History of Mathematics

The art of conjecturing: counting the chances

To conjecture about something is to measure its probability. Therefore we define the art of conjecturing, or stochastics, as the art of measuring the probabilities of things as exactly as possible, so that in our judgements and actions we may always choose or follow what seems to us better, more satisfactory, safer or more carefully considered. — Jakob Bernoulli, Ars conjectandi (1713), part IV In 1654, Pascal and Fermat exchanged letters on how to divide the stakes of an interrupted game. In 1657, the Dutchman Christiaan Huygens published a fifteen-page booklet on the calculations of dice games. These were parlour problems: dice, cards, bets.

October 14, 2026 Read

The exponential calculus: differentiating x to the power x

Leibniz’s calculus knew how to differentiate powers with a fixed exponent, like $x^3$ or $x^{1/2}$: the derivative of $x^n$ is $nx^{n-1}$. But in 1694, in a letter to Leibniz, Johann Bernoulli put a strange creature on the table: the expression $$y=x^x,$$in which the base and the exponent vary at the same time. The rule for $x^n$ does not work, because the exponent is not fixed. Nor does the rule for $a^x$, because the base is not fixed. In 1697 Johann published in the Acta Eruditorum the “principles of the exponential calculus”, with the method for handling these powers, which he called percurrent, because the exponent runs through values.

October 14, 2026 Read

The Nash bargaining solution

A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. — John Nash, The Bargaining Problem (1950) In 1948, still an undergraduate at the Carnegie Institute of Technology, Nash took a course in international economics, the only economics course of his life. He came out of it with a question: when two people negotiate, how do they share the gains from a deal? Economists thought the answer depended on each side’s psychology, skill and bluffing, and that there was nothing to say in general. Nash showed that, if the deal must satisfy a few reasonable requirements, there is only one possible answer. The paper, written before he arrived at Princeton, was published in 1950 in the journal Econometrica.

October 14, 2026 Read

The Sophie Germain identity: factoring what looks prime

Some numbers look prime and are not. 629, for example, is not even, does not end in 5 and is not a multiple of 3. But $629=5^4+4$, and that form gives it away: it factors by an identity that bears Sophie Germain’s name. The identity For any numbers $a$ and $b$, $$\begin{aligned}a^4+4b^4={}&\left(a^2+2b^2+2ab\right)\\&\times\left(a^2+2b^2-2ab\right).\end{aligned}$$The trick for finding it is completing the square, as in the quadratic formula. The sum $a^4+4b^4$ is almost the square of $a^2+2b^2$; only the middle term, $4a^2b^2$, is missing. Adding and subtracting that term,

October 14, 2026 Read

The Pascaline: a sum made of gears

In 1640 Étienne Pascal was appointed tax commissioner for Normandy. The job meant adding up column after column of figures late into the night, and his son Blaise, then 17, helped. To make matters worse, French money was not decimal: $$1\ \text{livre}=20\ \text{sols},\qquad 1\ \text{sol}=12\ \text{deniers}.$$Adding $3$ livres, $15$ sols and $8$ deniers to $2$ livres, $7$ sols and $9$ deniers needs three kinds of carry: deniers: $8+9=17$, which is $1$ sol and $5$ deniers; sols: $15+7+1=23$, which is $1$ livre and $3$ sols; livres: $3+2+1=6$. Result: $6$ livres, $3$ sols and $5$ deniers. Do that a thousand times a night and you will make mistakes. In 1642 Blaise began building a machine that would not.

October 11, 2026 Read

Fermat numbers: a conjecture and a counterexample

Mathematicians have always dreamed of a formula that produces only prime numbers. In 1640 Fermat thought he had found one. Five primes in a row Fermat studied numbers of the form $$F_n=2^{2^n}+1.$$The exponent is a power of 2: $1,2,4,8,16,\ldots$ Computing the first few: $$F_0=3,\quad F_1=5,\quad F_2=17,\quad F_3=257,\quad F_4=65{,}537.$$All five are prime. Checking $65{,}537$ by hand, trying divisors up to its square root, was already hard work. Fermat wrote to Frénicle de Bessy that he was convinced all numbers of this form were prime, although he admitted he had no proof.

