculus consider them.—On account of the peculiar nature of variable magnitudes, in this point of view it would have been well done to have introduced a name for them and also symbols other than those used for ordinary unknown magnitudes in any finite equation determinate or indeterminate; and this because of their essential difference from such merely unknown magnitudes, which in themselves are wholly determinate Quanta, or a determinate range of determinate Quanta.—And it is only because the peculiarity of that which constitutes the interest of the higher analysis, and has led to the need for and invention of the differential calculus, is not fully understood, that functions of the first degree, like the equation of the straight line, have been included by themselves in the treatment of this calculus. This formalism results, further, from the misconception which thinks to fulfil the just demand for the universalization of a method by neglecting the specific determinateness on which the need is based, so that here the assumption is made that in this sphere only variable magnitudes in general are dealt with. Much formalism in the consideration and treatment of these objects would have been spared if it had been understood that it dealt, not with variable magnitudes as such, but with determinations of powers.

But the mathematical infinite shows itself in its peculiar nature at yet a further stage. In an equation where x and y are posited primarily as determined through a ratio of powers, x and y as such are still meant to denote Quanta: now this meaning is entirely lost in the so-called infinitesimal differences. dx and dy are no longer Quanta and are not supposed to signify such; they have a significance only in their relation, a meaning merely as moments. They no longer are Something (Something being taken as Quantum), nor are they finite differences; but also they are not Nothing or the indeterminate nil. Apart from their relation they are pure nil; but they are meant to be taken only as moments of the relation, as determinations of the differential coefficient dx/dy.

In this concept of the infinite, Quantum veritably is perfected into a qualitative Determinate Being; it is posited as actually infinite, and is transcended not only as being this or that Quantum, but as Quantum in general. And yet the quantitative determinateness as element of Quanta remains the principle, or, as has also been said, it remains part of their primary concept.

All the attacks which have been made upon the fundamental determination of the mathematics of this infinite, that is, on the differential and integral calculus, have been directed against this concept. The mistaken ideas of mathematicians themselves prevented it from being justly recognized; but mainly these attacks are due to the impossibility of justifying this object as a concept. But, as has been mentioned above, mathematics cannot here evade the concept; for, being the mathematics of the infinite, it does not confine itself to the finite determinateness of its objects (as does genuine mathematics, where space and number and their determinations are considered and related only according to their finitude); but, on the contrary, it transplants into identity with its opposite a determination already taken from that study and treated by it, as for instance it makes a curve into a straight line, or a circle into a polygon, and so on. The operations which it presumes to undertake as differential or integral calculus are, therefore, quite contradictory to the nature of merely finite determinations and their relations; their only justification, therefore, would be in the concept.

Now when the mathematics of the infinite still maintained that these quantitative determinations were vanishing magnitudes, that is, magnitudes which no longer are any Quantum, but also are not nothing, being merely a determinateness as opposed to Other, it seemed abundantly clear that such an intermediate state, as it was called, between Being and Nothing did not exist.—The nature of this objection and of the so-called intermediate state has been indicated above under the category of Becoming, Observation 4. The unity of Being and Nothing is not indeed a state’, for a state would be a determination of Being and Nothing such as might have been reached by these moments only contingently, as it were through disease or external influence, and through erroneous thinking; but, on the contrary, this mean and unity, this vanishing and, equally, Becoming, is, in fact, their only truth.

It has further been said that infinities cannot be compared as greater or less, and that, therefore, there could be no ratio between infinite and infinite in their ranks and orders, as in the distinctions of the infinite differences that occur in the science which concerns itself with them.—This objection, already referred to, is based upon the idea that Quanta are here in question, and are being compared as such; and that determinations which are Quanta no longer, have no longer the relation of ratio. But, on the contrary, that which is only in such a relation is no Quantum: Quantum is a determination which is supposed to have a perfectly indifferent existence apart from its ratio, and its distinctness from an Other is supposed to be indifferent to it: while that which is qualitative is that only which it is in its distinctness from an Other. These infinite magnitudes, therefore, are not only comparable, but exist only as moments of the comparison or ratio.

I proceed to enumerate the chief determinations which have been offered in mathematics about this infinite; it will appear that, although they are based upon the thought of the real matter, agreeing with the concept here developed, its authors did not fully probe it as concept, and in applying it were again forced to have recourse to means dissonant with their better cause.

