This is the first English translation of Thomas Harriot's seminal Artis
Analyticae Praxis, 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. 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.