Karl Marx mathematical manuscripts : together with a special supplement 9788186210000, 8186210008

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KARLMARX

MATHEMATICAL MANUSCRIPTS . TOGETHER WITH A SPECIAL SUPPLEMENT

VISWAKOS PARISAD CALCUTTA

PUBLISHED IN INDIA BY VISWAKOS PARISAD. 73A, AMI-IERST ROW. CALCUlTA - 700009. Cl VISWAKOS PARISAD 1994. COMPOSED AT NEO COMPACT SYSTEMS PVT. LTD., 161.B.K.PI\UL AVENUE, CALCUlTA - 700 005. DATA ENTRY BY MR1TYUNJOY DAS. PAGE MAKE UP AND CORREcnON BY B1PASA ClI AUD HUR t AND PRADIP CHATERlEE. PROOF R"EADERS: SABYASACHJ CHAKRABARTI AND Dr. S. KUMAR. PR INTED AT SONA PRINTERS. 6711n, NIMTALLA Gl-IAT STREET, . CALCUlTA - 700 006.

CA TALOG I NG IN PUBLIC ATIO N DA T A : MARX, KARL. MATHEMATfCAL MANUSCR fPTS. TRANSLATfON OF, K.MARKS, MATEMATfCHESKIE RUKOPISI ("NAU KA", M .. 1968) EDITED BY SOFYA ALEKSAN DROVNA YANOVSKAYA, TOGETHER WfTH A SPECIAL SUPPLEMENT: M ARX AND MATHEMATICS. T RANS LATOR OF MARX'S M ATHEMATICA L MANUSCRIPTS (M., 1968) AND EDITOR OF TH E SPECIAL SUPPLEMENT, PRADIP BAKS!. I. MARXISM . 2. MATHEMA TICS ( DfFFERENTIAL CALCULUS, ALGEBRA). 3. MATHEMATfCS, HISTORY OF. ISBN 81-86210-00-8 PRICE: Rs. 1000.00 (U S $ 80.00)

4. MATHEMATICS, PH fLOSOPHY OF.

INTRODUCTION

The massive propaganda blitzkrieg denigratin g sociali sm nnd the voluminolls materinl brou ght out by imperialism and reactionaries notwithstanding, nobody so far has sllcceeded in cha llenging the basic theory that Marx placed before the world. The profundity of Marx's works lies in his sc ientific analysis of the evolution of society and the fo resight of the path that it will charier. It is based on such a scientific analys is that Marx concl uded that society will reach a s tage where the state will wither away. T hat capitalism has failed to prove its superiority or eternity, as postulated by its protagonists, even in the wake o f the so ca lled "demise of sociali s m" - the continuing recession and intens ified exploitati on, vindicate thi s analysis. Marx, unfortunately. could not co mplete the work that he set before himself. After the second volume of Capita l had come out in print. it was len to Frederick Engels to go thro ugh the ma nuscri pts facil itating it to be made available to the whole of mankind. In the preface to Volume lIt of Capital. Engels writes: "As regards the first part, the main manuscript was serviceable o nly with substantial limitat ions. The entire mathematical calcu lation of the re lation between the r' •.:

,TS

where the co ns tant a fignrcs as the Iimit 5 0f the ratio of [inilc differences of the variables.

Since a is' a constant, neither it, nor, consequen tly, the right hand side of the equation, reduced 10 it, mCl Y undergo any change. In that case, the differential process runs its course in the left hand side: y, - Y 6." - - or xl-x

=

I1x'

and this is a cha'racleristic of slIch simple functions as ax.

Let. in the denominator of this ratio, r the variahle] x\ decrease, approaching x; the limit of its decrease is att.:lincd, as soon as x\ turns il1lo x. With th is the diITcrcncc x\ -xwill become equa l to x\ -x ... 0, and hence also YI - Y "" Y - y ""' O.

W~ thus obta in , ~ "" a. Since in the expression

*

every trace of its origin and its meaning has been wiped out, we

change it into ~, where the finite differences x, - x or 6 x and y, - y o r 6yappearsymbolized, . ~ . t!J. Th us, as removed or vanished differences, so that " , 6x turns mto dx'

1x -

(I,

*'

The closely hcld consolation of some ralionnlizing m;lthcma ticians, that the quantities dyand

dx are in fact only infinitely small (lnd Ith:lt their ratio] merely approaches as will be shown tangibly, further, sub It) , 11 requ ires

to

be

mentioned

is a chimera, fnrther,

115

a

'charac'teristic of the instance under consideration, that as EL = a, likewise El.. = a i:e., the , '. 6x ~ limit [of the ratio] of finite differences is at the same time nlso Ihe limit lof the mtioJ of the differentials,

2) The following may serve as the second example of the same case:

y-x

y, - y or ~"" 1 6x

x, -x

o

o

or

d" dx""1.

