c) Continuation of [p.] 25* We have x1 - x = x from the beginning for the expression of the difference x1 - x; the difference exists here only in its form as a difference (as, if y is dependent on x, y1 - y is written for the most part). Since we set x1 - x = x, we already give the difference an expression different from itself. We express, if only in indeterminate form, the value of this difference as a quantity distinct from the difference itself. For example, 4 - 2 is the pure expression of the difference between 4 and 2; but 4 - 2 = 2 is the difference expressed in 2 (on the right-hand side): a) in positive form, so no longer as the difference; b) the subtraction is completed, the difference is calculated, and 4 - 2 = 2 gives me 4 = 2 + 2. The second 2 appears here in the positive form if the increment of the original 2. Therefore in a form directly opposite to the difference form (einer der Differenzform entgegengesetzten Form). Just as a - b = c, a = b + c, where c appears as the increment of b, so in x1 - x = x, x1 = x + x, where x enters immediately as the increment of x. The simple original setting x1 - x = x = anything therefore puts in place of the difference form another form, indeed that of a sum, x1 = x + x, and at the same time simply expresses the difference x1 - x as the equivalent of the value of this difference, the quantity x. It s just the same in x1 - x = x, x1 - x = x. We have the difference form again here on the left-hand side, but this time as the difference between the increased x1 and its own increment, standing independent next to it. The difference between it and the increment of x( = x) is a difference which now already expresses a defined, if also indeterminate, value of x. If however one leaves the mystical differential calculus, where x1 - x enters immediately as x1 - x = dx, and one first of all*2 corrects dx to x, then one begins from x1 - x = x ; thus from x1 = x + x; but this in turn may then be turned round to x + x = x1 , so that the increase of x again attains the undefined form x1, and as such enters directly into the calculus. This is the starting point of our applied algebraic method. d) From this simple distinction of form there immediately results a fundamental difference in the treatment of the calculus which we demonstrate it detail (see the enclosed loose sheets)72 in the analysis of d Alembert s method. Here we have only to remark in general: 1) If the difference x1 - x (and thus y1 - y) enters immediately as its opposite, as the sum x1 = x + x with its value therefore immediately in the positive form of the increment x, then, if x is replaced by x + x everywhere in the original function in x, a binomial of definite degree is developed and the development of x1 is resolved into an application of the binomial theorem. The binomial theorem is nothing but the general expression which results from a binomial of the first degree multiplied by itself m times. Multiplication therefore becomes the method of development of x1 [or] (x + x) if from the beginning we interpret the difference as its opposite, as a sum. 2) Since in the general form x1 = x + x the difference x1 - x, in its positive form x, in the form, that is of the increment, is the last or second term of the expression, thus x becomes the first and x the second term of the original function in x when this is presented as a function in x + x. We know from the binomial theorem, however, that the second term only appears next to the first term as a factor raised to increasing power, as a multiplier, so that the factor of the first expression in x (which is determined by the degree of the binomial) is ( x)0 = 1, the multiplier of the second term is ( x)1, that of the third is ( x) , etc. The difference, in the positive form of the increment, therefore only comes in as a multiplier, and then for the first time, really (since( x)0 = 1), as the multiplier of the second term of the expanded binomial (x + x)m. 3) If on the other hand we consider the development of the function in x itself, the binomial theorem then gives us for this first term, here x, the series of its derived functions. For example, if we have (x + h)4, where h is the known quantity in the binomial and x the unknown, we then have x4 + 4x h + etc. f(x + x) = (x + x)4 = x4 + 4x x + etc. 4) Furthermore: the second term of the binomial expansion, 4x h, provides us immediately ready-made (fix und fertig) with the first derived function of x4, namely 4x . Thus this derivation has been obtained by the expansion of f(x + x) = (x + x)4 ; It is thus the binomial expansion of f(x + x), or y1, which f(x) has become by its increase, which gives us the first derivative, the coefficient of h (in the binomial series); and indeed right at the beginning of the binomial expansion, in its second term. The derivative is thus in no way obtained by differentiation but instead simply by the expansion of f(x + h) or y1 into a defined expression obtained by simple multiplication. The crucial point (Angelpunkt) of this method is thus the development from the undefined expression y1 or f(x + h) to defined binomial form, but using not at all the development of x1 - x and therefore as well of y1 - y or f(x + h) - f(x) as differences. 5) The only difference equation which comes out in this method is the one which we obtain immediately: f(x + x) = (x + x)4 = x4 + 4x x + 6x x + 4x x + x4, x4 + 4x x + 6x x + 4x x + x4 , 4x x + 6x x + 4x x + x4 , y or f(x) = x4 . So that Newton also writes immediately: dy, to him y. = 4x x. + etc. 6) The entire remaining development now consists of the fact that we have to liberate the ready-made derivative 4x from its factor x and from its neighbouring terms, to prise it loose from its surroundings. So this is no method of development, but rather a method of separation. e) The differentiation of f(x) (as [a] general expression) Let us note first of all (d abord) that the concept of the derived function , for the successive real equivalents of the symbolic differential coefficients, which was completely unknown to the original discoverers of differential calculus and their first disciples, was in fact first introduced by Lagrange. To the former the dependent variable, y for example, appears only as a function of x, corresponding completely to the original algebraic meaning of function, first applied to the so-called indeterminate equations where there are more unknowns than equations, where therefore y, for example, takes on different values as different values are assumed for x. With Lagrange, however, the original function is the defined expression of x which is to be differentiated; so if y or f(x) = x4, the x4 is the original function, 4x is the first derivative, etc. In order to lessen the confusion, then the dependent y or f(x) is to be called the function of x in contrast to the original function in the Lagrangian sense, the original function in x, corresponding to the derived functions in x. In the algebraic method, where we first develop f1, the preliminary derivative or [the ratio of] finite differences, and where we first develop from it the definitive derivative, f , we know from the very beginning: f(x) = y, so that a) f(x) = y, and therefore turned round y = f(x). What is developed next is just f(x), the value of the finite difference of f(x). We find f1(x) = y/ x, so that y/ x = f1(x) . y = f1(x) x , f(x)= f1(x) x . df(x) = f (x)dx , In the usual method dy or df(x) = f (x)dx
Marx's Mathematical Manuscripts 1881