## Méchanique analitique. [Bound with:] Théorie des fonctions analytiques.

Paris; Paris: Veuve Desaint; de l’Imprimerie de la République, 1788; An V [1797].

An exceptional volume, in a fine contemporary binding, containing the first edition of Lagrange’s masterpiece, the *Méchanique,* “one of the outstanding landmarks in the history of both mathematics and mechanics” (Sarton, p. 470) and “perhaps the most beautiful mathematical treatise in existence, together with the corrected second printing of the *Théorie, *containing Lagrange’s formulation of calculus in terms of infinite series, which provided the basis for Augustin-Louis Cauchy’s development of complex function theory in the first decades of the next century. The *Méchanique* contains the discovery of the general equations of motion, the first epochal contribution to theoretical dynamics after Newton’s *Principia*” (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). “The year 1797 … saw the appearance of the famous work of Lagrange, *Théorie des fonctions analytiques* … This book developed with care and completeness the characteristic definition and method in terms of ‘fonctions dérivées,’ based upon Taylor’s series, which Lagrange had proposed in 1772 … Lagrange’s *Théorie des fonctions* was only one, but by far the most important, of many attempts made about this time to furnish the calculus with a basis which would logically modify or supplant those given in terms of limits and infinitesimals” (Cajori).

“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. 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*).

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, 445)” (Pulte, p. 220).

“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).

“By 1790 a critical attitude had developed both within mathematics and within general scientific culture. As early as 1734 Bishop George Berkeley in his work *The Analyst *had called attention to what he perceived as logical weaknesses in the reasonings of the calculus arising from the employment of infinitely small quantities. Although his critique was somewhat lacking in mathematical cogency, it at least stimulated writers in Britain and the Continent to explain more carefully the basic rules of the calculus. In the 1780s a growing interest in the foundations of analysis was reflected in the decisions of the academies of Berlin and Saint Petersburg to devote prize competitions to the metaphysics of the calculus and the nature of the infinite. In philosophy Immanuel Kant’s *Kritik der reinen Vernunft *(1787) set forth a penetrating study of mathematical knowledge and initiated a new critical movement in the philosophy of science …

“The full title of the *Théorie *explains its purpose: ‘Theory of analytical functions containing the principles of the differential calculus disengaged from all consideration of infinitesimals, vanishing limits or fluxions and reduced to the algebraic analysis of finite quantities’. Lagrange’s goal was to develop an algebraic basis for the calculus that made no reference to infinitely small magnitudes or intuitive geometrical and mechanical notions …

“In Lagrange’s conception of analysis, one is given a universe of functions, each expressed by a formula *y* = *f*(*x*) and consisting of a single analytical expression involving variables, constants and algebraic and transcendental operations. During the 18th century such functions were called continuous, and the *Théorie *is devoted exclusively to functions that are continuous in this sense. (Mathematicians were aware of the possibility of other sorts of functions, but alternate definitions never caught on.) Such functions were naturally suited to the usual application of calculus to geometrical curves. In studying the curve the calculus is concerned with the connection between local behaviour, or slope, and global behaviour, or area and path-length. If the curve is represented by a function *y* = *f*(*x*) given by a single analytical expression then the relation between *x* and *y* is permanently established in the form of *f*. Local and global behaviour become identified in this functional relation.

“It is also necessary to call attention to the place of infinite series in Lagrange’s system of analysis. Each function has the property that it may be expanded as [a] power series. Nevertheless, an infinite series as such is never defined to be a function. The logical concept of an infinite series as a functional object defined *a priori *with respect to some criterion such as convergence or summability was foreign to 18th-century analysis. Series expansions were understood as a tool for obtaining the derivative, or a way of representing functions that were already given.

“For the 18th-century analyst, functions are things that are given ‘out there’, in the same way that the natural scientist studies plants, insects or minerals, given in nature. As a general rule, such functions are very well-behaved, except possibly at a few isolated exceptional values. It is unhelpful to view Lagrange’s theory in terms of modern concepts (arithmetical continuity, differentiability, continuity of derivatives and so on), because he did not understand the subject in this way …

“Part One of the *Théorie *begins with some historical matters and examines the basic expansion of a function as a Taylor power series. There is considerable discussion of values where the expansion may fail, and a derivation of such well-known results as l’Hôpital’s rule. Lagrange then turned to methods of approximation and an estimation of the remainder in the Taylor series, followed by a study of differential equations, singular solutions and series methods, as well as multi-variable calculus and partial differential equations. He outlined and supplemented topics explored in some detail in memoirs of the 1760s and 1770s.

