After the introduction of algebra in the sixteenth century, mathematical discoveries flooded Europe. One of the most important being the fundamentals of the calculus,-a means for calculating the way quantities vary with each other, rather than just the quantities themselves. 2 Like most discoveries, calculus was the combination of centuries of work rather than an instant discovery. Brilliant mathematicians all over the world contributed to its development, but the two most well known pioneers of calculus are Sir Isaac Newton and Baron Gottfried Wilhelm Leibniz.

Although now the credit is given to both men, there was a time when the debate over which of them truly deserved the recognition was of huge controversy. Isaac Newton was born in the manor house of Hamlet of Woolsthorpe in the city of Lincolnshire on December 25th, 1642. His father, a yeoman farmer died shortly before he was born. His mother Hannah Ayscough was remarried to Reverend Barnabus Smith three years after Newtons birth and he was sent to live with his Grandmother. Newton lived very well off as a young boy, never having to suffer poverty.

As a child Newton was not robust and was forced to give up the rough games played by other boys his age. Instead, he had to find other ways to spend his time, in which his genius first showed up. His experimental ingenious shone thorough in his various creations of kites with lanterns, and perfectly constructed mechanical toys that he made entirely by himself. He created things such as waterwheels, a mill that grounded wheat into flour and other such inventions and toys. He was educated at Trinity College and Cambridge, and lived there from 1661 till 1696, during which time he produced the majority of his work in mathematics.

Isaac Newton died of old age at Kensington, London on March 20th 1727. Gottfried Wilhelm Leibniz was born on June 21st, 1646. His father died before he was six. Leibniz attended a school in Leipzig and the education there at the time that he attended was inadequate to say the least. Leibniz however being the genius that he was taught himself to read Latin and Greek by the time he was twelve. Leibniz attended the University of Leipzig studying law at that age of fifteen. Before Leibniz was twenty he mastered the fundamentals of mathematics, philosophy, theology, literature, logic, metaphysics, and law.

At the age of twenty Leibniz was fully ready for his doctors degree in law but was refused a degree by those at Leipzig who were jealous of him and so he moved to Nuremberg. Leibniz was a true genius living under unfortunate circumstances unlike Newton. Gottfried Wilhelm Leibniz died at Hanover on November 14th 1716. Now that a brief biography of each mathematician is presented, the true root of the conflict can emerge. The true conflict of who first discovered calculus lies in who published their works first.

With the mass of historical evidence having been evaluated historians have determined that Newton developed his infinitesimal calculus while secluded in his estate in Woolsthorpe between 1664 and 1666. Trying to avoid the bubonic plague Newton did not actually publish his works until 1687 and later. Some people were aware of what he was doing and the progress of his work, through letters and papers that Newton shared with colleagues and friends. A manuscript dated May 20th, 1665, shows that Newton at the very young age of twenty-three had sufficiently developed the principals of calculus.

He was able to find the tangent and curvature at any point of any continuous curve. He called his method fluxions from the idea of flowing. He also discovered what he called the binomial theorem. Leibniz first studied calculus around 1672 to1676, some ten years after Newton and published in 1684 and 1686. Leibniz published his own infinitesimal methods in a scholarly journal circulated around Europe, called the Acta Eruditorum. Leibniz’ methods appeared in the Acta in two different articles, one in 1682 and the other in 1684.

Thus, while Newton’s techniques were developed first, Leibniz was the first to publish. When Leibniz first began exploring infinitesimal limits, almost two decades had passed since Newton began. Newtons own techniques did not appear until his publication of Optiks. However, in 1710 Leibniz was accused formally of plagiarism by the Royal Society of England, but, with increased debate over who had discovered calculus first, in 1713 Newton too was accused. In actuality Newton and Leibniz ideas were very similar but they thought of the fundamental concepts in very different ways.

The differences in Newtonian and Leibnizian calculus arise mostly out of the context from which each arrived at calculus. Leibniz was a metaphysics philosopher and abstract mathematician. Newton, on the other hand, was primarily concerned with physics, or what was then known as natural philosophy. Thus, Newton developed his systems in order to solve problems of gravitation and justify Keplers orbit theories based on planetary motion. His mathematical methods attempted to understand motion and force in terms of infinitesimal changes with respect to time.

Leibniz arrived at his conclusions from a purely abstract point. His philosophical thesis was, individually imperceptible metaphysical entities were the basis of existence and that humans experience the world as the sum of these entities. 4 While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between consecutive values of these sequences. Leibniz knew that dy/dx gives the tangent line but he did not use it as a defining property.

