Mathematical Works of

translated and annotated by
Ian Bruce


Notes will be added here gradually as time permits and the necessity arises. As indicated in the index of contents, rather than continuing with Book I, various later works of Lagrange are to be considered first, starting with his Treatise on the Resolution of Numerical Equations of all degrees.

Book 1

Article 1  : A general method of for determining the max and min of functions of several variables, with examples.

   Article 2  : On the integration of a differential equation by finite differences, which contains the theory of recurring differences.

   Article 3 :Researches into the nature and propagation of sound. Introduction, Ch. 1 & Ch. 2.

Ch.1 is a rather scathing attack on Newton, who was not, I believe, initially discussing sound waves but atmospheric waves. The sound waves are relegated to a Scholium. Ch. 2 is a discussion about waves in strings, referring to d'Alembert's work, as well as that of Euler and Daniel Bernoulli;

Book 8

Treatise on the Resolution of Numerical Equations of all degrees.

   Introductions :Two introduction are given, that by Poinsot setting the scene, as it were, and the other by Lagrange.

  Ch.1 : Method for finding, in some given numerical equation, the value of the nearest whole number to each of its real roots.

Ch.2,3&4 :  CH.2: Concerning the manner of having equal and imaginary roots of equations.

                       CH.3: A new method of approximating the roots of numerical equations.

                       CH.4: Application of the preceding methods to some examples.

Ch.5 :  On imaginary Roots. The four articles discuss here:

1. How to recognize an equation with imaginary roots ;

2. Where we are given in certain cases, the number of imaginary roots in a given equation;

3. Where we apply the above rules to equations of the second, third, and fourth degrees;

4. How to find the imaginary roots of an equation.


Chap. VI. On the manner of approaching the numerical value of the roots of equations by continued fractions:

Ch.6 art1&2 :  Art. I. On periodic continued fractions, Art. II. Where we give a very simple way of reducing the roots of equations of the second degree into continued fractions.

Ch.6 art3&4 :  Art. III. Generalisation of the theory of continued fractions, Art. IV. Where we propose different means for simplifying the calculation of roots by continued fractions.

notes1-5 :  Note 1 : On the demonstration of Theorem I.

                  Note 2 : On the demonstration of Theorem II.

                  Note 3 : On the equation which gives the differences between the roots of a given equation, taken in pairs.

                  Note 4: On the manner of finding a limit smaller than the smallest difference between the roots of a given


                  Note 5: On the method of approximation given by Newton.

notes 6-8 :  Note 6 : On the method of approximation expressed by recurring series.

                  Note 7 : On the method of Fontaine, for the resolution of equations.

                  Note 8 : On the bounds of roots of equations, and the nature of the reality of all their roots.

notes 9 :  Note 9 : On imaginary roots.

notes 10 :  Note 10 : On the decomposition of a polynomial of any degree into real factors.

notes 11 :  Note 11 : On formulas for the approximation of the roots of equations.

notes 12 & 13 :  Note 12 : On the way of transforming any equation, such that the terms which contain the unknown

                          may have the same sign, and that the wholly known term may have a different sign.


                            Note 13 : On the resolution of algebraic equations.


 notes 14etc :      Note 14 :Where we give the general resolution of equations in two terms.



E30 :      A Conjecture concerning the form of the roots of equations of any order. In this early commentary, Euler

provides two clever methods for deriving the roots of quadratic, cubic, and biquadratic equations. Finally, he  hints that the method may not be possible to be extended to higher orders, or that perhaps another method will be found        


E282 :      Concerning the resolution of equations of any order. In this commentary, Euler sets the stage as it were, for

                 future research on solving equations of higher degrees, by providing a method for solving equations of

                 degrees 2, 3, and 4, as well as some forms of the 5th degree in which some coefficients are absent.


vandermonde :      The Resolution of Equations. In this long paper, Vandermonde sets out a rather elaborate scheme whereby he hopes to resolve equations of any degree; he succeeds in establishing a common link for the quadratic, cubic, and fourth power, but the mechanism fails on the fifth and higher powers, and produces resolvents of higher rather than lesser powers. However, this work helped to establish a new way of looking at such equations, and was instrumental in establishing some aspects of what became group theory in later years.

Ian Bruce. May 2018 latest revision. Copyright : See notes at bottom of index page;  I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for personal or educational use only. If you have any useful information regarding these translation, I would appreciate hearing from you. IB.