“Part Two on geometry opens with an investigation of the geometry of curves. Here Lagrange examined in detail the properties that must hold at a point where two curves come into contact—the relationships between their tangents and osculating circles. Corresponding questions concerning surfaces are also investigated, and Lagrange referred to Gaspard Monge’s memoirs on this subject in the *Académie des Sciences*. He derived some standard results on the quadrature and rectification of curves. The theory of maxima and minima in the ordinary calculus, a topic Lagrange suggested could be understood independently of geometry as part of analysis, is taken up. Also covered are basic results in the calculus of variations, including an important theorem of Adrien-Marie Legendre in the theory of sufficiency …

“The third part on dynamics is somewhat anticlimactic, given the publication nine years earlier of his major work *Méchanique analitique*. In this part Lagrange presented a rather kinematically-oriented investigation of particle dynamics, including a detailed discussion of the Newtonian problem of motion in a resisting medium. He also derived the standard conservation laws of momentum, angular momentum and live forces. The book closes with an examination of the equation of live forces as it applies to problems of elastic impact and machine performance.

“… some of the major original contributions of this work: the formulation of a coherent foundation for analysis; Lagrange’s conception of theorem-proving in analysis; his derivation of what is today called the Lagrange remainder in the Taylor expansion of a function; his formulation of the multiplier rule in the calculus and calculus of variations; and his account of sufficiency questions in the calculus of variations” (Fraser, p. 260).

The Norman catalogue identifies two versions of the *Théorie des fonctions analytiques*, Version A with 276 pages, and Version B with 277, and states that no priority has been established. Craig Fraser (*Landmark Writings*, p. 258) states that version B is the first edition but does not mention version A. Ivor Grattan-Guinness (*Convolutions*, p. 129) states that Version A is the first edition. In our opinion Version B is certainly the true first edition, and Version A is actually the second edition (not just the second issue). The two versions are not merely different issues: the last several pages of text (and also a few earlier pages) are completely reset. In Version A the text ends on p. 276 and the following page is blank (reader.digitale-sammlungen.de/de/fs1/object/display/bsb10053590_00294.html); in Version B the last 6 lines of the main text appear on p. 277 (these lines are identical to the last 6 lines of Version A), followed by a list of errata on the same page and the imprint ‘Par les soins de P. D. Duboy-Laverne, directeur de l’Imprimerie de la République’ on the following unnumbered page (see books.google.fr/books?id=xCEOAAAAQAAJ&printsec=frontcover&hl=fr#v=onepage&q&f=false). A further difference is that in Version A the verso of the half-title is blank; in Version B it carries the text ‘Se trouve à Paris, Chez Bernard, libraire, quai des Augustins, no. 37’. The priority of Version B is revealed by the 8-page index that appears at the beginning of both versions; *in both versions this index refers to the ‘Conclusion’ on pp*. 276-277. Clearly, Version A cannot have been issued with an index that is appropriate only to a version printed later; thus, Version B must be first. Its priority is confirmed by the fact that, in Version A the errata listed in Version B have been corrected. As Norman notes, there is a third version of the *Théorie *which forms Vol. III of the ninth cahier of the *Journal de l’Ecole Polytechnique*, published in 1801. Norman states that this journal issue is Version A, but in fact it is Version B (see gallica.bnf.fr/ark:/12148/bpt6k86263h/f1.image). The fact that the uncorrected Version B was used for the journal issue suggests that Version A was not only issued later than Version B, but after the journal issue, i.e., no earlier than 1801. The inscription of this copy indicates that copies of Version B were still available in 1804.

[*Méchanique*:] 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.). Sarton, ‘Lagrange’s personality,’ *Proceedings of the American Philosophical Society* (1944), pp. 457-98. [*Fonctions analytiques*:] Norman 1258; Barchas 1198; Honeyman 1881; Stanitz 250. Fraser, ‘Joseph Louis Lagrange, *Théorie des fonctions analytiques * (1797),’ Chapter 19 in *Landmark Writings*. Grattan-Guinness, *Convolutions in French Mathematics 1800-1840*, 1990 (see especially pp. 129-133).

4to (248 x 200 mm), [Méchanique:] [i-v], vi-xii, [1], 2-512; [Théorie:] pp. [4], [i], ii-viii, [1], 2-276. Contemporary full calf, corners with slight wear, but very good and unrestored. Internally very fine and fresh.

Item #4300

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Price:
$15,000.00
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