Based on lectures delivered at the AMS meeting in 1901, this book describes the progress in calculus of variations made in the last 30 years of the nineteenth century. Among other topics, the author describes the landmark results of Weierstrass on sufficient conditions for the extremum of a functional in terms of the second variation. Also discussed are Kneser's sufficient conditions, Weierstrass' theory of the isoperimetric problem, and Hilbert's theorem on the existence of an extremum of an integral. Although the original book was written nearly 100 years ago, it remains very useful in learning about classical calculus of variations.