Mathematical manuscripts

Mathematical Manuscripts of KARL MARX NEW PARK PUBLICATIONS Published by New Park Publications Ltd., 2lb Old Town, Clapham, London SW4 OJT Fint published in German and Russian as Karl Marx, Machnrwlichalcie Rl!Mpsii Nauka Presl;, Moscow, 1968 Copyright Q Translation New Park Publications Ltd !983 Set up, Printed and Bound by Trade Union Labour Distributed in the United States by: Labor Publications Inc., GPO 1876 NY New Yock 10001 Printed in Great Britain by Astmoor Utbo Ltd (TU), 21·22 Arkwright Road, Astmoor, Runcorn, Cheshire I~Bl~ 86151 000 Contents Publisher's Note V Preface to the 1968 Russian edition Letter from Engels to Marx, August 10, 1881 Letter from Engels to Mar:x, November 21, 1882 Letter from Marx to Engds, November 22, 1882 Two Manuscripts on Differential Calculus 'On the Coo~ept of the Derived Function' VII XXVII XXIV XXX I 3 ll On the Differential IS I 15 ll M Drafts and Supplements on the work 'On the Differential' 35 First Draft 37 Second Draft 54 I 54 ll 59 'Third Draft' 65 Some Supplements 69 On the History of Differential Calculus 73 A Page included in Notebook 'B (Continuation of A) 11' 75 I First Drafn; 76 II The Historical Path of Development 91 Ill Continuation of ExtracnJ 101 Taylor's Theorem, MacLaur'in's Theorem and Lagrange's Theory of Derived Functions 107 I From the Manuscript 'Taylor's Theorem, MacLaurin's Themem, and Lagrange's Theory of Derived Functions 109 I 1~ 11 113 Ill Lagrange's Theory of Functions 113 From the Unfinished Manuscript 'Taylor's Theorem' Appendix to the Manuscript 'On the History of Differential Calculus' Analysis of d'Alembert's Method 116 121 On the Ambiguity of the terms 'Limit' and 'Limit Value' 123 Comparison of d'Alembert's Merhod to the Algebraic Method 127 Analysis of d'Alembert's Method by means of yet another example 131 Appendices by the editors of the 1968 Russian edition 141 I Concerning the Concept of 'Limit' in the Sources consulted by M.arx 143 Il On the Lemmas of Newton cited by Marx 156 Ill On the Calculus of Zeroes by Leonard Euler 160 IV John Landen's 'Residual Analysis' 165 V The Principles of Differential Calculus according to Boucharlat 173 VI Taylor's and MacLaurin's Theorems and Lagrange's theory of Analytic Functions in the source-books used by Marx 182 Notes Notes to the 1968 Russian edition Additional material 191 193 215 E Kol'man Karl Marx and Mathematics: on the 'Mathematical Manuscripts' of Marx 217 Hegel and Mathematics by Ernst Kol'man and Sonye Yanovskaya 235 Hegel, Marx and Calculus by C Smith 256 Index 271 PUBLISHERS' NOTE The major part of this volume has been translated from Karl Marx, Mathmlaticheskie Rukoprii, edited by Professor S.A Yanovsk.aya, Moscow 1968 (referred to in this volume as Yanovskaya, 1968) This contained the ftrst publication of Marx's mathematical writings in their original form, alongside Russian translation (Russian translation of parts of these manuscripts had appeared in 1933.) We have included the first English translation of Part I of the Russian edition, comprising the more or less finished manuscripts left by Marx on the differential calculus, and earlier drafts of these We have not translated Part 11 of the 1968 volume, which consisted of extracts from and comments on the mathematical books which Marx had studied Professor Yanovskaya, who had worked on these manuscripts since 1930, died just before the book appeared We include a translation of her preface, together with six Appendices, and Notes to Part I In addition, we include the following: a) extracts from two letters from Engels to Marx and one from Marx to Engels, discussing these writings; b) a review of Yanovskaya, 1968, translated from the Russian, by rhe Soviet mathematician E Kol'man, who died in Sweden in 1979, and who had also been associated with these manuscripts since their first transcription; c) an article by Yanovskaya and Kol'man on 'Hegel and Mathematics', which appeared in 1931 in the journal Pod znamenem MGrbi.rma This has been translated from the version which appeared ln the German magazine Unter dem Banner des Marxismus; d) an essay on 'Hegel, Marx and the Calculus' written for this volume by Cyril Smith The material from Yanovskaya 1%8 has been translated by C Aron1on and M Meo, who are also responsible for translating the review by E Kol'man The letters between Man and Engels, and the article by Yanov•kaya and Kol'man, are translated by R A Archer V S.A Yanovskaya PREFACE TO THE 1968 RUSSIAN EDITION Engels, in his introduction to the second edition of Anti-Diihring, revealed that among the manuscripts which he inherited from M.arx were some of mathematical content, to which Engels anached great imponance and intended to publish later Photocopies of these manuscripts (nearly I ,000 sheets) are kept in the archives of the Marx· Lenin Institute of the Central Committee of the Communist Party of the Soviet Union In 1933, fifty years after the death of M.arx, parts of these manuscripts, including Marx's reflections on the essentials of the differential calculus, which he had summarised for Engels in 1881 in two manuscripts accompanied by preparatory material, were published in Russian translation, the first in the journal Under the Banner of Marxism (1933, no I, pp.IS-73) and the second in the collection Marxism and Science (1933, pp.S-61) However, even these parts of the mathematical manuscripts have not been published in the original Jansuages until now In the present edition all of the mathematical manuscripts of Marx having a more or less finished character or containing his own observations on the concepts of the calculus or other mathematical ques· dons, are published in full Marx's mathematical manuscripts are of several varieties; some of them represent his own work in the differential calculus, its narure IDd history, while others contain outlines and annotations of books which M.arx used This volume is divided, accordingly, into two puts Marx's original works appear in the first part, while in the 1econd are found full expository outlines and passages of mathematical content.