The Derivative

There is no turning back from here...

This article is part of the mathematics sequence of the Tom Swift Academy.

Calculus is the first branch of mathematics unique to our own Western civilization. The ancients would come close but always stop somewhat short of developing it in its entirety. Archimedes used polygons to approximate the values of curved shapes in Quadrature of the Parabola. Elements thereof were also glimpsed in India and Japan. The enri, or circle method of Seki Takakazu also approached the principles of integration, as a method of approximating the area of a circle. Infinite series were understood and used in the Kerala School of Indian mathematics.

However, these attempts were stillborn for several reasons. Mathematics in the ancient world was seen solely as a tool for describing geometric reality; the study of motion as an abstract concept had no place in the Graeco-Roman conception of the world. In India, it was seen as a priestly pursuit, far removed from any description of the real world. In the Far East, it was a method of imperial administration and a form of mysticism. Hence, linear algebra and binary numbers were first employed there.

It was for these reasons that nowhere outside our own Faustian era of western civilization were these methods forged into the Fundamental Theorem of Calculus. The uniquely Western habit of mastering the physical world with the aid of mathematics provided the impetus to unite the various methods of dealing with infinitesimal components. This synthesis was to be discovered independently by Newton and Leibniz in the seventeenth century; it was then that the first truly analytical branch of mathematics came to be.

For the representation of physical quantities with algebraic functions made any phenomenon subject to graphical representation and mathematical analysis. No longer was the mathematician limited to the simple description of physical space, as was in the classical world. Nature was now subject to the will of man, and there would be no return …