After our emergency screening of pi, minfeub saw Inherent Vice. This one was picked by rachel last week, because sable had up until now only liked Adam Curtis documentaries, though at the time of the screening she still claimed to have liked Pi (you might be hard-pressed to get that confession out of her still, though). Regardless, rachel felt like this has a similar spirit as those docs
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.
1 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 mathe¬ maticians, Cavalicri1 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 {lintUes) of the sums and ratios. It will be
1 Cavalicri, Francesco Bona ventura, 1598 1647, Professor of Mathematics at Bologna: GtamX ia imiunsibilium contuunrum nova, 1635: Extrdlatioms pomttriem, 1647
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 con¬ cluded 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,1 Reflexions sur la Metaphysique du Calcul Infinitesimal) 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 con¬ cept of a Quantum which is one term of a ratio and yet also a Quantum apart from this relation) to vanish. — This purifica¬ tion 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 pro¬ duct or quotient, roots, rectangles, squares, and also the sides
lots of sex and drugs but put so matter of factly i had difficulty staying awake. which i guess fits with the movie’s themes of being blitzed out and unaware of machinations seeking to dismantle the hippie scene
* Carnot, Lazare Nicolas Marguerite, Count, 1 753-1823, “organizer of victory” in the republican armies, equally important as politician and soldier until his banishment in 1815, died at Magdeburg: Reflexions, tic., 1797.
of rectangles and squares, and, in general, any finite magni¬ tude. — “It is considered as variable in its incessant motion and flow of increase or decrease, and to its momentary augmen¬ tation and diminution he gives the name of Moments. These, however, must not be taken as particles of determinate mag¬ nitude ( 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 determina¬ tion. In the latter the quantitative differences, the infinitesimal incrementa or decrementa, arc 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 incre¬ menta 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 con¬ ception indicated. This idea supposes them to be of such a nature that they may be neglected in relation to finite mag¬ nitudes; 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 mag¬ nitude 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 popularising 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 pro¬ cedure of a surveyor who, in measuring the height of a moun¬ tain, 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. tmiv. 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 com¬ parison 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
1 Euler, Leopold, 1707-1783, Professor at S. Petersburg, at Berlin, and again atS. Petersburg : Inbmhtctio in anafysut ufimktrum, 1 748 ; InsttMimei calculi Jijfftrtntialis, *755» ftutit.ealt. inttgralis, 1768-1794.
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. — Lagrange1 ( Theorie des fonctions analytiqu.es, 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 quali¬ tative 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
1 Lagrange, Jos. Louis, 1736-1812, Euler’s successor at Berlin, then Professor at the Eeole Polytechnique in Paris : Thiorie des fonctions analytiques, 1 797.
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.o = o; and therefore n : 1 = 0: o. — But it is just because 2 : 1 or n : 1 is a ratio between Qpanta that no ratio nor designation like o : o is adequate to it.
-avery
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 mathe¬ maticians 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 cal¬ culus rests the burden of justifying this method, where first, drawing them down into this sphere, it treats them as incre- menta 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 mathe¬ matics 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,1 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 some¬ thing 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 dear, 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 itsdf belongs; and that the peculiarity of the simple rdation (to be more fully considered later) to which the actual concrete determination of the calculus reduces itself, namdy, the relation of the derivative to the original function, is not here emphasized.
The older of the modems (such as Fermat,1 Barrow, 3 and others), who first used the infinitesimally small in that appli¬ cation 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 elemen-
> Landen, John, English mathematician, 1719-1790 : Mathematical Lucubra¬ tions, 1 755, etc.
• Fermat, Pierre de, 1601-1665 : Varia opera mathematica, 1679.
1 Barrow, Isaac, 1630-1677, Professor at Cambridge: Lectioms geometrical, 1669 Lectioms optical, 1674.
tary 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 ele- ments 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 determina¬ tions 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 andy are each taken as smaller by half its infinitesimal difference, the product passes
this was a strange film
by the same amount, it passes over into xy H - f-- - 1 - •
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 pro¬ ducts ; it, therefore, is the differential of x y. — Clearly, in this procedure, the term which forms the chief difficulty, the pro¬ duct 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;
= (# -t </*) (j> -t dy) — *y. It can only have been the need to justify the fiuxional 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 approxi¬ mation, 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 , Aquations Numeriques, 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 ( Thiorie des fonctions amlytiques, 3 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 instru¬ ment. 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 re¬ mained faithful to the formal and superficial principle of omit¬ ting 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.1 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 deter¬ mination (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 dis¬ tinction generally between a function and its variation after its variable magnitude has received an increment. In this sense,
> Lagrange in a simple manner sets side by side the two considerations in the application to mechanics of the theory of functions, in the chapter on rectilinear motion ( Thiorit dts fonct., ym P., Ch. I, art. 4). The space passed through, con¬ sidered as a function of the time elapsed, gives the equation x —ft; this developed
Thus the space passed through in the period of time presents itself as 6> 0*
“ Of t 4- — /" < + - /*+... The motion, therefore, by means of which
nothing made sense; everything was silly despite the unsettling undertones, and as the viewer i always felt a step or two behind the plot.
