This week minfeub watched Alice, 1988 Czech adaptation of Alice in Wonderland. it was recommended by rachel, she had it on her letterboxd watchlist for a few years. she thinks it ended up there because a different film club she was in was going to screen it & she didn’t make it, but the film seemed interesting
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 con¬ tradiction, was allowed to stand) ; and, similarly, in practice some of these sophistical proofs have already been forgotten or replaced by others.
The Purpose of the Differential Calculus deduced from its Application
In the previous Observation we considered, partly the con¬ ceptual determinateness of the infinitesimally small, which is used in the differential calculus, and partly the reason for its introduction into this calculus ; both determinations are abstract and in themselves therefore easy. Their so-called application, however, offers greater difficulties but also a more interesting aspect: the elements of this concrete side are to be the object of this Observation. — The whole method of the differential calculus is complete in the proposition that
d (*“) = n x*-1 d x, or~^* — — = P , that is, that it is equal
to the coefficient of the first term of the binomial x + dx, x + t, if the latter is developed according to the powers of dx or i. No need to learn anything further : the derivation of the next forms, the differential of a product, an exponential magnitude and so on, results mechanically; and in a short time, perhaps a mere half-hour — for the deduction of the differential also gives us the reverse, namely, the deduction of the original func¬ tion, or integration, from these former — the whole theory may be had by heart. The only delay is due to the effort to understand and make it intelligible that that other part — the omission of the remaining terms of the series which arises, apart from the first terms — is valid, after the first part of the task, the deduction of the coefficient, was effected so easily by analytical, that is, by purely arithmetical means, through the development of the
function of the variable magnitude, after an increment had given it the form of a binomial. And if it were the case that we needed only the coefficient, then, as was said, once it was determined the entire theoretical part would be done with in less than half an hour, and the omission of further terms of the series would cause no difficulty if only because they would not even come into question as terms of the series ; being second, third, or other functions, they are determined with the determination of the first, and now are quite beside the issue.
And first we may observe that probably the method of the differential calculus shows on the face of it that it was neither invented nor constructed as an end in itself. Not only was it not founded for its own sake as a new kind of analytical process, but the arbitrary neglect of terms resulting from the development of a function is contrary to every mathematical principle; — arbitrary, because it is assumed that the whole of this development entirely belongs to the matter in hand — the matter being looked at as the difference of the developed function of a variable magnitude (after it has been given the form of a binomial) from the original function. The need for such a mode of procedure and its internal lack of justification immediately point to the fact that its origin and basis must be elsewhere. It happens in other sciences, too, that that which is placed first, as being elementary, and which is the source from which are ostensibly derived the propositions of the science, is not self-evident, and that it eventually appears that its reason and foundation lie rather in what follows. In the history of the differential calculus the course of things makes it plain that, especially in the various so-called tangential methods, the matter began in special artifices. The method of procedure, later becoming extended to further objects, reached conscious¬ ness and was cast into abstract formulae, which it was then attempted to raise to the rank of first principles.
We have shown that the qualitative quantity-determinateness of entities, which, primarily, are related to one another as Qpanta, is the conceptual determinateness of the so-called infinitesimally small; from this there started the empirical investigation which attempted to demonstrate this conceptual determinateness in the descriptions or definitions of the in¬ finitesimally small, in so far as it is taken as infinitesimal
difference or something similar. — This was done only in the interest of abstract conceptual determinateness as such; it might further be asked, what is the nature of the transition thence to mathematical formation and application? To this end the theoretic part (the conceptual determinateness) must further be examined, and this will prove not wholly unfruitful in itself; next we must consider its relation to Application; and with both we must demonstrate, as far as here is possible, that the general conclusions are adequate to the end of the differential calculus and to the manner in which it brings that end about.
It must first be remembered that the mathematical form of the conceptual determinateness which is under discussion has already been mentioned in passing. The qualitative deter- minateness of the quantitative is first and generally stated in Quantitative Ratio, but already in the demonstration of the various so-called rules of arithmetic (see Observation on this subject) it was said that it was in the Ratio of Powers (to be considered later in its proper place) that Number is posited by the equation of Unit and Amount (the moments of its concept) as returned to itself, and thus acquires in itself the moment of Infinity or Being-for-Self, that is, of self-deter¬ minedness. Thus, the express qualitative magnitudinal deter¬ minateness essentially (as has also been mentioned already) refers to determinations of powers ; and since it is the specific characteristic of the differential calculus that it operates with qualitative forms of magnitude, its peculiar object in mathe¬ matics must be the treatment of forms of powers ; and all the problems, with their solutions, for which the differential calculus is used, show that all the interest lies only in the treatment of determinations of powers as such.
despite the disturbing nature of many of the props, relying a lot on taxidermy, it was less scary of a movie than I anticipated. It was rather whimsical, but kind of dawdled on the famous plot beats of alice in wonderland for most of the playtime in an incoherent manner. I did really like the way the movie switched from actor to doll in order to imply scale, both as an animation technique and a stylistic choice. I also really liked how liminal the plaster basements and timber laundry attics were, as brief a feature those were in the film.
