Quaternionic Eulerian Calculus (QEC) is a new field of calculus wherein differential and integral calculus are based upon Euler's formula. Given the definition of the limit, as defined and proven by the delta-epsilon proof, the Newtonian derivative, f'(x), is not and cannot be a limit as defined. The formal delta-epsilon definition of a limit proscribes precision. The derivative, contrarily, obtains a precise value symbolically. As a limit, it is proven that a precise value of the difference quotient is obtained when h=versine=0, absent any reliance upon the delta-epsilon proof. It is also proven that f'(x) is an instantaneous center of revolution, that f'(x) is equivalent to the quaternion rotator operator, *h[ ]h, and lastly that i=square root of -1, is a quaternion and functions as a derivative. These concepts are the essence of QEC. Several practical applications are provided including a solution to the Riemann Hypothesis and a QEC analysis of DNA translation.