1.1. From the Preface to the English translation by Craig G Fraser.

Lagrange's Mécanique Analytique appeared early in 1788 almost exactly one century after the publication of Newton's Principia Mathematica. It marked the culmination of a line of research devoted to recasting Newton's synthetic, geometric methods in the analytic style of the Leibnizian calculus. Its sources extended well beyond the physics of central forces set forth in the Principia. Continental authors such as Jakob Bernoulli, Daniel Bernoulli, Leonhard Euler, Alexis Clairaut and Jean d'Alembert had developed new concepts and methods to investigate problems in constrained interaction, fluid flow, elasticity, strength of materials and the operation of machines. The Mécanique Analytique was a remarkable work of compilation that became a fundamental reference for subsequent research in exact science.

During the eighteenth century there was a considerable emphasis on extending the domain of analysis and algorithmic calculation, on reducing the dependence of advanced mathematics on geometrical intuition and diagrammatic aids. The analytical style that characterises the Mécanique Analytique was evident in Lagrange's original derivation in 1755 of the δ-algorithm in the calculus of variations. It was expressed in his consistent attempts during the 1770s to prove theorems of mathematics and mechanics that had previously been obtained synthetically. The scope and distinctiveness of his 1788 treatise are evident if one compares it with an earlier work of similar outlook, Euler's Mechanica sive Motus Scientia Analytice Exposita of 1736. Euler was largely concerned with deriving the differential equations in polar coordinates for an isolated particle moving freely and in a resisting medium. Both the goal of his investigation and the methods employed were defined by the established programme of research in Continental analytical dynamics. The key to Lagrange's approach by contrast was contained in a new and rapidly developing branch of mathematics, the calculus of variations. In applying this subject to mechanics he developed during the period 1755-1780 the concept of a generalised coordinate, the use of single scalar variables (action, work function), and standard equational forms (Lagrangian equations) to describe the static equilibrium and dynamical motion of an arbitrary physical system. The fundamental axiom of his treatise, a generalisation of the principle of virtual work, provided a unified point of view for investigating the many and diverse problems that had been considered by his predecessors.

In what was somewhat unusual for a scientific treatise, then or now, Lagrange preceded each part with an historical overview of the development of the subject. His study was motivated not simply by considerations of priority but also by a genuine interest in the genesis of scientific ideas. In a book on the calculus published several years later he commented on his interest in past mathematics.

He suggested that although discussions of forgotten methods may seem of little value, they allow one "to follow step by step the progress of analysis, and to see how simple and general methods are born from complicated and indirect procedures."

Lagrange's central technical achievement in the Mécanique Analytique was to derive the invariant-form of the differential equations of motion

         $\Large \frac{\partial T}{\partial q_i} - \frac{d}{dt}\frac{\partial T}{\partial \dot{q_i}}\normalsize = {\partial V}{\partial q_i}$

for a system with $n$ degrees of freedom and generalised coordinates $q_{i} (i = 1,...,n)$. The quantities $T$ and $V$ are scalar functions denoting what in later physics would be called the kinetic and potential energies of the system. The advantages of these equations are well known: their applicability to a wide range of physical systems; the freedom to choose whatever coordinates are suitable to describe the system; the elimination of forces of constraint; and their simplicity and elegance.
...

Lagrange derived his general equations from a fundamental relation that originated with the principle of virtual work in statics. The latter was a well-established rule to describe the operation of such simple machines as the lever, the pulley and the inclined plane. The essential idea in dynamics - due to d'Alembert - was to suppose that the actual forces and the inertial reactions form a system in equilibrium or balance; the application of the static principle leads within a variational framework to the desired general axiom. Historian Norton Wise has called attention to the pervasiveness of the image of the balance in Enlightenment scientific thought. Condillac's conception of algebraic analysis emphasised the balancing of terms on each side of an equation. The high-precision balance was a central laboratory instrument in the chemical revolution of Priestley, Black and Lavoisier. A great achievement of eighteenth-century astronomy, Lagrange and Laplace's theory of planetary perturbations, consisted in establishing the stability of the various three-body systems within the solar system. The Mécanique Analytique may be viewed as the product of a larger scientific mentality characterised by a neo-classical sense of order and, for all its intellectual vigour, a restricted consciousness of temporality.

