Article｜Newton’s Method of Fluxions: What Did Calculus Look Like Before Leibniz’s Notation?

When we study calculus today, the notation for derivatives that we are probably most familiar with is $\frac{dx}{dy}$. If we go on to study higher mathematics, we also encounter forms such as $\frac{d^2x}{dy^2}$ and $\frac{d^3x}{dy^3}$. This notation originated with the German mathematician Gottfried Wilhelm Leibniz and, after several centuries of development, has become one of the most familiar languages of modern mathematics.

But when calculus was first emerging, mathematicians had not yet developed the unified system of notation we use today. When Newton developed his own approach to calculus, he used a completely different language. He did not write $\frac{dx}{dy}$, but instead preferred to place a small dot above a letter, such as $\dot{x}$.

What exactly did this little dot mean? Why did Newton write it this way? And what exactly was his “method of fluxions”?

The answer is actually much simpler than we might imagine.

Why Did Newton Imagine Variables as “Flowing”?#

When Newton developed his method of fluxions, he was particularly concerned with one question: how could mathematics be used to describe motion?

Suppose an object is moving. Its position changes continuously over time. If we use $x$ to represent the position of the object, then $x$ is not a fixed number. It is constantly changing as time passes.

Newton called a quantity that continually changes in this way a fluent, which can be translated as a “flowing quantity” or “quantity in flux.”

The word “flowing” does not mean that the quantity literally flows like water. It is a vivid way of describing a quantity that is continuously generated and continuously changing over time.

For example, the position of an object can be represented by $x$. As time passes, $x$ keeps changing, and therefore it is a fluent.

This leads to the next question: if $x$ is changing, we naturally want to know how quickly it is changing.

This is what Newton called a fluxion.

What Is a “Fluxion”?#

Newton called the rate of change of a fluent a fluxion, which corresponds to what we call a derivative today.

If $x$ is a quantity that is changing, Newton placed a small dot above $x$:

This $\dot{x}$ represents the fluxion of $x$, or the rate at which $x$ changes over time.

For example, if $x$ represents the position of an object, then $\dot{x}$ represents the velocity of that object.

If $x$ itself represents velocity, then $\dot{x}$ represents the rate at which velocity changes, or acceleration.

Thus, in Newton’s notation:

represents a quantity that changes over time, while:

represents how quickly that quantity changes.

If we add another dot:

then it represents the fluxion of $\dot{x}$, or the second-order rate of change of $x$.

In modern notation, we would usually write:

and:

The equals sign here is only meant to help us understand the correspondence between the two systems of notation. Newton’s mathematical language was not simply a rewritten version of Leibniz’s notation. He developed a system of calculus with its own distinctive way of expressing mathematical ideas.

Why Was It Called the “Method of Fluxions”?#

If a quantity that continually changes is called a fluent, and its rate of change is called a fluxion, then Newton’s method for studying these quantities was called the Method of Fluxions, commonly translated as the “method of fluxions.”

The name itself is quite revealing, because it reflects an important characteristic of Newton’s way of thinking about calculus: he liked to understand mathematics through motion and change.

If an object is moving, its position is a quantity that continually changes. If we study how quickly its position changes, we get velocity. If we then study how quickly its velocity changes, we get acceleration.

Thus:

In Newton’s language, these are not three unrelated symbols. They form a continuous chain of change.

A quantity changes, and that change itself can change. This is the most intuitive part of the concept of a fluxion.

What Is the Difference Between Newton’s $\dot{x}$ and Today’s $\frac{dx}{dt}$?#

At this point, it is easy to wonder: if $\dot{x}$ is essentially equivalent to $\frac{dx}{dt}$, was Newton simply using a different symbol?

From the perspective of modern mathematical results, the two are certainl[118;1

Today, we usually define a derivative using a limit. For example:

This rigorous definition of the derivative was developed through the work of mathematicians over a much longer period. Newton’s early method of fluxions had its own conceptual framework and methods of derivation.

Newton tended to begin with continuous motion. He treated quantities such as position and velocity as quantities that continually changed with time, and then studied their rates of change.

So, if we simply want a basic correspondence, we can understand:

as the modern:

But if we want to understand Newton historically, we should recognize that although the two symbols can be related in modern mathematics, the ways of expressing the underlying ideas were not exactly the same.

How Did Leibniz Write It?#

At roughly the same time, another founder of calculus, Gottfried Wilhelm Leibniz, developed a very different system of notation.

He used:

to represent differentials, and used:

to represent the rate at which $x$ changes with respect to $y$.

This notation proved extraordinarily influential and eventually became the most common way of expressing calculus.

So today we see:

while Newton’s language was closer to:

If we place the two systems of notation side by side, their difference becomes immediately visible:

The two describe closely related mathematical concepts, but their ways of expressing them are quite different.

Why Did Leibniz’s Notation Survive?#

If both Newton and Leibniz developed their own approaches to calculus, why do modern mathematics textbooks mostly use Leibniz’s notation rather than Newton’s dot notation?

One important reason is that Leibniz’s notation is better suited to expressing general mathematical relationships.

Newton’s dot notation is closely tied to time. $\dot{x}$ represents the rate at which $x$ changes with time. This is extremely convenient when studying motion, because the time variable is already explicitly present in the problem.

But mathematics is not limited to physical quantities that change over time. If we simply want to study the relationship between two variables, Leibniz’s notation is much more flexible.

For example:

This notation directly tells us that we are studying the rate of change of $x$ with respect to $y$. It does not require us to designate one of the variables as time, so it can naturally be applied to more general mathematical problems.