passage through this space takes place, is composed, as has been said, of various partial motions (and this, because analytical development produces a plurality and in fact an infinite plurality of terms), the spaces of which— corresponding 0* 0*
to the time — will be Oft, — f" t, - f" t, etc. When the motion is known,
the first partial motion is formally uniform and has a velocity determined by f t, the second uniformly accelerated and derived from an accelerative force proportional to /" (. “Now the remaining terms are related to no known simple motion, and, therefore, they need not be specially taken into account; and it will be shown that we can abstract from them in determining the motion at the beginning of the point of time.” There now follows the demonstration, which, however, is only a comparison of the series all of whose terms are re¬ quired to determine the magnitude of the space passed through in the period of time, with the equation given in art. 3 (for the motion of falling), namely x *= a * + 4 **, in which only these two terms occur. But this equation has itself received this form only because the explanation given to the terms evolved by analytical development is presupposed ; and this presupposition is, that uniformly accelerated motion is composed of a formally uniform motion, continued at the velocity reached in the previous period of time, and of an increment (which is a in #«= «<*, that is, the empirical coefficient) to which the force of gravity is ascribed; — a distinction which has no existence nor reason in the nature of the matter, and is only the expression— erroneously physieixed — of the result of the analytical treatment assumed.
it is clear that the first term of the series resulting from the development of (x 4- dx)n quite exhausts the differential of x". 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 cal¬ culus. A ratio, and not a sum, is here in question ; and, there¬ fore, with the first term the differential is folly 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 difficul¬ ties 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 aug-
mentation 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 intro¬ duce (dx and dy) must be regarded simply as moments of d y d x
—, and — itself as a single and indivisible symbol. We may d x dy
disregard the fact that the practical application of the mechan¬ ism 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
it was good though, and made me constantly want to say “go white boy go”, so i give it an 11/12; would watch again maybe on a cute date with someone
and not given magnitude (as when we say “an infinite mul¬ titude,” “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 inter¬ fere in the treatment of it nor in the use made of this deter¬ mination 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 inves¬ tigate 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_? passes over into 4- k, is to change into fx + ph-j-q/P + rh* +
and thus k — p h + q ha 4- . . and -r — p + qk + trhx+...
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 - is neverthe-
less not supposed to equal -, 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
but only the determinate value to which the ratio may approxi¬ mate 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 defpr-
initiation, 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
~ = -• If, on the other hand, ^ is assumed to be equal ax o ax
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 = o ; and by this assumption alone k k
-recima from minfeub
- =p is found. Again, grant that - = o; — and when h = o, h h
then, in fact, automatically k — o ; 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 infini¬ tesimal 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,1 and those which elsewhere (like nisus, or becoming,
1 The category of continuous or flowing magnitude arises in the consideration of external and empirical variation of magnitudes, — which, by an equation, are brought into this relation, that one is a function of the other. Now the scientific object of the differential calculus is a certain ratio (generally expressed by the differential coefficient) — a determinateness which might also be called Law ; and, therefore, for this specific determinateness mere continuity is partly a foreign aspect, and partly and in any case an abstract and here empty category, since nothing is here stated about the Law of Continuity. — The ingenious General Exposition of the fundamental determinations used in the deduction of the differential calculus by my respected colleague. Professor Dirksen,* which forms
- Dirksen, Enno Heeren, 1790-1850, Professor of Mathematics at Berlin: Analjitischi Darstellung dir Variathnunchumg, 1823.
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 mathe¬ matical 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 con¬ version 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
an appendix to a criticinn of certain new treatises on this science ( Jahrb . f. wissmsch. Kritik, 1897, Nr. 153 sqq.), will show what kind of formal definitions have been resorted to in this matter. The following definition (op. cit., p. 1951) b actually quoted: “Every magnitude which b considered as in the state of becoming, provided that this becoming does not take place by leaps, but in an uninterrupted progress, b a constant or continuous magnitude, a Continuum." Surely thb b a tautology, and the same as the dtfimtum.
