- Delaunay publishes the first volume of La Théorie du mouvement de la lune which is the result of 20 years work. Delaunay solves the three-body problem by giving the longitude, latitude and parallax of the Moon as infinite series.
- Weierstrass discovers a continuous curve that is not differentiable any point.
- Maxwell proposes that light is an electromagnetic phenomenon.
- Jevons reads General Mathematical Theory of Political Economy to the British Association.
- Listing publishes Der Census raumlicher Complexe oder Verallgemeinerung des Euler'schen Satzes von den Polyedern which discusses extensions of "Euler's formula".
- Weierstrass gives a proof in his lecture course that the complex numbers are the only commutative algebraic extension of the real numbers.
- Bertrand publishes Treatise on Differential and Integral Calculus.
- London Mathematical Society founded. (See this Article.)
- Benjamin Peirce presents his work on Linear Associative Algebras to the American Academy. It classifies all complex associative algebras of dimension less than seven using the, now familiar, tools of idempotent and nilpotent elements.
- Plücker makes further advances in geometry when he defines a four dimensional space in which straight lines rather than points are the basic elements.
- Hamilton's Elements of Quaternions is unfinished on his death but the 800 page work which took seven years to write is published posthumously by his son.
- Moscow Mathematical Society is founded.
- Lueroth discovers the "Lueroth quartic".