11 When y - f(x}, "'::'.kJi'·t:~in at the rig~t _hand side of the equation the function x is situated in its expallded'l1lgebfafc expression,6,. ihen, we c.:a ll it the expression (or the initia! function of x ; its first modification, obtai ned through the postulation of a ditTerence, [is calledJ the

ON 'IlIE CONCEPT OF '1'1 rE DErHVI]) FUNCT!ON

21

preliminary "derivative" of the fll nc tio n x ; llnd the rim!! rorm wh ic h i t tn kcs as a result o f the differential p r ocess I'is called I the "derived" jU llctio" o f x 7 1) y_ax 3 +bx 2 +cx_e. I ( x increases to Xl' then

Yt -

(u:/ + bx t 2 + cX

Y, - Y - a (XI) -

(l

j

-

e,

x 3) + b (X 12 - x 2) + C (Xl -x) '"

-

(XI -x) (X12 +XIX+X2) + b (Xl -x) (XI +x) +c (XI - x).

He nce,

Y

- - - or ~ = a (X, 2 +x, +X2) + b(x, +x) + c. !:t.x Xl -x )'1 -

The prelimiIlGr)' "deriwlliv(''' .

a(.\/+x\x+x2)+h(xI+x)+c

is herc Ihe limil of the ralio of finite differences , i.e., however sm '(x).

Let us multiply the equations 3) and 4), then : dydu dy l l , , 5) Tu' Ox or Ox - f (11). q> (x), and

this is what was req uired to be demonstrated ,

N.lII The end of this second installment will follow after I look over John L1nden in the Museum32.

,.

'

...

,

.

THE DRAFTS OF AND ADDITIONS TO "ON THE DIFFERENTIAL,,33

*FIRST DRAFT" . As soon as wc set ahout diffcrcntialing [(II, z) l = uz 1. whe re the va riables It a nd z arc both. functions of x ,wc gct-as dislinctfrom thccn rlicr instances,where th ere was o nl y o nc dependent variable, namely y - the di fferent ia l express ion on both skies, namel y:

ill Ihe first jllstance : dv

du

dz

=·Z - +IIdx clx ill: ill the second, briefly:

dy .. Zdfl + udz . The Jalter has not yel aW.incd thal form, whic h is ob ta ined in the case wilh OIlC de pe ndent

va ri,lble ; for example, in dy _ lr/aX"' - 1dx. Si nce here !!l'..dd l it once g ives us [,(x) .. max", -I , x which. is free from dilTc rcn tia l s ymho ls. Such :t form has no place in dy .. zdu + udz. In the eq uatio ns with o ne dependent va riable wc have seen once and for a ll , how the functions de rived fro m Ilhe fun ctions in I x - as in the above men tioned ins llmce of x'" - x.

Further, the unfo lding of the diffe rent ia l expressio n, which in the rinal count gives us

dl(x) - J'(x)dx, is simpl y the differe ntial expressio n of the fini te difference unfo lded earlie r. In the usual method

dy

0'

dl(x) - J'(x)dx

is no t a t a ll expa nded, but . scc above , the full y rcad y-made!'(x) furnished by the bino mia l (x + tJ. x) or (x + dx) o r, is o nly disengaged from its multiplier and fr0':fl the accompanying terms

*THEOREMS OF TA YLOR AND MACLAURIN LAGRANGE'S THEORY OF ANALYTICAL FUNCTIONS

1. FROM THE MANUSCRIPT "TAYLOR 'S THEOREM, MACLAURIN'S THEOREM AND LAGRANGIAN THEORY OF ANALYTICAL FUNCTlONS,,73 I Discovery of the binom ial theorem by Newton ( in its applicatio n to tbe po ly no mial), also gave rise to a revol utio nary trans formation in the whole o f algeb ra - firs t of all, because it made the genera/theory of equalions poss ible. But th e binomial theorem is al so an imporklnl fo undati on of th e differential ca lculus and lhis is defi nitel y recognised by the ma thcma lici from t!lIess is ultimately embodied Fltlch aljons ill the price of Foreign b~lIs Illterp~(!/atioll of Foreign Exchanges So called eorreetives of Foreign Exchanges Ca lClllalioll of Bills of Exchange. CalculalUm of Bills Of Exchange jn general CalClllatioll of Parily {COil version illto hard currency! COl/versioll of the Bills of Exchange illlo other rates. Direct Indirect Calculation of Arhilratiolls. D irect In direct Rule of three. Complex rule of three Chain rule Partnership rule RlIle of M ix III re Calculation ofpercelltages Ca/cilia lion of il/teresls Calculatioll of d~rcollllt {rebate} Calculalion of terms

