Joseph-Louis Lagrange (1736–1813) was born Giuseppe Lodovico Lagrangia in Turin, in what is now northern Italy. Lagrange contributed to several branches of mathematics, including the beginnings of abstract algebra. Below is an image of the title page from the 1808 edition of his *Traité de la Résolution des Equations Numérique de tous les degrés*.

Originally published in 1798, the 1808 “noevelle/new” edition contained revisions and more notes, making it the more widely used edition. The title can be translated as *A Treatise on the solution of numerical equations of all degrees*, indicating that the book deals with numerical solutions to equations. However, exact solutions to polynomial equations were also discussed, including a summary of Lagrange’s groundbreaking 1770 paper "Réflexions sur la résolution algébrique des équations" (published in the memoirs of the Berlin Academy of Sciences), which noted historian Israel Kleiner called one of the “four major sources in the evolution of group theory*.*” The first page of this summary is below.

The mathematical notation is interesting in this work. Below is a picture of page 2, which shows Lagrange representing general polynomial equations as a product of linear factors.

In the next picture, from Note 5 on Newton’s Method for approximating roots of polynomial equations, a general polynomial is shown in expanded form.

The pictures below are from Note 7 on the method of Alexis Fontaine for finding solutions of cubics, which shows notation for nonreal complex roots; Note 10 where the Greek capital sigma was used for summations; and from Note 12, depicting sequences of fractions starting with 1/0.

A complete digital scan of the 1808 edition of Lagrange’s *Traité de la Résolution des Equations Numérique* is available in the Linda Hall Library Digital Collections. The call number is QA218.L33 1808.

Janet Barnett has created a project for students based on this primary source as part of the TRIUMPHS project.

*Images in this article are courtesy of the Linda Hall Library of Science, Engineering & Technology and used with permission. The images may be downloaded and used for the purposes of research, teaching, and private study, provided the Linda Hall Library of Science, Engineering & Technology is credited as the source. For other uses, check out the LHL **Image Rights and Reproductions** policy.*

Kleiner, Israel. “The Evolution of Group Theory: A Brief Survey.” *Mathematics Magazine* 59, no. 4 (1986): 195-215. doi:10.2307/2690312.

Wussing, Hans. *The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory*. Dover Publications, 2007.