History of Mathematics

The marvellous spiral and the lemniscate

With Leibniz’s calculus in hand, the Bernoulli brothers went hunting for curves. For each one they asked: what is the tangent, what is the length, what is the area, what is the curvature? Many of these curves had been known since the Greeks, but only now could these questions be answered by a general rule. Two of them kept Jakob’s name forever. The spiral that does not change shape The logarithmic spiral had already been studied by Descartes and by Torricelli. In polar coordinates, where each point is given by its distance $r$ from the centre and by the angle $\theta$, it is

October 11, 2026 Read

The Chevalier de Méré's dice and the fair split

Antoine Gombaud, who styled himself the Chevalier de Méré, was a writer at the French court and a passionate gambler. Around 1654 he brought Pascal two questions about bets. Pascal found them so good that he wrote to Pierre de Fermat, and probability theory was born from those letters. About his friend, Pascal remarked to Fermat: He has a very good mind, but he is not a geometer; and that, as you know, is a great defect. — Blaise Pascal, letter to Fermat (29 July 1654)

October 8, 2026 Read

Fermat's little theorem: powers and remainders

Take the number 2 and raise it to larger and larger powers. Divide each result by 7 and note the remainder: Power $2^1$ $2^2$ $2^3$ $2^4$ $2^5$ $2^6$ Value 2 4 8 16 32 64 Remainder mod 7 2 4 1 2 4 1 At $2^6$ the remainder is 1. Now try 3 and the prime 5: $3^4=81$, which leaves remainder 1 when divided by 5. And 10 and the prime 13: $10^{12}=1{,}000{,}000{,}000{,}000$, which also leaves remainder 1 when divided by 13.

October 7, 2026 Read

The ∫ sign: from sum to accumulation

On 29 October 1675 Leibniz was in Paris working on a problem about areas. Until then he had written omn., short for the Latin omnia, “all”, to mean the sum of all the lines of a figure, as the Italian Cavalieri did. Halfway down the page he wrote: It will be useful to write ∫ instead of omn. — Leibniz, manuscript of 29 October 1675 The symbol $\int$ is a long S, for summa, sum. At the time the long s was an ordinary letter in handwriting. It has never been replaced since.

October 6, 2026 Read

Areas and integration: the inverse of fluxions

Measuring the area of curved figures is one of the oldest problems in mathematics. Around 250 BC Archimedes showed that the area of a parabolic segment is $\frac43$ of the area of a triangle inscribed in it. It was a stroke of genius, but made to measure: every new curve needed a new idea. By the 17th century, Fermat, Cavalieri, Wallis and others could already compute the area under $y=x^n$ for many values of $n$. What was missing was a general rule. And what was missing was the realization that Newton and Leibniz each reached in their own way: computing areas and computing rates of change are inverse operations.

October 6, 2026 Read

Clock arithmetic

If it is 15h now, what time will it be in 10 hours? It’s not “25 h”, but 1 am: on the clock, after 24 the count starts again. Gauss gave a name and symbol to this idea in Disquisitiones: $$25 \equiv 1 \pmod{24}$$ It reads “25 is congruous to 1, modulo 24”: both leave the same remainder when divided by 24. The symbol $\equiv$ works almost like an equals sign: you can add and multiply on both sides.

October 1, 2026 Read

The rule of signs

How many positive roots can a polynomial have? Descartes gave a simple rule: the number of roots positive is at most the number of sign changes between consecutive coefficients (and, as proved then differs from it by an even number). See with $x^3 - 6x^2 + 11x - 6$. The signs of the coefficients are $+\ -\ +\ -$: three changes. So there is 3 or 1 positive roots. In fact,

September 30, 2026 Read

The number $e$ and compound interest

1 2 3 4 Explore the scene: tap the numbers. With a keyboard, use Tab and Enter or the arrow keys. 1. Customer: initial capital The person in front of the counter represents someone bringing money for a transaction. In this article’s example, we start with 100 units of money: this is the initial capital. 2. Service: checking the accounts The person behind the counter represents service and checking amounts. To follow an investment in our model, we need the capital, the rate and the number of periods. 3. Coins: interest on interest In the mathematical example at 100% a year, one annual payment turns 100 into 200. With two half-year periods at 50%, 100 becomes 150, then 150 becomes 225: the second payment also earns interest on the earlier interest. 4. Cash box: saving is not compounding Stored money does not grow by itself. The interest in this example follows a mathematical payment rule. Explore the chart below to compare annual, monthly and increasingly frequent payments. Try compound interest ↓ Enlarge the engraving Banca, trapeza, mensa: the table of the first bankers, between 1450 and 1650.Illustrative reconstruction with AI based on an engraving from the Euler’s Eyes collection

September 29, 2026 Read

The convergence of series

What is $1 - 1 + 1 - 1 + \cdots$? Grouped one way, $(1 - 1) + (1 - 1) + \cdots = 0$. Grouped another, $1 - (1 - 1) - (1 - 1) - \cdots = 1$. Euler and Leibniz thought the “right” answer was $\frac{1}{2}$, the average. In the eighteenth century such sums were handled freely, and the habit sometimes led to absurd results. Partial sums In chapter 6 of the Cours d’analyse, Cauchy cut the problem off at the root. A series $u_0 + u_1 + u_2 + \cdots$ has partial sums

October 31, 2026 Read

The hypotheses on which geometry is founded

The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. — Bernhard Riemann, On the Hypotheses which lie at the Bases of Geometry (1854) On 10 June 1854, Riemann presented to the Göttingen faculty the lecture On the Hypotheses which lie at the Bases of Geometry. He wrote it for an audience that included philosophers and historians, almost without formulas. The text was only published in 1868, after his death, and it changed geometry.

October 31, 2026 Read

Laplace's equation

To know how a planet attracts a point, one must add up the attraction of each of its pieces. Each piece pulls in a different direction, and adding forces is laborious: they are vectors, with three components each. Eighteenth-century mathematicians discovered a shortcut, and Laplace turned it into a tool. Adding numbers instead of arrows For a mass $M$ concentrated at a point, consider the number $$V = \frac{M}{r},$$where $r$ is the distance to the mass (we leave out the constant $G$). This number, which later received the name potential, holds all the information about the force: the components of the attraction in each direction are the derivatives of $V$ in that direction. And the great advantage is that, for several masses, the potentials simply add like numbers. First one adds the potential of all the pieces of the planet; then, with a derivative, one gets the force.

October 30, 2026 Read

The quadrature of the parabola

To “square” a figure, for the Greeks, was to find a square, or another rectilinear figure, with the same area. Squaring the circle was the impossible problem par excellence. Archimedes managed to square a curved figure: the segment of a parabola, the region between a parabola and a chord. Every segment bounded by a straight line and a section of a right-angled cone is four thirds of the triangle that has the same base and the same height as the segment. — Archimedes, Quadrature of the Parabola, letter to Dositheus

October 29, 2026 Read