Cartesian Coordinate System (Edited from Wikipedia) SUMMARY A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as distances from the origin. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. HISTORY The adjective “Cartesian” refers to the French Mathematician and Philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his 1 students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Frans van Schooten was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes. Van Schooten's father was a professor of mathematics at the University of Leiden, having famous students like Christiaan Huygens. Van Schooten met Descartes in 1632 and read his Géométrie while it was still unpublished. Finding it hard to understand, he went to France to study the works of other important mathematicians of his time, such as François Viète and Pierre de Fermat. When Frans van Schooten returned to his home in Leiden in 1646, he inherited his father's position. Van Schooten's 1649 Latin translation of and commentary on Descartes' Géométrie was valuable in that it made the work understandable to the broader mathematical community, and thus was responsible for the spread of analytic geometry to the world. Over the next decade he enlisted the aid of other mathematicians of the time and expanded the commentaries to two volumes, published in 1659 and 1661. This edition and its extensive commentaries were far more influential than the 1649 edition. It was this edition that Gottfried Leibniz and Isaac Newton knew. Van Schooten was one of the first to suggest, in exercises published in 1657, that Descartes’ ideas be extended to three-dimensional space. Van Schooten's efforts also made Leiden the centre of the mathematical community for a short period in the middle of the seventeenth century. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three- dimensional space. The development of the Cartesian coordinate system would play a fundamental role in the development of the Calculus by Isaac Newton and Gottfried Wilhelm Leibniz. RENE DESCARTES René Descartes (31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist. Dubbed the father of modern western philosophy, much of subsequent Western philosophy is a response to his writings, which are studied closely to this day. He spent about 20 years of his life in the Dutch Republic. 2 Descartes's Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes's influence in mathematics is equally apparent; the Cartesian coordinate system—allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two- or three-dimensional coordinate system (and conversely, shapes to be described as equations)—was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, used in the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the scientific revolution. Descartes refused to accept the authority of previous philosophers. He frequently set his views apart from those of his predecessors. In the opening section of the Passions of the Soul , a treatise on the early modern version of what are now commonly called emotions, Descartes goes so far as to assert that he will write on this topic "as if no one had written on these matters before". He was also suspicious of his own senses: his best known philosophical statement is "Cogito ergo sum" (French: Je pense, donc je suis), which means “I think, therefore I am.” Descartes brought the question of how reliable knowledge may be obtained (which is called epistemology ) to the fore of philosophical enquiry. Many consider this to be Descartes' most lasting influence on the history of philosophy. Cartesianism is a form of rationalism because it holds that scientific knowledge can be derived a priori from 'innate ideas' through deductive reasoning. This is the idea that we have some knowledge even before we encounter new experiences. Cartesianism is opposed to both Aristotelianism and what is called “empiricism,” with their emphasis on sensory experience as the source of all knowledge of the world. For Descartes, the faculty of deductive reason is supplied by God and may therefore be trusted because God would not deceive us. 3.