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 * but to dy, is in the background. Relatively to d x, d 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 ; d x 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 per¬ missible 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
right
and ordinates, versed sine and cosine, and so forth, when (i 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 ks 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 quanti¬ tative determination. But when it is taken as intensive mag¬ nitude, 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 deter¬ minations, 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 opera¬ tion 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. philosopfuae naturalis lib. /., Sect. II., Prop. I., and cf. Schubert’s1 * 3 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
1 Schubert, Friedrich -Theodor von, 1 758-1895, Director of the Observatory
at S. Petersburg: Lthrbuch dir thmretischtn Astronomit, 1798; Populate Astronomii,
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
so
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.
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.
the nazi-supported dentist cabal operate a vertical operation of smuggling heroin from “indochina” and getting the addicts cleaned up in extremely expensive rehab anti-commie brainwash clinics, fixing their broken calcium-stripped teeth so they can go back out in the world and inevitably flail around and fuck up and come right back to the good good yummy stuff, for the purposes of being utterly rinsed of all the cash they can steal or beg from family and strangers, which is all run by the federal government but don’t worry about that, and we bear witness to the layers of slime being gently power washed away through the mutton chop master magnum pi mr larry “doc” sportello who is terminally baked and bumbling through all of his own shit happening concurrently, like 1. his ex gf who got embroiled with wealthy real-estate sex communist (on his way to the aforementioned dentist rehab anti-commie brainwash clinic (to be anti-commie brainwashed)), b. former renowned surf-jazz-tenor-cum-government-snitch owen wilson who must be freed from his prison of receiving stiches from the aryan brotherhood, and iii. renaissance lapd cryptid detective who love love loves chocolate covered bananas (they are best friends). joanna newsom also narrates lots of it but i think she might have been like, fake or a figment of someone’s imagination or something, nobody seemed to really interact with her. maybe that’s what a good narrator does. i like joanna newsom go listen to her hit 2010 album “have one on me” it’s like two hours long you’ll love it. anyway. everyone drinks Burgle! brand beer and smokes joints and snorts smack and you the viewer will watch along in turn as if you are smoking, like, a huge reefer man haha. it’s crazy. she even got her shastas out!!! don’t worry about the government it’s fine. the movie made sense and i enjoyed it.
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
becomes — = c, then, equally, a — — is the determination of at x
the tangents, or - = c the determination of plain velocity. dy
rating: dancing exclamation point dot gif © 2002 ARG! Cartoon Animation http://artiestick.com A division of CityStar Group, Inc. Colorado Springs, CO USA (719) 559-1945.USSPCMT fair use no copyright intended thanks for 100 likes
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
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.
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
-grombo
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
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
my favorite scene is the loony bin with the Burke Stodger marathons and where all is well, but especially the marathons. it’s not that deep, but it very artfully breaks the fourth wall, just a lil nod to the fact that this film is a part of the same vertically integrated escapism economy that it is critiquing. made me quite paranoid the first time i watched it, and still icked me this second time
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
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
offscreen: “This place is a lie!…”
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
(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 =
Dr. Threeply: “…This is particularly popular with the patients…”
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.
offscreen: “…Everything on this table is as phony as the town!”
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
the function of the ordinate given by the equation of the curve is taken to be the derivative function, then the (relatively) original function expresses the magnitude of the area of the curve which is cut off by this ordinate ; and, when a certain tangential determination is regarded as derivative function, then its original function expresses the magnitude of the arc belonging to this tangential determination, and so on: but the method which uses the infinitesimally small and operates with it does not take the trouble to recognize and to demon¬ strate that these ratios (firstly that which subsists between original and derivative function, and secondly that between the magnitudes of two parts, or conditions, of the mathematical object) together form one proportion. It is the peculiar merit of intellectual insight to have discovered from results already known from without, that certain and specified sides of a mathematical object stand in the relation one to the other of original and derivative function.
In this calculus, of these two the derivative function, or (as it has been determined) the function of potentiation, is the one which is given relatively to the original function ; the latter still remains to be discovered from the former by integration. But it is not given immediately, nor is it given for itself which part or determination of the mathematical object is to be looked at as derivative function, in order to find, through tracing it back to the original function, the other part or determination, whose magnitude the problem requires. The ordinary method, as has been stated, immediately represents certain parts of the object as infinitesimally small, in the form of derivative functions which can be determined from the original equation of the object by differentiation in general ( — thus for the rectification of a curve it takes the infinitesimal abscissae and ordinates) ; and this method takes instead such as can be brought into a connexion with the object of the problem (in the example, with the arc, which also is imagined as infinitesimal) which is fixed and determined by elementary mathematics, by which procedure, these parts being known, that part also of which the magnitude was to be found, is determined. Thus, for rectification, the three infinitesimals we mentioned are connected in the equation of the right-angled triangle, while for quadrature, the ordinate and the infinitesimal
i also love all the lighting and the sense of humor and the scene where Doc does a line and walks straight through the dentist office and ends up where he started and people keep showing up, the music is soooo great, aaaaa i love this film
abscissa are connected in a product, a plane being taken, in the general arithmetical manner, as a product of straight lines. The transition from such a so-called element of plane or arc, and so forth, to the magnitude of plane or arc itself now only counts as ascent from the infinitesimal to the finite expression or to the sum of an infinite number of elements, of which the required magnitude is supposed to consist.