This foundation is important and immediately puts in the forefront something determinate instead of the merely formal categories of variable, continuous, or infinite magnitudes and the like, or of functions in general ; but for all that it is still too general; other operations too are concerned here; the elevation to a power and the extraction of a root, the treat¬ ment of exponential magnitudes and of logarithms and series and the equations of higher orders, all are interested in and operate upon ratios only which are based upon powers. No
doubt they must between them constitute a system of treatment of powers; which, however, of the various ratios wherein determinations of powers can be posited is the one which is the proper object and interest of the differential calculus, this can only be elicited from itself, that is, from its so-called applications. These in fact are the thing itself, the actual procedure in the mathematical solution of a certain group of problems; and this procedure was earlier than the theory or general part, and has been called application only with refer¬ ence to the later created theory whose aim was to set up the general method of procedure and also to produce its first principles, that is, its justification. It has been shown in the previous Observation how idle an attempt it has been to find principles for the former manner of comprehending the pro¬ cedure, — principles which really solved the contradiction which there was found, instead of excusing it or hiding it behind the insignificance of that which was requisite to the mathematical procedure (which here meant that which was to be omitted), or behind the possibility (which comes to the same thing) of infinite or arbitrary approximation, and the like. If the general part of the procedure were to be abstracted, in a manner different from that which has hitherto been followed, from that real part of mathematics which is called differential calculus, then these principles and all the trouble taken over them would prove superfluous, inasmuch as they reveal in themselves a distortion and an abiding contradiction.
If we search out this peculiarity by simply gathering what we find in this part of mathematics, we discover as its object : — (a) Equations in which any number of magnitudes (we may, however, here confine ourselves to two) are combined into one determinate whole in such a manner that, firstly, they have their determinateness in empirical magnitudes which are their fixed limits, and, moreover, in the particular kind of union with them and with one another, which, indeed, is the case with equations generally; but, since there is only one equation for both magnitudes (or several equations for several magnitudes, but always fewer than the magnitudes), these equations belong to the class of indeterminate equations ; — and, secondly, that one aspect of them (the determinateness of these magnitudes here being what it is) is that they are, or at least
one of them is, present in the equation in a higher power than the first.
Certain observations may be made about this; and first, that according to the first determination mentioned these magnitudes are wholly of the character of such variable mag¬ nitudes as occur in the problems of indeterminate analysis. Their value is indeterminate, but is so in such a manner that when the one gets a perfectly determinate value (a numerical value) from without, then the other too is determined: one is a function of the other. The categories of variable magnitudes, functions, and the like, are, therefore, merely formal for that specific determinateness of magnitude with which we are dealing here, as was said above; for they are of a generality not yet containing that specific factor which is the one aim of the differential calculus; nor can that factor be explained from them analytically. In themselves they are simple, unim¬ portant, easy determinations, which are made difficult only when what they do not properly contain is put into them in order to be then deduced, I mean the specific determination of the differential calculus. — With regard to the so-called constant, we may remark that it exists first as an indifferent empirical magnitude, determining the variable magnitudes only with respect to their empirical Quantum, as Limit of their minimum and maximum; while the nature of the connexion between the constants and the variable magnitudes is itself one of the moments of the nature of that special function which these magnitudes are. But, conversely, the constants also are functions; for instance, in so far as a straight line has the
meaning that it is the parameter of a parabola, then its meaning
is this, that it is the function - ; and, generally, in the develop¬
ment of the binomial, the constant which is the coefficient of the first term of the development is the sum of the roots; the coefficient of the second is the sum of their products, in pairs, and so on : thus the constants here are just functions of the roots; where, in the integral calculus, the constant is determined from the given formula, it is in this respect treated as a function of the latter. We shall elsewhere consider these coefficients, in another determination, as functions, whose meaning in the concrete is the only matter of interest.
The peculiarity, however, by which the consideration of the variable magnitudes is distinguished, in the differential calcu¬ lus, from its nature in the indeterminate problems, must be attributed to what has been mentioned, namely that at least one, and possibly every one, of these magnitudes is in a higher power than the first; and here again it is indifferent whether all of them are of the same higher power, or of unequal powers ; the specific indeterminateness which they here have lies solely in the fact that they are functions of one another standing in such a ratio of powers. This gives a qualitative determination to the variation of the variable magnitudes : it is thus continuous, and this continuousness (which in itself is merely the formal category of an identity in general, a determinateness persisting unchanged in variation) here has its determinate meaning, and that only in the ratio of powers, which is supposed to have no Quantum for exponent and to constitute the non-quanti- tative and permanent determinateness of the ratio of variable magnitudes. On the other hand we may object to another formalism, that the first power is power only in relation to higher power: for itself, * is only any indeterminate Quantum. There is thus no meaning in differentiating in themselves the equations y — a x -f- b (the equation of the straight line) or s = ct (the equation of plain uniform velocity) ; if the
equation y — ax, or y — a x + b, becomes a — or s — ct
it’s rather comparable to “MAD GOD” in that it kind of felt like an animator doing whatever/doing stunts for the entire screentime instead of building up a coherent narrative, though at least in this one the one identifiable “main character” didn’t get off’d 40% of the way through and leave the camera lost and homeless
becomes — = c, then, equally, a — — is the determination of at x
the tangents, or - = c the determination of plain velocity. dy
The latter, as -f-, is exhibited in connexion with what is dx
asserted to be the development of uniformly accelerated motion ; but it has been remarked above that it is an empty assumption, based upon the routine of method alone, that a moment of simple, merely uniform velocity (that is, velocity not deter¬ mined by the higher power of one of the moments of motion), has a place in the system of such motion. The method proceeds from the idea of the increment to which the variable magnitude is subject; naturally, therefore, only a magnitude which is a function of a first power can be subject to an increment : when
now, in order to discover the differential, the difference be¬ tween the given equation and that which has thereby arisen, has to be found, the empty nature of this operation manifests itself; for, as we remarked, after as well as before it the equation is the same for the so-called increments as for the variable magnitudes themselves.
