Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture.

The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.

On 7 June 1742, the German mathematician Christian Goldbach ( originally of Brandenburg-Prussia ) wrote a letter to Leonhard Euler ( letter XLIII ) in which he proposed the following conjecture:

In the letter dated 30 June 1742, Euler stated:

* Goldbach's original letter to Euler — PDF format ( in German and Latin )

Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x < sup > 4 </ sup > − 4x < sup > 3 </ sup > + 2x < sup > 2 </ sup > + 4x + 4, but he got a letter from Euler in 1742 in which he was told that his polynomial happened to be equal to

This universal law of nature embraces laws of motion as well, for an object moving others by its own force in fact imparts to another object the force it loses " ( first articulated in a letter to Leonhard Euler dated 5 July 1748, rephrased and published in Lomonosov's dissertation " Reflexion on the solidity and fluidity of bodies ", 1760 ).

In 1754, in his letter to Leonard Euler, he wrote that his three years of experiments on the effects of chemistry of minerals on their colour led to his deep involvement in the mosaic art.

Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach ( but he used a different sign convention from the above ).

However, many scholars, including Leonhard Euler, believe it originates from the letter " r ", the first letter of the Latin word " radix " ( meaning " root "), referring to the same mathematical operation.

* The use of the Greek letter zeta ( ζ ) for a function previously mentioned by Euler

It was one of Landau's conjectures and goes back to Euler, who observed in a letter to Goldbach in 1752 that n < sup > 2 </ sup >+ 1 is often prime for n up to 1500.

The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772.

Although Le Sage published not many papers in his life, he had an extensive letter exchange to people like Jean le Rond d ' Alembert, Leonhard Euler, Paolo Frisi, Roger Joseph Boscovich, Johann Heinrich Lambert, Pierre Simon Laplace, Daniel Bernoulli, Firmin Abauzit, Lord Stanhope etc ..

By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Euler.

He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal Latin squares of order 4n + 2 for every n.

A rotation represented by Euler angles with ( φ, θ, ψ )=(− 60 °, 30 °, 45 °) using the 3-1-3 ( Z-X-Z ) co-moving axes rotations

Euler ascertained that 2 < sup > 31 </ sup > − 1 = 2147483647 is a prime number ; and this is the greatest at present known to be such, and, consequently, the last of the above perfect numbers 2 < sup > 30 </ sup >( 2 < sup > 31 </ sup > − 1 ), which depends upon this, is the greatest perfect number known at present, and probably the greatest that ever will be discovered ; for as they are merely curious, without being useful, it is not likely that any person will attempt to find one beyond it.

* Janet Heine Barnett ( 2004 ) " Enter, stage center: the early drama of the hyperbolic functions ", available in ( a ) Mathematics Magazine 77 ( 1 ): 15 – 30 or ( b ) chapter 7 of Euler at 300, RE Bradley, LA D ' Antonio, CE Sandifer editors, Mathematical Association of America ISBN 0-88385-565-8.

The three lines meet at the Euler infinity point, X ( 30 ).

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia.

* June 8 – Leonhard Euler writes to James Stirling describing the Euler – Maclaurin formula, providing a connection between integrals and calculus.

* June 8-Leonhard Euler writes to James Stirling describing the Euler – Maclaurin formula, providing a connection between integrals and calculus.

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

Among the foreign scholars invited to work at the academy were the mathematicians Leonhard Euler, Anders Johan Lexell, Christian Goldbach, Georg Bernhard Bilfinger, Nicholas and Daniel Bernoulli, botanist Johann Georg Gmelin, embryologists Caspar Friedrich Wolff, astronomer and geographer Joseph-Nicolas Delisle, physicist Georg Wolfgang Kraft, and historian Gerhard Friedrich Müller.

An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function ( alternating zeta function )

In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler.

Actually this simple use of " quaternions " was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares.

In St. Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician Anders Johan Lexell about the newly discovered Uranus and its orbit, Euler suffered a brain hemorrhage and died a few hours later.

When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces — the Coriolis force and the Euler force — will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the vector co-rotates with the planet.

De Camp and Ley have claimed ( in their Lands Beyond ) that Leonhard Euler also proposed a hollow-Earth idea, getting rid of multiple shells and postulating an interior sun across to provide light to advanced inner-Earth civilization but they provide no references ; indeed, Euler did not propose a hollow-Earth, but there is a slightly related thought experiment.

In other words, they satisfy the Euler – Lagrange equations

Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock.

In equilateral triangles, these four points coincide, but in any other triangle they do not, and the Euler line is determined by any two of them.

* The eigenvalues of Frobenius on H ( U, E ) can now be estimated as they are the zeros of the zeta function of the sheaf E. This zeta function can be written as an Euler product of zeta functions of the stalks of E, and using the estimate for the eigenvalues on these stalks shows that this product converges for | T |< q < sup >− d / 2 − 1 / 2 </ sup >, so that there are no zeros of the zeta function in this region.

This similarity between linear coordinates and angular coordinates makes Euler angles very intuitive, but unfortunately they suffer from the gimbal lock problem.

Note that the gimbal lock problem does not make Euler angles " wrong " ( they always play at least their role of a well-defined coordinates system ), but it makes them unsuited for some practical applications.

They are not the three dimensional instance of a general approach, like matrices ; nor are they easily related to real world models, like Euler angles or axis angles.

They are also referred to as the Euler – Lagrange equations of quantum field theories, since they are the equations of motion of the corresponding Green's function.

Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work of Gottfried Leibniz, Johann Bernoulli, Leonhard Euler, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome real number-based arguments developed by Georg Cantor, Richard Dedekind, and Karl Weierstrass, which were perceived as being more rigorous by Weierstrass's followers.

In Euler, arrays, procedures, and switches are not quantities which are declared and named by identifiers: they are not ( as opposed to ALGOL ) quantities which are on the same level as variables, rather, these quantities are on the level of numeric and boolean constants.

( It should be noted that Euler and Lagrange applied this method to nonlinear differential equations and that, instead of varying the coefficients of linear combinations of solutions to homogeneous equations, they varied the constants of the unperturbed motions of the celestial bodies.

This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation and then show that they are independent of the choice of presentation.

Such L-functions are called ' global ', in that they are defined as Euler products in terms of local zeta functions.

Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane.

In aerospace engineering they are usually referred to as Euler angles.

The second two names are somewhat anachronistic, but since the integrals of the last two are related to the instanton number and Euler characteristic respectively, they have some justification.