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This volume showcases the influential work of the Swiss mathematician Jacob Bernoulli in differential geometry, a field where he made significant contributions. Between 1690 and 1700, Bernoulli published twelve treatises in Acta Eruditorum, focusing on infinitesimal methods for solving geometrical problems. Included are preparatory notes from his scientific diary, Meditationes, featuring twenty-nine texts published for the first time. The work explores various curves, such as isochrones, parabolic spirals, loxodromes, cycloids, tractrices, and the logarithmic spiral, known as Bernoulli's spira mirabilis, which is also on his tombstone. The analysis of these curves through differential equations and geometrical constructions, alongside their rectification, quadrature, evolutes, and caustics, presented Bernoulli and his contemporaries with numerous complex challenges, many applicable to mechanical and optical fields. André Weil, a prominent figure in 20th-century mathematics and a founding member of the Bourbaki group, provides an introduction that situates Bernoulli's work within a historical continuum from Descartes and Huygens to Newton and Leibniz, culminating in Euler's codification of the subject. Martin Mattmüller, editor of the Bernoulli Edition at Basel, offers commentaries that contextualize Bernoulli's contributions within the late 17th-century landscape of differential geometry.