On a separate sheet attached to his draft sketch of the course of historical development of mathematical calculus, Marx referred to the Scholium of Lemma XI of Book One and the Lemma II of Book Two of Newton s Principia, devoted to two fundamental concepts used by Newton throughout his mathematical analysis, the concept of limit and moment . In the commentary (scholium) to Lemma XI of the first book to Principia mathematica de philosophiae naturalis Newton attempts to explain the concept of ultimate (limiting) ratio and ultimate sum by means of a not very transparent comparison: a metaphysical, not mathematical assumption, Marx characterised it. Indeed, Newton writes: In present-day mathematics the velocity of a body at the given moment t0 is defined with the help of the mathematical concept of limit, and the use by science of such a definition may lead to a variety of considerations, including those of an ontological character. However, the scientific definition of the velocity of a body at a given moment by means of a certain limit of the ratio of vanishing quantities can serve neither as a demonstration of the existence of such a limit nor, a fortiori as a justification for the definition of this limit as the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish, that is, as some sort of ratio of zeroes, the value of which is somehow compared to the speed which a body must have at the very moment when it reaches a place where its movement end. Clearly, however, from such a definition it is impossible to extract by mathematical calculations any corresponding limit, and we are essentially in a logical circle: velocity at the moment t0 is factually described as a certain limit, the limit, itself, however, is then defined by means of the velocity t the moment t0, the existence of which in this case now really seems to be some sort of metaphysical, not mathematical, assumption .* Lemma II of the second book of Principia mathematica contains the following explanation of the concept of moment (or infinitely small): It is natural that this explanation - in which Newton once again employs a metaphysical, not mathematical assumption , this time with respect to the existence of differentials ( moments ) - should have interested Marx first of all. But his lemma might also have attracted his attention insofar as in it Newton attempts to show the formula for the differentiation of the product of two functions without resorting to the suppression of the infinitesimals of higher order. This (unsuccessful) attempt proceeds in the following way: Let A - (1/2) a be the value of the function f(t) at the point t0, B - (1/2) b be the value of the function g(t) at the same point t0, and a and b increments of the respective functions f and g on the interval [t0, t1]. (Lower we denote these f and g respectively.) Then the increment of the product f(t) g(t) on the segment [t0, t1] is: (A + (1/2) a) (B + (1/2) b) - (A - (1/2) a) (B - (1/2) b) , (f(t0) + (1/2) f) g + (g(t0) + (1/2) g) f , As is apparent from the first drafts of the piece on the differential (see, for instance p.76), Marx at first wanted to elucidate the historical path of the development of differential calculus by the use of the example of the history of the theorem of derivative. Therefore it is not surprising that Lemma II should have drawn Marx s attention in this connection. Since the textbooks from which Marx made extracts do not specifically refer to Lemma XI of Book One or Lemma II of Book Two of the Principia, thee is every reason to believe that Marx selected them, having already immediately rejected Newton s work. Since the definition of the limit of the ratio of vanishing quantities by means of the velocity of a body at a given moment t0 contains no means for the calculation of this limit, Newton actually employs for the performance of such calculation, rather than this definition, certain hypothetical properties of limits sufficient to reduce the calculation of the limits of ratios of vanishing quantities to the calculation of the limits themselves, the numerical value of which is supposed to be completely and rigorously defined. Newton states these hypothetical properties first of all in Lemma I of the first section of Book One of Principia: The method of first and last ratios of quantities, by the help of which we demonstrate the propositions that follow. In his notes on the history of differential calculus Marx refers to this lemma together with the scholium to Lemma XI (see pp.75 and 76). Lemma I states: Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of first time approach nearer to each other than by any given difference, become ultimately equal. (Newton s Principia revised by Florion Cajori, Univ of Calif. Press, 1934, p.29) However, in the demonstration of this limit the existence of a limit as actually reached at the end of the period of time in question is implicitly assumed. Actually, the demonstration is composed of a denial that the value of the quantities obtained t the end of this time can be distinguished from each other. Thus, limit is always understood by Newton in an actual sense and therefore hardly surpasses - in mathematical precision and validity - Leibnitz s actually infinitely small differentials and their corresponding moments, which, as is well known, Newton used in practice. ____________
Marx's Mathematical Manuscripts 1881