A comparison of Lagrange's general equations with the various laws and special relations that had appeared in earlier treatises indicates the degree of formal sophistication mechanics had reached by the end of the century. The Mécanique Analytique contained as well many other significant innovations. Notable here were the use of multipliers in statics and dynamics to calculate the forces of constraint; the method of variation of arbitrary constants to analyse perturbations arising in celestial dynamics (added in the second edition of 1811); an analysis of the motion of a rigid body; detailed techniques to study the small vibrations of a connected system; and the Lagrangian description of the flow of fluids.

In addition to presenting powerful new methods of mechanical investigation Lagrange also provided a discussion of the different principles of the subject. The Mécanique Analytique would be a major source of inspiration for such nineteenth-century researchers as William Rowan Hamilton and Carl Gustav Jacobi. The seminal character of Lagrange's theory is evident in the way in which they were able to use it to derive new ideas for organising and extending the subject. Combining results from analytical dynamics, the calculus of variations and the study of ordinary and partial differential equations Hamilton and Jacobi constructed on Lagrange's variational framework a mathematical-physical theory of great depth and generality. Within the calculus of variations itself the Hamilton-Jacobi theory would become a source for Weierstrassian field theory at the end of the century; within physics it took on new importance with the advent of quantum mechanics in the 1920s.

Beyond its historical and scientific interest the Mécanique Analytique is a work of considerable significance in the philosophy of science. It embodies a type of empirical investigation which emphasises the abstract power of mathematics to link and to coordinate observational variables. The concepts of an idealised constraint, a generalised coordinate and a scalar functional allow one to describe the system without detailed hypotheses concerning its internal physical structure and working. In the third part of his Treatise on Electricity and Magnetism James Clerk Maxwell (1892) stressed this aspect of Lagrange's theory as he used it to create a "dynamical" theory of electromagnetism. Beginning with Auguste Comte and continuing with such later figures as Ernst Mach and Pierre Duhem, Lagrange's analytical mechanics has attracted the attention of leading positivist philosophers of physics. In 1883 Mach praised Lagrange for having brought the subject to its "highest degree of perfection" through his introduction of "very simple, highly symmetrical and perspicuous schema."

Lagrange's book remains valuable today as an exposition of subjects of ongoing utility to engineering physics and applied mathematics. Its value to the historian of mechanics, its intrinsic interest to the practising scientist and its contribution to the philosophy of physics ensure its place as an enduring classic of exact science.

1.2. Lagrange's Preface to the First Edition.

There already exist several treatises on mechanics, but the purpose of this one is entirely new. I propose to condense the theory of this science and the method of solving the related problems to general formulas whose simple application produces all the necessary equations for the solution of each problem. I hope that my presentation achieves this purpose and leaves nothing lacking.

In addition, this work will have another use. The various principles presently available will be assembled and presented from a single point of view in order to facilitate the solution of the problems of mechanics. Moreover, it will also show their interdependence and mutual dependence and will permit the evaluation of their validity and scope.

I have divided this work into two parts: Statics or the Theory of Equilibrium, and Dynamics or the Theory of Motion. In each part, I treat solid bodies and fluids separately.

No figures will be found in this work. The methods I present require neither constructions nor geometrical or mechanical arguments, but solely algebraic operations subject to a regular and uniform procedure. Those who appreciate mathematical analysis will see with pleasure mechanics becoming a new branch of it and hence, will recognise that I have enlarged its domain.

1.3. Lagrange's Preface to the Second Edition.

There already exist several treatises on mechanics, but the purpose of this one is entirely new. I propose to condense the theory of this science and the method of solving the related problems to general formulas whose simple application produces all the necessary equations for the solution of each problem.

