Portrait of L Euler

Elements of Algebra

Leonard Euler

Leonhard Euler (1707--1783) is one of the most influential and prolific mathematicians of all time. His Elements of Algebra is one of the first books to set out algebra in the modern form we would recognize today. However, it is sufficiently different from most modern approaches to the subject to be interesting for contemporary readers. Indeed, the choices made for setting out the curriculum, and the details of the techniques Euler employs, may surprise even expert readers. It is also the only mathematical work of Euler which is genuinely accessible to all. The work opens with a discussion of the nature of numbers and the signs + and -, before systematically developing algebra to a point at which polynomial equations of the fourth degree can be solved, first by an exact formula and then approximately. Euler's style is unhurried, and yet rarely seems long winded.

This volume is an edited reprint of Part I of J. Hewlett's 1822 English translation and has been made available to celebrate the three hundredth anniversary of his birth. You can order online direct from Tarquin, from Amazon and a hardback edition is also available here.

Editor
Christopher James Sangwin
School of Mathematics
University of Birmingham
United Kingdom
B15 2TT

ISBN 978-1-89961-873-6
© 2006 C J Sangwin
All Rights Reserved
First published 2006, Tarquin Publications
Cover image
 

Part I: Containing the Analysis of Determinate Quantities

Preface

This book is an edited reprint of Part I of J. Hewlett's 1822 English translation of Leonhard Euler's Elements of Algebra.

Leonhard Euler (1707-1783) was one of the most prolific mathematical authors of all time. Indeed, volumes of his vast Opera Omnia continue to appear since the inception of the project in 1911. And yet, much of the work of this great mathematician is either not available in English, is too obscure to be interesting to the general mathematical reader, or is simply difficult to obtain. Two exceptions to this are the recent translations by J. D. Blanton of Introduction to Analysis of the Infinite and Foundations of Differential Calculus. Although, even here, some advanced mathematics is needed as background to really appreciate their significance. On the other hand, almost anyone could read, appreciate and enjoy Elements of Algebra. The purpose of this volume is to make an original work of one of the great mathematicians available to the wider public and, I sincerely hope, inspire some to look at his other works.

Euler's Elements of Algebra is an elementary algebra text book. He wrote Elements towards the end of his career, when he was blind. From the editor of the German edition we learn that he dictated the book to a young servant, and through this taught him some mathematics. In this way educating a new amanuensis.

This work appears in Gustav Eneström's list of Euler's works as volumes 387 and 388, and has a complex bibliographic history. Euler's original was written about 1765 in German. However, the first published edition, by the Royal Academy of Sciences in St Petersburg, appeared in 1770 as a Russian translation. A German edition followed swiftly, as did further translations into other European languages. The German edition was translated into French by Johann III Bernoulli. At this stage approximately one hundred pages of Additions were added by Joseph Louis Lagrange (= La Grange). It is via the French edition that this English translation derives. Originally this translation began as a student project of Francis Horner. He died before it was completed and it was left to John Hewlett to complete the translation and editing and as a result the translation is attributed to him. The third edition of this English translation was published by Longman in 1822, although further reprints and editions followed. The 1822 volume consists of the following parts:

  1. English Translator's note, by J. Hewlett (6 pages)
  2. Memoir of the life of Euler, by F. Horner (15 pages)
  3. German editor's note, (1 page)
  4. French translator's note, by M. Bernoulli (2 pages)
  5. Elements of Algebra, by L. Euler
  6. Additions, by La Grange (120 pages)
This volume consists only of this short Preface and Part I of Euler's actual text. Bibliographic historians will easily be able to consult the original publications in research libraries whereas to reprint all the parts here would only constitute a distraction. In this spirit, many of the footnotes which had been added at various stages have also been omitted from this volume where the length seemed to exceed the value.

Part II, Containing the Analysis of Indeterminate Quantities, consists of a further 150 pages of work on simultaneous equations and more advanced algebraic techniques for algebraic expressions in more than one variable. These also include methods, eg Chapter 8, for determining when forms such as {a+bx+cx2+dx3} is rational. The Additions by La Grange are another 120 pages of work on Diophantine equations. This material may be the subject of a future volume should demand make this worthwhile.

I have seen fit to update the spelling, so that for example shew in the original becomes show in the present. The word series has been changed to sequence to conform with more modern usage. In the 1822 English edition the material was grouped into lengthy Sections, containing large numbers of small Chapters. In accordance with modern usage this has also been changed to Chapters containing Sections. The numbering of paragraphs has been preserved, although the original 1771 German edition divided our Part I into two Parts, each of which begins numbering the paragraphs. Hence, after § 562 the numbers do not correspond with the German original. The purpose of these changes was to produce a volume which is sensitive to the original material and the modern reader. While it was tempting to alter the sentence structure, this would have destroyed the charm of the early nineteenth century English and constituted yet another layer of "translation".

I would like to thank the Special Collections Department of the Library at the University of Birmingham for granting permission to reproduce the figures, and to Chantal Jackson for her help in preparing these figures and the cover design.

There is no need to comment on the substance of Euler's work here, since the text speaks for itself. While there are many parts which are interesting from a historical point of view, I have added only one footnote at a point where Euler's explanation does not conform with contemporary mathematical ideas in a way which might confuse a student. The original 1771 German edition has been consulted to confirm that this paragraph is not the result of a mistranslation, as have other points of detail where there was some doubt. I sincerely hope that the reader enjoys this book as much as I have while editing it, and in the process reflects on the curriculum Euler sets out here and that we use today. While I have quietly corrected typographical mistakes from the 1822 edition, I sincerely hope that few new errors have been introduced.


Christopher James Sangwin
chris@sangwin.com
Birmingham, England. 2006