This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. In the 1680s, Leibniz had given the first public hints of his "new method", differential and integral calculus. Together with Huygens, Tschirnhaus and L'Hopital, Jacob Bernoulli and his younger brother Johann belonged to the small group of scientists who developed these cryptic ideas and made use of them in their research. In the intense, sometimes competitive intellectual exchange that followed in their publications and correspondence, Leibnizian infinitesimal calculus evolved into the analytical method that was to dominate mathematical thinking for two centuries. Between 1690 and 1700, Jacob Bernoulli published 12 treatises in the scientific journal "Acta Eruditorum" on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary "Meditationes", from which 29 texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix and the logarithmic spiral (Bernoulli's "spira mirabilis", which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of the evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many of them relevant for mechanical or optical applications. At the same time, questions of a more general nature that were to shape the development of calculus in the 18th century gradually came into focus: the local geometry of general planar curves, their curvature and singularities; elliptic integrals and their transformation properties; and the search for algorithms for integrating algebraic functions. Through mutual suggestions, challenges and critiques, the small group of the first champions of infinitesimal calculus developed the techniques that shape differential geometry to this day. The volume concludes with Bernoulli's previously unpublished work on the classification of planar cubic curves as presented by his students Jacob Hermann and Nicolaus I Bernoulli. Though it was surpassed by Newton's contemporary research, the background of Bernoulli's contribution is of no small interest.