In addition, this work will have another use. The various principles presently available will be assembled and presented from a single point of view in order to facilitate the solution of the problems of mechanics. Moreover, it will also show their interdependence and mutual dependence, and will permit the evaluation of their validity and scope.

I have divided this work into two parts: Statics or the Theory of Equilibrium, and Dynamics or the Theory of Motion. In each part, I treat solid bodies and fluids separately.

No figures will be found in this work. The methods I present require neither constructions nor geometrical or mechanical arguments, but solely algebraic operations subject to a regular and uniform procedure. Those who appreciate mathematical analysis will see with pleasure mechanics becoming a new branch of it and hence, will recognise that I have enlarged its domain.

This is the purpose which I tried to fulfil in the first edition of this work published in 1788. The present edition is in many respects a new work based on the same outline but augmented. I have further developed the principles and general formulas and I have introduced numerous additional applications in which the solutions to the major problems in the domain of mechanics will be found.

I have kept the ordinary notation of the differential calculus because it fits the system of infinitesimals adopted in this treatise. Once the spirit of this system has been grasped well and the accuracy of its results established by either geometrical methods or by the analytical method of derived functions, the infinitesimal calculus can then be applied as a certain and manageable tool to shorten and simplify the demonstrations. It is in this way by using the method of indivisibles that the demonstrations of the Ancients are shortened.

We are now going to point out the principle additions which distinguish this edition from the First Edition.

SECTION I of Part I contains a more complete analysis of three principles of statics with new remarks on the nature and relation of these principles. The section ends with a direct demonstration of the Principle of Virtual Velocities which is completely independent of the two other principles.

In SECTION II, it is demonstrated in a more rigorous manner that the Principle of Virtual Velocities for an arbitrary number of forces can be deduced from the case where there are no more than two forces, which leads this principle back directly to the principle of the lever. The equations which result from this principle are then reduced to a more general form and the necessary conditions are given for a system of forces to be equivalent to and able to replace another system.

In SECTION III, the formulas for instantaneous rotational motion and for the composition of these types of motion are established in a more straight-forward manner. In addition, the theory of moments and their composition is deduced from the preceding development. Also, a little known property of the centre of gravity is presented and a new demonstration is given for maxima and minima in the state of equilibrium.

SECTION IV contains more general and simpler formulas for the solution of problems which depend on the calculus of variations and from the comparison of these formulas with those for the equilibrium of bodies of variable shape, it is shown how the problems relative to their equilibrium belong to the category of those which are known under the title of General Problems of Isoperimetrics and which are solved in the same manner.

SECTION V presents some new problems and some important comments on some of the solutions already given in the first edition.

In SECTION VI, some details are added to the historical analysis of the principles of hydrostatics.

In SECTION VII, the calculation of the variations associated with the molecules of a fluid have been treated more rigorously and with more generality. The analysis of the terms which refer to the limits of the fluid mass have been greatly simplified. From these terms, the theory of the action of fluids on the solids which they cover or on the walls of vessels which contain them is deduced and a direct demonstration is given of the following theorem: In the case of equilibrium between a solid and a fluid, the forces which act on the solid are the same as if the fluid and solid formed a solid mass. Much more has been added to this section and to the following section which treats the equilibrium of elastic fluids and presents some applications of the general formulas of the equilibrium of fluids.

Part II, which treats dynamics, has also been considerably augmented.

There is an important addition in SECTION II. It is shown for which cases the general formula of dynamics and consequently, the equations which result for the motion of a system of bodies, is independent of the position of the coordinate axes in space. This demonstration gives a means of completing a solution by the introduction of three new arbitrary constants where some constants would have otherwise been assumed to be equal to zero.

In SECTION III, more development is given to the properties relative to the motion of the centre of gravity and to the areas described by a system of bodies. There are additions to the theory of principal axes or of uniform rotation, deduced from the consideration of the instantaneous motion of rotation by an analysis which is different from the one used earlier.