We can, therefore, make only the superficial observation that the integral calculus is just the problem of the differential calculus, but inverted, and, in general, more difficult; while the real interest of the integral calculus concerns itself entirely with the relation to each other of the original and the derivative function in concrete objects.
Lagrange nowhere banished the difficulty of any problem in the facile manner of these direct assumptions, nor did he consent to do so in this part of the calculus. It will help the elucidation of the nature of the matter if here too we indicate the detail of his procedure in a few examples. For the task of his method is to demonstrate in itself that a relation of original to derivative function subsists between separate determinations of a mathematical whole, for instance a curve. But in this sphere, because of the nature of the relation itself, this cannot be done in a direct manner, since in the mathematical object the relation connects curved with straight lines, linear dimen¬ sions and their functions with plane dimensions and their function, and so on, — connects, that is, terms which are quali¬ tatively different; the determination thus can only be taken as the mean between a greater and a less. Thus here again there enters spontaneously the form of increment with its plus and minus, and that vigorous “ Developpons ” is here in place; but we have already discussed that purely arithmetical and finite meaning which here is all that belongs to the increments. When we develop the condition that the magnitude which is to be determined must be greater than one and less than another limit (itself easily determinable), we derive such facts as that the function of die ordinate stands in the relation of first derivative function to the function of the area.
Lagrange’s exposition of the rectification of curves starts from Archimedes’ principle, and is interesting, therefore, as affording an insight into the translation of the Archimedean
method into the principle of modem analysis, which allows us to look upon the inner and true meaning of an affair which by the other mode is pursued but mechanically. The procedure is necessarily analogous to that which has just been indicated: no direct equation results from the principle of Archimedes, which is that the arc of a curve is greater than its chord and less than the sum of two tangents drawn between the ends of the arc and their point of intersection. This fundamental determination of Archimedes is translated into the modem analytic form when an expression is found which is in itself a simple fundamental equation, whereas the form of Archi¬ medes only postulates that there must be an infinite progress between two elements, respectively too great and too small, which each time determine themselves, — this progress ever giving but a new pair of great and small, with an ever narrower limit of inaccuracy. The formalism of the infinitesimally small immediately gives us the equation d t* — d x® + djP. But Lagrange's demonstration, starting from the foundation which has been indicated, shows that the magnitude of the arc stands in the relation of original to a derivative function, in which the characteristic member itself is a function which comes from the relation of a derivative to the original function of the ordinate.
The method of Archimedes, and, at a later date, Kepler’s treatment of stereometrical objects, employ the idea of the infinitesimally small, — a fact which has often been quoted as authorizing the we made of this idea in the differential cal¬ culus, while its peculiarity and distinguishing quality were not sufficiently emphasized. The infinitesimally small means, first, the negation of Quantum as such, that is, of a so-called finite expression, or of that perfect determinateness which belongs to Qpantum as such. And, similarly, the fundamental deter¬ mination in the subsequent famow methods of Valeriw,1 Cavalieri, and others, which are based on the consideration of the relations of geometrical objects, is, that the Qpantum (as such) of determinations which so far are being considered merely as related terms, is to be neglected for this purpose, and consequently they are to be taken as non-magnitudinal.
1 Valerius, Lucas, died 1618 at Rome, called by Galileo die Archimedes of his time : dr quadrature parabolae per simplex faltum.
But here the general affirmative element which is latent in the merely negative determination remains unrecognized and unnoticed — an element which, abstractly, proved to be quali¬ tative magnitudinal determinateness based, more precisely, upon the ratio of powers; — and also, since this relation itself includes a number of more closely determinate relations like powers and the functions of their development, attempts have been made to base these too on the general and negative determination of the same infinitesimally small, and to derive them thence. In Lagrange’s exposition which we have just examined, the determinate affirmative which is contained in Archimedes’ manner of developing the problem has been dis¬ covered, and thus the procedure, which was infected with an unlimited overpassing, has been given its correct limit. The importance of the modern invention in itself, and its capacity to solve problems hitherto intractable, and to treat in a simple manner those which were not insoluble before, is due solely to the discovery of the relation of the original to the so-called derivative, and of the parts which, in a mathematical whole, stand in this relation.
