1) Mystical Differential Calculus. x1 = x + x from the beginning changes into x1 = x + dx or x + x., where dx is assumed by metaphysical explanation. First it exists, and then it is explained. Then, however, y1 = y + dy or y1 = y + y.. From the arbitrary assumption the consequence follows that in the expansion of the binomial x + x or x + x., the terms in x and x which are obtained in addition to the first derivative, for instance, must be juggled away in order to obtain the correct result etc. etc. Since the real foundation of the differential calculus proceeds from this last result, namely from the differentials which anticipate and are not derived but instead are assumed by explanation, then dy/dx or y./x. as well, the symbolic differential coefficient, is anticipated by this explanation. If the increment of x = x and the increment of the variable dependent on it = y, then it is self-evident (versteht sich von selbst) that y/ x represents the ratio of the increments of x and y. This implies, however, that x figures in the denominator, that is the increase of the independent variable is in the denominator instead of the numerator, not the reverse; while the final result of the development of the differential form, namely the differential, is also given in the very beginning by the assumed differentials.* If I assume the simplest possible (allereinfachste) ratio of the dependent variable y to the independent variable x, then y = x. Then I know that dy = dx or y. = x.. Since, however, I seek the derivative of the independent [variable] x, which here = x., I therefore have to divide64 both sides by x. or dx, so that: dy/dx or y./x. = 1. Beginning, however, with functions of x in the second degree, the derivative is found immediately by means of the binomial theorem [which provides an expansion] where it appears ready made (fix und fertig) in the second term combined with dx or x.; that is with the increment of the first degree + the terms to be juggled away. The sleight of hand (Eskamotage), however, is unwittingly mathematically correct, because it only juggles away errors of calculation arising from the original sleight-of-hand in the very beginning. x1 = x + x is to be changed to x1 = x + dx or x + x. , The only question which still could be raised: why the mysterious suppression of the terms standing in the way? That specifically assumes that one knows they stand in the way and do not truly belong to the derivative. The answer is very simple: this is found purely by experiment. Not only have the true derivatives been known for a long time, both of many more complicated functions of x as well as of their analytic forms as equations of curves, etc., but they have also been discovered by means of the most decisive experiment possible, namely by the treatment of the simplest algebraic function of second degree for example: y = x y + dy = (x + dx) = x + 2xdx + dx , y + y. = ( x + x.) = x + 2xx. +x. . If we subtract the original function, x (y = x ), from both sides, then: dy = 2xdx + dx y. = 2xx. + x.x. ; dy = 2xdx, y. = 2xx. , dy/dx = 2x , y./x. = 2x . We know, however, that the first term out of (x + a) is x ; the second 2xa ; if I divide this expression by a, as above 2xdx by dx or 2xx. by x., we then obtain 2x as the first derivative of x , namely the increase in x,65 which the binomial has added to x . Therefore the dx or x.x. had to be suppressed in order to find the derivative; completely neglecting the fact that nothing could begin with dx or x.x.*2 in themselves. In the experimental method, therefore, once comes - right at the second step - necessarily to the insight that dx or x.x. has to be juggled away, not only to obtain the true result but any result at all. Secondly, however, we had in 2xdx + dx or 2xx. + x.x. the true mathematical expression (second and third terms) of the binomial (x + dx) or (x + x.) . That this mathematically correct result rests on the mathematically basically false assumption that x1 - x = x is from the beginning x1 - x = dx or x., was not known.66 In other words, instead of using sleight of hand, one obtained the same result by means of an algebraic operation of the simplest kind and presented it to the mathematical world. Therefore: mathematicians (man ... selbst) really believed in the mysterious character of the newly-discovered means of calculation which led to the correct (and, particularly in the geometric application, surprising) result by means of a positively false mathematical procedure. In this manner they became themselves mystified, rated the new discovery all the more highly, enraged all the more greatly the crowd of old orthodox mathematicians, and elicited the shrieks of hostility which echoed even in the world of non-specialists and which were necessary for the blazing of this new path. 2) Rational Differential Calculus. D Alembert starts directly from the point de d part (sic) of Newton and Leibnitz: x1 = x + dx. But the immediately makes the fundamental correction: x1 = x + x, that is, x and an undefined but prima facie finite increment which he calls h. The transformation of this h or x into dx (he uses the Leibnitz notation, like all Frenchmen) is first found as the final result of the development or at least just before the gate swings shut (vor Toresschluss), while in the mystics and the initiators of the calculus it appears as the starting point (d Alembert himself begins with the symbolic side,*3 but first transforms it symbolically). By this means he immediately succeeds in two ways.67 a) The ratio of differences (f(x + h) - f(x))/h = (f(x + h) - f(x))/(x1 - x) 1) [the difference] f(x + h) - f(x), corresponding to the given algebraic function in x, stands out as soon as you replace x itself with its increment x + h in the original function in x, for example, in x . This form ( = y1 - y, if y = f(x)) is that of the difference of the function, whose transformation into a ratio of the increment of the function to the increment of the independent variable now requires a development, so that it plays a real role instead of a merely