# Chronology

### 1840 - 1850

#### 1840

- Cauchy publishes the first volume of the four volume work
*Exercises d'analyse et de physique mathematique*.

#### 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.
- Jacobi writes a long memoir
*De determinantibus functionalibus*devoted to the functional determinant now called the Jacobian. - Quetelet establishes the Belgium Central Statistical Bureau.

#### 1842

- Hesse introduces the "Hessian determinant" in a paper which investigates cubic and quadratic curves.
- 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.
- Liouville announces to the AcadÃ©mie des Sciences in Paris that he had found deep results in Galois's unpublished work and promises to publish Galois's papers together with his own commentary.
- Kummer invents "ideal complex numbers" in his study of unique factorisation. This leads to the development of ring theory.
- 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.
- 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. - 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*. - 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. - De Morgan proposes two laws of set theory that are now known as "de Morgan's laws".
- 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.