What has been said may serve to make clear what is characteristic in that proportion of magnitudes which is the object of the particular kind of calculus under discussion. Our exposition was able to confine itself to simple problems and the methods of solving them ; and it would neither have been suitable for the determination of the concept (which here alone was our object), nor would it have been in the power of the author, to examine the whole scope of the so-called application of the differential and integral calculus, and to complete the induction that they are based upon the principle which we discovered, by tracing back to it all their problems and their solutions. But our contribution has sufficiently shown that as every other calculus has for object a separate determinateness or relation of magnitude, and addition, multiplication, the raising to powers and extraction of roots, operations with logarithms and with series, and so on, constitute such objects, so too the differential and integral calculus; and the name of relation of a function of powers and of the function of its development or potentiation, is perhaps fittest for whatever is akin to this calculus, for this name places us nearest to an
insight into the nature of the matter. Only, operations accord¬ ing to other magnitudinal relations (addition and so on) are also generally used in this calculus, and so also logarithmic, circular, and serial relations are applied, especially in order to make more tractable expressions for the purpose of the necessary operations of the derivation of original functions from functions of development. The differential and integral calculus has this point of interest in common with the form of the series, that it determines the functions of development which, in series, are called coefficients of the terms; but the interest of this calculus is directed only upon the relation of the original function to the next coefficient of its development, and thus the series tries to represent a sum in the multitude of terms arranged according to powers which have these coefficients. The infinite of the infinite series — that indeter¬ minate expression of the negative element of Quantum in general — has nothing in common with the affirmative deter¬ mination which is contained in the infinite of this calculus. And also the infinitesimally small (whicn appears under the shape of increment), which gives to development the form of series, is a merely external means to this end, and the only meaning of its so-called infinity is to have no meaning except as this methodological device : the series, which in fa't is not what is wanted, produces an excess the removal of which causes the unnecessary trouble. Lagrange’s method too — he took up the serial form again by preference — is hampered by this trouble; although it is in this method that the true peculiarity stands revealed in what is called the application, for, without forcing the forms of d x, dy, and so on, into the objects, that part is directly indicated to which the deter¬ minateness of the derived function (function of development) belongs in them: so that it is clear that here the real matter in hand is not the form of the series.1
-rachel
* In the above-mentioned criticism ( Jahri . fir wissensch. Krit., II Bd. 1827, Nr. 115, 6 sqq.), interesting views of a sound scholar in this science, Mr. Spehr,* may be found; they are quoted from his Aims Prinzipien da FlutnUnkalkuls, Brunswick, 1826, relating to a fact which he asserts materially to contribute to the obscurities and unscientific parts of the differential calculus, and they agree
* Spehr, Friedrich Wilhelm, 1799-1833, a mathematician of Brunswick: Volls t&nd&gn Lthrbegriff der rtinen Kombinatiorulehre, 1824.
Further Forms connected with the Qualitative Determinateness of Magnitude
In the differential calculus the infinitesimally small appears in its affirmative meaning as qualitative magnitudinal deter¬ minateness; and of this it has further been shown that it is present in this calculus not only as determinateness of powers in general, but more especially as the determinateness of the ratio of a function of powers to the power of development. The qualitative determinateness is also present in a wider and, so to say, weaker form, which, together with the connected employment of the infinitesimally small and its meaning in this employment, will be considered in this Observation.
We begin from what precedes, and must first remark in this regard that the various determinations of powers, in the analytic aspect, manifest themselves as purely formal and quite homogeneous, since they denote numerical magnitudes, which, as such, do not possess this mutual relation of qualitative difference. But when applied to objects of space, the analytic relation shows itself in its qualitative determinateness as the transition from linear to plane determinations, from deter¬ minations of straight lines to determinations of curves, and so
with what has been said about the general nature of the theory of this calculus. “Purely arithmetical investigations,” he says, “(admittedly related more closely than any others to the differential calculus) were not distinguished from the differential calculus proper, and have ever been confused (as by Lagrange) with the matter itself, which latter was regarded as a mere application of them. These arithmetical investigations include the rules of differentiation, the derivation of Taylor’s theorem, etc., and even the various methods of integration. But the reverse is the case, and these applications are the object of the real differential calculus, which, from the analysis, presupposes all these arithmetical develop¬ ments and operations.” — We have shown how with Lagrange the separation of the so-called application from the procedure of the general part, which starts from series, serves only to emphasize the peculiar nature of the differential calculus for itself. And it is strange that the author, having this interesting under¬ standing that the so-called applications are just what constitutes the object of the differential calculus proper, should enter upon the formal metaphysics (there quoted) of Continuous Magnitude, Becoming, Flow, and so on, and should even desire to add new ballast to this old. These determinations ar c formal because they are only general categories, which do not indicate what is specific in the matter which was to be learned and abstracted from the concrete doctrines, the applications.