October 11, 2026 Read

Binary arithmetic: writing numbers with 0 and 1

Why do we count in tens? Probably because we have ten fingers. There is nothing special about the number 10: any number can be written in any base. Leibniz took this idea to the extreme and asked: what is the smallest number of digits that can write everything? The answer is two: 0 and 1. Each place is worth double The numbers 0 to 32 written in binary, in the table from the paper Explanation of Binary Arithmetic, which Leibniz published in 1703 in the memoirs of the Paris Academy of Sciences.G. W. Leibniz (1703) · Wikimedia Commons · public domain In the decimal system the places are worth 1, 10, 100, 1000… and each digit says how many times we use that value. In the binary system the places are worth 1, 2, 4, 8, 16, 32… and each digit can only say yes (1) or no (0).

October 11, 2026 Read

Newton's method: approximating roots with tangents

Some equations have no formula. There is no quadratic formula for $e^x+x-3=0$, and the formula for cubic equations is so laborious that hardly anyone uses it. In these cases the way forward is to approximate: guess a value, see how far off we are, and correct the guess. The question is how to correct it well. Newton’s example In De analysi, of 1669, Newton chose the equation $$y^3-2y-5=0.$$By trial he saw that the root is near $2$, since $2^3-2\cdot2-5=-1$. So he wrote $y=2+p$, where $p$ is the missing correction, and substituted:

October 11, 2026 Read

The fundamental theorem of algebra

In his doctoral thesis, in 1799, Gauss proved that every polynomial of degree $n$ has $n$ roots, as long as complex numbers are accepted. For example, $x^2 + 1 = 0$ has no real solution, but has two complex ones: $$x^2 + 1 = (x - i)(x + i).$$ Gauss also popularized the way of drawing complex numbers as points on a plane, the plane complex: $a + bi$ is the point $(a, b)$. Norwegian Caspar Wessel (1797) and Swiss Jean-Robert Argand (1806) had the same idea, but almost no one read their work; was the fame of Gauss, from 1831, who spread the drawing. That’s why it is also called Argand-Gauss plane. And then a nice connection appears with the polygon with 17 sides. The three roots of $x^3 = 1$ are

October 10, 2026 Read

The angles of polyhedra and Euler's formula

At one vertex of a cube there are three squares, and their angles add up. $3 \times 90° = 270°$. There are $90°$ left to complete one turn: this is the defect of the vertex. By Around 1630, Descartes discovered that, in a convex polyhedron, the sum of the defects of all vertices is always 720°. Polyhedron Angles at a vertex Defect Vertices Sum Tetrahedron 3 × 60° = 180° 180° 4 720° Cube 3 × 90° = 270° 90° 8 720° Octahedron 4 × 60° = 240° 120° 6 720° Dodecahedron 3 × 108° = 324° 36° 20 720° Icosahedron 5 × 60° = 300° 60° 12 720° The faces around a vertex, open in the plane. The red slice is the defect, the angle left to complete 360°. Ative o JavaScript para ver esta figura interativa.

October 10, 2026 Read

Euler's formula and the "most beautiful equation"

The number $e$ can be written as a series: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots$$ Euler had the courage to put an imaginary number in the exponent. Changing $x$ to $i\theta$ and using $i^2 = -1$, $i^3 = -i$, $i^4 = 1$: $$e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\frac{\theta^5}{5!} - \cdots$$ Separating terms without $i$ from terms with $i$:

October 10, 2026 Read

The Enigma: 159 quintillion keys and one flaw

A three-rotor Enigma machine on display at the National Cryptologic Museum of the American security agency. At the top, the rotors; in the middle, the lamps; below, the keyboard; at the front, the plugboard.National Security Agency · Wikimedia Commons · public domain The Enigma looked like a typewriter inside a wooden box. The operator pressed a letter on the keyboard and another letter lit up on a panel of lamps. A second operator wrote down the letters that lit up, and that sequence was sent by radio, in Morse code. At the other end, an operator with an identical machine, set up the same way, typed in the enciphered message and read the original text on the lamps.

October 16, 2026 Read