No correcter determination of the thought can be made than that offered by Newton. I here set apart the determinations belonging to the idea of motion and velocity (from which latter chiefly he took the name of fluxions), for here the thought appears not in its due abstraction, but concrete and mixed with unessential forms. Newton explains these fluxions (Prim. Mathem. Phil. Nat. L. 1. Lemma XI. Schol.) by saying that he takes them not as indivisible (a form used by earlier mathematicians, Cavalieri and others: it contains the concept of a Quantum determinate in itself), but as vanishing divisibilia; and, further, not as the sums and ratios of determinate parts, but as the limits (limites) of the sums and ratios. It will be objected, he says, that vanishing magnitudes have no final ratio, because the ratio before they vanish is not the last, and, after they have vanished, no longer exists. But by the ratio between vanishing magnitudes must not be understood the ratio which exists either before or after, but that with which, they vanish (quacum evanescunt). And, similarly, the first ratio of becoming magnitudes is that with which they arise.

Newton did what was demanded by the stage which scientific method had reached in his day—he only explained what was to be understood by a certain expression; but really the demand for such a particular explanation is subjective, or historical, and it is not demonstrated that such a concept is necessary in itself and possesses inner truth. But what has been quoted shows that the concept as posited by Newton conforms with the development of infinite magnitude from the reflexion of Quantum, as it was set out above. He means magnitudes which are at the point of vanishing, that is, are Quanta no longer; and not ratios between definite parts, but limits of the ratio. Thus, the Quanta in themselves, the sides of the ratio, and also the ratio in itself (that is, in so far as it is Quantum), are supposed to vanish; the limit of the magnitudinal ratio is that point where it is and is not,—or, more precisely, where the Quantum has vanished, and the ratio, therefore, is preserved only as qualitative quantity-ratio, and its sides as qualitative quantity-moments.—Newton adds that it must not be concluded from the existence of final ratios between vanishing magnitudes that there are final magnitudes or indivisibilia. For this would be another leap from the abstract ratio to such sides of it as should have a value for themselves outside their relation, as being indivisibilia or something that is a One, a non-relational entity.

And further, to refute this error he reminds us that final ratios are not ratios between final magnitudes, but limits to which the ratios of the magnitudes decreasing without limit are nearer than any given (that is, finite) difference; but they never overstep this limit so as to become Nothing.—It has been said above that by final magnitudes indivisibilia, or Ones, might have been understood. But in the determination of final ratio the idea of indifferent One (non-proportional Quantum), as well as that of finite Quantum, is absent. But, if the suggested mode of determination had been developed into the concept of a form of magnitude which is simply a moment of ratio, there would be need neither of that decrease without limit into which Newton transposes Quantum (and this expresses only the progress to infinity), nor of the characteristic of divisibility, which here no longer has any immediate meaning.

In connexion with the preservation of the ratio at the vanishing of Quanta, the expression is elsewhere found (for instance in Carnot, Réflexions sur la Métaphysique du Calcul Infinitésimal) that, by virtue of the law of continuity, vanishing magnitudes still retain the ratio whence they arise before they vanish.—This idea expresses the true nature of the matter in so far as it is not the continuity of Quantum which is meant, for this continuity of the Quantum would consist in the endless process by which it continues itself into its own vanishing in such a manner that in its Beyond there only arises again a finite Quantum, a new term of the series; while a continuous progress is always imagined as such that the values, which still are finite Quanta, are passed through. But where the transition made is to the true infinite, it is the ratio that is continuous; and it is so completely continuous and self-supporting that the transition just consists in presenting the ratio in its complete purity and causing the non-relational factor (that is, the concept of a Quantum which is one term of a ratio and yet also a Quantum apart from this relation) to vanish.—This purification of quantitative ratio is exactly analogous to what happens when an empirically given existent is grasped by conceptual thought. Such an existent is thereby raised above itself in such a manner that its concept contains the same determinations as itself, but taken in their essentiality and in the unity of the concept, wherein they have lost their indifferent and non-conceptual persistence.

The other form of Newton’s exposition of the magnitudes in question—namely, generative magnitudes or principles—is equally interesting. A generated (genita) magnitude is a product or quotient, roots, rectangles, squares, and also the sides of rectangles and squares, and, in general, any finite magnitude.—“It is considered as variable in its incessant motion and flow of increase or decrease, and to its momentary augmentation and diminution he gives the name of Moments. These, however, must not be taken as particles of determinate magnitude (particulae finitae). They are not moments themselves, but magnitudes produced by moments; the generative principles or beginnings of finite magnitudes must here be understood.”—An internal distinction is here made in Quantum; it is taken first as product or Determinate Being, and next in its Becoming, in its beginning and principle, that is, as it is in its concept or (which is here the same thing) in its qualitative determination. In the latter the quantitative differences, the infinitesimal incrementa or decrementa, are moments only; and it is only in what has been generated that we have Quantum or that which has passed over into the indifference of determinate existence and into externality.—But, although the philosophy of the true concept must acknowledge those determinations of the infinite just mentioned in the consideration of the incrementa and decrementa, it must also be observed that the forms themselves of incrementa (and so on) fall within the category of absolute Quantum and of the continuous progress to which we have alluded; indeed, the ideas of increment, of addition, of growth of x by dx or i, and so on, must be considered the fundamental evil inherent in these methods,—as an enduring obstacle which makes it impossible clearly to disengage the determination of the qualitative moment of quantity from the idea of ordinary Quantum.