on. This application further involves that spatial objects which from their very nature are given in the form of continuous magnitudes, are taken as discrete, the plane as a multitude of lines, the line as a multitude of points, and so on. This solution is interesting only in one respect, which is that it itself determines the points into which the line is analysed and the lines into which the plane, and so forth, in order that from such a determination it may proceed analytically, which really means arithmetically; the starting-points for the mag- nitudinal determinations which are to be found are those elements whence are to be derived the function and equation for the concrete, that is, the continuous magnitude. In those problems which are interesting chiefly because they employ this procedure, something which is determinate in itself is de¬ manded, in the element, for the starting-point, — in opposition to the method which is indirect, because it can begin on the contrary only with limits between which is supposed to lie that entity, determinate for itself, which is its objective. In both methods the result is the same if only it is possible to discover the law of further determination and impossible to reach the perfect (that is, so-called finite) determination which is demanded. To Kepler is ascribed the honour of first having thought of this reversal of the progress, and of making the discrete its starting-point. He expresses this simply when he explains how he understands the first theorem in Archimedes’ cyclometry. Archimedes’ first theorem is, of course, that a circle is equal to a right-angled triangle having one catheter equal to the radius and one equal to the circumference of the circle. Kepler takes the meaning of the theorem to be this, that the circumference of the circle has as many parts as it has points, that is, infinitely many,- each of which may be considered as the base of an isosceles triangle, and so on: he thus gives expression to the dissolution of the continuous into the form of discreteness. The expression “infinite” which here occurs is still far distant from the determination which it is destined to have in the differential calculus. — A determinateness or func¬ tion having now been found for such discreta, they must next be united, and exist essentially as elements of the continuous. But a sum of points produces no line and a sum of lines no olane: the points are, therefore, immediately taken as linear
and the lines as plane in nature. But also these linear entities must not ye; be lines (which they would be if they were taken as quantum) : they are, therefore, imagined as infinitesimally small. But what is discrete can be united only externally, the moments retaining their meaning of discrete Ones; the analytic transition from them is made only to their sum, and is not also the geometric transition from point into line, line into plane, and so on; and, therefore, the element which has its determination as point or as line is given the quality of line with the former, and the quality of plane with the latter determination, so that the sum, being a sum of little lines, may become a line, and being a sum of little planes, a plane.
The need of acquiring this moment of qualitative transition and of recurring to the infinitesimally small to this end must be regarded as the source of all the ideas which, intended to overcome this difficulty, are the greatest difficulty themselves. In order that one might, dispense with this expedient, it would have to be possible to show that the analytic procedure itself, which appears as a mere summation, in fact already contains multiplication. But in this regard a new assumption enters, which constitutes the basis of this application of arithmetical relations to geometrical figurations. This assumption is,, that arithmetical multiplication is a transition to a higher dimension for the geometrical determination too, and that the arith¬ metical multiplication of magnitudes which, according to their spatial determinations, are lines, also extracts a plane from the linear determination. Three times four linear feet are twelve linear feet, but three linear feet times four linear feet are twelve plane feet (square feet), for the unit in both, since each is a discrete magnitude, is the same. The multiplication of lines by lines at first appears meaningless, since multiplication only deals with numbers; that is, it is a change of entities which are perfectly homogeneous with that into which they pass over (the product), and change only their magnitude. On the other hand, a process of multiplying a line as such by a line — which has been called ductus liruae in lineam , like plant in planum, and is also ductus puncti in lineam — is a change not only of the mag¬ nitude but also of the line as qualitative determination of spatiality, as a dimension; the transition of line into plane must be taken as its sclf-extemalization , which for the point
is a line, and for the plane, a volume. It is this process which is imagined when it is said that the movement of a point is a line, and so forth; but movement includes the determination of time, and in this idea, therefore, appears rather as a con¬ tingent and external variation of the condition ; whereas it is the conceptual determinateness (which was expressed as self- extemalization) which must be taken, — that qualitative change, which in arithmetic is the multiplication, of Unit (the point and so on) into Amount (the line and so on). — We may here make this further remark, that in the self-externalization of the plane (which would manifest itself as a multiplication of plane into plane) the appearance of a difference between arithmetical and geometrical products results when the self- externalization of the plane, as ductus plani in planum, would in arithmetic produce the multiplication of two determinations of the second degree, that is, a product of four dimensions, which in the geometrical determination is however reduced to three. Although on the one hand Number, because it has One for principle, provides the fixed determination for the external quantity, yet equally its productive power is formal. Taken as a numerical determination, 3x3, when it repro¬ duces itself, is 3 x 3 X3 x 3; but this same magnitude as plane determination is not allowed, when it reproduces itself, to proceed beyond 3x3x3, because space, imagined as a progress from the point, or the merely abstract limit, has its true limit as concrete determinateness after line in the third dimension. This difference might prove powerful with regard to free movement, wherein one (spatial) side is governed by geometrical determination (in Kepler’s law j3 : fi), and the other (temporal) side by arithmetical determination.
