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1840

Cauchy publishes the first volume of the four volume work

1841

Gauss publishes a treatise on optics in which he gives a formulae for calculating the position and size of the image formed by a lens with a given focal length.

1841

Jacobi writes a long memoir *De determinantibus functionalibus* devoted to the functional determinant now called the Jacobian.

1841

Quetelet establishes the Belgium Central Statistical Bureau.

1842

Hesse introduces the "Hessian determinant" in a paper which investigates cubic and quadratic curves.

1842

Stokes begins his research on fluids and publishes *On the steady motion of incompressible fluids*.

1843

Hamilton discovers quaternions, which generalise complex numbers to four dimensions.

1843

Liouville announces to the Académie des Sciences in Paris that he had found deep results in Galois's unpublished work and promises to publish Galois's papers together with his own commentary.

1843

Kummer invents "ideal complex numbers" in his study of unique factorisation. This leads to the development of ring theory.

1843

Cayley is the first person to investigate "geometry of *n* dimensions" which occurs in the title of his paper of that year. He uses determinants as the major tool.

1844

Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.

1844

Grassmann publishes *Die lineale Ausdehnundslehre, ein neuer Zweig der Mathematik* in which he develops the idea of an algebra in which the symbols representing geometric entities such as points, lines and planes, are manipulated using specific rules.

1845

Cayley publishes *Theory of Linear Transformations* in which he examines the composition of linear transformations.

1845

While examining permutation groups Cauchy proves a fundamental theorem of group theory which became known as "Cauchy's theorem". (See this History Topic.)

1846

Liouville publishes Galois' papers on the solution of algebraic equations in *Liouville's Journal*.

1846

Maxwell writes his first paper at the age of 14: *On the description of oval curves, and those having a plurality of foci*.

1847

Boole publishes *The Mathematical Analysis of Logic*, in which he shows that the rules of logic can be treated mathematically rather than metaphysically. Boole's work lays the foundation of computer logic.

1847

De Morgan proposes two laws of set theory that are now known as "de Morgan's laws".

1847

Von Staudt publishes *Geometrie der Lage*. It is the first work to completely free projective geometry from any metrical basis.

1848

Thomson (Lord Kelvin) proposes the absolute temperature scale now named after him.

1849

Hermite applies Cauchy's residue techniques to doubly periodic functions.

1850

Chebyshev publishes *On Primary Numbers* in which he proves new results in the theory of prime numbers. He proves Bertrand's conjecture there is always at least one prime between *n* and 2*n* for *n* > 1.

1850

In his paper *On a New Class of Theorems* Sylvester first uses the word "matrix". (See this History Topic.)

List of mathematicians alive in 1840.

List of mathematicians alive in 1850.

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