(/?) What has been said determines the nature of the equation which is to be treated ; we must now indicate upon what point of interest its treatment is directed. This consideration can furnish only known results, such (with respect to form) as are especially to be found in Lagrange’s conception; but I have, of course, made this exposition wholly elementary in order to remove all heterogeneous determinations involved in it. — The basis of the treatment of an equation of this kind is seen to be this, that the power within itself is taken as a ratio or system of determinations of ratios. We stated above that Power is Number in so far as it has reached that stage where its variation is determined by itself, and its moments, Unit and Amount, are identical; — which, as was shown above, is found perfectly in the square, and more formally (which here makes no difference) in the higher powers. Now power is number (though the expression “magnitude” be preferred as more general, yet in itself it always is number), and thus is a mul¬ titude, and can be represented as a sum: it can, therefore, be divided within itself into any multitude of numbers, which relatively to one another and to their sum are without any determination except that together they are equal to the sum. But the power can also be divided into a sum of such differ¬ ences as are determined by the form of the power. If the power is taken as sum, then its radical number or root too is taken as a sum, and, as regards multiplicity of division, is arbitrary, — which multiplicity, however, is the indifferent and empirical Quantitative. The sum, in which shape the root is supposed to exist, reduced to its simple determinateness, that is, its true universality, is the binomial ; and any further multiplication of the terms is a pure repetition of the same determination, and, therefore, an empty process.1 What matters is the determinate-
1 It is only a part of the formalism of that universality to which analysis per¬ force lays claim, when (a 4- 4) * is not taken for the development of the power, and (a+ i + c + d+ ...)’• is substituted. This is often done elsewhere too; and this
ness of the terms (which thus is qualitative) which results from the raising of the root, taken as sum, to a power, and this determinateness lies entirely within that change which this process is. These terms, thus, are entirely functions of power and of raising to a power. Now this representation of number as sum of a multitude of such terms, which are functions of this process, and further the eagerness to find the form of such functions and also this sum from the multitude of such terms, this it is (the discovery being entirely dependent on that form) which constitutes, as is well known, the special doctrine of series. But here it is important to distinguish the further point of interest, namely, the relation of the basic magnitude itself (the determinateness of which, in so far as it is a complex, that is, in this instance, an equation, therefore includes a power) to the functions of its potentiation. This relation (apart altogether from the above-mentioned interest in the sum) will show itself to be the only truly scientific point of view by which the differential calculus is guided.
But first another determination must be added to what has been said; or rather, a determination implied in it must be removed. It was said that the variable magnitude, in whose determination power is an element, is looked upon as a sum within itself, as being, in fact, a system of terms, in so far as these are functions of potentiation, and that thus the root too was considered as a sum, and, in its simply determinate form, as a binomial, namely
This presentation started from Sum as such upon the process of developing Power, that is, upon the process of achieving its potentiational functions; here, however, the aim is not a sum as such, nor the series which arises from it: the only element which must be taken up from Sum is Relation. Relation of magnitudes as such is on the one hand all that remains after abstraction has been made from the plus of a sum as such, and is all that is needed on the other hand in
-Avery
form, «o to speak, should be taken for an affectation of the appearance of univer¬ sality, for the matter is exhausted in the binomial; by the development of this the law is found, and it is the law which is the true universality, and not external and empty repetition of the law, which is all that this «+ * + e+d+ . . . effects.
order that the functions of development of Power may be found. But such a Relation is determined already by the fact that the object here is an equation {jT—af), that is, a complex of several (variable) magnitudes which contains their determination of powers. In this complex each of these mag¬ nitudes is posited as just being in relation with the other, with the meaning, as one might say, of a plus in itself, — as a function of the other magnitudes; their characteristic (that of being functions of one another) gives them this determination of a plus, which for that very reason is quite indeterminate and is not an addition, increment, or the like. This abstract point of view could, however, be neglected ; we may simply remain at that point where, the variable magnitudes being given in the equation as functions of one another in such a manner that this determinateness contains a relation of powers, the functions of potentiation too are now compared one with the other; — which second functions have no determination what¬ ever except that which comes from potentiation. So far we can assert that it is optional, or possible, to transpose an equation from the powers of its variable magnitudes to a relation of its functions of development ; and the utilitarian quality of such a transformation must be indicated by some further purpose, advantage, or use; the transformation has been made only because of its usefulness. Above we started from the presen¬ tation of these determinations of potentiation, and experimented on a magnitude which was stated to be a sum and, therefore was to be assumed as being differentiated within itself; but this was only done, partly to indicate the nature of such functions, and partly because this implies the mode by which they are found.
We have thus reached the ordinary analytical development, which for the purposes of the differential calculus is taken in this manner, that an increment, dx or i, is given to the variable magnitude, and then the power of the binomial is explained by means of the series of terms belonging to it. The so-called increment, however, is intended to be not a Quantum but a form, whose only value is that it assists the development; what is wanted (and this is admitted, and most explicitly by Euler and Lagrange) in the above-mentioned idea of Limit, is the resulting determinations of powers of the variable
magnitudes, the so-called coefficients of the increment (as is admitted) and its powers, according to which the series is ordered, and to which the different coefficients belong. To this we might also add that an increment (without Quantum) is only assumed for the sake of development, and that, therefore, it would be most convenient to take 1 (the One) for that purpose; for in the development this only occurs as factor; so that the factor One fulfils the purpose, which is, that the increment is not to imply the positing of any quantitative determinateness or change ; dx, on the other hand, is infected with the false idea of a quantitative difference, and other symbols, like t, with the show, useless here, of universality, so that they always have the appearance and pretension of a Quantum and its powers ; which pretension then involves the trouble that they must nevertheless be taken away and kept away. In order to preserve the form of a series developed on the principle of powers, the denominations of exponents (being indices) might equally well be placed behind a One. In any case abstraction must be made from the series and from the determination of coefficients according to their place in the series : the relation between all is the same; the second function is derived from the first in just the same manner as this from the original, and for the one which is counted second, the first, and derivative, function, is counted original. The essential point of interest is not, however, the series, but solely the determination of powers resulting from the development in relation to the magnitude which, to them, is immediate. They are not, therefore, deter¬ mined as being coefficients of the first term of the development, since one term is designated as first in relation to others follow¬ ing it in the series, and such a power as the power of an incre¬ ment, together with the series, is here out of place ; therefore, the plain expression of derivative function of a power or, as was said above, function of potentiation of a magnitude, would be preferable, — the knowledge being presupposed of the manner in which the derivation is taken as a development included within a power.
