# Formulas (1500-1799) 🧮

## 1500

\begin{aligned} &x^{3} + mx = n\ &\Rightarrow\ &x=\sqrt[3]{\sqrt{\frac{n^{2}}{4}+\frac{m^{3}}{27}}+\frac{n}{2}}-\sqrt[3]{\sqrt{\frac{n^{2}}{4}+\frac{m^{3}}{27}}-\frac{n}{2}} \end{aligned}

At some point in the early 1500’s, an Italian mathematician named Scipione del Ferro determined a general solution for what is known as the depressed cubic equation. This is cubic equation without any $x^{2}$ terms. The general form is : $x^{3} + mx = n$ . As it turns out, any cubic equation of the form $x^{3} + bx^{2} + cx + d = 0$ can be written as a depressed cubic, but that came later.

At the time, mathematicians didn’t publish their results, but, instead, kept them secret so that they could win the problem contests that were common at the time in Italy. As a result, del Ferro didn’t tell anyone about his discovery until shortly before his death in 1526. He then revealed the secret to a student of his named Antonio Maria Fior. In 1535, Fior used this knowledge to challenge a better mathematicain named Niccolo Fontana to a problem contest.

## 1636

$$a^{p} \equiv a \quad \bmod p$$

Fermat’s little theorem states that if p is a prime number, then for any integer a, the number a − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as aᵖ≡a mod p. For example, if a = 2 and p = 7, then 2⁷ = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, Fermat’s little theorem is equivalent to the statement that a − 1 is an integer multiple of p, or in symbols: aᵖ⁻¹≡1 mod p.

For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7.

Fermat’s little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the “little theorem” to distinguish it from Fermat’s last theorem.

## 1637

$$\begin{array}{c} x^{n}+y^{n}=z^{n} \quad n>2 \ => \ x y z=0 \end{array}$$

In number theory, Fermat’s Last Theorem (sometimes called Fermat’s conjecture, especially in older texts) states that no three positive integers x, y, and z satisfy the equation $x^{n} + y^{n} = z^{n}$ for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

## 1674

$$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$$

In mathematics, the Leibniz formula for ${\pi}$, named after Gottfried Leibniz. It is also called the Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1674. The series for the inverse tangent function, which is also known as Gregory’s series, can be given by:

$$\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots$$

The Leibniz formula for $\frac{\pi}{4}$ can be obtained by putting x = 1 into this series.

It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s = 1, and therefore the value β(1) of the Dirichlet beta function.

## 1735

$$\frac{\pi^{2}}{6}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots$$

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler’s solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper “On the Number of Primes Less Than a Given Magnitude”, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:

$$\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots$$

The sum of the series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be $\frac{\pi^{2}}{6}$ and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct. He produced a truly rigorous proof in 1741.

## 1737

$$\sum_{p_{n}} \frac{1}{p_{n}}$$

Euler achieved the first major advance beyond Euclid’s proof by combining his method of generating functions with another highlight of ancient Greek number theory, unique factorization into primes.

Theorem [Euler product for the zeta function]. The identity

$$\sum_{n=1}^{\infty} \frac{1}{n^{s}}=\prod_{p \text { prime }} \frac{1}{1-p^{-s}}$$

holds for all s such that the left-hand side converges absolutely.

Proof : Here and henceforth we adopt the convention:
The notation $\prod_{p}$ or $\sum_{p}$ means a product or sum over prime p.

## 1746

$$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$$

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

## 1750

$$V-E+F=2$$

Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically $V−E+F=2$. For instance, a tetrahedron has four vertices, four faces, and six edges; $4-6+4=2$.

A version of the formula dates over 100 years earlier than Euler, to Descartes in 1630. Descartes gives a discrete form of the Gauss-Bonnet theorem, stating that the sum of the face angles of a polyhedron is $2π(V−2)$, from which he infers that the number of plane angles is $2F+2V-4$. The number of plane angles is always twice the number of edges, so this is equivalent to Euler’s formula, but later authors such as Lakatos, Malkevitch, and Polya disagree, feeling that the distinction between face angles and edges is too large for this to be viewed as the same formula. The formula $V−E+F=2$ was (re)discovered by Euler; he wrote about it twice in 1750, and in 1752 published the result, with a faulty proof by induction for triangulated polyhedra based on removing a vertex and retriangulating the hole formed by its removal. The retriangulation step does not necessarily preserve the convexity or planarity of the resulting shape, so the induction does not go through. Another early attempt at a proof, by Meister in 1784, is essentially the triangle removal proof given here, but without justifying the existence of a triangle to remove. In 1794, Legendre provided a complete proof, using spherical angles. Cauchy got into the act in 1811, citing Legendre and adding incomplete proofs based on triangle removal, ear decomposition, and tetrahedron removal from a tetrahedralization of a partition of the polyhedron into smaller polyhedra. Hilton and Pederson provide more references as well as entertaining speculation on Euler’s discovery of the formula.