The idea of infinitely small magnitudes (latent also in increment and decrement) is far inferior to the mode of conception indicated. This idea supposes them to be of such a nature that they may be neglected in relation to finite magnitudes; and not only that, but also their higher orders relatively to the lower order, and the products of several relatively to one.—With Leibniz this demand to neglect (which previous inventors of methods referring to this kind of magnitude also bring into play) becomes more strikingly prominent. It is this chiefly which gives an appearance of inexactitude and express incorrectness, the price of convenience, to this calculus in the course of its operation.—Wolf has attempted to make it intelligible in his manner of popularizing a problem, which is to confuse the concept and to put in its place incorrect sensuous images. He compares the neglect of infinitesimal differences of higher orders relatively to lower with the procedure of a surveyor who, in measuring the height of a mountain, is none the less exact if meanwhile the wind has blown away a grain of sand from the top; or with the neglect of the height of houses or towers in a calculation of eclipses of the moon (Element. Mathes. univ. Tom. I. El. Analys. math. P. II. C. 1. s. Schol.).

Common sense in fairness admits such an inaccuracy; but all geometers have rejected this idea. It is perfectly self-evident that in the science of mathematics such empirical accuracy is not at all in question, and that mathematical measurement by means of the operations of the calculus or the constructions and proofs of geometry differs altogether from surveying, from the measuring of empirical lines, figures, and so on. And this apart, analysts (as mentioned above) demonstrate, by a comparison of the result as reached in a strictly geometrical way and by the method of infinitesimal difference, that one is the same as the other and that there is, absolutely, no more nor less of exactness. And it is obvious that an absolutely exact result could not be obtained by an inexact procedure. And yet, on the other hand, the procedure itself cannot do without this neglect of the “insignificant,” in spite of all protests against the method of justification just quoted. And this is the difficulty about which the analysts are at pains,—to make intelligible and to remove the inherent anomaly.

In this regard Euler’s idea especially must be cited. On the basis of Newton’s general definition, he insists that the differential calculus considers the ratios of the incrementa of a magnitude, while the infinitesimal difference as such is to be regarded wholly as nil (Institut Calc, different. P.I.C. III.).—It will be clear from the above how this is to be understood: the infinitesimal difference is nil only quantitatively, it is not a qualitative nil, but, as nil of quantum, it is pure moment of a ratio only. There is no magnitudinal difference; but for that reason it is, in a manner, wrong to express as incrementa or decrementa and as differences those moments which are called infinitely small magnitudes. This point of view has for basis the thought that something is joined to or taken from the finite magnitude first given, that subtraction or addition—an arithmetical external operation—takes place. But the transition of the function of the variable magnitude into its differential must be considered; it is of quite different nature, and in fact, as decided above, it must be regarded as the reduction of the finite functions to the qualitative ratio of their determinations of quantity.—On the other hand, the difficulty is self-evident when it is said that for themselves the incrementa are each nil, and that only their ratios are being considered; for a nil is altogether without determinateness. Thus this image, although it reaches the negative aspect of Quantum and expressly asserts it, yet does not simultaneously seize this negative in its positive meaning of qualitative determinations of quantity, which, if torn away from the ratio and treated as Quanta, would each be but a nil.—Lagrange (Théorie des fonctions analytiques, Introd.) judges, regarding the idea of limits or final ratios, that while the ratio between two magnitudes may easily be imagined so long as they remain finite, this ratio offers to understanding no clear and determinate concept when its terms simultaneously become nil.—And indeed understanding must pass beyond this merely negative side, where each member of the ratio is a nil as Quantum, and must take them positively, as qualitative moments.—But we cannot regard as satisfactory Euler’s further remarks (loc. cit. § 84 sqq.) with regard to this theory of his, by way of proving that two so-called infinitesimal magnitudes, which ought to be nothing but nil, yet stand in a ratio to one another; on which account they are commonly denoted, not by the symbol for nought, but by others. He attempts to base this upon the distinction between arithmetical and geometrical relations: in the former we consider the difference, in the latter the quotient, and, although the former is the same between any two noughts, this is not true of geometrical ratio; if 2 : 1 = 0 : 0, then it would follow from the nature of the proportion (since the first term is twice as great as the second) that the third term is twice as great as the fourth; thus, according to the proportion, o: o is to be taken as being the ratio of 2 : 1.—And even in common arithmetic n.0 = 0; and therefore n : 1 = 0 : 0.—But it is just because 2 : 1 or n : 1 is a ratio between Quanta that no ratio nor designation like o: o is adequate to it.