Also, some new theorems are demonstrated on the rotation of a solid body or a system of bodies, when they depend on an initial impulse.

There is very little difference between SECTION IV of the first edition and this edition.

SECTION V is entirely new. It contains the theory of the Method of the Variation of Arbitrary Constants which is the subject of three memoirs printed among those of the First Class of the Institute in 1808. It is presented here in a much simpler manner and as a general method of approximation for all the problems of mechanics, where there are perturbing forces which are small compared to the principle forces.

In order to extend this theory as far as possible, the function $V$, which depends on forces principally, can only be an exact function of the independent variables $\xi, \psi, \phi$, etc., and of the time $t$, but it is not necessary that the function denoted by $\Omega$ and which depends on perturbing forces, also possess the same nature. Whatever the forces, if they are resolved for each body $m$ of the system into three components $X, Y, Z$ in the positive directions of the coordinates $x, y, z$ and with the tendency to increase them, there remains only to reduce the coordinates to functions of the independent variables $\xi, \psi, \phi$, etc. and to substitute in place of the partial derivatives $d\Omega d\xi, d\Omega d\psi$, etc. their respective sums

         $\sum m (X \Large \frac{dx}{d\xi}\normalsize + Y\Large \frac{dy}{d\xi}\normalsize + Z\Large \frac{dz}{d\xi})\normalsize, \sum m (X \Large \frac{dx}{d\psi}\normalsize + Y\Large \frac{dy}{d\psi}\normalsize + Z\Large \frac{dz}{d\psi})$, ...

and as a consequence, the quantity

        $\sum m (X\Delta x + Y\Delta y + Z\Delta z)$

will be obtained in place of $\Delta \Omega$, where the operator $\Delta$ refers to the arbitrary constants in such a way that the derivative $\Large\frac{d\Omega }{d\alpha}\normalsize$ can be changed to

          $\sum m (X \Large\frac{dx}{d\alpha}\normalsize + Y\Large\frac{dy}{d\alpha}\normalsize + Z\Large\frac{dz}{d\alpha})$

and so on for the other partial derivatives of $\Omega$. In this fashion, the method is applicable to perturbing forces represented by arbitrary variables.

Finally, SECTION VI, which is the last section of this volume and which corresponds to the first paragraph of SECTION V of the first edition, is augmented by various remarks and above all, by the solution of some problems on the small vibrations of bodies. It ends with the theory of vibrating strings which I presented earlier in the first volume of the Mémoires of the Académie de Turin and which is presented here in a very simple manner and free of the objections which d'Alembert made against this theory, in the first volume of his Opuscules.

1.4. Review by: Antonino Drago.
Mathematical Reviews MR1784335 (2001j:01060).

The English translation of Lagrange's main work on mechanics has to be welcome. The present translation will be a stimulus to commit a large audience of scholars to the task of throwing more light on what represents the major mathematical root of present mathematical physics. Its modern notation and two detailed prefaces - by specialist historian (C Fraser) and translators-editors (two professional scientists, A Boissonnade and V N Vagliente), respectively - greatly will help in introducing a reader to understanding an ancient book, whose relevance has increased in the past two centuries.

The book represents a milestone in the history of theoretical physics. Although Lagrange's innovation was accepted very slowly, at present the work constitutes the main formulation not only of theoretical mechanics but also theoretical physics as a whole. Moreover, it is mainly through the Lagrangian formulation that theoretical physics gives a correspondence between modern theories and classical mechanics. The appreciation of the nature of his innovation has changed in the course of the centuries. The original purpose was declared in the first page of the Preface: "No figures will be found in this work", just Lagrange's program to subordinate geometry to calculus; indeed: "The method I present requires neither constructions nor geometrical or mechanical arguments, but solely algebraic operations..." [we rather say, in modern terms, operations belonging to infinitesimal analysis], in such a way as to achieve the ideal of "... a regular and uniform procedure. Those who appreciate mathematical analysis will see with pleasure mechanics becoming a new branch of it and hence, will recognise that I have enlarged its domain." Then, in the 19th century this revolutionary program was rather recognised as a theoretical scheme which, making intensive use of calculus, is capable of including any branch of theoretical physics whatsoever. In the 20th century Noether's theorem on invariants changed Lagrange's original purpose; this theorem allows one to consider the Lagrangian formulation as the best framework for computing symmetries through group theory (ironically, this modern theory had inherited that geometrical tradition that Lagrange considered out-of-date). If not for other reasons, in the development of mathematical physics the Lagrangian formulation played a central role in the changing relationships among mechanics, geometry (then group theory) and calculus.

