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In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
Descartes' rule of signs. In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's ...
Snell's law (also known as the Snell–Descartes law, the ibn-Sahl law, [1] and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.
He then called the logarithm, with this number as base, the natural logarithm. As noted by Howard Eves, "One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use." Carl B. Boyer wrote, "Euler was among the first to treat logarithms as exponents, in the manner now so familiar."
The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation. and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting In the plane the equation is. If we stretch the curve in the direction by a factor of this becomes.
Logarithmic spiral ( pitch 10°) A section of the Mandelbrot set following a logarithmic spiral. A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").
René Descartes ( / deɪˈkɑːrt / day-KART or UK: / ˈdeɪkɑːrt / DAY-kart; French: [ʁəne dekaʁt] ⓘ; [note 3] [11] 31 March 1596 – 11 February 1650) [12] [13] [14] : 58 was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was ...
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.