(1' 1'. 87-89)

(90) (90) (90-93) (93-99) (99- 104) (104-109) (l09) (109-110) ( Il O- 1l 2) (11 2-114) ( 1l4-118) (135-138) ( 118-119) (118- 121) ( 121- 123) (123-125) (1 25- 127) (127-13.1) (131-l32) ( 131-134)

From this list wc find that at first Mlloe delllt only with certain "IYpeS· of ~ rithmet ical problems, for [he solution of which, special rules have been proj)Qsed for each "type" . S.U.N.2400

11 is n big nole book. consisting of l25 sheets with Engels ' SUpct1iCriplion :

1869 1) Co mmcrci:tl C1lcu lations, Note book 11, End. pp. 2) Foster, Co mmerc ial Exchanges, 37·51.

1~36.

11&

OESCRII'110N 0 1' THE Ml\nmMl\llCI\L MI\NUSCRWI'S

3) Haus ner, Comp. Statistics, 1865. 4) Sadler, Ireland, 1829. The first 36 pages (ss. 3-38) He continuation of manuscripl2388. lIere, firsl of all the notes from chapter XI V come 10 an cnd and, noles arc taken from chapler XV of the book by Fcller and Oderman n §§ 413·426, pp. 382·400 -1I00U! the calculation of Bills of Exchange, calculations of values of shares and of olher governmcnt papers. Further on, MHX relurns 10 Ihe chaplers XI .XIII. §§ 317-380, pp. 246-318 of this very book - which he skipl>cd earlier - on [he gold and silver contents of the currencies of dif(ercrll counlries. This nole comes 10 an e nd with Ihe notes from chapters XV I - XVIII, §§ 428-471, pp. 402·481. These arc aoou t ClIlculations of weights ,l nd measures, estimll le of commodities and calculation of losses in cases of shipwreck.

MANUSCRIPTS OF THE 18705 THE MAN USC RIPTS ON TH E THEORY OF CONI C SECTI ONS Here the following manuscripts are in view : S.U.N. 2760, 2761

~nd

2762.

S.U.N.2760 It consists of 9 sheets (55.1-9) of notes, taken from: 1. lIymers, "A treatise on conic sections and the application of algebra 10 geometry·, 3rd cd., Cambridge, 1845. 111;5 book was found in Marx 's pcrsonallibrllry. Marx look notcs from rhe first 12 pages of it. These pages nre related 10 the introduction of coordinates; the problem of finding the distance between two poi nts. given their coordinates; the eq uation of the straight line and, the problems of: determining the equatio n of a s traight line , in te rms of Ihe segments cut of[from it by the axes of coordinates and, the equal io n of a straight line passing through one amI two points,tlleir coordinates being given. S.U.N.276 t 5.5 double page sheets of rough notes, on the theory of conic sections , from Sauri's book cited above, volume 2, pp. 2·27, in French and English. S.U.N.2762 4 double page sheets in Frenc.h . Fair notes on the theo ry o f conic sections from the same book: by Sauri. volume 2. pp. 2-2.7.

THE FIRST NOTE S ON THE DI FFERENTIAL CALCU LUS S.U.N.3704 4 sheets of photocopies; lhe beginning is not there, we have only rp. 3-6 in Mane's pagination. From the content il is clear,that to all appearance, this manuscript is the very firsl note taken from the initial paragraphS o f Boucharlat's text-hook: (" An Elementary Treatise on the Different ial and Integral Calculus" by J .-L l3oucharlal. Trans la ted from French By R. Blakelock:, B.A., Cathar;na Hall , Cambridge-London, 1828). i.e., (this note] is relaled 10 that lime, when Marx, having gal acquainted with the b~sics of diffe rentiftl calculus according 10 Sauri's course, turned 10 8 newe r Irelltise on this C/l leulus by Boue harlal. The preserved sheets of lhe manuscript contain notes from II 5· 18 this text book. The following beginning of page 3 of Marx '5 manuscript indicales thaI, in the missing pages 1-2 notes were lake n from the paragraphs preceeding the fifth onc. I1 reads :

He wou ld say (x+dxP - xl +3x2dx+3x(th-)2 + (dxp. Now, if we sub tract the gfven quantity x 3, there remains 3x 2tb: + 3x(dxF + (dxP ; the two lalter term s disa ppear as in f in ities of the second [and thi rd ] order [s1 and, wc ge t d (x 3) _ 3x2dx, which is the differe ntial of Xl, e.g. =d (x 3), and there is less to be said aga inst this; as in the other equa lion y = etc., x changes independently of y and the changes of y are onl y correlative to lhose of x.

