History of Mathematics

Maxima and minima: the rectangle with the largest area

You have 24 metres of fencing and want to enclose a rectangular plot with the largest possible area. Which dimensions should you choose? A long thin rectangle wastes fencing; a square seems better. But how do you prove it? Fermat invented a method for questions like this around 1636, more than thirty years before Newton and almost fifty before Leibniz published calculus. Fermat’s problem Fermat’s first example was exactly this: divide a segment of length $b$ into two parts, $x$ and $b-x$, so that the product $x(b-x)$ is as large as possible. That is our plot: with 24 m of fencing, two adjacent sides add up to $12$, and the area is

October 7, 2026 Read

The product rule: two changes in one calculation

If you know how $u$ changes and how $v$ changes, how does the product $uv$ change? The natural guess is to multiply the changes: $d(uv)=du\cdot dv$. It looks so symmetrical, so elegant… Leibniz made the same guess. Leibniz tests and fails In the same manuscript of 11 November 1675 in which he began using the letter $d$, Leibniz asked himself whether $d(xy)$ would equal $dx\,dy$, and whether $d\!\left(\frac xy\right)$ would equal $\dfrac{dx}{dy}$. Instead of trusting elegance, he tested it with an example and found it did not. A few days later he found the right rule.

October 6, 2026 Read

Fluxions: how to measure an instantaneous change

A car leaves one city and reaches another, 90 km away, in one hour. Its average speed was 90 km/h. But at every instant the speedometer showed a different number. What exactly does it measure? Speed is distance divided by time, but in an instant no distance is covered and no time passes. Dividing zero by zero gives no answer. That was the question Newton wanted to answer in 1665. His answer was the method of fluxions.

October 6, 2026 Read

The 17 sided polygon

Since the Greeks, it was known how to construct regular polygons with 3, 4, 5 and 15 sides with a ruler and compass, and all that come out of them doubling the number of sides (6, 8, 10, 12…). For 2000 years, no one found another. Gauss showed that constructing the $n$ sided polygon is the same as calculating $\cos\frac{2\pi}{n}$ using only integers, the four operations and square roots. And ruler and compass know how to take root square: it is exactly the construction that we saw in the chapter on Descartes. For 17 sides, he reached

October 1, 2026 Read

Square root with ruler and compass

Geometry begins by showing that arithmetic operations can be done with segments. The most beautiful is the square root. To find $\sqrt{a}$: Draw on a straight line a segment $AB$ of length 1 and then a segment $BC$ of length $a$. Draw a semicircle with diameter $AC$. Lift a perpendicular through $B$ until you find the semicircle at point $D$. Then $BD = \sqrt{a}$. Change the value of a: the height BD is always √a. For a = 4, 9 or 2,25 the root is exact. Ative o JavaScript para ver esta figura interativa.

September 30, 2026 Read

The Basel problem

In 1650, Pietro Mengoli asked how much the sum of the inverse squares is worth: $$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n^2} = \ ?$$ The Bernoullis tried and failed. The sum grows slowly: with 10 terms it gives $1{,}5498$, with 100 terms gives $1{,}6350$, with 1000 terms gives $1{,}6439$. No one recognized the number to which she walked. See the figure: each bar is the sum up to $N$ terms.

September 29, 2026 Read

Riemann surfaces

Riemann’s doctoral thesis, of 1851, has a modest title: Foundations for a General Theory of Functions of a Complex Variable. Inside it are two ideas that changed mathematics: the conditions a complex function must satisfy in order to have a derivative, and a geometric way of dealing with functions that have several values. Functions that can be differentiated A complex function $f(z)$, with $z = x + iy$, can be written as $f = u + iv$, where $u$ and $v$ are real functions of $x$ and $y$. For $f$ to have a derivative in the complex sense, that is, for the limit $\frac{f(z + h) - f(z)}{h}$ to be the same whatever the direction in which $h$ goes to zero, the parts $u$ and $v$ must satisfy the Cauchy–Riemann equations:

October 31, 2026 Read

Limits and continuity

In the eighteenth century calculus was a formidable and somewhat mysterious machine. Euler summed divergent series and drew correct results from them; Lagrange tried to base everything on power series; books spoke of “infinitely small” quantities that were zero when convenient and non-zero when not. It worked, but nobody could explain exactly why. The Cours d’analyse In 1821 Cauchy published the Cours d’analyse de l’École royale polytechnique, with the lessons he gave to future engineers. In the preface he announced his intention of giving analysis “all the rigour that is demanded in geometry”. Everything begins with a definition:

October 30, 2026 Read

The dark stars

A luminous star with the same density as the Earth and a diameter 250 times that of the Sun would not, because of its attraction, allow any of its rays to reach us. It is therefore possible that the largest luminous bodies in the universe are, for that very reason, invisible. — Pierre-Simon Laplace, Exposition of the System of the World (1796) These sentences are in the first edition of the Exposition of the System of the World. Laplace wrote them at a time when people believed, like Newton, that light was made of particles. If light is a swarm of corpuscles, gravity must pull on them as it pulls on a stone. And then, if a star is compact enough, the light it emits will fall back, like a stone thrown upward.

October 30, 2026 Read

The measurement of the circle and the number π

That every circle has the same ratio between circumference and diameter was known before Archimedes. What was uncertain was the value of this ratio, which we now call $\pi$: the Babylonians sometimes used $3\frac{1}{8}$, the Egyptians something close to $3.16$. In the short treatise Measurement of a Circle, Archimedes did two things: he related the area of the circle to its circumference, and he showed how to compute $\pi$ to any desired precision.

October 29, 2026 Read

Maxima, minima and points of inflection

The second book of the Instituzioni analitiche is devoted to differential calculus, and one of its main applications is the one every student learns: finding maxima and minima. Agnesi teaches the method with dozens of examples, of curves and of geometric problems, always in the same sequence: write the quantity, differentiate, set the differential equal to zero and check. The highest point At a maximum or a minimum, the curve is momentarily horizontal: the tangent line has slope zero. Since the slope of the tangent is the derivative, the candidates for maximum and minimum are the points where $y' = 0$.

October 29, 2026 Read

Hamilton's mechanics

In 1788 Lagrange had turned Newton’s mechanics into a set of equations valid in any coordinates. Hamilton admired that book, which he called “a kind of scientific poem”. In 1834 and 1835, in the two essays On a General Method in Dynamics, he took the idea further, bringing into mechanics what he had learned from rays of light. Position and momentum Lagrange described a system by its positions $q$ and velocities $\dot q$. Hamilton replaced the velocity with the momentum $p$ (for a simple particle, mass times velocity, $p = m\dot q$) and put at the centre of everything a single function, the Hamiltonian $H(q, p)$, which in most cases is the total energy, kinetic plus potential:

October 27, 2026 Read