Throughout history, there have been scientists, inventors, and many brilliant minds that have helped shape the world into what it is today. Advancements and discoveries in fields like engineering, biology, physics, mathematics, and many others play an important role in our daily lives. Regarding the field of mathematics, there are two important mathematicians whose contributions were essential to connect mathematics with physics.

Isaac Newton and Gottfried Wilhelm Leibniz are both well known for conceiving the ideas of differential and integral calculus. From completely different educational backgrounds and nations, these two minds came with similar ideas of calculus during the seventeenth century. During the time of the discovery of calculus, there was a dispute between Newton and Leibniz over who was the one who discovered calculus first. Prestige and acknowledgement over new discoveries was very important during those years as it is now.

Independently from their dispute, it is important to mention what similarities and differences in their life were the key factors for them to discover calculus. Their early life, background, education, inspiration, or approaches to their discoveries may not have been exactly the same, but their minds managed to come to the same ideas and at the end of the day their polemic was unnecessary; contributions from both of these minds were a gift for humanity.

On the one hand, Isaac Newton (1643-1727) was born in Woolsthorpe, Lincolnshire, England. He was not only a mathematician, but also an astronomer, philosopher, and physicist. He was pulled out of school at the age of twelve because his mother wanted him to work as a farmer, a job in which Newton would not succeed because he found it simple, monotonous, and uninteresting at all.

As mentioned in an article by Biography.com, “Perhaps sensing the young man’s innate intellectual abilities, his uncle, a graduate of the University of Cambridge’s Trinity College, persuaded Newton’s mother to have him enter the university.” He enrolled in Trinity College in 1661, and while he was there “Isaac paid his way through college by part time work such as waiting tables and cleaning rooms for the faculty and the wealthier students.” Among the important books he read are Euclid’s Elements, Descartes’ Geometry, and Wallis’ Arithmetica infinitorum.

The study of these works was of great significance to him because it was the foundations from which he came up with many discoveries in mathematics. When discovering calculus, Newton used what he called the Method of Fluxions, a method which he approached and described in geometric terms. As stated in an article by Dorothy V. Schrader, “The direct method of fluxions can be summarized in the solution of the mechanical problem: “The length of the Space described being continually given, to find the Velocity of the Motion at any time proposed,” or “The relation of the flowing Quantities being given, to determine the relation of their Fluxion.””

On the other hand, we have Gottfried Leibniz (1646-1716.) He was born in Leipzig, Germany. One of his many remarkable attributes was his self-learning ability. As stated in the textbook History of Mathematics, by professor Shanyu from the University of Houston, “Before he was twenty he had mastered the ordinary text-books on mathematics, philosophy, theology and law.”

Hence, Leibniz was very competent in many fields. His discovery of calculus was strongly influenced by the study of Descartes and Pascal. Unlike Newton, Leibniz had a more analytical approach to understand calculus, he developed differentials and integrals with the notations we use today.

Eventually after both mathematicians had conceived calculus, a big dispute over the title of the discoverer of calculus arose. According to the article When Lions Battle, by Nicholas Tasaday, “The battle over priority in the discovery of calculus is arguably the most well-studied and bitter scientific dispute in history.” This same article mentions an important event that would later cause the famous controversy.

In 1676, while visiting London, Leibniz had access to draft publications by Isaac Newton about calculus. Even though these documents were shown to Leibniz by his own correspondent, Isaac Newton would later accuse Leibniz for plagiarizing his works. Nonetheless, and as mentioned also by Castro Ramos in his article The Discovery of Calculus: Leibniz Vs. Newton, “Their differences in approach suggests that their discoveries were simultaneous but independent.”

Moreover, in his first edition of Principia in 1687, Newton himself recognized that they were both working on similar methods to determine maxima and minima. During the upcoming years, “The unfortunate intellectual controversy, over who first discovered calculus, began in 1699 and was not ended even by Leibniz’s death in 1716.”

Despite the controversy over the title of discoverer of calculus, today Newton and Leibniz are generally considered co-discoverers. They both had different approaches, but ultimately came to the same idea to represent motion with calculus. Calculus as known today is in a way a mixture of their contributions. Calculus is often visualized using Newton’s ideas and laws of motion to calculate velocity and acceleration of objects.

To symbolize these ideas in mathematics language, we use Leibniz’s notation like the signs dy/dx and ∫ used for differentiation and integration respectively. With different contributions made to mathematics, it is clear that both Newton and Leibniz were crucial in the development of a mathematical area which would allow us to link mathematics to physics in order to have a better understanding of motion. We were lucky to have not one, but two brilliant minds who made calculus possible.

More importantly, calculus is very helpful in many other fields such as engineering, biology, architecture, astronomy, and even business and economics. Despite their dispute and controversy, we all won because at the end of the day the contributions in calculus made by Newton and Leibniz are still present with us and they are both acknowledge equally. Undoubtedly, a world without them or calculus would not be what it is today.

## Works Cited

- Biography.com editors. “Isaac Newton Biography.” The Biography.com https://www.biography.com/people/isaac-newton-9422656 (Accessed November 10, 2018)
- Castro Ramos, Sebastian. “The Discovery of Calculus: Leibniz Vs. Newton.” Last modified November 3, 2017. https://www.stmuhistorymedia.org/the-discovery-of-calculus-leibniz-vs-newton/?fbclid=IwAR1cE6aXmECH55CW2plRa1reLSDYotGbPOrUXVV-w1o1gSyRBH_hp61HmPE
- Cirillo, Michelle. ‘Humanizing Calculus.’ The Mathematics Teacher 101, no. 1 (2007): 23- 27. http://www.jstor.org.ezproxy.lib.uh.edu/stable/20876025.
- Ji, Shanyu. “History of Mathematics.” Schrader, Dorothy V. ‘The Newton-Leibniz Controversy concerning the Discovery of the Calculus.’ The Mathematics Teacher 55, no. 5 (1962): 385-96. http://www.jstor.org.ezproxy.lib.uh.edu/stable/27956626.
- Tasaday, Nicholas. ‘When Lions Battle.’ Math Horizons 14, no. 4 (2007): 8-31. http://www.jstor.org.ezproxy.lib.uh.edu/stable/25678687.
- Whitrow, G. J. ‘Newton’s Role in the History of Mathematics.’ Notes and Records of the Royal Society of London 43, no. 1 (1989): 71-92. http://www.jstor.org.ezproxy.lib.uh.edu/stable/531719.