Unfortunately, Lagrange's works have not been so much analysed by historians in past times, maybe also because its archaic French language obstructed a full comprehension of the text. The mathematical language too creates difficulties for historians. Oddly enough, in mechanics Lagrange put aside his innovation - presented in Fonctions analytiques - of basing calculus upon Taylor expansions, viewed as generalisations of the binomial theorem. Furthermore, both in calculus and his new calculus of variations, Lagrange made use of infinitesimals in a self-confident way, although his notation and his techniques appear to be cumbersome to the reader. His hope to obtain universality led him to apply his equations to "all problems of mechanics" - friction too (Part II, Section VII) - and to consider the case of conservative forces as "properly the case in nature" (pp. 55, 62 in the third French edition; in [R Dugas, Histoire de la mécanique, 1950] "properly" is incorrectly changed to "probably"); an appraisal then reiterated as "ce qui est proprement le cas de la nature" (pp. 289-290 in the French edition; incorrectly translated as "which is ... properly a property of its nature", p. 229).

Moreover, historians do not agree on the role played by Lagrange in the history of mechanics. Contrarily to the traditional appraisal of Lagrange's work as the top in the development of theoretical mechanics, C A Truesdell, III [An idiot's fugitive essays on science, 1984] depreciated Lagrange's work as a mere extension of Euler's work, even in what concerns the invention of the calculus of variations. A related question - not mentioned in the two introductions to this translation - concerns the origins of Lagrange's Mécanique analitique.

Over all, it is Lagrange's theoretical point that is not apparent, owing to the following three problems: What does T-V mean in either physics or mathematics, with the total energy being T+V? To what extent does his new formalism generalise the principle of virtual velocities (which some years previously was declared by him as a new possible basis for the whole theoretical physics)? To what extent does this same principle not depend on Newtonian mechanics? Lagrange claimed to have solved the latter question; by exploiting an old idea, he offered a theorem on pulleys (pp. 24-25); which is clearly inconsistent, owing to the contemporary assumption of null mass of pulleys. One further question is to throw more light upon the foundations of Lagrange's calculus according to the present pluralistic view on the several kinds of mathematics, in particular constructive calculus, rigorous calculus, nonstandard analysis, and even reverse mathematics.

The editors translated the third edition (1853) of the original book (1788). They added 245 short footnotes (in 17 pages; "T. N." is an acronym for "Translator's Note"), in part quoting some notes by the editor of this third edition, J Bertrand. These notes give indispensable historical references, make more comprehensible Lagrange's formalism and justify difficult mathematical proofs. Incorrectly, note 20 of Part II identifies the notion of an inelastic body with the dismissed notion of a "hard" body, defined as a body whose shape does not change under any impact whatsoever. Both a subject index and an author index, which could be very helpful in exploring this long book, are lacking.

1.5. Review by: Massimo Galuzzi.
Isis 89 (1) (1998), 140-141.

The first edition of Joseph Louis Lagrange's Mécanique analytique, one of his masterpieces, was published in 1788, one year after Lagrange had left Berlin for Paris. Only many years later did a second edition appear: its first volume came out in 1811, the second reached the printer after Lagrange had died. With the help of Prony, Garnier, Binet, and Lacroix, the work was completed in 1815. It is the text of this second edition - rather than that of the third edition of 1853-1855, by Joseph Bertrand, the one reproduced in Lagrange's Oeuvres - that Auguste Boissonnade and Victor N Vagliente have chosen to use for the first English version of the Mécanique analytique.