* Both Marx's own writings and his observations located in the surveys are published in the original language tad in Russian translation Thl1 volume contains a t:canslation of the firs! part only VII VIII MATHEMATICAL MANUSCRIPTS AJthough Marx's own work, naturally, is separated from the outlines and long passages quoting the works of others, a full understanding of Marx's thought requires frequent acquaintance with his surveys of the literature Only from the entire book, therefore, can a true presentation of the contents of Marx's mathematical writings be made complete Marx developed his interest in mathematics in connection with his work on Capital In his letter to Engels dated January 11, 1858, Marx writes: 'I am so damnedly held up by mistakes in calculation in the working out of the economic principles that out of despair I intend to master algebra promptly Arithmetic remains foreign to me But I am again shooting my way rapidly along the algebraic route.' (K.Marx to F.Engels, Work.s, Vol.29, Berlin, 1963, p.256.) Traces of Marx's first studies in mathematics are scattered in passages in his first notebooks on political economy Some algebraic expositions had already appeared in notebooks, principally those dated 1846 It does not follow, however, that they could not have been done on loose notebook sheets at a much later time Some sketches of elementary geometry and several algebraic expositions on series and logarithms can be found in notebooks containing preparatory material for Critique of PoliliaJl Ecre', 7th edition, Paris (1863), 171 LAGRANGE,J.L., 'Theoriedesfonctionsanalytiques' ,Paris, 1813, Oeuvres Lagrange, Voi.IX, Paris (1881), 75, 76, 99, 154-155, 172 LANDEN, J., 'A discourse of the residual analysis', London (1758), 172 LANDEN, J., 'The residual analysis', London (1764), 113,155, 165-172 LHUILIER, S., 'Principioru.m calculi differentialis et integralis exposito dementaris', Tlibingen (1795), 182 273 274 MATHEMATICAL MANUSCRIPTS MACLAURIN, C., 'A treatise of algebra in parts', 1st ed (1748), 6th ed., London (1796), 186 MOIGNO, F., '~os de calcul differentiel et de calcul integral, rblig~ d'apres les methodes et les ouvrages publi~ ou in~ts de M L.A Caucby', Vol., Paris (1840 and 1844), 75 NEWTON, 1., 'Philosophiae naturalis principia mathematica', London (1687), 75, 76, 15~159 NEWTON, 1., 'Arithmetica universalis, sive de compositione et resolutione ari tbmetica liber', Cambridge (1707), 112 NEWTON, I., 'Analysis per quantitarum series flu.xiones et differ en tias, cum enumeratione linearum tertii ordinis' ( 1711), 75, 76 TAYLOR, B., 'Methodus incrementorum directa et inversa', London (1715), 75, 182 INDEX OF SUBJECTS -starting point of, 20, 26,27, 50, 69, 94, 97, 102, 113, 118 - wk of, 22, 59 -three stages of, 20, 54, 91-100 Absolute minimum expression/Quantity, (see Minimum expression) Algebra, 29, 98, 109, ll5, 117 Algebraic method, 25-32, 54, 64}-64, 70, 84-90, 97-100, 102, 118 -compared with d'Alembert's method, 127-131 - foundation for c:alc:ulus, SO, H, 18 -and' mystical' differential method, 21, 24, 54, 58, 68, 69, 128 Alias, 24, 65 Approach to limit, 7, 29, 67, 124 Appronmate, 68, 83 Change of value ofvarub1e, 7, 9, 10, 78, 86, 88, 101, 103, 128 -variable (see Variable, change of) Childishness, 126 Chimera, S Circular functioos, I 09 Constant function, 110, 112 Coup detat, 131 Ballast, 98 Binomial, 86, 87, 91, 93, 102, 128 -theorem,81,92,97,98,102-104, 109, 112, 113, 116, 128, 129, 134 - theorem of undefined degree, 99, 113, 117 Boucha.rlat, Jean-Louis (1775-1848), 24, 65 Calct1lus, content of, S4 ' founden of, 50, 54, 78, 105, 118 -history of, 54, 63, 64, 67, 73-90, 91-106 -operating on its own ground, 20, 27, 38, 41, so, 78, 113 -opposite methods of, 21, 54, 68 -special manneJ: of calculation of, 20, 26, 27, 84, 113 D'Alembert, jean Le Rood (1717-1783), 75, 77, 94-99, 102, 127-131, 132-139 Deception, 117 Delta (see Increment) Dependent and independent variables (see Variables) Derivative, 7, 8, 88, 92, 110 -assumed known, 49, 50, 56, Ill, ll8 -ready-made, 88, 92, 97, 98, 104, 106, 129, 134, 137 -as second tenniDpowerseries, 87, 92, 96, 98, 103, 104, 129, 132, 134, 136 Derived function, 8-14, 80, 84, 98, 105, 109, 115, 116, ll9, 139 Devdopmem, 83, 84, 88, 91, 96, 102, 103, 104, llO, 116, 128, 137, 138 [}Uferenc:e,86,88,95,10l,l02,104,118 - fmite, 4, ll, 29, 67, 78, 90, 106, 128 277 278 Diff~ential, 15-33,50,51,52,511, 65, 113, 91, 98, 106 -arbitrary introduction of, 67, 79 -coefficient, 24, 2S, 52, Sl! - equatioDII, 15, 19, 22, 25, 27, 40, 45, 79 - inde~ndent existence of, 79, 83 -method of calculus, 20, 24, 37,91-94, 97, 128 -operations, 3, 109, 112 -presumed in derivation, 67, 91 - prO