Newton, Leibniz y and the infinitesimal calculus


Gottfried Wilhelm Leibniz and Sir Isaac Newton

Science needs mathematics to formally and firmly describe its observations, to describe how the universe is.

As Galileo Galilei said:

“Mathematics is the language with which God has written the universe”

Maths is so big that listing all its areas of study would be large and complex. Basically all these areas are used by all kinds of science, including social sciences. Maybe the most widely used area is the infinitesimal calculus, which is split into integral calculus, which basis are integrals, and differential calculus, which basis are derivatives. A derivative is the change in the rate of a function as a function of a variable when it is small, what it is known as an infinitesimal. For example, when we are in a car, at every moment we are accelerating what means that the speed is changing with time. This change is represented as the derivative of the speed with respect to the time. An integral is the opposite operation, to establish the speed of the car at every moment from the acceleration.

But, where do derivatives and integrals come from? Someone had to invent them, is it right? Well, yes. Derivatives have an origin and their inventor is well known: Sir Isaac Newton. Yes, the same Newton of the apple, the gravity and the responsible of the theory that allows us to have astronauts orbiting above our heads (or below, depending where they are).

Newton was one of the biggest minds in history and, to develop his theories of gravity or optics, he used the maths that existed at that time, and when they were not available, he invented them. So big was Newton.

Sometimes, the big discoveries are no made by one single person, and even less nowadays when the research is made in large groups made of various scientists (even hundreds), and very often they are made by different persons at the same time independently, what leads to arguments about who was the first.

This was the case of infinitesimal calculus, Newton invented it, but Gottfried Wilhelm Leibniz, who was also working in the same field, arrived to the same conclusions at the same time as Newton, what means that there was a mess to win the battle of who had discovered calculus first.

Newton started to set the basis for infinitesimal calculus in his work of 1669 De analysi per aequationes numero terminorum infinitas. However Newton was reluctant to publish anything, as he was, somehow, afraid of the comments of his colleagues about his work. The same happened with every Newton’s work. However, in 1671 when he published De methodis serierum et fluxionum, it was when he developed in the detail the concepts established in De analysi and introduced the concept of fluent, which is something similar to our speed in the above example (i.e. something that depends on time), and the concept of fluxion of fluent, which is similar to our acceleration (i.e. the derivative with respect to time). It is here where the concept of derivative, that makes us to consider Newton as one of the parents of calculus, appears.

However, the fluxions of fluent from Newton are not the derivatives we use nowadays, but the most ‘comfortable’ version developed by Leibniz.

Leibniz was searching for a universal language, that used to a deductive system enabled to make reasoning so tangible as mathematics are so that we can discover easily any error, and when there is any argument between people we could simply say, let’s ‘calculate’ to see who is right. Leibniz also worked in the field of infinitesimal calculus, independently from Newton, and also worked in the development of a more simple and easy to use notation to make the calculus. In fact, he called his calculus with the Latin word ‘differentia’ and that is why we know it as differential calculus.

Leibniz published his articles frequently in magazines, while Newton, due to its reluctance to publish anything, made it in the form of a book, what enabled him to delay the publication as much as he could.

This different way of working was what led Newton and Leibniz to initiate a dispute to find out who was the first that invented the infinitesimal calculus. Newton and Leibniz, exchange a number of letters, often through intermediaries and colleagues. The dispute begun when Leibniz let Newton know about his method in 1676 and Newton send him back part of what he had written (but not published) in his De analysi and De methodis.

At that moment, Leibniz had not published anything yet and Newton, who had already written his books, should have realised that it was the moment for publishing. At the beginning of these communications everything were praises, but maybe not really deep considering later communications that begun when letters from their collaborators and assistants started to arrive positioning themselves in favour of one or another. Newton and Leibniz excused themselves several time, but these excuses were not useful to calm down the      tension between them.

It may be thought that a dispute of this kind could end at some moment, but it was not like this. Leibniz died in 1816 and Newton carried on filling pages and pages about his right to be considered as the inventor of infinitesimal calculus for ten years more, until the day of his dead.

Who was the inventor of infinitesimal calculus, could remain unknown and probably one day historians will find the proof that will give the reason to one or another, but the most important thing is not who invented infinitesimal calculus, but what was invented that is out there for the benefit of all of us


La verdad está en el límite. El cálculo infinitesimal. Antonio J. Durán. Colección El mundo es matemático

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