Thābit's formula was rediscovered by Fermat ( 1601 – 1665 ) and Descartes ( 1596 – 1650 ), to whom it is sometimes ascribed, and extended by Euler ( 1707 – 1783 ).

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

It was Euler ( presumably around 1740 ) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him.

The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 ( and later generalized as Darboux's formula ).

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

This view was further elaborated by Belidor ( representation of rough surfaces with spherical asperities, 1737 ) and Leonhard Euler ( 1750 ) who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction ..

Euler was born on April 15, 1707, in Basel to Paul Euler, a pastor of the Reformed Church.

Paul Euler was a friend of the Bernoulli family — Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard.

As a result, it was made especially attractive to foreign scholars like Euler.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731.

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece.

This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy.

Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes.

For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

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.

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.

This direct relationship between curved streamlines and pressure differences was derived from Newton's second law by Leonard Euler in 1754:

It is named for the Dutch-Swiss mathematician and scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others.

Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

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.

The most valuable result of her labours was the Instituzioni analitiche ad uso della gioventù italiana, a work of great merit, which was published at Milan in 1748 and " was regarded as the best introduction extant to the works of Euler.

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.

This function as a function of a real argument was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time.

The values of the Riemann zeta function at even positive integers were computed by Euler.

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

#: Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles.

Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:

* Euler Function and Theorem at cut-the-knot

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

Solovay and Strassen showed that for every composite n, for at least n / 2 bases less than n, n is not an Euler – Jacobi pseudoprime.

If one bends and deforms the surface, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will.

* The center of any nine-point circle ( the nine-point center ) lies on the corresponding triangle's Euler line, at the midpoint between that triangle's orthocenter and circumcenter.

* Nine-point conic and Euler line generalization at Dynamic Geometry Sketches Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.

Beginning with the equation of motion for a fluid ( say, the Euler equations or the Navier-Stokes equations without viscosity ) and taking the curl, one arrives at the equation of motion for the curl of the fluid velocity, that is to say, the vorticity.

Note that, with the opposite sign convention, this is the potential generated by a pointlike sink ( see point particle ), which is the solution of the Euler equations in two-dimensional incompressible flow.

If a point P is chosen on the Euler line HN of the reference triangle ABC with a position vector < u > p </ u > such that < u > p </ u >

Other notable points that lie on the Euler line are the de Longchamps point, the Schiffler point, the Exeter point and the far-out point.

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

Another particularly useful way to represent the Euler line is in terms of a parameter t. Starting with the circumcenter ( with trilinears ) and the orthocenter ( with trilinears, every point on the Euler line, except the orthocenter, is given as

* Euler infinity point

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin.

In formal language, gimbal lock occurs because the map from Euler angles to rotations ( topologically, from the 3-torus T < sup > 3 </ sup > to the real projective space RP < sup > 3 </ sup >) is not a covering map – it is not a local homeomorphism at every point, and thus at some points the rank ( degrees of freedom ) must drop below 3, at which point gimbal lock occurs.

A triangle ABC with the incenter I has its Schiffler point at the point of concurrence of the Euler lines of the four triangles BCI, CAI, ABI, and ABC.

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