Thomas Harriot's "Artis analyticae praxis" is an essential work in the history of algebra. This is the first English translation of his work, first published in Latin in 1631. It has recently become clear that Harriot's editor substantially rearranged the work, and omitted sections beyond his comprehension. Basing their work on manuscripts in the British Library, Pentworth House, and Lambeth Palace, the commentary contains some of Harriot's most novel and advanced mathematics, very little of which has been published in the past. Commentary included with this translation relates to corresponding pages in the manuscript papers, enabling exploration of Harriot's novel and advanced mathematics. This publication provides the basis for a reassessment of the development of algebra.