Summary This paper investigates the fate of Thomas Harriot's algebra after his death in 1621 and, in particular, the largely unsuccessful efforts of seventeenth-century mathematicians to promote it. The little known surviving manuscripts of Nathaniel Torporley have been used to elucidate the roles of Torporley and Walter Warner in the preparation of the Praxis, and a partial translation of Torporley's important critique of the Praxis is offered here for the first time. The known whereabouts of Harriot's mathematical papers, both originals (...) and copies, during the seventeenth century and later are summarised. John Wallis's controversial 1685 account of Harriot's algebra is examined in detail and it is argued that John Pell's influence on Wallis was far more significant than has previously been realised. The paper ends with a reassessment of Harriot's underrated and important contribution to the development of modern algebra. (shrink)