The Calculus of Variations from Euler to Lagrange:As a Result of Formalized Improvement

The calculus of variations was established as a distinct branch of analysis with the publication in 1744 of Euler's Methodus inveniendi curvas lineas maximi minimive proprietate gaudentes sive solution problematis isoperimetrici latissimo sensu accepti. The invention in 1755 of the δ -algorithm of Lagrange ushered in a new epoch in the calculus of variations. From Euler to Lagrange the subject had gone through great innovations. Based on some original literature and the detailed comparison and analysis between methods of Euler and Lagrange,the background and genesis and influence of Lagrange's algorithm are considered and explored in this paper. Finally,the paper concludes that Lagrange's improvement in the calculus of variations is essentially the result of formalization,and that the success of the formal improvement produced great effects on the development of the 18th calculus,and to some extent reinforced the trust of mathematicians in mathematical symbol.