[permalink] [id link]
Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle ( whence right triangles become meaningless ) and of equality of length of line segments in general ( whence circles become meaningless ) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments ( so line segments continue to have a midpoint ).
from
Wikipedia
Some Related Sentences
Euler and discussed
The concept was introduced by Leonhard Euler in his 1765 book Theoria motus corporum solidorum seu rigidorum ; he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.
Lambert first considered the related Lambert's Transcendental Equation in 1758, which led to a paper by Leonhard Euler in 1783 that discussed the special case of we < sup > w </ sup >.
However, scholarship indicates that this claim of priority is not so clear ; Leonhard Euler discussed the principle in 1744, and there is evidence that Gottfried Leibniz preceded both by 39 years.
They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736.
For example the difference between a counterexample to a lemma ( a so-called ' local counterexample ') and a counterexample to the specific conjecture under attack ( a ' global counterexample ' to the Euler characteristic, in this case ) are discussed.
* Various manifold decompositions, as discussed for Euler characteristic.
This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760.
Euler and generalization
* 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.
# Gauss – Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ ( M ) where χ ( M ) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.
* 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.
* A further Euler line generalization at Dynamic Geometry Sketches Generalizes the Euler line further by disassociating it from the nine-point conic ( see above ).
This is a generalization of the Euler – Lagrange equation: indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics ( 18th century ).
Euler is a programming language created by Niklaus Wirth and Helmut Weber, conceived as an extension and generalization of ALGOL 60.
In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity.
Instanton Floer homology is viewed as a generalization of the Casson invariant because the Euler characteristic of Floer homology is identified with Casson invariant.
Euler and Euclidean
The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder, Möbius strip, torus, and Klein bottle.
Euler and geometry
Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.
The Gauss – Bonnet theorem or Gauss – Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry ( in the sense of curvature ) to their topology ( in the sense of the Euler characteristic ).
* Euler centre and maltitudes of cyclic quadrilateral at Dynamic Geometry Sketches, interactive dynamic geometry sketch.
In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral ; it passes through several important points determined from the triangle.
Issues that cause deviation from the pure Euler strut behaviour include imperfections in geometry in combination with plasticity / non-linear stress strain behaviour of the column's material.
Euler and called
If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the Euler force.
The component with a period of about 435 days is identified with the 8 month wandering predicted by Euler and is now called the Chandler wobble after its discoverer.
where, by definition, the left hand side is ζ ( s ) and the infinite product on the right hand side extends over all prime numbers p ( such expressions are called Euler products ):
In number theory, an odd composite integer n is called an Euler – Jacobi pseudoprime to base a, if a and n are coprime, and
In the case of 3 × 3 matrices, three such rotations suffice ; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices ( though not uniquely ) in terms of the three angles used, often called Euler angles.
An Euler probable prime which is composite is called an Euler – Jacobi pseudoprime to base a.
Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial.
In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called ( in 1766 ) the calculus of variations.
Such a walk is now called an Eulerian path ( oy • lɛr • i • ən ) or Euler walk in his honor.
Such a walk is called an Eulerian circuit or an Euler tour.
In arithmetic, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by
Such infinite products are today called Euler products.
Euler continued to write on the topic ; in his Reflexions sur quelques loix generales de la nature ( 1748 ), he called the quantity " effort ".
It is called the Euler characteristic of.
In mathematics, the generalized Gauss – Bonnet theorem ( also called Chern – Gauss – Bonnet theorem ) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature.
One of them is called " proper " Euler angles and the other Tait – Bryan angles.
Sometimes Tait-Bryan angles are called Euler angles.
is sometimes called the energy or action of the curve ; this name is justified because the geodesic equations are the Euler – Lagrange equations of motion for this action.
1, 2, and 3 are not of the required form, so the Heegner numbers that work are, yielding prime generating functions of Euler's form for ; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.
Bifurcation buckling is sometimes called Euler buckling even when applied to structures other than Euler columns.
As such, if spoke tension is increased beyond a safe level, the wheel spontaneously fails into a characteristic saddle shape ( sometimes called a " taco " or a " pringle ") like a three-dimensional Euler column.
1.591 seconds.