nominal one, as it does with the mystics; for, if I have in these authors f(x) = x , f(x + h) = (x + h) = x + 3x h + 3xh + h , f(x + h) - f(x) = x + 3x h + 3xh + h - x , In d Alembert it is necessary to hold fast to this difference because the steps of the development (Entwicklungsbewegungen) are to be executed upon it. In place of the positive expression of the difference, namely the increment, the negative expression of the increment, namely the difference, and thus f(x + h) - f(x), therefore comes to the fore on the left-hand side. And this emphasis on the difference instead of the increment ( fluxion in Newton) is foreshadowed at least in the dy of Leibnitzian notation as opposed to the Newtonian y.. 2) f(x + h) - f(x) = 3x h + 3xh + h . When both sides have been divided by h, we obtain (f(x + h) - f(x))/h = 3x + 3xh + h . (f(x + h) - f(x))/h = ((x + h) - f(x))/(x1 - x) 3) Now when in (f(x + h) - f(x))/h = (f(x + h) - f(x))/(x1 - x) 4) 0/0 or dy/dx = 3x = f (x) . Just as with the mystics, this already existed as given, as soon as x became x + h, for (x + h) in place of x produces x + 3x h + etc., where 3x already appears in the second term of the series as the coefficient of h to the first power. The derivation is therefore essentially [the] same as in Leibnitz and Newton, but the ready-made derivative 3x is separated in a strictly algebraic manner from its other companions. It is no development but rather a separation of the f (x) - here 3x - from its factor h and from the neighbouring terms marching in closed ranks in the series. What has on the other hand really been developed is the left-hand, symbolic side, namely dx, dy, and their ratio, the symbolic differential coefficient dy/dx = 0/0 (rather the inverse 0/0 = dy/dx), which in turn once more generates certain metaphysical shudderings, although the symbol has been mathematically derived. D Alembert stripped the mystical veil from the differential calculus and took an enormous step forward. Although his Trait des fluides appeared in 1744 (see p.15*4), the Leibnitzian method continued to prevail for years in France. It is hardly necessary to remark that Newton prevailed in England until the first decades of the 19th century. But here as in France earlier d Alembert s foundation has been dominant until today, with some modifications. 3) Purely Algebraic Differential Calculus. Lagrange, Th orie des fonctions analytiques (1797 and 1813). Just as under I) and 2), the first starting point is the increased x; if y or f(x) = etc., xm + mxm-1h + etc., a) When x + h replaces x in a given original function of x, f(x + h) is related to the series expansion (Entwicklungsreihe) opposite it in exactly the same way that the undeveloped general expression in algebra, in particular the binomial, is related to its corresponding series expansion, as (x + h) , for example in (x + h) = x + 3x h + etc., a/(a - x) = 1 + x/a + x /a + x /a + etc., sin(x + h) = sin x cos h + cos x sin h D Alembert merely algebraicised (x + dx) or (x + x.) into (x + h), and thus f(x + h) from y + dy, y + y. into f(x + h). But Lagrange reduces the entire expression (Gesamtausdruck) to a purely algebraic character, since he places it, as a general underdeveloped expression, opposite the series expansion to be derived from it. b) In the first method 1), as well as the rational one 2), the real coefficient sought is is fabricated ready-made by means of the binomial theorem; it is found at once in the second term of the series expansion, the term which therefore is necessarily combined with h . All the rest of the differential process then, whether in 1) or in 2), is a luxury. We therefore throw the needless ballast overboard. From the binomial expansion we know once and for all that the first real coefficient is the factor of h , the second that of h , and so on. The real differential coefficients are nothing other than those of the binomially developed series of derived functions of the original function in x (and the introduction of this category of derived function one of the most important). As for the separate differential forms, we know that x is transformed into dx, y into dy, and that the symbolic figure of dy/dx represents the first derivative, the symbolic figure d y/dx represents the second derivative, the coefficient of (1/2) h , etc. We may thus allow the symmetry of half of our purely algebraically obtained result to appear at the same time in these its differential equivalent quantities (Differential quivalenten) - a matter of nomenclature alone, all that remains from differential calculus proper. The whole problem is then resolved into finding (algebraic) methods of developing all kinds of functions of x + h in integral ascending powers of h, which in many cases cannot be effected without great prolixity of operation .68 Until this point there is nothing in Lagrange which could not be a direct result of d Alembert s method (since this includes also the entire development of the mystics, only corrected). c) While the development, therefore, of y1 or f(x + h) = etc. steps into the place of the differential calculus up to now [and thereby, in fact, clarifies the mystery of the methods proceeding form y + dy or y + y., x + dx or x + x. , Here Lagrange takes as his immediate starting point for the algebraicisation of the differential calculus the theorem of Taylor outlived by Newton and the Newtonians69 which in fact is the most general, comprehensive theorem and at the same time operational formula of differential calculus, namely the series expansion, expressed in symbolic differential coefficients, of y1 or f(x + h), viz: = y ( or f(x)) + (dy/dx) h + (d y/dx ) h / + (d y/dx ) h / + (d y/dx ) h / + etc. d) Investigation of Taylor s and MacLaurin s theorem to be added here.70 e) Lagrange s algebraic expansion of f(x + h) into a equivalent series, which Taylor s dy/dx etc. replaces, and it may only still be the symbolic differential expression of the algebraically derived functions of x. (This is to be continued from here on.71)
Marx's Mathematical Manuscripts 1881