The present work is the first ever English translation of the original text of Thomas Harriot's Artis Analyticae Praxis, first published in 1631 in Latin. Thomas Harriot's Praxis is an essential work in the history of algebra. Even though Harriot's contemporary, Viete, was among the first to use literal symbols to stand for known and unknown quantities, it was Harriot who took the crucial step of creating an entirely symbolic algebra. This allowed reasoning to be reduced to a quasi-mechanical manipulation of symbols. Although Harriot's algebra was still limited in scope (he insisted, for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern. While Harriot's book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriot's achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond comprehension, such as negative and complex roots of equations. The commentary included with this translation relates the contents of the Praxis to the corresponding pages in his manuscript papers, which enables much of Harriot's most novel and advanced mathematics to be explored. This publication will become an important contribution to the history of mathematics, and it will provide the basis for a reassessment of the development of algebra.