- De Witt completes writing Elementa curvarum linearum. It is the first systematic development of the analytic geometry of the straight line and conic. It is not published, however, until 1661 when it appears as an appendix to van Schooten's major work.
- Nicolaus Mercator publishes three works on trigonometry and astronomy, Trigonometria sphaericorum logarithmica, Cosmographia and Astronomica sphaerica. He gives the well known series expansion of .
- Pascal publishes Treatise on the Arithmetical Triangle on "Pascal's triangle". It had been studied by many earlier mathematicians.
- Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they exchange during the summer.
- Pascal publishes his Treatise on the Equilibrium of Liquids on hydrostatics. He recognizes that force is transmitted equally in all directions through a fluid, and gives Pascal's law of pressure.
- Wallis publishes Arithmetica infinitorum which uses interpolation methods to evaluate integrals.
- Huygens patents the first pendulum clock.
- Huygens publishes De ratiociniis in ludi aleae (On Reasoning in Games of Chance). It is the first published work on probability theory, outlining for the first time the concept called mathematical expectation based on the ideas in the letters of Fermat and Pascal from 1654.
- Neile becomes the first to find the arc length of an algebraic curve when he rectified the cubical parabola. (See this Famous curve.)
- Frenicle de Bessy publishes Solutio duorm problematum ... which gives solutions to some of Fermat's number theory challenges.
- Rahn publishes Teutsche algebra which contains (the division sign) probably invented by Pell.
- De Sluze discusses spirals, points of inflection and the finding of geometric means in his works. He studies curves which Pascal names the "pearls of Sluze". (See this Famous curve.)
- Hooke discovers Hooke's law of elasticity.
- Viviani measures the velocity of sound. He determines the tangent to a cycloid. (See this Famous curve.)
- Van Schooten publishes the second and final volume of Geometria a Renato Des Cartes. This work establishes analytic geometry as a major mathematical topic. The book also contains appendices by three of his disciples, de Witt, Hudde, and Heuraet.
- The Royal Society of London is founded. Brouncker becomes its first President. (See THIS LINK.)
- Graunt and Petty publish Natural and Political Observations made upon the Bills of Mortality. It is one of the first statistics books.
- Barrow becomes the first Lucasian Professor of Mathematics at the University of Cambridge in England. (See THIS LINK.)
- The Académie des Sciences in Paris is founded.
- James Gregory publishes Vera circuli et hyperbolae quadratura which lays down exact foundations for the infinitesimal geometry.
- James Gregory publishes Geometriae pars universalis which is the first attempt to write a calculus textbook.
- Pell gives a table of factors of all integers up to 100000.
- Wren publishes his result that a hyperboloid of revolution is a ruled surface.
- Barrow resigns the Lucasian Chair of Mathematics at Cambridge University to allow his pupil Newton to be appointed.
- Wallis publishes his Mechanica (Mechanics) which is a detailed mathematical study of mechanics.
- Barrow publishes Lectiones Geometricae which contains his important work on tangents which forms the starting point of Newton's work on the calculus.
- De Witt publishes A Treatise on Life Annuities. It contains the idea of mathematical expectation.
- James Gregory discovers Taylor's Theorem and writes to Collins telling him of his discovery. His series expansion for arctan() gives a series for \large \frac π 4 .
- Mengoli publishes The Problem of Squaring the Circle which studies infinite series and gives an infinite product expansion for π/2.
- Mohr publishes Euclides danicus in which he shows that all Euclidean constructions can be carried out with compasses alone.
- Leibniz demonstrates his incomplete calculating machine to the Royal Society. It can multiply, divide and extract roots.
- Huygens publishes Horologium Oscillatorium sive de motu pendulorum. As well as work on the pendulum he investigates evolutes and involutes of curves and finds the evolutes of the cycloid and of the parabola.