I refrain from citing any further attempts. Those which we have considered have shown sufficiently that they contain the true concept of the infinite, but that it is not disengaged nor seized upon in its determinateness. Therefore, when mathematicians proceed to practice, it is impossible for the true conceptual determination to assert itself therein: the finite determinateness of quantity returns, and the operation can no longer do without the idea of a Quantum which is merely relatively small. The calculus makes it necessary to subordinate the so-called infinite magnitudes to the ordinary arithmetical operations of addition and so on (which are based on the nature of finite magnitudes), and thus to count and treat them for a moment as finite magnitudes. And on the calculus rests the burden of justifying this method, where first, drawing them down into this sphere, it treats them as incrementa or differences, and then neglects them as Quanta after having applied to them the forms and laws of finite magnitudes.

I now recount the chief heads of the attempts made by the geometers to remove these difficulties.

The older analysts had small scruples in this matter; the efforts of the moderns were chiefly directed towards bringing back the infinitesimal calculus to the evidence of a “properly geometrical method,” and thereby attaining in mathematics to the “strictness of proof of the ancients.” (These are Lagrange’s expressions.) But since the principle of the analysis of the infinite is of higher nature than the principle of the mathematics of finite magnitudes, the former immediately was forced to renounce this kind of evidence; just as philosophy cannot claim the kind of obviousness proper to the sciences of the sensuous, such as natural history,—and indeed eating and drinking are counted a more intelligible business than thinking and understanding. We shall thus deal only with the attempts to reach the strictness of proof of the ancients.

Many have tried altogether to discard the concept of the infinite, and to effect their aim while dispensing with what seemed indispensable.—Lagrange speaks for example of the method invented by Landen, and says that it is purely analytical and does not employ infinitesimal differences, but at first introduces different values of the variable magnitudes, and, in the course of the demonstration, makes them equal. But he considers that in other respects the advantages peculiar to the differential calculus—simplicity of method and ease in operation—are here lost.—Probably this procedure has something which corresponds to the starting point of Descartes’ tangential method, of which further mention must be made below. But we may here remark that this at least is clear, that the general procedure which assumes different values of the variable magnitudes and afterwards makes them equal, belongs to a different sphere of mathematical treatment from that to which the method of the differential calculus itself belongs; and that the peculiarity of the simple relation (to be more fully considered later) to which the actual concrete determination of the calculus reduces itself, namely, the relation of the derivative to the original function, is not here emphasized.

The older of the moderns (such as Fermat, Barrow, and others), who first used the infinitesimally small in that application which was later developed into differential and integral calculus, and, subsequently, Leibniz and his successors, with Euler, always undisguisedly believed that they could neglect the products and the higher powers of infinitesimal differences only because they vanish relatively to the lower order. It is on this alone that with them the fundamental proposition is based, namely, the determination of what the differential of a product or a power is; for to this all their theoretical doctrine reduces itself. The rest is partly the mechanism of development and partly also application, in which, however (as must further be considered), in fact lies the higher or rather the only interest.—With respect to our present subject, only the elementary statement need here be made that, for the same reason of insignificance as caused the acceptance of the capital proposition, it is assumed, with regard to curves, that the elements of curves, the incrementa of abscissae and ordinates, are in the relation of subtangents and ordinates to one another; and for the purpose of obtaining similar triangles, the arc which constitutes the third side of a triangle together with the two incrementa of the characteristic triangle (as it used rightly to be called formerly) is regarded as a straight line and as part of the tangent, and thus one of the incrementa is regarded as reaching the tangent. By these assumptions those determinations are, on the one hand, raised above the nature of finite magnitudes; on the other hand a procedure is applied to the moments (now called infinitesimal) which is valid only for finite magnitudes; and where this is applied nothing may be neglected on the ground of insignificance. With such a mode of procedure, the difficulty which obsesses the method remains in all its force.