The pure mathematical beginning in this part of the analysis is just the discovery of the function determined by the develop¬ ment of powers; it is a further question, what is to be done with the ratio thus obtained, where it is to be applied and usedf
or, in fact, for what purpose such functions are looked for. It is the process of discovering proportions in concrete objects which may be referred back to those abstract analytic pro¬ portions, that has given the differential calculus its great interest.
But as regards applicability, the immediate outcome of the nature of the matter — merely in virtue of the form which we have shown the moments of powers to possess, and no con¬ clusion being yet drawn from any instances of application — is as follows. The development of powers whence result the functions of their potentiation contains (abstraction being made from closer determination) the reduction of magnitude to the next lowest power. Thus it is that this operation is applicable to such objects as also contain this distinction of determinations of powers. If now we consider spatial determinateness, we find that it contains the three dimensions which we may call the concrete, in order to distinguish them from the abstract distinc¬ tions of height, length, and breadth, — namely, line, surface, and solid space; and, when they are taken in their simplest forms and with reference to self-determination and, therefore, to analytic dimensions, we arrive at the straight line, the plane surface and surface taken as square, and the cube. The straight line has an empirical Quantum, but with the plane we reach the qualitative element — determination of power; we may neglect closer modifications, — the reflection, for instance, that this also happens to plane curves ; for here we are only dealing with the distinction in general. Herewith also the need arises to pass from a higher to a lower determination of power, and conversely, — the attempt being for instance to derive linear determinations from given equations of plane and so on, or the other way about. — In motion, further, the magnitudinal proportion of the space passed through with its elapsed time must be considered, and motion manifests itself in various determinations, as simply uniform, as uniformly accelerated, and as alternately accelerated uniformly and retarded uni¬ formly, returning upon itself; these different kinds of motion are expressed according to the magnitudinal proportion of their moments, space and time; and thus there result for it equations having different determinations of powers; and, in so far as it may be necessary to determine one kind of motion or of
spatial magnitude to which one kind is tied, from another kind of these, the operation also involves the transition of one function of power to another, either higher or lower. — The examples of these two objects should suffice for the purpose for which they are cited.
The appearance of contingency which the differential cal¬ culus presents in its applications would certainly be simplified if the nature of the spheres where the application can take place, and the peculiar need for and condition of this applica¬ tion, were clearly understood. But now it is necessary further to know, within these spheres, between what parts of the objects of the mathematical problem such a relation takes place as is posited peculiarly by the differential calculus. And first it must be remarked that two kinds of relation are to be observed. The operation of the depotentiation of any equation (the equation considered according to the derivative functions of its variable magnitudes) gives a result which in itself really no longer is an equation but a proportion: this proportion is the object of the differential calculus proper. And precisely in this fact we are also presented with a second proportion, that between the higher determination of power (the original equation) itself and the lower (the derivative). This second proportion we must here provisionally neglect: it will prove to be the peculiar object of the integral calculus.
We will then consider the first relation, and, for the deter¬ mination of the moment (which must be taken from the so- called application and contains the interesting part of the operation) we will take the simplest example, curves such as are determined by an equation of the second degree. The relation of the coordinates is, of course, given immediately through the equation in a determination of powers. From the fundamental determination there follow determinations of other straight lines connected with the coordinates — tangents, subtangents, normals, and so on. The equations, however, between these lines and the coordinates are linear equations; and the wholes, of which these lines are determined to be parts, are right-angled triangles of straight lines. Now the transition from the fundamental equation, which contains the determination of the powers, to these linear equations, con¬ tains the above transition from the original function (that is.
This was fun! It was rather surreal, more of a fever dream than what I’d call a story, but the animation was pleasant, and it wasn’t too disturbing. The narration provided by Alice after any of the other characters “spoke” also felt nice. No idea what was happening most of the time, but I still don’t feel like I missed out on much. I ate some stale bread during the beginning of the film, and it really matches the mood; aged but pleasant in it’s own unique way. It’s a very screen-shootable film, lot’s of iconic little critters doing… whatever they’re doing. Had some pleasant morals too, like not eating random cakes you find off the ground, and not drinking ink at the risk of growth or shrink. Ten out of twelve, makes me want to cut off someone’s head :3
the function which is an equation) to the derivative function (which is a relation — a relation subsisting between certain lines contained in the curve). What must now be discovered is the connexion between the relation of these lines and the equation of the curve.
