If we now consider the differential of y in its general form, dy = f (x)dx, then we already have before us a purely symbolic operational equation, even in the case where f (x) from the very beginning is a constant, as in dy = d(ax) = adx. This child of 0/0 or dy/dx = f (x) looks suspiciously like its mother. For in dy/dx = 0/0 numerator and denominator are inseparably bound together; in dy = f (x)dx they are obviously separated, so that one is forced to the conclusion: dy = f (x)dx is only a masked expression for 0 = f (x) 0, thus 0 = 0 with which nothing s to be done ( nichts zu wolle ). Looking more closely, analysts in our century, such as, for example, the Frenchman Boucharlat, smell a rat here too. He says:* In dy/dx = 3x , for example, 0/0 alias dy/dx, or even more its value 3x , is the differential coefficient of the function y. Since dy/dx is thus the symbol which represents the limit 3x , dx must always stand under dy but, in order to facilitate algebraic operation we treat dy/dx as an ordinary fraction and dy/dx = 3x as an ordinary equation, and thus by removing the denominator dx from the equation obtain the result dy = 3x dx, which expression is called the differential of y. 43 In order to facilitate algebraic operation , we thus introduce a false formula. In fact the thing (Sache) doesn t behave that way. In 0/0 (usually written (0/0)), the ratio of the minimal expression (Minimalausdrucks) of y1 - y, or of f(x1) - f(x), or of the increment of f(x), to the minimal expression of x1 - x, or to the increment of the independent variable quantity x, possesses a form in which the numerator is inseparable from the denominator. But why? In order to retain 0/0 as the ratio of vanished differences. As soon, however, as x1 - x = 0 obtains in dx a form which manifests it as the vanished difference of x, and thus y1 - y = 0 appears as dy as well, the separation of numerator and denominator becomes a completely permissible operation. Where dx now stands its relationship with dy remains undisturbed by this change of position. dy = df(x), and thus = f (x)dx, is only another expression for dy/dx [ = f (x) ], which must lead to the conclusion that f (x) is obtained independently. How useful this formula dy = df(x) immediately becomes as an operational formula (Operationsformel), however, is shown, for example, by: y = ax , d(y ) = d(ax) , 2ydy = adx ; y = ax, [thus] = 2ax/a = 2x ; However, if dy = df(x) serves as the first point of departure (Ausgangspunkt), which only later is developed into dy/dx itself, then, for this differential of dy to have any meaning at all, these differentials dy, dx must be assumed to be symbols with a defined meaning. Had such assumptions not originated from mathematical metaphysics but instead been derived quite directly from a function of the first degree, such as y = ax, then, as seen earlier, this leads to (y1 - y)/(x1 - x) = a, which is transformed to dy/dx = a. From here as well, however, nothing certain is to be got a priori. For since y/ x is just as much = a as dy/dx = a, and the, x, y, although finite differences or increments, are yet finite differences or increments of unlimited capacity to contract (Kontraktionsf higkeit), one then may just as well represent dx, dy as infinitely small quantities capable of arbitrarily approaching 0, as if they originate from actually setting the equality x1 - x = 0, and thus as well y1 - y = 0. The result remains identical on the right-hand side in both cases, because there in itself there is no x1 at all to set = x, and thus as well no x1 - x = 0. This substitution = 0 on the other side consequently appears just as arbitrary an hypothesis as the assumptions that dx, dy are arbitrarily small quantities. Under (sub) IV) I will briefly indicate the historical development through the example of d(uz), but yet prior to that will give an example under (sub) III)44 which is treated the first time on the ground of symbolic calculus, with a ready-made operational formula (fertigen Operationsformel), and is demonstrated a second time algebraically. Enough (soviel) has been shown under (sub) II), so that the latter method alone, by means of its application to so elementary a function as the product of two variables, using its own results, necessarily leads to starting points (die Ausgangspunkte) which are the opposite pole as far as operating method goes. To (ad) IV. Finally (following Lagrange) it is to be noted that the limit or the limit value, which is already occasionally found in Newton for the differential coefficients and which he still derives from purely geometric considerations (Vorstellungen), still to this day always plays a predominant role, whether the symbolic expressions appear (figurieren) as the limit of f (x) or conversely f (x) appears as the limit of the symbol or the two appear together as limits. This category, which Lacroix in particular analytically broadened, only becomes important as a substitute for the category minimal expression , whether it is of the derivative as opposed to the preliminary derivative , or of the ratio (y1 - y)/(x1 - x), when the application of calculus to curves is treated. It is more representable (vorstellbarer) geometrically and is already found therefore among the old geometricians. Some contemporaries (Modernen) still hide behind the statement that the differentials and differential coefficients merely express very approximate values.45
Marx's Mathematical Manuscripts 1881