Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. This began when Fermat and Descartes developed analytic geometry, which is the precursor to modern calculus. The modern foundations of mathematical analysis were established in 17th century Europe. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. Alongside his development of Taylor series of trigonometric functions, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series, of functions such as sine, cosine, tangent and arctangent. In the 12th century, the Indian mathematician Bhāskara II gave examples of derivatives and used what is now known as Rolle's theorem. Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In Indian mathematics, particular instances of arithmetic series have been found to implicitly occur in Vedic Literature as early as 2000 BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis. Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides.
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