It is not without interest to introduce this much of historicity, as to remark that the first discoverers can indicate their dis¬ covery only in a wholly empirical manner, without being able to render an account of the operation, which has remained quite external. In this regard I content myself with a reference to Barrow, who was Newton’s master. In his led. Opt. et Geom., in which he treats problems of higher geometry according to the method of the indivisibles (which, in the first place, differs from the characteristic feature of the differential calculus), he also indicates his mode of determining the tangents, “because his friends had urged him” {led. X.). The nature of this indication must be read in his own words if we would properly understand how this method is given quite as an external rule — in the same style in which formerly in arithmetical school¬ books the “rule of three,” or, better still, the so-called “test by casting out the nines,” was enunciated. He enumerates the minute lines, which later were called the increments in the characteristic triangle of a curve, and then gives the instruction — which is a mere rule — to cast off as superfluous the terms which appear when the equations are developed as powers of these increments or products ( etenim isti termini nihilum valebant) ; adding that the terms which contain only magnitudes determined by the original equation must also be cast off ( — which means that the original equation is sub¬ tracted afterwards from the equation formed with the incre¬ ments), and that the ordinates themselves must be substituted for the increment of the ordinates, and the subtangents for the increment of the abscissae. The method could be set forth (if one may say so) in no more school-masterly manner; — the second substitution is the assumption of the proportionality of the increments of ordinates and abscissae to the ordinates and subtangents, which is the basis of the procedure for deter¬ mining tangents in the ordinary differential method : in Barrow’s rule this assumption appears in all its naive naked¬ ness. A simple way of determining subtangents had been
found: Roberval’s1 and Fermat’s methods come to the same issue; — the method for finding maximal and minimal values, from which the latter started, is based on the same foundation and the same procedure. It was a mathematical mania of that period to discover so-called methods, that is, rules of this kind, and to make a mystery of them ; which was not only easy, but also, in one respect, necessary, and for the same reason for which it was easy, — namely that the inventors had found only an empirical and external rule and no method, that is, nothing derived from acknowledged principles. Such so-called methods Leibniz absorbed from his period, and Newton from it and, immediately, from his master: the generalization implied in their form and applicability opened new roads to the Sciences, but also brought the need of forcing the method out of the shape of merely external rules, and attempted to give it the justification which it demanded.
If now the method be more closely analysed, the true process is this. First the determinations of powers (powers, of course, of variable magnitudes) contained in the equation, are reduced to their first derivatives. But this changes the value of the members of the equation : hence no equation remains, and a proportion has arisen which subsists between the first derivative of one variable magnitude and the first derivative of the other; for fix =jp we have p : 2 y, or for 2 a x — xt —y% we have a — x :y, which later used to be called the proportion dy dx
on the other hand, wholly dependent on it and wholly derived from it (derived in the process described above in accordance with a bare rule), is linear, certain lines here being proportional one to the other; p : ay or a—x:y are themselves relations of straight lines of the curve, or of coordinates or parameters ; but so far knowing this we know nothing. The aim is to know, with regard to other lines which occur in connexion with the curve, that a certain one of these relations holds between them, — that is, to identify two relations. — Thus, secondly, the question arises, which are the straight lines determined by the nature of the curve which are thus related? — But this was known already — namely, that a relation thus reached is the
relation between ordinates and subtangents. The ancients had found this by an ingenious geometrical method ; what modem inventors discovered was the empirical mode of arranging the equation of a curve in such a manner that the first relation should result, of which it is already known that it is equal to a relation which contains the line (here the subtangent) which it is desired to determine. Now, in part, this arrangement of the equation has been taken methodically and made methodi¬ cally (Differentiation), — and, in part, the imaginary increments of the coordinates and the imaginary characteristic triangle (which is formed of these and of a similar increment of the tangents) have been invented, — simply in order that the pro¬ portionality between the relation found by the depotentiation of the equation and the relation between ordinates and sub¬ tangents should not be represented as taken up empirically from old acquaintance, but as a demonstrated truth. And yet old acquaintance proves itself, generally and most unmistak¬ ably in the form of rules (such as we quoted), as the only occasion and justification (where needed) of the assumption of the characteristic triangle and of that proportionality.
It was Lagrange who rejected this pretence and followed the true scientific course: thanks to his method we know the real point at issue, for it consists in a separation of the two transitions which must be made in order that the problem may be solved, and in a separate treatment and proof of each of these sides. — In our more detailed explanation of the pro¬ cedure we will retain as example the elementary problem of finding the subtangents. — Now one part of this solution, the theoretical or general part, which is the finding of the first function from the given equation of curves, is regulated by itself: this part gives us a linear relation, a relation, that is, of straight lines, which occur in the system of curve-deter¬ mination. The other part of the solution now is the finding of those lines in the curve which are in this relation. This is done directly ( Theorie des Fonct. Anal. II. P. II. Chap.), that is, without the characteristic triangle, which means that no assumption is made of infinitely small arcs, ordinates, or abscissae, nor are the determinations of dy and d x (that is, of being sides of this relation) attributed to them, which would n^:an immediately that it was equated with the ordinate and
