JUBILEE NOTES

JEAN LE ROND D'ALEMBERT

OF HIS BIRTH

K . A . B r e u s

ON T H E 2 5 0 t h ANNIVERSARY

The name of Jean Le Rond d ' A l e m b e r t (16,XI,1717 - 29.X.1783) the F rench ma themat i c i an and phi lo- sopher and enl ightener of the per iod of p r epa ra t i on of the bourgeo is revolut ion in F rance , holds a p rominen t p lace in the i l l u s t r i ous cons te l la t ion of names of eminent s c i en t i s t s of the world. D ' A l e m b e r t ' s sc ient i f ic works made a fundamental contr ibut ion in the development of many b ranches of m a t he m a t i c s , mechan ics and a s t ronomy. D ' A l e m b e r t ' s e lect ion as a m e m b e r of the Academies of P a r i s , P e t e r s b u r g and other c i t i e s was a recogni t ion of his g rea t sc ient i f ic s e r v i c e s .

Together with D. Diderot d ' A l e m b e r t c r e a t e d the "Encyclopedia of the Sciences , Ar t s and Craf t s" ; in the "Encyclopedia" he was r e spons ib l e for the sec t ions of ma thema t i c s and phys ics . In i ts f i r s t volume he publ ished impor t an t a r t i c l e s such as "Limi t , " "Different ia l , " "Equation, " "Dynamics , " and "Geomet ry . " Of his phi losophica l works the mos t impor t an t was his in t roductory a r t i c l e to the encyclopedia - "Sketch of the or ig in and development of the Sciences" - in which, following F. Bacon, he gave a c l a s s i f i ca t i on of the Sciences , and a l so his e s say "Elements of Philosophy" (1759).

In his phi losophica l works d ' A l e m b e r t c r i t i c i z e d t h e f e u d a l a n d e l e r i c a l v i e w of the world, but he did not a t ta in to the m a t e r i a l i s m and a the i sm of the F rench encyc loped is t s . The l imi ta t ions of d ' A l e m b e r t ' s ph i lo - sophy were mos t c l e a r l y e x p r e s s e d in his t heo r e t i c a l - c onc e p t ua l views; incl ining to s ensua l i sm , he turned away f r o m a m a t e r i a l i s t i c solution of the fundamental question of philosophy, and recogn ized the p r e s e n c e of a s e p a r a t e subs tance in man which was d i s t inc t f rom m a t t e r - a consc ience .

The bas ic ma thema t i ca l inves t iga t ions of d ' A l e m b e r t were concerned with the theory of d i f fe ren t ia l equations, though the r e s u l t s he obtained in many other b ranches of ma thema t i c s were a lso of f i r s t i m p o r - tance . In a l geb r a d ' A l e m b e r t gave the f i r s t , if not s t r i c t l y r i go rous proof of the fundamental t heo rem of the ex is tence of a root for any a lgeb ra i c equation; he s igni f icant ly fu r the red the pene t ra t ion into m a t h e -

,? m a t i c s of the concept of an i r r a t i o n a l number . In Cons idera t ions on the genera l causes of winds" (1747) and a lso somewhat l a t e r , d ' A l e m b e r t showed that any a lgeb ra i c expres s ion fo rmed of an a r b i t r a r y number of imag ina ry quanti t ies can be r educed to the fo rm p+ iq , where p and q a r e r ea l quant i t ies . F u r t h e r m o r e , he a s s e r t e d that the d i f fe ren t ia l f(x + iy) d(x + iy) can a lways be r e p r e s e n t e d in the fo rm dp + idq. Applying loga r i t hmic d i f fe rent ia t ion of a power with imag ina ry r ad ix and exponent, d ' A l e m b e r t t r e a t e d the r ad ix as a va r i ab le i th is was, pe rhaps , one of the f i r s t c a s e s of the appea rance of a complex va r i ab l e .

In his e s s a y s on ma thema t i c s which a p p e a r e d in the "Encyc loped ia" d' A lember t made the f i r s t a t tempt , which was subsequent ly b r i l l i an t l y pe r fec ted by Cauchy, to put the theory of l imi t s at the foundation of

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 20, No.2, pp.228-231, March-April, 1968.

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infinitesimal analysis . For Euler, the infinitesimal calculus was a method for the determination of the ra t io of the vanishingly small increments acquired by functions when their arguments grow by vanishingly small increments . He considered vanishingly small increments as absolute zeros, so that differential calculus was concerned not so much with them as with their ra t ios . D'Alembert pointed out that, in con- t r a s t to the generally held opinion, in real i ty "in differential calculus one is not at all concerned with in- finitely small quantities but only with the l imits of finite quantities. The words infinite and small a re used only to abbreviate expressions. " Here d 'Alember t was not speaking about a limit rat io of vanishing quan- t i t ies, but about the limit rat io of finite quantities, which represented a considerable advance over the approach of Newton, who completely retained the infinitely small in the actual sense in an explicit or masked form. D'Alembert refra ined f rom myst ica l points of view concerning the foundations of the infinitesimal calculus and s tr ived to rat ionalize this calculus. He added to the independent var iable a quantity which was not in the l i teral sense infinitely small but ra ther a finite increment, formed the rat io of the increment of the function to the increment of the argument and then considered what happened when the increment of the argument tends to zero. Thus, d 'Alember t defined a derivative by means of the operation of a transit ion to a limit, but at that time the theory of l imits itself did not have a r igorous logical justification, and the transit ion to the limit had m o r e of the charac te r of a jump than of a natural development. D 'Alember t ' s method only slightly differed f rom Newton's method of f i rs t and last rat ios, but d 'Alember t brought this concept c loser to the mathematicians of the continent of Europe, especially as he himself used Leibniz 's symbolism.

