Marx's Mathematical Manuscripts 1881

Written: August, 1881;

Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;

First published: in Russian translation, in *Pod znamenem marksizma*, 1933.

*Newton*, born 1642, †1729. *‘Philosophiae naturalis principia mathematica’*, pub. 1687.

L. I. *Lemma* XI, *Schol*. Lib. II.

L. II. *Lemma* II, *from Proposition* VII.^{51}

*‘Analysis per quantitatum series, fluxiones etc.’*, composed 1665, publ. 1771.^{52}

2) *Leibnitz*.

3) *Taylor* (J. Brook), born 1685, †1731, published 1715-17: *‘Methodus incrementorum etc.’*

4) *MacLaurin (Colin)*, born 1698, †1746.

5) *John Landen* .

6) *D’Alembert*, born 1717, †1783. *‘Traité des fluides’*, 1744.^{53}

7) *Euler* (Léonard), [born] 1707, †1783. *‘Introduction à l’analyse de l’infini’*, Lausanne, 1748. *‘Institutions du calcul différentiel’*, 1755 (p.I, c.III).^{54}

8) *Lagrange*, born 1736. *‘Théorie des fonctions analytiques’* (1797 and 1813) *(see Introduction)*.

9) *Poisson* (Denis, Siméon), born 1781, †1840.

10) *Laplace* (P. Simon, marquis de), born 1749, †1827.

11) *Moigno’s, ‘Leçons de Calcul Différentiel et de calcul intégral’*.^{55}

^{50}
The bibliography which Marx presents in this list is accompanied in many cases by indications of the exact passages in the sources cited where the fundamental concepts and methods of differential calculus are discussed. These were not indicated in the textbooks at Marx’s disposal. There is therefore every reason to suppose that Marx chose these passages by consulting the corresponding works (in the library of the British Museum, apparently). The fact that Marx especially distinguished (placed in a panel) the name John Landen is obviously related to the fact that he had decided to acquaint himself particularly well with J.Landen’s *Residual Analysis*. For more details on this see Appendix IV. The sources for Marx’s notation of the dates of birth and death on the list are unknown. It is only clear that the sources did not have the date of death of Lagrange.

^{51}
In the scholium (lesson) to Lemma XI of the first book of *Principia Mathematica* and in Lemma II of the second book, Newton explains the fundamental concepts of differential calculus which correspond to our concepts ‘derivative’ and ‘differential’. For more details on these lemmas of Newton see Appendix II, pp.156-159.

^{52}
See Marx’s outlines of these works (with his critical commentaries) on pp.272-280 [Yanovskaya, 1968].

^{53}
D’Alembert’s *Traité des fluides* does not contain any material on the fundamentals of differential calculus. D’Alembert’s views on the fundamental concepts of differential calculus were presented in his articles in the *Encyclopédie* and in his *Opuscules mathématiques*. It is not known what attracted Marx’s attention to the *Traité des fluides* of d’Alembert.

^{54}
The third chapter of part one of L[eonhard] Euler’s *Institutiones calculi differentialis* deals with the question ‘Of Infinity and the Infinitely Small’. For more details see Appendix III. pp160-164.

^{55}
This book was assembled by the Abbé Moigno ‘following the methods and works of Cauchy, published and unpublished’. The first volume of Moigno’s *Lectures* appeared in 1840, the second in 1844.