Fluxion

The method of fluxions and infinite series p.20

Fluxion is a term historically used in mathematics to denote the rate of change of a quantity. It was introduced in the context of calculus by Isaac Newton in the late 17th century. The concept of fluxions is foundational in the development of differential calculus, where it represents the instantaneous rate of change or the derivative of a function. Newton's notation and conceptualization of fluxions laid the groundwork for modern calculus, although the notation and terminology have evolved. Today, the term "fluxion" is largely historical, with the concept being referred to as the derivative or differential of a function.

History

The history of fluxions dates back to the work of Isaac Newton in the 1660s and 1670s. Newton developed his method of fluxions as a means to solve problems of motion and change, which he referred to as the "science of fluxions." This was Newton's version of what is today known as differential and integral calculus. Newton's work on fluxions was part of his broader contributions to mathematics and physics, including the laws of motion and universal gravitation.

Concept

In Newton's calculus, a "fluent" was a quantity that flows or changes with time, such as the distance traveled by a moving object. The fluxion, denoted by a dot over the variable (e.g., \( \dot{y} \)), represented the rate of change of the fluent with respect to time. This is analogous to the modern concept of the derivative \( \frac{dy}{dt} \), where \( y \) is a function of \( t \).

Mathematical Formulation

The mathematical formulation of fluxions involves the limit process, where the fluxion of a quantity is the limit of the average rate of change as the interval of time becomes infinitesimally small. In modern terms, if \( y = f(t) \) is a fluent, its fluxion \( \dot{y} \) is given by the derivative \( \frac{dy}{dt} \).

Impact and Legacy

The introduction of fluxions was a pivotal moment in the history of mathematics, marking the beginning of calculus as a systematic discipline. Newton's method of fluxions, alongside Gottfried Wilhelm Leibniz's independent development of calculus, provided the mathematical tools necessary for advances in physics, engineering, and other sciences. The controversy over priority between Newton and Leibniz, known as the calculus controversy, overshadowed the mathematical and scientific achievements involved for many years.

Despite the historical significance of fluxions, the term and Newton's notation have been superseded by the modern notation and terminology of calculus developed in the 18th and 19th centuries. However, the principles underlying fluxions remain central to the study of calculus and its applications in various fields.

See Also

• Calculus
• Differential calculus
• Integral calculus
• History of calculus
• Isaac Newton
• Gottfried Wilhelm Leibniz