subtangent. A line (and also a point) has its determination in so far as it constitutes the side of a triangle, and the deter¬ mination of a point too lies only in this. This (as may be mentioned in passing) is the fundamental proposition of analytical geometry, which introduces the coordinates as (what is the same thing) in mechanics it does the parallelogram of forces, — which for that very reason does not require all the efforts that are made to find a proof. — The subtangent is now made the third side of a triangle, the other sides of which are the ordinate and the relative tangent. The latter is a straight line, and, therefore, its equation is p — aq, (the addition of b adds nothing to the determination and is made only for the sake of the fetish of universality) ; — the determination of
the ratio - falls within a, the coefficient of q, which in turn
is the first derivative of the equation, but need be considered
-reci from minfeub
only as a — -, — being, as has been said, the essential deter-
mination of the straight line which is applied as tangent to the curve. Now, further, the first derivative of the curve- equation is taken, and, therefore, it is also the determination of a straight line ; further, it is assumed that the coordinate p of the first straight line, and y, the ordinate of the curve, are identical, and, therefore, that the point where that first straight line (which is taken to be a tangent) touches the curve is also the beginning of the straight line which is determined by the first function of the curve : the task, therefore, is to show that this second straight line coincides with the first and, therefore, is a tangent; which may be thus algebraically expressed: — since y —fx and p = F q, and it is assumed that y = p and hence that fx—Fq, therefore f'x — F'q. In order to prove that the straight line which is applied as tangent and the straight line of the equation which is determined as being its first function coincide, and, therefore, that the latter is a tangent, recourse is had to the increment t of the abscissa and to the increment of the ordinate which is determined during the development of the function. Thus here too that ill-famed increment is introduced ; but its introduction for this purpose, and the development of the function under its guidance, must carefully be distinguished from the use (mentioned above) ,of
the increment in the discovery of the differential equation and in the characteristic triangle. The use here made is justified and necessary; it falls within the scope of geometry, for it is part of the geometrical determination of a tangent as such that between it and the curve with which it has a point in common no other straight line can pass and pass also through that point. In this determination the quality of tangent or not-tangent is reduced to a magnitudinal difference : that line is the tangent of which simply greater smallness with respect to the essential determination is predicated. There is no empirical element whatever in this smallness, which appears to be only relative, — nothing, that is, which depends on a Quantum as such; it is posited as qualitative by the nature of the formula when the difference of the moment on which the magnitude which is to be compared depends, is a difference of powers; this difference is reduced to i and i*, and i (which after all must signify a number) must then be imagined as a fraction, and thus il in itself is smaller than i; so that the idea of an arbitrary magnitude as which i might be taken is here superfluous and even out of place. And for this very reason the demonstration of a greater smallness has nothing to do with an infinitesimally small quantity, the introduction of which is thus here by no means necessary.
I will now mention— if only for its beauty and for its fame, mostly forgotten now, but well deserved — the tangential method of Descartes; it moreover has a bearing on the nature of equations, on which an observation remains to be made. Descartes unfolds this independent method, where the linear determination required is discovered from the same derivative function, in his geometry, which has proved so fruitful in other respects too (liv. II. p. 357 sqq. Oeuvres Compl. ei. Cousin Tom. V.) ; for in it he teaches the great basis of the nature of equations and their geometrical construction, and of the Analysis of Geometry, the sense of which he had thereby so greatly ex¬ tended. With him the form of the problem was this, to draw straight lines at right angles to given points of a curve, by which method the subtangent (and so forth) is determined; and it is easy to understand the satisfaction which he expresses there at his discovery, which concerned an object of general scientific interest at that period, and is purely geometrical and
thereby stands high above the mere methods of rule (mentioned above) used by his rivals: “ J'ose dire que c'est ceci le probleme le plus utile et le plus general, non seulerrunt que je sache, mais mime que foie jamais disire de savoir en giometrie." — He bases the solution upon the analytic equation of the right-angled triangle which is formed by (1) the ordinate of the point of the curve to which the straight line demanded in the problem is to be perpendicular, by (2) this straight line itself (the normal), and by (3) that part of the axis which is cut off by the ordinate and the normal (the sub-normal). Now the equation of a curve is known, and from this equation the value of ordinate or abscissa is substituted into this equation of the triangle. Thus an equa¬ tion of the second degree results ; and Descartes shows also how curves whose equations contain higher degrees are reduced to this. In this equation only one of the variable magnitudes occurs, either as square or as first power ; — a quadratic equation which at first appears as a so-called impure equation. On this Descartes reflects as follows : — If the point taken in the curve is imagined as the point of intersection of this and of a circle, then this circle will intersect the curve at another point, so that for the two unequal x’s which will thus arise there will be two equations with the same constants and having the same form; — or else there will be only one equation with unequal values of x. But the equation is one only for the one triangle, in which the hypotenuse is perpendicular to the curve (or is normal), — which is imagined in this way, that the two points of intersection of the curve and the circle are allowed to coin¬ cide, so that the curve is allowed to touch the circle. But then also the fact that the x or y of the quadratic equation has unequal roots, disappears. But in a quadratic equation of two equal roots, the coefficient of the member contained by the unknown in the first power is twice the single root ; and from this there results an equation by which the required deter¬ minations are discovered. This method must be considered the brilliant device of a true analytic mind, compared with which the arbitrarily assumed proportionality of subtangent and ordinate, together with the so-called increments of abscissa and ordinate (supposed to be infinitely small), is vastly inferior.
