## Méchanique analitique.

Paris: Veuve Desaint, 1788.

First edition, and a fine copy. “Perhaps the most beautiful mathematical treatise in existence; called by Hamilton ‘a kind of scientific poem’. It contains the discovery of the general equations of motion, the first epochal contribution to theoretical dynamics after Newton’s Principia, next to which it has been ranked” (Evans). “Lagrange's masterpiece, the Méchanique Analitique (Paris, 1788), laid the foundations of modern mechanics, and occupies a place in the history of the subject second only to that of Newton’s Principia” (Wolf). “With the appearance of the Mécanique Analytique in 1788, Lagrange proposed to reduce the theory of mechanics and the art of solving problems in that field to general formulas, the mere development of which would yield all the equations necessary for the solution of every problem ... [it] united and presented from a single point of view the various principles of mechanics, demonstrated their connection and mutual dependence, and made it possible to judge their validity and scope” (DSB). “In the preface of the book La Grange proudly points to the complete absence of diagrams, so lucid is his presentation. He regarded mechanics (statics and dynamics) as a geometry of four dimensions and in this book set down the principle of virtual velocities as applied to mechanics” (Dibner). “With the appearance of the Mécanique analytique in 1788, Lagrange proposed to reduce the theory of mechanics and the art of solving problems in that field to general formulas, the mere development of which would yield all the equations necessary for the solution of every problem. The Traité united and presented from a single point of view the various principles of mechanics, demonstrated their connection and mutual dependence, and made it possible to judge their validity and scope. It is divided into two parts, statics and dynamics, each of which treats solid bodies and fluids separately. There are no diagrams. The methods presented require only analytic operations, subordinated to a regular and uniform development. Each of the four sections begins with a historical account which is a model of the kind” (DSB).

“In [Méchanique Analitique] he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalized co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D’Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation … Amongst other minor theorems here given I may mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action. All the analysis is so elegant that Sir William Rowan Hamilton said the work could only be described as a ‘scientific poem’” (Rouse Ball, A Short Account of the History of Mathematics).

“Lagrange introduces the principle of virtual velocities in the first edition as ‘a kind of axiom for mechanics’ (p. 12) for statics, where it ‘has all the simplicity one might desire in a fundamental principle’ (p. 10). By statics he means the ‘science of equilibrium of forces’ (p. 1), as he says right at the beginning. If one now considers a system of mass-points in a static equilibrium acted on at any given time by forces P, Q, R, . . . and gives it a small perturbation, then the individual masses experience ‘virtual’ displacements, that is, displacements compatible with any connections that may exist between the masses. Let δp, δq, δr, … be their projections on the forces P, Q, R, ..., with the sense of direction of the projection indicated by a suitable choice of sign. Lagrange labels these displacements as ‘virtual velocities’ by appealing to a fixed time element dt. The principle of virtual velocities (or displacements) now asserts that a system is in equilibrium if the sum of the ‘moments of force’ vanishes (p. 15):

Pδp + Qδq + Rδr + · · · = 0.

He then applies this relation, from ‘Section III’ of the Méchanique analitique, in the treatment of general properties of the equilibrium of point systems (Section III), methods for solving the resulting equations (Section IV), special problems in statics (Section V), hydrostatics (Section VI), problems of equilibrium of incompressible fluids (Section VII) and problems of equilibrium of compressible and elastic fluids (Section VIII).

“Lagrange constructs dynamics in an entirely analogous way. He first extends the principle of virtual velocities to problems of motion in that, as well as the external forces P, Q, R, ..., he also takes into account on the individual point masses their accelerations, which must be compatible with the connections within the system. Multiplication by the instantaneous masses yields the forces that the same accelerations would produce in free masses. His claim is then that under a virtual displacement the ‘moments of the forces’ P, Q, R, … must be equal to the moments of these forces of acceleration” (Pulte, p. 213). Using the method of ‘Lagrange multipliers’, Lagrange deduces from this equality the ‘Lagrange equations’ of motion, which reduce all dynamical problems to the determination of the two functions kinetic and potential energy. This has proved to be enormously influential right up to the present day, when modern quantum field theories are presented in terms of their ‘Lagrangian.’

The Méchanique analitique “was certainly regarded as the most important unification of rational mechanics at the turn of the 18th century and as its ‘crowning’ (Dugas). This achievement of unification and the abstract-formal nature of the work, physically reflected in immediate applications, earned the extravagant praise of Ernst Mach: ‘Lagrange […] strove to dispose of all necessary considerations once and for all, including as many as possible in one formula. Every case that arises can be dealt with according to a very simple, symmetric and clearly arranged scheme […] Lagrangian mechanics is a magnificent achievement in respect of the economy of thought’ (Mach, Die Mechanik in ihrer Entwicklung (1933), p. 445).

“Lagrange produced the Méchanique analitique during his time in Berlin. He referred as early as 1756 and 1759 to an almost complete textbook of mechanics, now lost; a later draft first saw the light of day in 1764. But it was not until the end of 1782 that Lagrange seems to have put the textbook into an essentially complete form, and the publication of the book was delayed a further six years” (Pulte, p. 209).

Grolier/Horblit 61; Evans 10; Dibner 112; Sparrow 120; Norman 1257; Wolf II, 69. Pulte, ‘Joseph Louis Lagrange, Méchanique analitique (1788),’ Chapter 16 in Landmark Writings in Western Mathematics 1640–1940, Grattan-Guinness (ed.).

4to (260 x 196 mm), , pp. [i-v], vi-xii, [1], 2-512. Contemporary speckled calf, spine gilt and with red lettering-piece (a little rubbed at extremities). A fine, untouched copy.

Item #5849

Price: \$12,500.00