Isaac Newton stood at the gateway between an age when the tools of the geometer were predominant, and the age of analysis using the tools of calculus.
The dividing line is not sharply drawn, and encompassed many individuals. Among them was Leibniz who independently developed a calculus whose notations we use today. With respect to the calculation of pi, Francois Viete, John Wallis, and James Gregorie, were among those who replaced geometric methods for the visualization and calculation of pi with essentially abstract analytic methods, utilizing such techniques as infinite products and infinite expansions of inverse trigonometric functions. This not only enabled pi to ultimately be calculated to any degree of accuracy, but to be expressed as the product of rational numbers.
Newton's method stands out as the symbolic cusp because it somewhat elegantly combined both the historic tools of the geometer, and the new tool which Newton termed fluxions and which we today refer to as calculus. Newton started with a semi-circle having its center c at (1/2,0), with a radius of 1!2, and using a combination of geometry and fluxions derived a remarkably precise 16 decimal approximation of pi. He set this out in his Methodus Fluxionum et Serierum Infinitarum, written in 1671 but published only decades later. Newton also recorded for posterity another problem familiar to those intoxicated with the fascination of pi:
"I am ashamed to tell you to how may places of figures I carried these computations, having no other business at the time".