It may seem strange that a book of such extraordinary scientific relevance has never been translated into English before, especially when one considers that it has been translated into German, Portuguese, and Russian. But at the time Lagrange's text appeared, French had a role in scientific communication comparable to that of English today. Even if many authors still wrote in Latin (and there were also authors who wrote in their native languages), every scientist understood French perfectly. Hence there was really no need for a translation. But is such a translation necessary even now? Isn't such a venerable book a document in the history of science, a field in which every scholar is supposed to read French well enough to understand (at least) Lagrange's language?

Besides the obvious relevance of such a text for the history and philosophy of science, many aspects of Lagrange's mathematics, and of eighteenth-century mathematics more generally, are closely linked to modern research. The first complete English translation of Leonhard Euler's Introductio in analysin infinitorum, strongly supported by André Weil, appeared a few years ago and rapidly became a source of inspiration for modern combinatorics. I have reason to believe that, given a careful rereading, Lagrange's texts may become similarly influential.

Mécanique analytique appeared at the end of a period in which the domain of analysis was greatly extended and geometrical intuition and diagrammatic aids were progressively eliminated from the domain of mathematics. Lagrange's statements in the preface of the first edition, reproduced unchanged in the second edition, are fairly well known, but as they constitute a sort of manifesto, it is appropriate to look at them once again: "No figures will be found in this work. The methods I present require neither constructions nor geometrical or mechanical arguments, but solely algebraic operations subject to a regular and uniform procedure." Is such a tenet still meaningful? Modern mathematics is firmly bound to the computer and to computer algebra. The possibility of visualising figures, shapes, and movements on the computer screen, so generously offered by mathematical software, appears to be a vital help for many mathematicians. But isn't this possibility the product of "logical operations" subjected to "regular and uniform procedures"? In some sense, computer algebra seems a metaphor for Lagrange's ideas.

The neoclassical sense of order (which pervades every page of Lagrange's book), his austere and blunt style, is only one aspect of his work. The invariant form of the differential equations of motion

         $\Large \frac{\partial T}{\partial q_i} - \frac{d}{dt}\frac{\partial T}{\partial \dot{q_i}}\normalsize = {\partial V}{\partial q_i}$

for a system with $n$ degrees of freedom and generalised coordinates $q_{i}$ - Lagrange's central technical achievement is the basis for an astounding number of analyses of physical situations that juxtapose Lagrange's severe mathematics with our real world, with all its shapes, sounds, and colours.

Although every historian and philosopher of science is a potential reader of this book, will it appeal to other readers as well? Like every enduring classic of exact science, the Mécanique analytique is certainly hard reading, but, as Craig Fraser says in his preface, "Lagrange's book remains valuable today as an exposition of subjects of ongoing utility to engineering physics and applied mathematics."

As for the translation itself: Boissonnade and Vagliente have rendered Lagrange's style with clarity and precision. No difficulties are left to the reader other than the mathematical ones, which no translation can eliminate. A few of the footnotes added by Bertrand to the third edition have been retained, but most of his footnotes have been omitted, "since the text should be clear to the modern reader without them." In their introduction, which takes advantage of all the recent scholarly work, Boissonnade and Vagliente describe the main events of Lagrange's career and place the Mécanique analytique in its proper historical and mathematical context - a great help for the modern reader. They also offer a clear-cut judgment on Lagrange's Historiques. "These summaries were not intended in any sense to serve as histories of the subject, although some investigators have viewed them in this fashion. They simply fulfilled the Enlightenment ideal that a system be demonstrably derived from indubitably understood and known premises." As Boissonnade and Vagliente frankly admit, not all investigators share this opinion. But in fact the long historical passages that Lagrange sometimes adds to his major works reveal many facets of his topics, and even if they are hardly "histories of the subject," they are nevertheless closely connected to its history and are thus rightfully part of this book.