Calculus

Wallis's ∞: a sign that is not a number

The first known use of the sign ∞, in John Wallis’s treatise De sectionibus conicis (1655). In the middle of the text: “esto enim ∞ nota numeri infiniti”, let ∞ be the sign of an infinite number. Two lines above, the height of each parallelogram is 1/∞ of the total height.John Wallis · Wikimedia Commons · public domain In 1655 the Englishman John Wallis, a professor at Oxford, divided a plane figure into infinitely many very thin parallelograms, each with a height equal to an infinitely small part of the total height. To write that part, he needed a sign for infinity, and invented one:

October 18, 2026 Read

The fundamental theorem: when d and ∫ cancel

Before calculus, tangents and areas were two unrelated problems. Finding the tangent to a curve was one art; finding the area beneath it, another. The great discovery of the seventeenth century, by Isaac Barrow, Newton and Leibniz, was that one is the inverse of the other. This is the fundamental theorem of calculus, and Leibniz’s notation writes it in a way that makes it look almost obvious. Leibniz: like sums and differences Leibniz thought of $\int$ as a sum and of $d$ as a difference. And adding up differences undoes the difference, as on a staircase: the sum of the steps is the total height. He wrote that $\int$ and $d$ are inverse to each other, “like powers and roots in ordinary calculation”. In symbols:

October 18, 2026 Read

The integral read sign by sign

The expression $$\int_0^2 x^2\,dx$$has five pieces: the sign $\int$, the numbers $0$ and $2$, the function $x^2$ and the $dx$. Each does a different job, and each came from somewhere. Chapter 1 told the passage from Archimedes’ exhaustion to the integral. Here the integral is read slowly, the way one reads a sentence in a new language. A sentence in five pieces The $\int$ is an elongated S, for summa, sum. Leibniz first wrote it in a manuscript dated 29 October 1675. It is an iconic sign in a curious way: it does not look like an area, but like the word that describes how the area is made. The limits $0$ and $2$, below and above, say from where to where one sums. They are not Leibniz’s: it was the Frenchman Joseph Fourier who, around 1820, began writing them attached to the sign. Before that, they were written out in words, alongside. The function $x^2$ is the height of each strip. The $dx$ is the width of each strip, infinitely thin. And it is also the marker of which letter varies: if it said $dt$, the variable would be $t$. The letter $x$ is a dummy letter: $\int_0^2x^2\,dx$ and $\int_0^2t^2\,dt$ have the same value. The result, $\frac83$, does not depend on $x$. Read as a whole, the sentence says: add up the heights $x^2$ times the widths $dx$, for all $x$ from $0$ to $2$.

October 18, 2026 Read

The D that factorises: the derivative as an operator

Write $D$ for “differentiate with respect to $x$”. Then $D(x^3)=3x^2$, $D(\sin x)=\cos x$, and differentiating twice is $D(D(f))$, abbreviated $D^2f$. So far it is just an abbreviation. Things get interesting when you take the shape of the notation seriously: if $D^2$ looks like a square, why not treat $D$ as a number? A differential equation becomes a polynomial Consider the equation $$y''-3y'+2y=0.$$With $D$, it is written $D^2y-3Dy+2y=0$, or, taking $y$ out as a factor,

October 18, 2026 Read

Is dy/dx a fraction?

Calculus teachers tend to say two things at the same time: “$\frac{dy}{dx}$ is not a fraction” and “you can cancel the $du$ as in a fraction”. Both are right, and that is precisely the strength and the danger of Leibniz’s notation. It looks like a fraction and, in most calculations, it behaves like one. Knowing where the resemblance holds and where it breaks is one of the subtlest skills in reading mathematics.

October 18, 2026 Read

From Δ to d: the difference that shrinks

A car passes kilometre 120 at 2:00 p.m. and kilometre 210 at 3:00 p.m. Its average speed was $$\frac{\Delta s}{\Delta t}=\frac{210-120}{15-14}=90\ \text{km/h}.$$The Greek letter $\Delta$, capital delta, is the Greek $D$, for difference. $\Delta s$ is not “delta times $s$”: it is a single sign meaning “how much $s$ has changed”. It is a calculation with real numbers, one you can do with a ruler and a clock. But the speedometer does not show the average speed over an hour. It shows the speed now. And “now” has no duration: $\Delta t=0$, and dividing by zero makes no sense. Calculus resolves this impasse with a passage to the limit, and its notation follows the passage without changing shape.

October 18, 2026 Read

Four ways to write the derivative

If $y=x^3$, the derivative is $3x^2$. Everyone writes that the same way. What changes is the name given to that derivative. In a physics book you will read $\dot y$; in a calculus book, $\frac{dy}{dx}$ or $f'(x)$; in a book on differential equations, $Dy$. They are four signs with the same object, and none of them has beaten the others. The four ways of writing Notation Author When How it reads $\dot y$ Isaac Newton 1660s, published in 1704 the fluxion of $y$, the speed at which $y$ flows in time $\dfrac{dy}{dx}$ Gottfried Leibniz 1675, published in 1684 the ratio of two infinitely small differences $f'(x)$ Joseph-Louis Lagrange 1797 the derived function of $f$, evaluated at $x$ $Df$ Louis Arbogast 1800 the operator $D$ applied to the function $f$ Chapter 1 has already told the quarrel between Newton’s dots and Leibniz’s $d$s, in the text on Leibniz and the calculus. Here another question matters: what each way of writing shows and what it leaves out.

October 18, 2026 Read

Leibniz against the dots: active and sterile notation

On 29 October 1675, in Paris, Leibniz was adding up infinitely many thin slices under a curve and got tired of writing omnia, “all”, in front of each sum. He noted in his notebook: Utile erit scribi ∫ pro omnia. (It will be useful to write ∫ instead of “all”.) — Leibniz, manuscript of 29 October 1675 The $\int$ is a stretched S, for summa. Two weeks later, in the same notebook, the $d$ appeared, for differentia, to denote infinitely small differences. With these two signs Leibniz had in his hands a language for the whole of calculus. Isaac Newton had reached the same ideas ten years earlier, in Cambridge, but wrote them differently. This is history’s best case for seeing how the shape of a sign changes what can be done with it.

October 10, 2026 Read