/

DESCRII~nON

120

OF'111E MKl1lEMAl1CAL MANUSCRWI~

Here Marx co nsiders 1111 eX~lllple, which BOllclmrlnt investigated in § 3 (for llie fllillcxt of Ihis paragraph sce PV, 326·327, \)ot he diffe rentiates it according to Sauri, i.e., using the inf'inilcsimat method of Newton-Leibnit1.. Mllrx's objections 10 il are sliIIlo be met with. It ilppear.>. th:tt the words Mhe would s~y" [Hbovel refer to wh .. a (x + hY'. This example (sheets 52-63, pp. 48, 46, 44, 45-52, 52 according 10 MMx) is ve ry convenient, beaJuse in ilthe expansion inlo a series according to the powers of 11 , is given by Ihe hi,nom;a l theorem of Newton, Le., without the help of diffe rential c.1Iculus. IIcre MRTX verifies in detail, 3111h81 he has said above. Thus, for example. having shown sub 1) • that the coefficient of h ( i.e., ma,y ..-I) is actually (the first) deriva tive of ox" in the usual senSe (i.e., it is the limit (lf the ralio [(:c + Ir) - [(x) as Ir

"

-+

0). Ma rl( writes further (sheets 53.p. 46, the second 46-111 page IIccording 10

Marx):

max "'-\ is a pure function of x, 'Yhere in no h ente rs ,just lIS it was the case with ax"'. In both of these expressions x is also the yar iablc, ,lIld th at is why it also admi ts of varia ti on in the second, just as in the first. Thai is why, sub 1 ) we could j us t as well take max"'-' as the starling point, asf(x), as we did in the case ofax'". After Ihis, M~rx goes through all these details, also for seeking the second deriva tive, stressing , that the sC(:Ond derivative of f(x) - considered as the fitsl derivative o f f'(:c) - herein tlctually tu rns oulto be the second derivDtive of f(:c) and that too in the Lagrangian sense (i.e., the coefficient of 11 212) • spcci~lly

In connection with this, and in addition 10 the foregoing, Marx dwells upon yet another definition of the second , that is, also of the higher order derivatives (and differentials) : through the differences of higher orders. Some of the manuals at Marx's disposal are known to us. In them derivativcs of higher orders have not been defined through finite differences of highe r orders (usually this i~ not done in modern cOurses too). If it is done. liS for examplc. in Lacroix's book (S.F.Lacroix, "Traite du calcul diff\!rentiel et du calcul integral", voU- III , Paris, 1810-1819), then only in Ihe section on "Finite ,Jjtrerences". which Marx did not study at all. (Marx's notes on mathematical analysis - with the exception oflhose from the works of Newton ,seellbove pp.127-l31 - are related [ 0 the section entitled -Differential Calculus".) If we use that definition of the "second difference or A'y ~ . which to all appearance belongs 10 Marx himself, and which he inlrouuced above (see p.136 ). Ihen we get thllt f"(x) is the limit of the ra tio

2~oX2 Y

when Ax -O(here " is uesignaled through •

Ilx). meanwhile he re for Marx (sheel57, p. 47 according

10

Mllrx,second lillle "47" for Marx)

f"(:c) is thal ve ry limit of the ratiog, which is cited in the mouem manuals, where Ox

lJ,~ y is

the

usual difference of second order. The calculations Through which Marx I/lys the founda tion of Ihis conclusion, also indicate the same. These calculations - if carried out with more appropriate notations (Marx's no tation fo r the augmented value of the derivative y' is inappropriale, because, • was adopted to designate the derivative) -look as unde r : Let us have the {ollowing no tations (the symbol:::: Y ::::

reads : designates):

f (.1'),

Y, :::: /(.1' + Ill.

