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When the "Cartesian coordinate system" was discovered By René Descartes, Algebra and Geometry were connected. How exactly did that affect Newton and Leibniz in the invention of what we know as "Calculus"?

Is the concept of graph the way of showing the function geometrically, by using "Cartesian coordinate system"?

If that is true, could we also have connection between "Calculus" and "Geometry" in graphing functions with coordinate system?

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    $\begingroup$ Descartes invented an algebraic version of calculus that preceded Newton-Leibniz's analytic one, see Is there a 'lost calculus'?. This, his fixation of algebraic notation, and analytic translations of geometry were indispensable in transforming methods of the Greeks and building on them what became calculus. $\endgroup$
    – Conifold
    Feb 24 '19 at 3:08
  • $\begingroup$ It is not clear what exactly are you asking about. "How - historically?" As a student, Newton was not satisfied with what was taught in Cambridge, and started to study himself, mostly by reading Descartes. Source: Westfall's scientific biography of Newton. $\endgroup$ Feb 24 '19 at 14:59
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    $\begingroup$ The fundamental contribution of Descartes was its improved algebraic symbolysm and its systematic use in analyzing geometrical problems, first of all the problem of tangents. Thus, bith Newton and Leibniz studied it. $\endgroup$ Feb 24 '19 at 16:40
  • $\begingroup$ Its all about the tangent $\endgroup$
    – Milan
    Mar 13 '19 at 20:50
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Calculus is based upon functions that are defined in a coordinate system. So Descartes' work of Cartesian coordinates lay the foundation for Newton and Leibniz to invent the Calculus. For example, in order to calculate the area below a curve, you must have a function $f(x)$ to express the curve based on Cartesian coordinate before integration and the area is $$ A=\int_a^b f(x)\:dx $$ Furthermore, this fundamental contribution of Descartes goes far beyond geometry because the physical quantities such as force, distance, time, velocity, work and so on, are also based on Cartesian coordinates. This means that many notions in physical sciences like mechanics, electricity, thermodynamics, chemistry and so on, can be based on Cartesian coordinates as well.

For example, the acceleration that is proportional to the force is the derivative of the distance function with respect to time $$ F=m\cdot a=m\cdot s''(t) $$ To calculate work by force, we need first to express force as a function with respect to distance and then integrate it as $$ W=\int_C F(s)\:ds $$ Thus in order to do Calculus, you need functions expressed in the Cartesian coordinate system first.

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