A remarkable procedure of Newton’s may here be noted (Princ. Math. phil. nat. Lib. II. Lemma II. after Propos. VII.)—the invention of an ingenious trick designed to remove the neglect, arithmetically invalid, of the products of infinitesimal differences or of higher orders of these in the process of finding differentials. He finds the differential of the product (whence the differentials of quotients, powers and so on are then easily derived) in the following manner. When x and y are each taken as smaller by half its infinitesimal difference, the product passes over into xy − xdy/2 − ydx/2 + dxdy/4; but if x and y are increased by the same amount, it passes over into xy + xdy/2 + ydx/2 + dxdy/4. If the first product be subtracted from this second product, there remains over ydx + xdy as surplus, and this is said to be the surplus due to an augmentation by a whole dx and dy, for this augmentation is the difference between the two products; it, therefore, is the differential of xy.—Clearly, in this procedure, the term which forms the chief difficulty, the product of the two infinitesimal differences, dxdy, drops out by itself. But, in spite of the name of Newton, it must be said that such an operation, though very elementary, is incorrect; it is not true that (x + dx/2)(y + dy/2) − (x − dx/2)(y − dy/2) = (x + dx)(y + dy) − xy. It can only have been the need to justify the fluxional calculus in its importance which could bring a Newton to deceive himself with such a proof.

Other forms which Newton employed in the derivation of the differential are bound up with concrete meanings (having reference to motion) of the elements and their powers.—In the case of the serial form, which elsewhere characterizes his method, the statement suggests itself that it is always possible to take the magnitude as exactly as required by the addition of further terms, and that those which are left out are relatively insignificant and the result only an approximation; and not that here too he was satisfied with that ground on which, in his method of solving equations of higher degrees by approximation, he omits the higher powers (which arise when any value, just found and as yet inexact, is substituted in the given equation)—namely for the crude reason that they are small; see Lagrange, Équations Numériques, p. 125.

The error into which Newton fell in the solution of a certain problem through neglecting higher powers which were essential to the problem, gave his opponents an opportunity of triumph for their method over his: Lagrange, in his recent investigation of it (Théorie des fonctions analytiques, me P., Ch. IV,), has shown its true origin; and this mistake proves the formality and unreliability then still inherent in the use of this instrument. Lagrange shows that Newton made the mistake because he neglected the term of the series which contained that power which was essential in the given problem. Newton had remained faithful to the formal and superficial principle of omitting terms because of their relative smallness.—For it is known that in mechanics there is attributed a definite meaning to the terms of a series in which the function of a motion is developed, so that the first term (or the first function) refers to the moment of velocity, the second to accelerating force, and the third to the resistance of forces. Thus here the terms of the series must not only be regarded as part of a sum, but as qualitative moments of a conceptual whole. For this reason, if here the other terms of the series (infinite in the bad sense) are omitted, this has a significance wholly different from that which belongs to their omission on the ground of relative smallness. The Newtonian solution contained this mistake, not because in it terms of the series were taken account of only as parts of a sum, but because the term containing the qualitative determination (and this was the essential matter) was neglected.

In this example, the procedure is made dependent upon the qualitative meaning. And here the general assertion can be made that the whole difficulty of the principle would be removed, if the qualitative meaning of the principle were indicated and the operation made dependent upon it, in place of the formalism which identifies the determination of the differential with the problem which gives it its name, the distinction generally between a function and its variation after its variable magnitude has received an increment. In this sense, it is clear that the first term of the series resulting from the development of (x + dx)ⁿ quite exhausts the differential of xⁿ. Thus the neglect of the other terms is not due to their relative smallness;—and no inexactitude, no mistake nor error is here assumed which is supposed to be compensated or rectified by another error,—a point of view from which many, and notably Carnot, justify the ordinary method of the infinitesimal calculus. A ratio, and not a sum, is here in question; and, therefore, with the first term the differential is fully found; and where further terms and differentials of higher orders are required, their determination does not imply the prolongation of a series as sum, but the repetition of one and the same ratio, which alone is aimed at and is complete already in the first term. The need for the serial form, its summation, and all therewith connected, must be quite separated from this interest in ratio.

The explanations of Carnot concerning the method of infinitesimal magnitudes contain, most lucidly exposed, all that is soundest in the ideas cited above. But in the transition to practice the usual ideas about the infinite smallness of the terms omitted, relatively to the others, more or less emerge. And Carnot justifies his method by the fact that the results turn out correct, and by the advantage coming from the introduction of such imperfect equations (as he calls them—meaning those where there has been such an arithmetically incorrect omission) in the simplification and abbreviation of the calculus: he does not justify it from the nature of the matter itself.