On the other hand, Newton used quantities x’ and y’, which were finite velocities, to compute the tangent. Leibniz and Newton had very different views of calculus in that Newtons was based on limits and concrete reality, while Leibniz focused more on the infinite and the abstract. Neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton calculus was geometrical while Leibniz took it towards analysis. Because of these distinct approaches, most people consider Leibniz as the inventor of integral calculus, while Newton is typically seen as the inventor of differential calculus.

This is very much a simplification, however, as both Leibniz and Newton developed versions of both integration and differentiation and the way in which both mathematicians arrived at their conclusions required both operations. In more closely analyzing the notation involved in Newton and Leibniz calculus many differences arise. Leibniz specifically set out to develop a more specific notation system than Newtons, and he developed the integral sign ( I ) and the d sign, which are still used today. Leibniz sent letters to Newton outlining his own presentation of his own methods, and these letters focused upon the subject of tangents and curves.

Because Newton had been approaching calculus primarily in regards to its applications to physics, he purported curves to be the creation of the motion of points while perceiving velocity to be the primary derivative. 1 Conversely, the calculus of Leibniz was applied more to discoveries in geometry. As a result of these fundamental differences, Leibniz notation involved discrete undefined variables such as, y, x, dy, dx. He used them to describe areas under curves, and integration of many infinitesimally small units.

Newton, on the other hand, based his differentiation on force with respect to time, and thusly represented differentials with a dot over its respective variable such as, x-dot, y-dot. It is interesting to note that Leibniz was very conscious of the importance of good notation and put a lot of thought into the symbols that he used. Newton, on the other hand, wrote more for himself than anyone else. Consequently, he tended to use whatever notation he thought of on that day. Leibniz’ notation was better suited to generalizing calculus to multiple variables and in addition it highlighted the aspect of the derivative and integral.

Since Leibniz’ approach was geometrical, the notation of the differential calculus and many of the general rules for calculating derivatives are still used today, while Newton’s approach, was “primarily cinematical”. Although, Newtons symbols are still used in physics classrooms, for the most part Leibnizs notation is universal, because it was popularized in Europe before Newton had a chance to publish his works and discoveries, but also, and more importantly, because it is more clear and generalized It is also known that Leibniz and Newton corresponded by letter quite regularly, and they most often discussed the subject of mathematics.

In fact, Newton first described his methods, formulas and concepts of calculus, including his binomial theorem, fluxions and tangents, in letters he wrote to Leibniz. However an examination of Leibniz’ unpublished manuscripts provided evidence that despite his correspondence with Newton, he had come to his own conclusions about calculus already. The letters may then, have merely helped Leibniz to expand upon his own initial ideas. What is also important to note about Newton and Leibniz was that even in mathematics Leibniz broad intelligence differs with Newtons direction to a unique end.

Newtons career was focused on physics and calculus. Aside from his development of calculus was his theory of the universal law of gravitation. Leibniz on the other hand could be considered more well rounded, a master of all trades so to speak. Leibniz had a lifelong interest in the pursuit of the idea that the principles of reasoning could be reduced to a formal symbolic system, an algebra or calculus of thought, in which controversy would be settled by calculations. But other then his discoveries concerning calculus Leibniz occupies a large space in the history of philosophy so much as he does in the history of mathematics.

Leibniz is known among philosophers for his wide range of thought about fundamental philosophical ideas and principles, including truth, necessary and contingent truths, possible worlds, the principle of sufficient reason (that nothing occurs without a reason), the principle of pre-established harmony (that God constructed the universe in such a way that corresponding mental and physical events occur simultaneously), and the principle of noncontradiction (that any proposition from which a contradiction can be derived is false).

Leibniz system of philosophy was based on the fact that he regarded the ultimate elements of the universe as individual percipient beings whom he called monads. According to him monads are centers of force, and substance is force, while space, matter, and motion, are merely phenomenal. Finally, the existence of God is inferred from the existing harmony among the monads. His influence on literature was almost that of his on philosophy.

His most prevalent theory was his overthrow of the then prevalent belief that Hebrew was the primeval language of the human race. The debate over who founded the calculus had been evaluated by historians long after Newtons and Leibniz deaths. This controversial debate over who discovered calculus may seem somewhat trivial by modern standards. However at the time, this was a serious issue that not only involved matters of mathematical invention but also matters of national pride and allegiance.

What is important to understand is that no matter who actually discovered calculus first, both Newton and Leibniz made great contributions to the advancement of mathematical processes, and both deserve credit for that. In the end Leibniz died poor and dishonored, while Newton was given a state funeral. Yet history validates Leibniz, and both men remain two of the greatest minds in mathematics and the co-founders of the brilliance of calculus.