Book: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes

Title and context
Leonhard Euler's 1744 treatise "Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes" lays out a systematic method for finding curves that maximize or minimize functionals, inaugurating what is now called the calculus of variations. The pamphlet appears in the mid-18th century mathematical revival of analytic methods, where problems about optimal shapes and paths demanded a general technique rather than case-by-case ingenuity. Euler formulates a clear variational approach that turns many classical extremal problems into a unified differential equation.

Core idea
Euler begins by considering quantities defined as definite integrals that depend on an unknown curve, and studies how small changes in the curve alter the integral. By examining the first variation and requiring it to vanish for an extremum, he obtains a necessary condition connecting the integrand and its derivatives. This condition becomes a differential equation that any extremal curve must satisfy. The approach replaces ad hoc geometric reasoning with a systematic analytic procedure amenable to calculation.

Derivation of the Euler-Lagrange equation
The method isolates a general integrand F(x, y, y') and computes the change when the curve y(x) is perturbed by a small function that vanishes at the endpoints. Integrating by parts and discarding boundary terms under fixed-endpoint conditions, Euler shows that the vanishing of the first variation implies the celebrated differential relation d/dx (∂F/∂y') - ∂F/∂y = 0. He also identifies simple conserved quantities that arise when the integrand lacks explicit dependence on one of the variables, leading to first integrals that simplify integration in special cases.

Examples and problem types
Euler demonstrates the method on several classical problems, bringing clarity and calculational power to questions such as shortest paths, curves of quickest descent and other extremal curve problems. He treats problems with fixed endpoints and hints at conditions when endpoints are free, showing how boundary conditions enter the resulting differential equations. The treatise addresses isoperimetric constraints by introducing Lagrange multipliers-like constants, foreshadowing later formalism for constrained variational problems.

Techniques and innovations
Beyond the principal differential condition, Euler emphasizes useful manipulations: using first integrals when variables are absent from the integrand, recognizing invariants, and reducing higher-order complexities. He frames the variational argument in analytically precise terms, using approximating functions and limiting processes to justify formal manipulations. This rigor and the clear algebraic expression of variational conditions mark a significant step toward modern analytic methods in mechanics and geometry.

Impact and legacy
The treatise established the Euler-Lagrange equation as the central tool of the calculus of variations and laid groundwork for later formalism by Lagrange and others. Its influence extends into classical mechanics through the principle of least action, into differential geometry via geodesics and minimal surfaces, and into modern physics and optimization theory. Euler's systematic reduction of varied extremal problems to a common differential condition transformed how mathematicians and scientists formulate and solve problems of optimization and dynamics.