- La Hire publishes Sectiones conicae which is a major work on conic sections.
- Leibniz uses the modern notation for an integral for the first time.
- Leibniz discovers the differentials of basic functions independently of Newton.
- Leibniz discovers the rules for differentiating products, quotients, and the function of a function.
- Giovanni Ceva publishes De lineis rectis containing "Ceva's theorem".
- Cocker's Arithmetic is published two years after Cocker's death. It would run to more than 100 editions over a period of about 100 years.
- Leibniz introduces binary arithmetic. It was not published until 1701.
- Cassini studies the "Cassinian curve" which is the locus of a point the product of whose distances from two fixed foci is constant. (See this Famous curve.)
- Tschirnhaus studies catacaustic curves, being the envelope of light rays emitted from a point source after reflection from a given curve.
- Seki Kowa publishes a treatise that first introduces determinants. He considers integer solutions of where are integers.
- Leibniz publishes details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus. In contains the familiar notation, and the rules for computing the derivatives of powers, products and quotients.
- Wallis publishes De Algebra Tractatus (Treatise of Algebra) which contains the first published account of Newton's binomial theorem. It made Harriot's remarkable contributions known.
- Kochanski gives an approximate method to find the length of the circumference of a circle.
- Newton publishes The Principia or Philosophiae naturalis principia mathematica (The Mathematical Principles of Natural Philosophy). In this work, recognised as the greatest scientific book ever written, Newton presents his theories of motion, gravity, and mechanics. His theories explain the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon.
- Jacob Bernoulli uses the word "integral" for the first time to refer to the area under a curve.
- Rolle publishes Traité d'algèbre on the theory of equations.
- Jacob Bernoulli invents polar coordinates, a method of describing the location of points in space using angles and distances.
- Rolle publishes Méthods pour résoudre les égalités which contains Rolle's theorem. His proof uses a method due to Hudde.
- Leibniz introduces the term "coordinate".
- Halley publishes his mortality tables for the city of Breslau (now Wroclaw) in Poland. His attempts to relate mortality and age in a population and proves highly influential in the future production of actuarial tables in life insurance.