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1820

Brianchon publishes

1821

Navier gives the well known "Navier-Stokes equations" for an incompressible fluid.

1821

Cauchy publishes *Cours d'analyse* (*A Course in Analysis*), which sets mathematical analysis on a formal footing for the first time. Designed for students at the Ecole Polytechnique it was concerned with developing the basic theorems of the calculus as rigorously as possible.

1822

Poncelet develops the principles of projective geometry in *Traité des propriétés projectives des figures* (*Treatise on the Projective Properties of Figures*). This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity.

1822

Fourier's prize winning essay of 1811 is published as *Théorie analytique de la chaleur* (*Analytical Theory of Heat*). It makes widely available the techniques of Fourier analysis, which will have widespread applications in mathematics and throughout science.

1822

Feuerbach publishes his discoveries on the nine point circle of a triangle.

1823

János Bolyai completes preparation of a treatise on a complete system of non-Euclidean geometry. When Bolyai discovers that Gauss had anticipated much of his work, but not published anything, he delays publication. (See this History Topic.)

1823

Babbage begins construction of a large "difference engine" which is able to calculate logarithms and trigonometric functions. He was using the experience gained from his small "difference engine" which he constructed between 1819 and 1822.

1824

Sadi Carnot publishes *Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance* (*Thoughts on the Motive Power of Fire, and on Machines Suitable for Developing that Power*). A book on steam engines, it will be of fundamental importance in thermodynamics. The "Carnot cycle" which forms the basis of the second law of thermodynamics also appears in the book.

1824

Abel proves that polynomial equations of degree greater than four cannot be solved by radicals. He publishes it at his own expense as a six page pamphlet.

1824

Bessel develops "Bessel functions" further while undertaking a study of planetary perturbations.

1824

Steiner develops synthetic geometry. He publishes his theories on the topic in 1832.

1825

Gompertz gives "Gompertz's Law of Mortality" which shows that the mortality rate increases in a geometric progression so when death rates are plotted on a logarithmic scale, a straight line known as the "Gompertz function" is obtained.

1826

Ampère publishes *Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience*. It contains a mathematical derivation of the electrodynamic force law and describes four experiments. It lays the foundation for electromagnetic theory.

1826

Crelle begins publication of his *Journal für die reine und angewandte Mathematik* which will become known as *Crelle's Journal*. The first volume contains several papers by Abel.

1826

Poncelet's work on the pole and polar lines associated with conics lead him to discover the principle of duality. Gergonne, who introduced the word polar, discovers independently the principle of duality.

1827

Jacobi writes a letter to Legendre detailing his discoveries on elliptic functions. Abel was independently working on elliptic functions at this time.

1827

Möbius publishes *Der barycentrische Calkul* on analytical geometry. It becomes a classic and includes many of his results on projective and affine geometry. In it he introduces homogeneous coordinates and also discusses geometric transformations, in particular projective transformations.

1827

Feuerbach writes a paper which, independently of Möbius, introduces homogeneous coordinates.

1828

Gauss introduces differential geometry and publishes *Disquisitiones generales circa superficies*. This paper arises from his geodesic interests, but it contains such geometrical ideas as "Gaussian curvature". The paper also includes Gauss's famous *theorema egregrium*.

1828

Green publishes *Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnets*, in which he applies mathematics to the properties of electric and magnetic fields. He introduces the term potential, develops properties of the potential function and applies them to electricity and magnetism. The formula connecting surface and volume integrals, now known as "Green's theorem", appears for the first time in the work, as does the "Green's function" which would be extensively used in the solution of partial differential equations.

1828

Abel begins a study of doubly periodic elliptic functions.

1828

Plücker publishes *Analytisch-geometrische* which develops the "Plücker abridged notation". He, independently of Möbius and Feuerbach one year earlier, discovers homogeneous coordinates.

1829

Galois submits his first work on the algebraic solution of equations to the Académie des Sciences in Paris.

1829

Lobachevsky develops non-euclidean geometry, in particular hyperbolic geometry, and his first account of the subject is published in the Kazan Messenger. When it was submitted for publication in the St Petersburg Academy of Sciences Ostrogradski rejects it. (See this History Topic.)

About 1830

Babbage creates the first accurate actuarial tables for use in insurance calculations.

1830

Poisson introduces "Poisson's ratio" in elasticity which involves stresses and strains on materials.

1830

Peacock publishes his *Treatise on Algebra* which attempts to give algebra a logical treatment comparable to Euclid's *Elements*.

List of mathematicians alive in 1820.

List of mathematicians alive in 1830.

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