- Poincaré publishes important results on automorphic functions.
- Venn introduces his "Venn diagrams" which become a useful tools in set theory.
- Gibbs develops vector analysis in a pamphlet written for the use of his own students. The methods will be important in Maxwell's mathematical analysis of electromagnetic waves.
- Lindemann proves that π is transcendental. This proves that it is impossible to construct a square with the same area as a given circle using a ruler and compass. The classic mathematical problem of squaring the circle dates back to ancient Greece and had proved a driving force for mathematical ideas through many centuries.
- Mittag-Leffler founds the journal Acta Mathematica.
- Reynolds publishes An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. The "Reynolds number" (as it is now called) used in modelling fluid flow appears in this work.
- Poincaré publishes a paper which initiates the study of the theory of analytic functions of several complex variables.
- The Edinburgh Mathematical Society is founded. (See THIS LINK.)
- Volterra begins his study of integral equations.
- Frege publishes The Foundations of Arithmetic.
- Hölder discovers the "Hölder inequality".
- Mittag-Leffler publishes Sur la représentation analytique fes fonctions monogènes uniformes d'une variable indépendante which gives his theorem on the construction of a meromorphic function with prescribed poles and singular parts.
- Frobenius proves Sylow's theorems for abstract groups.
- Ricci-Curbastro begins work on the absolute differential calculus.
- Circolo Matematico di Palermo is founded.
- Weierstrass shows that a continuous function on a finite subinterval of the real line can be uniformly approximated arbitrarily closely by a polynomial.
- Edgeworth publishes Methods of Statistics which presents an exposition of the application and interpretation of significance tests for the comparison of means.
- Reynolds formulates a theory of lubrication
- Peano proves that if is continuous then the first order differential equation has a solution.
- Dedekind publishes Was sind und was sollen die Zahlen? (The Nature and Meaning of Numbers). He puts arithmetic on a rigorous foundation giving what were later known as the "Peano axioms".
- Galton introduces the notion of correlation.
- Engel and Lie publish the first of three volumes of Theorie der Transformationsgruppen (Theory of Transformation Groups) which is a major work on continuous groups of transformations.
- Peano publishes Arithmetices principia, nova methodo exposita (The Principles of Arithmetic) which gives the Peano axioms defining the natural numbers in terms of sets.
- FitzGerald suggests what is now called the FitzGerald-Lorentz contraction to explain the "Michelson-Morley experiment".