It is well known that Lagrange resumed Newton’s original method, the method of series, in order to be rid of the difficulties inherent in the idea of the infinitesimally small and in the method of first and last ratios and limits. The advantages of his functional calculus for exactitude, abstractness, and universality are generally acknowledged: here it is relevant to mention that it rests on the fundamental proposition that the difference, without becoming nil, may yet be taken as so small that any term of the series is greater than the sum of all the following terms.—In this method too a beginning is made from the categories of augmentation and the difference of the function, whose variable magnitude receives the augmentation from the original function, thereby introducing that tiresome series; while in the sequel the terms of the series which must be omitted are important only in the respect that they constitute a sum,—the reason of their omission being placed in the relative nature of their Quantum. Here too, therefore, the omission is not in a general way reconducted to that point of view which partly occurs in some applications, where (as was recalled above) the terms of the series are supposed to have a definite qualitative meaning, and terms are neglected because they are qualitatively insignificant, and not because they are insignificant in magnitude; but then partly the omission itself finds no place in the essential point of view, which, for the so-called differential coefficient, manifests itself only with Lagrange in the so-called application of the calculus,—as will be explained more fully in the next Observation.

That element of the form of magnitude under discussion which is called the infinitesimally small has been proved to partake of a qualitative character; and this is found most immediately in the category of the Limit of Ratio cited above, the consistent use of which in the calculus has become a characteristic and peculiar method. Lagrange criticizes this method as lacking ease in application and offering no definite idea in the expression “limit”: we will take up the second criticism and consider more closely what is stated about its analytical significance. For the idea of limit does contain the valid category mentioned of qualitative determination of the ratio of variable magnitudes; for the forms which these introduce (dx and dy) must be regarded simply as moments of dy/dx, and dx/dy itself as a single and indivisible symbol. We may disregard the fact that the practical application of the mechanism of the calculus loses the advantage which it derives from the separation of the sides of the differential coefficient. Now this limit is supposed to be Limit of a given function;—to indicate a certain value with reference to it, which value determines itself according to its derivation. The mere category of limit, however, would not take us beyond the scope of this Observation, which is to show that the infinitesimally small (which appears as dx and dy in the differential calculus) has not merely the negative and empty meaning of a not finite and not given magnitude (as when we say “an infinite multitude,” “to infinity,” and the like), but also the definite meaning of a qualitative determinateness of the Quantitative, or a moment of ratio as such. As yet, however, this category has no relation to what a given function is, and does not interfere in the treatment of it nor in the use made of this determination in it; and thus the idea of limit too, if confined to this demonstrated determinateness, would lead nowhere. The term “limit” itself, however, implies that it is the limit of something, that is, that it expresses a certain value contained in the function of variable magnitude; and we must here investigate the nature of this concrete behaviour.—It is supposed to be the limit of the ratio between the two increments by which the two variable magnitudes connected in an equation (one being regarded as a function of the other) were taken to be increased; the augmentation is here taken as just indeterminate, and, in so far, the infinitesimally small is not employed. But, first, the approach by which this limit is found involves the same inconsistencies as are implied in all the other methods. The approach is as follows. If y = fx, then fx, when y passes over into y + k, is to change into fx + ph + qh² + rh³ + . . ., and thus k = ph + qh² + . . ., and k/h = p + qh + rh² + . . . Now if k and h vanish, the second member, except p, also vanishes: p thus is the limit of the ratio of the two increments. Clearly h (as Quantum) is equated with o, while k/h is nevertheless not supposed to equal 0/0, but still to remain a ratio. Now the idea of the limit is supposed to afford the advantage of averting the implied inconsistency; and p at the same time is supposed to be, not the actual ratio (which would be = 0/0), but only the determinate value to which the ratio may approximate infinitely, that is, in such a manner that the difference can become smaller than any given difference. The closer meaning of approximation (with respect to the terms which are supposed to approximate) will be considered below.—But if anything can be self-evident in mathematics, then this is self-evident, that a quantitative difference which has the determination, not only that it can, but that it shall, be smaller than any given difference, has ceased to be a quantitative difference; and thus no progress has been made beyond dy/dx = 0/0. If, on the other hand, dy/dx is assumed to be equal to p, that is, is assumed as a determinate quantitative ratio—which in fact is the case—then there are difficulties for the assumption which makes h = 0; and by this assumption alone k/h = p is found. Again, grant that k/h = 0;—and when h = 0, then, in fact, automatically k = 0; for the increase of k into y takes place only on the condition that the increment be h. But now the question arises, what p—which is a perfectly determinate quantitative value—is to be. A simple, barren answer immediately offers itself: it is a coefficient, and the derivation whence it arises is this,—the first function, derived in a determinate manner, of a primary function. It is possible to remain satisfied with this; in practice Lagrange did so; and then the general part of the science of the differential calculus, and, immediately, that part of it which is called Theory of Limits, would be freed from the incrementa, from their infinitesimal or arbitrary smallness, from the difficulty of removing again, beyond the first term, or rather only the coefficient of the first term, the further terms of a series which inevitably arrive with the introduction of these incrementa; it would be purged, too, of what follows further, the formal categories (chiefly that of the Infinite), of infinite approximation, and of the further categories (here equally void) of continuous magnitude, and those which elsewhere (like nisus, or becoming, or opportunity for change) are deemed necessary. But now it would be necessary to show what p is apart from the bare determination (quite adequate for the theory) that it is nothing but a function derived from the development of a binomial, and what is its further meaning and value, that is, how it is connected with and employed for further mathematical requirements: this will be the subject of the second Observation.—But first we add an explanation of the confusion which has been introduced into the view of the proper and qualitative determinateness of Ratio by the common use, already exemplified, of the idea of approximation.