This flexibility became increasingly important as calculus developed. Mathematicians began studying increasingly complex functional relationships, dealing with changes between different variables, working with multiple variables, and building increasingly general mathematical systems. In these contexts, Leibniz’s $d$ notation could be extended quite naturally.

For example, for a function of several variables, we can write:

to represent the partial derivative of $f$ with respect to $x$. If we need to represent a second-order derivative, we can write:

These notations all continue the expressive tradition established by Leibniz.

Leibniz’s notation also has another important advantage: it allows differentials, derivatives, and integrals to exist within a single interconnected language.

Today we see:

and it is easy to recognize that these symbols are somehow related. In particular, the integral sign $\int$ itself was also created by Leibniz, and together with his differential notation, it forms a remarkably complete language for calculus.

By contrast, Newton’s dot notation, although extremely well suited to expressing derivatives with respect to time, is much harder to extend naturally to all of these mathematical situations.

If we are studying how one quantity changes with respect to another rather than with respect to time, $\dot{x}$ cannot express the relationship between the variables as directly as $\frac{dx}{dy}$ can.

As calculus gradually developed from a tool for studying motion into a more general branch of mathematics, Leibniz’s notation therefore proved more adaptable.

Of course, this does not mean that Leibniz’s notation “defeated” Newton’s notation from the beginning, nor did mathematicians hold some kind of formal vote to decide which notation should be used.

The fate of mathematical notation is usually determined through a long historical process. Different mathematicians continually adopt, modify, and spread different forms of notation. Eventually, the expressions that are most convenient, easiest to generalize, and best suited to new problems tend to survive.

Today, Leibniz’s notation has become the primary language of modern calculus, while Newton’s dot notation has survived mainly in physics, where it is particularly useful for representing derivatives with respect to time.

Why Does Newton’s Dot Notation Survive in Physics?#

Although Newton’s dot notation did not become the dominant notation of modern mathematical calculus, it survived in physics. Even today, in classical mechanics and dynamics, we frequently encounter $\dot{x}$ and $\ddot{x}$.

What survived, however, was primarily the notation rather than Newton’s original terminology. Modern physics usually describes $\dot{x}$ as the derivative with respect to time, or, in specific mechanics problems, simply as velocity. Likewise, $\ddot{x}$ corresponds to acceleration. In other words, what Newton once called a “fluxion” has been given a different language in modern physics, but the little dot itself has remained.

Why Is a Little Dot So Convenient?#

Newton’s notation has a very intuitive advantage.

If $x$ is position, then:

is velocity.

If $\dot{x}$ is velocity, then:

is acceleration.

If we continue adding dots, we can continue to represent higher-order derivatives with respect to time.

In this way, a physicist only needs to look at how many dots appear above the letter to know which order of time derivative is being discussed.

For example, in classical mechanics, the position of an object can be written as $x(t)$, its velocity as:

and its acceleration as:

This notation has survived to the present day.

So when we encounter a small dot above a letter in a physics textbook, we are not merely looking at a modern mathematical symbol. We are seeing a notation tradition that has continued for several centuries.

The Method of Fluxions Is Closely Connected to Motion#

Newton’s development of the method of fluxions was closely connected to his study of motion.

Suppose a point moves along a curve. Its position can be represented by two coordinates:

As time changes, $x$ and $y$ both change, so each has its own fluxion:

The direction in which this moving point travels along the curve can then be described through their rates of change.

For example, the slope of the tangent to the curve can be written as:

Using the Leibniz notation we are more familiar with today, this becomes:

This is why Newton’s method of fluxions could be used not only to deal with velocity and acceleration, but also to study tangents to curves, areas, and other problems in calculus.

From our modern perspective, it is easy to feel that these concepts simply belong to calculus. But in the seventeenth century, there was no mature, unified method for dealing with all of these problems.

One of the important things Newton and Leibniz accomplished was to gradually bring these previously separate problems into a new mathematical framework.

If We Returned to Newton’s Time, Calculus Could Look Like This#

Today, we are accustomed to:

When we see it, we immediately recognize it as a derivative.

But if we went back to the seventeenth century, Newton might have written:

If he were studying a second derivative, he might write:

If he were describing the process by which a quantity changes, he would think of the quantity as a fluent, and its rate of change as a fluxion.

Thus, the “derivative” familiar to modern readers acquires a much more vivid interpretation in Newton’s language: a quantity is flowing, and the fluxion describes the speed of that flow.

This is what makes the name “method of fluxions” so interesting.

It sounds like a mathematical term that disappeared into history, but whenever we see $\dot{x}$ and $\ddot{x}$ in physics, we can still see traces of that old language.

Understanding Newton’s Mathematics Through a Little Dot#

Modern calculus has developed into an enormous mathematical system. The notation we use has gone through centuries of selection and evolution, and is now very different from what Newton used.

Understanding the method of fluxions is therefore not about relearning Newton’s calculus, nor is it about replacing modern calculus notation with his.

What makes it interesting is that it allows us to temporarily step away from the mathematical language we have become so accustomed to, return to the moment when calculus was just beginning, and see how Newton tried to express the idea of “change.”

Today we write:

Newton wrote:

Today we speak of a “derivative.” Newton used the concept of a “fluxion.”

The same mathematical relationship can have completely different forms of expression in different historical periods. And the little dot Newton left behind can still be seen in physics today.

Its meaning and name have changed as modern mathematics developed, but this notation from several centuries ago has survived.