# Mathematical Works of LAGRANGE

### 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:

### 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.