It has been shown that the so-called infinitesimal differences express the vanishing of the sides of the ratio as Quanta, and that what remains is their quantitative ratio, purely in so far as it is qualitatively determined; so little is the qualitative ratio here lost, that it is precisely the very result of the conversion of finite magnitude into infinite. And we have seen that this is the very nature of the matter.—Thus, for instance, in the ultimate ratio the Quanta of abscissae and of ordinates vanish; but the sides of this ratio essentially remain, the one an element of the ordinates, and the other of the abscissae. When that form of image is used in which one ordinate is allowed infinitely to approach the other, the one ordinate formerly distinct passes over into the other, and the previously distinct abscissa into the other abscissa; but it is an essential fact that ordinate does not pass over into abscissa nor abscissa into ordinate. The element of ordinate (to go no further than this example of variable magnitudes) must not be taken as the difference between one ordinate and another, but rather as the difference (or qualitative magnitudinal determination) relatively to the element of abscissa: the principle of one variable magnitude relatively to that of the other—these two are here in ratio with one another. The difference, no longer being a difference between finite magnitudes, has ceased to be a manifold within itself: it has collapsed into simple intensity, into the determinateness of one qualitative moment of the ratio relatively to the other.

This aspect of the matter is, however, obscured when that which we have just called element (element, for instance, of the ordinate) is taken as difference or increment in the sense that it is only the numerical difference of the Quantum of one ordinate from the Quantum of another ordinate. Thus Limit has not here the meaning of ratio: it counts only as the ultimate value which another and similar magnitude steadily approaches in such a manner, that the difference between them may be as small as desired, and the ultimate relation be a relation of equality. Thus the infinitesimal difference is the ghost of a difference between one Quantum and another, and, when it is thus imagined, the qualitative nature, according to which dx is related essentially as a determination of ratio not to x but to dy, is in the background. Relatively to dx, dx² is allowed to vanish; but dx vanishes still more relatively to x; and this, justly considered, means that it is related only to dy.—In this kind of demonstration, geometers are chiefly at pains to make intelligible the approximation of a magnitude to its limit, clinging to this aspect of the difference between Quantum and Quantum, where it is no difference and yet still is a difference. But in any case Approximation is a category which in itself means and makes intelligible nothing; dx has already passed through approximation, it is neither near nor is it nearer; and “infinitely near” itself means the negation of nearness and of approximation.

The process thus has been this, that the incrementa or infinitesimal differences have been considered only from the side of the Quantum which vanishes in them, and as the limit of this: they are, then, taken as unrelated moments. From this the inadmissible idea would follow, that it would be permissible to equate in the ultimate ratio abscissae and ordinates, or else sines and cosines, tangents, and versed sines,—anything, in fact.—This idea seems to be operating when a curve is treated as a tangent; for the curve too is incommensurable with the straight line, and its element of a different quality from the element of the straight line. And it seems even more irrational and less permissible than the confusion of abscissae and ordinates, versed sine and cosine, and so forth, when (quadrata rotundis) a part—though infinitely small—of a curve is taken for part of a tangent and thus is treated as a straight line.—However, this treatment differs essentially from the confusion we have just denounced; and this is its justification,—in the triangle, which has for sides the element of a curve and the elements of its abscissae and ordinates, the relation is the same as though this element of the curve were the element of a straight line—the tangent: the angles, which constitute the essential relation (that is, that which remains in these elements after abstraction made from the finite magnitudes belonging to them), are the same.—We can also express ourselves in this matter as follows:—straight lines, as being infinitely small, have passed over into curves, and the ratio which subsists between them in their infinity is a ratio of curves. The straight line according to its definition is the shortest distance between two points, and therefore its difference from the curve is based upon the determination of amount, upon the smaller number of distinguishable steps on this route,—and this is a quantitative determination. But when it is taken as intensive magnitude, or infinite moment, or element, this determination vanishes in it, and with it vanishes the difference from the curve, which is based solely on a difference in Quantum.—Taken as infinitesimal, therefore, straight line and curve have no quantitative relation, and hence (on the basis of the accepted definition) no qualitative difference relatively to each other: the latter now passes over into the former.