D 'Alember t published a large number of investigations on the theory of ordinary l inear differential equations of f i r s t and second order with constant coefficients. D 'Alember t ' s f i rs t resul ts on the integration of l inear sys tems of differential equations were contained in his "Trac ts on Dynamics" (1743). D 'Alembert developed a very general method of solution of an inhomogeneous l inear differential equation, which con- sisted of the reduction of the solution of an equation of higher order to the solution of a system of s imul- taneous f i rs t order differential equations. In solving f i rs t order differential equations, d 'Alember t made great use of the method of an integrating factor and gave the f i rs t proof of the existence of an integrating factor for any f i rs t order differential equation. As applied to a l inear sys tem of two differential equations with constant coefficients of the form:

dg d~ = axg -t- blz + f (x)

dz d'-x = a2g + b,~z + (p (x)

d 'Alember t ' s method for finding the f i rs t integral consisted of the following. tion (1) by some number k and adding the resul t to the f i rs t equation, we obtain

o r

d(y + kz) dx (al + kay) g + (bl + kb2) z + f (x) -p k(p (x)

We choose k such that

d (g -F kz) .= (a~ -b kay) (g b~ q- kb2 \ d~ + ~-i~, ~E~ z) + f(x) + kco(x).

bl + kb2 = k (al q- kay) bl ~ kb~ - ~ ~ Or a 1 -p ka~

then equation (2) can be re-wr i t ten in the form

d (y + kz) dx - - (ai -P kay) (g -~- kz) -F f (x) + k~ (x),

(1)

Multiplying the second equa-

(2)

f rom which

If the roots of the equation (3) are different and real then, denoting them by k 1 and k 2 w e have two f i r s t integrals in the form:

(4)

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(5) + ~,zz = e !'+h~''~ {C~ + ~ if (x) + k~ r e-"'+k~dx} ;

solving the sys tem (5) with respec t to y and z, we find the general solution of the sys tem (1). D 'Alember t ' s method given here can also be extended to l inear sys tems which eontain derivat ives of higher o rders .

Attempts to apply the basic equations of mechanics to problems of the theory of planetary motion (the three-body problem) led to the development of approximate methods of the solution of differential equations. Cireular orbits were chosen as approximate solutions of these problems, and these were then cor rec ted by means of ser ies expanded in powers of small quantities (the eecentr ici ty of the planetary orbits, their in- clination to be elliptic and so on). D 'Alember t also developed a method of approximate solution which consis ted of introducing infinite ser ies with undetermined eoefficients which were then determined by a differential equation.

D 'Alember t per formed par t icular ly great serv ices in the development of methods of solution of part ial differential equations. Here one should mention f i rs t the reduction of the famous problem of an oseiIlating s tr ing to a seeond order partial differential equation (the wave equation) and the representa t ion of the solu- tion in the fo rm of the sum of two a rb i t r a ry functions. The solution of the problem of an oscil lating str ing played an important part in the works of d 'Alembert , Euler, D. Bernoulli, Lagrange and other mathe- maticians, tn investigations connected with the solution of this equation, the mathematicians f i r s t es tab- lished how many a rb i t r a ry funetions occur in the integral of a part ial differential equation and they also investigated the problem of the representa t ion of the a rb i t r a ry functions by means of the t r igonometr ica l ser ies which were later called Four ier se r ies . In solving an elliptic partial differential equation which is encountered in hydrodynamics, d 'Alember t and L. Euler were the f i rs t to find the basic equations whieh eonneet the real and imaginary parts of an analytic function, which were la ter called the Cauchy-Riemann equations.

D 'Alember t found the solution of a number of other part ial differential equations. Thus, in the paper "Investigations on Integral Calculus" he eonsidered the solution of l inear part ial differential equations with constant coefficients. In solving these equations, d 'Alember t made use of two methods: the f i rs t of these eonsisted of the reduction of the partial differential equations to ordinary differential equations and their solution by the method of an integrating faetor . The second method, which D'Alember t applied to f i rs t o rder part ial differential equations, consis ted of introducing a substitution which t ransformed the given equation into a more simple equation which could be integrated easily. Although not completely c lear ly formulated, D 'Alember t gave a classif icat ion of l inear and nonlinear equations, and of those with constant or var iable coefficients. One should also mention a widely used suffieient condition of eonvergence of ser ies which c a r r i e s the name of d 'Alembert .

D 'Alember t made a large number of investigations of many applied problems, in par t icular , hydro- dynamics, aerodynamies and the three-body problem. In "Trac t s on Dynamies" d 'Alember t introduced the method of the reduction of problems of the dynamics of solid bodies to statics, which is known under the name "d 'Alember t ' s prInciple. " D 'Alember t ' s principle can be considered as an asser t ion about the equi- l ibr ium between the given forces and the inertial forces . In this form d 'Alember t ' s principle is the basis of that branch of mechanics which provides methods of solution of dynamical problems by the methods of static s.

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