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1650

De Witt completes writing

1651

Nicolaus Mercator publishes three works on trigonometry and astronomy, *Trigonometria sphaericorum logarithmica*, *Cosmographia* and *Astronomica sphaerica*. He gives the well known series expansion of log(1 + *x*).

1653

Pascal publishes *Treatise on the Arithmetical Triangle* on "Pascal's triangle". It had been studied by many earlier mathematicians.

1654

Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they exchange during the summer.

1654

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.

1655

Brouncker gives a continued fraction expansion of ^{4}/_{π} . He also computes the quadrature of the hyperbola, a result he will publish three years later.

1656

Wallis publishes *Arithmetica infinitorum* which uses interpolation methods to evaluate integrals.

1656

Huygens patents the first pendulum clock.

1657

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.

1657

Neile becomes the first to find the arc length of an algebraic curve when he rectified the cubical parabola. (See this Famous curve.)

1657

Frenicle de Bessy publishes *Solutio duorm problematum ...* which gives solutions to some of Fermat's number theory challenges.

1658

Wren finds the length of an arc of the cycloid. (See this Famous curve.)

1659

Rahn publishes* Teutsche algebra* which contains (the division sign) probably invented by Pell.

1660

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.)

1660

Hooke discovers Hooke's law of elasticity.

1660

Viviani measures the velocity of sound. He determines the tangent to a cycloid. (See this Famous curve.)

1661

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.

1662

The Royal Society of London is founded. Brouncker becomes its first President. (See this Article.)

1662

Graunt and Petty publish *Natural and Political Observations made upon the Bills of Mortality*. It is one of the first statistics books.

1663

Barrow becomes the first Lucasian Professor of Mathematics at the University of Cambridge in England. (See this Article.)

1665

Newton discovers the binomial theorem and begins work on the differential calculus.

1666

The Académie des Sciences in Paris is founded.

1667

James Gregory publishes *Vera circuli et hyperbolae quadratura* which lays down exact foundations for the infinitesimal geometry.

1668

James Gregory publishes *Geometriae pars universalis* which is the first attempt to write a calculus textbook.

1668

Pell gives a table of factors of all integers up to 100000.

1669

Wren publishes his result that a hyperboloid of revolution is a ruled surface.

1669

Barrow resigns the Lucasian Chair of Mathematics at Cambridge University to allow his pupil Newton to be appointed.

1669

Wallis publishes his *Mechanica* (*Mechanics*) which is a detailed mathematical study of mechanics.

1670

Barrow publishes *Lectiones Geometricae* which contains his important work on tangents which forms the starting point of Newton's work on the calculus.

1671

De Witt publishes *A Treatise on Life Annuities*. It contains the idea of mathematical expectation.

1671

James Gregory discovers Taylor's Theorem and writes to Collins telling him of his discovery. His series expansion for arctan(*x*) gives a series for ^{π}/_{4}.

1672

Mengoli publishes *The Problem of Squaring the Circle* which studies infinite series and gives an infinite product expansion for π/2.

1672

Mohr publishes *Euclides danicus* in which he shows that all Euclidean constructions can be carried out with compasses alone.

1673

Leibniz demonstrates his incomplete calculating machine to the Royal Society. It can multiply, divide and extract roots.

1673

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.

1675

La Hire publishes *Sectiones conicae* which is a major work on conic sections.

1675

Leibniz uses the modern notation for an integral for the first time.

List of mathematicians alive in 1650.

List of mathematicians alive in 1675.

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