**Mathematics and History: Integration**

More than two thousand years ago, Archimedes discovered the formulas for solids but his method was refused due to lack of knowledge of algebra, the function concept and decimal representation of numbers. He applied his discovered concepts in calculating curved surface areas and volumes of solids by using the method of exhaustion. However, his outcomes were not expressed as the volumes of these solids were being compared.

For example, he compared a cylinder and a sphere regarding their radii and heights and made a concrete conclusion based on the comparison (Boyer, 127). Between 1642 and 1727, both Newton and Leibniz independently discovered integration and differentiation. Newton found out that the two components of calculus lie in the opposite ends even though they are related. Newton is known for his Fluxional Calculus. He applied calculus during his investigations in physics and geometry. He looked at calculus as the scientific narrative of the generation of motion and their sizes. His concepts were widely applied in solving mathematical, physics and astronomical problems. On the other hand, Leibniz emphasized on solving tangent problems and concluded that calculus is a metaphysical description of the change. Just like Newton, his central insight was the behaviors of the universe and focused on integral and the differential of a function. The insights have assisted in understanding the process of universal algorithm thus contributing to the world of mathematics (Zill et al., 341). During the years of mathematical discoveries, Leibniz struggled in developing calculus. While Newton started studying mathematics at an early age, Leibniz began his mathematical studies at a mature age. Newton was a fountain of knowledge in the mathematical world and his workings revolved around metaphysics, logics, and mathematics.

He applied metaphysics in which he viewed the world as an infinite collection of unseen objects prompting him to form a logical reasoning in understanding the universe. Leibniz’s motivation to study mathematics was heightened 1672 when he met the mathematician Huygens, who convinced him to dedicate his time to studying mathematics. The rise of calculus is a great memory in the early discoveries of mathematics (Simmons, 98). Calculus is the mathematics of motion and change, and its discovery needed the development of a novel mathematical system. Both Newton and Leibniz used different basis in creating mathematical systems to solve variable quantities. While Newton viewed change as a variable quantity, Leibniz viewed it as the difference stretching over a sequence of infinite close values.

The differences of thoughts in calculus and its invention brought about a controversy referred as Leibniz and Newton calculus controversy linking Leibniz, from Germany and Newton from England. This led to a rift in the European mathematical community. Although Leibniz was the first mathematician to publish his findings, Newton was the first one to begin inventions in calculus and consequently came up with tangents theory. It is argued that Newton started his investigations in 1666 while Leibniz started his in 1673 and his subsequent visits …