The final equation reached in this manner, in which the coefficient of the second member of the quadratic equation
is equated with the double root or unknown, is the same as is found by the method of the differential calculus. When #* — ax — b — o is differentiated, there results the new equation sx — a — o; or, again, 3 x* — p = o results from ** — p x -r- q = o. And here we may observe that it is by no means self-evident that such a derivative equation is also correct. We have already considered the fact that in an equation with two variable magnitudes which never lose their quality of being unknown magnitudes just because they are variable, only a proportion results; and this for the simple reason indicated, that, when the functions of the potentiation are substituted for the powers themselves, the value of the two members of the equation is altered, and it remains as yet unknown whether an equation subsists between them with
their values thus altered. The equation -j- — P expresses
nothing except that P is a proportion, and no other real
meaning can be ascribed to And also it is still not known
i thought it was neat, but it didn’t fully resonate with me, i guess. it was missing something. perhaps it’s just tough because alice in wonderland story is very well known, so at least in terms of situations, there’s a bit less of a sense of openness and wonder that the story itself demands. pan’s labyrinth of course comes to mind as a fairy tale that feels much more imaginative, but it’s not a very fair comparison since that one exists in both the magical and mundane worlds. looking at the similar films on letterboxd… eraserhead is a more fair comparison - since it’s not portraying a story we all know, that film feels like anything can happen in it, which better puts you into the appropriate position as viewer
of this proportion = P, to what other proportion it is equal; it is only this equation, or proportionality, which gives value and meaning to the proportion. — It was mentioned that this meaning — and this was what was called application — was introduced empirically and from without; and, similarly, as the equations here under discussion are derived by differentia¬ tion, we must know from some other source whether they have equal roots, in order to know whether the equation reached remains correct. But this fact is not expressly brought to notice in the manuals; although it must be admitted that it is got out of the way when an equation with an unknown, reduced to nought, is straightway equated with y, whereupon
tion. It is true that the functional calculus is supposed to deal with functions of potentiation, and the differential calculus with differentials ; but it by no means follows immediately that magnitudes whose differentials or functions of potentiation are taken, are themselves only to be functions of other magnitudes. In any case, in the theoretic part, that is, where the instruc¬ tion is given to derive the differential (or the functions of
potentiation), no thought is yet given to the intention that the magnitudes, of which the treatment according to such a derivation is there taught, are themselves to be functions of other magnitudes.
With regard to the omission of the constant in the process of differentiation, this further observation may be made, that differentiation here means that the constant is indifferent for the determination of the roots when they arc equal, the deter¬ mination being exhausted by the coefficient of the second member of the equation. In the example quoted from Descartes the constant is itself the square of the roots, so that it can be determined from the constant as well as from the coefficients ; for, generally, like the coefficients, it is a function of the roots of the equation. In the ordinary exposition the omission of the so-called constants (which are connected with the other members only by plus or minus) is brought about by the mere mechanism of the method, — when, in order that the differential of a composite expression may be found, only the variable magnitudes receive an increment, and the expression thus formulated is subtracted from the original expression. The meaning of the constants and of their omission, the question how far they are themselves functions and serve or do not serve in this determination, finds no expression here.
In connexion with the omission of constants, an observation may be made about the names of Differentiation and Inte¬ gration, similar to the observation which was made above about the expressions Finite and Infinite, — namely, that their deter¬ mination contains the opposite of what is denoted by the terms. To differentiate denotes that differences are posited, whereas differentiation in fact reduces an equation to lesser dimensions, and the omission of the constant removes one moment of the determinateness ; for, as we remarked, the roots of the variable magnitude were placed upon an equality, and thus the dif¬ ference between them was cancelled. And in integration the constant is supposed to be reintroduced; and though by this process the equation is integrated, it is so in this sense, that the difference of the roots, which had just been cancelled, is reconstructed, so that the equalization is differentiated once more. — The ordinary expression adds its share in obscuring the essential nature of the matter and in setting everything
in a point of view which is subordinate and even alien to the main issue : I mean the point of view of the infinitely small difference, the increment, and the like, and also of the bare difference generally between the given and the derivative function, no designation being made of the specific, namely the qualitative, difference.
Mechanics is another important field where the differential calculus is applied; mention has already been made inciden¬ tally of the different functions of powers that result from the elementary equations of its object, which is motion, and of their significance. I will admit these directly here. The equation (that is, the mathematical expression) for motion which is s
simply uniform, c = - or s — c t, where the spaces passed
through are proportionate to the times elapsed according to an empirical unit c (the magnitude of velocity), offers no meaning for differentiation: the coefficient c is already fully determined and known, and no further development of powers can take place. — The analysis of s = a P, the equation of the motion of a falling body, has already been recorded; — the first
member of the analysis ^ = 2 a t is translated into language
-rachel
(or into fact, as the case may be) when it is postulated that a member of a sum (an idea which we banished long ago) must be one part of this motion ; and that, further, this part must be added to the force of inertia (a merely uniform velocity) in such a manner that the motion is uniform in infinitesimally small parts of time, and not uniform in finite parts of time, that is, in those which actually exist. It is true that f s = 2 at; and the meaning of a and t is known already, together with the fact that this suffices to posit the deter-
mination of the uniform velocity of a motion ; for since a =
then 2 at — — universally ; but having this we are no wiser
at all, and only the deceitful assumption that 2 a I is part of the motion regarded as a sum gives the deceitful appearance of a physical proposition. The factor itself (a, the empirical unit, a Quantum as such) is ascribed to gravity; but if the category of force of gravity is employed at all, it should rather be said
that the whole s — aP is the effect, or, better, the law, of gravity. — Similar to this is the proposition derived from is
— — 2at, which enunciates that if gravity ceased to act, the
body, moving at the velocity reached at the end of its fall, would describe twice the space it has already passed through in a time equal to that occupied by its fall. — This contains a metaphysic which in itself is unsound : the end of the fall, or the end of a period of time during which the body has been falling, still is itself a period of time; if it were not, a state of rest, which excludes velocity, would be assumed; velocity can be introduced only according to the space passed through in a period of time and not at the end of a period. — When the differential calculus is actually applied in other spheres of physics where there is no motion at all, for instance in the behaviour of light (apart from its so-called propagation in space) or in the magnitudinal determinations of colours, the first derivative of a quadratic function being here too called velocity, then we must regard this as a still more illegitimate example of the formalism which feigns real existence. —
The motion (says Lagrange) which is represented by the equation s = a P is empirically given in falling bodies ; the next simplest motion after this equation would be that whose equation was s — c P, but nature knows no such motion ; and we do not know what meaning the coefficient c could have. This is true ; but there is a motion whose equation is j* = a P, and this is Kepler’s law of the motion of the bodies of the solar system; — and indeed it would appear an interesting task to show the intended meaning here of the first derivative function
and so on, to treat this equation further and directly by
means of differentiation, and to develop the laws and deter¬ minations of that absolute motion from this starting point, — a task in which analysis might display a brilliance most worthy of itself.
