Carl Gustav Jacob Jacobi (1804-51) was one of the nineteenth century's greatest mathematicians, as attested by the diversity of mathematical objects named after him. His early work on number theory had already attracted the attention of Carl Friedrich Gauss, but his reputation was made by his work on elliptic functions. Elliptic integrals had been studied for a long time, but in 1827 Jacobi and Niels Henrik Abel realised independently that the correct way to view them was by means of their inverse functions - what we now call the elliptic functions. The next few years witnessed a flowering of the subject as the two mathematicians pushed ahead. Adrien-Marie Legendre, an expert on the old theory, wrote: 'I congratulate myself that I have lived long enough to witness these magnanimous conflicts between two equally strong young athletes'. This Latin work, first published in 1829, is Jacobi's pioneering account of the new theory.