The Artis analyticae praxis was published in 1631 in Latin. Until relatively - cently, an English edition would not have been considered necessary since most of the people who might have been expected to be interested in the text would have been able to read it in Latin. That is no longer the case and it is ?tting that an English edition should be published. It is also the case that a considerable proportion of the readership of the present volume will not be professional mathematicians, so we have tried to produce a translation that makes the mathematical content accessible to the modern reader. This is not because the algebra is intrinsically dif?cult but because it is not the kind of mathematics which is a part of today's secondary school curriculum. A further problem lies in the fact that the book in the form in which it was - blished in 1631 may very well not conform to Harriot's intentions for the publi- tion of his mathematical manuscript papers. This is why the book is accompanied by a Commentary which attempts to compare it with the appropriate passages in the surviving manuscript papers. The present work is a translation of the original text and not intended as a facsimile. The original has well over 300 errors (there may very well be more) and we have listed these at the end. In the interests of mathematical accessibility we have tried to produce a mathematically "clean" copy.