6y ::: Yt- Y. 6y t ::: Y2 - Yt

62y

1 let

:;:: 6 Y t -6 Y I

6x .. 11 bea

constant.

lWO DIfFERENT WAYS OF DETEnMINING Til E DEltlVKllVE

.47

And let x .. "lID f'(Hh)-f'(x) J • f "() h_O I

(' )

Let us substitute for the derivative f'(x) its approximatc (Hprc-limit") value: [(x +

';> -/(.ll .. ~. Thcn analogously f'(x + 11) is substituted by 6.x I

[(x + 211) -[Col + 11) .. Yl-YI .. 6.YI and the ratio f' (x +~) -['(x) by h h 6x 6YI-6y 6 2y 6x /lx 6 2y 6.x .. 6.l .. 6. .';" whence Marx draws the conclusion [ see (I)

1: 2

_l .. ,.Im ~2 f "() d~_06.l

Thus it is natural 10 think that here Marx used - not in the form of an extract, but a proper account (of) - some other source still unnoticed by us, where the differential calculus has been set forth in closer connection with the calculus of finite differences (as it was done, for example, in Euler, whose "Differential CalculusH even begins with a chapter on "The Finite Diffcrences" ; sce, Institutioncs calculi differentialis eum ejus usu in analysi [initorum ae doetrina serierum, al/ctore Lconhardo Eulero, impensis ~cademiae imperialis scientiarum, J>etropolitanae, 1755), ( there il> a Russian translation: L. Euler, "Differentsial noe ischislcnie M • Moskva-Leningrad, 1949, ch. I ), It is true that, 10 all appearance, Marx could not gel acquainted with Euler's book, though apparently, he'intended to do so. Later on (sheet 57-58, pp, 47-48 according 10 Marx) havi ng done (excluding the definitions through finite differences), what was done for the second derivative, also for the derivatives of third and fourth orders, Marx still fUrther concretises his e)Cample, assuming m .. 3 in the formulae obtained by him, Namely, he writes (sheeI59,p, 49) :

For example, if ins tead of the indeterm inate expone nt or index m we subs titute 3, then we shaH get [ .... ]. In this example, Marx is able 10 complete all the calculations through 10 the cnd (upto obtaining the fourth derivative, equal 10 zero), which he does (sheets 59-63, pp. 49-52, once more 52) first by listing the formulae obtained earlier in a general [arm (in terms of m), and then by assuming in them m=3. Herein Matx especially highlights three circumstances : that 1) the initial function ~nd all its derivatives arc different fUnctions (sheet 62,p. 52),

bUI all the terms, which later on develop independently. and arc differentiated, arc includ ed in the initial function, so that Ihe initial function a lready contains these derivatives in embryo. 2) The successive derivatives are not simply coefficients o( h, h2,

11" . .. in

[(.l +,11), but arc distinguished from the eocftlcicnt of II~ by the multiplier

the expansion of

1'2'~ ... n (sheet 62,Marx's

p.52) . • In the modern courses of mathematical analysis (see, for example, F. Franldin, "Mathematical Analysis", Part I, Moscow, IL, 1950,§93, Finite Differences, pp,150-160) this is proved in a more general form and more strictly (though not constructively"b). -Ed.

148

DESCRIPTION OFTHE MATIIEMA'I1CAL MANlISCRwfS 3) While obtllining the coefficients of lhe cXJl~n~ion for f(x + 11) (if wc do not have them yet ) along the path of successive differentiation of the illrcady obl" incd terms of fhe cxp,lnsion, it should be remembered,fhM h is considered to' be 11 conslllol (sheet 63,p. 52, MilTX'S second 52-nd page).

In other words, M:lTX knew beforehllnd lhat,lhc mistakes of calculalion impeding understa nding mlly be connected with !he lack of auention towards the ex;!c! formulation of the theorem llboul the connection between the derived f UllClions of /(x) and lhe cocf(icicnts of expansion of f(x + It) into a series according 10 the powers of fr.

Following Ihis, lhe notes on sheets 64-68 (pp. 53-57 according 10 Mar,,,) have been str uck out in pe ncil. 111CY contain the following extracts: extracts on ug range's mcthod from Boucharlat's book, pp. 168-17 1 (of the 5th ed. )ancl, exlracts o n Ta ylor 's theorem from lIi nd -s book, pp_ 83-85. He did not strike off the no tes on sheets 69 -77 (1'1'.58-66 according 10 Marx). However, since they contain only the extracts from Hi nd's book, here we limit ourselves to indicating the corresponding pages of the book and lhe titles, under which these extracts have been cited by Ma ne. Sheets 68-7 1 (pp. 57-60 according 10 Marx). Extracts from Hind's book, pp, 86-92. Marx ' s heading: • A. Finding certain limits or Taylor's theorem in its application ,,] 12, Sheets 71-73(pp. 60-62 acco rding to Marx) . Extracts from Hind's book, pp. 96-98, Marx 's ti tle

:"B . Further manipulations with Tay/or's theorem" . . Sheets 73-77 (pp. 62-66 according to Marx) . EXlracts from Hind's book, pp. 92-96, unde r Marx's heading ;"c. Failure of Tay/or's theorem". Sheets 78 and 79 (upper part) (pp. 67-68 acco rding 10 Marx) contain his own obsctV