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
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.
CH.3: A new method of approximating the roots of
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 ;
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 Therorem I.
Note 2 : On the demonstration
of Therorem 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
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
Ian Bruce. March 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.