translated and annotated by
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
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;
Treatise on the Resolution
of Numerical Equations of all degrees.
: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.
new method of approximating the roots of numerical equations.
CH.4: Application of the preceding methods to some
Ch.5 : On imaginary Roots. The four articles discuss here:
How to recognize an equation with imaginary roots ;
Where we are given in certain cases, the number of imaginary roots in a given
Where we apply the above rules to equations of the second, third, and fourth
How to find the imaginary roots of an equation.
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
Note 4: On the manner of
finding a limit smaller than the smallest difference between the roots of a
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
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
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.
: 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
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.