## 1760

For a non-conservative force which depends on velocity, it may be possible to find a potential energy function V that depends on positions and velocities. If the generalized forces $Q_{i}$ can be derived from a potential V such that

$$Q_{i}=\frac{\mathrm{d}}{\mathrm{d} t}\frac{\partial V}{\partial \dot{q} _{i}}-\frac{\partial V}{\partial q _{i}}$$

equating to Lagrange’s equations and defining the Lagrangian as $L = T − V$ obtains Lagrange’s equations of the second kind or the Euler–Lagrange equations of motion

$$\frac{\partial L}{\partial q _{i}}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q} _{i}}=0$$

However, the Euler–Lagrange equations can only account for non-conservative forces if a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange’s equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations.

The Euler–Lagrange equations also follow from the calculus of variations. The variation of the Lagrangian is

$$\delta L=\sum_{i=1}^{n} \left( \frac{\partial L}{\partial q _{i}} \delta q _{i}+\frac{\partial L}{\partial \dot{q} _{i}} \delta \dot{q} _{i} \right) \ , \quad \delta \dot{q} _{i} \equiv \delta \frac{\mathrm{d} q _{i}}{\mathrm{d} t} \equiv \frac{\mathrm{d} \left( \delta q _{i} \right) }{\mathrm{d} t} \ ,$$

which has a form similar to the total differential of L, but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An integration by parts with respect to time can transfer the time derivative of $\delta q_{i}$ to the $\partial L / \partial(\mathrm{d} q_{i} / \mathrm{d} t)$, in the process exchanging $\mathrm{d}\left(\delta q_{i}\right) / \mathrm{d} t$ for $\delta q_{i}$, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,

$$\int_{t_{1}} ^ {t_{2}} {\delta L} {\mathrm{~d} t} = \int_{t_{1}} ^ {t_{2}} \sum_{i=1} ^ {n} \left( \frac{\partial L}{\partial q _{i}} {\delta q _{i}} + \frac{\mathrm{d}}{\mathrm{d} t} \left( \frac{\partial L}{\partial \dot{q} _{i}} {\delta q _{i}} \right) - \frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q} _{i}} {\delta q _{i}} \right) {\mathrm{d} t}$$

$$= \sum_{i=1} ^ {n} \left[ \frac{\partial L} {\partial \dot{q} _{i}} {\delta q _{i}} \right] _{t _{1}} ^ {t _{2}} +$$

$$\int_{t_{1}}^{t_{2}} \left( \frac{\partial L}{\partial q _{i}} - \frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q} _{i}} \right) {\delta q _{i}} {\mathrm{~d} t} \ .$$

Now, if the condition $\delta q_{i}(t_{1}) = \delta q_{i}(t_{2}) = 0$ holds for all i, the terms not integrated are zero. If in addition the entire time integral of $\delta L$ is zero, then because the $\delta q_{i}$ are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of $\delta q_{i}$ must also be zero. Then we obtain the equations of motion. This can be summarized by Hamilton’s principle;

$$\int_{t_{1}}^{t_{2}} \delta L \mathrm{~d} t=0 \ .$$

The time integral of the Lagrangian is another quantity called the action, defined as

$$S=\int_{t_{1}}^{t_{2}} L \mathrm{~d} t \ ,$$

which is a functional; it takes in the Lagrangian function for all times between $t_{1}$ and $t_{2}$ and returns a scalar value. Its dimensions are the same as [angular momentum], [energy]·[time], or [length]·[momentum]. With this definition Hamilton’s principle is

$$\delta S=0 \ .$$

Thus, instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton’s principle is sometimes referred to as the principle of least action, however the action functional need only be stationary, not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of the action.

## 1799

$$\begin{array}{c} \forall p(x) \in \mathbb{C}[x] \ \exists z \in \mathbb{C}: p(z)=0 \end{array}$$

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.