Editing Joseph-Louis Lagrange (section)

=== Differential calculus and calculus of variations ===
[[File:Langrange portrait.jpg|thumb|right|Joseph-Louis Lagrange]]
Lagrange's lectures on the [[differential calculus]] at École Polytechnique form the basis of his treatise ''Théorie des fonctions analytiques'', which was published in 1797. This work is the extension of an idea contained in a paper he had sent to the Berlin papers in 1772, and its object is to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series, relying in particular on the principle of the [[generality of algebra]].

A somewhat similar method had been previously used by [[John Landen]] in the ''Residual Analysis'', published in London in 1758. Lagrange believed that he could thus get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of [[Taylor's theorem]], the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics.

Another treatise on the same lines was his ''Leçons sur le calcul des fonctions'', issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of [[Lagrange multipliers]], in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as the starting point for the researches of [[Augustin Louis Cauchy|Cauchy]], [[Carl Gustav Jakob Jacobi|Jacobi]], and [[Karl Weierstrass|Weierstrass]].<gallery>
File:Lagrange-6.jpg|1813 copy of "Theorie des fonctions analytiques"
File:Lagrange-7.jpg|Title page to "Theorie des fonctions analytiques"
File:Lagrange-9.jpg|Introduction to "Theorie des fonctions analytiques"
File:Lagrange-10.jpg|First page of "Theorie des fonctions analytiques"
</gallery>[[File:Lagrange-2.jpg|alt=Title page of volume I of Lagrange's "Mécanique Analytique" (1811)|thumb|Title page of volume I of Lagrange's "Mécanique Analytique" (1811)|231x231px]]