Different from the equation of heterogeneous determinations (although analogous) is the assumption, indeterminate in itself and quite indifferent, that infinitely small parts of the same whole are equal to one another; but when this is applied to an object heterogeneous within itself (that is, infected with essential non-uniformity of magnitudinal determination), it produces the peculiar inversion contained in that proposition of higher mechanics which asserts that infinitely small parts of a curve are passed through in equal (and infinitely small) periods of time at a uniform rate: and this is asserted of that kind of motion where, in equal finite (that is, existing) periods of time, finite (that is, existing) and unequal parts of a curve are passed through; the assertion is thus made of a ‘movement which as existent is, and is assumed to be, non-uniform. This proposition is the verbal expression of the intended meaning of an analytic term which results from the development (quoted above) of the formula covering a motion which is non-uniform but, otherwise, conforms to some law. Older mathematicians attempted to express in words and sentences, and to exhibit in geometrical tables, the results of the newly invented infinitesimal calculus (which in any case had to do solely with concrete objects), chiefly in order to use them in theorems in accordance with the ordinary method of proof. The terms of a mathematical formula into which analytic treatment decomposed the magnitude of an object (for instance, motion) there received an objective meaning such as velocity, accelerated force, and so on; according to this meaning they were to produce correct propositions and physical laws; their objective connexions and relations, too, were to be determined by analytic means; for instance, it was said that in a uniformly accelerated motion, there existed a special velocity proportional to the periods of time, while besides an accretion was added uniformly from the force of gravity. In the modern analytical form of mechanics such propositions are regularly cited as results of the calculus, regardless of whether they had in themselves a real meaning—that is, a meaning having a corresponding existent—or whether such a meaning could be proved; and it is thought that the difficulty of making intelligible the connexion of such determinations, when they are taken in the express real meaning (like the transition from that plain uniform velocity to a uniform accelerated velocity), has been totally removed by the analytic treatment, since in it this connexion follows directly from the authority, now regarded as established, of the operation of the calculus. We are told that it is a triumph of science when supra-empirical laws—that is, existential propositions having no existence—are discovered by the pure calculus. But in the first and naive stage of the infinitesimal calculus the task was to indicate a real meaning of these determinations and propositions, which were represented in geometrical diagrams: this meaning was to be made plausible, and with this meaning they were to be applied in the proof of the capital propositions which were in question. (See Newton’s proof of his fundamental proposition of the theory of gravitation in Princ. mathem. philosophiae naturalis lib. I., Sect. II., Prop. I., and cf. Schubert’s Astronomy—first Ed., Vol. III.,§ 20—, where it is admitted that things are not exactly as Newton assumes, that is, at that point which is the nerve of the proof.)

It will be impossible to deny that in this sphere much has been accepted as proof—chiefly veiled under the kindly mist of the infinitesimally small—on no other ground than that the result was always already known beforehand, and the proof, which was arranged in such a manner as to produce the result, at least effected the illusion of a framework of proof,—which illusion was preferred to mere belief or empirical knowledge. But I do not hesitate to regard this method as no better than demonstrational jugglery and counterfeiting; and I include even some of Newton’s demonstrations, and especially such as belong to those just mentioned, for which Newton has been extolled to the skies and above Kepler, because what Kepler had discovered empirically he demonstrated mathematically.

The useless framework of such proofs was erected in order to prove physical laws. But mathematics is altogether unable to prove the magnitudinal determinations of physics in so far as they are laws based upon the qualitative nature of the moments, for this simple reason that this science is not philosophy and does not start from the concept, so that the qualitative element (in so far as it is not taken lemmatically from experience) lies outside its sphere. The assertion of the honour of mathematics, which demands the strict proof of all its propositions, often allowed it to forget its limit; thus it seemed against its honour to acknowledge simply experience as source and sole proof of empirical propositions. Thought at a later period achieved a more instructed view of this matter; and, until it clearly understands the difference between what is mathematically demonstrable and what can only come from another source, between the things which are terms of an analytical development and those which are physical existents, scientific method cannot achieve a strict and pure attitude.—But justice will no doubt be done to that framework of Newtonian proof as it was to another baseless Newtonian structure of optical experiments and conclusions connected with them. Applied mathematics still seethes with a similar brew of experience and reflection; but long ago one part after another of that optics began to be ignored by science in practice (with this inconsistency—that all the rest, though in contradiction, was allowed to stand); and, similarly, in practice some of these sophistical proofs have already been forgotten or replaced by others.