It will now be evident without further remark how the qualitative here considered differs from the subject of the previous Observation. There the qualitative element consisted in the determinateness of powers : here, like the infinitesimally small, it stands in the mere arithmetical relation of factor to product, or as point to line, line to plane, and so on. And the qualitative transition which must be made to the continuous from the discrete (into which continuous magnitude is imagined as dissolved) is effected by a process of summation.
a multiplication and, therefore, a transition from linear to plane determination; and this appears most simply in the manner in which (for instance) it is shown that the area of a trapezium is equal to the product of the sum of the two opposite parallel lines and half the height. This height is imagined simply as the amount of a multitude of discrete mag¬ nitudes which must be summed up. These magnitudes are lines which lie parallel between those two limiting parallels; their number is infinite, for they must constitute the surface, and also they are lines which, therefore, in order to be of plane nature, must be posited together with negation. In order to escape the difficulty that a sum of lines is to produce a plane, lines are immediately assumed to be planes, but infinitely narrow, for their only determination lies in the linear element of the parallel limits of the trapezium, being parallel, and being limited by the other pair of rectilinear sides of the trapezium, they can be imagined as terms of an arithmetical progression, having a uniform difference which, however, need not be determined, and having those two parallels for first and last terms ; the sum of this series is of course the product of the parallels and half the amount of the terms. This latter quantum is called Amount only relatively to the idea of the in finitely many lines: it is the magnitudinal deter¬ minateness of something continuous, namely height. Clearly that which is called sum is also a ductus lineae in lineam, multi¬ plication of linear by linear element, and, therefore, the produc¬ tion (according to the above determination) of something of plane nature. In the simplest case of any rectangle A B, each of the two factors is a simple magnitude; already in the next example (itself elementary) of the trapezium, only one factor is once half the height, while the other is determined by a progression : it too is of linear nature, but the determinateness of its magnitude is more complex : and since it can be expressed only by a series, the endeavour to sum it up is called analytical, that is, arithmetical ; while multiplication here is the geomet¬ rical moment, the qualitative part of the transition from the dimension of line into plane : one factor is taken, discretely, only for the arithmetical determination of the other, and, like the other, is itself the magnitude of a linear something.
The method in which planes are imagined as sums of lines
is also frequently used when multiplication as such is not used in order to find the result. This happens where the object is not to indicate the magnitude in the equation as Quantum, but as a proportion. For instance, it is commonly proved that the area of a circle is to the area of an ellipse, of which the major axis is equal to the diameter of the circle, as the major axis is to the minor axis, by assuming that each of these areas is the stun of the relative ordinates. Each ordinate of the ellipse is to the corresponding ordinate of the circle as the minor is to the major axis : it is concluded that, therefore, the sums of the ordinates too (that is, the areas) are in the same proportion. Those who here wish to avoid the idea that an area is a sum of lines have recourse to the common and quite gratuitous makeshift of making the ordinates into trapezia of infinitely small breadth: the equation is only a proportion, and there¬ fore only one of the two linear elements of a plane enters into the comparison. The other element (the axis of the abscissa) is assumed to be equal in ellipse and in circle, and therefore as = x in so far as it is a factor of arithmetical magnitudinal determination: the proportion therefore depends solely upon the relation of the one determining moment. The two dimen¬ sions are essential to the idea of area: but the magnitudinal determination, as it is required to be indicated in this pro¬ portion, concerns itself only with one moment ; and when the idea of sum is added to this one moment (which is a surrender to, or attempted propping of, the idea), then this means that the real point demanded by mathematical determinateness is here missed.
