# Mathematician:Joseph Louis Lagrange

## Contents

## Mathematician

Italian-born French mathematician who made big advances in the fields of the calculus of variations and analytical mechanics.

Contributed to number theory and algebra.

Extended a lot of the fields established by Euler, and in turn laid down the groundwork for later explorations by Gauss and Abel.

Played a leading part in establishing the metric system of weights and measures.

He did the following:

- Author of
*Réflexions sur la Résolution Algébrique des Equations*($1770$), a complete restudy of all the known methods of solving the cubic and quartic equations. - Proposed a prime number as the universally adopted number base. Thus every systematic fraction would be reducible and represent the number in a unique way.
- Established some very general theorems on whether a number is prime from examining its digits.
- Tried in vain to prove Fermat's Last Theorem.
- One of the few exceptions of his time who was doubtful that a polynomial equation of degree greater than four was capable of a formal solution by means of radicals.
- Gave an insufficient proof of the Fundamental Theorem of Algebra.
- Demonstrated in $1794$ that $\pi^2$ is irrational.
- Proved Wilson's Theorem.
- Completed the solution of the partial differential equation which defines the form of a vibrating string.
- Between $1772$ and $1785$ he also addressed the partial differential equation of the first order.
- He established the general solution of the linear equation, and classified the various kinds of non-linear equation.

## Nationality

Italian-born, of mixed Italian and French parentage, living mainly in France and Prussia.

## History

- Born: 25 January 1736, Turin, Italy
- 1755: Appointed Professor at Royal Artillery School at Turin
- 1766: Moved to Berlin to take over position of Euler, who had moved to St. Petersburg
- 1786: Moved to Paris after death of Frederick the Great
- Died: 10 April 1813, Paris, France

## Theorems and Definitions

- Lagrange's Theorem (Number Theory)
- Lagrange's Formula
- Lagrange's Identity
- Lagrange Interpolation Formula
- Lagrange's Four Square Theorem
- Lagrange's Method of Multipliers
- Euler-Lagrange Equation (with Leonhard Paul Euler)
- Lagrangian
- Lagrange Basis Polynomial
- Lagrange Form of Remainder of Taylor Series

- Proved Wilson's Theorem, which as a result is sometimes referred by some sources as the Wilson-Lagrange Theorem.

- Lagrange's Theorem (Group Theory) was named after him, although he did not prove the general form. What he actually proved was that if a polynomial in $n$ variables has its variables permuted in all $n!$ ways, the number of different polynomials that are obtained is always a divisor of $n!$.

Results named for **Joseph Louis Lagrange** can be found here.

Definitions of concepts named for **Joseph Louis Lagrange** can be found here.

## Publications

- 1770:
*Réflexions sur la Résolution Algébrique des Equations*: a complete restudy of all the known methods of solving the cubic and quartic equations. - 1788:
*Mécanique Analytique* - 1797:
*Théorie des fonctions analytiques* - 1798:
*Résolution des équations numériques*: Includes a method of approximating to the real roots of an equation by means of continued fractions. - 1800:
*Leçons sur le calcul des fonctions*

## Critical View

*The "generalized coordinates" of our mechanics of today were conceived and installed by Lagrange, and this was an achievement of unmatchable magnitude.*

## Also known as

Some sources render his name as **Joseph-Louis Lagrange**.

He was born **Giuseppe Lodovico Lagrangia**, or **Giuseppe Ludovico de la Grange Tournier**.

He is also reported as **Giuseppe Luigi Lagrange**, and also **Giuseppe Luigi Lagrangia**.

Some prepend his title **Comte**.

## Sources

- John J. O'Connor and Edmund F. Robertson: "Joseph Louis Lagrange": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{X}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Introduction - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 40$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): A List of Mathematicians in Chronological Sequence - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Lagrange, Joseph Louis, Comte de**(1736-1813) - 1991: David Wells:
*Curious and Interesting Geometry*... (previous) ... (next): A Chronological List Of Mathematicians - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.22$: Lagrange ($\text {1736}$ – $\text {1813}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): A List of Mathematicians in Chronological Sequence - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Lagrange, Joseph Louis, Comte**(1736-1813) - 2004: Ian Stewart:
*Galois Theory*(3rd ed.) ... (previous) ... (next): Historical Introduction: Polynomial Equations - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Lagrange, Joseph Louis, Comte**(1736-1813) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Lagrange, Joseph-Louis**(1736-1813)