Thus the application of the differential calculus to the elementary equations of motion offers in itself no real interest : formal interest is derived from the general mechanism of the calculus. But another significance is gained by the analysis of motion with respect to the determination of its trajectory : if
this is a curve and its equation contains higher powers, then the transition is necessary of rectilinear functions (as functions of potentiation) to the powers themselves ; the former must be extracted from the original equation of motion, which contains the factor of time, and time must be eliminated ; and so at the same time this factor must be reduced to lower functions of development, from which these equations of linear determina¬ tions can be worked out. This aspect leads us to the interesting element of the other part of the differential calculus.
What has been said so far was said with the purpose of emphasizing and fixing the simple and specific determination of the differential calculus, and of demonstrating it in a few elementary examples. It was seen that this determination consisted in the following process : the coefficient of the member of development (the so-called first derivative) is found from an equation of functions of powers ; this function is a propor¬ tion which is demonstrated in moments of the concrete object; an equation results which determines these moments themselves as between the two proportions. And also we must briefly consider the principle of the integral calculus, and what results from its application for its specific concrete determination. The consideration of this calculus has been simplified and more correctly determined through the fact that it is no longer taken as a method of summation, as it was called in opposition to differentiation ; the increment was there considered the essential ingredient, and with this it appeared in essential connexion with the serial form. — The task of this calculus, as of the differential, is, first, theoretical or rather formal, but, as is well known, it is the converse of the differential. Here a beginning is made from a function which is considered as derivative and as the coefficient of the next member, which originates from the development of an equation as yet un¬ known; and from it the original function of power is to be calculated. The function which in the natural order of develop¬ ment must be regarded as primary is here considered derivative, and that which before was considered derivative here is con¬ sidered as given and in fact as beginning to exist. But it appears that the formal part of this operation has already been per¬ formed by the differential calculus; since in it the transition and relation in general between original function and function
of development is established. In order to apply the function from which we must start, and also in order to effect the transition from it to the original function, it is necessary in many cases to have recourse to the serial form; but it must be remembered that this form as such has nothing to do directly with the peculiar principle of integration.
It appears next that the other part of the task of the calculus, with respect to its formal operation, is the application of the latter. This is now itself the task, namely, to know the meaning (in the sense indicated above), as a separate object, of the original function of the given function, which is considered the first derivative. It might appear that this doctrine was quite done with in the differential calculus; but a further circumstance enters into play which does not allow the matter to be so simple. For it resulted from this calculus that the proportion — which is linear — was obtained from the first derivative of the equation of a curve ; and, therefore, knowing this we also know that the integration of this proportion gives us the equation of the curve in the proportion of abscissa and ordinate; or, if the equation were given for the plane of a curve, then it would be the case that the differential calculus ought already to have taught, with respect to the meaning of the first derivative of such an equation, that this function exhibited the ordinate as function of the abscissa, and, there¬ fore, the equation of a curve.
But the question is, which of the determining moments of the object is itself given in the equation ; for the analytic treat¬ ment can proceed only from what is given, and pass thence to the remaining determinations of the object. Thus what is given is not the equation of an area of the curve, nor of the body arising from its revolution, nor of an arc of the curve; but only the proportion of the abscissae and ordinates in the equation of the curve itself is given. Therefore, the transitions from those determinations to this equation itself cannot already be treated in the differential calculus: it is reserved for the integral calculus to find these proportions.
But further it has been shown that the differentiation of an equation of more than one variable magnitude gives the power of development or differential coefficient, not as an equation, but only as a proportion : the task, then, is to indicate in the
moments of the object a second proportion that shall be equal to this first proportion, which is the derivative function. In the integral calculus, on the other hand, the object is the proportion itself of the original function to the derivative (which is here supposed to be given) ; and the task is, to indicate the meaning of the original function (which is to be discovered) in the object of the given first derivative, or rather, since this meaning has already been declared to be the problem (the meaning is, for instance, the plane of a curve, or the curve, imagined as rectilinear, which remains to be rectified; and so on), to demonstrate that such a determination is found by the original function, and to show which is the moment of the object that must be taken for this purpose as initial function of the derivative function.
Now the ordinary method, which uses the idea of the dif¬ ference as equivalent to the infinitely small, makes its task simple : thus, for the quadrature of curves, an infinitely small rectangle, a product of ordinate and element, that is, the infinitely small part of the abscissa, is taken for the trapezium which is supposed to have for one of its sides the infinitesimally small arc opposite to that infinitesimally small part of the abscissa. The product is integrated in this sense, that the integral gives the sum of the infinitely great number of trapezia, the plane whose determination is required, that is, the finite magnitude of this element of the plane. And, similarly, it forms a right-angled triangle out of the infinitesimally small elements of the arc and the ordinates and abscissae belonging to them; in this the square of that arc is supposed to be equal to the sum of the squares of the other two infinitesimally small elements, the integration of which presents us with this arc as a finite arc.
This procedure is based upon the general discovery which is the foundation of this part of Analysis; and here is based upon it in that the quadrated curve, the rectified arc, and so on, stand to a certain function which is given by the equation of the curve, in the relation of so-called original to derivative function. The question now is this: when a certain part of a mathematical object (for instance, of a curve) is assumed to be the derivative function, what other part of it is expressed by the corresponding original function? We know that, when