KMShannon's_Calculus

Introduction
The Calculus student of today has the opportunity to stand on the shoulders of giants, to learn in a few semesters the culmination of two and a half millennia of the work of many of the best minds in the civilized world. The countless mathematicians, philosophers and scientists whose inquiries eventually gave birth to the Calculus could not have imagined where it would lead us. We can send people to the moon and bring them back safely. We can build bridges across huge spans of water and traverse them at incredible speeds in vehicles designed using techniques entirely dependent on the tools the Calculus has helped to bring us. We can communicate instantly with people half a world away. It is incredible to think about the changes that have occurred since the Calculus gained its sure footing in the 19th century. Compare them to the changes that occurred during the two and a half millennia that brought us the Calculus. It is impossible to determine what exactly has caused the acceleration of change but it is certain that the vast majority of technological advances during this century have relied at least to some extent on the mathematical methods encompassed in the Calculus. How can we appreciate where we are today without appreciating the mathematics that has helped to bring us here?

Calculus is an incredibly powerful tool which has facilitated technological triumphs ... and a beautiful monument to two and a half millennia of human collaboration, invention and imagination.
No one today who wishes to be a well rounded, liberally educated citizen should remain ignorant of its beauty and utility.
In fact, by the 17th century, . . . accusations of plagiarism were rampant. . The most famous dispute was between Newton and Leibnitz
Newton was perfectly correct in claiming that he stood "on the shoulders of giants." Newton and Leibnitz gathered a myriad of different methods and approaches and molded them into a single logical framework which Newton called Fluxions and we call the Calculus.
Newton & Leibnitz were hampered by :

the lack of a clear, rigorous formulation of the real number system,
infinity and infinitesimals
the difficulties inherent in blending geometry, algebra and arithmetic

The proofs, etc. for the Calculus are mostly algebraic and arithmetic but the inspiration and intuition were primarily geometric. It is not surprising that it took an entire century after Leibnitz and Newton to completely formulate the logical, rigorous foundations.
In the 17th century Viete "realized the facility to be gained in the handling of geometric problems by their reduction to the solution of algebraic equations . . "
Descartes is generally credited with the development of Analytic Geometry published in his famous Geometrie of 1637 and the primary coordinate system used throughout mathematics is called Cartesian.

Fermat, in his solution of tangent, area, and maximization problems very nearly approached the Calculus but he did not discover the Fundamental Theorem of Calculus.

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