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The normal curve, Gauss's bell

If measurement errors are random, how are they distributed? In Theoria motus, Gauss started from the idea that the mean is the best estimate and arrived at the shape of the error curve: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}},$$ where $\mu$ is the mean and $\sigma$ is the standard deviation. It is the normal curve, or Gaussian bell: high in medium, symmetrical, and increasingly lower away from the mean. It appears everywhere, from the heights of the people to the errors of a machine.

October 10, 2026 Read
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Descartes and the idea of ​​function

Descartes never used the word “function”. But it was he who made the idea possible: in Geometry, a curve stops being just a drawing and becomes a relationship between two numbers, $x$ and $y$. When one changes, the other changes along with it, following a rule. In this part, we tell you how this idea gained name, symbol and graphic. The following five topics develop this reading and the subsequent history of the idea of ​​function.

October 10, 2026 Read
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$V - A + F = 2$ and the soccer ball

In 1750, in a letter to mathematician Christian Goldbach, Euler noted a relationship between the number of vertices ($V$), edges ($A$) and faces ($F$) of a convex polyhedron: $$V - A + F = 2.$$ Rotate the polyhedra with your finger or mouse and check the calculation: V, A and F are counted by the program from the drawing itself. In all cases it gives 2. Ative o JavaScript para ver esta figura interativa.

October 10, 2026 Read
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Leibniz's legacy

The calculus war Leibniz published calculus in 1684 and 1686. Newton had reached the same ideas in 1665–1666 but kept everything in manuscripts. For fifteen years nobody disputed anything. In 1699 the Swiss mathematician Nicolas Fatio de Duillier, a friend of Newton, hinted in a book that Leibniz had drawn on Newton. In 1708 the Scot John Keill went further and accused Leibniz of plagiarism. The argument: in 1676 Leibniz had visited London and seen a copy of Newton’s De analysi.

October 11, 2026 Read
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Newton's legacy

Newton was the greatest genius who ever lived, and the most fortunate, for there is only one universe to discover, and it can be discovered only once. — Joseph-Louis Lagrange, as reported by his contemporaries The calculus war Newton created the method of fluxions in 1665–1666 but did not publish it. Leibniz reached calculus independently in 1675 and published in 1684. For some years nobody quarrelled. In 1676 the two even exchanged cordial letters through Henry Oldenburg, and Newton hid his ideas in an anagram.

October 11, 2026 Read
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The Egregium Theorem and curvature

Measuring Hannover, Gauss thought a lot about how to make maps of a curved Earth. In 1827 published the result he called Theorema Egregium, the “remarkable theorem”: the curvature of a surface can be measured by those who live inside it, without going out to look from the outside. A consequence: in a plane, the angles of a triangle add up to 180°. In a sphere, they add up to more. Take the triangle with one vertex at the north pole and the other two at the equator:

October 10, 2026 Read
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A reading of the Cogito: listening and speaking

Cogito ergo sum drawn on the Cartesian plane.Euler’s Eyes Collection Here at Euler’s Eyes we like to read “I think, therefore I am” as a little machine. The thought hears something (in Latin, audire) and speaks another (fabulari). The rule that binds what What you hear is the function. Method words physis: nature; methode: the reasoning, the path; graphiko’s: the lines, the drawing; máthema: what is learned, the act of deciphering; functus: who fulfills a task; dominus: the lord, where the word dominion comes from. An example: “say what you hear twice”. Empty parentheses act as a waiting ear a number:

October 10, 2026 Read
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Infinitely many prime numbers from a sum

Euclid had already proven, around 300 BC, that there are infinitely many prime numbers. In 1737 Euler found a completely different proof, which linked the primes to calculus. For each prime $p$, the sum of a geometric progression gives $$\frac{1}{1 - \frac{1}{p}} = 1 + \frac{1}{p} + \frac{1}{p^2} + \frac{1}{p^3} + \cdots$$ Multiply these expressions to $p = 2, 3, 5, 7, \dots$ Each product of terms has the form $\frac{1}{2^a 3^b 5^c \cdots}$. As every natural number is written in a single way as a product of primes, each fraction $\frac{1}{n}$ appears exactly once:

October 10, 2026 Read
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Fermat's legacy

The father of number theory Title page of the 1670 edition of Diophantus’s Arithmetica, printed in Toulouse “with the observations of P. de Fermat”. It was through this book, published by his son five years after his death, that the margin notes reached Euler.Bernard Bosc, Toulouse (1670) · Wikimedia Commons · public domain Before Fermat, problems about whole numbers were seen as riddles, Diophantus’s puzzles. Fermat turned them into an area of mathematics, with deep questions and methods of its own. For almost a century hardly anyone followed him. It was Euler, from 1730 onwards, who read Fermat’s notes, proved many of them and made number theory a central branch of mathematics. Then came Lagrange, Legendre and Gauss.

October 11, 2026 Read
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Systems of equations: the determinant and Gaussian elimination

To calculate orbits using least squares, Gauss needed to solve systems of equations with many unknowns. Two names that every student knows come from this story: the determinant and the Gaussian elimination. The determinant. Take a system with two equations: $$\begin{cases} a\,x + b\,y = e \\ c\,x + d\,y = f \end{cases} \quad\Longrightarrow\quad x = \frac{e\,d - b\,f}{a\,d - b\,c}, \qquad y = \frac{a\,f - e\,c}{a\,d - b\,c}.$$ It all depends on the number $ad - bc$ in the denominator: if it is not zero, the system has a single solution. This number “determines” the system.

October 10, 2026 Read
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Descartes and the problem of cannons

Cannons and artillerymen in an old engraving.Euler’s Eyes Collection Descartes experienced war up close. In 1618 he served as a volunteer in the army of Maurice of Nassau and, according to his first biographer, Adrien Baillet, he was in 1628 at the siege of La Rochelle, led by Cardinal Richelieu. On the battlefields, a practical problem occupied the artillerymen: with what angle tilt the cannon to hit a target at a certain distance?

October 10, 2026 Read
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Fermat's number that was not prime

Pierre de Fermat believed that all numbers of the form $2^{2^n} + 1$ were prime. For $n = 0, 1, 2, 3, 4$ works: 3, 5, 17, 257 and 65 537 are primes. The next one is $$2^{32} + 1 = 4\,294\,967\,297.$$ Testing all dividers by hand is impractical. In 1732, Euler showed that this number is divisible by 641. An elegant way to look at this uses two ways of writing 641:

October 10, 2026 Read