This exposition also contains the criterion of Cavalieri’s method of the indivisible (which has been mentioned above) : this, too, is therefore justified and need not have recourse to the infinitesimally small. These indivisibilia are lines when he is considering a plane, and squares or plane circles when he is considering a pyramid, cone, and so on. The basic line or plane (which he takes as determinate) is called the regula ; it is the constant, and with reference to a series it is its first or last term; and to it these indivisibilia are considered parallel, that is, as having the same determination with regard to the figure. Now Cavalieri’s general fundamental proposition is ( Exerc . Geometr. VI., in the later work Exerc. /., p. 6), “that all
figures, both plane and solid, are proportionate to all their indivisibilia, and compare these collectively or, when there is a common proportion in them, distributively.”— For this pur¬ pose he compares, in figures of the same base and height, the proportions between lines drawn parallel to the former and at equal distance from it; all such lines in a figure have one and the same determination and constitute its whole content. In this manner, too, Cavalieri proves, for instance, the elemen¬ tary proposition that parallelograms of equal height are pro¬ portionate each to its base : any two lines drawn at an equal distance from the base and parallel to it in the two figures are in the same proportion to the bases; and so, therefore, are the whole figures. In fact, the lines do not constitute the content of the figure in so far as it is continuous , but only in so far as it is to be determined arithmetically; the linear factor is its element, and through it alone its determinateness must be seized.
And here we are led to reflect upon the difference which exists with respect to that element in which the determinateness of a figure consists; it is either of the same nature as is here the height of the figure, or it is external limit. In so far as it is determinateness as being external limit, it is admitted that the continuity of the figure follows, so to speak, upon the equality or the proportion of the limit; thus the equality of figures which coincide with one another follows from the feet that the limiting lines coincide with one another. But when parallelograms are of equal height and base, only the latter determinateness is external limit; height, and not parallelism (and on this the second capital determination of figures, that is their ratio, depends), introduces a second principle which determines external limits. Euclid’s proof that parallelograms of equal height and base are equal, reduces them to triangles, that is, to continua limited externally: in Cavalieri’s proof, and first in his proof of the proportionality of parallelograms, the limit is magnitudinal determinateness as such in general, which is explained to be taken as applied to each pair of lines drawn at an equal distance in the two figures. These lines, which are either equal or in an equal ratio with the base, being „ taken collectively, produce the figures which also are in the same ratio. The idea of ai> aggregate of lines is incompatible
with the continuity of a figure; the mere consideration of lines quite exhausts the essential deterxninateness. Cavalieri frequently answers the objection that the idea of the indivisible necessarily involves the comparison of lines or planes infinite in amount ( Geom . Lib. II. Prop. I. Schol.) ; and makes the just distinction, that it is not their amount, which we do not know ( — it is, as has been observed, an empty auxiliary idea — ), but only magnitude, that is quantitative determinateness as such, equal to the space enclosed by these lines, which he compares. This space is contained in limits, and, therefore, its magnitude too is contained within these limits : the continuous is nothing other than the indivisibilia themselves , says Cavalieri : if it were outside them it would be incommensurable: but it would be absurd to say that limited continua were incommensurable.
Clearly, thus, Cavalieri attempts to distinguish what belongs to the external existence of the continuous from that which constitutes its determinateness and needs to be emphasized only for comparison and for theorems dealing with it. It is true that the categories which he uses for this purpose (the fact that the continuous is composed or consists of the indivisibilia, and the like) are inadequate, since they also involve the intuition, or, as was said before, the external existence, of the continuous ; instead of saying that “the continuous is nothing but the indivisibilia themselves,” it would be more correct and therefore also immediately more clear to say that the mag- nitudinal determinateness of the continuous is the same as that of the indivisibilia themselves. — Cavalieri does not care for the faulty conclusion that there are greater and less infinites, which (he says) is drawn by the schools from the idea that the indivisibilia constitute the continuous ; and, further, (Geom. Lib. VII. Preej.) he expresses the more definite knowledge that his method of proof by no means forces him into the idea of the composition of the continuous from indivisibilia : the con¬ tinua only follow the proportion of the indivisibilia. He claims to have taken the aggregates of indivisibilia, not in that manner in which, for the sake of an infinite multitude of lines or planes, they appear to fall into the determination of infinity, but in so far as they contain a determinate condition and nature of limitedness. But in order finally to remove this 1 tumbling-block, he does not spare himself the pains of proving
the capital propositions of his geometry (in the seventh book, specially added for this purpose) in a manner designed to be uninfected with infinity. — This manner reduces the proof to the ordinary form (mentioned before) of the coincidence of figures, that is, as was observed, to the idea of determinateness as external spatial limit.