Mathematics of infinite - Essay Example

Calculus passed along a dramatic path, with a history worth reminiscing and value worth appreciating. Introduction Calculus, is a branch of mathematics that deals with rates of changes of quantities, area, volume, length and motion of objects. It is also called analysis, real analysis or infinitesimal analysis. Calculus is divided into two branches: differential calculus -concerned with derivatives and the integral calculus- that deal with integrals (Calculus 2013). The invention of Calculus is basically accredited to Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716).

Newton and Leibniz’s breakthrough in mathematics had triggered a lot of debates and arguments from their successors which somehow contributed to the total development of the modern calculus that is existent today. The Birth: Calculus by Newton and Leibniz It is amazing to note that Newton and Leibniz discovered the concepts of calculus independently in separate periods of time, in years 1665 and 1673 respectively. Newton first discovered the subject although Leibniz was the first to publish his works in 1684.

Leibniz uses symbols and notations and developed formulas obeying certain rules that transformed the theoretical methods of calculus. Leibniz’s study of the relationships of sequences of sums and differences has contributed greatly in concrete problem solving making it the essence of the Fundamental Theorem of Calculus. He also established the harmonic triangle forming sequences of sums and differences. Leibniz’ another major contribution is the idea of “characteristic triangle” where a triangle with a curve running along it has infinitesimal sides at every point of the curve (Leibniz’s Fundamental Theorem of Calculus n.

d : 133-136). Newton established calculus based on three methods which are the infinitesimal, method of fluxions and the ultimate ratios. He also introduced “moments of fluxions”, as the amount of increase of a fluxion in an infinitely small period of time. The “ultimate ratios” is his attempt to lay the foundation of calculus with the concept of limits (The Calculus of Leibniz and Newton n.d). Newton uses fluxion and fluent instead of derivative and integral. He uses infinitesimals for computations and provided more concepts about differentiation.

Newton and Leibniz were not the first mathematicians that made the evolution of calculus possible. They were the first to define “algorithmic processes” and to set general notations. They formulated the inverse relationship of integration and differentiation in the most logical manner. Their ideas though individually and independently discovered by each one but resulted to be very related has made calculus to stand on firm foundations of knowledge which until now has been used and applied.

As a general statement: “Newton’s legacy is more about the sorts of scientific problems that calculus has considered during the past three to four centuries, while Leibniz’ legacy is more about the way such problem are studied” (Burton n.d) Gray Period: Berkeley’s Criticism Newton and Leibniz’ profound usage of “infinitesimals” has disturbed many mathematicians succeeding them. Lord Bishop Berkeley is one of those who hurled stinging and serious criticisms to this idea. In his book, The Analyst, he exposed his judgment about the validity of calculus comparing it with religion.

He derided the idea of “