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.